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Have the Elusive Progenitors of Supernovae Type Ia Been Discovered?
Mario Livio and Adam Riess
Space Telescope Science Institute
3700 San Martin Drive
Baltimore, MD 21218
email: mlivio@stsci.edu; ariess@stsci.edu
Abstract
The recent detection of H$\alpha$ emission in the supernova Type Ia SN 2002ic could be taken to mean that the elusive progenitor systems of Type Ia supernovae have finally been identified. At first glance, the observation appears to support a single-degenerate scenario, in which the white dwarf accretes from a normal companion. In this Letter we show that the opposite may be true, and the observations may support the merger of two white dwarfs as the cause for Type Ia supernovae.
cosmology: observations – supernovae: general
1 Introduction
The recent detection of H$\alpha$ emission in the spectrum of the supernova Type Ia (SN Ia) SN 2002ic (Hamuy et al. 2003) is a landmark discovery. While there is very little doubt that SNe Ia represent the thermonuclear disruption of mass accreting white dwarfs (WDs), the precise nature of the progenitor systems remains uncertain (Branch et al. 1995; Livio 2001). Given that SNe Ia are the tool of choice for confirming the acceleration of cosmic expansion (Riess et al. 1998; Perlmutter et al. 1999), the importance of identifying the progenitors cannot be overemphasized. The two main scenarios that have been proposed involve either the merger of two white dwarfs (the double-degenerate scenario; Iben & Tutukov 1984; Webbink 1984), or a single white dwarf accreting from a normal companion (the single-degenerate scenario; Whelan & Iben 1973; Nomoto 1982). Recently it has been argued theoretically, that single-degenerate progenitors are favored (even though it is very difficult for hydrogen-accreting WDs to reach the Chandrasekhar limit; Piersanti et al. 2000), and that double WD mergers may lead to accretion-induced collapses rather than to SNe Ia (Livio 2001; Nomoto et al. 2000). The tentative discovery (if confirmed) of an enhanced SN Ia rate near jets in active galactic nuclei (Livio, Riess, & Sparks 2002; Capetti 2002) appears to support this conclusion. Nevertheless, until SN 2002ic the “smoking gun”—the presence of hydrogen in the spectrum—was missing. The clear detection of a broad (FWHM $\sim 1800$ km s${}^{-1}$) H$\alpha$ component in SN 2002ic appears on the face of it to demonstrate that at least some SNe Ia result from single-degenerate progenitors. In the present letter we show that this conclusion may be premature.
2 Why Now?
One of the key questions posed by the observations of Hamuy et al. (2003) is: Why was hydrogen not detected before? This becomes particularly puzzling when we realize that there exist about 100 spectra of SNe Ia in which a signature of the strength of that seen in SN 2002ic would have been detected (T. Matheson, private communication), had it been there. In fact, Hamuy et al. noted that the amount of shock-heated circumstellar material needed to produce the observations of SN 2002ic is totally unexpected for a SN Ia. Accordingly, they suggested that the progenitor system was a binary consisting of a C/O white dwarf and a massive (3–7 M${}_{\odot}$) asymptotic giant branch (AGB) star. The presence of the latter was necessitated by the need to have an integrated circumstellar mass of at least a few solar masses.
The main problem with this scenario is that one would expect to observe a range of strengths of H$\alpha$ lines in SNe Ia, depending on the amount of circumstellar material (in turn, determined primarily by the mass of the AGB star), rather than detecting a relatively strong line in one case only (it is also hard to believe that this is the first progenitor system containing an AGB star).
We propose instead that the total absence of H$\alpha$ lines in all the pre-SN 2002ic SNe Ia observed to date argues that SN 2002ic represents rather rare circumstances, and not a white dwarf accreting from the wind of an AGB star.
3 A Supernova Ia in a Common Envelope?
All the evolutionary scenarios leading to the formation of close double white dwarf systems involve a stage in which an AGB star fills its Roche lobe and transfers mass onto a white dwarf companion (e.g. Yungelson & Livio 2000). Under these conditions, the mass transfer process is unstable, and the system evolves rapidly into a common envelope (CE) configuration, inside which the white dwarf and the core of the AGB star spiral-in (e.g. Rasio & Livio 1996; Taam & Sandquist 2000). Typically, the CE phase lasts a few hundred to a few thousand years, and results in the ejection of the envelope and the emergence of a double white dwarf system (e.g. Sandquist et al. 1998; Taam & Sandquist 2000 and references therein). I propose that SN 2002ic represents one of those rare cases in which the explosion occurs during (or immediately following) the CE phase, and in which some part of the envelope has not been previously ejected. This raises two immediate questions: (i) Is this possible at all? and (ii) Does this support a single-degenerate or a double-degenerate scenario?
For the white dwarf to actually reach the Chandrasekhar mass via accretion of hydrogen-rich material during the CE phase is extraordinarily unlikely. Steady burning occurs for a narrow range of accretion rates of order (Paczyński & Żytkow 1978; Nomoto, Nariai & Sugimoto 1979; the limits are determined: at the low end by the requirement that the pressure at the time of ignition be sufficiently low to prevent a shell flash, and at the high end by the accretor expanding to supergiant dimensions)
$$0.4\ \dot{M}_{\mathrm{RG}}\lesssim\dot{M}\lesssim\dot{M}_{\mathrm{RG}}~{}~{}.$$
(1)
Here $\dot{M}_{\mathrm{RG}}$ is the rate at which the white dwarf expands to giant dimensions and is given by
$$\dot{M}_{\mathrm{RG}}\simeq 8.5\times 10^{-7}(M_{\mathrm{WD}}/\mathrm{M}_{%
\odot}-0.52)\ \mathrm{M}_{\odot}/\mathrm{yr}~{}~{}.$$
(2)
Even assuming that the accretion rate could be regulated to the rate given by equation (1) [most likely it would settle on the Eddington rate of $\dot{M}_{\mathrm{EDD}}\simeq 1.7\times 10^{-5}(R_{\mathrm{WD}}/10^{9}~{}%
\mathrm{cm})\ \mathrm{M}_{\odot}/\mathrm{yr}$ at which mass would not be retained], the WD would increase in mass by at most $\sim$0.001 M${}_{\odot}$ during the CE phase. This would require the WD to be within 0.001 M${}_{\odot}$ of the Chandrasekhar mass upon entering the CE—a very unlikely situation, even taking into account the rarity of H$\alpha$ detection (e.g. only 2 out of a sample of 130 WDs were found to have masses higher than 1.2 M${}_{\odot}$; Bergeron, Saffer, & Liebert 1992; although see Hachisu & Kato 1999).
A second possibility is that the WD spirals-in all the way to the center and merges with the AGB star’s core. Interestingly, a scenario for SNe of similar type was suggested almost 30 years ago by Sparks & Stecher (1974), but has long since been discarded due to the absence of hydrogen in the spectra. What I propose here is that the spiraling-in process unbinds most, but not all of the envelope, so that coalescence becomes inevitable. At the time of merger, most of the envelope will be at a distance of
$$d\simeq 3\times 10^{15}\left(\frac{V}{10~{}\mathrm{km~{}s}^{-1}}\right)\left(%
\frac{\tau_{\mathrm{CE}}}{100~{}\mathrm{yr}}\right)\ \mathrm{cm}~{}~{}.$$
(3)
from the core. Here $V$ is the ejection velocity and $\tau_{\mathrm{CE}}$ is the duration of the CE phase. The condition for a merger to occur (as opposed to ejection of the entire envelope and the formation of a binary WD system) is given by the requirement that the binding energy of the CE be larger than the gravitational energy available from orbital shrinkage (Livio 1996; deKool 1990)
$$\frac{M_{\mathrm{AGB}}(M_{\mathrm{AGB}}-M_{C})}{\lambda\,a_{0}\,r_{L}}>\alpha_%
{\mathrm{CE}}\left(\frac{M_{C}M_{\mathrm{WD}}}{2R_{C}}-\frac{M_{\mathrm{AGB}}M%
_{\mathrm{WD}}}{2a_{0}}\right)~{}~{}.$$
(4)
Here $a_{0}$ is the initial separation, $r_{L}$ is the Roche lobe radius of the AGB star (in units of the separation), $M_{C}$ and $R_{C}$ are the mass and radius of the core, respectively, $\alpha_{\mathrm{CE}}$ is the CE efficiency parameter (Livio & Soker 1988; Iben & Tutukov 1984), and $\lambda\sim 0.5$ depends on the stellar density profile. The value of $\alpha_{\mathrm{CE}}$ is not known even to within a factor 10 (e.g. Livio 1996). However, for reasonable values ($\alpha_{\mathrm{CE}}\sim 0.1$–1) condition (4) requires relatively massive AGB stars [since the condition can be approximated as $(M_{\mathrm{AGB}}/M_{\mathrm{WD}})^{2}\gtrsim 1/8\,\alpha_{\mathrm{CE}}(a_{0}/%
R_{C})$; and $a_{0}/R_{C}\sim 10^{4}$] and can be expected to be satisfied only in a fraction of a percent of all systems (e.g. Yungelson & Livio 1998). The observed H$\alpha$ emission would result from the interaction of the explosion with the previously-ejected envelope. This would be consistent with the rarity of H$\alpha$ detections. Most importantly, however, if this scenario is correct, the H$\alpha$ detection by Hamuy et al. results from a double-degenerate scenario!
4 Conclusions
One might have thought that the detection of hydrogen in the spectrum of a SN Ia would have finally revealed the elusive progenitor to be a single-degenerate system. In this Letter we suggest that this may not be the case. Paradoxically, the H$\alpha$ detection could result from a double-degenerate scenario! To be sure, the actual result of the merger process remains as uncertain as ever, and it may lead to an accretion-induced collapse rather than to a SN Ia. Other exotic possibilities, such as the explosion of the core of an AGB star (“type 1.5” event; Iben & Renzini 1983) may exist (as already suggested by Hamuy et al. 2003). However, the latter would require some other mechanism to place (at least a part of) the envelope at $\sim 10^{15}$ cm. Future, more sensitive, observations will reveal whether the detection of H$\alpha$ is a very rare, but relatively clear event, or whether a range of line strengths is detected. The latter case would clearly support a single-degenerate interpretation.
We would like to thank David Branch and Tom Matheson for helpful discussions.
References
(1)
Bergeron, P., Saffer, R. A., & Liebert, J. 1992, ApJ, 394, 228
(2)
Branch, D., Livio, M., Yungelson, L. R., Boffi, F. R., & Baron, E. 1995, PASP, 107, 717
(3)
Capetti, A. 2002, ApJ, 574, L25
(4)
deKool, M. 1990, ApJ, 358, 189
(5)
Hachisu, I. & Kato, M. 1999, ApJ, 517, L47
(6)
Hamuy, M., et al. 2003, Nature, in press
(7)
Iben, I. Jr. & Renzini, A. 1983, AR&A, 21, 271
(8)
Iben, I. Jr. & Tutukov, A. V. 1984, ApJS, 54, 355
(9)
Livio, M. 1996, in Evolutionary Processes in Binary Stars, eds. R. A. M. J. Wijers, M. B. Davies, & C. A. Tout (Dordrecht: Kluwer), 141
(10)
Livio, M. 2001, in Supernovae and Gamma-Ray Bursts (Cambridge: Cambridge University Press), 334
(11)
Livio, M., Riess, A., & Sparks, W. 2002, ApJ, 571, L99
(12)
Livio, M. & Soker, N. 1988, ApJ, 329, 764
(13)
Nomoto, K., Nariai, K., & Sugimoto, D. 1979, PASJ, 31 287
(14)
Nomoto, K., Umeda, H., Kobayashi, C., Hachisu, I., & Tsujimoto, T. 2000, in AIP Conf. Proc. 522, Cosmic Explosions, ed. S. S. Holt & W. W. Zheng (Melville: AIP), 35
(15)
Paczyński, B. & Żytkow, A. N. 1978, ApJ, 222, 604
(16)
Perlmutter, S., et al. 1999, ApJ, 517, 565
(17)
Piersanti, L., Cassisi, S., Iben, I. Jr., & Tornambé, A. 2000, ApJ, 535, 932
(18)
Rasio, F. A. & Livio, M. 1996, ApJ, 471, 366
(19)
Riess, A. G., et al. 1998, AJ, 116, 1009
(20)
Sandquist, E. L., Taam, R. E., Chen, X., Bodenheimer, P., & Burkert, A. 1998, ApJ, 500, 909
(21)
Sparks, W. M. & Stecher, T. P. 1974, ApJ, 188, 149
(22)
Taam, R. E. & Sandquist, E. L. 2000, ARA&A, 38, 113
(23)
Webbink, R. F. 1984, ApJ, 227, 355
(24)
Whelan, J. & Iben, I. Jr. 1973, ApJ, 186, 1007
(25)
Yungelson, L. & Livio, M. 1998, ApJ, 497, 168
(26)
Yungelson, L. & Livio, M. 2000, ApJ, 528, 108 |
Formulation and properties of a divergence used to compare probability measures without absolute continuity
Paul Dupuis and Yixiang Mao
Research supported in part by the National Science Foundation (NSF-DMS-1904992).Research supported in part by the Air Force Office of Scientific Research (FA-9550-18-1-0214).
Abstract
This paper develops a new divergence that generalizes relative entropy and can be used to compare probability measures
without a requirement of absolute continuity.
We establish properties of the divergence, and in particular derive and exploit a representation
as an infimum convolution of optimal transport cost and relative entropy.
Also included are examples of computation and approximation of the divergence,
and the demonstration of properties that are useful when one quantifies model uncertainty.
1 Introduction
To compare different probabilistic models for a given application, one
needs a notion of “distance” between
the distributions. The specification of this distance is a subtle
issue. Probability models are typically large or infinite dimensional, and the
usefulness of the distance will depend on its mathematical properties. Is it
convenient for analysis and optimization? Does it scale well with system size?
For situations that require an analysis of (probabilistic) model form uncertainly, the
quantity known as relative entropy (or Kullback-Leibler divergence)
is the most widely used such distance. This is true because relative entropy
has all the attractive properties asked for in the last paragraph, and
many more. (Relative entropy is not a true metric since it is not symmetric in
its arguments, but owing to its other attributes it is more widely
used for these purposes than any legitimate metric.)
The definition of relative entropy is as follows. Suppose $S$ is a Polish
space with metric $d(\cdot,\cdot)$ and associated Borel $\sigma$-algebra
$\mathcal{B}$. Let $\mathcal{P}(S)$ be the space of probability measures over
$(S,\mathcal{B})$. If $\mu,\nu\in\mathcal{P}(S)$ and $\mu$ is absolutely continuous with respect to $\nu$ (denoted $\mu\ll\nu$), then
$$R(\mu\lVert\nu)\doteq\int_{S}\left(\log\frac{d\mu}{d\nu}\right)d\mu$$
(even though $\log{d\mu}/{d\nu}$ can take both positive and negative values,
as we discuss in the beginning of section 2, the definition is never ambiguous). Otherwise, we set $R(\mu\lVert\nu)=\infty$.
While we cannot go into
all the reasons why relative entropy is so useful, it is essential that we
describe why it is convenient for the analysis of
model form uncertainty. This is due to a
dual pair of variational formulas which relate $R(\mu\lVert\nu)$, integrals
with respect $\mu$, and what are called risk-sensitive integrals with respect
to $\nu$. Let $M_{b}(S)$ denote the set of bounded and measurable functions on $S$. Then
[8, Proposition 1.4.2 and Lemma 1.4.3] gives
$$R(\mu\left\|\nu\right.)=\sup_{g\in M_{b}(S)}\left\{\int_{S}gd\mu-\log\int_{S}e%
^{g}d\nu\right\},$$
(1.1)
and for any $g\in M_{b}(S)$,
$$\log\int_{S}e^{g}d\nu=\sup_{\mu\in\mathcal{P}(S)}\left\{\int_{S}gd\mu-R(\mu%
\left\|\nu\right.)\right\}.$$
It is immediate from either of these that for $\mu,\nu\in\mathcal{P}(S)$
and $g\in M_{b}(S)$,
$$\int_{S}gd\mu\leq R(\mu\left\|\nu\right.)+\log\int_{S}e^{g}d\nu.$$
If we interpret $\nu$ as the nominal or design model
(chosen perhaps on the basis of data or for computational tractability) and
$\mu$ as the true model (or at least a more accurate model),
then according to the last display one obtains a bound on an integral with respect to the true model.
(In fact by introducing a parameter one can obtain bounds that are in some sense optimal[10].)
We typically interpret the integral $\int_{S}gd\mu$ as a performance measure, and so we have a bound
on the performance of the system under the true distribution in terms of the
relative entropy distance $R(\mu\left\|\nu\right.)$, plus a
risk-sensitive performance measure under the design model. From this elementary
but fundamental inequality, and by exploiting the helpful qualitative and
quantitative properties of relative entropy, there has emerged a set of tools
that can be used to answer many questions where probabilistic model form
uncertainty is important, including
[2, 6, 7, 9, 10, 11, 12, 15, 16, 17, 18].
However, relative entropy has one important shortcoming: for the bound to be
meaningful we must have $R(\mu\left\|\nu\right.)<\infty$, which imposes
the requirement of absolute continuity of the true model with respect to the
design model. For various uses, such as model building and model
simplification, this restriction can be significant. In the context of model
building, it can happen that one attempts to fit distributions to data by
comparing an empirical measure constructed using data with the elements of a
parameterized family, such as a collection of Gaussian distributions. In this
case the two distributions one would compare are singular, and so relative
entropy cannot be used. A second example, and one that occurs frequently in
the physical sciences, operations research and elsewhere,
is that a detailed model (such as the population process of a
chemical reaction network, which takes values in a lattice) is approximated by
a simpler process that takes values in the continuum (for example a diffusion
process). For exactly the same reason as in the previous example, these
processes, as well as their corresponding stationary distributions, are not
absolutely continuous.
Because relative entropy is not directly applicable to such problems,
significant effort has been put into investigating alternatives ([3, 4] and references therein).
A class that has attracted some
attention (e.g., in the machine learning community) are the Wasserstein or, more generally,
optimal transport distances [13, 19, 23]. These distances, which are true
metrics, have certain attractive properties but also some
shortcomings. The most important shortcomings are: (a) Wasserstein distances
do not in general scale well with respect to system dimension, and (b) such
distances do not have an interpretation as the dual of a strictly convex function. To
be a little more concrete about point (b), it is
the strict concavity of the mapping
$$g\rightarrow\int_{S}gd\mu-\log\int_{S}e^{g}d\nu$$
in the variational representation for $R(\mu\left\|\nu\right.)$ that
leads to tight bounds when applied to problems of control or optimization of
stochastic uncertain systems. In contrast, the analogous variational representation for
Wasserstein type distances involves the mapping $g\rightarrow\int_{S}gd\mu-\int_{S}gd\nu$. Point (a) is an issue in applications
to problems from the physical sciences, where large time horizons and large
dimensions are common.
Rather than give up entirely the attractive features of the dual pair
$(R(\mu\left\|\nu\right.),\log\int_{S}e^{g}d\nu)$, an alternative is to
be more restrictive regarding the class of costs or performance measures for
which bounds are required. Indeed, the requirement of absolute
continuity in relative entropy is entirely due to the very large class of
functions, $M_{b}(S)$, appearing in (1.1). For a collection
$\Gamma\subset M_{b}(S)$ one can consider in lieu of $R(\mu\left\|\nu\right.)$ what we call the $\Gamma$-divergence, which is defined by
$$G_{\Gamma}(\mu\left\|\nu\right.)\doteq\sup_{g\in\Gamma}\left\{\int_{S}gd\mu-%
\log\int_{S}e^{g}d\nu\right\}.$$
By imposing regularity conditions on $\Gamma$ (e.g., Lipschitz continuity,
additional smoothness) one generates (under mild additional conditions on $\Gamma$)
divergences which relax the absolute continuity condition. Thus one is trading
restrictions on the class of performance measures or observables for which
bounds are valid, for the enlargement of the class of distributions to which
the bounds apply. These divergences are of course not as nice as relative
entropy, but one can prove that they retain versions of its most important
properties. In addition, the dual function (which serves as the cost to be
minimized when considering problems of optimization or control) remains
$\log\int_{S}e^{g}d\nu$. This is important because the corresponding risk-sensitive
optimization and optimal control problems are well studied in the literature.
In our formulation of the $\Gamma$-divergence the underlying idea is that to extend the range of probability measures that can be compared, one must restrict the class of integrands that will be considered.
However, this leads directly to an interesting connection with the Wasserstein distance
mentioned previously, which is that for suitable collections $\Gamma$ we will
prove the inf-convolution expression
$$\displaystyle G_{\Gamma}(\mu\left\|\nu\right.)$$
$$\displaystyle=\inf_{\gamma\in\mathcal{P}(S)}\left\{W_{\Gamma}(\mu-\gamma)+R(%
\gamma\left\|\nu\right.)\right\},$$
where $W_{\Gamma}$ is the Wasserstein metric whose dual (sup) formulation uses
the set of functions $\Gamma$. Moreover one recovers relative entropy by taking the
limit $b\rightarrow\infty$ in $G_{b\Gamma}(\mu\left\|\nu\right.)$, which
may be useful if one wants to allow relatively small violations of the absolute
continuity restriction, while at the same time taking advantage of simple
approximations for the Wasserstein distance in the high transportation cost limit.
The organization of the paper is as follows.
In Section 2 we define the $\Gamma$-divergence, and prove the first main result of this paper, which is the inf-convolution formula described above (Theorem 2.4). In Section 3, we show several properties of the
$\Gamma$-divergence, and establish a convex duality formula for the $\Gamma$-divergence.
Section 4 investigates the $\Gamma$-divergence for a special choices of $\Gamma$, which are sets of bounded Lipschitz continuous functions.
We establish a relation between $\Gamma$-divergence and optimal transport cost, and prove existence and uniqueness for optimizers of variational representations of $\Gamma$-divergence (Theorem 4.8), and also a formula for directional derivatives of the $\Gamma$-divergence (Theorem 4.14).
Section 5 considers limits for the $\Gamma$-divergence, and in Section 6 there is a preliminary discussion on how one can apply the $\Gamma$-divergence to obtain uncertainty quantification bounds.
As a last remark we note that the paper [1] defines a “relaxation” of
Wasserstein distance by putting in an entropy term of the mass-transfer
matrix. The new divergence so defined is easier to compute than the original
Wasserstein distance, but is not the same as the divergences we develop here.
2 Definition of the ${\Gamma}$-divergence
Throughout this section, $S$ is a Polish space with metric $d(\cdot,\cdot)$
and associated Borel $\sigma$-alegra $\mathcal{B}$. $C_{b}(S)$ denotes the
space of all bounded continuous functions from $S$ to $\mathbb{R}$, and
$M_{b}(S)$ denotes the space of all bounded measurable functions from $S$ to
$\mathbb{R}$. Let $\mathcal{P}(S)$ be the space of probability measures over
$(S,\mathcal{B})$, $\mathcal{M}(S)$ be the space of finite signed (Borel)
measures over $(S,\mathcal{B})$, and $\mathcal{M}_{0}(S)$ be the subspace of
$\mathcal{M}(S)$ whose total mass is $0$. $\overline{\mathbb{R}}\doteq\mathbb{R}\cup\{\infty\}$ is the extended real numbers. Throughout this
section, we consider $C_{b}(S)$ equipped with weak topology induced by
$\mathcal{M}(S)$. Thus for $f_{n},f\in C_{b}(S)$, $f_{n}\rightarrow f$ if $\int_{S}f_{n}d\mu\rightarrow\int_{S}fd\mu$ for all $\mu\in\mathcal{M}(S)$.
We recall that
$$R(\mu\lVert\nu)\doteq\int_{S}\left(\log\frac{d\mu}{d\nu}\right)d\mu$$
whenever $\mu$ is absolutely continuous with respect to $\nu$. For $t\in\mathbb{R}$ define $t^{-}\doteq-(t\wedge 0)$. Since the function $s(\log s)^{-}$ is bounded for
$s\in[0,\infty)$, whenever $\mu\ll\nu$,
$$\int_{S}\left(\log\frac{d\mu}{d\nu}\right)^{-}d\mu=\int_{S}\frac{d\mu}{d\nu}%
\left(\log\frac{d\mu}{d\nu}\right)^{-}d\nu<\infty.$$
Thus $R(\mu\lVert\nu)$ is always well defined.
We recall the Donsker-Varadhan variational representation (1.1)
for relative
entropy.
We will use equation (1.1) as an equivalent
characterization of $R(\cdot\lVert\nu)$ on $\mathcal{P}(S)$, and consider an extension to
$\mathcal{M}(S)$ in the following lemma. With an abuse of notation, we will also call the extended function $R$.
To set up the functionals of interest on a space with the proper structure (locally convex Hausdorff space),
we will use that
$$\sup_{g\in C_{b}(S)}\left\{\int_{S}gd\mu-\log\int_{S}e^{g}d\nu\right\}=\sup_{g%
\in M_{b}(S)}\left\{\int_{S}gd\mu-\log\int_{S}e^{g}d\nu\right\}$$
(2.1)
[8, Lemma 1.4.3(a)](It is worth notating in this reference, (2.1) is only proved for $\mu,\nu\in\mathcal{P}(S)$. However, the exact same argument applies for $\mu,\nu\in\mathcal{M}(S)$, and we are using the latter version here). The fact that one obtains the same value
when supremizing over the smaller class $C_{b}(S)$ is closely related to the fact that
$R(\mu\lVert\nu)$ is finite only when $\mu\ll\nu$.
Lemma 2.1
Consider $R:\mathcal{M}(S)\times\mathcal{P}(S)\rightarrow(-\infty,\infty]$ defined by (1.1). Then
1.
$R(\mu\lVert\nu)\geq 0$ and $R(\mu\lVert\nu)=0$ if and only if $\mu=\nu$,
2.
$R(\cdot\lVert\cdot)$ is convex,
3.
$R(\mu\lVert\nu)=\infty$ if $\mu\in\mathcal{M}(S)\backslash\mathcal{P}(S)$.
Proof. If we prove item 3, then items 1 and 2 will follow from the corresponding
statements when $\mu$ is restricted to $\mathcal{P}(S)\leavevmode\nobreak\ $[8]. If
$m=\mu(S)\neq 1$, then taking $g(x)\equiv c$ a constant,
$$\int_{S}gd\mu-\log\int_{S}e^{g}d\nu=c\mu(S)-c=c(m-1).$$
Since $m\neq 1$ and $c\in\mathbb{R}$, the right hand side of equation
(1.1) is $\infty$.
Suppose next that $\mu(S)=1$ but $\mu\in\mathcal{M}(S)\backslash\mathcal{P}(S)$. Then there exist sets $A,B\in\mathcal{B}$ such that $A\cap B=\varnothing,A\cup B=S,\mu(A)<0$ and $\mu(B)>0$. For $c>0$, let $g(x)=-c$ for
$x\in A$ and $g(x)=0$ for $x\in B$. Then
$$\int_{S}gd\mu-\log\int_{S}e^{g}d\nu=c\left|\mu(A)\right|-C_{c},$$
where $C_{c}\in(\log\nu(B),0)$ for all $c$. Letting $c\rightarrow\infty$ and
using (2.1) shows $R(\mu\lVert\nu)=\infty$.
Though relative entropy has very attractive regularity and optimization
properties, as noted $R(\mu\lVert\nu)$ is finite if and only if $\mu\ll\nu$.
As such, it cannot be used to give a meaningful notion of “distance” without this absolute continuity restriction. In
order to define a meaningful divergence for a pair of probability measures
that are not mutually absolute continuous, but at the same time not to lose the
useful properties of the “dual” function
$g\rightarrow\log\int_{S}e^{g}d\nu$ appearing in (1.1), a natural
approach is to restrict the set of test functions in the variational formula.
We define a criterion for the classes of “admissible” test functions we want to use.
Definition 2.2
Let $\Gamma$ be a subset of $C_{b}(S)$ endowed with the inherited weak topology. We call $\Gamma$
admissible if the following hold.
1) $\Gamma$ is convex and closed.
2) $\Gamma$ is symmetric in that $g\in\Gamma$ implies $-g\in\Gamma$, and
$\Gamma$ contains all constant functions.
3) $\Gamma$ is determining for $\mathcal{P}(S)$, i.e., for any $\mu,\nu\in\mathcal{P}(S)$ with $\mu\neq\nu$, there exists $g\in\Gamma$ such that
$$\int_{S}gd\mu\neq\int_{S}gd\nu.$$
We next define a new divergence by restricting the class of test functions in
the definition of relative entropy.
Definition 2.3
Fix $\nu\in\mathcal{P}(S)$. For $\mu\in\mathcal{M}(S)$, we define the
$\mathbf{\Gamma}$-divergence associated with the admissible set $\Gamma$ by
$$G_{\Gamma}(\mu\lVert\nu)\doteq\sup_{g\in\Gamma}\left\{\int_{S}gd\mu-\log\int_{%
S}e^{g}d\nu\right\}.$$
We also define the following related quantity. For $\eta\in\mathcal{M}(S)$ let
$$W_{\Gamma}(\eta)\doteq\sup_{g\in\Gamma}\left\{\int_{S}gd\eta\right\}=\sup_{g%
\in\mathcal{C}_{b}(S)}\left\{\int_{S}gd\eta-\infty 1_{\{g\in\Gamma^{c}\}}%
\right\}.$$
When $\Gamma$ is clear based on context, we will drop the subscript from
$G_{\Gamma}$ and $W_{\Gamma}$. Using a similar argument as in Lemma
2.1, one can show that $G_{\Gamma}(\mu\lVert\nu)=\infty$ if $\mu(S)\neq 1$. The next theorem
states an important property of the ${\Gamma}$-divergence, which is that it
can be written as a convolution involving relative entropy and $W_{\Gamma}$.
Theorem 2.4
Assume $\Gamma$ is an admissible set. Then for $\mu\in\mathcal{M}(S)$, $\nu\in\mathcal{P}(S)$,
$$G_{\Gamma}(\mu\lVert\nu)=\inf_{\gamma\in\mathcal{P}(S)}\left\{R(\gamma\lVert%
\nu)+W_{\Gamma}(\mu-\gamma)\right\}$$
Remark 2.5
The theorem tells us that by restricting the set of test functions in the
variational representation of relative entropy, we get a quantity which is an
inf-convolution of relative entropy and a metric. It will be pointed out in
Section 4 that by restricting $\Gamma$ to Lipschitz functions
with respect to a cost function $c(x,y)$ that satisfies some specified
conditions, $W_{\Gamma}(\mu-\nu)$ will be the corresponding optimal transport
cost from $\mu$ to $\nu$.
The rest of this section is focused on the proof of Theorem 2.4. In
order to do this, we need a few definitions and also will find it convenient
to consider a more general setting.
Definition 2.6
Points $x$ and $y$ in a topological space $Y$ can be separated if
there exists an open neighborhood $U$ of $x$ and an open neighborhood $V$ of
$y$ such that $U$ and $V$ are disjoint ($U\cap V=\varnothing$). $Y$ is a
$\mathbf{Hausdorff}$ space if all distinct points in $Y$ are pairwise separable.
Definition 2.7
A subset $C$ of a topological vector space $Y$ over the number field
$\mathbb{R}$ is
1. $\mathbf{convex}$ if for any $x,y\in C$ and any $t\in[0,1]$,
$tx+(1-t)y\in C$,
2. $\mathbf{balanced}$ if for all $x\in C$ and any $\lambda\in\mathbb{R}$ with
$|\lambda|\leq 1$, $\lambda x\in C$,
3. $\mathbf{absorbant}$ if for all $y\in Y$, there exists $t>0$ and $x\in C$
such that $y=tx$.
A topological vector space $Y$ is called $\mathbf{locally}$ $\mathbf{convex}$
if the origin has a local topological basis of convex, balanced and absorbent sets.
Definition 2.8
For a topological vector space $Y$ over the number field $\mathbb{R}$, its
$\mathbf{topological\ dual\ space}$ $Y^{*}$ is defined as the space of all
continuous linear functionals ${\displaystyle\varphi:Y\to{\mathbb{R}}}$.
The $\mathbf{weak^{\ast}\ topology}$ on $Y^{\ast}$ is the topology induced by
$Y$. In other words, it is the coarsest topology such that functional
$y:Y^{\ast}\rightarrow\mathbb{R}$, $y(\varphi)=\varphi(y)$ is continuous in
$Y^{\ast}$.
For $y\in Y$ and $\varphi\in Y^{\ast}$, we also write $\langle y,\varphi\rangle\doteq\varphi(y)=y(\varphi)$.
Now let $Y$ be a Hausdorff locally convex space with $Y^{\ast}$ being its
topological dual space and endowed with the weak* topology.
Definition 2.9
For a function $f:Y\rightarrow\overline{\mathbb{R}}$, its $\mathbf{convex}\ \mathbf{dual}$ $f^{\ast}:Y^{\ast}\rightarrow\overline{\mathbb{R}}$ is
defined by
$$f^{\ast}(z)=\sup_{y\in Y}\left\{\langle y,z\rangle-f(y)\right\}.$$
Definition 2.10
Let $f_{1},f_{2}:Y\rightarrow\overline{\mathbb{R}}$ be two functions. We
define the inf-convolution of $f_{1}$ and $f_{2}$ by
$$\left[f_{1}\Box f_{2}\right](y)\doteq\inf_{y_{1}\in Y}\{f_{1}(y_{1})+f_{2}(y-y%
_{1})\}.$$
Definition 2.11
For a function $f:Y\rightarrow\overline{\mathbb{R}}$ the
$\mathbf{lower\ semicontinuous\ hull}$ $\overline{f}$ is defined by
$$\overline{f}(x)\doteq\sup\{g(x):g\leq f,g:Y\rightarrow\overline{\mathbb{R}}\ %
is\ continuous\}.$$
Definition 2.12
A convex function $f:Y\rightarrow\overline{\mathbb{R}}$ is proper if
there exists $y\in Y$ such that $f(y)<\infty$. The domain of a
convex, proper funciton $f$ is defined by
$$\mathrm{dom}(f)\doteq\{y\in Y:f(y)<\infty\}.$$
Now let us introduce an important lemma.
Lemma 2.13
[5, Theorem 2.3.10] Let $f_{i}:Y\rightarrow\overline{\mathbb{R}}$ be convex, proper and lower-semicontinuous functions
fulfilling $\bigcap_{i=1}^{m}\mathrm{dom}(f_{i})\neq\varnothing$. Then one has
$$\left(\sum_{i=1}^{m}f_{i}\right)^{\ast}=\overline{f_{1}^{\ast}\Box\cdots\Box f%
_{m}^{\ast}}.$$
In our use we take $Y=C_{b}(S)$ equipped with topology induced by
$\mathcal{M}(S)$, i.e., the topological basis around $g\in Y$ is taken as sets
of the form
$$\left\{f\in Y:\int_{S}fd\mu_{k}\in\left(\int_{S}gd\mu_{k}-\epsilon_{k},\int_{S%
}gd\mu_{k}+\epsilon_{k}\right),k=1,2,\dots,m\right\},$$
where $m\in\mathbb{N},\{\mu_{k}\}_{k=1,2,\dots,m}\subset\mathcal{M}(S)$ and
$\epsilon_{k}>0,k=1,2,\dots,m$ are arbitrary. It can be easily verified that
under this topology, $C_{b}(S)$ is a Hausdorff locally convex space, with
$C_{b}(S)^{\ast}=\mathcal{M}(S)$ [21, Theorem 3.10].
For $g\in C_{b}(S)$ and $\mu\in\mathcal{M}(S)$, we define the bilinear form
$$\langle g,\mu\rangle\doteq\int_{S}gd\mu.$$
We are now ready to prove the main theorem.
Proof of Theorem 2.4. Define $H_{1},H_{2}:C_{b}(S)\rightarrow\overline{\mathbb{R}}$ by
$$H_{1}(g)\doteq\log\int_{S}e^{g}d\nu\text{ and }H_{2}(g)\doteq\infty 1_{\Gamma^%
{c}}(g).$$
Then
$$\displaystyle G_{\Gamma}(\mu\lVert\nu)$$
$$\displaystyle=\sup_{g\in\Gamma}\left\{\int_{S}gd\mu-\log\int_{S}e^{g}d\nu\right\}$$
$$\displaystyle=\sup_{g\in C_{b}(S)}\left\{\int_{S}gd\mu-\log\int_{S}e^{g}d\nu-%
\infty 1_{\Gamma^{c}}(g)\right\}$$
$$\displaystyle=\left(H_{1}+H_{2}\right)^{\ast}(\mu).$$
Notice that $\{0\}\in\mathrm{dom}(H_{1})\cap\mathrm{dom}(H_{2})\neq\varnothing$, and both $H_{1}$ and $H_{2}$ are proper and convex. For
lower-semicontinuity, under the topology induced by $\mathcal{M}(S)$, $H_{1}$ is
actually continuous, and $H_{2}$ is lower semicontinuous since $\Gamma$ is
closed. Thus,
by Lemma 2.13
$$G_{\Gamma}(\mu\lVert\nu)=(H_{1}+H_{2})^{\ast}(\mu)=[\overline{H_{1}^{\ast}\Box
H%
_{2}^{\ast}}](\mu).$$
By equation (1.1) and the definition of $W_{\Gamma}$, we know that
$$R(\mu\lVert\nu)=H_{1}^{\ast}(\mu)\text{ and }W_{\Gamma}(\eta)=H_{2}^{\ast}(%
\eta).$$
In the following display, the first equality is due to the definition of
inf-convolution, and the second is since $R(\gamma\lVert\nu)<\infty$ only when
$\gamma\in\mathcal{P}(S)$:
$$\displaystyle H_{1}^{\ast}\Box H_{2}^{\ast}(\mu)$$
$$\displaystyle=\inf_{\gamma\in\mathcal{M}(S)}\left\{R(\gamma\lVert\nu)+W_{%
\Gamma}(\mu-\gamma)\right\}$$
$$\displaystyle=\inf_{\gamma\in\mathcal{P}(S)}\left\{R(\gamma\lVert\nu)+W_{%
\Gamma}(\mu-\gamma)\right\}.$$
Thus the last thing we need to prove is that $H_{1}^{\ast}\Box H_{2}^{\ast}$
is lower semicontinuous. Note that relative entropy is lower semicontinuous in
the first argument in the weak topology [8, Lemma 1.4.3 (b)], and
$W_{\Gamma}$ is lower semicontinuous in the weak topology since
it is the supremum of a collection of linear functionals. Let
$$F(\mu)\doteq H_{1}^{\ast}\Box H_{2}^{\ast}(\mu)=\inf_{\gamma\in\mathcal{P}(S)}%
\left\{R(\gamma\lVert\nu)+W_{\Gamma}(\mu-\gamma)\right\}.$$
Consider any sequence $\mu_{n}\Rightarrow\mu$ with $\mu_{n},\mu\in\mathcal{M}(S)$. Here “$\Rightarrow$” means convergence in the weak${}^{*}$ topology, i.e., for any $f\in C_{b}(S)$, $\int fd\mu_{n}\to\int fd\mu$. Let $\varepsilon>0$, and
for each $\mu_{n}$ let $\gamma_{n}$ satisfy
$$R(\gamma_{n}\lVert\nu)+W_{\Gamma}(\mu_{n}-\gamma_{n})\leq F(\mu_{n})+\varepsilon.$$
We want to show that
$$\liminf_{n\rightarrow\infty}F(\mu_{n})\geq F(\mu).$$
(2.2)
If $\liminf_{n\rightarrow\infty}F(\mu_{n})=\infty$, the inequality above holds
automatically. Assuming $\liminf_{n\rightarrow\infty}F(\mu_{n})<\infty$, let
$n_{k}$ be a subsequence such that
$$\lim_{k\rightarrow\infty}F(\mu_{n_{k}})=\liminf_{n\rightarrow\infty}F(\mu_{n}).$$
Notice that
$$R(\gamma_{n_{k}}\lVert\nu)\leq R(\gamma_{n_{k}}\lVert\nu)+W_{\Gamma}(\mu_{n_{k%
}}-\gamma_{n_{k}})\leq F(\mu_{n_{k}})+\varepsilon.$$
Since $\{F(\mu_{n_{k}})\}_{k\geq 1}$ is bounded, we know that $\{\gamma_{n_{k}}\}_{k\geq 1}$ is tight [8, Lemma 1.4.3(c)]. Then we can take a further
subsequence that converges weakly. For simplicity of notation, let $n_{k}$
denote this subsequence, and let $\gamma_{\infty}$ denote the weak limit of
$\gamma_{n_{k}}$. Then using the lower semicontinuity of $R(\cdot\lVert\nu)$
on $\mathcal{P}(S)$ and the lower semicontinuity of $W_{\Gamma}$ on
$\mathcal{M}(S)$,
$$\displaystyle\liminf_{n\rightarrow\infty}F(\mu_{n})+\varepsilon$$
$$\displaystyle=\lim_{k\rightarrow\infty}F(\mu_{n_{k}})+\varepsilon$$
$$\displaystyle\geq\lim_{k\rightarrow\infty}\left[R(\gamma_{n_{k}}\lVert\nu)+W(%
\mu_{n_{k}}-\gamma_{n_{k}})\right]$$
$$\displaystyle\geq R(\gamma_{\infty}\lVert\nu)+W(\mu-\gamma_{\infty})$$
$$\displaystyle\geq\inf_{\gamma\in\mathcal{P}(S)}\left\{R(\gamma\lVert\nu)+W_{%
\Gamma}(\mu-\gamma)\right\}$$
$$\displaystyle=F(\mu).$$
Since $\varepsilon>0$ is arbitrary this establishes (2.2), and thus
$F$ is lower semicontinuous in $\mathcal{M}(S)$. The theorem is proved.
3 Properties of the $\Gamma$-divergence
Theorem 2.4 provides an interesting characterization of the $\Gamma$-divergence. Before we
continue to specific choices of $\Gamma$, we first state some general
properties associated with $\Gamma$-divergence. Throughout this section we fix an
admissible set $\Gamma$, and thus drop the subscript from $G_{\Gamma}$ and
$W_{\Gamma}$ in this section. Also, now that we have established the
expression for $G$ as an inf-convolution as in Theorem 2.4, we no
longer need to consider $G$ as a function on $\mathcal{M}(S)\times\mathcal{P}(S)$, and instead can consider it just on $\mathcal{P}(S)\times\mathcal{P}(S)$, since we want to use $G$ as a measure of how two probability distributions differ.
Lemma 3.1
For $(\mu,\nu)\in\mathcal{P}(S)\times\mathcal{P}(S)$ define
$G(\mu\lVert\nu)$ by Definition 2.3 and assume $\Gamma$ is
admissible. Then the following properties hold.
1) $G(\mu\lVert\nu)\geq 0$, with $G(\mu\lVert\nu)=0$ if and only if $\mu=\nu$.
2) $G(\mu\lVert\nu)$ is a convex and lower semicontinuous function of
$(\mu,\nu)$. In particular, $G(\mu\lVert\nu)$ is a convex, lower
semicontinuous function of each variable $\mu$ or $\nu$ separately.
3) $G(\mu\lVert\nu)\leq R(\mu\lVert\nu)$ and $G(\mu\lVert\nu)\leq W(\mu-\nu)$.
Remark 3.2
1) The first property justifies our calling $G$ a divergence as the term is used in information theory.
2) Relative entropy has the property that for each fixed $\nu\in\mathcal{P}(S)$,
$R(\cdot\lVert\nu)$ is strictly convex on $\{\mu\in\mathcal{P}(S):R(\mu\lVert\nu)<\infty\}$. However,
$G(\cdot\lVert\nu)$ in general is not strictly convex.
Proof of Lemma 3.1.
1) As noted in Lemma 2.1, $R(\cdot\lVert\cdot)$ is non-negative
[8, Lemma 1.4.1], and for any $\mu\in\mathcal{P}(S)$
$$W(\mu)=\sup_{g\in\Gamma}\left\{\int_{S}gd\mu\right\}\geq\int_{S}0d\mu=0.$$
Thus
$$G(\mu\lVert\nu)=\inf\{R(\mu_{1}\lVert\nu)+W(\mu_{2}):\mu_{1}+\mu_{2}=\mu\}\geq
0.$$
Also by Lemma 2.1, $R(\mu_{1}\lVert\nu)=0$ if and only if
$\mu_{1}=\nu$. Thus $G(\mu\lVert\nu)=0$ if and only if
$$W(\mu-\nu)=\sup_{g\in\Gamma}\left\{\int_{S}gd(\mu-\nu)\right\}=0,$$
which tells us $\mu=\nu$ since $\Gamma$ is admissible.
2) This is a straightforward corollary of Theorem 2.4, since the
supremum of a collection of linear and continuous functionals is both convex
and lower semicontinuous.
3) This follows from Theorem 2.4 and that $R(\nu\lVert\nu)=W(0)=0$.
For relative entropy we have the following lemma [8, Proposition 1.4.2].
Lemma 3.3
For all $g\in C_{b}(S)$
$$\log\int_{S}e^{g}d\nu=\sup_{\mu\in\mathcal{P}(S)}\left\{\int_{S}gd\mu-R(\mu%
\lVert\nu)\right\},$$
where the supremum is achieved uniquely at $\mu_{0}$ satisfying
$$\frac{d\mu_{0}}{d\nu}(x)\doteq\frac{e^{g(x)}}{\int_{S}e^{g}d\nu}.$$
A similar duality formula holds for the $\Gamma$-divergence when $g\in\Gamma$.
Theorem 3.4
If $\Gamma$ is admissible then for $g\in\Gamma$
$$\log\int_{S}e^{g}d\nu=\sup_{\mu\in\mathcal{P}(S)}\left\{\int_{S}gd\mu-G(\mu%
\lVert\nu)\right\}.$$
Proof. Using the definition of $G$ divergence
$$\displaystyle\sup_{\mu\in\mathcal{P}(S)}\left\{\int_{S}gd\mu-G(\mu\lVert\nu)\right\}$$
$$\displaystyle=\sup_{\mu\in\mathcal{P}(S)}\left\{\int_{S}gd\mu-\sup_{f\in\Gamma%
}\left\{\int_{S}fd\mu-\log\int_{S}e^{f}d\nu\right\}\right\}$$
$$\displaystyle\leq\sup_{\mu\in\mathcal{P}(S)}\left\{\int_{S}gd\mu-\left\{\int_{%
S}gd\mu-\log\int_{S}e^{g}d\nu\right\}\right\}$$
$$\displaystyle=\log\int_{S}e^{g}d\nu.$$
On the other hand, we know for relative entropy that
$$\log\int_{S}e^{g}d\nu=\sup_{\mu\ll\nu}\left\{\int_{S}gd\mu-R(\mu\lVert\nu)%
\right\}.$$
Since $G(\mu\lVert\nu)\leq R(\mu\lVert\nu)$,
$$\displaystyle\log\int_{S}e^{g}d\nu$$
$$\displaystyle=\sup_{\mu\ll\nu}\left\{\int_{S}gd\mu-R(\mu\lVert\nu)\right\}$$
$$\displaystyle\leq\sup_{\mu\ll\nu}\left\{\int_{S}gd\mu-G(\mu\lVert\nu)\right\}$$
$$\displaystyle\leq\sup_{\mu\in\mathcal{P}(S)}\left\{\int_{S}gd\mu-G(\mu\lVert%
\nu)\right\}.$$
The statement of the theorem follows from the two inequalities.
The last theorem has two important implications.
The first is related to the fact that Lemma 3.3 implies bounds for $\int_{S}gd\mu$ when $R(\mu\lVert\nu)$ is bounded,
and observation that has served as the basis for the analysis of various aspects of model form uncertainty [7, 10]. Using Theorem 3.4, we obtain analogous bounds on $\int_{S}gd\mu$ for $g\in\Gamma$ when $G(\mu\lVert\nu)$ is bounded. Applications of these bounds will be further developed elsewhere.
The second is that for $g\in\Gamma$, if we take $\mu_{0}$ as defined in Lemma 3.3, then
$$\displaystyle\log\int_{S}e^{g}d\nu$$
$$\displaystyle=\int_{S}gd\mu_{0}-R(\mu_{0}\lVert\nu)$$
$$\displaystyle\leq\int_{S}gd\mu_{0}-G(\mu_{0}\lVert\nu)$$
$$\displaystyle\leq\sup_{\mu\in\mathcal{P}(S)}\left\{\int_{S}gd\mu-G(\mu\lVert%
\nu)\right\}$$
$$\displaystyle=\log\int_{S}e^{g}d\nu,$$
where the first inequality comes from $G(\mu_{0}\lVert\nu)\leq R(\mu_{0}\lVert\nu)$. Since both
inequalities above must be equalities, we must have
$$R(\mu_{0}\lVert\nu)=G(\mu_{0}\lVert\nu).$$
The next lemma gives a more detailed picture of $G(\mu\lVert\nu)$ when $\mu\ll\nu$.
Lemma 3.5
For $\mu,\nu\in\mathcal{P}(S)$, if $\mu\ll\nu$ then
$$G(\mu\lVert\nu)=\sup_{\gamma\in\mathcal{A}(S)}\left\{\int_{S}\log\left(\frac{d%
\gamma}{d\nu}\right)d\mu\right\},$$
where
$$\mathcal{A}(S)\doteq\left\{\gamma\in\mathcal{P}(S):\gamma\ll\nu,\exists g\in%
\Gamma\mbox{ such that }\frac{d\gamma}{d\nu}(x)=e^{g(x)}\mbox{ for }x\in%
\mathrm{supp}(\nu)\right\}.$$
Proof. We use the definition
$$G(\mu\lVert\nu)=\sup_{g\in\Gamma}\left\{\int_{S}gd\mu-\log\int_{S}e^{g}d\nu\right\}$$
to prove this lemma.
For any $g\in\Gamma$, we define $\gamma_{g}\in\mathcal{P}(S)$ by the relation
$$\frac{d\gamma_{g}}{d\nu}(x)=\frac{e^{g(x)}}{\int_{S}e^{g}d\nu}$$
for $x\in\mathrm{supp}(\nu)$, and $\gamma_{g}(\mathrm{supp}(\nu)^{c})=0$. Then for $x\in\mathrm{supp}(\nu)$,
$$\log\left(\frac{d\gamma_{g}}{d\nu}(x)\right)=g(x)-\log\int_{S}e^{g}d\nu.$$
Since $\mu\ll\nu$, we have
$$\int_{S}\log\left(\frac{d\gamma_{g}}{d\nu}\right)d\mu=\int_{S}gd\mu-\log\int_{%
S}e^{g}d\nu,$$
and thus
$$G(\mu\lVert\nu)=\sup_{g\in\Gamma}\left\{\int_{S}gd\mu-\log\int_{S}e^{g}d\nu%
\right\}\leq\sup_{\gamma\in\mathcal{A}(S)}\left\{\int_{S}\log\left(\frac{d%
\gamma}{d\nu}\right)d\mu\right\}.$$
On the other hand, for any $\gamma\in\mathcal{A}(S)$, by definition, we can find a $g_{\gamma}\in\Gamma$ such that
$$g_{\gamma}(x)=\log\left(\frac{d\gamma}{d\nu}(x)\right)$$
for $x\in\mathrm{supp}(\nu)$. Then
$$\int_{S}g_{\gamma}d\mu-\log\int_{S}e^{g_{\gamma}}d\nu=\int_{S}\log\left(\frac{%
d\gamma}{d\nu}\right)d\mu.$$
Thus
$$\sup_{\gamma\in\mathcal{A}(S)}\left\{\int_{S}\log\left(\frac{d\gamma}{d\nu}%
\right)d\mu\right\}\leq\sup_{g\in\Gamma}\left\{\int_{S}gd\mu-\log\int_{S}e^{g}%
d\nu\right\}=G(\mu\lVert\nu).$$
Combining the two inequalities completes the proof.
Remark 3.6
When $\mu\in\mathcal{A}(S)$ we always have $G(\mu\lVert\nu)=R(\mu\lVert\nu)$.
This is because if $\gamma\in\mathcal{A}(S)$ then $\mu\ll\gamma$, and therefore
$$\int_{S}\log\left(\frac{d\mu}{d\nu}\right)d\mu-\int_{S}\log\left(\frac{d\gamma%
}{d\nu}\right)d\mu=\int_{S}\log\left(\frac{d\mu}{d\gamma}\right)d\mu=R(\mu%
\lVert\gamma)\geq 0.$$
Rearranging gives
$$\int_{S}\log\left(\frac{d\gamma}{d\nu}\right)d\mu=R(\mu\lVert\nu)-R(\mu\lVert%
\gamma),$$
and so
$$G(\mu\lVert\nu)=\sup_{\gamma\in\mathcal{A}(S)}\left\{\int_{S}\log\left(\frac{d%
\gamma}{d\nu}\right)d\mu\right\}=R(\mu\lVert\nu).$$
This statement is not valid when $\mu\ll\nu$ does not hold, since then $\log(d\gamma/d\nu)$ is not defined in $\mathrm{supp}(\mu)\backslash\mathrm{supp}(\nu)$, thus
$$\int_{S}\log\left(\frac{d\gamma}{d\nu}\right)d\mu$$
is not well defined.
4 Connection with optimal transport theory
In the proceeding sections, we discussed general properties for the $\Gamma$-divergence
with an admissible set $\Gamma\subset C_{b}(S)$. In this section, we
discuss specific choices of $\Gamma$ which relate the $\Gamma$-divergence with
optimal transport theory. First we state some well known results in
optimal transport theory.
4.1 Preliminary results from optimal transport theory
The results in this section are from [19, Chapter 4]. The
general Monge-Kantorovich mass transfer problem with given marginals $\mu,\nu\in\mathcal{P}(S)$ and cost function $c:S\times S\rightarrow\mathbb{R}_{+}$ is
$$\mathcal{C}(c;\mu,\nu)\doteq\inf_{\pi\in\Pi(\mu,\nu)}\left\{\int_{S\times S}c(%
x,y)\pi(dx,dy)\right\},$$
where $\Pi(\mu,\nu)$ denotes the collection of all probability
measures on $S\times S$ with first and second marginals being $\mu$ and $\nu$, respectively.
A natural dual problem with respect to this is
$$\mathcal{B}(c;\rho)\doteq\sup_{f\in\mathrm{Lip}(c,S;C_{b}(S))}\left\{\int_{S}f%
(x)\rho(dx)\right\},$$
where $\rho=\mu-\nu$, $C_{b}(S)$ denotes the set of bounded continuous functions mapping $S$ to $\mathbb{R}$ and
$$\displaystyle\mathrm{Lip}(c,S;C_{b}(S))\doteq\left\{f\in C_{b}(S):f(x)-f(y)%
\leq c(x,y)\mbox{ for all }x,y\in S\right\}.$$
(4.1)
We want to know when
$$\mathcal{C}(c;\mu,\nu)=\mathcal{B}(c,\rho)$$
(4.2)
holds. The following is a necessary and sufficient condition.
As with many results in this section,
one can extend in a trivial way to the case where costs are bounded from below, rather than non-negative.
Recall that $S$ is a Polish space.
Condition 4.1
There is a nonempty
subset $Q\subset C_{b}(S)$ such that the cost
$c:S\times S\rightarrow[0,\infty]$ has the representation
$$c(x,y)=\sup_{u\in Q}\left(u(x)-u(y)\right)\quad\text{for\ all}\ (x,y)\in S%
\times S.$$
(4.3)
Theorem 4.2
[19, Theorem 4.6.6] Under Condition 4.1,
(4.2) holds.
Remark 4.3
Condition 4.1 implies that $c$ satisfies the triangle inequality, i.e., for all $x,y,z\in S$
$$c(x,z)\leq c(x,y)+c(y,z).$$
This follows easily from
$$\displaystyle\sup_{u\in Q}\left(u(x)-u(z)\right)$$
$$\displaystyle=\sup_{u\in Q}\left((u(x)-u(y))+(u(y)-u(z))\right)$$
$$\displaystyle\leq\sup_{u\in Q}\left(u(x)-u(y)\right)+\sup_{u\in Q}\left(u(y)-u%
(z)\right).$$
On the other hand, Condition 4.1 also allows for a wide range of choices of
$c(x,y)$. For example, suppose that $c$ is a continuous metric on $S$, where
continuity is with respect to the underlying metric
of $S$. Then we can choose
$$Q=\left\{\min(c(x,x_{0}),n):x_{0}\in S,n\in\mathbb{N}\right\}.$$
It is easily verified that $Q\subset C_{b}(S)$, and that with this choice of
$Q$ (4.3) holds.
4.2 $\Gamma$-divergence with the choice $\Gamma=\mathrm{Lip}(c,S;C_{b}(S))$
Suppose $\Gamma=\mathrm{Lip}(c,S;C_{b}(S))$, with $c:S\times S\rightarrow[0,\infty]$ satisfying
Condition 4.1. To make the presentation simple, we have assumed that $c$ is non-negative,
and further assume it is symmetric, meaning $c(x,y)=c(y,x)\geq 0$ for any $x,y\in S$.
To distinguish from $W_{\Gamma}(\mu-\nu)$ for general $\Gamma$,
we denote the transport cost for $\mu,\nu\in\mathcal{P}(S)$ by
$$W_{c}(\mu,\nu)\doteq\sup_{g\in\mathrm{Lip}(c,S;C_{b}(S))}\left\{\int_{S}gd(\mu%
-\nu)\right\}.$$
Then by Theorem 4.2
$$W_{c}(\mu,\nu)=\sup_{g\in\mathrm{Lip}(c,S;C_{b}(S))}\left\{\int_{S}gd(\mu-\nu)%
\right\}=\inf_{\pi\in\Pi(\mu,\nu)}\left\{\int_{S\times S}c(x,y)\pi(dx,dy)%
\right\}.$$
Condition 4.4
Suppose $\mathrm{Lip}(c,S;C_{b}(S))$ is measure determining, i.e.,
for all $\mu,\nu\in\mathcal{P}(S)$, $\mu\neq\nu$, there exists $f\in\mathrm{Lip}(c,S;C_{b}(S))$ such that
$$\int_{S}fd\mu\neq\int_{S}fd\nu.$$
Under Condition 4.4, $\Gamma$ is admissible (see Definition 2.2), and by
Theorem 2.4
$$G_{\Gamma}(\mu\lVert\nu)=\sup_{g\in\Gamma}\left\{\int_{S}gd\mu-\log\int_{S}e^{%
g}d\nu\right\}=\inf_{\gamma\in\mathcal{P}(S)}\left\{W_{c}(\mu,\gamma)+R(\gamma%
\lVert\nu)\right\}.$$
(4.4)
Hence by choosing $\Gamma$ properly, we get that the $\Gamma$-divergence is an
infimal convolution of relative entropy, which is a convex function of likelihood ratios, and an
optimal transport cost, which depends on a cost structure on the space $S$.
Natural questions to raise here are the following.
i) Do there exist optimizers $\gamma^{\ast}$ and $g^{\ast}$ in the variational
problem (4.4)? If so, are they unique?
ii) How can one characterize $\gamma^{\ast}$ and $g^{\ast}$?
iii) For a fixed $\nu\in\mathcal{P}(S)$, what is the effect of a perturbation of
$\mu$ on $G_{\Gamma}(\mu\lVert\nu)$?
We will address these questions sequentially in this section. From now on, we will drop the subscript $\Gamma$ in this section for the simplicity of writing. We consider the case where $G(\mu\lVert\nu)<\infty$. To
impose additional constraints on $\mu$ and $\nu$ such that $G(\mu\lVert\nu)<\infty$ holds, we make a further
assumption on $c$.
Condition 4.5
There exist $a:S\rightarrow\mathbb{R}_{+}$ such that
$$c(x,y)\leq a(x)+a(y).$$
Now consider $\mu,\nu\in L^{1}(a)\doteq\{\theta\in\mathcal{P}(S):\int_{S}a(x)\theta(dx)<\infty\}$. Then
$$\displaystyle G(\mu\lVert\nu)$$
$$\displaystyle=\inf_{\gamma\in\mathcal{P}(S)}\left\{W_{c}(\mu,\gamma)+R(\gamma%
\lVert\nu)\right\}$$
$$\displaystyle\leq W_{c}(\mu,\nu)$$
$$\displaystyle=\inf_{\pi\in\Pi(\mu,\nu)}\left\{\int_{S\times S}c(x,y)\pi(dx,dy)\right\}$$
$$\displaystyle\leq\inf_{\pi\in\Pi(\mu,\nu)}\left\{\int_{S\times S}\left[a(x)+a(%
y)\right]\pi(dx,dy)\right\}$$
$$\displaystyle=\int_{S}a(x)\mu(dx)+\int_{S}a(y)\nu(dy)$$
$$\displaystyle<\infty.$$
We will assume the following mild conditions on the space $S$ and cost $c$ to make $\mathrm{Lip}(c,S;C_{b}(S))$ precompact.
Condition 4.6
There exists $\left\{K_{m}\right\}_{m\in\mathbb{N}}$ such that $K_{m}\subset S$ is compact, $K_{m}\subset K_{m+1}$ for all $m\in\mathbb{N}$, and $S=\cup_{m\in\mathbb{N}}K_{m}$. For each $m$, there exists $\theta_{m}:\mathbb{R}_{+}\to\mathbb{R}_{+}$, such that $\lim_{a\to 0}\theta_{m}(a)=0$, and $\delta_{m}>0$, such that for any $x,y\in K_{m}$ satisfying $d(x,y)\leq\delta_{m}$,
$$c(x,y)\leq\theta_{m}(d(x,y)).$$
Recalling the definition (4.1),
we define the unbounded version as follows
$$\mathrm{Lip}(c,S)\doteq\left\{f\in C(S):f(x)-f(y)\leq c(x,y)\mbox{ for all }x,%
y\in S\right\},$$
where $C(S)$ is the set of continuous functions mapping $S$ to $\mathbb{R}$. Before we proceed, we state the following lemma, which will be used repeatedly in this section.
Lemma 4.7
If $g\in\mathrm{Lip}(c,S)$ and $\theta,\nu\in P(S)$ satisfy $\int_{S}|g|d\theta<\infty$,
then
$$\int_{S}gd\theta-\log\int_{S}e^{g}d\nu\leq G(\theta\lVert\nu)\leq R(\theta%
\lVert\nu).$$
Proof. We use a standard truncation argument. Since by Lemma 3.1 we already have $G(\theta\lVert\nu)\leq R(\theta\lVert\nu)$,
we only need to prove the first inequality in the statement of the lemma.
If $\int_{S}e^{g}d\nu=\infty$, then
$$\int_{S}gd\theta-\log\int_{S}e^{g}d\nu=-\infty<0\leq G(\theta\lVert\nu).$$
Hence we only need consider the case $\int_{S}e^{g}d\nu<\infty$. Let $g_{n}=\min(\max(g,-n),n)\in\mathrm{Lip}(c,S;C_{b}(S))=\Gamma$ for $n\in\mathbb{N}$. We have
$|g_{n}(x)|\leq|g(x)|$
and
$$\lim_{n\to\infty}g_{n}(x)=g(x)\quad x\in S.$$
Thus by the dominated convergence theorem
$$\lim_{n\to\infty}\int_{S}g_{n}d\theta=\int_{S}gd\theta.$$
Also we have
$$e^{g_{n}(x)}\leq e^{g(x)}+1\mbox{ and }\lim_{n\to\infty}e^{g_{n}(x)}=e^{g(x)}.$$
Then again using the dominated convergence theorem,
$$\lim_{n\to\infty}\int_{S}e^{g_{n}}d\nu=\int_{S}e^{g}d\nu.$$
Together with (1.1), this gives
$$\displaystyle\int_{S}gd\theta-\log\int_{S}e^{g}d\nu$$
$$\displaystyle=\lim_{n\to\infty}\left(\int_{S}g_{n}d\theta-\log\int_{S}e^{g_{n}%
}d\nu\right)$$
$$\displaystyle\leq\sup_{f\in\Gamma}\left\{\int_{S}fd\theta-\log\int_{S}e^{f}d%
\nu\right\}$$
$$\displaystyle=G(\theta\lVert\nu).$$
Now we are ready to state the first main theorem of this section.
Theorem 4.8
Suppose Conditions 4.1, 4.4, 4.5 and 4.6 are
satisfied. Fix $\mu,\nu\in L^{1}(a)$.
Then the following conclusions hold.
1) There exists a unique optimizer $\gamma^{\ast}$ in the expression
(4.4).
2) There exists an optimizer $g^{\ast}\in\mathrm{Lip}(c,S)$ in the
expression (4.4), which is unique up to an additive constant in
$\mathrm{supp}(\mu)\cup\mathrm{supp}(\nu)$.
3) $g^{\ast}$ and $\gamma^{\ast}$ satisfy the following
conditions:
i)
$$\frac{d\gamma^{\ast}}{d\nu}(x)=\frac{e^{g^{\ast}(x)}}{\int_{S}e^{g^{\ast}(y)}d%
\nu},\quad\nu-a.s.$$
ii)
$$W_{c}(\mu,\gamma^{*})=\int_{S}g^{*}d(\mu-\gamma^{*}).$$
Remark 4.9
With many analogous expressions related to relative entropy, one can only conclude the uniqueness of $\gamma^{*}$ and $g^{*}$ (up to constant addition) almost everywhere according to either the measure $\mu$ or $\nu$. Moreover, because of the regularity condition $g^{*}\in\mathrm{Lip}(c,S;C(S))$ and Condition 4.6, the uniqueness of $g^{*}$ (up to constant addition) on $\mathrm{supp}(\mu)\cup\mathrm{supp}(\nu)$ will follow.
Proof. For $n\in\mathbb{N}$ consider $\gamma_{n}\in\mathcal{P}(S)$ that satisfies
$$R(\gamma_{n}\lVert\nu)+W_{c}(\mu,\gamma_{n})\leq G(\mu\lVert\nu)+\frac{1}{n}.$$
Then by [8, Lemma 1.4.3(c)] $\{\gamma_{n}\}_{n\geq 1}$ is precompact in the weak topology, and thus
has a convergent subsequence $\left\{\gamma_{n_{k}}\right\}_{k\geq 1}$. Denote $\gamma^{\ast}\doteq\lim_{k\rightarrow\infty}\gamma_{n_{k}}$. Then by the lower semicontinuity of
both $R(\cdot\lVert\nu)$ and $W_{c}(\mu,\cdot)$, we have
$$R(\gamma^{\ast}\lVert\nu)+W_{c}(\mu,\gamma^{\ast})\leq\liminf_{k\rightarrow%
\infty}\left(R(\gamma_{n_{k}}\lVert\nu)+W_{c}(\mu,\gamma_{n_{k}})\right)\leq G%
(\mu\lVert\nu).$$
Since
$$G(\mu\lVert\nu)=\inf_{\gamma\in\mathcal{P}(S)}\left\{R(\gamma\lVert\nu)+W_{c}(%
\mu,\gamma)\right\}\leq R(\gamma^{\ast}\lVert\nu)+W_{c}(\mu,\gamma^{\ast})$$
it follows that
$$G(\mu\lVert\nu)=R(\gamma^{\ast}\lVert\nu)+W_{c}(\mu,\gamma^{\ast}),$$
which shows that $\gamma^{\ast}$ is an optimizer in expression (4.4).
If there exist two optimizers $\gamma_{1}\neq\gamma_{2}$, the strict convexity
of $R(\cdot\lVert\nu)$ and convexity of $W_{c}(\mu,\cdot)$ imply that for
$\gamma_{3}=\frac{1}{2}(\gamma_{1}+\gamma_{2})$
$$\displaystyle R(\gamma_{3}\lVert\nu)+W_{c}(\mu,\gamma_{3})$$
$$\displaystyle<\frac{1}{2}\left(\left(R(\gamma_{1}\lVert\nu)+W_{c}(\mu,\gamma_{%
1})\right)+\left(R(\gamma_{2}\lVert\nu)+W_{c}(\mu,\gamma_{2})\right)\right)$$
$$\displaystyle=G(\mu\lVert\nu)\leq R(\gamma_{3}\lVert\nu)+W_{c}(\mu,\gamma_{3}),$$
a contradiction. Thus the existence and uniqueness of an optimizer
$\gamma^{\ast}$ of (4.4) is proved, which establishes 1) in the statement of the theorem. Before proceeding, we establish the following lemma.
Lemma 4.10
If $g\in\mathrm{Lip}(c,S)$, then
$$\int_{S}gd\gamma^{*}<\infty.$$
Proof. This can be shown by contradiction. Assume there exists $h\in\mathrm{Lip}(c,S)$ such that $\int_{S}|h|d\gamma^{*}=\infty$. By symmetry, we can just consider $h$ to be non-negative, since $\max(h,0)\in\mathrm{Lip}(c,S)$ and $h=\max(h,0)-\max(-h,0)$. Thus we can assume there exists non-negative $h\in\mathrm{Lip}(c,S)$ satisfying
$$\int_{S}hd\gamma^{*}=\infty,$$
and by the fact that $u\in L^{1}(a)$ together with Condition 4.5,
$$\displaystyle\int_{S}hd\mu$$
$$\displaystyle\leq\int_{S}\left[h(0)+c(x,0)\right]\mu(dx)$$
$$\displaystyle=h(0)+a(0)+\int_{S}a(x)\mu(dx)<\infty.$$
Then
$$\displaystyle W_{c}(\mu,\gamma^{*})$$
$$\displaystyle=\sup_{g\in\mathrm{Lip(c,S)}}\int_{S}gd(\mu-\gamma^{*})$$
$$\displaystyle\geq\limsup_{n\to\infty}\int_{S}\max(-h,-n)d(\mu-\gamma^{*})$$
$$\displaystyle=\limsup_{n\to\infty}\left[\int_{S}\max(-h,-n)d\mu+\int_{S}\min(h%
,n)d\gamma^{*}\right]$$
$$\displaystyle=\int_{S}-hd\mu+\int_{S}hd\gamma^{*}$$
$$\displaystyle=\infty,$$
where the second to last equation comes from dominated and monotone convergence theorems applied to the first and second terms respectively. However, since $\gamma^{*}$ is the optimizer, we have
$$W_{c}(\mu,\gamma^{*})\leq W_{c}(\mu,\gamma^{*})+R(\gamma^{*}\lVert\nu)=G(\mu%
\lVert\nu)<\infty.$$
This contradiction shows the integrability of $\gamma^{*}$ with respect to any $\mathrm{Lip}(c,S)$ function.
Now we consider the other variational representation of $G(\mu\lVert\nu)$, which is
$$G(\mu\lVert\nu)=\sup_{g\in\mathrm{Lip}(c,S;C_{b}(S))}\left\{\int_{S}gd\mu-\log%
\int_{S}e^{g}d\nu\right\}.$$
Take $g_{n}\in\mathrm{Lip}(c,S;C_{b}(S))$ such that
$$G(\mu\lVert\nu)-1/n\leq\int_{S}g_{n}d\mu-\log\int_{S}e^{g_{n}}d\nu\leq G(\mu%
\lVert\nu).$$
Without loss of generality, we can assume $g_{n}(x_{0})=0$ for some fixed $x_{0}\in K_{0}\subset S$. Since for any $m\in\mathbb{N}$ $K_{m}\subset S$ is compact, we have that $\left\{g_{n}\right\}_{n\in\mathbb{N}}$ is bounded and equicontinuous on $K_{m}$ by Condition 4.6. By the Arzelà-Ascoli theorem, there exists a subsequence of $\left\{g_{n}\right\}_{n\in\mathbb{N}}$ that converges uniformly in $K_{m}$. Using diagonal argument, by taking subsequences sequentially along $\left\{K_{m}\right\}_{m\in\mathbb{N}}$, where the next subsequence is a subsequence of the former one, and take one element from each sequence, we conclude there exists a subsequence $\left\{g_{n_{j}}\right\}_{j\in\mathbb{N}}$, that converges uniformly in any $K_{m}$. Since $S=\cup_{m\in\mathbb{N}}K_{m}$, we conclude that $\left\{g_{n_{j}}\right\}_{j\in\mathbb{N}}$ converges pointwise in $S$. Denotes its limit by $g^{*}$. It can be easily verified that $g^{*}\in\mathrm{Lip}(c,S)$.
Since $g_{n_{j}}(x)\leq g_{n_{j}}(x_{0})+c(x_{0},x)\leq a(x_{0})+a(x)$ and $\int_{S}\left(a(x_{0})+a(x)\right)d\mu<\infty$, by the dominated convergence theorem
$$\lim_{j\to\infty}\int_{S}g_{n_{j}}d\mu=\int_{S}g^{*}d\mu.$$
By Fatou’s lemma, we have
$$\liminf_{j\to\infty}\int_{S}e^{g_{n_{j}}}d\nu\geq\int e^{g^{*}}d\nu,$$
and therefore
$$-\log\int e^{g^{*}}d\nu\geq\limsup_{j\to\infty}-\int_{S}e^{g_{n_{j}}}d\nu.$$
Putting these together, we have
$$\displaystyle G(\mu\lVert\nu)$$
$$\displaystyle=\sup_{g\in\mathrm{Lip}(c,S;C_{b}(S))}\left\{\int_{S}gd\mu-\log%
\int_{S}e^{g}d\nu\right\}$$
$$\displaystyle\leq\limsup_{j\to\infty}\left\{\int_{S}g_{n_{j}}d\mu-\log\int_{S}%
e^{g_{n_{j}}}d\nu\right\}$$
$$\displaystyle\leq\int_{S}g^{*}d\mu-\log\int_{S}e^{g^{*}}d\nu$$
$$\displaystyle=\left(\int_{S}g^{*}d\mu-\int_{S}g^{*}d\gamma^{*}\right)+\left(%
\int_{S}g^{*}d\gamma^{*}-\log\int_{S}e^{g^{*}}d\nu\right).$$
We can add and subtract $\int_{S}g^{*}d\gamma^{*}$ because we have proved in Lemma 4.10 that $\gamma^{*}$ is integrable with respect to functions in $\mathrm{Lip}(c,S)$, and $g^{*}\in\mathrm{Lip}(c,S)$.
By Lemma 4.7 we have
$$\int_{S}g^{*}d\gamma^{*}-\log\int_{S}e^{g^{*}}d\nu\leq R(\gamma^{*}\lVert\nu).$$
We also have
$$\int_{S}g^{*}d\mu-\int_{S}g^{*}d\gamma^{*}\leq W_{c}(\mu,\gamma^{*}),$$
which is due to
$$\displaystyle W_{c}(\mu,\gamma^{*})$$
$$\displaystyle=\sup_{g\in\mathrm{Lip}(c,S;C_{b}(S))}\int_{S}gd(\mu-\gamma^{*})$$
$$\displaystyle\geq\limsup_{n\to\infty}\int_{S}\max(\min(g^{*},n),-n)d(\mu-%
\gamma^{*})$$
$$\displaystyle=\int_{S}g^{*}d(\mu-\gamma^{*}),$$
where the last equality is because of the dominated convergence theorem and integrability of $|g^{*}|$ with respect to $\mu$ and $\gamma^{*}$ (Lemma 4.10).
We can therefore continue the calculation above as
$$\displaystyle\left(\int_{S}g^{*}d\mu-\int_{S}g^{*}d\gamma^{*}\right)+\left(%
\int_{S}g^{*}d\gamma^{*}-\log\int_{S}e^{g^{*}}d\nu\right)$$
$$\displaystyle\qquad\leq W_{c}(\mu,\gamma^{*})+R(\gamma^{*}\lVert\nu)$$
$$\displaystyle\qquad=G(\mu\lVert\nu).$$
Since both the upper and lower bounds on the inequalities coincide, we must have all inequalities to be equalities, and therefore
$$G(\mu\lVert\nu)=\int_{S}g^{*}d\mu-\log\int_{S}e^{g^{*}}d\nu,$$
$$\int_{S}g^{*}d\mu-\int_{S}g^{*}d\gamma^{*}=W_{c}(\mu,\gamma^{*}),$$
and
$$\int_{S}g^{*}d\gamma^{*}-\log\int_{S}e^{g^{*}}d\nu=R(\gamma^{*}\lVert\nu).$$
The last equation gives us the relationship
$$\frac{d\gamma^{*}}{d\nu}(x)=\frac{e^{g^{*}(x)}}{\int_{S}e^{g^{*}}d\nu}\quad\nu%
-a.s.$$
Thus we have shown the existence of optimizer $g^{*}\in\mathrm{Lip}(c,S)$ and its relationship with $\gamma^{*}$. Lastly, for any other optimizer $\bar{g}\in\mathrm{Lip}(c,S)$ the analogous argument shows
$$\frac{d\gamma^{*}}{d\nu}(x)=\frac{e^{\bar{g}(x)}}{\int_{S}e^{\bar{g}}d\nu}%
\quad\nu-a.s.$$
Hence uniqueness of the optimizer $g^{*}$ in $\mathrm{supp}(\nu)$ up to $\nu-a.s.$ is also proved.
To determine the uniqueness of the optimizer $g^{*}$ in $\mathrm{supp}(\mu)$, we take an optimal transport plan between $\mu$ and $\gamma^{*}$, $\pi^{*}\in\Pi(\mu,\gamma^{*})$ for $W_{c}(\mu,\gamma^{*})$, which means
$$W_{c}(\mu,\gamma^{*})=\inf_{\pi\in\Pi(\mu,\gamma^{*})}\left\{\int_{S\times S}c%
(x,y)\pi(dx,dy)\right\}=\int_{S\times S}c(x,y)\pi^{*}(dx,dy).$$
(Note that $c$ satisfying Condition 4.1 is lower semicontinuous, and therefore [14, Theorem 1.5] shows the existence of an optimal transport plan $\pi^{*}$.)
Since $g^{*}(x)-g^{*}(y)\leq c(x,y)$,
$$\displaystyle W_{c}(\mu,\gamma^{*})$$
$$\displaystyle=\int_{S\times S}c(x,y)\pi^{*}(dx,dy)$$
$$\displaystyle\geq\int_{S\times S}\left[g^{*}(x)-g^{*}(y)\right]\pi^{*}(dx,dy)$$
$$\displaystyle=\int_{S}g^{*}(x)(\mu-\gamma^{*})(dx)$$
$$\displaystyle=W_{c}(\mu,\gamma^{*}).$$
Then the only inequality above must be equality, which implies that for $(x,y)\in\mathrm{supp}(\gamma^{*})$, $g^{*}(x)-g^{*}(y)=c(x,y)$, $\pi^{*}-a.s.$ This is also true for any other optimizer $\bar{g}\in\mathrm{Lip}(c,S)$ for (4.4). Thus we are able to determine $g^{*}$ uniquely in $\mathrm{supp}(\mu)$ $\mu-a.s.$ with the help of $\pi^{*}$ and data of $g^{*}$ in $\mathrm{supp}(\nu)$. Lastly, since $g^{*}\in\mathrm{Lip}(c,S)$ and by Condition 4.6, we conclude the uniqueness of $g^{*}$ in $\mathrm{supp}(\mu)\cup\mathrm{supp}(\nu)$ by the continuity of $g^{*}$.
Remark 4.11
When $\mu\ll\nu$ Theorem 4.8 implies that for some constant $c_{0}$
$$g^{\ast}(x)=\log\left(\frac{d\gamma^{\ast}}{d\nu}(x)\right)-c_{0}\quad\nu-a.s.$$
Hence
$$G(\mu\lVert\nu)=\int_{S}g^{\ast}d\mu-\log\int_{S}e^{g^{\ast}}d\nu=\int_{S}\log%
\left(\frac{d\gamma^{\ast}}{d\nu}(x)\right)d\mu,$$
and so the $\Gamma$-divergence of $\mu$ with respect to $\nu$ looks like a “modified” version of relative entropy.
The next theorem tells us that 3) of Theorem 4.8 is not only a
description of of the pair of optimizer $(g^{*},\gamma^{*})$, but also a
characterization of it.
Theorem 4.12
Suppose Conditions 4.1, 4.4, 4.5 and 4.6 are
satisfied. Fix $\mu,\nu\in L^{1}(a)$. If $g_{1}\in$ Lip$(c,S)$ and
$\gamma_{1}\in\mathcal{P}(S)$ satisfy condition 3) in Theorem
4.8, then $(g_{1},\gamma_{1})$ are optimizers in the corresponding
variational problem (4.4).
Proof. The theorem follows from the two variational characterization of $\Gamma$-divergence
in (4.4).
Condition 3) of Theorem 4.8 implies
$$R(\gamma_{1}\lVert\nu)=\int_{S}g_{1}d\gamma_{1}-\log\int_{S}e^{g_{1}}d\nu\mbox%
{ and }W_{c}(\mu,\gamma_{1})=\int_{S}g_{1}d(\mu-\gamma_{1}),$$
and therefore
$$R(\gamma_{1}\lVert\nu)+W_{c}(\mu,\gamma_{1})=\int_{S}g_{1}d\mu-\log\int_{S}e^{%
g_{1}}d\nu.$$
This implies
$$\displaystyle G(\mu\lVert\nu)$$
$$\displaystyle=\inf_{\gamma\in\mathcal{P}(S)}\left\{R(\gamma\lVert\nu)+W_{c}(%
\mu,\gamma)\right\}$$
$$\displaystyle\leq R(\gamma_{1}\lVert\nu)+W_{c}(\mu,\gamma_{1})$$
$$\displaystyle=\int_{S}g_{1}d\mu-\log\int_{S}e^{g_{1}}d\nu$$
$$\displaystyle\leq\sup_{g\in\Gamma}\left\{\int_{S}gd\mu-\log\int_{S}e^{g}d\nu\right\}$$
$$\displaystyle=G(\mu\lVert\nu).$$
The first inequality comes from the fact that $\gamma_{1}\in\mathcal{P}(S)$, while the second needs a little more discussion, which will be given below. Assuming this, the last display shows that
$(g_{1},\gamma_{1})$ are optimizers. The second inequality follows from Lemma 4.7 and the fact that
$$\displaystyle\int_{S}|g_{1}(x)|\mu(dx)$$
$$\displaystyle\leq\int_{S}|g_{1}(0)|+c(0,x)\mu(dx)$$
$$\displaystyle\leq\int_{S}|g_{1}(0)|+a(0)+a(x)\mu(dx)<\infty.$$
The proof is complete.
The last theorem answers questions i) and ii) raised earlier in this section, now we want to answer iii), which is to characterize the directional derivative of $G(\mu\lVert\nu)$ in the first variable when fixing the second one, i.e.,
$$\lim_{\varepsilon\to 0^{+}}\frac{1}{\varepsilon}\left(G(\mu+\varepsilon\rho%
\lVert\nu)-G(\mu\lVert\nu)\right)$$
for $\rho\in\mathcal{M}_{0}(S)$ which satisfies certain conditions. From Theorem 4.8 and remarks following it we know that any optimizer $g^{*}$ of expression (4.4) is unique in $\mathrm{supp}(\mu)\cup\mathrm{supp}(\nu)$. However, there is still freedom to choose $g^{*}$ in $S\backslash\left\{\mathrm{supp}(\mu)\cup\mathrm{supp}(\nu)\right\}$, since the variational problem in (4.4) does not take into account of the information of $g^{*}$ outside $\mathrm{supp}(\mu)\cup\mathrm{supp}(\nu)$, other than requiring that $g^{*}$ belong to $\mathrm{Lip}(c,S)$. We will define a special $g^{*}$ that is uniquely defined not only in $\mathrm{supp}(\mu)$ and $\mathrm{supp}(\nu)$, but also on $S\backslash\left\{\mathrm{supp}(\mu)\cup\mathrm{supp}(\nu)\right\}$. For $x\in S\backslash\left\{\mathrm{supp}(\mu)\cup\mathrm{supp}(\nu)\right\}$, set
$$\displaystyle g^{*}(x)\doteq\inf_{y\in\mathrm{supp}(\nu)}\left\{g^{*}(y)+c(x,y%
)\right\}.$$
(4.5)
From now on we will use the notation $g^{*}$ for the function defined in (4.5). The following lemma
confirms that this construction of $g^{*}$ still lies in $\mathrm{Lip}(c,S)$.
Lemma 4.13
The following two statements hold.
1) For $x\in\mathrm{supp}(\mu)$, the expression $(\ref{the_opt})$ also holds. In other words, for $x\in S\backslash\mathrm{supp}(\nu)$, we have
$$g^{*}(x)=\inf_{y\in\mathrm{supp}(\nu)}\left\{g^{*}(y)+c(x,y)\right\}.$$
2) $g^{*}$ defined by equation (4.5) is in $\mathrm{Lip}(c,S)$. In addition,
$$\displaystyle g^{*}(x)=\sup\{h(x):h\in\mathrm{Lip}(c,S),h(y)=g^{*}(y)\ \mathrm%
{for}\ y\in\mathrm{supp}(\nu)\}$$
(4.6)
Proof.
1) For $x\in\mathrm{supp}(\mu)$, from an optimal transport plan between $\mu$ and $\gamma^{*}$, $\pi^{*}\in\Pi(\mu,\gamma^{*})$ for $W_{c}(\mu,\gamma^{*})$, we know there exists $y_{x}\in\mathrm{supp}(\nu)$ such that $(x,y_{x})\in\mathrm{supp}(\pi^{*})$. Thus by [14][Remark 1.15],
$$g^{*}(x)=g^{*}(y_{x})+c(x,y_{x}).$$
On the other hand, by Theorem 4.8, $g^{*}|_{\mathrm{supp}(\nu)\cup\mathrm{supp}(\mu)}\in\mathrm{Lip}(c,S)$. Thus, for other $y\in\mathrm{supp}(\nu)$, $g^{*}(x)\leq c(x,y)+g^{*}(y)$, which in turn gives
$$g^{*}(x)\leq\inf_{y\in\mathrm{supp}(\nu)}\left\{g^{*}(y)+c(x,y)\right\}.$$
By combining the two expressions above, we have for $x\in\mathrm{supp}(\mu)$, (4.5) also holds. In other words, $g^{*}$ is totally characterized by $g^{*}|_{\mathrm{supp}}(\nu)$ and (4.5).
2) Since $c\geq 0$, it is easily checked that for any $x\not\in\mathrm{supp}(\nu)$ and any $y\in\mathrm{supp}(\nu)$,
$$g^{*}(y)\leq g^{*}(x)\leq g^{*}(y)+c(x,y).$$
For $x\in\mathrm{supp}(\nu)$, since we already know $g^{*}|_{\mathrm{supp}(\nu)}$ is uniquely determined and the optimizer constructed in Theorem 4.8 is in $\mathrm{Lip}(c,S)$, we conclude that for any $y\in\mathrm{supp}(\nu)$,
$$g^{*}(y)-c(x,y)\leq g^{*}(x)\leq g^{*}(y)+c(x,y).$$
Hence to show $g^{*}\in\mathrm{Lip}(c,S)$ we only need to check for $x_{1},x_{2}\not\in\mathrm{supp}(\nu)$ the Lipschitz constrait is satisfied.
From the definition (4.5),
we know for any $n<\infty$
there exists $y_{1}\in\mathrm{supp}(\nu)$ such that
$$c(x_{1},y_{1})-1/n\leq g^{*}(x_{1})-g^{*}(y_{1}).$$
Also, because $y_{1}\in\mathrm{supp}(\nu)$,
$$g^{*}(x_{2})-g^{*}(y_{1})\leq c(x_{2},y_{1}).$$
Therefore
$$\displaystyle g^{*}(x_{2})-g^{*}(x_{1})$$
$$\displaystyle\leq(c(x_{2},y_{1})-c(x_{1},y_{1}))+1/n$$
$$\displaystyle\leq c(x_{1},x_{2})+1/n,$$
where the last inequality uses the triangle inequality property of $c$. Since $n>0$ is arbitrary and we can swap the roles of $x_{1}$ and $x_{2}$, we have proved the Lipschitz condition of $g^{*}$ for $x_{1},x_{2}\not\in\mathrm{supp}(\nu)$. Thus the statement that $g^{*}\in\mathrm{Lip}(c,S)$ is proven.
For (4.6), notice that for $h\in\mathrm{Lip}(c,S)$, $x\in S$ and $y\in\mathrm{supp}(\nu)$,
$$h(x)\leq h(y)+c(x,y).$$
So if $h(y)=g^{*}(y)$ for $y\in\mathrm{supp}(\nu)$, then for $x\in S\backslash\mathrm{supp}(\nu)$,
$$h(x)\leq\inf_{y\in\mathrm{supp}(\nu)}\left\{h(y)+c(x,y)\right\}=\inf_{y\in%
\mathrm{supp}(\nu)}\left\{g^{*}(y)+c(x,y)\right\}=g^{*}(x).$$
Since $g^{*}$ is also in $\mathrm{Lip}(c,S)$, this proves (4.6).
Then based on this construction, we have the following result.
Theorem 4.14
Take $\Gamma=\mathrm{Lip}(c,S;C_{b}(S))$ where $c$ satisfies the conditions of Theorem 4.8 and $\mu,\nu\in L^{1}(a)$. Take $\rho=\rho_{+}-\rho_{-}\in\mathcal{M}_{0}(S)$ where $\rho_{+},\rho_{-}\in\mathcal{P}(S)$ are mutually singular probability measures, $\rho_{+}\in L^{1}(a)$, and assume there exists $\varepsilon_{0}>0$ such that $\mu+\varepsilon\rho\in\mathcal{P}(S)$ for $0<\varepsilon\leq\varepsilon_{0}$. Then
$$\lim_{\varepsilon\to 0^{+}}\frac{1}{\varepsilon}\left(G(\mu+\varepsilon\rho%
\lVert\nu)-G(\mu\lVert\nu)\right)=\int_{S}g^{*}d\rho.$$
where $g^{*}$ is the optimizer found in (4.5).
Proof. We use the variational formula (4.4) for $G(\mu+\varepsilon\rho\lVert\nu)$, where $\mu+\varepsilon\rho\in\mathcal{P}(S)$ and $\rho_{+}\in L^{1}(a)$. Recall that $g^{*}$ is the optimizer for (4.4). Using Lemma 4.7 with $\theta=\mu+\varepsilon\rho$,
$$\displaystyle G(\mu+\varepsilon\rho\lVert\nu)$$
$$\displaystyle=\sup_{g\in\Gamma}\left\{\int_{S}gd(\mu+\varepsilon\rho)-\log\int%
_{S}e^{g}d\nu\right\}$$
$$\displaystyle\geq\int_{S}g^{*}d(\mu+\varepsilon\rho)-\log\int_{S}e^{g^{*}}d\nu$$
$$\displaystyle=\varepsilon\int_{S}g^{*}d\rho+\int_{S}g^{*}d\mu-\log\int_{S}e^{g%
^{*}}d\nu$$
$$\displaystyle=\varepsilon\int_{S}g^{*}d\rho+G(\mu\lVert\nu).$$
Thus
$$\liminf_{\varepsilon\to 0^{+}}\frac{1}{\varepsilon}\left(G(\mu+\varepsilon\rho%
\lVert\nu)-G(\mu\lVert\nu)\right)\geq\int_{S}g^{*}d\rho.$$
(4.7)
The other direction is more delicate. Take $f(\varepsilon)=G(\mu+\varepsilon\rho\lVert\nu)$. From Lemma 3.1 we know that $f$ is convex, lower semicontinuous and finite on $[0,\varepsilon_{0}]$. Using a property of convex functions in one dimension, we know $f$ is differentiable on $(0,\varepsilon_{0})$ except for a countable number of points. Take $\varepsilon\in(0,\varepsilon_{0})$ to be a place where $f$ is differentiable, and $\delta>0$ small. Take $g^{*}_{\varepsilon}\in\mbox{Lip}(c,S)$ to be the optimizer for $G(\mu+\varepsilon\rho\lVert\nu)$ satisfying $g^{*}_{\varepsilon}(0)=0$, so that
$$G(\mu+\varepsilon\rho\lVert\nu)=\int_{S}g^{*}_{\varepsilon}d(\mu+\varepsilon%
\rho)-\log\int_{S}e^{g^{*}_{\varepsilon}}d\nu.$$
Then using an argument that already appeared in this proof, we have
$$G(\mu+(\varepsilon+\delta)\rho\lVert\nu)-G(\mu+\varepsilon\rho\lVert\nu)\geq%
\delta\int_{S}g^{*}_{\varepsilon}d\rho,$$
and
$$G(\mu+(\varepsilon-\delta)\rho\lVert\nu)-G(\mu+\varepsilon\rho\lVert\nu)\geq-%
\delta\int_{S}g^{*}_{\varepsilon}d\rho.$$
It follows that
$$\displaystyle\int_{S}g^{*}_{\varepsilon}d\rho$$
$$\displaystyle\leq\lim_{\delta\to 0}\frac{1}{\delta}\left(G(\mu+(\varepsilon+%
\delta)\rho\lVert\nu)-G(\mu+\varepsilon\rho\lVert\nu)\right)$$
$$\displaystyle=f^{\prime}(\varepsilon)$$
$$\displaystyle=\lim_{\delta\to 0}\frac{1}{\delta}\left(G(\mu+\varepsilon\rho%
\lVert\nu)-G(\mu+(\varepsilon-\delta)\rho\lVert\nu)\right)$$
$$\displaystyle\leq\int_{S}g^{*}_{\varepsilon}d\rho.$$
and therefore
$$f^{\prime}(\varepsilon)=\int_{S}g^{*}_{\varepsilon}d\rho.$$
(4.8)
If we denote
$$f^{\prime}_{+}(0)=\lim_{\varepsilon\to 0^{+}}\frac{1}{\varepsilon}(f(%
\varepsilon)-f(0)),$$
then by a property of convex functions [20, Theorem 24.1],
for any sequence of $\left\{\varepsilon_{n}\right\}_{n\in\mathbb{N}}$ such that $\varepsilon_{0}>\varepsilon_{n}\downarrow 0$ and $f$ is differentiable at $\varepsilon_{n}>0$, we have
$$f^{\prime}_{+}(0)=\lim_{n\to\infty}f^{\prime}(\varepsilon_{n})=\lim_{n\to%
\infty}\int_{S}g^{*}_{\varepsilon_{n}}d\rho.$$
By the same argument used in the proof of Theorem 4.8 (paragraphs following Lemma 4.10), i.e., by applying the Arzelà-Ascoli theorem to $\{g_{\varepsilon_{n}}\}$ on each compact set $K_{m}\subset S$, and then doing a diagonalization argument,
there exists a subsequence of $\left\{n_{k}\right\}_{k\geq 0}\subset\left\{n\right\}_{n\geq 0}$, such that $g_{\varepsilon_{n_{k}}}^{*}$ converges pointwise to a function that we denote by $g_{0}^{*}\in\mathrm{Lip}(c,S)$. To simplify the notation, let $n$ denote the convergent subsequence.
Since $\rho=\rho_{+}-\rho_{-}$, where $\rho_{+}\in L^{1}(a)$ and $\mu+\varepsilon_{0}\rho\in P(S)$, $\mu\in L^{1}(a)$ implies $\rho_{-}\in L^{1}(a)$, therefore
$$\int_{S}ad|\rho|<\infty.$$
Here $|\rho|=\rho_{+}+\rho_{-}$. Recall that for any $\varepsilon\in(0,\varepsilon_{0})$, $g^{*}_{\varepsilon}(0)=0$. For any $x\in S$,
$$g^{*}_{\varepsilon}(x)\leq g^{*}_{\varepsilon}(0)+c(0,x)\leq a(0)+a(x).$$
Thus by the dominated convergence theorem
$$f^{\prime}_{+}(0)=\lim_{n\to\infty}\int_{S}g^{*}_{\varepsilon_{n}}d\rho=\int_{%
S}g^{*}_{0}d\rho.$$
Lastly, to connect $g_{0}^{*}$ back to $g^{*}$ defined in (4.5), note that by the lower semincontinuity of $G(\cdot\lVert\nu)$,
$$\displaystyle G(\mu\lVert\nu)$$
$$\displaystyle\leq\liminf_{n\to\infty}G(\mu+\varepsilon_{n}\rho\lVert\nu)$$
$$\displaystyle=\liminf_{n\to\infty}\left(\int_{S}g^{*}_{\varepsilon_{n}}d(\mu+%
\varepsilon_{n}\rho)-\log\int_{S}e^{g^{*}_{\varepsilon_{n}}}d\nu\right)$$
$$\displaystyle=\liminf_{n\to\infty}\int_{S}g^{*}_{\varepsilon_{n}}d(\mu+%
\varepsilon_{n}\rho)-\limsup_{n\to\infty}\log\int_{S}e^{g^{*}_{\varepsilon_{n}%
}}d\nu$$
$$\displaystyle\leq\int_{S}g^{*}_{0}d\mu-\log\int_{S}e^{g_{0}^{*}}d\nu$$
$$\displaystyle\leq G(\mu\lVert\nu).$$
The second inequality uses dominated convergence, (4.8),
and that by Fatou’s lemma
$$\limsup_{n\to\infty}\int_{S}e^{g^{*}_{\varepsilon_{n}}}d\nu\geq\liminf_{n\to%
\infty}\int_{S}e^{g^{*}_{\varepsilon_{n}}}d\nu\geq\int_{S}e^{g^{*}_{0}}d\nu.$$
The third inequality uses Lemma 4.7.
Since both sides of the inequality coincide, $g^{*}_{0}$ must be the optimizer for variational expression (4.4). By Theorem 4.8 and equation (4.6),
we have $g^{*}_{0}(x)\leq g^{*}(x)$ for all $x\in S$.
Thus
$$f^{\prime}_{+}(0)=\int_{S}g^{*}_{0}d\rho\leq\int_{S}g^{*}d\rho,$$
(4.9)
the other direction of the inequality is proved.
Combining (4.9) and (4.7) gives
$$\lim_{\varepsilon\to 0^{+}}\frac{1}{\varepsilon}\left(G(\mu+\varepsilon\rho%
\lVert\nu)-G(\mu\lVert\nu)\right)=\int_{S}g^{*}d\rho.$$
Remark 4.15
When $\rho\in\mathcal{M}_{0}(S)$ is taken such that there exists $\varepsilon_{0}>0$ such that for $\varepsilon\in[-\varepsilon_{0},\varepsilon_{0}]$, $\mu+\varepsilon\rho\in P(S)$, then by applying the above theorem to $\rho$ and $-\rho$ respectively, we can conclude $G(\mu+\varepsilon\rho\lVert\nu)$ as a function of $\varepsilon$ is differentiable at $\varepsilon=0$ with derivative $\int_{S}g^{*}d\rho$.
Remark 4.16
We call $g^{*}$ defined in (4.5) the unique potential associated with $G(\mu\lVert\nu)$. This $g^{*}$ is similar to the Kantorovich potential in the optimal transport literature. However, for the optimal transport cost $W_{c}(\mu,\nu)$ more conditions are needed(e.g. [22][Proposition 7.18]) to ensure the uniqueness of the Kantorovich potential. Here under very mild conditions we are able to confirm the uniqueness of the potential, and prove that it is the directional derivative of the corresponding $\Gamma$-divergence, as is case of the Kantorovich potential for optimal transport cost when its uniqueness is established.
5 Limits and Approximations of $\Gamma$-divergence
In this section, we consider limits that are obtained as the admissible set gets large or small,
and the $\Gamma$-divergence will be approximated by relative entropy or a transport distance, respectively. We also consider in special cases more informative expansions.
Throughout the section we assume the conditions of Theorem 4.8.
Fix an admissible set of $\Gamma_{0}\subset C_{b}(S)$, and take $\Gamma=b\Gamma_{0}=\left\{b\cdot f:f\in\Gamma_{0}\right\}$ for $b>0$. Then the following proposition holds.
Proposition 5.1
For $\mu,\nu\in\mathcal{P}(S)$,
$$\lim_{b\to\infty}G_{b\Gamma_{0}}(\mu\lVert\nu)=R(\mu\lVert\nu).$$
Proof. We separate the proof into two cases, $R(\mu\lVert\nu)<\infty$ and $R(\mu\lVert\nu)=\infty$.
1) If $R(\mu\lVert\nu)<\infty$, then for any $b>0$,
$$\displaystyle G_{b\Gamma_{0}}(\mu\lVert\nu)=\inf_{\gamma\in\mathcal{P}(S)}%
\left\{W_{b\Gamma_{0}}(\mu,\gamma)+R(\gamma\lVert\nu)\right\}\leq R(\mu\lVert%
\nu)<\infty.$$
(5.1)
From Theorem 4.8 we know there exists a unique optimizer $\gamma^{*}$ for each $b$, which we write as $\gamma_{b}^{*}$. Note that
$$R(\gamma_{b}^{*}\lVert\nu)\leq R(\mu\lVert\nu)<\infty,$$
and therefore $\left\{\gamma_{b}^{*}\right\}_{b>0}$ is precompact in the weak topology [8, Lemma 1.4.3(c)].
Given any subsequence $b_{k}$, there exists a further subsequence (again denoted by $b_{k}$) and $\gamma_{\infty}^{*}\in\mathcal{P}(S)$ such that $\gamma^{*}_{b_{k}}\Rightarrow\gamma^{*}_{\infty}$. On the other hand,
$$\displaystyle W_{b\Gamma_{0}}(\mu,\gamma^{*}_{b})$$
$$\displaystyle=\sup_{f\in b\Gamma_{0}}\left\{\int_{S}fd(\mu-\gamma_{b}^{*})\right\}$$
$$\displaystyle=b\sup_{f\in\Gamma_{0}}\left\{\int_{S}fd(\mu-\gamma_{b}^{*})%
\right\}=bW_{\Gamma_{0}}(\mu,\gamma_{b}^{*}),$$
and
$W_{b\Gamma_{0}}(\mu,\gamma_{b}^{*})\leq G_{b\Gamma_{0}}(\mu\lVert\nu)\leq R(%
\mu\lVert\nu)<\infty.$
Thus
$$\displaystyle W_{\Gamma_{0}}(\mu,\gamma_{\infty}^{*})$$
$$\displaystyle\leq\liminf_{k\to\infty}W_{\Gamma_{0}}(\mu,\gamma_{b_{k}}^{*})=%
\liminf_{k\to\infty}\frac{1}{b_{k}}W_{b_{k}\Gamma_{0}}(\mu,\gamma_{b_{k}}^{*})$$
$$\displaystyle\leq\liminf_{k\to\infty}\frac{1}{b_{k}}R(\mu\lVert\nu)=0,$$
and since $\Gamma_{0}$ is admissible,
$\gamma_{\infty}^{*}=\mu$. We thus conclude that
$$\displaystyle\liminf_{k\to\infty}G_{b_{k}\Gamma_{0}}(\mu\lVert\nu)$$
$$\displaystyle=\liminf_{k\to\infty}\left(W_{b_{k}\Gamma_{0}}(\mu,\gamma_{b_{k}}%
^{*})+R(\gamma_{b_{k}}^{*}\lVert\nu)\right)$$
$$\displaystyle\geq\liminf_{k\to\infty}R(\gamma_{b_{k}}^{*}\lVert\nu)$$
$$\displaystyle\geq R(\mu\lVert\nu),$$
and since the original subsequence was arbitrary
$$\liminf_{b\to\infty}G_{b\Gamma_{0}}(\mu\lVert\nu)\geq R(\mu\lVert\nu).$$
On the other hand, we have by (5.1) that
$$\limsup_{b\to\infty}G_{b\Gamma_{0}}(\mu\lVert\nu)\leq R(\mu\lVert\nu),$$
and the statement is proved.
2) $R(\mu\lVert\nu)=\infty.$ For this case, we want to prove that
$$\liminf_{b\to\infty}G_{b\Gamma_{0}}(\mu\lVert\nu)=\infty.$$
If not, then there exists a subsequence $\left\{b_{k}\right\}_{b\in\mathbb{N}}$ such that
$$\lim_{k\to\infty}G_{b_{k}\Gamma_{0}}(\mu\lVert\nu)<\infty.$$
For this subsequence, we can apply the argument used in part 1) to conclude there exists $\gamma_{b_{k}}^{*}$ such that
$$G_{b_{k}\Gamma_{0}}(\mu\lVert\nu)=W_{b_{k}\Gamma_{0}}(\mu,\gamma_{b_{k}}^{*})+%
R(\gamma_{b_{k}}^{*}\lVert\nu).$$
Moreover there exists a further subsequence of this sequence, which for simplicity we also denote by $\left\{b_{k}\right\}_{k\in\mathbb{N}}$, which satisfies $\gamma_{b_{k}}^{*}\Rightarrow\mu$. Then by the same argument as in 1), we would conclude
$$\lim_{k\to\infty}G_{b_{k}\Gamma_{0}}(\mu\lVert\nu)\geq R(\mu\lVert\nu)=\infty.$$
This contradiction proves the statement.
On the other hand, if $\Gamma=\delta\Gamma_{0}$ for small $\delta>0$, we can approximate the $\Gamma$-divergence in terms of the $W_{\Gamma_{0}}$.
Proposition 5.2
For $\mu,\nu\in\mathcal{P}(S)$
$$\lim_{\delta\to 0}\frac{1}{\delta}G_{\delta\Gamma_{0}}(\mu\lVert\nu)=W_{\Gamma%
_{0}}(\mu,\nu).$$
Proof. For any $\delta>0$, Jensen’s inequality implies
$$\displaystyle\frac{1}{\delta}G_{\delta\Gamma_{0}}(\mu\lVert\nu)$$
$$\displaystyle=\frac{1}{\delta}\sup_{g\in\delta\Gamma_{0}}\left\{\int_{S}gd\mu-%
\log\int_{S}e^{g}d\nu\right\}$$
$$\displaystyle\leq\frac{1}{\delta}\sup_{g\in\delta\Gamma_{0}}\left\{\int_{S}gd%
\mu-\int_{S}gd\nu\right\}$$
$$\displaystyle=\sup_{g\in\Gamma_{0}}\left\{\int_{S}gd\mu-\int_{S}gd\nu\right\}$$
$$\displaystyle=W_{\Gamma_{0}}(\mu,\nu),$$
and therefore
$$\limsup_{\delta\to 0}\frac{1}{\delta}G_{\delta\Gamma_{0}}(\mu\lVert\nu)\leq W_%
{\Gamma_{0}}(\mu,\nu).$$
For the reverse inequality we consider two cases.
1) $W_{\Gamma_{0}}(\mu,\nu)<\infty.$ For $0<\delta<1$ the argument used above shows
$$G_{\delta\Gamma_{0}}(\mu\lVert\nu)\leq\delta W_{\Gamma_{0}}(\mu,\nu)\leq W_{%
\Gamma_{0}}(\mu,\nu)<\infty.$$
By Theorem 4.8, we know there exists $\gamma^{*}_{\delta}\in\mathcal{P}(S)$, such that
$$G_{\delta\Gamma_{0}}(\mu\lVert\nu)=W_{\delta\Gamma_{0}}(\mu,\gamma^{*}_{\delta%
})+R(\gamma^{*}_{\delta}\lVert\nu).$$
Since $R(\gamma^{*}_{\delta}\lVert\nu)<G_{\delta\Gamma_{0}}(\mu\lVert\nu)\leq W_{%
\Gamma_{0}}(\mu,\nu)$ for $\delta\in(0,1)$, for any sequence $\delta_{k}\subset(0,1)$ there a further a subsequence (again denoted $\delta_{k}$) such that $\delta_{k}$ is decreasing, $\lim_{k\to\infty}\delta_{k}=0$, and $\gamma^{*}_{\delta_{k}}$ converges weakly to a probability measure, which we denote as $\gamma_{0}^{*}$. Then
by the lower semicontinuity of $R(\cdot\lVert\nu)$
$$R(\gamma_{0}^{*}\lVert\nu)\leq\liminf_{k\to\infty}R(\gamma_{\delta_{k}}^{*}%
\lVert\nu)\leq\liminf_{k\to\infty}G_{\delta_{k}\Gamma_{0}}(\mu,\nu)\leq\lim_{k%
\to\infty}\delta_{k}W_{\Gamma_{0}}(\mu,\nu)=0.$$
Since $R(\gamma_{0}^{*}\lVert\nu)\geq 0$ with equality if and only if $\gamma_{0}^{*}=\nu$, we conclude $R(\gamma_{0}^{*}\lVert\nu)=0$ and $\gamma_{0}^{*}=\nu$. Therefore
$$\displaystyle\liminf_{k\to\infty}\frac{1}{\delta_{k}}G_{\delta_{k}\Gamma_{0}}(%
\mu\lVert\nu)$$
$$\displaystyle\geq\liminf_{k\to\infty}\frac{1}{\delta_{k}}W_{\delta_{k}\Gamma_{%
0}}(\mu,\gamma^{*}_{\delta_{k}})$$
$$\displaystyle=\liminf_{k\to\infty}W_{\Gamma_{0}}(\mu,\gamma^{*}_{\delta_{k}})$$
$$\displaystyle\geq W_{\Gamma_{0}}(\mu,\gamma^{*}_{0})=W_{\Gamma_{0}}(\mu,\nu),$$
and since the original sequence was arbitrary
$$\liminf_{\delta\to 0}\frac{1}{\delta}G_{\delta\Gamma_{0}}(\mu\lVert\nu)\geq W_%
{\Gamma_{0}}(\mu,\nu).$$
2) $W_{\Gamma_{0}}(\mu,\nu)=\infty.$
If $\liminf_{\delta\to 0}\frac{1}{\delta}G_{\delta\Gamma_{0}}(\mu\lVert\nu)<\infty$, then there is a subsequence $\left\{\delta_{l}\right\}_{l\in\mathbb{N}}\subset(0,1)$ that achieves this $\liminf$.
From essentially the same proof above applied to this subsequence, it can be shown there exists a further subsequence (again denoted $\left\{\delta_{l}\right\}$) and $\gamma_{0}^{*}\in\mathcal{P}(S)$ such that
$$G_{\delta_{l}\Gamma_{0}}(\mu\lVert\nu)=W_{\delta_{l}\Gamma_{0}}(\mu,\gamma^{*}%
_{\delta})+R(\gamma^{*}_{\delta_{l}}\lVert\nu),$$
and
$$\gamma^{*}_{l}\Rightarrow\gamma_{0}^{*}.$$
Denote $M\doteq\liminf_{\delta\to 0}\frac{1}{\delta}G_{\delta\Gamma_{0}}(\mu\lVert\nu)%
=\lim_{l\to\infty}\frac{1}{\delta_{l}}G_{\delta_{l}\Gamma_{0}}(\mu\lVert\nu)<\infty$.
Since for $l$ large enough
$$R(\gamma_{\delta_{l}^{*}}\lVert\nu)\leq G_{\delta_{l}\Gamma_{0}}(\mu\lVert\nu)%
\leq\delta_{l}(M+1),$$
we have
$$R(\gamma_{0}^{*}\lVert\nu)\leq\liminf_{l\to\infty}R(\gamma_{\delta_{l}}^{*}%
\lVert\nu)\leq\lim_{l\to\infty}\delta_{l}(M+1)=0,$$
and thus $\gamma_{0}^{*}=\nu$. However this leads to
$$\displaystyle M=\lim_{l\to\infty}\frac{1}{\delta_{l}}G_{\delta_{l}\Gamma_{0}}(%
\mu\lVert\nu)$$
$$\displaystyle\geq\lim_{l\to\infty}\frac{1}{\delta_{l}}W_{\delta_{l}\Gamma_{0}}%
(\mu,\gamma_{\delta_{l}}^{*})$$
$$\displaystyle=\lim_{l\to\infty}W_{\Gamma_{0}}(\mu,\gamma_{\delta_{l}}^{*})\geq
W%
_{\Gamma_{0}}(\mu,\nu)=\infty.$$
This contradiction implies
$$\liminf_{\delta\to 0}\frac{1}{\delta}G_{\delta\Gamma_{0}}(\mu\lVert\nu)=\infty%
=W_{\Gamma_{0}}(\mu,\nu).$$
We now consider more refined approximations when $b$ is large.
Previously we described the limiting behavior when we vary the size of $\Gamma$. From Proposition 5.1, we know that when $\mu\not\ll\nu$, $\lim_{b\to\infty}G_{b\Gamma_{0}}(\mu\lVert\nu)=\infty$. In some applications one might use a large transport cost as “penalty” so that while allowing non-absolutely continuous perturbations, control on $G_{\Gamma}(\mu\lVert\nu)$ will ensure that $\mu$ is not too far away from $\nu$.
In the rest of this section, we investigate the behavior when $b\to\infty$, and in particular how $G_{b\Gamma_{0}}(\mu\lVert\nu)$ will behave for fixed $\mu$ and $\nu$. We only consider the case that $\Gamma_{0}=\mathrm{Lip}(c,S;C_{b}(S))$ for some function $c$ satisfies the condition of Theorem 4.2, Assumption 4.4 and Assumption 4.5, and $\mu,\nu\in L^{1}(a)$ with $a$ in Assumption 4.5. We separate the cases depending on whether $\mu$ and $\nu$ are discrete or continuous. The results presented here are only for special cases,
and further development of these sorts of expansions would be useful.
5.1 Finitely supported discrete measures
We will consider the case where $\mathrm{supp}(\nu)$ has finite cardinality, and $\mu$ is also discrete with finite support.
Theorem 5.3
Suppose $\nu$ and $\mu$ are discrete with finite support, where $\mathrm{supp}(\nu)=\left\{x_{i}\right\}_{1\leq i\leq N}$ and $\mathrm{supp}(\mu)=\left\{y_{j}\right\}_{1\leq j\leq M}$. Then there exists $\gamma^{*}\in\mathcal{P}(S)$ with $\gamma^{*}\ll\nu$ such that
$$G_{b\Gamma_{0}}(\mu\lVert\nu)=bW_{\Gamma_{0}}(\mu,\gamma^{*})+R(\gamma^{*}%
\lVert\nu)+o(b),$$
(5.2)
where $o(b)\leq 0$ satisfies $o(b)\rightarrow 0$ as $b\rightarrow\infty$.
Furthermore, we can characterize $\gamma^{*}$ as the measure that minimizes $R(\gamma\lVert\nu)$ over the collection of $\gamma\in P(S)$ that satisfy the constraint
$$\displaystyle W_{\Gamma_{0}}(\mu,\gamma)=\inf_{\theta\ll\nu}W_{\Gamma_{0}}(\mu%
,\theta).$$
(5.3)
If we further assume that
$$c(y_{j},x_{i})\neq c(y_{j},x_{l})$$
for $1\leq j\leq M$ and $1\leq i\neq l\leq N$, then $\gamma^{*}$ has the following form.
Let $S_{i}$ be the indicies $j$ in $\{1,\ldots,M\}$ for which $x_{i}$ is the point in $\left\{x_{l}\right\}_{1\leq l\leq N}$ closest to $y_{j}$.
Then for $1\leq i\leq N$,
$$\gamma^{*}(\{x_{i}\})=\sum_{j\in S_{i}}\mu(\{y_{j}\}).$$
Remark 5.4
In discrete case, is easily checked that the infimum in (5.3) is achieved. Take a sequence of $\theta_{n}\ll\nu$ such that
$$W_{\Gamma_{0}}(\mu,\theta_{n})\leq\inf_{\theta\ll\nu}W_{\Gamma_{0}}(\mu,\theta%
)+1/n.$$
Since $\theta_{n}$ is supported on the compact set $\mathrm{supp}(\nu)=\left\{x_{i}\right\}_{1\leq i\leq N}$
$\left\{\theta_{n}\right\}_{n\in\mathbb{N}}$ is compact, and hence there exist $\theta^{*}\ll\nu$
and a subsequence $\left\{\theta_{n_{k}}\right\}_{k\in\mathbb{N}}$
that converges to $\theta^{*}$ weakly.
By the lower semicontinuity of $W_{\Gamma_{0}}$
$$W_{\Gamma_{0}}(\mu,\theta^{*})\leq\liminf_{n\to\infty}W_{\Gamma_{0}}(\mu,%
\theta_{n})\leq\inf_{\theta\ll\nu}W_{\Gamma_{0}}(\mu,\theta),$$
and therefore $\theta^{*}$ achieves the infimum of (5.3).
Proof. We use the representation $G_{\Gamma}(\mu\lVert\nu)=\inf_{\gamma\in\mathcal{P}(S)}\left\{R(\gamma\lVert%
\nu)+W_{\Gamma}(\mu,\gamma)\right\}$.
First note that
$$\displaystyle G_{b\Gamma_{0}}(\mu\lVert\nu)$$
$$\displaystyle=\inf_{\gamma\in\mathcal{P}(S)}\left\{R(\gamma\lVert\nu)+W_{b%
\Gamma_{0}}(\mu,\gamma)\right\}$$
$$\displaystyle\leq R(\gamma^{*}\lVert\nu)+W_{b\Gamma_{0}}(\mu,\gamma^{*})$$
$$\displaystyle=R(\gamma^{*}\lVert\nu)+bW_{\Gamma_{0}}(\mu,\gamma^{*}).$$
Next, fix any $\varepsilon>0$, and take a near optimizer $\gamma_{b}$, so that for each $b$
$$G_{b\Gamma_{0}}(\mu\lVert\nu)\geq R(\gamma_{b}\lVert\nu)+W_{b\Gamma_{0}}(\mu,%
\gamma_{b})-\varepsilon.$$
We must have $\gamma_{b}\ll\nu$. By (5.3), we know
$$W_{b\Gamma_{0}}(\mu,\gamma_{b})=bW_{\Gamma_{0}}(\mu,\gamma_{b})\geq bW_{\Gamma%
_{0}}(\mu,\gamma^{*})=W_{b\Gamma_{0}}(\mu,\gamma^{*}).$$
Thus
$$\displaystyle R(\gamma^{*}\|\nu)+W_{b\Gamma_{0}}(\mu,\gamma^{*})$$
$$\displaystyle\geq\inf_{\gamma\in P(S)}\left\{R(\gamma\lVert\nu)+W_{b\Gamma_{0}%
}(\mu,\gamma)\right\}$$
$$\displaystyle=G_{b\Gamma_{0}}(\mu,\nu)$$
$$\displaystyle\geq R(\gamma_{b}\lVert\nu)+W_{b\Gamma_{0}}(\mu,\gamma_{b})-\varepsilon$$
$$\displaystyle\geq R(\gamma_{b}\lVert\nu)+W_{b\Gamma_{0}}(\mu,\gamma^{*})-\varepsilon.$$
(5.4)
Since $W_{b\Gamma_{0}}(\mu,\gamma^{*})$ is finite we can subtract it on both sides, and get
$$R(\gamma_{b}\lVert\nu)\leq R(\gamma^{*}\lVert\nu)+\varepsilon$$
for any $b<\infty$. Then by [8, Lemma 1.4.3(c)] $\left\{\gamma_{b}\right\}_{b\in(0,\infty)}$ is tight. Take a convergent subsequence $\left\{\gamma_{b_{k}}\right\}$, and denote its limit by $\gamma_{\infty}$. It is easily checked that $\gamma_{\infty}\ll\nu$, so $W_{\Gamma_{0}}(\mu,\gamma_{\infty})\geq W_{\Gamma_{0}}(\mu,\gamma^{*})$. On the other hand, by (5.4)
$$\displaystyle W_{\Gamma_{0}}(\mu,\gamma_{\infty})-W_{\Gamma_{0}}(\mu,\gamma^{*})$$
$$\displaystyle\leq\liminf_{k\to\infty}W_{\Gamma_{0}}(\mu,\gamma_{b_{k}})-W_{%
\Gamma_{0}}(\mu,\gamma^{*})$$
$$\displaystyle=\liminf_{k\to\infty}\frac{1}{b_{k}}(W_{b_{k}\Gamma_{0}}(\mu,%
\gamma_{b_{k}})-W_{b_{k}\Gamma_{0}}(\mu,\gamma^{*}))$$
$$\displaystyle\leq\liminf_{k\to\infty}\frac{1}{b_{k}}(R(\gamma^{*}\lVert\nu)-R(%
\gamma_{b_{k}}\lVert\nu)+\varepsilon)$$
$$\displaystyle\leq\liminf_{k\to\infty}\frac{1}{b_{k}}(R(\gamma^{*}\lVert\nu)+\varepsilon)$$
$$\displaystyle=0.$$
Thus we conclude that $W_{\Gamma_{0}}(\mu,\gamma_{\infty})=W_{\Gamma_{0}}(\mu,\gamma^{*})$. By the definition of $\gamma^{*}$ we must have $R(\gamma_{\infty}\lVert\nu)\geq R(\gamma^{*}\lVert\nu)$. Choose $k_{0}$ such that $b_{k_{0}}\geq 1$.
Then
$$\displaystyle\liminf_{k\to\infty}\left(G_{b_{k}\Gamma_{0}}(\mu\lVert\nu)-[R(%
\gamma^{*}\lVert\nu)+b_{k}W_{\Gamma_{0}}(\mu,\gamma^{*})]\right)$$
$$\displaystyle\quad\geq\liminf_{k\to\infty}\left(R(\gamma_{b_{k}}\lVert\nu)+b_{%
k}W_{\Gamma_{0}}(\mu,\gamma_{b_{k}})-\varepsilon-(R(\gamma^{*}\lVert\nu)+b_{k}%
W_{\Gamma_{0}}(\mu,\gamma^{*}))\right)$$
$$\displaystyle\quad\geq\liminf_{k\to\infty}(R(\gamma_{b_{k}}\lVert\nu)-R(\gamma%
^{*}\lVert\nu))+\liminf_{k\to\infty}b_{k}(W_{\Gamma_{0}}(\mu,\gamma_{b_{k}})-W%
_{\Gamma_{0}}(\mu,\gamma^{*}))-\varepsilon$$
$$\displaystyle\quad\geq(R(\gamma_{\infty}\lVert\nu)-R(\gamma^{*}\lVert\nu))+%
\liminf_{k\to\infty}(W_{\Gamma_{0}}(\mu,\gamma_{b_{k}})-W_{\Gamma_{0}}(\mu,%
\gamma^{*}))-\varepsilon$$
$$\displaystyle\quad\geq 0+(W_{\Gamma_{0}}(\mu,\gamma_{\infty})-W_{\Gamma_{0}}(%
\mu,\gamma^{*}))-\varepsilon$$
$$\displaystyle\quad\geq-\varepsilon$$
where the fourth inequality is because $R(\gamma_{\infty}\lVert\nu)\geq R(\gamma^{*}\lVert\nu)$ and the lower semi-continuity of $W_{\Gamma_{0}}(\mu,\cdot)$. Since $\varepsilon>0$ is arbitrary, this establishes (5.2) along the given subsequence. For any other sequence $\{b_{k}\}_{k\in\mathbb{N}}$ along which $\lim_{k\to\infty}\left(G_{b_{k}\Gamma_{0}}(\mu\lVert\nu)-[R(\gamma^{*}\lVert%
\nu)+b_{k}W_{\Gamma_{0}}(\mu,\gamma^{*})]\right)$ has a limit, we can also take a subsequence from it according to the discussion above. Thus the statement is proved.
The proof of the claimed form for $\gamma^{*}$ is straightforward and omitted.
5.2 An example with $\nu$ is continuous
To illustrate an interesting scaling phenomenon, here we consider the example with
$S=\mathbb{R}$, $c(x,y)=|x-y|$, $\nu=\mbox{Unif}([0,1])$, $\mu=\delta_{0}$.
Consider $\gamma^{*}(dx)=c_{0}e^{-bx}dx$ and $g^{*}(x)=-bx$ for $0\leq x\leq 1$, where $c_{0}$ is the normalizing constant.
For this example $\Gamma_{0}=\mathrm{Lip}(c,S;C_{b}(S))$ is the set of bounded functions over $\mathbb{R}$ with Lipschitz constant 1. It is easily checked using Theorem 4.12 that $\gamma^{*}$ and $g^{*}$ are the optimizers in
$$G_{b\Gamma_{0}}(\mu\lVert\nu)=\inf_{\gamma\in\mathcal{P}(S)}\left\{W_{b\Gamma_%
{0}}(\mu,\gamma)+R(\gamma\lVert\nu)\right\}=\sup_{g\in b\Gamma_{0}}\left\{\int%
_{S}gd\mu-\log\int_{S}e^{g}\nu\right\}.$$
Thus we have
$$G_{b\Gamma_{0}}(\mu\lVert\nu)=-\int_{0}^{1}bxd\mu-\log\int_{0}^{1}e^{-bx}d\nu=%
-\log\int_{0}^{1}e^{-bx}dx=\log\left(\frac{b}{1-e^{-b}}\right),$$
and in this case, $G_{b\Gamma_{0}}(\mu\lVert\nu)$ scales as $\log(b)+o(\log(b))$.
For comparison we consider the optimal transport cost between $\mu$ and $\nu$.
We have
$$\displaystyle W_{bc}(\mu,\nu)$$
$$\displaystyle\doteq\sup_{g\in b\Gamma_{0}}\left\{\int_{S}gd\mu-\int_{S}gd\nu\right\}$$
$$\displaystyle=b\sup_{g\in\Gamma_{0}}\left\{\int_{S}gd\mu-\int_{S}gd\nu\right\}%
=bW_{c}(\mu,\nu)$$
and one can calculate that $W_{c}(\mu,\nu)=1/2.$ Thus $W_{bc}(\mu,\nu)=b/2$, and so $G_{b\Gamma_{0}}(\mu\lVert\nu)$ gives a much smaller divergence between non absolutely continuous measures $\mu$ and $\nu$ than the corresponding optimal transport cost when the admissible $\Gamma=b\Gamma_{0}$ is becoming large.
6 Application to Uncertainty Bounds
6.1 Extension to unbounded functions
From
$$G_{\Gamma}(\mu\lVert\nu)\doteq\sup_{g\in\Gamma}\left\{\int_{S}gd\mu-\log\int_{%
S}e^{g}d\nu\right\}$$
we get for all $g\in\Gamma$,
$$\int_{S}gd\mu\leq G_{\Gamma}(\mu\lVert\nu)+\log\int_{S}e^{g}d\nu.$$
The inequality above with relative entropy in place of $G_{\Gamma}(\mu\lVert\nu)$ is the key to uncertainty bounds in [10]. We would like to extend this inequality to unbounded functions. Define
$$\hat{\Gamma}_{+}=\left\{f:\text{ there exist }g_{i}\in\Gamma\text{ with }c\leq
g%
_{i}(x)\uparrow f(x)\text{ for }x\in S\right\},$$
and
$$\hat{\Gamma}_{-}=\left\{f:\text{ there exist }g_{i}\in\Gamma\text{ with }c\geq
g%
_{i}(x)\downarrow f(x)\text{ for }x\in S\right\}.$$
Proposition 6.1
For $g\in\hat{\Gamma}_{+}\cup\hat{\Gamma}_{-}$, we have
$$\int_{S}gd\mu\leq G_{\Gamma}(\mu\lVert\nu)+\log\int_{S}e^{g}d\nu.$$
(6.1)
Proof. The proof is straightforward. Take $g\in\hat{\Gamma}_{+}$. Then there exist $g_{i}\in\Gamma$, which are bounded below, and increase to $g$ pointwise in $S$. By monotone convergence theorem,
$$\lim_{i\to\infty}\int_{S}g_{i}d\mu=\int_{S}gd\mu,$$
and
$$\lim_{i\to\infty}\int_{S}e^{g_{i}}d\nu=\int_{S}e^{g}d\nu.$$
Since $g_{i}\in\Gamma$, for all $i$
$$\int_{S}g_{i}d\mu\leq G_{\Gamma}(\mu\lVert\nu)+\log\int_{S}e^{g_{i}}d\nu.$$
Taking $i\to\infty$ in the last display gives (6.1).
For $g\in\hat{\Gamma}_{-}$ the reasoning is essentially the same.
In the case when $\Gamma=\mathrm{Lip}(c,S;C_{b}(S))$, where $c$ satisfies the conditions introduced in Section 4, we can get a stronger version of the result.
The proof is essentially the same as in Lemma 4.7, and is omitted.
Proposition 6.2
Assume $c:S\times S\to\mathbb{R}\cup\{+\infty\}$ satisfies Conditions 4.1, 4.4, 4.5 and 4.6. Fix $\mu,\nu\in L^{1}(a)$. Then for $g\in\mathrm{Lip}(c,S)$
$$\int_{S}gd\mu\leq G_{\Gamma}(\mu\lVert\nu)+\log\int_{S}e^{g}d\nu.$$
6.2 Decomposition and scaling properties
A property of great importance in applications of relative entropy is the chain rule.
When probability measures can be decomposed,
such as when Markov measures on a path space are written as the repeated
integration with respect to transition kernels,
the chain rule allows one to decompose the relative entropy of two such measures on path space in terms of the simpler relative entropies of the transition kernels.
This decomposition also exhibits important scaling properties of relative entropy, e.g.,
that for such Markov measures on path space the relative entropy scales proportionate to the number of time steps.
Except in special circumstances, optimal transport metrics do not possess a property like the chain rule,
and it is therefore not to be expected that $\Gamma$-divergence would either.
However,
if one considers certain classes of functions on path space, then one can show there are analogous decomposition and scaling properties.
In this section we will discuss a setting relevant to many applications,
though the results have many analogues and possible generalizations.
As usual, we assume that $S$ is a Polish space, and let $p:S\times\mathcal{B}(S)$ be a probability transition kernel:
•
for every $A\in\mathcal{B}(S)$ the map $x\rightarrow p(x,A)$ is Borel measurable, and
•
for every $x\in S$, $p(x,\cdot)$ is in $\mathcal{P}(S)$.
The quantities of interest are large and infinite time averages,
both with respect to time and the underlying distribution,
and we wish to bound in a tight fashion the error in such quantities due to model misspecification.
Thus if $q$ is some other transition kernel, then we seek useful bounds on differences of the form
$$\frac{1}{cT}\log E^{\gamma,p}\left[e^{c\sum_{i=1}^{T}f({X}_{i})}\right]-E^{%
\theta,q}\left[\frac{1}{T}\sum_{i=1}^{T}f({X}_{i})\right],$$
where $E^{\gamma,p}$ indicates that the chain uses transition kernel $p$ and initial distribution $\gamma$, and similarly for $E^{\theta,q}$.
Under conditions, relative entropy can provide useful bounds when $q(x,\cdot)\ll p(x,\cdot)$ for a suitable set of $x\in S$.
One question then is under what conditions will the $\Gamma$-divergence allow one to weaken the absolute continuity restriction.
It is also worth noting that even when $q(x,\cdot)\ll p(x,\cdot)$ the bounds obtained using the $\Gamma$-divergence (when applicable) are tighter,
since it is never greater than relative entropy,
and in some cases the improvement can be dramatic.
These issues will be explored in greater detail elsewhere.
It follows directly from discussion in earlier sections that even in the setting of product measures that one must restrict the class of functions $f$ under consideration.
When considering Markov measures, the following definition is relevant.
Definition 6.3
For a transition kernel $p$, let
$$\mathcal{R}(\Gamma,p)=\left\{-\log\int_{S}e^{-g(y)}p(x,dy)-g(x)+a:g\in\Gamma%
\mbox{ and }a\in\mathbb{R}\right\}.$$
Then $\mathcal{R}(\Gamma,p)$ will determine the set of costs $f$ such that bounds can be obtained using the $\Gamma$-divergence.
In particular, we have the following.
Theorem 6.4
Suppose that $f\in\mathcal{R}(\Gamma,p)$ for some $g$ and $a$. Consider any transition kernel $q$ on $S$ and any stationary probability measure $\pi_{q}$ of $q$.
Then
$$\displaystyle\int_{S}f(x)\pi_{q}(dx)$$
$$\displaystyle\leq\int_{S}G_{\Gamma}(q(x,\cdot)\lVert p(x,\cdot))\pi_{q}(dx)+a.$$
Remark 6.5
If $p$ is ergodic then we recognize
$$f(x)=-\log\int_{S}e^{-g(y)}p(x,dy)-g(x)+a$$
as the equation that uniquely characterizes the multiplicative cost
$$a=\lim_{N\rightarrow\infty}\frac{1}{N}\log E^{p}e^{-\sum_{i=0}^{N-1}f(X_{i})},$$
with $g$ a type of cost potential.
Note that for a given $f$ the function $g$ plays no role in the bound.
We need to check that $f$ is in the range of $\Gamma$ (which of course imposes restrictions on $f$),
but the bound does not depend on knowing the specific form of $g$.
Proof. Since $g\in\Gamma$
$$\displaystyle g(x)$$
$$\displaystyle=-f(x)-\log\int_{S}e^{-g(y)}p(x,dy)+a$$
$$\displaystyle=-f(x)+\inf_{q(x,dy)}\left[G_{\Gamma}(q(x,\cdot)\lVert p(x,\cdot)%
)+\int_{S}g(y)q(x,dy)\right]+a.$$
For the given transition kernel $q$
$$g(x)\leq-f(x)+\left[G_{\Gamma}(q(x,\cdot)\lVert p(x,\cdot))+\int_{S}g(y)q(x,dy%
)\right]+a,$$
and integrating both sides with respect to $\pi_{q}(dx)$ and using $\int_{S}q(x,dy)\pi_{q}(dx)=\pi_{q}(dy)$ gives the result.
We next consider two examples to illustrate Definition 6.3.
Example 6.1
$S=\mathbb{R}$, $p(x,\cdot)\sim N(\alpha x,\sigma^{2})$ is normal distribution with mean $\alpha x$ and variance $\sigma^{2}$, where $0<\alpha<1$. Let $g(x)=-bx^{2}-cx-d$, for $b,c,d\in\mathbb{R}$.
Then direct computation gives that when $1-2b\sigma^{2}>0$
$$\displaystyle-\log\int_{S}e^{-g(y)}p(x,dy)-g(x)+a$$
$$\displaystyle\quad=-\frac{b\alpha^{2}x^{2}+c\alpha x+c^{2}\sigma^{2}/2}{1-2b%
\sigma^{2}}+bx^{2}+cx+a$$
$$\displaystyle\quad=b\left(1-\frac{\alpha^{2}}{1-2b\sigma^{2}}\right)x^{2}+c%
\left(1-\frac{\alpha}{1-2b\sigma^{2}}\right)x+a.$$
Letting $k(b)=b(1-\frac{\alpha^{2}}{1-2b\sigma^{2}})$,
$$k^{\prime}(b)=1-\frac{\alpha^{2}}{(1-2b\sigma^{2})^{2}}.$$
Since $1-2b\sigma^{2}>0$, we can conclude $k$ reaches its maximum at $1-2b\sigma^{2}=\alpha$, i.e., $b=\frac{1-\alpha}{2\sigma^{2}}$, where $k(b)=\frac{(1-\alpha)^{2}}{2\sigma^{2}}$.If $b\to\frac{1}{2\sigma^{2}}$ then $k(b)\to-\infty$. Also notice that when $b\neq\frac{1-\alpha}{2\sigma^{2}}$, we can pick $c$ to make the coefficient of $x$ to be any given number. Thus with $p(x,\cdot)\sim N(\alpha x,\sigma^{2})$ and $\Gamma=\left\{bx^{2}+cx+d:b,c,d\in\mathbb{R}\right\}$,
$$R(\Gamma,p)=\left\{bx^{2}+cx+d:b<\frac{(1-\alpha)^{2}}{2\sigma^{2}},c,d\in%
\mathbb{R}\right\}\cup\left\{\frac{(1-\alpha)^{2}}{2\sigma^{2}}x^{2}+d:d\in%
\mathbb{R}\right\}.$$
Example 6.2
$S=\left\{x_{1},x_{2},\dots,x_{n}\right\}$ is a finite space, and there is
a cost function $c:S\times S\rightarrow\mathbb{R}_{+}$ associated with this
space. Take $\Gamma=\mathrm{Lip}(c,S;C_{b}(S))$. Since $p$ is a transition
matrix we denote $p_{ij}=p(x_{i},x_{j})$ and $P=(p_{ij})_{1\leq i,j\leq n}\in\mathbb{R}^{n\times n}$.
A question we ask here is whether there exists $\sigma>0$ such that
$\sigma\Gamma\in R(\Gamma,p)$. In other words, does there exist $\sigma>0$
such that for any $f\in\sigma\Gamma$ we can find $g\in\Gamma$ and
$a\in\mathbb{R}$ such that
$$f(x_{i})=-g(x_{i})-\log\sum_{j=1}^{n}p(x_{i},x_{j})e^{-g(x_{j})}+a,\quad i=1,2%
,\dots,n.$$
If $R(\Gamma,p)$ includes such a neighborhood of zero, then when combined with
Theorem 6.4 it would allow for sensitivity bounds, i.e., bounds on quantities of the
form
$$\frac{d}{d\theta}\sum_{x\in S}\pi(\theta,x)f(x),$$
where $f\in\Gamma$, $\pi(\theta,\cdot)$ is the stationary distribution of
$P(\theta)$, $P(0)=P$, and $P(\theta)$ depends smoothly on a vector of
parameters $\theta$ (see [10]). In contrast with [10], we would not need that
the transition matrices be mutually absolutely continuous.
Since $S$ is finite we write $f_{i}$ for $f(x_{i})$ and let $f=(f_{1},\ldots,f_{n})$, and similarly for $g$. Then the relation above defines a
mapping from $(g,a)$ to $f$, which we denote it by $f=\varphi(g,a)$. Note that
$$(0,0,\dots,0)=\varphi((0,0,\dots,0),0),$$
The $(n,n+1)$ dimensional matrix of partial derivatives takes the form
$$J=\left[\left(P-I\right),\boldsymbol{1}\right],$$
where $I$ is the $n\times n$ identity matrix and $\boldsymbol{1}$ is a column
vector of ones. If we can show that $J$ is of full rank then the range of the
mapping defined by $J$, i.e., the linearization of $\varphi$ will be onto
$\mathbb{R}^{n}$. Then by the implicit function theorem there will be an open
neighborhood $U$ of $\mathbf{0}\in\mathbb{R}^{n}$ and a continuous function
$\gamma:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ such that for all $f\in U$,
$$f=\varphi(0,\gamma(f)).$$
Since $O\doteq\left\{(y_{1},y_{2},\dots,y_{n})|(0,y_{1},\dots,y_{n-1})\in\mathrm{int}%
(\mathrm{Lip}(c,S)),y_{n}\in\mathbb{R}\right\}$ is open,
$\mathbf{0}\in U\cap\gamma^{-1}(O)\subset\mathbb{R}^{n}$ is also open. Thus we
can pick $\sigma>0$ such that $\mathbf{0}\in\sigma\Gamma\subset U\cap\phi^{-1}(O)$. So we have shown the existence of $\sigma>0$ such that
$\sigma\Gamma\in R(\Gamma,p)$.
Whether or not $J$ is of full rank will depend on the structure of $P$. We
have the following lemma.
Lemma 6.6
Suppose that $S=\bar{S}\cup M$, where $M$ consists of the transient states, and
that when restricted to $\bar{S}$, $P$ is ergodic. Then $J$ is of full rank.
Proof. Let $\pi$ denote the stationary distribution of $P$. Then interpreting $\pi$
as a column vector, it is the unique vector in the null space of $(P-I)^{T}$.
According to the Fredholm alternative, the range of $(P-I)$ is the $n-1$
dimensional collection of vectors $b\in\mathbb{R}^{n}$ such that $\left\langle b,\pi\right\rangle=0$. Now $\left\langle\boldsymbol{1},\pi\right\rangle>0$,
which shows that $\boldsymbol{1}$ is not in the range of $(P-I)$.
Therefore the range of $J$ is all of $\mathbb{R}^{n}$.
To give a simple example of how the $\Gamma$-divergence could be used for
model simplification, consider the situation where we are given an ergodic
chain $P$ with state space $\bar{S}$, and would like to replace $P$ by a chain
$Q$ with state space $S=\bar{S}\cup M$, where the new states are intended to
replace a (possibly large) number of states in $\bar{S}$, with the goal being
to maintain good approximation of certain functionals of the stationary
distribution. If $\pi_{q}$ denotes the stationary distribution of $Q$ on $S$
and $\pi_{p}$ that of $P$ on $\bar{S}$, then one could not use relative
entropy to obtain any bounds. Suppose we were to extend $P$ to $\bar{S}\cup M$
(while keeping $P$ as the transition matrix), by making all states in $M$
transient. Then one could use the $\Gamma$-divergence as long as the
functionals of interest are in $R(\Gamma,p)$ (with respect to the extended
transition probabilities). Note that the location of the new states would be
relevant to this question, since the costs $f$ depend on these locations.
Similarly, one could do sensitivity bounds for non-absolutely continuous
transitions by using such a device.
7 Conclusion
In this paper, we defined a new divergence by starting with a variational representation for relative entropy and placing additional restrictions on the collection of test functions used in the representation,
so as to relax the requirement of absolute continuity.
Basic qualitative properties of the divergence were investigated, as well as its relationship with optimal transport metrics.
Future work will use the divergence to develop uncertainty quantification bounds, sensitivity bounds and methods for model approximation and simplification for stochastic for models without the absolute continuity requirement.
Also needed is further investigation of qualitative and computational aspects of the $\Gamma$-divergence.
References
[1]
S.-I. Amari, R. Karakida, and M. Oizumi.
Information geometry connecting Wasserstein distance and
Kullback–Leibler divergence via the entropy-relaxed transportation
problem.
Information Geometry, 1(1):13–37, Sep 2018.
[2]
G$\ddot{u}$zin Bayraksan and David K. Love.
Data-Driven Stochastic Programming Using Phi-Divergences,
chapter 1, pages 1–19.
2015.
[3]
Jose Blanchet, Yang Kang, and Karthyek Murthy.
Robust Wasserstein profile inference and applications to machine
learning.
Journal of Applied Probability, 56, 10 2016.
[4]
Jose Blanchet and Karthyek Murthy.
Quantifying distributional model risk via optimal transport.
SSRN Electronic Journal, 04 2016.
[5]
R.I. Bot, S.-M. Grad, and G. Wanka.
Duality in Vector Optimization.
Springer-Verlag, Berlin Heidelberg, 2009.
[6]
T. Breuer and I. Csisz$\acute{a}$r.
Measureing distribution model risk.
Mathematical Finance, 26:395–411, 2013.
[7]
K. Chowdhary and P. Dupuis.
Distinguishing and integrating aleatoric and epistemic variation in
uncertainty quantification.
ESAIM: Mathematical Modelling and Numerical Analysis,
47:635–662, 2013.
[8]
P. Dupuis and R.S. Ellis.
A Weak Convergence Approach to the Theory of Large Deviations.
John Wiley & Sons, New York, 1997.
[9]
P. Dupuis, M. R. James, and I. R. Petersen.
Robust properties of risk–sensitive control.
Math. Control Signals Systems, 13:318–332, 2000.
[10]
P. Dupuis, M.A. Katsoulakis, Y. Pantazis, and Petr Plechac.
Path-space information bounds for uncertainty quantification and
sensitivity analysis of stochastic dynamics.
SIAM/ASA J. Uncertainty Quantification, 4:80–111, 2016.
[11]
Paul Glasserman and Xingbo Xu.
Robust risk measurement and model risk.
Quantitative Finance, 14(1):29–58, 2014.
[12]
Lars Peter Hansen and Thomas J. Sargent.
Robust control and model uncertainty.
The American Economic Review, 91(2):60–66, 2001.
[13]
S. Kolouri, S. Park, M. Thorpe, D. Slepcev, and G. Rohde.
Optimal mass transport: Signal processing and machine-learning
applications.
IEEE Signal Processing Magazine, 34:43–59, 07 2017.
[14]
Ambrosio L. and Gigli N.
A User’s Guide to Optimal Transport.
Springer, Berlin, Heidelberg, 2013.
[15]
Henry Lam.
Robust sensitivity analysis for stochastic systems.
Mathematics of Operations Research, 41(4):1248–1275, 2016.
[16]
A. E. B. Lim, J. G. Shanthikumar, and T. Watewai.
Robust intensity control with multiple levels of model uncertainty
and the dual risk-sensitive problem.
In 49th IEEE Conference on Decision and Control (CDC), pages
4305–4310, Dec 2010.
[17]
Arnab Nilim and Laurent El Ghaoui.
Robust control of markov decision processes with uncertain transition
matrices.
Operations Research, 53(5):780–798, 2005.
[18]
I.R. Petersen, M.R. James, and P. Dupuis.
Minimax optimal control of stochastic uncertain systems with relative
entropy constraints.
Automatic Control, IEEE Transactions on, 45(3):398–412, 2000.
[19]
Svetlozar T. Rachev and Ludger Rüschendorf.
Mass Transportation Problems.
Probability and Its Applications. Springer-Verlag New York, 1998.
[20]
R.T. Rockafellar.
Convex Analysis.
Princeton University Press, Princeton, 1970.
[21]
W. Rudin.
Functional Analysis.
McGraw-Hill, New York, 1991.
[22]
F. Santambrogio.
Optimal Transport for Applied Mathematicians.
Progress in Nonlinear Differential Equations and Their Applications.
Birkhauser Basel, 2015.
[23]
C. Villani.
Optimal Transport: Old and New.
Springer-Verlag, Berlin Heidelberg, 2009.
P. Dupuis
Division of Applied Mathematics
Brown University
Providence, RI 02912, USA
email:
Paul_Dupuis@brown.edu
Y. Mao
Department of Mathematics
Harvard University
Cambridge, MA 02138, USA
email: mao@math.harvard.edu |
Allocation of Divisible Goods
under Lexicographic Preferences
Leonard J. Schulman
Caltech, MC305-16, Pasadena CA 91125, USA.
schulman@caltech.edu
and
Vijay V. Vazirani
College of Computing, Georgia Institute of
Technology, Atlanta GA 30332, USA.
vazirani@cc.gatech.edu
(Date:: December 2, 2020)
Abstract.
We present a simple and natural non-pricing mechanism for
allocating divisible goods among strategic agents having
lexicographic preferences. Our mechanism has favorable
properties of incentive compatibility (strategy-proofness),
Pareto efficiency, envy-freeness, and time efficiency.
1. Introduction
The study of principled ways of allocating divisible goods
among agents has long been a central topic in mathematical
economics. The method of choice that emerged from this
study, the Arrow-Debreu market model [1], provides a
powerful approach based on pricing and leads to the
fundamental welfare theorems. However, these market-based
methods have limitations when agents are assumed to be
strategic, e.g., these methods are not incentive
compatible. Issues of the latter kind have been studied
within the area of mechanism design for the last four
decades, and have played a large role in the last decade in
algorithmic game theory [18].
In this paper our primary focus is a particular simple and
natural non-pricing mechanism, the Synchronized
Greedy (SG) mechanism, for allocating divisible goods. The
SG mechanism generalizes a mechanism introduced by Crès
and Moulin [6] in the context of a job scheduling
problem, and studied further by Bogomolnaia and
Moulin [5] for the allocation of indivisible goods.
For the setting defined below, we show that SG has favorable
efficiency, incentive compatibility, and fairness
properties. Our setting assumes that each agent has a
lexicographic preference relation over goods. We note that
this preference relation is rational in the sense that
it is complete and transitive. It does not, on the other
hand, satisfy
the continuity condition that preferences between
allocations are preserved under limits; a rational preference
relation that also satisfies this continuity condition
is known to be representable by a utility function,
see [15].
However, the simplicity of the SG mechanism suggests the
possibility of obtaining related mechanisms that achieve
approximate versions of the above properties, when agents’
preferences are representable by utility functions.
In detail, the allocation problem we consider is this.
There are $m$ distinct divisible goods which need to be allocated
among $n$ agents. Good $j$ ($1\leq j\leq m$)
is available in the amount $q_{j}>0$, and agent $i$ $(1\leq i\leq n$) is
to receive a specified total of $r_{i}>0$ across all goods;
the parameters satisfy $\sum_{j}q_{j}=\sum_{i}r_{i}$. An
allocation of goods is a list of
numbers $a_{ij}\geq 0$, with $\sum_{j}a_{ij}=r_{i}$ and
$\sum_{i}a_{ij}=q_{j}$, indicating that agent $i$ receives
quantity $a_{ij}$ of good $j$. The vector
$a_{i*}=(a_{i1},\ldots,a_{im})$ is referred to as agent
$i$’s (share of the) allocation. Each agent $i$ has a
preference list, which is a permutation $\pi_{i}$ of
the goods; $(a_{i\pi_{i}(1)},\ldots,a_{i\pi_{i}(m)})$ is
agent $i$’s sorted allocation. Agent $i$’s
preference among allocations is induced by
lexicographic order. That is to say, agent $i$
lexicographic-prefers $a_{i*}$ to $b_{i*}$, $a_{i*}>_{i}b_{i*}$, if the leftmost nonzero coordinate of
$(a_{i\pi_{i}(1)},\ldots,a_{i\pi_{i}(m)})-(b_{i\pi_{i}(1)},\ldots,b_{i\pi_{i}(m%
)})$ is positive.
Furthermore, we will say that agent $i$
majorization-prefers $a_{i*}$ to $b_{i*}$ if
$$\mbox{for all}\ k=1,\ldots,m:\ \ \sum_{\ell=1}^{k}a_{i\pi_{i}(\ell)}\geq b_{i%
\pi_{i}(\ell)},$$
with at least
one of the inequalities being strict. Observe that
“lexicographic-prefers” is a complete preference relation
without indifference contours (since it is antisymmetric for
distinct allocation shares), and that
“majorization-prefers” is an incomplete preference
relation; moreover the lexicographic order is a refinement
of the majorization order, i.e., majorization-prefers
implies lexicographic-prefers. The phrase “agent $i$
weakly X-prefers” will be used to include the possibility
that agent $i$’s share is identical in the two allocations.
The SG mechanism has the following attributes
w.r.t. the relation lexicographic-prefers.
(1)
The allocation produced by the SG mechanism in response
to truthful bids is Pareto efficient.
(2)
If all $r_{i}$’s are equal, the allocation produced by
the SG mechanism in response to truthful bids is envy-free
in the following sense: each agent weakly
majorization-prefers her allocation to that of any other
agent.
(3)
Incentive compatibility: The SG mechanism is
strategy-proof if $\min_{j}q_{j}\geq\max_{i}r_{i}$. We give a
counterexample in the absence of this condition.
(4)
More generally: the SG mechanism is group strategy-proof
against coalitions of $\ell$ agents if $\min_{j}q_{j}\geq\max_{S:|S|=\ell}\sum_{i\in S}r_{i}$. Again, we give a
counterexample in the absence of this condition.
(5)
The running time to implement the SG mechanism is
$\tilde{O}(mn)$.
(6)
An appropriate extension of the SG mechanism characterizes all
Pareto efficient allocations. (However, in general, the
extension does not possess the rest of the properties listed
above.)
The SG mechanism is deterministic and treats all agents
symmetrically.
It is also interesting to consider whether a mechanism is
equitable—minimizing, in some measurable sense, the
disparity in the welfare of the players. In spite of being
deterministic and treating all agents symmetrically, the SG
mechanism is not particularly equitable, except regarding
the allocation of each agent’s most preferred good. We
provide an example showing that even the allocations of each
agent’s two most preferred goods may be quite
inequitable. However, we describe a
time-efficient algorithm that, for any given $1\leq k\leq m$, equitably allocates the top $k$ goods for each agent. We
further define the notion of a lexicographically
most equitable allocation and give a time-efficient
algorithm to find one.
Since most of our paper deals with the relation
“lexicographic-prefers”, we subsequently abbreviate it to
“prefers”.
1.1. Literature
There has been considerable work on the strategy-proof
allocation of divisible goods in Arrow-Debreu economies,
starting with the seminal work of Hurwicz [10],
e.g., see
[7, 12, 20, 21, 22, 24].
Most of these results are negative, among the recent ones
being Zhou’s result showing that in a 2-agent, $n$-good pure
exchange economy, there can be no allocation mechanism that
is efficient, non-dictatorial (i.e., both agents must
receive non-zero allocations) and strategy-proof
[24].
The paper that is most closely related to our work is that
of Bogomolnaia and Moulin [5]. In their setting
there are $n$ agents and $n$ indivisible goods, each agent
having a total preference ordering over the goods; the
desired outcome is a matching of goods with agents. A
straightforward mechanism for allocating one good to each
agent is random priority (RP): pick a uniformly random
permutation of the agents and ask each agent in turn to
select a good among those left. It is easy to see that this
mechanism is ex post efficient, i.e., the
allocation it produces can be represented as a probability
distribution over Pareto efficient deterministic
allocations, and it is strategy-proof. However, it is not ex
ante efficient.
A random allocation is said to ex ante efficient if
for any profile of von Neumann-Morgenstern utilities that
are consistent with the preferences of agents, the expected
utility vector is Pareto efficient. It is easy to see that
ex ante efficiency implies ex post efficiency.
Solving a conjecture of Gale [8], Zhou [23]
showed that no strategy-proof mechanism that elicits von
Neumann-Morgenstern utilities and achieves Pareto efficiency
can find a “fair” solution even in the weak sense of equal
treatment of equals. He further showed that the solution
found by RP may not be efficient if agents are endowed with
utilities that are consistent with their preferences. Hence,
ex ante efficiency had to be sacrificed, if
strategy-proofness and fairness were desired.
In the face of these choices, the work of Bogomolnaia and
Moulin gave the notion of ordinal efficiency that
is intermediate between ex post and ex ante efficiency; an
allocation $a$ is ordinally efficient if there is no other
allocation $b$ such that every agent majorization-prefers
$b$ to $a$. They went on to show that the mechanism called
probabilistic serial (PS), introduced in Crès
and Moulin [6], yields an ordinally efficient
allocation. Further they show that PS is envy-free and
weakly strategy-proof, defined appropriately for the partial
order “majorization-prefers”. Finally, Bogomolnaia and
Moulin define an extension of PS by introducing different
“eating rates” and show that this set of mechanisms
characterizes the set of all ordinally efficient
allocations.
Katta and Sethuraman [13] generalize the setting of
Bogomolnaia and Moulin to the “full domain”, i.e., agents
may be indifferent between pairs of goods.
Thus, each agent partitions the goods by equality and
defines a total order on the equivalence classes of her
partition (the agent is equally happy with any good received
from an equivalence class). For this setting, they give a
randomized mechanism that is a generalization (different
from ours) of PS and achieves the same game-theoretic
properties as PS.
A mechanism that probabilistically allocates indivisible
goods can also be viewed as one that fractionally allocates
divisible goods. Under the latter interpretation, the SG
mechanism is equivalent to PS for the case that $m=n$, and
the quantity of each good and the requirement of each agent
is one unit. An important difference is that Bogomolnaia
and Moulin analyze PS under an incomplete preference
relation (majorization) in which “most” allocation shares
are incomparable; whereas we analyze SG under a complete
preference relation (lexicographic) that is a refinement of
majorization. The statement that a mechanism’s allocation
is Pareto optimal w.r.t. lexicographic preferences is
considerably stronger than the same statement w.r.t. majorization preferences, because each agent’s share is
dominated by more alternative shares in the lexicographic
order, than it is in the majorization order; so, fewer
allocations are Pareto optimal in the lexicographic than in
the majorization order. Our results should be viewed
therefore as demonstrating that the PS mechanism and its
natural generalization, SG, have far stronger game-theoretic
properties than even envisioned in [5].
Another setting in which incentive compatible, efficient
mechanisms for allocating divisible goods have been studied
is that of of Leontief utilities (after imposing additional
rules). Nicolo [17] gave a mechanism for the case
of two goods and two agents; however, he could not
generalize to an arbitrary number of goods and agents. This
was achieved by Ghodsi et al. [9] under their
non-wasteful rule. In general, under Leontief
utilities, some of the resources may be redundant and will
not improve any agent’s welfare. The non-wasteful rule
requires these excess resources to not be assigned to any
agent and be removed (for possible use outside the
mechanism). This removes the possibility of strategic
manipulation, and in fact Ghodsi et al. gave an incentive
compatible, efficient mechanism. A substantial
generalization of this result was achieved by Li and
Xue [14]; their mechanism is group strategy-proof and
also satisfies fairness conditions.
Finally, we remark only that the problem of allocating a
single divisible good among multiple agents with
known privileges is considerably different; the principal
issue studied in that problem is how to make the division in
a manner that is fair w.r.t. the given privileges. This is
known as the bankruptcy problem and has a long history,
e.g., see [19, 2]. Despite an interesting
resemblance between the PS mechanism and some of the
mechanisms used in the solutions of that
problem [11], the issues at stake in the
bankruptcy literature are distinct from those in our paper
and its predecessors.
2. The Synchronized Greedy Mechanism
The mechanism is simple. Each agent $i$ submits a preference
list $\sigma_{i}$, which is a permutation of the goods; the
submitted list may or may not agree with his true preference
list, called $\pi_{i}$. A simple, representative case, to keep
in mind while reading the mechanism, is that of $m=n$ and
all $q_{j}=r_{i}=1$.
The mechanism simulates the following physical process.
Considering good $j$ as a “liquid”, and each agent as a
receptacle of capacity $r_{i}$, the mechanism starts out at
time $0$ by (for all $i$ in parallel) pouring good
$\sigma_{i}(1)$ into receptacle $i$ at rate $r_{i}$ units of
liquid per unit time. A good may be be simultaneously poured
into several receptacles. This continues until the first
good is exhausted. Then instantaneously all agents whose
favorite good ran out, switch to the next available good on
their preference list, and begin receiving it at the same
rate $r_{i}$. In general, whenever a good is exhausted, all
agents who were in the process of receiving it, switch to
the good which is highest in their preference permutation
and has not yet been exhausted.
Each agent $i$ is, at any time, receiving exactly one type
of good, at rate $r_{i}$. At time $1$ all agents
simultaneously complete their full allocation.
This continuous process can easily be converted into an
$\tilde{O}(mn)$-time discrete algorithm: maintain a priority
queue of goods, keyed by termination times.
3. Properties of the Synchronized Greedy Mechanism
3.1. Pareto Optimality
Let $a^{\sigma}_{ij}$ be the allocation created by
the SG mechanism in response to bids $\sigma$ declared by the
agents. Let $\pi$ denote the bids corresponding to the true
preferences $\pi_{i}$.
Theorem 1.
The allocation produced by the SG mechanism in response to
truthful bids is Pareto efficient: For all $a\neq a^{\pi}$
there is an $i$ such that $a^{\pi}_{i*}>_{i}a_{i*}$.
Proof.
For a collection of bids $\sigma$ let $T^{\sigma}_{j}$ be the
time at which good $j$ is exhausted if the mechanism is run
with bids $\sigma$. In particular $T^{\pi}_{j}$ is the time at
which good $j$ is exhausted if the mechanism is run with the
true preferences $\pi$. Let $\tau_{1}$ be the first time at
which any good is exhausted in $\pi$ and let
$\tau_{k}$, $k\geq 2$, be the least time $t>\tau_{k-1}$ at
which some good is exhausted. Let $R_{k}$ be the set of goods
which are exhausted at time $\tau_{k}$.
By assumption $a\neq a^{\pi}$. For each $j$ let $C_{j}^{+}=\{i:a_{ij}<a^{\pi}_{ij}\}$ and let $C_{j}^{-}=\{i:a_{ij}>a^{\pi}_{ij}\}$. Let $J=\{j:C_{j}^{+}\neq\emptyset\}=\{j:C_{j}^{-}\neq\emptyset\}$ be the set of goods
that are allocated differently in $a$ than in $a^{\pi}$.
Let $k$ be least such that $R_{k}\cap J\neq\emptyset$; that is, $\tau_{k}$ is the first time at which a
good of $J$ is exhausted. Fix $j\in R_{k}\cap J$ and let $i\in C_{j}^{+}$. Then in order that $a_{i*}\geq_{i}a^{\pi}_{i*}$, there
must be a $j^{\prime}$, $\pi_{i}^{-1}(j^{\prime})<\pi_{i}^{-1}(j)$,
$a_{ij^{\prime}}>a^{\pi}_{ij^{\prime}}$. Then $i\in C_{j^{\prime}}^{-}$ and so $j^{\prime}\in J$. Moreover, since $a^{\pi}_{ij}>0$, good $j$ was available
at the time that agent $i$ requested it,
which can only be after time $T^{\pi}_{j^{\prime}}$, so
$T^{\pi}_{j^{\prime}}<T^{\pi}_{j}=\tau_{k}$. Letting $k^{\prime}$ be such that
$\tau_{k^{\prime}}=T^{\pi}_{j^{\prime}}$, we have that $k^{\prime}<k$ and $j^{\prime}\in R_{k^{\prime}}\cap J$, a contradiction.
∎
3.2. Strategy-Proofness
A mechanism is said to be strategy-proof if
no agent can obtain a strictly improved allocation by lying,
provided the rest of the agents submit truthful bids.
Theorem 2.
The SG mechanism is strategy-proof if $\min q_{j}\geq\max r_{i}$.
This is a special case of
Theorem 7/Corollary 8.
However for convenience we provide a stand-alone proof,
since not all the complications of Theorem 7
arise.
Proof.
Without loss of generality focus on agent $1$. We need to
show that for any bid $\sigma_{1}$ (with
$\sigma=(\sigma_{1},\pi_{2},\ldots,\pi_{n})$), $a^{\sigma}_{1*}\leq_{1}a^{\pi}_{1*}$. The theorem is trivial if
$a^{\sigma}=a^{\pi}$.
The theorem is also trivial if agent $1$, bidding
truthfully, receives only his top choice. So we may suppose
that agent $1$ does not receive the entire allocation of any
one good.
We may also suppose that if $a^{\sigma}_{1j}=0$ and
$a^{\sigma}_{1j^{\prime}}>0$, then $\sigma_{1}(j)>\sigma_{1}(j^{\prime})$. In
other words, all the requests in $\sigma_{1}$ that come up
empty may as well be deferred to the end.
Let $\pi_{1}^{-1}(j)$ be the $s$ such that
$\pi_{1}(s)=j$. Let $G(j)=\{j^{\prime}:\pi_{1}^{-1}(j^{\prime})\leq\pi_{1}^{-1}(j)$ and $a^{\pi}_{1}(j^{\prime})>0\}$.
Say that agent $1$ sacrifices good $j$ in $\sigma$
if:
(1)
$a^{\pi}_{1j}>0$,
(2)
$\sigma_{1}(j)>\left|G(j)\right|$, and
(3)
$\pi_{1}^{-1}(j)<\pi_{1}^{-1}(j^{\prime})$ if $j^{\prime}$ also satisfies
(1),(2).
That is to say, $j$ is the most-preferred good which agent
$1$ receives a positive quantity in $\pi$, but requests
later in $\sigma$ than in $\pi$.
Agent $1$ must sacrifice some good, call it $B$, since
otherwise the allocation will not change. See
Figure 1.
Lemma 3.
If $D$ is a good and
$T^{\pi}_{D}<T^{\pi}_{B}$, then $T^{\sigma}_{D}\leq T^{\pi}_{D}$.
Proof.
Supposing the contrary, let $D$ be a counterexample
minimizing $T^{\pi}_{D}$. Since $T^{\pi}_{D}<T^{\pi}_{B}$,
$D\neq B$.
Now let $i$ be any agent (who may or may not be agent $1$)
for whom $a^{\pi}_{iD}>0$. Due to the minimality of $D$, each
of the goods $j$ which $i$ truthfully prefers to $D$, has
$T^{\sigma}_{j}\leq T^{\pi}_{j}$. Therefore $i$ requests $D$ at a
time in $\sigma$ that is at least as soon as the time $i$
requests it in $\pi$.
Since this holds for all $i$ who received a positive
allocation of $D$ in $\pi$, the lemma follows.
∎
Let $N_{B}$ be the set of agents $i\neq 1$ for whom
$a^{\pi}_{iB}>0$. Due to the lemma,
for each agent in
$\{1\}\cup N_{B}$, the request time for $B$ in $\sigma$ is
weakly earlier than it is in $\pi$.
Now let $C$ be the good such that $\pi_{1}^{-1}(C)$ is maximal
subject to $\pi_{1}^{-1}(C)<\pi_{1}^{-1}(B)$ and
$a^{\pi}_{1}(C)>0$. Due to the lemma, all goods $j^{\prime}$ such that
$\pi_{1}^{-1}(j^{\prime})\leq\pi_{1}^{-1}(C)$ have $T^{\sigma}_{j^{\prime}}\leq T^{\pi}_{j^{\prime}}$. Next we show:
Proposition 4.
If $\pi_{1}^{-1}(j^{\prime})\leq\pi_{1}^{-1}(C)$, then
$a^{\sigma}_{1j^{\prime}}=a^{\pi}_{1j^{\prime}}$.
Proof.
Supposing the contrary, let
$\pi_{1}^{-1}(j^{\prime})$ be minimal such that $\pi_{1}^{-1}(j^{\prime})\leq\pi_{1}^{-1}(C)$ and
$a^{\sigma}_{1j^{\prime}}\neq a^{\pi}_{1j^{\prime}}$. There are two possibilities to consider.
(a) $a^{\sigma}_{1j^{\prime}}<a^{\pi}_{1j^{\prime}}$. This is not possible
because then $a^{\sigma}_{1*}<_{1}a^{\pi}_{1*}$.
(b) $a^{\sigma}_{1j^{\prime}}>a^{\pi}_{1j^{\prime}}$. Note:
Lemma 5.
Let $j_{1},j_{2}$ be such that
$\pi_{1}^{-1}(j_{1})\leq\pi_{1}^{-1}(B)$, $\pi_{1}^{-1}(j_{2})\leq\pi_{1}^{-1}(B)$, $a^{\pi}_{1j_{1}}>0$, and
$\pi_{1}^{-1}(j_{1})<\pi_{1}^{-1}(j_{2})$. Then
$\sigma_{1}^{-1}(j_{1})<\sigma_{1}^{-1}(j_{2})$.
Proof.
Consider the least $j_{1}$ that is part of a
pair $j_{1},j_{2}$ violating
the lemma. Then $j_{1}$ satisfies conditions
(1),(2) above, contradicting that $B$ is
the good sacrificed by agent $1$. ∎
It follows that $T^{\sigma}_{j^{\prime}}\geq\sum_{j^{\prime\prime}:\pi_{1}^{-1}(j^{\prime\prime%
})\leq\pi_{1}^{-1}(j^{\prime})}a^{\sigma}_{1j^{\prime\prime}}$.
Due to the minimality of $j^{\prime}$, this means that if $a^{\sigma}_{1j^{\prime}}>a^{\pi}_{1j^{\prime}}$, then $T^{\sigma}_{j^{\prime}}>T^{\pi}_{j^{\prime}}$,
contradicting our earlier conclusion. This completes
demonstration of the Proposition. ∎
A consequence of the Proposition is that $T^{\sigma}_{C}=T^{\pi}_{C}$.
Since agent $1$ sacrifices $B$, his request time for $B$ in
$\sigma$ is strictly greater than his request time
for $B$ in $\pi$.
Recall that $N_{B}$ is nonempty. At time $T^{\pi}_{B}$, the agents
of $N_{B}$ have received as least as much of $B$ in $\sigma$
as they have in $\pi$, and the latter is positive. On the
other hand, at the same time $T^{\pi}_{B}$, agent $1$ has
received strictly less of $B$ in $\sigma$ than he has in
$\pi$.
In order for agent $1$
to receive at least as much of $B$ in $\sigma$ as in
$\pi$, he would have to receive all of $B$ that is
allocated after time $T^{\pi}_{B}$; however, that is not
possible, because the set of agents receiving $B$ after $T^{\pi}_{B}$
includes $N_{B}$. Thus $a^{\sigma}_{1*}<_{1}a^{\pi}_{1*}$.
∎
3.3. Necessity of a Hypothesis on $\{r_{i}\},\{q_{j}\}$
We next provide an example in which strategy-proofness
fails in the absence of the condition $\max r_{i}\leq\min q_{j}$. For convenience now let $r_{1}\geq\ldots\geq r_{n}$ and $q_{1}\leq\ldots\leq q_{m}$.
Example 6.
Let $n=2$ and $m=3$. Let $r_{1}=r_{2}=3/2$; label the goods
$A,B,C$, let $q_{A}=q_{B}=q_{C}=1$, and let the
preference lists be $\pi_{1}=(A,B,C)$, $\pi_{2}=(B,C,A)$. If
agent $1$ bids truthfully he receives the sorted allocation
$(1,0,1/2)$. If instead he bids $(B,A,C)$ (while agent
$2$ bids truthfully), he receives the improved sorted
allocation $(1,1/2,0)$. See Figure 2.
This example does not limit the theorem sharply, because it
uses $r_{1}=(3/2)q_{1}$ rather than $r_{1}$ arbitrarily close to
$q_{1}$. Jeremy Hurwitz has pointed out that one may construct
similar examples with whenever $r_{1}\geq q_{1}/(1-q_{2}/\sum q_{j})$; this would appear to be a tight bound.
3.4. Group Strategy-Proofness
A mechanism is group strategy-proof against a
family $F$ of subsets of agents if for every “coalition”
$S\in F$, if all the agents outside $S$ submit truthful
bids, then the agents of $S$ cannot obtain an improved
allocation by lying, where by “improved allocation” we
mean that no agent of $S$ obtains a worse allocation and at
least one obtains a strictly better allocation.
Theorem 7.
The SG mechanism is group strategy-proof against the family of subsets
$S$ for which $\min_{j}q_{j}\geq\sum_{i\in S}r_{i}$.
Corollary 8.
The SG mechanism is group strategy-proof against coalitions of
$\ell$ agents if $\min_{j}q_{j}\geq\max_{S:|S|=\ell}\sum_{i\in S}r_{i}$.
Proof.
Let $S$ be a coalition as in the theorem statement. We need
to show there is no list of bids for the agents in $S$ such
that all do at least as well as in $\pi$, and some do
strictly better.
The structure of the proof is, as promised, the same as that
of Theorem 2, but what makes
this theorem more involved is that different agents in $S$
can sacrifice different goods, and some of them may be
better off due to the untruthful bids, as they can benefit
from the results of others’ lies; the proof needs to
effectively “chase through” the complicated transfer of
goods (relative to $a^{\pi}$), and show that some
agent in the coalition is worse off than in
$\pi$. Fortunately this “chasing” will not require any
actual iteration.
If $\alpha^{\pi}_{i1}=1$ for all $i\in S$, that is, with
truthful bids these agents receive only their top choices,
then none of them can be strictly rewarded by submitting a
different bid.
Otherwise (i.e., if $\alpha^{\pi}_{i1}<1$ for some $i\in S$),
then thanks to the hypothesis, under the truthful bids
$\pi$, every good has a positive allocation outside $S$.
We need to show that if some members of $S$ submit
untruthful bids, while the agents outside $S$ bid
truthfully, and if the resulting allocation is different
than in $\pi$, then some agent in $S$ does strictly worse
than in $\pi$. Let $\sigma$ be the alternate bids. (Note
$\sigma_{i}=\pi_{i}$ for $i\notin S$.) We use the phrase “$i$
is a willing participant in the coalition $S$” to mean that
$i\in S$ and $a^{\sigma}_{i*}\geq_{i}a^{\pi}_{i*}$.
Without loss of generality we may suppose that every agent
$i\in S$ bids untruthfully and that this has an effect,
i.e., if the agent reverts to a truthful bid then
the allocation is different than in $\sigma$.
Moreover, we may simplify the argument slightly by supposing
that for each agent $i\in S$, if $a^{\sigma}_{ij}=0$ and
$a^{\sigma}_{ij^{\prime}}>0$, then $\sigma_{i}(j)>\sigma_{i}(j^{\prime})$. In
other words, all the requests that come up empty may as
well be deferred to the end.
Let $\pi_{i}^{-1}(j)$ be the $s$ such that
$\pi_{i}(s)=j$. Let $G(i,j)=\{j^{\prime}:\pi_{i}^{-1}(j^{\prime})\leq\pi_{i}^{-1}(j)$ and $a^{\pi}_{i}(j^{\prime})>0\}$.
Say that agent $i$ sacrifices good $j$ in $\sigma$ if:
(1)
$a^{\pi}_{ij}>0$,
(2)
$\sigma_{i}(j)>\left|G(i,j)\right|$, and
(3)
$\pi_{i}^{-1}(j)<\pi_{i}^{-1}(j^{\prime})$ if $j^{\prime}$ also satisfies
(1),(2).
Some good must be sacrificed by some agent, since otherwise the
allocation will not change. (However, while every agent in
$S$ is untruthful, not every $i\in S$ necessarily sacrifices a
good; setting $\sigma_{i}(j)>\pi_{i}(j)$ might have an effect
even if $a^{\pi}_{i}(j)=0$ because of increased availability of
$j$ due to bidding changes of other agents.)
Of all the sacrificed goods let $B$ be one for which $T^{\pi}_{B}$
is minimal.
Lemma 9.
If $D$ is a good and
$T^{\pi}_{D}<T^{\pi}_{B}$, then $T^{\sigma}_{D}\leq T^{\pi}_{D}$.
Proof.
Supposing the contrary, let $D$ be a counterexample
minimizing $T^{\pi}_{D}$.
By the minimality of $B$, $D$ cannot be a sacrificed good.
Now let $i$ be any agent (inside or outside of $S$) for whom
$a^{\pi}_{iD}>0$. Due to the minimality of $D$, each of the
goods $j$ which $i$ truthfully prefers to $D$, has
$T^{\sigma}_{j}\leq T^{\pi}_{j}$. Therefore $i$ requests $D$ at a
time in $\sigma$ that is at least as soon as the time $i$
requests it in $\pi$.
Since this holds for all $i$ who
received a positive allocation of $D$ in $\pi$, the lemma
follows.
∎
Let $O_{B}\subseteq S$ be the set of agents who sacrifice
$B$, and let $N_{B}$ be the set of agents $i$ for whom
$a^{\pi}_{iB}>0$ but who do not sacrifice $B$. Due to the
lemma, for each agent in $O_{B}\cup N_{B}$, the request time
for $B$ in $\sigma$ is weakly earlier than it is in
$\pi$. Now consider an agent $i\in O_{B}$. Let $C$ be the
good such that $\pi_{i}^{-1}(C)$ is maximal subject to
$\pi_{i}^{-1}(C)<\pi_{i}^{-1}(B)$ and $a^{\pi}_{i}(C)>0$. Due to the
lemma, all goods $j^{\prime}$ such that $\pi_{i}^{-1}(j^{\prime})\leq\pi_{i}^{-1}(C)$ have $T^{\sigma}_{j^{\prime}}\leq T^{\pi}_{j^{\prime}}$. Next we
show:
Proposition 10.
If $\pi_{i}^{-1}(j^{\prime})\leq\pi_{i}^{-1}(C)$, then
$a^{\sigma}_{ij^{\prime}}=a^{\pi}_{ij^{\prime}}$.
Proof.
Supposing the contrary, let
$\pi_{i}^{-1}(j^{\prime})$ be minimal such that $\pi_{i}^{-1}(j^{\prime})\leq\pi_{i}^{-1}(C)$ and
$a^{\sigma}_{ij^{\prime}}\neq a^{\pi}_{ij^{\prime}}$. There are two possibilities to consider.
(a) $a^{\sigma}_{ij^{\prime}}<a^{\pi}_{ij^{\prime}}$. This is not possible
because $i$ is a willing participant in the coalition.
(b) $a^{\sigma}_{ij^{\prime}}>a^{\pi}_{ij^{\prime}}$. Note:
Lemma 11.
Let $j_{1},j_{2}$ be such that
$\pi_{i}^{-1}(j_{1})\leq\pi_{i}^{-1}(B)$, $\pi_{i}^{-1}(j_{2})\leq\pi_{i}^{-1}(B)$, $a^{\pi}_{ij_{1}}>0$, and
$\pi_{i}^{-1}(j_{1})<\pi_{i}^{-1}(j_{2})$. Then
$\sigma_{i}^{-1}(j_{1})<\sigma_{i}^{-1}(j_{2})$.
Proof.
Identical to the proof of
Lemma 5 with agent $i$ in place of agent
$1$.
∎
It follows that $T^{\sigma}_{j^{\prime}}\geq\sum_{j^{\prime\prime}:\pi_{i}^{-1}(j^{\prime\prime%
})\leq\pi_{i}^{-1}(j^{\prime})}a^{\sigma}_{ij^{\prime\prime}}$.
Due to the minimality of $j^{\prime}$, this means that if $a^{\sigma}_{ij^{\prime}}>a^{\pi}_{ij^{\prime}}$, then $T^{\sigma}_{j^{\prime}}>T^{\pi}_{j^{\prime}}$,
contradicting our earlier conclusion. This completes
demonstration of the Proposition. ∎
A consequence of the Proposition is that $T^{\sigma}_{C}=T^{\pi}_{C}$.
Since agent $i$ sacrifices $B$, his request time for $B$ in
$\sigma$ is strictly greater than his request time
for $B$ in $\pi$.
Since we are in the case that every good has a positive
allocation outside $S$, $N_{B}$ is nonempty. At time
$T^{\pi}_{B}$, the agents of $N_{B}$ have received as least as
much of $B$ in $\sigma$ as they have in $\pi$, and the
latter is positive. On the other hand, at the same time
$T^{\pi}_{B}$, the agents of $O_{B}$ have received strictly less
of $B$ in $\sigma$ than they have in $\pi$. In order for the
agents of $O_{B}$ to receive collectively at least as much of
$B$ in $\sigma$ as in $\pi$, they would have to receive all
of $B$ that is allocated after time $T^{\pi}_{B}$; however, that
is not possible, because the set of agents receiving $B$
after $T^{\pi}_{B}$ includes $N_{B}$. Therefore there is some $i\in O_{B}$ for whom $a^{\sigma}_{iB}<a^{\pi}_{iB}$. This
contradicts the requirement that $i$ be a willing
participant in the coalition $S$.
∎
Example 12.
Example 6, in which strategy-proofness failed
absent the hypothesis of
Theorem 2, can be extended in a
straightforward manner to one in which the group
strategy-proof property fails to hold absent the hypothesis
of Corollary 8. Again use $m=3$, but
instead of two agents, use $n=2\ell$ agents, the first half
having the same preference order $(A,B,C)$ as agent $1$ in
the earlier example, and the second half having the same
preference order $(B,C,A)$ as agent $2$ in the earlier
example. If all agents bid truthfully, then the first $\ell$
agents each receive the sorted allocation $(1,0,1/2)$;
however if they lie and bid $(B,A,C)$, while the remainder
bid truthfully, then each lying agent receives the improved
sorted allocation $(1,1/2,0)$.
3.5. Envy-Freeness
Theorem 13.
Suppose all $r_{i}$ are equal. Under truthful
bidding, every agent $i$ weakly majorization-prefers his
allocation $a^{\pi}_{i*}$ to the allocation $a^{\pi}_{i^{\prime}*}$of
any other agent $i^{\prime}$.
Proof.
Fix any $1\leq t\leq m$. We
are to show that
$$\sum_{\ell=1}^{k}a^{\pi}_{i\pi_{i}(\ell)}\geq\sum_{\ell=1}^{k}a^{\pi}_{i^{%
\prime}\pi_{i}(\ell)}.$$
Let $t=\max_{1\leq\ell\leq k}T^{\pi}_{\pi_{i}(\ell)}$. Then $t/\sum q_{j}$ is the time that
agent $i$ stops receiving his top $k$ goods. So
$t=\sum_{\ell=1}^{k}a^{\pi}_{i\pi_{i}(\ell)}$. No other agent
can receive any of these goods after time $t$, so $t\geq\sum_{\ell=1}^{k}a^{\pi}_{i^{\prime}\pi_{i}(\ell)}$.
∎
(If the $r_{i}$’s are not equal, the same statement applies to
the relative allocations; see Sec. 5.)
4. Characterizing All Pareto Efficient Allocations
Bogomolnaia and Moulin [5] extended their mechanism
by allowing players to receive
goods at time-varying rates. Specifically, for each agent
$i$ there is a speed function $\eta_{i}$ mapping
the time interval $[0,1]$ into the nonnegative reals, such
that for all $i$
$$\int_{0}^{1}\eta_{i}(t)\;dt=r_{i}.$$
Subject to these speeds, goods flow to agents in order of
the preference lists they bid, just as before. They showed
that this extension characterizes all ordinally efficient
allocations. In this section, we obtain an analogous
characterization of all Pareto efficient allocations by a
similar extension of our mechanism. Specifically, we prove
that for any Pareto efficient allocation of goods,
there exist speeds such that the extended SG mechanism produces
that allocation. We prove this after first noting that the
extended SG mechanism always results in Pareto efficient
allocations.
4.1. Pareto Efficiency
Theorem 14.
Let $\eta_{i}$, $1\leq i\leq n$, be
speed functions. The allocation $a^{\pi}$ produced by the
extended SG mechanism with truthful bids and these speeds, is
Pareto optimal.
Proof.
Let $T^{\pi}_{j}$ be the time at
which good $j$ is exhausted with bids $\pi$ and
speeds $\eta_{i}$. The proof is then, word for word, the
proof of Theorem 1.
∎
4.2. Characterizing All Pareto Efficient Allocations
If the last result mirrored the First Welfare Theorem, the
next mirrors the Second Welfare Theorem:
Theorem 15.
Let $\pi$ be the collection of agent
preference lists and let $a$ be a Pareto-efficient
allocation. There exist speed functions $\eta_{i}$,
$1\leq i\leq n$, such that $a=a^{\pi}$.
Proof.
Construction of the $\eta_{i}$ is simple. Let a “partial
allocation” be $\alpha_{ij}\geq 0$ such that $\sum_{j}\alpha_{ij}\leq r_{i}$ and $\sum_{i}\alpha_{ij}\leq q_{j}$.
Initialize $t=0$ and initialize each agent $i$ with the
empty partial allocation
$\alpha_{i1}=\ldots=\alpha_{im}=0$. Initialize also
$c_{j}=q_{j}$ for all $j$. Then repeat the following until all
$t=1$.
Find an agent $i$ for whom there is an $\ell$ such that
$\alpha_{i\pi_{i}(\ell)}<a_{i\pi_{i}(\ell)}$, and such that for
all $\ell^{\prime}<\ell$, $c_{\pi_{i}(\ell^{\prime})}=0$. Set $\delta=(a_{i\pi_{i}(\ell)}-\alpha_{i\pi_{i}(\ell)})/\sum q_{j}$. For $t<t^{\prime}<t+\delta$, make the settings $\eta_{i}(t^{\prime})=\sum q_{j}$
and, for $i^{\prime}\neq i$, $\eta_{i^{\prime}}(t^{\prime})=0$. Then increment
$\alpha_{i\pi_{i}(\ell)}$ by $\delta\sum q_{j}$, and decrement
$c_{\pi_{i}(\ell)}$ by the same amount. Finally, increment $t$
by $\delta$.
This process can only fail to complete if there comes a time
$t$ at which every agent $i$ falls into one of the following
sets $S_{1}$ and $S_{2}$, and $S_{1}$ is nonempty:
(1)
$S_{1}=\{i$ such that $\sum_{j}\alpha_{ij}<r_{i}$, and
the minimal $\ell$ for which $c_{\pi_{i}(\ell)}>0$ also
satisfies $\alpha_{i\pi_{i}(\ell)}=a_{i\pi_{i}(\ell)}\}$.
(2)
$S_{2}=\{i$ such that $\sum_{j}\alpha_{ij}=r_{i}\}$.
If there is such a $t$, then
for each $i\in S_{1}$, define $\ell_{i}$ to be the $\ell$
identified in the definition of $S_{1}$.
Then for each $i\in S_{1}$, a small positive amount of the good
$\pi_{i}(\ell_{i})$
can be added to $i$’s current partial allocation; the new
partial allocation, no matter how it is completed to an
allocation, improves strictly on $a$ for all $i\in S_{1}$,
and is unchanged for $i\in S_{2}$. Therefore $a$ is not
Pareto efficient.
∎
Examination of the above proof reveals:
Corollary 16.
There is a polynomial time algorithm for checking whether a given
allocation is Pareto efficient.
4.3. No Incentive Compatibility for the Variable Speeds
Variant
We note that the synchrony imposed among agents by the SG
mechanism is key to its incentive compatibility and
envy-freeness properties (indeed, the properties hold even
if the basic mechanism is extended with the same speed
function for all agents). If different agents have
different speed functions under the extended SG mechanism,
Theorems 2 and
7, showing incentive compatibility, fail to
hold. The argument breaks down as soon as it uses
termination times, in Lemma 3. Below is a
counter-example for strategy-proofness; a similar idea gives
counter-examples for group strategy-proofness and
envy-freeness.
Example 17.
Assume
$m=n=4$ and that all $r_{i}=q_{j}=1$. Let the speed function for
agent $1$ be $1$ over the interval $[0,1]$. The speeds of
agents 2, 3, and 4 equal $1$ over the interval $[0,1/2]$,
$0$ over the interval $(1/2,5/6]$, and $3$ over the interval
$(5/6,1]$. The preference orders of agents $1$ and $2$ are
$(1,2,3,4)$, and the preference orders of agents $3$ and
$4$ are $(2,4,3,1)$. If all agents bid truthfully, agent
$1$ receives the sorted allocation $(1/2,0,1/2,0)$. On
the other hand, if agent $1$ bids $(2,1,3,4)$ while the
rest bid truthfully, then agent $1$ receives the better
sorted allocation $(1/2,1/3,1/6,0)$.
5. Equitable Allocations
Given an allocation $a$, let $\bar{a}$ denote the
relative allocation, where
$\bar{a}_{ij}=a_{ij}/r_{i}$. For any $k,\ 1\leq k\leq m$,
say that an allocation is equitable w.r.t. agents’
top $k$ choices if it belongs to
$$\arg\!\max_{a}\min_{i}{(\bar{a}_{i\pi_{i}(1)}+\ldots+\bar{a}_{i\pi_{i}(k)})},$$
where the max is over all allocations $a$.
It is easy to see that the allocation produced by the SG
mechanism is equitable for $k=1$. However, as the
following example illustrates, it is not equitable for $k=2$, or
larger values of $k$.
Example 18.
Let $n=2$, $m=3$, $r_{1}=r_{2}=1$,
$q_{1}=1/2$, $q_{2}=5/6$, and $q_{3}=2/3$. Let the preference
list of the first agent be $(1,2,3)$ and that of the
second agent $(2,3,1)$. Then the SG mechanism gives sorted
allocations of $(1/2,1/6,1/3)$ and $(2/3,1/3,0)$
respectively to the agents, so each receives $2/3$ of his
total allocation from his top two choices. On the other
hand, the sorted allocations $(1/2,1/2,0)$, $(1/3,2/3,0)$ are also feasible, and in this case each agent receives
his entire allocation from his top two
choices.
Next, we show that there is a polynomial-time algorithm
which, given $k$, $(r_{i})$, $(q_{j})$ and the list of agent
preferences, obtains an allocation that is equitable
w.r.t. agents’ top $k$ choices. In fact this allocation
$x=(x_{ij})$ is the solution to the linear program given
below, together with $t$, the minimum over agents of the
relative allocation from the agent’s top $k$ goods.
(5.1)
Maximize
$$\displaystyle t$$
Such that
$$\displaystyle\forall i:\;t\leq\frac{1}{r_{i}}\left(\sum_{\ell=1}^{k}{x_{i\pi_{%
i}(\ell)}}\right)$$
$$\displaystyle\forall i:\;\sum_{j=1}^{m}{x_{ij}}=r_{i}$$
$$\displaystyle\forall j:\;\sum_{i=1}^{n}x_{ij}=q_{j}$$
$$\displaystyle\forall i\ \forall j:\;x_{ij}\geq 0$$
Finally, let us define the notion of the
lexicographically most equitable allocation, which
intuitively is an allocation that simultaneously optimizes
for each $k$, to the extent possible. For any allocation
$a$, and each $k,\ 1\leq k\leq m$, define
(5.2)
$$\beta_{k}=\min_{i}{(\bar{a}_{i\pi_{i}(1)}+\ldots+\bar{a}_{i\pi_{i}(k)})}.$$
Now, define a lexicographically most equitable
allocation to be one that lexicographically maximizes
$(\beta_{1},\ldots,\beta_{m})$.
We now give a polynomial-time algorithm to find a
lexicographically most equitable allocation—it involves
solving $m$ LPs derived from LP (5.1). The
first LP simply computes $\beta_{1}$ by solving LP
(5.1) for $k=1$. Next, for each $k,\ 2\leq k\leq m$, add the following constraints to LP
(5.1) and solve it to determine $\beta_{k}$:
$$\forall i,\forall 1\leq h\leq k-1:\ \frac{1}{r_{i}}\left(\sum_{\ell=1}^{h}{x_{%
i\pi_{i}(\ell)}}\right)\geq\beta_{h}.$$
Clearly, the last LP will yield a most equitable allocation.
Example 19.
For the agents in
Example 18, the
lexicographically most equitable allocation is (given as a
sorted allocation): $(1/2,1/3,1/6)$ for agent $1$ and
$(1/2,1/2,0)$ for agent $2$. This is different from both
the SG allocation and the allocation that is equitable w.r.t. agents’ top $2$ choices.
Although equitability seems to be a desirable property, it
must be noted that an equitable allocation need not be even
Pareto optimal:
Example 20.
Let $n=3$, $m=4$, $r_{1}=r_{2}=r_{3}=2$,
$q_{1}=q_{2}=1$, $q_{3}=q_{4}=2$. Let the preference lists be
$\pi_{1}=(1,2,3,4)$, $\pi_{2}=(3,4,1,2)$,
$\pi_{3}=(4,3,1,2)$. For any $0\leq x\leq 1$, the
following allocation is lexicographically most equitable,
and even stronger, it simultaneously optimizes all
$\beta_{k}$ in Eqn. 5.2:
$$a_{1}=(1,1,0,0),a_{2}=(0,0,1,x),a_{3}=(0,0,1-x,1).$$
Yet this allocation is Pareto optimal only in the
single case $x=0$.
6. Discussion
Our main open problem is the one mentioned in the
Introduction, i.e., achieving approximate versions of the
properties of the SG mechanism but when agents’ preferences
are representable by utility functions.
In [4], Bogomolnaia and Heo show that efficiency
(under the majorization relation), envy-freeness, and a
property they call bounded invariance characterize the PS
mechanism. This leads to the question of appropriately
characterizing the SG mechanism. Towards this end we ask if
efficiency, under the more stringent lexicographic relation,
and envy-freeness suffice. Clearly, a first step would be to
characterize the PS mechanism in this manner. A mechanism is
said to have the bounded invariance property if for
any agent $i$ and any good $j$, changing $i$’s preference
order for goods she likes less than $j$ does not change the
amount (equivalently, probability) of good $j$ each agent
gets.
A natural open question concerns the existence of mechanisms
to produce lexicographically most equitable allocations,
having favorable algorithmic and game-theoretic properties
(e.g., incentive compatibility).
Consider the generalization of our setting so agents may be
indifferent between pairs of goods. Thus, each agent
partitions the goods by equality and defines a total order
on the equivalence classes of her partition (the agent is
equally happy with any good received from an equivalence
class). Preferences are again defined lexicographically over
classes, i.e., by considering the total amount of goods
received in each class. Is there a generalization of our
mechanism to this setting?
Finally, consider the setting in which agents’ preferences
are random permutations of $1,\ldots m$. This leads to many
interesting questions, e.g., what is the distribution of
$\beta_{k}$ as in Eqn. 5.2,
for the allocation $a$ given by the SG mechanism; and,
what is the distribution of the maximized value of $t$ as in LP
(5.1), both for various values of $k$. Regarding
the latter question, for $n=m$ and all $r_{i}=q_{j}=1$, there is
a correspondence in the case $k=1$ with the collision
statistics of random pointers, and so it is known that $t\to(\log\log n)/(\log n)$; for larger $k$ there is a rough
correspondence with the “power of two choices”
literature [3, 16], suggesting likely
asymptotics of $(\log k)/(\log\log n)$ for fixed $k$,
although the correspondence between the problems is not
close enough for us to state this with certainty.
7. Acknowledgments
We are indebted to Hervé Moulin for generously sharing his deep
understanding of this research domain. Thanks also to
Jeremy Hurwitz for stimulating discussions and to Amin
Saberi for pointing us to useful references.
Schulman was supported in part by the NSF. Vazirani was
supported in part by NSF Grant CCF-0914732 and a Google
Research Grant. Vazirani would like to thank the Social and Informational
Sciences Laboratory at Caltech for their hospitality; this work was done
while he was Distinguished SISL Visitor during 2011-12.
References
[1]
K. Arrow and G. Debreu.
Existence of an equilibrium for a competitive economy.
Econometrica, 22:265–290, 1954.
[2]
R. Aumann and M. Maschler.
Game theoretic analysis of a bankruptcy problem from the Talmud.
J. Economic Theory, 36:195–213, 1985.
[3]
Y. Azar, A. Z. Broder, A. R. Karlin, and E. Upfal.
Balanced allocations.
SIAM J. Comput., 29(1):180–200, 1999.
(STOC 1994:593-602).
[4]
A. Bogomolnaia and E. J. Heo.
Probabilistic assignment of objects: Characterizing the serial rule.
Journal of Economic Theory, 2012.
To appear.
[5]
A. Bogomolnaia and H. Moulin.
A new solution to the random assignment problem.
Journal of Economic Theory, 100:295–328, 2001.
[6]
H. Crès and H. Moulin.
Scheduling with opting out: improving upon random priority.
Operations Research, 49(4):565–577, 2001.
[7]
P. Dasgupta, P. Hammond, and E. Maskin.
The implementation of social choice rules.
Review of Economic Studies, 46:153–170, 1979.
[8]
D. Gale.
College course assignments and optimal lotteries.
Mimeo, University of California, Berkeley, 1987.
[9]
A. Ghodsi, M. Zaharia, B. Hindman, A. Konwinski, S. Shenker, and I. Stoica.
Dominant resource fairness: fair allocation of multiple resource
types.
In Proc. 8’th USENIX conference on networked systems design and
implementation, 2011.
[10]
L. Hurwicz.
On informationally decentralized systems.
In C. B. McGuire and R. Radner, editors, Decision and
Organization, pages 297–336. North-Holland, Amsterdam, 1972.
[11]
M. Kaminski.
Hydraulic rationing.
Mathematical Social Sciences, 40:131–155, 2000.
[12]
M. Kato and S. Ohseto.
Toward general impossibility theorems in pure exchange economies.
Social Choice and Welfare, 19:659–664, 2002.
[13]
A. Katta and J. Sethuraman.
A solution to the random assignment problem on the full preference
domain.
Journal of Economic Theory, 131:231–250, 2006.
[14]
J. Li and J. Xue.
Egalitarian division under Leontief preferences.
Manuscript, 2012.
[15]
A. Mas-Colell, M. D. Whinston, and J. R. Green.
Microeconomic theory.
Oxford University Press, 1995.
[16]
M. Mitzenmacher, A. Richa, and R. Sitaraman.
The power of two random choices: A survey of techniques and results.
In P. Pardalos, S. Rajasekaran, and J. Rolim, editors, Handbook
of Randomized Computing: Volume 1, pages 255–312.
To appear.
[17]
A. Nicolo.
Efficiency and truthfulness with Leontief preferences.
Review of Economic Design, 8(4):373–382, 2004.
[18]
N. Nisan.
Introduction to mechanism design (for computer scientists).
In N. Nisan, T. Roughgarden, E. Tardos, and V. V. Vazirani, editors,
Algorithmic Game Theory, pages 209–241. Cambridge University Press,
2007.
[19]
B. O’Neill.
A problem of rights arbitration from the Talmud.
Mathematical Social Sciences, 2(4):345–371, 1982.
[20]
M. Satterthwaite and H. Sonnenschein.
Strategy-proof allocation mechanisms at differentiable points.
Review of Economic Studies, 48:587–597, 1981.
[21]
S. Serizawa.
Inefficiency of strategy-proof rules for pure exchange economies.
Journal of Economic Theory, 106:219–241, 2002.
[22]
S. Serizawa and J. Weymark.
Efficient strategy-proof exchange and minimum consumption guarantees.
Journal of Economic Theory, 109:246–263, 2003.
[23]
L. Zhou.
On a conjecture by Gale about one-sided matching problems.
J. Econ. Theory, 52:123–135, 1990.
[24]
L. Zhou.
Inefficiency of strategy-proof allocation mechanisms in pure exchange
economies.
Social Choice and Welfare, 8(3):247–254, 1991. |
Measurement of charm fragmentation ratios and fractions in
photoproduction at HERA
ZEUS Collaboration
(August 2005)
Abstract
The production of $D^{\ast+}$, $D^{0}$, $D^{+}$, $D_{s}^{+}$ and $\Lambda_{c}^{+}$
charm hadrons and their antiparticles in
$ep$ scattering
at HERA was measured with the ZEUS detector using an
integrated luminosity of $79\,\text{pb}^{-1}$. The measurement has been performed
in the photoproduction regime
with the exchanged-photon virtuality $Q^{2}<1{\,\text{Ge}\kern-0.6667pt\text{V\/}}^{2}$ and
for photon-proton centre-of-mass energies in the range $130<W<300{\,\text{Ge}\kern-0.6667pt\text{V\/}}$.
The charm hadrons were
reconstructed in the range
of transverse momentum $p_{T}(D,\Lambda_{c})>3.8{\,\text{Ge}\kern-0.6667pt\text{V\/}}$
and pseudorapidity $|\eta(D,\Lambda_{c})|<1.6$.
The production cross sections were
used to determine the ratio of neutral and charged $D$-meson
production rates, $R_{u/d}$, the strangeness-suppression factor, $\gamma_{s}$,
and the fraction of charged $D$ mesons produced in a vector state,
$P^{d}_{\rm v}$.
The measured $R_{u/d}$ and $\gamma_{s}$ values agree
with those obtained in deep inelastic scattering and in $e^{+}e^{-}$ annihilations.
The measured $P^{d}_{\rm v}$ value is smaller than, but consistent with,
the previous measurements.
The fractions of $c$ quarks hadronising as a particular charm hadron,
$f(c\rightarrow D,\Lambda_{c})$, were derived
in the given kinematic range.
The measured open-charm fragmentation fractions are consistent
with previous results,
although the measured $f(c\rightarrow D^{*+})$ is smaller
and $f(c\rightarrow\Lambda_{c}^{+})$ is larger than those
obtained in $e^{+}e^{-}$ annihilations.
These results generally support the hypothesis that fragmentation
proceeds independently of the hard sub-process.
\prepnum
DESY–05–147
\makezeustitle
The ZEUS Collaboration
S. Chekanov,
M. Derrick,
S. Magill,
S. Miglioranzi${}^{1}$,
B. Musgrave,
J. Repond,
R. Yoshida
Argonne National Laboratory, Argonne, Illinois 60439-4815, USA ${}^{n}$
M.C.K. Mattingly
Andrews University, Berrien Springs, Michigan 49104-0380, USA
N. Pavel, A.G. Yagües Molina
Institut für Physik der Humboldt-Universität zu Berlin,
Berlin, Germany
P. Antonioli,
G. Bari,
M. Basile,
L. Bellagamba,
D. Boscherini,
A. Bruni,
G. Bruni,
G. Cara Romeo,
L. Cifarelli,
F. Cindolo,
A. Contin,
M. Corradi,
S. De Pasquale,
P. Giusti,
G. Iacobucci,
A. Margotti,
A. Montanari,
R. Nania,
F. Palmonari,
A. Pesci,
A. Polini,
L. Rinaldi,
G. Sartorelli,
A. Zichichi
University and INFN Bologna, Bologna, Italy ${}^{e}$
G. Aghuzumtsyan,
D. Bartsch,
I. Brock,
S. Goers,
H. Hartmann,
E. Hilger,
P. Irrgang${}^{2}$,
H.-P. Jakob,
O.M. Kind,
U. Meyer,
E. Paul${}^{3}$,
J. Rautenberg,
R. Renner,
M. Wang,
M. Wlasenko
Physikalisches Institut der Universität Bonn,
Bonn, Germany ${}^{b}$
D.S. Bailey${}^{4}$,
N.H. Brook,
J.E. Cole,
G.P. Heath,
T. Namsoo,
S. Robins
H.H. Wills Physics Laboratory, University of Bristol,
Bristol, United Kingdom ${}^{m}$
M. Capua,
S. Fazio,
A. Mastroberardino,
M. Schioppa,
G. Susinno,
E. Tassi
Calabria University,
Physics Department and INFN, Cosenza, Italy ${}^{e}$
J.Y. Kim,
K.J. Ma${}^{5}$
Chonnam National University, Kwangju, South Korea ${}^{g}$
M. Helbich,
Y. Ning,
Z. Ren,
W.B. Schmidke,
F. Sciulli
Nevis Laboratories, Columbia University, Irvington on Hudson,
New York 10027 ${}^{o}$
J. Chwastowski,
A. Eskreys,
J. Figiel,
A. Galas,
M. Gil,
K. Olkiewicz,
P. Stopa,
D. Szuba,
L. Zawiejski
The Henryk Niewodniczanski Institute of Nuclear Physics, Polish Academy of Sciences, Cracow,
Poland ${}^{i}$
L. Adamczyk,
T. Bołd,
I. Grabowska-Bołd,
D. Kisielewska,
J. Łukasik,
M. Przybycień,
L. Suszycki,
J. Szuba${}^{6}$
Faculty of Physics and Applied Computer Science,
AGH-University of Science and Technology, Cracow, Poland ${}^{p}$
A. Kotański${}^{7}$,
W. Słomiński
Department of Physics, Jagellonian University, Cracow, Poland
V. Adler,
U. Behrens,
I. Bloch,
K. Borras,
G. Drews,
J. Fourletova,
A. Geiser,
D. Gladkov,
P. Göttlicher${}^{8}$,
O. Gutsche,
T. Haas,
W. Hain,
C. Horn,
B. Kahle,
U. Kötz,
H. Kowalski,
G. Kramberger,
H. Lim,
B. Löhr,
R. Mankel,
I.-A. Melzer-Pellmann,
C.N. Nguyen,
D. Notz,
A.E. Nuncio-Quiroz,
A. Raval,
R. Santamarta,
U. Schneekloth,
H. Stadie,
U. Stösslein,
G. Wolf,
C. Youngman,
W. Zeuner
Deutsches Elektronen-Synchrotron DESY, Hamburg, Germany
S. Schlenstedt
Deutsches Elektronen-Synchrotron DESY, Zeuthen, Germany
G. Barbagli,
E. Gallo,
C. Genta,
P. G. Pelfer
University and INFN, Florence, Italy ${}^{e}$
A. Bamberger,
A. Benen,
F. Karstens,
D. Dobur,
N.N. Vlasov${}^{9}$
Fakultät für Physik der Universität Freiburg i.Br.,
Freiburg i.Br., Germany ${}^{b}$
P.J. Bussey,
A.T. Doyle,
W. Dunne,
J. Ferrando,
J.H. McKenzie,
D.H. Saxon,
I.O. Skillicorn
Department of Physics and Astronomy, University of Glasgow,
Glasgow, United Kingdom ${}^{m}$
I. Gialas${}^{10}$
Department of Engineering in Management and Finance, Univ. of
Aegean, Greece
T. Carli${}^{11}$,
T. Gosau,
U. Holm,
N. Krumnack${}^{12}$,
E. Lohrmann,
M. Milite,
H. Salehi,
P. Schleper,
T. Schörner-Sadenius,
S. Stonjek${}^{13}$,
K. Wichmann,
K. Wick,
A. Ziegler,
Ar. Ziegler
Hamburg University, Institute of Exp. Physics, Hamburg,
Germany ${}^{b}$
C. Collins-Tooth${}^{14}$,
C. Foudas,
C. Fry,
R. Gonçalo${}^{15}$,
K.R. Long,
A.D. Tapper
Imperial College London, High Energy Nuclear Physics Group,
London, United Kingdom ${}^{m}$
M. Kataoka${}^{16}$,
K. Nagano,
K. Tokushuku${}^{17}$,
S. Yamada,
Y. Yamazaki
Institute of Particle and Nuclear Studies, KEK,
Tsukuba, Japan ${}^{f}$
A.N. Barakbaev,
E.G. Boos,
N.S. Pokrovskiy,
B.O. Zhautykov
Institute of Physics and Technology of Ministry of Education and
Science of Kazakhstan, Almaty, Kazakhstan
D. Son
Kyungpook National University, Center for High Energy Physics, Daegu,
South Korea ${}^{g}$
J. de Favereau,
K. Piotrzkowski
Institut de Physique Nucléaire, Université Catholique de
Louvain, Louvain-la-Neuve, Belgium ${}^{q}$
F. Barreiro,
C. Glasman${}^{18}$,
M. Jimenez,
L. Labarga,
J. del Peso,
J. Terrón,
M. Zambrana
Departamento de Física Teórica, Universidad Autónoma
de Madrid, Madrid, Spain ${}^{l}$
F. Corriveau,
C. Liu,
M. Plamondon,
A. Robichaud-Veronneau,
R. Walsh,
C. Zhou
Department of Physics, McGill University,
Montréal, Québec, Canada H3A 2T8 ${}^{a}$
T. Tsurugai
Meiji Gakuin University, Faculty of General Education,
Yokohama, Japan ${}^{f}$
A. Antonov,
B.A. Dolgoshein,
I. Rubinsky,
V. Sosnovtsev,
A. Stifutkin,
S. Suchkov
Moscow Engineering Physics Institute, Moscow, Russia ${}^{j}$
R.K. Dementiev,
P.F. Ermolov,
L.K. Gladilin,
I.I. Katkov,
L.A. Khein,
I.A. Korzhavina,
V.A. Kuzmin,
B.B. Levchenko,
O.Yu. Lukina,
A.S. Proskuryakov,
L.M. Shcheglova,
D.S. Zotkin,
S.A. Zotkin
Moscow State University, Institute of Nuclear Physics,
Moscow, Russia ${}^{k}$
I. Abt,
C. Büttner,
A. Caldwell,
X. Liu,
J. Sutiak
Max-Planck-Institut für Physik, München, Germany
N. Coppola,
G. Grigorescu,
A. Keramidas,
E. Koffeman,
P. Kooijman,
E. Maddox,
H. Tiecke,
M. Vázquez,
L. Wiggers
NIKHEF and University of Amsterdam, Amsterdam, Netherlands ${}^{h}$
N. Brümmer,
B. Bylsma,
L.S. Durkin,
A. Lee,
T.Y. Ling
Physics Department, Ohio State University,
Columbus, Ohio 43210 ${}^{n}$
P.D. Allfrey,
M.A. Bell, A.M. Cooper-Sarkar,
A. Cottrell,
R.C.E. Devenish,
B. Foster,
C. Gwenlan${}^{19}$,
T. Kohno,
K. Korcsak-Gorzo,
S. Patel,
V. Roberfroid${}^{20}$,
P.B. Straub,
R. Walczak
Department of Physics, University of Oxford,
Oxford United Kingdom ${}^{m}$
P. Bellan,
A. Bertolin, R. Brugnera,
R. Carlin,
R. Ciesielski,
F. Dal Corso,
S. Dusini,
A. Garfagnini,
S. Limentani,
A. Longhin,
L. Stanco,
M. Turcato
Dipartimento di Fisica dell’ Università and INFN,
Padova, Italy ${}^{e}$
E.A. Heaphy,
F. Metlica,
B.Y. Oh,
J.J. Whitmore${}^{21}$
Department of Physics, Pennsylvania State University,
University Park, Pennsylvania 16802 ${}^{o}$
Y. Iga
Polytechnic University, Sagamihara, Japan ${}^{f}$
G. D’Agostini,
G. Marini,
A. Nigro
Dipartimento di Fisica, Università ’La Sapienza’ and INFN,
Rome, Italy ${}^{e}\leavevmode\nobreak\ $
J.C. Hart
Rutherford Appleton Laboratory, Chilton, Didcot, Oxon,
United Kingdom ${}^{m}$
H. Abramowicz${}^{22}$,
A. Gabareen,
S. Kananov,
A. Kreisel,
A. Levy
Raymond and Beverly Sackler Faculty of Exact Sciences,
School of Physics, Tel-Aviv University, Tel-Aviv, Israel ${}^{d}$
M. Kuze
Department of Physics, Tokyo Institute of Technology,
Tokyo, Japan ${}^{f}$
S. Kagawa,
T. Tawara
Department of Physics, University of Tokyo,
Tokyo, Japan ${}^{f}$
R. Hamatsu,
H. Kaji,
S. Kitamura${}^{23}$,
K. Matsuzawa,
O. Ota,
Y.D. Ri
Tokyo Metropolitan University, Department of Physics,
Tokyo, Japan ${}^{f}$
M. Costa,
M.I. Ferrero,
V. Monaco,
R. Sacchi,
A. Solano
Università di Torino and INFN, Torino, Italy ${}^{e}$
M. Arneodo,
M. Ruspa
Università del Piemonte Orientale, Novara, and INFN, Torino,
Italy ${}^{e}$
S. Fourletov,
J.F. Martin
Department of Physics, University of Toronto, Toronto, Ontario,
Canada M5S 1A7 ${}^{a}$
J.M. Butterworth${}^{24}$,
R. Hall-Wilton,
T.W. Jones,
J.H. Loizides${}^{25}$,
M.R. Sutton${}^{4}$,
C. Targett-Adams,
M. Wing
Physics and Astronomy Department, University College London,
London, United Kingdom ${}^{m}$
J. Ciborowski${}^{26}$,
G. Grzelak,
P. Kulinski,
P. Łużniak${}^{27}$,
J. Malka${}^{27}$,
R.J. Nowak,
J.M. Pawlak,
J. Sztuk${}^{28}$,
T. Tymieniecka,
A. Ukleja,
J. Ukleja${}^{29}$,
A.F. Żarnecki
Warsaw University, Institute of Experimental Physics,
Warsaw, Poland
M. Adamus,
P. Plucinski
Institute for Nuclear Studies, Warsaw, Poland
Y. Eisenberg,
D. Hochman,
U. Karshon,
M.S. Lightwood
Department of Particle Physics, Weizmann Institute, Rehovot,
Israel ${}^{c}$
E. Brownson,
T. Danielson,
A. Everett,
D. Kçira,
S. Lammers,
L. Li,
D.D. Reeder,
M. Rosin,
P. Ryan,
A.A. Savin,
W.H. Smith
Department of Physics, University of Wisconsin, Madison,
Wisconsin 53706, USA ${}^{n}$
S. Dhawan
Department of Physics, Yale University, New Haven, Connecticut
06520-8121, USA ${}^{n}$
S. Bhadra,
C.D. Catterall,
Y. Cui,
G. Hartner,
S. Menary,
U. Noor,
M. Soares,
J. Standage,
J. Whyte
Department of Physics, York University, Ontario, Canada M3J
1P3 ${}^{a}$
${}^{\ 1}$ also affiliated with University College London, UK
${}^{\ 2}$ now at Siemens VDO/Sensorik, Weissensberg
${}^{\ 3}$ retired
${}^{\ 4}$ PPARC Advanced fellow
${}^{\ 5}$ supported by a scholarship of the World Laboratory
Björn Wiik Research Project
${}^{\ 6}$ partly supported by Polish Ministry of Scientific Research and Information
Technology, grant no.2P03B 12625
${}^{\ 7}$ supported by the Polish State Committee for Scientific Research, grant no.
2 P03B 09322
${}^{\ 8}$ now at DESY group FEB, Hamburg, Germany
${}^{\ 9}$ partly supported by Moscow State University, Russia
${}^{10}$ also affiliated with DESY
${}^{11}$ now at CERN, Geneva, Switzerland
${}^{12}$ now at Baylor University, USA
${}^{13}$ now at University of Oxford, UK
${}^{14}$ now at the Department of Physics and Astronomy, University of Glasgow, UK
${}^{15}$ now at Royal Holloway University of London, UK
${}^{16}$ also at Nara Women’s University, Nara, Japan
${}^{17}$ also at University of Tokyo, Japan
${}^{18}$ Ramón y Cajal Fellow
${}^{19}$ PPARC Postdoctoral Research Fellow
${}^{20}$ EU Marie Curie Fellow
${}^{21}$ on leave of absence at The National Science Foundation, Arlington, VA, USA
${}^{22}$ also at Max Planck Institute, Munich, Germany, Alexander von Humboldt
Research Award
${}^{23}$ Department of Radiological Science
${}^{24}$ also at University of Hamburg, Germany, Alexander von Humboldt Fellow
${}^{25}$ partially funded by DESY
${}^{26}$ also at Łódź University, Poland
${}^{27}$ Łódź University, Poland
${}^{28}$ Łódź University, Poland, supported by the KBN grant 2P03B12925
${}^{29}$ supported by the KBN grant 2P03B12725
$${}^{a}$$
supported by the Natural Sciences and Engineering Research Council of Canada (NSERC)
$${}^{b}$$
supported by the German Federal Ministry for Education and Research (BMBF), under
contract numbers HZ1GUA 2, HZ1GUB 0, HZ1PDA 5, HZ1VFA 5
$${}^{c}$$
supported in part by the MINERVA Gesellschaft für Forschung GmbH, the Israel Science
Foundation (grant no. 293/02-11.2), the U.S.-Israel Binational Science Foundation and
the Benozyio Center for High Energy Physics
$${}^{d}$$
supported by the German-Israeli Foundation and the Israel Science Foundation
$${}^{e}$$
supported by the Italian National Institute for Nuclear Physics (INFN)
$${}^{f}$$
supported by the Japanese Ministry of Education, Culture, Sports, Science and Technology
(MEXT) and its grants for Scientific Research
$${}^{g}$$
supported by the Korean Ministry of Education and Korea Science and Engineering
Foundation
$${}^{h}$$
supported by the Netherlands Foundation for Research on Matter (FOM)
$${}^{i}$$
supported by the Polish State Committee for Scientific Research, grant no.
620/E-77/SPB/DESY/P-03/DZ 117/2003-2005 and grant no. 1P03B07427/2004-2006
$${}^{j}$$
partially supported by the German Federal Ministry for Education and Research (BMBF)
$${}^{k}$$
supported by RF Presidential grant N 1685.2003.2 for the leading scientific schools and
by the Russian Ministry of Education and Science through its grant for Scientific
Research on High Energy Physics
$${}^{l}$$
supported by the Spanish Ministry of Education and Science through funds provided by
CICYT
$${}^{m}$$
supported by the Particle Physics and Astronomy Research Council, UK
$${}^{n}$$
supported by the US Department of Energy
$${}^{o}$$
supported by the US National Science Foundation
$${}^{p}$$
supported by the Polish Ministry of Scientific Research and Information Technology,
grant no. 112/E-356/SPUB/DESY/P-03/DZ 116/2003-2005 and 1 P03B 065 27
$${}^{q}$$
supported by FNRS and its associated funds (IISN and FRIA) and by an Inter-University
Attraction Poles Programme subsidised by the Belgian Federal Science Policy Office
1 Introduction
Charm quark production has been extensively studied at HERA
using $D^{\ast\pm}$ and $D_{s}^{\pm}$
mesons [1, 2, 3, 4, 5].
The data have been compared with theoretical predictions
by assuming the universality of charm fragmentation and using
the charm fragmentation characteristics
obtained in $e^{+}e^{-}$ annihilation
for the calculations of charm production in $ep$ scattering.
However, the charm production mechanisms are not the same
in different collisions. In particular, $c{\bar{c}}$ pairs
in $e^{+}e^{-}$ annihilation are produced dominantly
in a colour-singlet state, which is
not the case for $ep$ scattering. Thus, it is important to test
the charm-fragmentation universality by measuring the charm
fragmentation characteristics at HERA.
In this paper, the measurement of the production
of the weakly decaying charm ground states, the $D^{0}$, $D^{+}$,
$D_{s}^{+}$ pseudo-scalar mesons and the $\Lambda_{c}^{+}$ baryon, is presented.
The production of the charm vector meson $D^{\ast+}$ has also been studied.
The antiparticles of these charm hadrons have been measured as
well111Hereafter, charge conjugation is implied..
The measurement has been performed in
$ep$ scattering at HERA
in the photoproduction regime
with exchanged-photon virtuality, $Q^{2}$, close to zero and for
photon-proton centre-of-mass energies in the range $130<W<300\,$GeV.
The measured production cross sections have
been used to determine the ratio of neutral and charged $D$ meson
production rates, $R_{u/d}$, the strangeness-suppression factor, $\gamma_{s}$,
and the fraction of charged $D$ mesons produced in
a vector state, $P^{d}_{\rm v}$.
The fractions of $c$ quarks hadronising as a particular charm hadron,
$f(c\rightarrow D,\Lambda_{c})$, have been calculated
in the accepted kinematic range.
The open-charm fragmentation fractions in photoproduction
are reported here for the first time.
The results have been compared with the previous HERA
measurements
of the charm fragmentation characteristics in
photoproduction [4]
and in deep inelastic scattering (DIS)
with $Q^{2}>2\,$GeV${}^{2}$ [6].
To compare the results with those obtained in charm production
in $e^{+}e^{-}$
annihilations,
the $f(c\rightarrow D,\Lambda_{c})$ fractions compiled
previously [7] have been updated using recent
values [8] of the relevant branching ratios.
2 Experimental set-up
The analysis was performed with data taken
by the ZEUS Collaboration
from 1998 to 2000.
In this period, HERA collided electrons or positrons222
From now on, the word “electron” is used as a generic term
for electrons and positrons.
with energy $E_{e}=27.5{\,\text{Ge}\kern-0.6667pt\text{V\/}}$ and protons with energy $E_{p}=920{\,\text{Ge}\kern-0.6667pt\text{V\/}}$.
The results are based on a sum of the $e^{-}p$ and $e^{+}p$ samples corresponding
to a total integrated luminosity of
$78.6\pm 1.7\,\text{pb}^{-1}$.
Due to trigger considerations,
$D^{+}$ and $\Lambda_{c}^{+}$ production was measured using only
the $e^{+}p$ sample
corresponding
to an integrated luminosity of
$65.1\pm 1.5\,\text{pb}^{-1}$.
A detailed description of the ZEUS detector can be found
elsewhere [9]. A brief outline of the
components most relevant to this analysis is given
below.
Charged particles are tracked in the central tracking detector (CTD) [10, *npps:b32:181, *nim:a338:254],
which operates in a magnetic field of $1.43\,\text{T}$ provided by a thin
superconducting solenoid. The CTD consists of 72 cylindrical drift chamber
layers, organized in nine superlayers covering the polar-angle333The ZEUS coordinate system is a right-handed Cartesian system, with the $Z$
axis pointing in the proton beam direction, referred to as the “forward
direction”, and the $X$ axis pointing left towards the centre of HERA.
The coordinate origin is at the nominal interaction point. region
$15^{\circ}<\theta<164^{\circ}$. The transverse-momentum resolution for
full-length tracks is $\sigma(p_{T})/p_{T}=0.0058p_{T}\oplus 0.0065\oplus 0.0014/p_{T}$,
with $p_{T}$ in GeV. To estimate the energy loss per unit length, $dE/dx$, of particles in
the CTD [4, 13],
the truncated mean of the anode-wire pulse heights was calculated,
which
removes the lowest $10\%$ and at least the highest $30\%$
depending on the number of saturated hits.
The measured $dE/dx$ values were normalised to
the $dE/dx$ peak position for tracks
with momenta $0.3<p<0.4\,$GeV,
the region of minimum ionisation for pions.
Henceforth $dE/dx$ is quoted in units of minimum
ionising particles (mips).
The resolution of the $dE/dx$ measurement
for full-length tracks is about $9\%$.
The high-resolution uranium–scintillator calorimeter (CAL) [14, *nim:a309:101, *nim:a321:356, *nim:a336:23] consists
of three parts: the forward (FCAL), the barrel (BCAL) and the rear (RCAL)
calorimeters. Each part is subdivided transversely into towers and
longitudinally into one electromagnetic section (EMC) and either one (in RCAL)
or two (in BCAL and FCAL) hadronic sections (HAC). The smallest subdivision of
the calorimeter is called a cell. The CAL energy resolutions, as measured under
test-beam conditions, are $\sigma(E)/E=0.18/\sqrt{E}$ for electrons and
$\sigma(E)/E=0.35/\sqrt{E}$ for hadrons, with $E$ in GeV.
The luminosity was determined from the rate of the bremsstrahlung process
$ep\rightarrow e\gamma p$, where the photon was measured with a
lead–scintillator calorimeter [18, *zfp:c63:391, *acpp:b32:2025]
located at $Z=-107\,\text{m}$.
3 Event simulation
Monte Carlo (MC) samples of charm and beauty events
were produced with
the Pythia 6.156 [21],
Rapgap 2.0818 [22]
and Herwig 6.301 [23, *jhep:0101:010]
event generators.
The generation, based on leading-order matrix elements,
includes direct photon processes,
in which the photon couples
as a point-like object in the hard scatter,
and resolved photon processes, where the photon acts as a source
of partons, one of which participates in the hard scattering process.
Initial- and final-state parton showering is added to simulate
higher-order processes.
The CTEQ5L [25] and GRV LO [26] parametrisations
were used for the proton and photon structure functions, respectively.
The charm and bottom quark masses were set to $1.5\,$GeV and $4.75\,$GeV,
respectively.
Events for all processes were generated in proportion to the predicted
MC cross sections.
The Lund string model [27]
as implemented in Jetset [21]
was used for hadronisation in Pythia and Rapgap.
The Bowler modification [28]
of the Lund symmetric fragmentation function [29]
was used for the charm and bottom quark fragmentation.
In Herwig,
the cluster model [30]
was used for hadronisation.
The fraction of charged $D$ mesons produced in
a vector state was set to $0.6$ for all MC samples.
The Pythia and Rapgap generators
were tuned to describe the photoproduction and DIS regimes,
respectively.
Consequently,
the Pythia events, generated with $Q^{2}<0.6{\,\text{Ge}\kern-0.6667pt\text{V\/}}^{2}$,
were combined with the Rapgap events, generated with $Q^{2}>0.6{\,\text{Ge}\kern-0.6667pt\text{V\/}}^{2}$.
Diffractive events, characterised by
a large rapidity gap
between the proton at high rapidities and the centrally-produced
hadronic system,
were generated using the Rapgap generator in the diffractive mode
and combined with the non-diffractive MC sample.
The contribution of diffractive events was estimated by fitting
the $\eta_{\rm max}$ distribution444
The quantity $\eta_{\rm max}$ is defined as the pseudorapidity
of the CAL energy deposit with the lowest polar angle and an energy
above $400{\,\text{Me}\kern-0.6667pt\text{V\/}}$.
of the data with a linear
combination of the non-diffractive and diffractive MC samples.
The combined sample was used to evaluate the nominal acceptances.
The Herwig MC sample,
generated over the full range of $Q^{2}$ values,
was used to estimate the model dependence
of the acceptance corrections.
To ensure a good description of the data,
the transverse momenta, $p_{T}(D,\Lambda_{c})$, and pseudorapidity,
$\eta(D,\Lambda_{c})$, distributions were reweighted
for both combined Pythia+Rapgap and Herwig MC samples.
The reweighting
factors were tuned using a large $D^{*\pm}$ sample [31].
The effect of the reweighting on the measured fragmentation ratios and
fractions was small; the reweighting uncertainty was included
when estimating the model dependence of the acceptance corrections.
The generated events were passed through a full simulation
of the detector using Geant 3.13 [32]
and processed with the same reconstruction program as used for the data.
4 Event selection
A three-level trigger system was used
to select events online [9, 33].
The first- and second-level trigger used CAL and CTD
data to select $ep$ collisions and to reject beam-gas events.
At the third level,
where the full event information was available,
at least one
reconstructed
charm-hadron candidate was required.
The efficiency of the online charm-hadron reconstruction,
determined relative to the efficiency of the offline reconstruction,
was above $95\%$.
Photoproduction events were selected by requiring that no scattered
electron was identified in the CAL [34].
The Jacquet-Blondel [35] estimator of $W$,
$W_{\rm JB}=\sqrt{2E_{p}(E-p_{Z})}$,
was used, where $E-p_{Z}={\Sigma_{i}(E-p_{Z})_{i}}$
and the sum $i$ runs over all final state energy-flow objects [36] produced from
charged tracks,
as measured in the CTD, and energy clusters measured in the CAL.
After correcting for detector effects, the most important of which were
energy losses in inactive material in front of the CAL and particle losses
in the beam pipe [34, 37],
events were selected in the interval $130<W<300\,$GeV.
The lower limit was set by the trigger requirements, while the upper
limit was imposed to suppress remaining DIS events
with an unidentified scattered
electron in the CAL [34].
Under these conditions, the
photon virtuality lies below $1\,$GeV${}^{2}$.
The median $Q^{2}$ value was estimated from a
Monte Carlo simulation to be about $3\times 10^{-4}\,$GeV${}^{2}$.
5 Reconstruction of charm hadrons
The production of
$D^{\ast+}$, $D^{0}$, $D^{+}$, $D_{s}^{+}$ and $\Lambda_{c}^{+}$
charm hadrons
was measured in
the range of transverse momentum $p_{T}(D,\Lambda_{c})>3.8{\,\text{Ge}\kern-0.6667pt\text{V\/}}$
and pseudorapidity $|\eta(D,\Lambda_{c})|<1.6$.
Charm hadrons were reconstructed using tracks measured
in the CTD and assigned to the reconstructed event vertex.
To ensure good momentum resolution, each track was required
to reach at least the third superlayer of the CTD.
The combinatorial background was significantly reduced by requiring
$p_{T}(D)/E_{T}^{\theta>10^{\circ}}>0.2$
and $p_{T}(\Lambda_{c})/E_{T}^{\theta>10^{\circ}}>0.25$
for charm mesons and baryons, respectively.
The transverse energy was calculated as
$E_{T}^{\theta>10^{\circ}}={\Sigma_{i,\theta_{i}>10^{\circ}}(E_{i}\sin\theta_{i%
}})$,
where the sum runs over all energy deposits in the CAL
with the polar angle
$\theta$ above $10^{\circ}$.
Further background reduction was achieved by imposing cuts
on the transverse momenta and decay angles
of the charm-hadron decay products.
The cut values were tuned using
MC simulation
to enhance signal over background ratios
while keeping acceptances high.
The details of the
reconstruction of the five charm-hadron samples
are given in the next sub-sections.
5.1 Reconstruction of $D^{0}$ mesons
The $D^{0}$ mesons were reconstructed
from the decay
$D^{0}\rightarrow K^{-}\pi^{+}$.
In each event, tracks with opposite charges and
$p_{T}>0.8{\,\text{Ge}\kern-0.6667pt\text{V\/}}$
were combined in pairs to form $D^{0}$ candidates.
The nominal kaon and pion masses were assumed
in turn for each track and
the pair invariant mass, $M(K\pi)$, was calculated.
The distribution of the cosine of the $D^{0}$ decay angle
(defined as the angle $\theta^{*}(K)$ between the kaon
in the $K\pi$ rest frame
and the $K\pi$ line of flight in the laboratory frame) is flat,
whereas the combinatorial background peaks
in the forward and backward directions.
To suppress the background,
$|\cos\theta^{*}(K)|<0.85$ was required.
For selected $D^{0}$ candidates, a search was performed for a track
that could be a “soft” pion ($\pi_{s}$) in a $\mbox{$D^{\ast+}$}\rightarrow\mbox{$D^{0}$}\pi^{+}_{s}$
decay.
The soft pion was required to have $p_{T}>0.2{\,\text{Ge}\kern-0.6667pt\text{V\/}}$ and a charge opposite
to that of the particle taken as a kaon.
The $p_{T}$ cut was raised to $0.25{\,\text{Ge}\kern-0.6667pt\text{V\/}}$ for a data subsample,
corresponding to an integrated luminosity of
$16.9\pm 0.4\,\text{pb}^{-1}$, for which the low-momentum track reconstruction
efficiency was smaller due to the operating conditions
of the CTD [38].
The corresponding $D^{0}$ candidate was assigned to a class of candidates
“with $\Delta M$ tag” if the mass difference,
$\Delta M=M(K\pi\pi_{s})-M(K\pi)$, was in the range
$0.143<\Delta M<0.148{\,\text{Ge}\kern-0.6667pt\text{V\/}}$.
All remaining $D^{0}$ candidates were assigned
to a class of candidates “without $\Delta M$ tag”.
For $D^{0}$ candidates with $\Delta M$ tag,
the kaon and pion mass assignment was fixed
by the track-charge requirements.
For $D^{0}$ mesons without $\Delta M$ tag, the mass
assignment is ambiguous.
The pion and kaon masses can therefore be
assigned to two tracks either correctly, producing a signal peak,
or incorrectly, producing a wider reflected signal.
To remove this reflection,
the mass distribution, obtained for $D^{0}$ candidates with $\Delta M$ tag
and an opposite mass assignment to the kaon and pion tracks,
was subtracted from the $M(K\pi)$ distribution for all $D^{0}$ candidates
without $\Delta M$ tag. The subtracted mass distribution was normalised
to the ratio of numbers of $D^{0}$ mesons without and with
$\Delta M$ tag obtained from a fit described below.
Figure 1 shows the $M(K\pi)$ distribution for $D^{0}$ candidates
without $\Delta M$ tag, obtained after the reflection subtraction,
and the $M(K\pi)$ distribution for $D^{0}$ candidates
with $\Delta M$ tag.
Clear signals are seen at the nominal value of $M(\mbox{$D^{0}$})$ in both
distributions.
The distributions were fitted simultaneously assuming the same shape for
signals in both distributions. To describe the shape, a “modified”
Gaussian function was used:
$${\rm Gauss}^{\rm mod}\propto\exp[-0.5\cdot x^{1+1/(1+0.5\cdot x)}],$$
(1)
where $x=|[M(K\pi)-M_{0}]/\sigma|$.
This functional form described both data and MC signals well.
The signal position, $M_{0}$,
and width, $\sigma$, as well as the numbers of $D^{0}$ mesons in each signal
were free parameters of the fit.
Monte Carlo studies showed that background shapes
in both distributions are compatible with being
linear in the mass range above the signals.
For smaller $M(K\pi)$ values, the background
shapes exhibit an exponential enhancement due to contributions from
other $D^{0}$ decay modes and other $D$ mesons.
Therefore the background shape in the fit was described by the form
$[A+B\cdot M(K\pi)]$ for $M(K\pi)>1.86{\,\text{Ge}\kern-0.6667pt\text{V\/}}$ and
$[A+B\cdot M(K\pi)]\cdot\exp\{C\cdot[M(K\pi)-1.86]\}$
for $M(K\pi)<1.86{\,\text{Ge}\kern-0.6667pt\text{V\/}}$.
The free parameters $A$, $B$ and $C$ were assumed to be independent for the
two $M(K\pi)$ distributions.
The numbers of $D^{0}$ mesons yielded by the fit were
$N^{\rm untag}(\mbox{$D^{0}$})=11430\pm 540$ and
$N^{\rm tag}(\mbox{$D^{0}$})=3259\pm 91$ for selections without and with $\Delta M$ tag,
respectively.
5.2 Reconstruction of additional $D^{\ast+}$ mesons
The $\mbox{$D^{\ast+}$}\rightarrow\mbox{$D^{0}$}\pi^{+}_{s}$
events
with $p_{T}(\mbox{$D^{\ast+}$})>3.8{\,\text{Ge}\kern-0.6667pt\text{V\/}}$ and $|\eta(\mbox{$D^{\ast+}$})|<1.6$
can be considered as a sum of two subsamples:
events with the $D^{0}$ having $p_{T}(\mbox{$D^{0}$})>3.8{\,\text{Ge}\kern-0.6667pt\text{V\/}}$ and $|\eta(\mbox{$D^{0}$})|<1.6$,
and events with the $D^{0}$ outside
of that kinematic range.
The former sample is represented by
$D^{0}$ mesons reconstructed with $\Delta M$ tag, as discussed in the previous
section. The latter sample of “additional” $D^{\ast+}$ mesons
was obtained using
the same $D^{0}\rightarrow K^{-}\pi^{+}$ decay channel
and an independent selection described below.
In each event, tracks with opposite charges and $p_{T}>0.4{\,\text{Ge}\kern-0.6667pt\text{V\/}}$
were combined in pairs to form $D^{0}$ candidates.
To calculate the pair invariant mass, $M(K\pi)$,
kaon and pion masses were assumed in turn for each track.
Only $D^{0}$ candidates which satisfy $1.81<M(K\pi)<1.92{\,\text{Ge}\kern-0.6667pt\text{V\/}}$
were kept.
Moreover, the $D^{0}$ candidates were required to have
either
$p_{T}(\mbox{$D^{0}$})<3.8{\,\text{Ge}\kern-0.6667pt\text{V\/}}$ or $|\eta(\mbox{$D^{0}$})|>1.6$.
Any additional track, with $p_{T}>0.2{\,\text{Ge}\kern-0.6667pt\text{V\/}}$
and a charge opposite to that of the kaon track,
was assigned the pion mass and combined with the $D^{0}$ candidate
to form a $D^{\ast+}$ candidate
with invariant mass $M(K\pi\pi_{s})$.
Here again the $p_{T}$ cut was raised to $0.25{\,\text{Ge}\kern-0.6667pt\text{V\/}}$ for the data subsample
for which the low-momentum track reconstruction
efficiency was smaller.
Figure 2 shows
the $\Delta M$ distribution
for the
$D^{\ast+}$ candidates after all cuts.
A clear signal is seen at the nominal value of
$M(\mbox{$D^{\ast+}$})-M(\mbox{$D^{0}$})$.
The combinatorial background was estimated from
the mass-difference distribution for wrong-charge combinations,
in which both tracks forming the $D^{0}$ candidate have the same charge
and the third track has the opposite charge.
The same tracks from a wrong-charge combination
can produce two $D^{0}$ candidates
due to an ambiguity in the kaon and pion mass assignment to tracks
with the same charge.
To exclude double counting, the multiple combinations of the same tracks
which passed all cuts,
including the $M(K\pi)$ requirement,
were included
with a weight $1/2$.
The number of reconstructed additional $D^{*+}$ mesons
was determined by subtracting
the wrong-charge $\Delta M$ distribution after normalising it
to the distribution of $D^{\ast+}$ candidates
with the appropriate charges in the range
$\,0.15<\Delta M<0.17{\,\text{Ge}\kern-0.6667pt\text{V\/}}$.
The subtraction, performed in the signal range
$0.143<\Delta M<0.148{\,\text{Ge}\kern-0.6667pt\text{V\/}}$, yielded
$N^{\rm add}(\mbox{$D^{\ast+}$})=826\pm 40$.
The $\Delta M$ distribution was also fitted
to a sum of the modified
Gaussian function
(Eq. (1))
describing the signal
and a threshold function
describing the non-resonant background.
The threshold function had a form
$A\cdot(\Delta M-m_{\pi})^{B}$,
where $m_{\pi}$ is the pion mass [8]
and $A$ and $B$ were free parameters.
The results obtained using the fit instead of the subtraction
procedure were used to estimate the systematic uncertainty
of the
signal extraction procedure.
5.3 Reconstruction of $D^{+}$ mesons
The $D^{+}$ mesons were reconstructed
from the decay
$D^{+}\rightarrow K^{-}\pi^{+}\pi^{+}$.
In each event, two tracks with the same charges and
$p_{T}>0.5{\,\text{Ge}\kern-0.6667pt\text{V\/}}$ and
a third track with opposite charge
and $p_{T}>0.7{\,\text{Ge}\kern-0.6667pt\text{V\/}}$
were combined to form $D^{+}$ candidates.
The pion masses were assigned to the two tracks
with the same charges and the kaon mass was assigned to the third track,
after which the candidate invariant mass, $M(K\pi\pi)$, was calculated.
To suppress the combinatorial background,
a cut of
$\cos\theta^{*}(K)>-0.75$ was imposed, where $\theta^{*}(K)$
is the angle between the kaon in the $K\pi\pi$ rest frame and the $K\pi\pi$
line of flight in the laboratory frame.
To suppress background from $D^{\ast+}$ decays, combinations with
$M(K\pi\pi)-M(K\pi)<0.15{\,\text{Ge}\kern-0.6667pt\text{V\/}}$ were removed.
The background from
$\mbox{$D_{s}^{+}$}\rightarrow\phi\pi^{+}$ with $\phi\rightarrow K^{+}K^{-}$
was suppressed by requiring that the invariant mass of any two
$D^{+}$ candidate tracks
with opposite charges
was not within $\pm 8{\,\text{Me}\kern-0.6667pt\text{V\/}}$ of the $\phi$ mass [4]
when the kaon mass was assigned to both tracks.
Figure 3 shows the $M(K\pi\pi)$ distribution for
the $D^{+}$ candidates after all cuts.
Reflections from $D_{s}^{+}$ and $\Lambda_{c}^{+}$
decays to three charged particles
were subtracted using the simulated reflection shapes
normalised to the measured $D_{s}^{+}$ and $\Lambda_{c}^{+}$ production rates.
A clear signal is seen at the nominal value of $D^{+}$ mass.
The mass distribution was fitted to a sum of a modified
Gaussian function
(Eq. (1))
describing the signal
and a linear function describing the non-resonant background.
The number of reconstructed $D^{+}$ mesons yielded by the fit was
$N(\mbox{$D^{+}$})=8950\pm 600$.
5.4 Reconstruction of $D_{s}^{+}$ mesons
The $D_{s}^{+}$ mesons were reconstructed
from the decay
$\mbox{$D_{s}^{+}$}\rightarrow\phi\pi^{+}$ with $\phi\rightarrow K^{+}K^{-}$.
In each event, tracks with opposite charges and
$p_{T}>0.7{\,\text{Ge}\kern-0.6667pt\text{V\/}}$
were assigned the kaon mass and
combined in pairs to form $\phi$ candidates.
The $\phi$ candidate was kept if its invariant mass, $M(KK)$,
was within $\pm 8{\,\text{Me}\kern-0.6667pt\text{V\/}}$ of the $\phi$ mass [4].
Any additional track with $p_{T}>0.5{\,\text{Ge}\kern-0.6667pt\text{V\/}}$
was assigned the pion mass and combined with the $\phi$ candidate
to form a $D_{s}^{+}$ candidate
with invariant mass $M(KK\pi)$.
To suppress the combinatorial background,
the following requirements were applied:
•
$\cos\theta^{*}(\pi)<0.85$, where $\theta^{*}(\pi)$
is the angle between the pion in the $KK\pi$ rest frame and the $KK\pi$
line of flight in the laboratory frame;
•
$|\cos^{3}\theta^{\prime}(K)|>0.1$, where $\theta^{\prime}(K)$
is the angle between one of the kaons and the pion in
the $KK$ rest frame.
The decay of the pseudoscalar $D_{s}^{+}$ meson to the $\phi$ (vector)
plus $\pi^{+}$ (pseudoscalar) final state results in an alignment of
the spin of the $\phi$ meson with respect to the direction of
motion of the $\phi$ relative to $D_{s}^{+}$.
Consequently, the distribution of $\cos\theta^{\prime}(K)$
follows a $\cos^{2}\theta^{\prime}(K)$ shape, implying a flat
distribution for $\cos^{3}\theta^{\prime}(K)$.
In contrast, the $\cos\theta^{\prime}(K)$ distribution of
the combinatorial background is flat and its $\cos^{3}\theta^{\prime}(K)$
distribution peaks at zero. The cut suppressed the background significantly
while reducing the signal by $10\%$.
Figure 4 shows the $M(KK\pi)$ distribution for
the $D_{s}^{+}$ candidates after all cuts.
Reflections from $D^{+}$ and $\Lambda_{c}^{+}$
decays to three
charged particles
were subtracted using the simulated reflection shapes
normalised to the measured $D^{+}$ and $\Lambda_{c}^{+}$ production rates.
A clear signal is seen at the nominal $D_{s}^{+}$ mass.
There is also a smaller signal around the nominal $D^{+}$ mass
as expected from the decay
$\mbox{$D^{+}$}\rightarrow\phi\pi^{+}$ with $\phi\rightarrow K^{+}K^{-}$.
The mass distribution was fitted to a sum of two modified
Gaussian functions
(Eq. (1))
describing the signals
and an exponential function describing the non-resonant background.
To reduce the number of free parameters,
the width of the $D^{+}$ signal was constrained to $8/9$ of the $D^{+}_{s}$
signal width; the constraint was verified by MC studies.
The number of reconstructed $D_{s}^{+}$
mesons yielded by the fit was
$N(\mbox{$D_{s}^{+}$})=1102\pm 83$
555
The number of $D^{+}$ mesons, $239\pm 63$, was not used further in the analysis..
5.5 Reconstruction of $\Lambda_{c}^{+}$ baryons
The $\Lambda_{c}^{+}$ baryons were reconstructed
from the decay
$\mbox{$\Lambda_{c}^{+}$}\rightarrow K^{-}p\pi^{+}$.
In each event, two same-charge tracks
and a third track with opposite charge
were combined to form $\Lambda_{c}^{+}$ candidates.
Due to the large difference between the proton and pion masses, the proton
momentum
is typically larger than that of
the pion.
Therefore, the proton (pion) mass was assigned to those of the two tracks with
the same charges which had larger (smaller) momentum.
The kaon mass was assigned to the third track and
the candidate invariant mass, $M(Kp\pi)$, was calculated.
Only candidates with $p_{T}(K)>0.75{\,\text{Ge}\kern-0.6667pt\text{V\/}}$, $p_{T}(p)>1.3{\,\text{Ge}\kern-0.6667pt\text{V\/}}$
and $p_{T}(\pi)>0.5{\,\text{Ge}\kern-0.6667pt\text{V\/}}$ were kept.
To suppress the combinatorial background,
the following requirements, motivated by MC studies, were applied:
•
$\cos\theta^{*}(K)>-0.9$, where $\theta^{*}(K)$
is the angle between the kaon in the $Kp\pi$ rest frame and the $Kp\pi$
line of flight in the laboratory frame;
•
$\cos\theta^{*}(p)>-0.25$, where $\theta^{*}(p)$
is the angle between the proton in the $Kp\pi$ rest frame and the $Kp\pi$
line of flight in the laboratory frame;
•
$p^{*}(\pi)>90{\,\text{Me}\kern-0.6667pt\text{V\/}}$, where $p^{*}(\pi)$
is the pion momentum in the $Kp\pi$ rest frame.
To suppress the combinatorial background further,
the measured $dE/dx$ values of the three $\Lambda_{c}^{+}$ candidate
tracks were used. The parametrisations of the $dE/dx$ expectation values
and the $\chi^{2}_{1}$ probabilities $l_{p}$, $l_{K}$ and $l_{\pi}$ of the proton, kaon
and pion hypotheses, respectively, were obtained in the same way as described
in a previous publication [31].
The $l_{p}$, $l_{K}$ and $l_{\pi}$ distributions
for the $\Lambda_{c}^{+}$ candidate tracks
show sharp peaks around
zero and become relatively flat towards one.
To maximise the ratios of the numbers of correctly assigned protons,
kaons and pions to the square roots of the numbers of background
particles, the cuts $l_{p}>0.15$, $l_{K}>0.03$ and $l_{\pi}>0.01$ were
applied. The cuts rejected those ranges where the $l_{p}$, $l_{K}$ and $l_{\pi}$
distributions were at least twice as high as in
the range $0.8-1$.
Figure 5 shows the $M(Kp\pi)$ distribution for
the $\Lambda_{c}^{+}$ candidates after all cuts.
Reflections from $D^{+}$ and $D_{s}^{+}$
decays to three
charged particles were subtracted using the simulated reflection shapes
normalised to the measured $D^{+}$ and $D_{s}^{+}$ production rates.
A clear signal is seen at the nominal $\Lambda_{c}^{+}$ mass.
The mass distribution was fitted to a sum of a modified
Gaussian function
(Eq. (1))
describing the signal
and a linear function describing the non-resonant background.
The number of reconstructed $\Lambda_{c}^{+}$ baryons yielded by the fit was
$N(\mbox{$\Lambda_{c}^{+}$})=1440\pm 220$.
6 Charm-hadron production cross sections
The charm-hadron cross sections were calculated
for the process $ep\rightarrow eD(\Lambda_{c})X$
in the kinematic region $Q^{2}<1{\,\text{Ge}\kern-0.6667pt\text{V\/}}^{2}$,
$\,\,\,130<W<300{\,\text{Ge}\kern-0.6667pt\text{V\/}}$,
$p_{T}(D,\Lambda_{c})>3.8{\,\text{Ge}\kern-0.6667pt\text{V\/}}$ and $|\eta(D,\Lambda_{c})|<1.6$.
The cross section for a given charm hadron was calculated from
$$\sigma(D,\Lambda_{c})=\frac{N(D,\Lambda_{c})}{\mbox{$\cal A$}\cdot\mbox{$\cal L%
$}\cdot\mbox{$\cal B$}}\,,$$
where $N(D,\Lambda_{c})$ is the number of reconstructed charm hadrons,
$\cal A$ is the acceptance for this charm hadron,
$\cal L$ is the integrated luminosity
and $\cal B$ is the branching ratio or the product of the
branching ratios [8] for the decay channel
used in the reconstruction.
The third uncertainties quoted below for
the measured cross sections and charm fragmentation ratios and
fractions are due to
the branching-ratio uncertainties666Contributions from uncertainties of
different branching ratios were added in quadrature..
The combined Pythia+Rapgap MC sample
was used to evaluate the nominal acceptances.
Small admixtures to the reconstructed signals from
other decay modes were taken into account
in the acceptance correction procedure.
To correct from $N^{\rm tag}(\mbox{$D^{0}$})$ ($N^{\rm untag}(\mbox{$D^{0}$})$) to
the production cross sections for $D^{0}$ mesons originating (not originating)
from $D^{\ast+}$ decays, small migrations between the two samples
were taken into account.
The $b$-quark relative contributions, predicted by the MC simulation
using branching ratios of $b$-quark decays to the charmed hadrons
measured at LEP [39, 40],
were subtracted from all measured cross sections777The branching
ratios of the $b$-quark decays were updated using recent
values [8] of the relevant charm-hadron decay branching ratios..
Subtraction of the $b$-quark contribution reduced the measured cross
sections by $3-7\%$ and
changed the measured charm fragmentation ratios and fractions
by less than $4\%$.
Using the reconstructed signals (see Section 5)
the following cross sections
for the sum of each charm hadron and its antiparticle
were calculated.
The systematic uncertainties are discussed in Section 8:
•
the production cross section for $D^{0}$ mesons
not originating from the $\mbox{$D^{\ast+}$}\rightarrow\mbox{$D^{0}$}\pi^{+}_{s}$ decays:
$$\sigma^{\rm untag}(\mbox{$D^{0}$})=8.49\pm 0.44({\rm stat.})^{+0.47}_{-0.48}({%
\rm syst.})^{+0.20}_{-0.19}({\rm br.})\,{\rm nb};$$
•
the production cross section for $D^{0}$ mesons originating
from the $\mbox{$D^{\ast+}$}\rightarrow\mbox{$D^{0}$}\pi^{+}_{s}$ decays:
$$\sigma^{\rm tag}(\mbox{$D^{0}$})=2.65\pm 0.08({\rm stat.})^{+0.11}_{-0.10}({%
\rm syst.})\pm 0.06({\rm br.})\,{\rm nb}.$$
The ratio $\sigma^{\rm tag}(\mbox{$D^{0}$})/\mbox{${\cal B}_{D^{\ast+}\rightarrow D^{0}%
\pi^{+}}$}$ gives
the $D^{\ast+}$
cross section, $\sigma(\mbox{$D^{\ast+}$})$,
corresponding to $D^{0}$ production in the kinematic range
$p_{T}(\mbox{$D^{0}$})>3.8{\,\text{Ge}\kern-0.6667pt\text{V\/}}$ and $|\eta(\mbox{$D^{0}$})|<1.6$
for the $\mbox{$D^{\ast+}$}\rightarrow\mbox{$D^{0}$}\pi^{+}_{s}$ decay.
Here $\mbox{${\cal B}_{D^{\ast+}\rightarrow D^{0}\pi^{+}}$}=0.677\pm 0.005$ [8]
is the branching ratio of the $\mbox{$D^{\ast+}$}\rightarrow\mbox{$D^{0}$}\pi^{+}_{s}$ decay;
•
the production cross section for additional $D^{\ast+}$ mesons:
$$\sigma^{\rm add}(\mbox{$D^{\ast+}$})=1.05\pm 0.07({\rm stat.})^{+0.09}_{-0.04}%
({\rm syst.})\pm 0.03({\rm br.})\,{\rm nb}.$$
The sum $\sigma^{\rm add}(\mbox{$D^{\ast+}$})+\sigma^{\rm tag}(\mbox{$D^{0}$})/\mbox{${%
\cal B}_{D^{\ast+}\rightarrow D^{0}\pi^{+}}$}$
gives the production cross section for $D^{\ast+}$ mesons
in the kinematic range
$p_{T}(\mbox{$D^{\ast+}$})>3.8{\,\text{Ge}\kern-0.6667pt\text{V\/}}$ and $|\eta(\mbox{$D^{\ast+}$})|<1.6$:
$$\sigma^{\rm kin}(\mbox{$D^{\ast+}$})=4.97\pm 0.14({\rm stat.})^{+0.23}_{-0.18}%
({\rm syst.})^{+0.13}_{-0.12}({\rm br.})\,{\rm nb};$$
•
the production cross section for $D^{+}$ mesons:
$$\sigma(\mbox{$D^{+}$})=5.07\pm 0.36({\rm stat.})^{+0.44}_{-0.23}({\rm syst.})^%
{+0.34}_{-0.30}({\rm br.})\,{\rm nb};$$
•
the production cross section for $D_{s}^{+}$ mesons:
$$\sigma(\mbox{$D_{s}^{+}$})=2.37\pm 0.20({\rm stat.})\pm 0.20({\rm syst.})^{+0.%
72}_{-0.45}({\rm br.})\,{\rm nb};$$
•
the production cross section for $\Lambda_{c}^{+}$ baryons:
$$\sigma(\mbox{$\Lambda_{c}^{+}$})=3.59\pm 0.66({\rm stat.})^{+0.54}_{-0.66}({%
\rm syst.})^{+1.15}_{-0.70}({\rm br.})\,{\rm nb}.$$
7 Charm fragmentation ratios and fractions
7.1 Ratio of neutral to charged $D$-meson production rates
Neglecting influences from decays of heavier excited $D$ mesons,
the ratio of neutral to charged $D$-meson production rates
is given by the ratio of the sum of $D^{*0}$ and direct $D^{0}$
production cross sections to the sum
of $D^{\ast+}$ and direct $D^{+}$
production cross sections:
$$R_{u/d}=\frac{\sigma(D^{*0})+\sigma^{\rm dir}(D^{0})}{\sigma(D^{*+})+\sigma^{%
\rm dir}(\mbox{$D^{+}$})},$$
where $\sigma^{\rm dir}(D^{0})$ and $\sigma^{\rm dir}(\mbox{$D^{+}$})$ are those parts of the $D^{0}$ and $D^{+}$
inclusive cross sections which
do not originate from $D^{*0}$ and $D^{\ast+}$ decays.
Since all $D^{*0}$ decays produce a $D^{0}$ meson [8],
the sum of $\sigma(D^{*0})$ and $\sigma^{\rm dir}(D^{0})$ is the production
cross section for $D^{0}$ mesons not originating from $D^{\ast+}$ decays:
$$\sigma(D^{*0})+\sigma^{\rm dir}(D^{0})=\sigma^{\rm untag}(\mbox{$D^{0}$}).$$
(2)
Subtracting from $\sigma(\mbox{$D^{+}$})$ the contribution from $D^{\ast+}$ decays gives
$$\sigma^{\rm dir}(\mbox{$D^{+}$})=\sigma(\mbox{$D^{+}$})-\sigma(D^{*+})\cdot(1-%
\mbox{${\cal B}_{D^{\ast+}\rightarrow D^{0}\pi^{+}}$}).$$
(3)
Thus, the ratio of neutral and charged $D$-meson production rates
can be calculated as
$$R_{u/d}=\frac{\sigma^{\rm untag}(D^{0})}{\sigma(D^{+})+\sigma(D^{*+})\cdot%
\mbox{${\cal B}_{D^{\ast+}\rightarrow D^{0}\pi^{+}}$}}=\frac{\sigma^{\rm untag%
}(D^{0})}{\sigma(D^{+})+\sigma^{\rm tag}(D^{0})}.$$
Using the measured cross sections, the ratio of neutral to charged $D$-meson production rates,
obtained for the kinematic region $Q^{2}<1{\,\text{Ge}\kern-0.6667pt\text{V\/}}^{2}$,
$130<W<300{\,\text{Ge}\kern-0.6667pt\text{V\/}}$,
$p_{T}(D)>3.8{\,\text{Ge}\kern-0.6667pt\text{V\/}}$ and $|\eta(D)|<1.6$,
is
$$R_{u/d}=1.100\pm 0.078\,({\rm stat.})^{+0.038}_{-0.061}\,({\rm syst.})^{+0.047%
}_{-0.049}\,({\rm br.}).$$
The measured $R_{u/d}$ value agrees with unity, i.e.$\>$it is
consistent with isospin
invariance, which implies that $u$ and $d$ quarks are produced equally
in charm fragmentation.
Table 1 compares the measurement with the values obtained
in DIS [6] and
in $e^{+}e^{-}$ annihilations. The latter value
was calculated as
$$R_{u/d}=\frac{\mbox{$f(c\rightarrow D^{0})$}-\mbox{$f(c\rightarrow D^{\ast+})$%
}\cdot\mbox{${\cal B}_{D^{\ast+}\rightarrow D^{0}\pi^{+}}$}}{\mbox{$f(c%
\rightarrow D^{+})$}+\mbox{$f(c\rightarrow D^{\ast+})$}\cdot\mbox{${\cal B}_{D%
^{\ast+}\rightarrow D^{0}\pi^{+}}$}}$$
using fragmentation fractions
compiled previously [7]
and updated with the recent branching ratio values [8].
All measurements agree with unity within experimental uncertainties.
The branching ratio uncertainties
of all measurements
are highly correlated.
7.2 Equivalent phase-space treatment
In the subtraction of the $D^{\ast+}$ contribution to $D^{+}$ production
in Eq. (3),
the cross-section $\sigma(\mbox{$D^{\ast+}$})$,
corresponding to $D^{0}$ production in the kinematic range
$p_{T}(\mbox{$D^{0}$})>3.8{\,\text{Ge}\kern-0.6667pt\text{V\/}}$ and $|\eta(\mbox{$D^{0}$})|<1.6$ for
the $\mbox{$D^{\ast+}$}\rightarrow\mbox{$D^{0}$}\pi^{+}_{s}$ decay,
was used.
Replacing $\sigma(D^{*+})$ with $\sigma^{\rm tag}(\mbox{$D^{0}$})/\mbox{${\cal B}_{D^{\ast+}\rightarrow D^{0}%
\pi^{+}}$}$ gives
$$\sigma^{\rm dir}(\mbox{$D^{+}$})=\sigma(\mbox{$D^{+}$})-\sigma^{\rm tag}(\mbox%
{$D^{0}$})\cdot(1-\mbox{${\cal B}_{D^{\ast+}\rightarrow D^{0}\pi^{+}}$})/\mbox%
{${\cal B}_{D^{\ast+}\rightarrow D^{0}\pi^{+}}$}\,.$$
To compare direct $D^{+}$ and $D^{\ast+}$ production,
the cross section $\sigma^{\rm kin}(\mbox{$D^{\ast+}$})$ for
$p_{T}(\mbox{$D^{\ast+}$})>3.8{\,\text{Ge}\kern-0.6667pt\text{V\/}}$ and $|\eta(\mbox{$D^{\ast+}$})|<1.6$
is used in Section 7.4.
To compare the inclusive $D^{+}$ and $D^{0}$ cross sections with each other
and with the inclusive $D^{\ast+}$ cross section
it is necessary
to take into account that
only a fraction of the parent $D^{*}$
momentum is transfered
to the daughter $D$ meson.
For such comparisons,
the “equivalent” $D^{+}$ and $D^{0}$ cross sections were defined
as the sums
of their direct cross sections and contributions from $D^{*}$ decays
calculated using $\sigma^{\rm kin}(\mbox{$D^{\ast+}$})$ and $\sigma^{\rm kin}(D^{*0})$:
$$\displaystyle\sigma^{\rm eq}(\mbox{$D^{+}$})$$
$$\displaystyle=$$
$$\displaystyle\sigma^{\rm dir}(\mbox{$D^{+}$})+\sigma^{\rm kin}(\mbox{$D^{\ast+%
}$})\cdot(1-\mbox{${\cal B}_{D^{\ast+}\rightarrow D^{0}\pi^{+}}$}),$$
$$\displaystyle\sigma^{\rm eq}(\mbox{$D^{0}$})$$
$$\displaystyle=$$
$$\displaystyle\sigma^{\rm dir}(\mbox{$D^{0}$})+\sigma^{\rm kin}(\mbox{$D^{\ast+%
}$})\cdot\mbox{${\cal B}_{D^{\ast+}\rightarrow D^{0}\pi^{+}}$}+\sigma^{\rm kin%
}(D^{*0}),$$
where $\sigma^{\rm kin}(D^{*0})$ is the inclusive $D^{*0}$ cross section
for $p_{T}(D^{*0})>3.8{\,\text{Ge}\kern-0.6667pt\text{V\/}}$ and $|\eta(D^{*0})|<1.6$.
This cross section can be written as the sum
$\sigma(D^{*0})+\sigma^{\rm add}(D^{*0})$,
where $\sigma(D^{*0})$ is
the part contributing to the $D^{0}$ production in the nominal kinematic
range (as in Eq. (2)) and
$\sigma^{\rm add}(D^{*0})$ is
the production
cross section for “additional” $D^{*0}$ mesons
producing $D^{0}$ mesons outside of that kinematic range.
The latter cross section was calculated using $\sigma^{\rm add}(D^{*+})$
and the expression for $R_{u/d}$:
$$\sigma^{\rm add}(D^{*0})=\sigma^{\rm add}(D^{*+})\cdot R_{u/d}=\sigma^{\rm add%
}(D^{*+})\cdot\frac{\sigma^{\rm untag}(D^{0})}{\sigma(D^{+})+\sigma^{\rm tag}(%
D^{0})}.$$
Using Eqs. (2) and (3) for
$\sigma^{\rm dir}(\mbox{$D^{0}$})$ and $\sigma^{\rm dir}(\mbox{$D^{+}$})$, respectively, and
the expressions for $\sigma^{\rm kin}(D^{*0})$ and $\sigma^{\rm kin}(D^{*+})$
gives
$$\displaystyle\sigma^{\rm eq}(\mbox{$D^{0}$})$$
$$\displaystyle=$$
$$\displaystyle\sigma^{\rm untag}(\mbox{$D^{0}$})+\sigma^{\rm tag}(\mbox{$D^{0}$%
})+\sigma^{\rm add}(D^{*+})\cdot(R_{u/d}+\mbox{${\cal B}_{D^{\ast+}\rightarrow
D%
^{0}\pi^{+}}$}),$$
$$\displaystyle\sigma^{\rm eq}(\mbox{$D^{+}$})$$
$$\displaystyle=$$
$$\displaystyle\sigma(\mbox{$D^{+}$})+\sigma^{\rm add}(\mbox{$D^{\ast+}$})\cdot(%
1-\mbox{${\cal B}_{D^{\ast+}\rightarrow D^{0}\pi^{+}}$}).$$
MC studies show that such “equivalent phase-space treatment”
for the non-strange $D$ and $D^{*}$ mesons
minimises differences between the fragmentation ratios and
fractions measured in the accepted $p_{T}(D,\Lambda_{c})$ and $\eta(D,\Lambda_{c})$
kinematic region and those in the
full phase space (see Section 7.6).
7.3 Strangeness-suppression factor
The strangeness-suppression factor for charm mesons
is given by the ratio of twice the production rate of charm-strange mesons
to the production rate of non-strange charm mesons.
All $D^{\ast+}$ and $D^{*0}$ decays produce either a $D^{+}$ or
a $D^{0}$ meson, while
all $D_{s}^{*+}$ decays produce a $D_{s}^{+}$ meson [8].
Thus, neglecting decays of heavier excited charm-strange mesons
to non-strange charm mesons,
the strangeness-suppression factor
can be calculated as
a ratio of twice the $D_{s}^{+}$ production cross section to the sum
of $D^{0}$ and $D^{+}$ production cross sections.
Using the equivalent $D^{0}$ and $D^{+}$ cross sections gives
$$\gamma_{s}=\frac{2\,\sigma(D^{+}_{s})}{\sigma^{\rm eq}(D^{+})+\sigma^{\rm eq}(%
D^{0})}=\frac{2\,\sigma(D^{+}_{s})}{\sigma(D^{+})+\sigma^{\rm untag}(D^{0})+%
\sigma^{\rm tag}(D^{0})+\sigma^{\rm add}(D^{*+})\cdot(1+R_{u/d})}.$$
Using the measured cross sections,
the strangeness-suppression factor,
obtained for the kinematic region $Q^{2}<1{\,\text{Ge}\kern-0.6667pt\text{V\/}}^{2}$,
$130<W<300{\,\text{Ge}\kern-0.6667pt\text{V\/}}$,
$p_{T}(D)>3.8{\,\text{Ge}\kern-0.6667pt\text{V\/}}$ and $|\eta(D)|<1.6$, is
$$\gamma_{s}=0.257\pm 0.024\,({\rm stat.})^{+0.013}_{-0.016}\,({\rm syst.})^{+0.%
078}_{-0.049}\,({\rm br.}).$$
Thus, charm-strange meson production is suppressed by a factor $\approx 3.9$
in charm fragmentation.
In simulations based on the Lund string fragmentation
scheme [41, *cpc:43:367], strangeness suppression is
a free parameter
which determines the ratio of probabilities to create $s$ to $u$ and
$d$ quarks during
the fragmentation processes.
In the absence of excited charm-strange meson decays to non-strange
charm mesons, the Lund strangeness-suppression parameter would be
effectively the observable, $\gamma_{s}$.
In fact, production rates of the excited charm-strange mesons
are poorly known; varying these rates
in wide ranges in the Pythia
simulation suggests that
the Lund strangeness-suppression parameter is $10-30\%$ larger than
the observable, $\gamma_{s}$.
Table 2 compares the measurement with the previous
ZEUS 96-97 result, calculated from the ratio of $D_{s}^{+}$ to $D^{\ast+}$
cross sections [4], and with the values obtained
for charm production
in DIS [6] and
in $e^{+}e^{-}$ annihilations. The $e^{+}e^{-}$ value
was calculated as
$$\gamma_{s}=\frac{2\mbox{$f(c\rightarrow D_{s}^{+})$}}{\mbox{$f(c\rightarrow D^%
{+})$}+\mbox{$f(c\rightarrow D^{0})$}}$$
using fragmentation fractions
compiled previously [7]
and updated with the recent branching ratio values [8].
All measurements agree within experimental uncertainties.
The large branching-ratio uncertainties are dominated by the common
uncertainty of the ${\mbox{$D_{s}^{+}$}}\rightarrow\phi\pi^{+}$ branching ratio.
This uncertainty can be ignored in the comparison with other
measurements using the same branching ratios.
7.4 Fraction of charged $D$ mesons produced in a vector state
Neglecting influences from decays of heavier excited $D$ mesons,
the fraction of $D$ mesons produced in a vector state
is given by the ratio of vector to (vector+pseudoscalar)
charm meson production cross sections. Only direct parts
of the production cross sections for pseudoscalar charm mesons
should be used.
Using the expressions for $\sigma^{\rm kin}(\mbox{$D^{\ast+}$})$
and $\sigma^{\rm dir}(\mbox{$D^{+}$})$,
the fraction for charged charm mesons is given by
$$P^{d}_{\rm v}=\frac{\sigma^{\rm kin}(D^{*+})}{\sigma^{\rm kin}(D^{*+})+\sigma^%
{\rm dir}(\mbox{$D^{+}$})}=\frac{\sigma^{\rm tag}(D^{0})/\mbox{${\cal B}_{D^{%
\ast+}\rightarrow D^{0}\pi^{+}}$}+\sigma^{\rm add}(D^{*+})}{\sigma(D^{+})+%
\sigma^{\rm tag}(D^{0})+\sigma^{\rm add}(D^{*+})}.$$
Using the measured cross sections,
the fraction of charged $D$ mesons produced in a vector state,
obtained for the kinematic region $Q^{2}<1{\,\text{Ge}\kern-0.6667pt\text{V\/}}^{2}$,
$130<W<300{\,\text{Ge}\kern-0.6667pt\text{V\/}}$,
$p_{T}(D)>3.8{\,\text{Ge}\kern-0.6667pt\text{V\/}}$ and $|\eta(D)|<1.6$,
is
$$P^{d}_{\rm v}=0.566\pm 0.025\,({\rm stat.})^{+0.007}_{-0.022}\,({\rm syst.})^{%
+0.022}_{-0.023}\,({\rm br.}).$$
The measured $P^{d}_{\rm v}$
fraction
is considerably smaller than the naive spin-counting prediction of $0.75$.
The predictions of the thermodynamical approach [43]
and the string fragmentation approach [44],
which both predict $2/3$ for the fraction, are closer to,
but still above, the measured value.
The BKL model [45, 46], based on
a tree-level perturbative QCD calculation with
the subsequent hadronisation
of the $(c,\bar{q})$ state, predicts $P^{d}_{\rm v}\approx 0.6$ for charm
production in $e^{+}e^{-}$ annihilations where only fragmentation diagrams
contribute.
For charm photoproduction, where both fragmentation and recombination
diagrams contribute, the BKL prediction is $P^{d}_{\rm v}\approx 0.66$
in the measured kinematic range.
Table 3 compares the measurement with the values obtained
in DIS [6] and
in $e^{+}e^{-}$ annihilations. The latter value
was calculated as
$$P^{d}_{\rm v}=\frac{\mbox{$f(c\rightarrow D^{\ast+})$}}{\mbox{$f(c\rightarrow D%
^{+})$}+\mbox{$f(c\rightarrow D^{\ast+})$}\cdot\mbox{${\cal B}_{D^{\ast+}%
\rightarrow D^{0}\pi^{+}}$}}$$
using fragmentation fractions
compiled previously [7]
and updated with the recent branching ratio values [8].
The measured $P^{d}_{\rm v}$ value is smaller than, but consistent with,
the previous measurements.
The branching-ratio uncertainties
of all measurements
are highly correlated.
7.5 Charm fragmentation fractions
The fraction of $c$ quarks hadronising as a particular charm hadron,
$f(c\rightarrow D,\Lambda_{c})$, is given by the ratio
of the production cross section for the hadron to the sum
of the production cross sections
for all charm
ground states that decay weakly.
In addition to the measured $D^{0}$, $D^{+}$, $D_{s}^{+}$ and $\Lambda_{c}^{+}$
charm ground states, the production cross sections of the charm-strange baryons
$\Xi^{+}_{c}$, $\Xi^{0}_{c}$ and $\Omega^{0}_{c}$
should be included in the sum.
The production rates for these baryons are expected
to be much lower than that of the $\Lambda_{c}^{+}$
due to strangeness suppression.
The relative rates for the
charm-strange baryons
which decay weakly
were estimated from the non-charm sector
following the LEP procedure [47].
The measured $\Xi^{-}/\Lambda$
and $\Omega^{-}/\Lambda$ relative rates are
$(6.65\pm 0.28)\%$ and
$(0.42\pm 0.07)\%$, respectively [8].
Assuming equal production of $\Xi^{0}$ and $\Xi^{-}$ states
and that a similar suppression is applicable to the charm baryons,
the total rate for the three charm-strange baryons relative to the
$\Lambda_{c}^{+}$ state is expected to be about $14\%$.
Therefore the $\Lambda_{c}^{+}$ production cross section was scaled by the
factor $1.14$ in the sum of the production cross sections.
An error of $\pm 0.05$ was assigned to the scale factor when evaluating
systematic uncertainties.
Using the equivalent $D^{0}$ and $D^{+}$ cross sections,
the sum
of the production cross sections
for all open-charm ground states (gs) is given by
$$\sigma_{\rm gs}=\sigma^{\rm eq}(D^{+})+\sigma^{\rm eq}(D^{0})+\sigma(D_{s}^{+}%
)+\sigma(\Lambda_{c}^{+})\cdot 1.14,$$
which can be expressed as
$$\sigma_{\rm gs}=\sigma(D^{+})+\sigma^{\rm untag}(D^{0})+\sigma^{\rm tag}(D^{0}%
)+\sigma^{\rm add}(D^{*+})\cdot(1+R_{u/d})+\sigma(D_{s}^{+})+\sigma(\Lambda_{c%
}^{+})\cdot 1.14.$$
For the measured cross sections,
$$\sigma_{\rm gs}=24.9\pm 1.0\,({\rm stat.})^{+1.7}_{-1.4}\,({\rm syst.})^{+1.6}%
_{-1.0}\,({\rm br.})\,{\rm nb}.$$
The fragmentation fractions for the measured charm ground states
are given by
$$\displaystyle f(c\rightarrow D^{+})$$
$$\displaystyle=$$
$$\displaystyle\sigma^{\rm eq}(D^{+})/\sigma_{\rm gs}=[\sigma(D^{+})+\sigma^{\rm
add%
}(D^{*+})\cdot(1-\mbox{${\cal B}_{D^{\ast+}\rightarrow D^{0}\pi^{+}}$})]/%
\sigma_{\rm gs},$$
$$\displaystyle f(c\rightarrow D^{0})$$
$$\displaystyle=$$
$$\displaystyle\sigma^{\rm eq}(D^{0})/\sigma_{\rm gs}$$
$$\displaystyle=$$
$$\displaystyle[\sigma^{\rm untag}(D^{0})+\sigma^{\rm tag}(D^{0})+\sigma^{\rm add%
}(D^{*+})\cdot(R_{u/d}+\mbox{${\cal B}_{D^{\ast+}\rightarrow D^{0}\pi^{+}}$})]%
/\sigma_{\rm gs},$$
$$\displaystyle f(c\rightarrow D_{s}^{+})$$
$$\displaystyle=$$
$$\displaystyle\sigma(D_{s}^{+})/\sigma_{\rm gs},$$
$$\displaystyle f(c\rightarrow\Lambda_{c}^{+})$$
$$\displaystyle=$$
$$\displaystyle\sigma(\Lambda_{c}^{+})/\sigma_{\rm gs}.$$
Using $\sigma^{\rm kin}(D^{*+})$,
the fragmentation fraction
for the $D^{\ast+}$ state is given by
$$f(c\rightarrow D^{*+})=\sigma^{\rm kin}(D^{*+})/\sigma_{\rm gs}=[\sigma^{\rm
tag%
}(D^{0})/\mbox{${\cal B}_{D^{\ast+}\rightarrow D^{0}\pi^{+}}$}+\sigma^{\rm add%
}(D^{*+})]/\sigma_{\rm gs}.$$
The open-charm fragmentation fractions,
measured in the kinematic
region $Q^{2}<1{\,\text{Ge}\kern-0.6667pt\text{V\/}}^{2}$,
$130<W<300{\,\text{Ge}\kern-0.6667pt\text{V\/}}$,
$p_{T}(D,\Lambda_{c})>3.8{\,\text{Ge}\kern-0.6667pt\text{V\/}}$ and $|\eta(D,\Lambda_{c})|<1.6$,
are summarised in
Table 4.
The results
are compared
with the values obtained
in DIS [6] and
with the combined fragmentation fractions
for charm production in $e^{+}e^{-}$ annihilations compiled
previously [7]
and updated with the recent branching-ratio values [8].
The branching-ratio uncertainties
of all measurements are highly correlated.
The measurements are consistent
although the measured $f(c\rightarrow D^{*+})$ is smaller and
$f(c\rightarrow\Lambda_{c}^{+})$ is larger than those
obtained in $e^{+}e^{-}$ annihilations.
About half of the difference in the $f(c\rightarrow D^{*+})$ values
is due to the difference in the $f(c\rightarrow\Lambda_{c}^{+})$ values.
The measurement may indicate an enhancement of $\Lambda_{c}^{+}$ production
in $ep$ collisions with respect to $e^{+}e^{-}$. However, this is unlikely to be
a consequence of
the baryon-number-flow effect [48, *hep-ph-0006325]
because no significant asymmetry between the $\Lambda_{c}^{+}$ and ${\bar{\Lambda}}_{c}^{{}^{-}}$
production rates was observed888Separate fits of the $M(K^{-}p\pi^{+})$ and $M(K^{+}{\bar{p}}\pi^{-})$ distributions
yielded $N(\Lambda_{c}^{+})/N({\bar{\Lambda}^{{}^{-}}_{c}})=0.8\pm 0.2$..
7.6 Discussion of extrapolation effects
The charm fragmentation ratios and fractions were measured
in the region
$p_{T}(D,\Lambda_{c})>3.8{\,\text{Ge}\kern-0.6667pt\text{V\/}}$ and $|\eta(D,\Lambda_{c})|<1.6$.
To minimise differences between the
values measured in the accepted $p_{T}(D,\Lambda_{c})$ and $\eta(D,\Lambda_{c})$
kinematic region and those in the
full phase space, the equivalent phase-space treatment
for the non-strange $D$ and $D^{*}$ mesons was used (see Section 7.2).
Table 5 shows estimates of extrapolation factors
correcting the values measured in the accepted
$p_{T}(D,\Lambda_{c})$ and $\eta(D,\Lambda_{c})$ region to the
full phase space.
The extrapolation factors were determined using three
different fragmentation schemes:
the Peterson parameterisation [50] of the charm fragmentation
function as implemented in Pythia,
the Bowler modification [28]
of the LUND symmetric fragmentation function [29]
as implemented in Pythia
and the cluster model [30] as implemented in Herwig.
The quoted uncertainties were obtained by varying relevant parameters
in the Pythia and Herwig MC generators.
The extrapolation factors obtained are generally close to unity.
The only exceptions are the factors
given by the cluster model
for $f(c\rightarrow\Lambda_{c}^{+})$
and, to a lesser extent, for $\gamma_{s}$ and $f(c\rightarrow D_{s}^{+})$.
This MC study suggests that the measured charm fragmentation
ratios and fractions are close to those in the full
$p_{T}(D,\Lambda_{c})$ and $\eta(D,\Lambda_{c})$ phase space.
8 Systematic uncertainties
The systematic uncertainties of the measured cross sections
and fragmentation ratios and fractions
were determined
by changing the analysis procedure
and repeating
all calculations.
The following groups of the systematic uncertainty sources
were considered:
•
$\{\delta_{1}\}$
the model dependence of the acceptance corrections was estimated using
the Herwig MC sample, varying
the $p_{T}(D,\Lambda_{c})$ and
$\eta(D,\Lambda_{c})$ distributions of the
reference MC sample
and by changing the MC fraction of charged $D$ mesons produced in a vector
state from 0.6 to $0.5$ or $0.7$;
•
$\{\delta_{2}\}$
the uncertainty of the beauty subtraction was determined by
varying the
$b$-quark cross section by a factor of two
in the reference MC sample and by varying
the branching ratios of
$b$-quarks to
charm hadrons by their uncertainties [39, 40];
•
$\{\delta_{3}\}$
the uncertainty of the tracking simulation was obtained
by varying all momenta by $\pm 0.3\%$
(magnetic field uncertainty),
varying the track-loss probabilities by $\pm 20\%$ of their values
and by changing the track momentum and angular resolutions
by ${}^{+20}_{-10}\%$ of their values.
The asymmetric resolution variations were used since the MC signals
typically had somewhat narrower widths than observed in the data;
•
$\{\delta_{4}\}$
the uncertainty of the CAL simulation was determined by
varying the CAL energy scale by $\pm 2\%$,
by changing the CAL energy resolution by $\pm 20\%$ of its value
and by varying
the first-level trigger CAL efficiencies;
•
$\{\delta_{5}\}$
the uncertainties related to the signal extraction procedures were
obtained as follows:
–
for the $D^{0}$ signals with and without $\Delta M$ tag:
the background parametrisation and the range used for the signal fits
were varied;
–
for the additional $D^{\ast+}$ signal:
the range used for the background normalisation
was varied or
the fit was used instead of the subtraction procedure;
–
for the $D^{+}$, $D_{s}^{+}$ and $\Lambda_{c}^{+}$ signals:
the background parametrisations,
ranges used
for the signal fits
and amounts of the mutual reflections were varied.
In addition, in the $D_{s}^{+}$ signal-extraction procedure,
the constraint used for the $\mbox{$D^{+}$}\rightarrow KK\pi$ signal width was varied.
In the $\Lambda_{c}^{+}$ signal extraction procedure,
an uncertainty in the $dE/dx$ simulation was estimated by changing the $dE/dx$
cut values in the MC and checking the effects with respect to changes
expected from the $\chi^{2}_{1}$ distribution.
•
$\{\delta_{6}\}$
the uncertainties of the luminosities of the $e^{-}p$ ($\pm 1.8\%$) and $e^{+}p$
($\pm 2.25\%$) data samples were included taking into account their
correlations;
•
$\{\delta_{7}\}$
the uncertainty in the rate
of the charm-strange baryons
(see Section 7.5).
Contributions from
the different systematic uncertainties were calculated and added
in quadrature separately for positive and negative variations.
The total and $\delta_{1}$-$\delta_{7}$ systematic uncertainties
for the charm-hadron
cross sections and charm fragmentation ratios and fractions
are summarised in
Table 6.
Correlated systematic uncertainties largely cancelled
in the calculation of
the fragmentation ratios and fractions.
To check the hadron-mass effects on the measured charm fragmentation
ratios and fractions, the analysis was repeated
using the charm-hadron transverse energy instead of the transverse momentum
in the definition of the kinematic range of the measurement;
the results obtained agreed with the reported values within statistical
errors.
The charm fragmentation ratios and fractions were also calculated separately
for two $W$ sub-ranges; no significant variations were observed.
9 Summary
The production of the charm hadrons
$D^{\ast+}$, $D^{0}$, $D^{+}$, $D_{s}^{+}$ and $\Lambda_{c}^{+}$
has been measured with the ZEUS detector
in the kinematic range
$p_{T}(D,\Lambda_{c})>3.8{\,\text{Ge}\kern-0.6667pt\text{V\/}}$, $|\eta(D,\Lambda_{c})|<1.6$,
$130<W<300{\,\text{Ge}\kern-0.6667pt\text{V\/}}$ and $Q^{2}<1{\,\text{Ge}\kern-0.6667pt\text{V\/}}^{2}$.
The cross sections have
been used to determine
the charm fragmentation ratios and fractions
with comparable precision to the $e^{+}e^{-}$ results.
The ratio of neutral to charged $D$-meson production rates
is
$$R_{u/d}=1.100\pm 0.078\,({\rm stat.})^{+0.038}_{-0.061}\,({\rm syst.})^{+0.047%
}_{-0.049}\,({\rm br.}).$$
The measured $R_{u/d}$ value agrees with unity, i.e.$\>$it is
consistent with isospin invariance,
which implies that $u$ and $d$ quarks are produced equally
in charm fragmentation.
The strangeness-suppression factor is
$$\gamma_{s}=0.257\pm 0.024\,({\rm stat.})^{+0.013}_{-0.016}\,({\rm syst.})^{+0.%
078}_{-0.049`}\,({\rm br.}).$$
Thus, $D_{s}$-meson production is suppressed by a factor $\approx 3.9$
in charm fragmentation.
The fraction of charged $D$ mesons produced in a vector state is
$$P^{d}_{\rm v}=0.566\pm 0.025\,({\rm stat.})^{+0.007}_{-0.022}\,({\rm syst.})^{%
+0.022}_{-0.023}\,({\rm br.}).$$
The measured fraction
is considerably smaller than the naive spin-counting prediction of $0.75$.
The predictions of the thermodynamical approach [43]
and the string fragmentation approach [44],
which both predict $2/3$ for the fraction,
and the BKL model [45, 46]
prediction ($\approx 0.66$)
are closer to,
but still above, the measured value.
The measured $R_{u/d}$ and $\gamma_{s}$ values agree
with those obtained in DIS [6] and in $e^{+}e^{-}$ annihilations.
The $e^{+}e^{-}$ values
were calculated
using fragmentation fractions
compiled previously [7]
and updated with the recent branching ratio values [8].
The measured $P^{d}_{\rm v}$ value is smaller than, but consistent with,
the previous
measurements.
The fractions of $c$ quarks hadronising as
$D^{\ast+}$, $D^{0}$, $D^{+}$, $D_{s}^{+}$ and $\Lambda_{c}^{+}$
hadrons have been calculated in the accepted kinematic range.
The measured open-charm fragmentation fractions are
consistent
with previous results
although the measured $f(c\rightarrow D^{*+})$ is smaller and
$f(c\rightarrow\Lambda_{c}^{+})$ is larger than those
obtained in $e^{+}e^{-}$ annihilations.
About half of the difference in the $f(c\rightarrow D^{*+})$ values
is due to the difference in the $f(c\rightarrow\Lambda_{c}^{+})$ values.
These measurements generally support the hypothesis that fragmentation
proceeds independently of the hard sub-process.
10 Acknowledgements
We thank the DESY Directorate for their strong support and encouragement.
The remarkable achievements of the HERA machine group were essential
for the successful completion of this work and
are greatly appreciated.
The design, construction and installation of the ZEUS detector
has been made possible by the efforts of many people who are
not listed as authors.
We thank A.V. Berezhnoy and A.K. Likhoded for providing us with
their predictions.
{mcbibliography}10
References
[1]
ZEUS Collab., J. Breitweg et al.,
Eur. Phys. J. C 6, 67 (1999)
[2]
H1 Collab., C. Adloff et al.,
Nucl. Phys. B 545, 21 (1999)
[3]
ZEUS Collab., J. Breitweg et al.,
Eur. Phys. J. C 12, 35 (2000)
[4]
ZEUS Collab., J. Breitweg et al.,
Phys. Lett. B 481, 213 (2000)
[5]
H1 Collab., C. Adloff et al.,
Phys. Lett. B 528, 199 (2002)
[6]
H1 Collab., A. Aktas et al.,
Eur. Phys. J. C 38, 447 (2005)
[7]
L. Gladilin,
Preprint hep-ex/9912064 (1999)
[8]
Particle Data Group, S. Eidelman et al.,
Phys. Lett. B 592, 1 (2004)
[9]
ZEUS Collab., U. Holm (ed.),
The ZEUS Detector.
Status Report (unpublished), DESY (1993),
available on
http://www-zeus.desy.de/bluebook/bluebook.html
[10]
N. Harnew et al.,
Nucl. Instr. and Meth. A 279, 290 (1989)
[11]
B. Foster et al.,
Nucl. Phys. Proc. Suppl. B 32, 181 (1993)
[12]
B. Foster et al.,
Nucl. Instr. and Meth. A 338, 254 (1994)
[13]
ZEUS Collab., J. Breitweg et al.,
Eur. Phys. J. C 18, 625 (2001)
[14]
M. Derrick et al.,
Nucl. Instr. and Meth. A 309, 77 (1991)
[15]
A. Andresen et al.,
Nucl. Instr. and Meth. A 309, 101 (1991)
[16]
A. Caldwell et al.,
Nucl. Instr. and Meth. A 321, 356 (1992)
[17]
A. Bernstein et al.,
Nucl. Instr. and Meth. A 336, 23 (1993)
[18]
J. Andruszków et al.,
Preprint DESY-92-066, DESY, 1992
[19]
ZEUS Collab., M. Derrick et al.,
Z. Phys. C 63, 391 (1994)
[20]
J. Andruszków et al.,
Acta Phys. Pol. B 32, 2025 (2001)
[21]
T. Sjöstrand,
Comp. Phys. Comm. 82, 74 (1994)
[22]
H. Jung,
Comp. Phys. Comm. 86, 147 (1995)
[23]
G. Marchesini et al.,
Comp. Phys. Comm. 67, 465 (1992)
[24]
G. Corcella et al.,
JHEP 0101, 010 (2001)
[25]
CTEQ Collab., H.L. Lai et al.,
Eur. Phys. J. C 12, 375 (2000)
[26]
M. Glück, E. Reya and A. Vogt,
Phys. Rev. D 46, 1973 (1992)
[27]
B. Andersson et al.,
Phys. Rep. 97, 31 (1983)
[28]
M. G. Bowler,
Z. Phys. C 11, 169 (1981)
[29]
B. Andersson, G. Gustafson and B. Söderberg,
Z. Phys. C 20, 317 (1983)
[30]
B. R. Webber,
Nucl. Phys. B 238, 492 (1984)
[31]
ZEUS Collab., S. Chekanov et al.,
Eur. Phys. J. C 38, 29 (2004)
[32]
R. Brun et al.,
geant3,
Technical Report CERN-DD/EE/84-1, CERN, 1987
[33]
W. H. Smith, K. Tokushuku and L. W. Wiggers,
Proc. Computing in High-Energy Physics (CHEP), Annecy, France,
Sept. 1992, C. Verkerk and W. Wojcik (eds.), p. 222.
CERN, Geneva, Switzerland (1992).
Also in preprint DESY 92-150B
[34]
ZEUS Collab., M. Derrick et al.,
Phys. Lett. B 322, 287 (1994)
[35]
F. Jacquet and A. Blondel,
Proceedings of the Study for an $ep$ Facility for Europe,
U. Amaldi (ed.), p. 391.
Hamburg, Germany (1979).
Also in preprint DESY 79/48
[36]
G.M. Briskin.
Ph.D. Thesis, Tel Aviv University, (1998).
(Unpublished)
[37]
ZEUS Collab., M. Derrick et al.,
Phys. Lett. B 349, 225 (1995)
[38]
D. Bailey and R. Hall-Wilton,
Nucl. Instr. and Meth. A 515, 37 (2003)
[39]
ALEPH Collab., D. Buskulic et al.,
Phys. Lett. B 388, 648 (1996)
[40]
OPAL Collab., K. Ackerstaff et al.,
Eur. Phys. J. C 1, 439 (1997)
[41]
T. Sjöstrand,
Comp. Phys. Comm. 39, 347 (1986)
[42]
T. Sjöstrand and M. Bengtsson,
Comp. Phys. Comm. 43, 367 (1987)
[43]
F. Becattini,
Z. Phys. C 69, 485 (1996)
[44]
Yi-Jin Pei,
Z. Phys. C 72, 39 (1996)
[45]
A. V. Berezhnoy, V. V. Kiselev and A. K. Likhoded,
Phys. Rev. D 62, 074013 (2000)
[46]
A. V. Berezhnoy and A. K. Likhoded.
Private communication
[47]
OPAL Collab., G. Alexander et al.,
Z. Phys. C 72, 1 (1996)
[48]
B. Kapeliovich and B. Povh,
Phys. Lett. B 446, 321 (1999)
[49]
G. T Garvey, B. Z. Kapeliovich and B. Povh,
Preprint hep-ph/0006325 (2000)
[50]
C. Peterson et al.,
Phys. Rev. D 27, 105 (1983) |
\setitemize
itemsep=0em,leftmargin=0.25in
\setenumerateitemsep=0em,leftmargin=0.25in
\bibinputreferences
Graph Neural Network Guided Local Search for the Traveling Salesperson Problem
Benjamin Hudson
Department of Computer Science and Technology
University of Cambridge
bh511@cam.ac.uk
&Qingbiao Li
Department of Computer Science and Technology
University of Cambridge
ql295@cam.ac.uk
\ANDMatthew Malencia
Department of Electrical and Systems Engineering
University of Pennsylvania
malencia@seas.upenn.edu
&Amanda Prorok
Department of Computer Science and Technology
University of Cambridge
asp45@cam.ac.uk
Abstract
Solutions to the Traveling Salesperson Problem (TSP) have practical applications to processes in transportation, logistics, and automation, yet must be computed with minimal delay to satisfy the real-time nature of the underlying tasks. However, solving large TSP instances quickly without sacrificing solution quality remains challenging for current approximate algorithms. To close this gap, we present a hybrid data-driven approach for solving the TSP based on Graph Neural Networks (GNNs) and Guided Local Search (GLS). Our model predicts the regret of including each edge of the problem graph in the solution; GLS uses these predictions in conjunction with the original problem graph to find solutions. Our experiments demonstrate that this approach converges to optimal solutions at a faster rate than three recent learning based approaches for the TSP. Notably, we reduce the mean optimality gap on the 100-node problem set from 1.534% to 0.705%, a 2$\times$ improvement. When generalizing from 20-node instances to the 100-node problem set, we reduce the optimality gap from 18.845% to 2.622%, a 7$\times$ improvement.
1 Introduction
Sixty years ago, the route of a delivery truck would have been fixed well before the truck departed the warehouse. Today, thanks to the availability of real-time traffic information, and the option to transmit instructions to the driver to add or remove delivery locations on-the-fly, the route is no longer fixed. Nevertheless, minimizing the length or duration of the route remains an important problem. This is an instance of the Traveling Salesperson Problem (TSP), and one of a growing number of practical applications which require solving combinatorial optimization problems in real time.
In these problems, there is often a cost attributed to waiting for an optimal solution or hard deadlines at which decisions must be taken.
For example, the driver cannot wait for a new solution to be computed – they may miss their deliveries, or the traffic conditions may change again.
There is a need for general, anytime combinatorial optimization algorithms that produce high-quality solutions under restricted computation time. This remains challenging for current approaches, as they are specialized for a specific problem (with specific assumptions and constraints), or fail to produce good solutions quickly.
Contributions
We present a hybrid data-driven approach for approximately solving the Euclidean TSP based on Graph Neural Networks (GNNs) and Guided Local Search (GLS), which we demonstrate converges to optimal solutions more quickly than three recent learning based approaches.
We provide the following contributions:
•
Our GLS algorithm is guided by the global regret of each edge in the problem graph. This allows the algorithm to differentiate edges that are very costly to include in the solution from ones that are not so costly, whether or not they are part of the optimal solution. Thus, using regret allows us to find high-quality, rather than optimal, solutions very quickly. We are the first to use a measure of global regret, which we define as the cost of enforcing a decision relative to the cost of an optimal solution.
•
We make this computationally tractable by approximating regret with a learned representation. Our GNN-based model operates on the line graph of the problem graph, which allows us to build a network without node features, focusing only on edge weight.
•
We present experimental results for our approach and several learning based approaches that are aligned with recent guidelines for computational testing of learning based approaches for the TSP. Our experiments emphasize the trade-off between solution quality and time.
•
We reduce the mean optimality gap on the 50-node problem set from 0.268% to 0.009%, a 30$\times$ times improvement, and on the 100-node problem set from 1.534% to 0.705%, a 2$\times$ improvement. When generalizing from 20-node instances to the 100-node problem set, we reduce the mean optimality gap from 18.845% to 2.622%, a 7$\times$ improvement.
2 Related Work
The Operations Research (OR) community is responsible for the majority of research in algorithms for solving combinatorial optimization problems.
Concorde (Applegate et al., 2006) is widely regarded as the best exact TSP solver.
As an exact solver, it can guarantee the level of optimality of the solutions it finds.
It uses cutting plane algorithms (Dantzig et al., 1954; Padberg & Rinaldi, 1990; Applegate et al., 2003) and branch-and-cut methods to iteratively prune away parts of search space that will not contain an optimal solution. LKH-3 (Helsgaun, 2017), and its predecessor, LKH-2 (Helsgaun, 2009), are approximate TSP and Capacitated Vehicle Routing Problem (CVRP) solvers based on the $\kappa$-opt heuristic (Lin & Kernighan, 1973). While these solvers do not provide any guarantees, experimentation has demonstrated that they are extremely effective.
Concorde, LKH-2 and LKH-3 are highly specialized and efficient solvers which can solve challenging TSPs in seconds.
However, practical real-time routing problems often impose constraints on top of the basic TSP or CVRP problem definitions. Highly-specialized solvers are difficult to adapt to these new constraints. Thus, there is a need for general algorithmic frameworks that can produce high-quality solutions with minimal computation time. To address this, Arnold & Sörensen (2019a) introduce KGLS, a GLS algorithm for the CVRP that is guided by three engineered features. They demonstrate that their algorithm can find solutions almost as good as those found by state-of-the-art metaheuristics in a fraction of the time. Our work also aims to address this gap.
Machine learning offers a way to construct flexible, yet effective combinatorial optimization algorithms.
This area of research has existed for more than three decades (Smith, 1999), yet new neural architectures, especially GNNs, have driven a surge of activity in this area.
As noted by Bengio et al. (2021); Cappart et al. (2021), classical combinatorial optimization approaches solve each problem instance in isolation, overlooking the fact that in practice, problems and their solutions stem from related data distributions. Machine learning offers a way to exploit this observation.
We classify learning based approaches for solving the TSP into the three categories identified by Bengio et al. (2021); our approach belongs to the second category, described below.
ML alone provides a solution.
These approaches use a machine learning model to output solutions directly from the problem definition.
Vinyals et al. (2015) propose PtrNet, a sequence-to-sequence model based on LSTMs, which iteratively constructs a solution by outputting the permutation of the input nodes. They train their model using expert solutions from Concorde. They use beam search (Reddy, 1977) to produce better solutions than those sampled from their model directly.
Bello et al. (2016) present a method of training PtrNet without supervision using policy gradients. Their “active search” method resembles early neural network-based approaches to combinatorial optimization, in which a model was trained online to solve a particular problem instance (Smith, 1999).
This method effectively solves each problem in isolation, thus it does not leverage expertise extracted from distribution of training problem instances.
Kool et al. (2018) take a similar approach to Bello et al. (2016) but use a Transformer architecture (Vaswani et al., 2017) rather than an LSTM. Their model can also be seen as a Graph Attention Network (GAT) (Veličković et al., 2017) applied to a complete graph. Kwon et al. (2021) also take a similar approach but leverage the symmetry of many combinatorial optimization problems to improve performance.
Joshi et al. (2019) train a model
to produce a heatmap of which edges in the TSP graph are likely to be part of the optimal solution, and reconstruct valid tours using beam search. Similar to Vinyals et al. (2015), they train their model using expert solutions from Concorde. Kool et al. (2021) extend the work of Joshi et al. (2019) to VRPs, and use a dynamic programming method to construct tours. Also in this category, Fu et al. (2021) train a model to output a heatmap for small TSP instances to and sample small sub-graphs from large instances to construct heatmaps for large instances. They train their approach using reinforcement learning, and use Monte Carlo tree search (MCTS) to construct valid tours.
ML provides information to an OR algorithm.
Machine learning may not be suitable to solve a combinatorial optimization problem alone. Instead, it can provide information to augment a combinatorial optimization algorithm. Deudon et al. (2018) use a model based on the Transformer architecture to construct solutions iteratively and train it using policy gradients (they take a similar approach to Kool et al., 2018). They apply a 2-opt local search heuristic to solutions sampled from their model.
In contrast to prior work, we are the first to use a machine learning model to approximate regret. Furthermore, many previous works that produce predictions on edges (Joshi et al., 2019; Kool et al., 2021; Fu et al., 2021) construct tours using heuristics, whereas to our knowledge, we are the first to use predictions to inform a metaheuristic (GLS).
ML makes decisions within an OR algorithm.
Finally, a machine learning model can be embedded inside a combinatorial optimisation algorithm. In this paradigm, a master algorithm controls the high-level procedure while repeatedly calling a machine learning model to make lower level decisions.
Dai et al. (2017) present a model that uses a GNN to learn an embedding of the current partial solution and a DQN (Mnih et al., 2015) to iteratively select nodes to insert using a cheapest insertion heuristic. They also use an “active search” method when solving the TSP.
Wu et al. (2020) and da Costa et al. (2020) present policy gradient algorithms to learn policies to apply 2-opt operations to existing feasible solutions. Wu et al. (2020) use a Transformer architecture. da Costa et al. (2020) use a combination of GCN and RNN modules, and acheive better results. These approaches can either be seen as end-to-end learning approaches belonging to the first category, or as local search procedures where an ML model decides where to apply an operator.
3 Preliminaries
Traveling Salesperson Problem
We focus on the Euclidean TSP, although our approach can be applied to other routing problems, such as the CVRP. A problem with $n$ cities, typically denoted TSP$n$, is represented as a complete, undirected, weighted graph $G=(V,E)$ with $n$ nodes.
The edges $E$ represent connections between cities and are weighted by the Euclidean distance between the adjacent nodes.
A solution, or tour, is a Hamiltonian cycle: a closed path through the graph that visits every node exactly once. An optimal solution is a cycle of minimum weight.
Regret
Regret measures the future cost of an action that also yields an immediate reward, and is typically used to make greedy combinatorial optimization algorithms less myopic. Generally, regret is computed over a fixed horizon. For example, Potvin & Rousseau (1993) evaluate the cost of inserting a node in the best versus second-best position when constructing a CVRP solution. Hassin & Keinan (2008) solve the TSP using regret, allowing a greedy construction heuristic to remove the most recently inserted node. It is not computationally feasible to calculate regret over a global horizon, for example, for all possible insertion sequences. However, if it were possible, a greedy algorithm could compute an optimal solution by selecting the lowest regret decision at each step.
Local Search
Local Search (LS) is a general improvement heuristic.
Starting from an initial solution, local search iteratively moves to neighboring solutions that are lower cost than the current solution according to the objective function $g(s)$.
Neighboring solutions are solutions that are reachable from a given solution by applying a certain function, or operator.
The set of all solutions reachable from another solution by applying an operator define the neighborhood of that operator.
The algorithm terminates when all neighboring solutions are inferior to the current solution, meaning the search has reached a local optimum.
Guided Local Search
Guided Local Search (GLS) is a metaheuristic that sits on top of LS and allows it to escape local optima (Voudouris & Tsang, 1996). To apply GLS, the designer must define some aspects of a solution. When trapped in a local optimum, the algorithm penalizes certain aspects of the current solution which are considered to be unfavorable. The underlying LS procedure searches using an objective function that is augmented by these penalties, thus it is incentivized to remove heavily penalized aspects from the solution. The augmented objective function $h(s)$ is
$$h(s)=g(s)+\lambda\sum_{i=0}^{M}p_{i}I_{i}(s),$$
(1)
where $s$ is a solution, $g(s)$ is the objective function, $\lambda$ is a scaling parameter, $i$ indexes the aspects, $M$ is the number of aspects, $p_{i}$ is the current number of penalties assigned to aspect $i$, and $I_{i}$ is an indication of whether $s$ exhibits aspect $i$, i.e.
$$I_{i}(s)=\begin{cases}1&\text{if }s\text{ exhibits aspect }i,\\
0&\text{otherwise.}\end{cases}$$
(2)
For the TSP, aspects of the solution are often defined as the edges in the problem graph. Therefore, $I_{i}(s)$ indicates if an edge is in the solution $s$.
Upon reaching a local optimum, which aspects are penalized is determined by a utility function. The utility of penalising aspect $i$, $\text{util}_{i}$, is defined as
$$\text{util}_{i}(s_{*})=I_{i}(s_{*})\frac{c_{i}}{1+p_{i}},$$
(3)
where $s_{*}$ is the solution at a local optimum, $I_{i}(s_{*})$ indicates if the solution exhibits aspect $i$, and $c_{i}$ is the cost of the aspect $i$.
The cost of an aspect measures how unfavorable it is. The higher the cost, the greater the utility of penalizing that aspect. In the context of the TSP, the cost can be the weight of the edge (Voudouris & Tsang, 1999), or a combination of various features (Arnold & Sörensen, 2019a).
Conversely, the more penalties assigned to that aspect, the lower the utility of penalising it again. The aspects with the maximum utility are always penalized, which means increasing $p_{i}$ by one. This penalization mechanism distributes the search effort in the search space, favoring areas where a promise is shown (Voudouris & Tsang, 1996).
We use a variation of the classical GLS algorithm (see Voudouris & Tsang, 1999) that applies alternating optimisation and perturbation phases (see Arnold & Sörensen, 2019a). During an optimization phase, the local search procedure is guided by the original objective function $g$. During a perturbation phase, it is guided by the augmented objection function $h$.
After an edge is penalized, the local search is applied only on the penalized edge. That is, only operations that would remove the penalized edge are considered. This differs from the local search during the optimization phase, which considers all solutions in the neighborhood of the given operator. The perturbation phase continues until $K$ operations (operations that improve the solution according to the augmented objective function $h$) have been applied to the current solution. These operations perturb the solution enough for the local search to escape a local minimum.
The alternating phases continue until the stopping condition is met.
4 Method
Our hybrid method, shown in Figure 1, combines a machine learning model and a metaheuristic.
Our GNN-based model learns an approximation of the global regret of including each edge of the problem graph in the solution. The metaheuristic, GLS, uses this learned regret conjunction with the original problem graph to quickly find high-quality solutions. The learned regret allows the algorithm to differentiate between edges which are costly to include in the solution and ones that are not so costly, thus improving its ability to steer the underlying local search procedure out of local minima and towards promising areas of the solution space.
4.1 Global Regret
We define global regret as the cost of requiring a certain decision to be part of the solution relative to the cost of a globally optimal solution. Unlike previous heuristics using regret, which calculate the cost of a decision relative to some fixed number of alternatives (for example, the next best, or two next best options), our regret is measured relative to a global optimal solution. Decisions that are part of an optimal solution have zero regret, and all other decisions have positive regret. Mathematically,
$$r_{i}=\frac{g(s_{i}^{*})}{g(s^{*})}-1,$$
(4)
where $r_{i}$ is the regret of decision $i$, $g$ is the objective function, $s_{i}^{*}$ is an optimal solution with $i$ fixed, and $s^{*}$ a globally optimal solution.
With perfect information, a greedy algorithm could construct an optimal solution by sequentially selecting the lowest-regret decisions.
In the TSP, decisions correspond to which edges are included in the solution, thus regret is defined as the cost of requiring that a certain edge be part of the solution.
We posit that using regret is preferable to directly classifying which edges are part of the optimal solution (which is the approach taken by Joshi et al., 2019). Where classification can produce a probability that the edge is part of optimal solution, regret can differentiate between edges that are very costly to have as part of the solution and ones that are not so costly, whether or not they are part of the optimal solution. Thus, using regret furthers our goal of finding high-quality, rather than optimal, solutions with minimal computation time.
4.2 Regret Approximation Model
Calculating the global regret of an edge in the TSP graph requires solving the TSP itself, which is computationally intractable. Instead, we aim to learn a function $\hat{r}_{ij}$ that approximates the regret of an edge $r_{ij}$.
We use GNNs to achieve this, as they are universal function approximators that operate on graph-structured data.
Input transformation
Typically, GNNs aggregate messages and store states on the nodes of a graph (Gilmer et al., 2017).
Instead, we input the line graph of the original problem graph into our model. The line graph $L(G)$ of an undirected graph $G$ is a graph such that there exists a node in $L(G)$ for every edge in $G$, and that for every two edges in $G$ that share a node, there exists an edge between their corresponding nodes $L(G)$. Figure 2 illustrates this transformation for a simple graph.
The result is that our model aggregates messages and stores states on the edges of the problem graph (the nodes of the line graph). Primarily, this allows us to build models with no node features, which is advantageous as the edge weights, not the specific node locations, are relevant when solving the TSP.
For a complete, undirected graph $G$ with $n$ nodes, there are $n(n-1)/2$ nodes and $n(n-1)(n-2)$ edges in $L(G)$. Thus, although $G$ is complete, $L(G)$ can be very sparse.
Model architecture
Our model consists of an embedding layer, several GNN layers, and an output layer. The embedding layer is an edge-wise fully connected layer that computes $d_{h}$-dimensional edge embeddings from $d_{x}$-dimensional edge features. Node features, if used, are concatenated onto the feature set of the adjacent edges. The forward pass of the embedding layer is written as
$$\mathbf{h}_{ij}^{0}=\mathbf{Wx}_{ij}+\mathbf{b},$$
(5)
where $\mathbf{h}_{ij}^{0}$ is the initial embedding of edge $ij$, $\mathbf{W}$ is a learnable weight matrix, $\mathbf{x}_{ij}$ are the input features of edge $ij$ (including any node features), and $\mathbf{b}$ is a set of learnable biases. We use $d_{x}=1$ and $d_{h}=128$.
The edge embeddings are updated using $T$ message passing layers. Inspired by the encoder layer of Kool et al. (2018), each layer consists of multiple sublayers. The forward pass is given by
$$\displaystyle\dot{\mathbf{h}}_{ij}^{t+1}$$
$$\displaystyle=f_{\text{BN}}^{t}(\mathbf{h}_{ij}^{t}+f_{\text{MHA}}^{t}(\mathbf{h}_{ij}^{t},L(G))),$$
(6)
$$\displaystyle\mathbf{h}_{ij}^{t+1}$$
$$\displaystyle=f_{\text{BN}}^{t}(\dot{\mathbf{h}}_{ij}^{t+1}+f_{\text{FF}}^{t}(\dot{\mathbf{h}}_{ij}^{t+1})),$$
(7)
where $f_{\text{MHA}}$ is a multi-headed GAT layer (Veličković et al., 2017), $f_{\text{FF}}$ is a feedforward layer, $f_{\text{BN}}$ is batch normalisation (Ioffe & Szegedy, 2015),
and $\dot{\mathbf{h}}_{u}^{t+1}$ is a hidden state. The layers do not share parameters. The GAT layer uses $M=8$ heads and dimensionality $d_{h}/{M}=16$, and the FF layer uses one hidden sublayer with dimensionality 512 and ReLU activation. Finally, the output layer is a single edge-wise fully connected layer that computes a one-dimensional output from the $d_{h}$-dimensional node embeddings computed by the final message passing layer. This is written as
$$\hat{r}_{ij}=\mathbf{Wh}_{ij}^{T}+\mathbf{b},$$
(8)
where $\hat{r}_{ij}$ is the output for edge $ij$ and $\mathbf{h}_{ij}^{T}$ is the final embedding of that edge.
4.3 Regret-Guided Local Search
We adapt GLS to use regret to solve the TSP, including how the initial solution is built, the local search procedure, and the perturbation strategy.
Our GLS uses alternating optimization and perturbation phases.
During an optimization phase, the local search procedure greedily accepts changes to the solution until it reaches a local minimum. During a perturbation phase, the algorithm penalizes and attempts to remove edges in the current solution with high regret, thus allowing it to escape local minima while simultaneously guiding it towards promising areas of the solution space, i.e., those with low regret. Effectively, the regret predictions produced by our model allow our GLS algorithm to undo costly decisions made during the greedy optimization phase.
Initial solution
We use a greedy nearest neighbor algorithm to construct an initial solution. Beginning from the origin node we iteratively select the lowest-regret edge leading to an unvisited node, until all nodes have been visited.
Local Search neighborhoods
Our LS procedure uses two solution operators for the TSP, relocate and 2-opt. It alternates between using either operator, and uses a “best improvement” strategy, meaning that it exhaustively searches the neighborhood corresponding with the current operator and accepts the solution that improves the objective function the most before continuing with the other operator. The algorithm terminates when no improvement can be found in either neighborhood.
The relocate operator simply changes the position of a single node in the tour.
The 2-opt operator selects two nodes in the tour to swap. This divides the tour into three segments: an initial segment, an intermediate segment, and a final segment. The tour is reassembled beginning with the initial segment, the intermediate segment in reverse order, and the final segment.
It is a special case of the $\kappa$-opt operator (Lin & Kernighan, 1973), although it was introduced earlier (Croes, 1958).
Perturbation strategy
We define the cost of an edge as the predicted regret $\hat{r}_{ij}$ of that edge.
The utility of penalizing edge $ij$, $\text{util}_{ij}$, is therefore defined as
$$\text{util}_{ij}(s_{*})=I_{ij}(s_{*})\frac{\hat{r}_{ij}}{1+p_{ij}},$$
(9)
where we remind the reader that $s_{*}$ is the solution at a local optimum, $I_{ij}(s_{*})$ indicates if the solution contains edge $ij$, and $p_{ij}$ is the number of penalties assigned to that edge.
The edges of maximum utility are penalized. Afterward, the local search is applied only on the penalized edge. That is, only operations that would remove the penalized edge are considered.
5 Experiments
We train our model using supervised learning. It uses a single input feature, edge weight (i.e. the Euclidean distance between adjacent nodes), to predict the regret of the edges in the problem graph. Further details of our implementation are found in Appendix A. While further input features could be considered, they come at a computational cost, as seen in Table 4.
In Appendix D, we compare 11 input features, and demonstrate that edge weight is the most important feature when predicting the regret of an edge.
Evaluation
We evaluate the trade-off between solution quality and computation time when solving randomly generated TSPs as well as TSPLIB instances111Downloaded from http://comopt.ifi.uni-heidelberg.de/software/TSPLIB95/. Where available, we use publicly available implementations and pre-trained models. There is no publicly available implementation of Arnold & Sörensen (2019a), so we use the GLS implementation described in section 4.3 with the guides described in their paper.
We compare our approach to the following methods, the first six of which are non-learning based, and the other three are learning based:
1.
Nearest Neighbour
2.
Farthest Insertion
3.
Local Search (as described in section 4.3)
4.
Concorde (Applegate et al., 2006)
5.
LKH-3 (Helsgaun, 2017)
6. \usebibentry
kglstitle (Arnold & Sörensen, 2019a)
7.
\usebibentryattentionlearntosolvetitle (Kool et al., 2018)
8.
\usebibentrygcntsptitle (Joshi et al., 2019)
9.
\usebibentrylearn2opttitle (da Costa et al., 2020)
Following \usebibentrycomp_guidelinestitle (Accorsi et al., 2021), we conduct two types of experiments.
1) we leave both computation time and solution quality unfixed. We measure the mean optimality gap and computation time per instance across the entire problem set. We use implementations as-described.
2) we fix computation time and measure the solution quality, in terms of mean optimality gap and the number of instances solved optimally. This allows for a direct comparison of approaches that was impossible in the first experiments and previous works. Where possible, we modified implementations to run continuously until the computation time limit is reached. We allow up to 10 seconds to solve each problem, including the time required to calculate input features, evaluate a model, or to construct an initial solution.
We evaluate the problem sets one instance at a time, which allows us to accurately measure computation time per instance. We allow parallelism in the GPU to sample multiple solutions to a single problem at once, where implementations are configured to do so.
We use a common, single CPU and single GPU setup for all experiments. More details on the experimental setup and the hyperparameters of different approaches are given in Appendix A.
We conduct both types of experiments on three problem sets, TSP20, TSP50, and TSP100, consisting of one thousand 20, 50, and 100-node 2D Euclidean TSP instances, respectively. We generate the instances by uniform-randomly sampling node locations in the unit square $[0,1]^{2}$, which is in line with the methods used by Kool et al. (2018); Joshi et al. (2019); da Costa et al. (2020).
In the second type of experiment (fixed computation time), we focus on the learning based approaches and keep Concorde as a benchmark.
Furthermore, we evaluate the generalization performance of the learning based approaches from smaller to larger problem sets, and from the randomly generated instances to TSPLIB instances, which are meant to represent real-world problems.
Results
The performance of each approach when computation time and solution quality are unfixed is shown in Table 1.
Approaches that are not dominated by others (i.e. they produce higher-quality solutions or are faster) are bolded. These values form a Pareto front for the trade-off between solution quality and computation time. Non-learning and learning approaches are considered separately. Our approach always lies on the Pareto front.
These results are visualized in Figure 3 in Section B.1. The results presented for Joshi et al. (2019) differ from the results presented by the authors, due to an implementation error we found in their code (see Appendix A for the correction).
To better evaluate this trade-off, the second experiment fixes computation time and measures the solution quality as it evolves over time.
Figure 4 in Section B.2 visualizes the performance as a function of time, which provides a rich understanding of the trade-off made by each approach.
Table 2 shows the performance of the learning based approaches after 10 seconds of computation time per instance.
Our approach finds better solutions on average than the other approaches for all problem sets. Notably, we reduce the mean optimality gap on the 50-node problem set from 0.268% to 0.009%, a 30$\times$ times improvement, and on the 100-node problem set from 1.534% to 0.705%, a 2$\times$ improvement.
Joshi et al. (2019) finds more optimal solutions to the TSP100 problem set, but has worse average performance than our approach. Our approach finds more optimal solutions to the TSP20 and TSP50 problem sets than the other approaches.
We evaluate the performance of our approach and other learning based approaches when generalizing from smaller instances to larger ones, and from randomly generated instances to TSPLIB instances, which are meant to represent real-world instances. These results are summarized in Section C.1 and Section C.2 respectively.
Our approach generalizes well. Notably, when generalizing from 20-node instances to the 100-node problem set, we reduce the mean optimality gap from 18.845% to 2.622%, a 7$\times$ improvement.
6 Discussion
Our approach generalizes to larger problems and to real-world problems well, which may be due to the unique input transformation we apply to the problem graph. We input the line graph $L(G)$ of the problem graph $G$, which allows the model to aggregate messages and store states on edges rather than nodes. This allows us to build models without node features, which is advantageous as the edge weights, not the specific node positions, are relevant when solving the TSP. Including the node positions as features, as done by Kool et al. (2018); Joshi et al. (2019); da Costa et al. (2020) may hinder the learned policy’s ability to generalize.
Deudon et al. (2018) apply PCA on the node positions before inputting them into their model so they are rotation invariant, yet they do not report generalization performance. For a Euclidean TSP problem graph with $n$ nodes, $L(G)$ has $n(n-1)/2$ nodes, meaning that although $G$ is complete, $L(G)$ can be very sparse, which may help learning (Cappart et al., 2021). However, the model consumes more GPU memory than models using the problem graph $G$.
Many approaches that learn construction heuristics (for example Kool et al., 2018) treat the tour as a permutation of the input nodes. By considering the solution as a sequence, these approaches miss out on its symmetry: a TSP solution is invariant to the origin city. Outputting the correct sequence becomes increasingly difficult as the number of cities increases. This has recently been addressed by Kwon et al. (2021), who leverage the symmetry of the TSP in their approach.
Our model learns the regret of including a given edge in the solution based on its features and those of its neighbors, which implicitly assumes the symmetry of a TSP solution.
We argue that our hybrid architecture is more general than others, as our global regret is defined for most routing problems. When applying our method to a new routing problem, the designer need only plug-and-play a local search procedure that is appropriate for the problem.
In contrast, methods which learn heuristics cannot always be transferred to other problems. For example, 2-opt cannot be used on routing problems that include pickup and drop-off constraints (Pacheco et al., 2021).
A drawback of our approach is that it relies on supervised learning.
In future work, our regret approximation model could be trained end-to-end.
7 Conclusion
We present a hybrid data-driven approach for solving the TSP based on Graph Neural Networks (GNNs) and Guided Local Search (GLS).
To inform GLS, we use a GNN-based model to compute fast, generalizable approximations of the global regret of each edge. Our model operates on the line graph of the TSP graph, which allows us to build a network that uses edge weight as the only input feature.
This approach allows our algorithm to escape local minima and find high-quality solutions with minimal computation time.
Finally, we present experimental results for our approach and several recent learning based approaches that emphasize the trade-off between solution quality and computation time. We demonstrate that our approach converges to optimal solutions at a faster rate than the evaluated learning based approaches.
Acknowledgements
We gratefully acknowledge the support of the European Research Council (ERC) Project 949940 (gAIa).
References
Accorsi et al. (2021)
Luca Accorsi, Andrea Lodi, and Daniele Vigo.
Guidelines for the computational testing of machine learning
approaches to vehicle routing problems, 2021.
Applegate et al. (2003)
David Applegate, Robert Bixby, Vašek Chvátal, and William Cook.
Implementing the Dantzig-Fulkerson-Johnson algorithm for large
traveling salesman problems.
Mathematical Programming, 97(1):91–153,
2003.
Applegate et al. (2006)
David Applegate, Robert Bixby, Vašek Chvatál, and William Cook.
The Traveling Salesman Problem: A Computational Study.
Princeton University Press, Princeton, NJ, 2006.
Arnold & Sörensen (2019a)
Florian Arnold and Kenneth Sörensen.
Knowledge-guided local search for the vehicle routing problem.
Computers & Operations Research, 105:32–46,
2019a.
Arnold & Sörensen (2019b)
Florian Arnold and Kenneth Sörensen.
What makes a VRP solution good? The generation of problem-specific
knowledge for heuristics.
Computers & Operations Research, 106:280–288,
2019b.
Bello et al. (2016)
Irwan Bello, Hieu Pham, Quoc V Le, Mohammad Norouzi, and Samy Bengio.
Neural combinatorial optimization with reinforcement learning.
arXiv preprint arXiv:1611.09940, 2016.
Bengio et al. (2021)
Yoshua Bengio, Andrea Lodi, and Antoine Prouvost.
Machine learning for combinatorial optimization: A methodological
tour d’horizon.
European Journal of Operational Research, 290(2):405–421, 2021.
Cappart et al. (2021)
Quentin Cappart, Didier Chételat, Elias Khalil, Andrea Lodi, Christopher
Morris, and Petar Veličković.
Combinatorial optimization and reasoning with graph neural networks.
arXiv preprint arXiv:2102.09544, 2021.
Croes (1958)
G. A. Croes.
A method for solving traveling-salesman problems.
Operations Research, 6(6):791–812, 1958.
da Costa et al. (2020)
Paulo R de O da Costa, Jason Rhuggenaath, Yingqian Zhang, and Alp Akcay.
Learning 2-opt heuristics for the traveling salesman problem via deep
reinforcement learning.
arXiv preprint arXiv:2004.01608, 2020.
Dai et al. (2017)
Hanjun Dai, Elias B. Khalil, Yuyu Zhang, Bistra Dilkina, and Le Song.
Learning combinatorial optimization algorithms over graphs.
In International Conference on Neural Information Processing
Systems, pp. 6351–6361, 2017.
Dantzig et al. (1954)
G. Dantzig, R. Fulkerson, and S. Johnson.
Solution of a large-scale traveling-salesman problem.
Journal of the Operations Research Society of America,
2(4):393–410, 1954.
Deudon et al. (2018)
Michel Deudon, Pierre Cournut, Alexandre Lacoste, Yossiri Adulyasak, and
Louis-Martin Rousseau.
Learning heuristics for the TSP by policy gradient.
In Integration of Constraint Programming, Artificial
Intelligence, and Operations Research, pp. 170–181, 2018.
Fu et al. (2021)
Zhang-Hua Fu, Kai-Bin Qiu, and Hongyuan Zha.
Generalize a small pre-trained model to arbitrarily large tsp
instances, 2021.
Gilmer et al. (2017)
Justin Gilmer, Samuel S Schoenholz, Patrick F Riley, Oriol Vinyals, and
George E Dahl.
Neural message passing for quantum chemistry.
In International Conference on Machine Learning, pp. 1263–1272. PMLR, 2017.
Gratiet et al. (2016)
Loïc Le Gratiet, Stefano Marelli, and Bruno Sudret.
Metamodel-based Sensitivity Analysis: Polynomial Chaos
Expansions and Gaussian Processes, pp. 1–37.
Springer International Publishing, Cham, 2016.
Hassin & Keinan (2008)
Refael Hassin and Ariel Keinan.
Greedy heuristics with regret, with application to the cheapest
insertion algorithm for the TSP.
Operations Research Letters, 36(2):243–246, 2008.
Helsgaun (2009)
Keld Helsgaun.
General k-opt submoves for the Lin–Kernighan TSP heuristic.
Mathematical Programming Computation, 1(2):119–163, 2009.
Helsgaun (2017)
Keld Helsgaun.
An extension of the Lin-Kernighan-Helsgaun TSP solver for
constrained traveling salesman and vehicle routing problems.
Technical report, Roskilde University, 2017.
Ioffe & Szegedy (2015)
Sergey Ioffe and Christian Szegedy.
Batch normalization: Accelerating deep network training by reducing
internal covariate shift.
In International Conference on Machine Learning, pp. 448–456. PMLR, 2015.
Joshi et al. (2019)
Chaitanya K Joshi, Thomas Laurent, and Xavier Bresson.
An efficient graph convolutional network technique for the travelling
salesman problem.
arXiv preprint arXiv:1906.01227, 2019.
Kingma & Ba (2014)
Diederik P Kingma and Jimmy Ba.
Adam: A method for stochastic optimization.
arXiv preprint arXiv:1412.6980, 2014.
Kool et al. (2018)
Wouter Kool, Herke Van Hoof, and Max Welling.
Attention, learn to solve routing problems!
arXiv preprint arXiv:1803.08475, 2018.
Kool et al. (2021)
Wouter Kool, Herke van Hoof, Joaquim Gromicho, and Max Welling.
Deep policy dynamic programming for vehicle routing problems, 2021.
Kwon et al. (2021)
Yeong-Dae Kwon, Jinho Choo, Byoungjip Kim, Iljoo Yoon, Youngjune Gwon, and
Seungjai Min.
Pomo: Policy optimization with multiple optima for reinforcement
learning, 2021.
Lin & Kernighan (1973)
S. Lin and B. W. Kernighan.
An effective heuristic algorithm for the traveling-salesman problem.
Operations Research, 21(2):498–516, 1973.
McKay et al. (1979)
M. D. McKay, R. J. Beckman, and W. J. Conover.
A comparison of three methods for selecting values of input variables
in the analysis of output from a computer code.
Technometrics, 21(2):239–245, 1979.
Mnih et al. (2015)
Volodymyr Mnih, Koray Kavukcuoglu, David Silver, Andrei A. Rusu, Joel Veness,
Marc G. Bellemare, Alex Graves, Martin Riedmiller, Andreas K. Fidjeland,
Georg Ostrovski, Stig Petersen, Charles Beattie, Amir Sadik, Ioannis
Antonoglou, Helen King, Dharshan Kumaran, Daan Wierstra, Shane Legg, and
Demis Hassabis.
Human-level control through deep reinforcement learning.
Nature, (508):529–533, 2015.
Pacheco et al. (2021)
Toni Pacheco, Rafael Martinelli, Anand Subramanian, Túlio A. M. Toffolo, and
Thibaut Vidal.
Exponential-size neighborhoods for the pickup-and-delivery traveling
salesman problem.
arXiv preprint arXiv:2107.05189, 2021.
Padberg & Rinaldi (1990)
Manfred Padberg and Giovanni Rinaldi.
A branch-and-cut algorithm for the resolution of large-scale
symmetric traveling salesman problems.
Society for Industrial and Applied Mathematics, 33(1):60–100, 1990.
Paleyes et al. (2019)
Andrei Paleyes, Mark Pullin, Maren Mahsereci, Neil Lawrence, and Javier
Gonzalez.
Emulation of physical processes with emukit.
In Second Workshop on Machine Learning and the Physical
Sciences. NeurIPS, 2019.
Paszke et al. (2019)
Adam Paszke, Sam Gross, Francisco Massa, Adam Lerer, James Bradbury, Gregory
Chanan, Trevor Killeen, Zeming Lin, Natalia Gimelshein, Luca Antiga, et al.
Pytorch: An imperative style, high-performance deep learning library.
In Advances in Neural Information Processing Systems, pp. 8026–8037, 2019.
Potvin & Rousseau (1993)
Jean-Yves Potvin and Jean-Marc Rousseau.
A parallel route building algorithm for the vehicle routing and
scheduling problem with time windows.
European Journal of Operational Research, 66(3):331–340, 1993.
Prim (1957)
Robert C. Prim.
Shortest connection networks and some generalizations.
The Bell System Technical Journal, 36(6):1389–1401, 1957.
Reddy (1977)
Dabbala Raj Reddy.
Speech understanding systems: Summary of results of the five-year
research effort at Carnegie-Mellon University.
Technical report, Carnegie Mellon University, 1977.
Saltelli (2002)
Andrea Saltelli.
Making best use of model evaluations to compute sensitivity indices.
Computer Physics Communications, 145(2):280–297, 2002.
Smith (1999)
Kate Smith.
Neural networks for combinatorial optimization: A review of more than
a decade of research.
INFORMS Journal on Computing, 11(1):15–34, 1999.
Vaswani et al. (2017)
Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones,
Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin.
Attention is all you need.
In Advances in Neural Information Processing Systems, pp. 5998–6008, 2017.
Veličković et al. (2017)
Petar Veličković, Guillem Cucurull, Arantxa Casanova, Adriana Romero,
Pietro Lio, and Yoshua Bengio.
Graph attention networks.
arXiv preprint arXiv:1710.10903, 2017.
Vinyals et al. (2015)
Oriol Vinyals, Meire Fortunato, and Navdeep Jaitly.
Pointer networks.
arXiv preprint arXiv:1506.03134, 2015.
Voudouris & Tsang (1996)
Chris Voudouris and Edward Tsang.
Partial constraint satisfaction problems and guided local search.
Technical report, Department of Computer Science, University of
Essex, 1996.
Voudouris & Tsang (1999)
Christos Voudouris and Edward Tsang.
Guided local search and its application to the traveling salesman
problem.
European Journal of Operational Research, 113(2):469–499, 1999.
Wang et al. (2019)
Minjie Wang, Da Zheng, Zihao Ye, Quan Gan, Mufei Li, Xiang Song, Jinjing Zhou,
Chao Ma, Lingfan Yu, Yu Gai, Tianjun Xiao, Tong He, George Karypis, Jinyang
Li, and Zheng Zhang.
Deep graph library: A graph-centric, highly-performant package for
graph neural networks.
arXiv preprint arXiv:1909.01315, 2019.
Wu et al. (2020)
Yaoxin Wu, Wen Song, Zhiguang Cao, Jie Zhang, and Andrew Lim.
Learning improvement heuristics for solving routing problems.
arXiv preprint arXiv:1912.05784, 2020.
Appendix A Implementation details
We conduct all computational experiments on virtual machines using a single core of an Intel Xeon E5-2650 v4 processor, 24 GB of RAM, and a single Tesla P100 GPU with 16 GB of VRAM.
We evaluate the various approaches one-at-a-time, to ensure they do not compete for computational resources.
When evaluating the various approaches, we record actual tours and use a common method of calculating tour cost and optimality gap, as we found that small discrepancies in cost calculations between implementations led to skewed results. We recommend this approach when comparing different approaches for solving the TSP.
Based on approximately 200,000 TSP solutions, we determined that an absolute threshold of $10^{-7}$ on the value of the objective function is sufficient to separate optimal and sub-optimal solutions.
We train our model on sets of one hundred thousand 20, 50, and 100-node 2D Euclidean TSP instances. We generate all instances by uniform-randomly sampling node locations in the unit square $[0,1]^{2}$.
We calculate the target output, the regret of including each edge in the solution, according to eq. 4. We use Concorde to find an optimal solution and LKH-3 with 100 trials and 10 runs to solve find an optimal solution with a given edge fixed.
We empirically validate that this configuration for LKH-3 is sufficient to find optimal solutions for problems with 100 nodes or less. We scale the input features and target outputs to a range of $[0,1]$.
We train our model by minimizing the $l^{2}$ loss to the target output using the Adam optimizer (Kingma & Ba, 2014) and an exponentially decaying learning rate $\gamma=10^{-3}\times(0.99)^{e}$, where $e$ is the epoch. We train the model for 100 epochs with early stopping.
We use a training batch size $B=32$ for the 20 and 50-node training sets, and a training batch size of $B=15$ for the 100-node training set due to limited GPU memory. Our model uses $T=3$ message passing layers and our GLS algorithm uses $K=20$ perturbation moves.
Our approach is implemented using Python 3.8.11, PyTorch 1.9.0 (Paszke et al., 2019), DGL 0.7.1 (Wang et al., 2019), and CUDA 10.2, and is open-source.
We use the open-source implementation of Kool et al. (2018) with the best configuration, which samples 1280 solutions in parallel. We use this implementation as-is for experiments where computation time is not fixed. We modified the implementation to continuously sample 1280 solutions until the computation time limit is reached for experiments where computation time is fixed. We use the open-source implementation of da Costa et al. (2020) with the best configuration, which uses 2000 iterations. We use this implementation as-is for experiments where computation time is not fixed. We modified the implementation to iterate until the computation time limit is reached for experiments where computation time is fixed.
We used the open-source implementation of Joshi et al. (2019).
We found a programming error in their beamsearch implementation that caused invalid tours to be reported. We modified their implementation to detect invalid tours and discarded them when they are produced. Our changes are shown in Listing LABEL:lst:fix. Therefore, our results may differ from those originally reported by the authors.
Appendix B Comparison to other approaches
B.1 Performance charts
Following the guidelines in Accorsi et al. (2021), we measure the mean optimality gap and computation time per instance, while both are unfixed. Figure 3 depicts the performance of the evaluated non-learning and learning based approaches averaged across the entire 20, 50, and 100-node problem sets. The axes are scaled so the plots are easily readable. Results that fall outside the plot limits are shown with an arrow pointing towards the horizontal axis at the mean computation time. These plots clearly identify Pareto optimal approaches and dominated ones.
B.2 Convergence profile charts
We evaluate the solution quality, in terms of mean optimality gap and number of problems solved optimally, as a function of computation time over a fixed computation time. The resulting convergence profile provides a detailed view of how each approach trades off between solution quality and computation time.
Figure 4 depicts convergence profiles for our approach and several recent learning based approaches, for models trained and evaluated on 20, 50, and 100-node problem instances.
Computation time required to compute input features, evaluate a model, or to construct an initial solution is visible as a gap between the trace and the vertical axis, and is especially noticeable for Joshi et al. (2019).
Appendix C Generalization performance
C.1 Generalization to larger instances
Figure 5 depicts convergence profiles for the learning based approaches when generalizing from smaller problem instances to larger ones. The plots are arranged in order of increasing difference in size between the training and test problem sets.
C.2 Generalization to real-world instances
In many cases there may be insufficient examples of real-world problem instances to train a model, meaning it must be trained on synthetic data. Therefore, it is important to evaluate this “sim-to-real” transfer. Table 3 shows mean optimality gap and computation time for the evaluated learning based methods using models trained on random 100-node TSPs and evaluated on TSPLIB instances with 50 to 200 nodes and 2D Euclidean distances (29 instances).
Appendix D Analysis of additional input features
We evaluate the potential benefit of adding additional features to our regret approximation model. While more features can help produce better predictions, they come at the cost of additional computation time. We consider a total of ten additional features, described in Table 4.
We use a Gaussian process-based surrogate model to conduct global sensitivity analysis (Gratiet et al., 2016) of the input features on the validation loss of the model after training. We assume that better regret predictions will equate to better guidance by the GLS algorithm, ultimately resulting in better performance to the problem sets.
We semi-randomly sample 100 input feature sets using Latin hypercube sampling (McKay et al., 1979). We train a model using each of these feature sets for 35 epochs without early stopping and record the final validation loss. Using these results, we fit a Gaussian process to emulate the mapping between the feature set $F=\{f_{0},f_{1},\dots,f_{10}\}$, and the validation loss $l$ achieved by our model after training, where $f_{n}$ indicates whether or not feature $n$ is used. We then compute Monte-Carlo estimates of the main and total effects of each input feature on the model’s validation loss (Saltelli, 2002). Our implementation uses Emukit (Paleyes et al., 2019). Figure 6 depicts the estimated total effect for each input feature. Edge weight is the most important feature, followed by neighbor rank, depot weight, edge width, and node width. The remaining features have little to no importance.
We train a model using these top five features on 20-node problems and compare its performance to the equivalent model using edge weight as the only feature when solving the 20, 50, and 100-node problem sets. The performance of both models at the computation time limit is shown in Table 5. While the model using additional input features produces slightly more accurate regret predictions, any benefit is cancelled out by the additional time required to compute these features. Note that the results are slightly different from those reported in Figure 5 due to different training hyperparameters. |
Relativistic Brownian motion:
From a microscopic binary collision model to the Langevin equation
Jörn Dunkel
joern.dunkel@physik.uni-augsburg.de
Peter Hänggi
Institut für Physik, Universität Augsburg,
Theoretische Physik I, Universitätstraße 1, D-86135 Augsburg, Germany
(December 5, 2020)
Abstract
The Langevin equation (LE) for the one-dimensional relativistic
Brownian motion is derived from a microscopic collision model. The model assumes that a heavy point-like Brownian particle interacts with the lighter heat bath particles via elastic hard-core collisions. First, the commonly known, non-relativistic LE is deduced from this model, by taking into account the non-relativistic conservation laws for momentum and kinetic energy.
Subsequently, this procedure is generalized to the relativistic case.
There, it is found that the relativistic stochastic force is still $\delta$-correlated (white noise) but does no longer correspond to a Gaussian
white noise process. Explicit results for the friction and momentum-space diffusion coefficients are presented and discussed.
pacs:
02.50.Ey, 05.40.-a, 05.40.Jc, 47.75.+f
I Introduction
The theories of the non-relativistic Brownian motion and special
relativity were introduced more than 100 years
ago Einstein (1905); Einstein and von Smoluchowski (1999); Uhlenbeck and Ornstein (1930); Chandrasekhar (1943); Wang and Uhlenbeck (1945); Hänggi and Jung (1995); Hänggi and Marchesoni (2005). Since
then, they have become cornerstones for our understanding of a wide
range of physical processes Frey and Kroy (2005); Hänggi et al. (2005); Lämmerzahl (2005); Reimann and Hänggi (2002); Bartussek et al. (1996).
This fact notwithstanding, the unification of both concepts poses a
theoretical challenge still nowadays (classical references are
Schay (1961); Hakim (1965); Dudley (1965); Guerra and Ruggiero (1978); Boyer (1979); Ben-Ya’acov (1981); recent contributions
include Morato and Viola (1995); Posilicano (1997); Debbasch et al. (1997); Kostädt and Liu (2000); Wolschin (2004); Debbasch (2004); Koide (2005); Zygadlo (2005); Rapoport (2005); Dunkel and Hänggi (2005a, b, 2006); Fingerle (2006); Fa (2006);
potential applications in high-energy physics and astrophysics are
considered
in Abdel-Aziz and Gavin (2004, 2005); van Hees
et al. (2006a, b); Dieckmann et al. (2006); Marti et al. (2006)).
The relatively slow progress in this field can be attributed to the
severe difficulties that arise when one tries to describe $N$-body
systems in a relativistically consistent
manner Wheeler and Feynman (1949); Van Dam and Wigner (1965); Komar (1978); Duviryuk et al. (2001); Lehmann (2006). Due to this reason, the derivation of
relativistic Langevin equations (LEs) from an underlying microscopic
model has remained an unsolved issue until now 111For the
non-relativistic Brownian motion, this problem was solved by
Bogolyubov Bogolyubov (1945), Magalinskii Magalinskii (1959), Ford et al. Ford et al. (1965) and Zwanzig Zwanzig (1973), who considered a
bath of harmonic oscillators.. However, in the present paper we aim
to provide a solution to this problem.
More precisely, by considering quasi-elastic, binary collisions
between the Brownian and heat bath particles Pechukas (1991); Tsonchev and Pechukas (2000) we are
able to treat the heat bath in a fully relativistic manner without
having to account for the exact details of the relativistic $N$-body
interactions. As shown in Sec. II, for a
non-relativistic framework this approach yields the well-known
non-relativistic LE with Gaussian white noise as well as the correct
fluctuation-dissipation theorem (the Einstein-Sutherland
relation Hänggi and Marchesoni (2005)). In Sec. III, the method is
transferred to the relativistic case, leading to the main result of
this paper, the relativistic LE (32). Remarkably,
the relativistic stochastic force is also $\delta$-correlated (‘white’) but no longer of Gaussian (or Wiener Wiener (1923)) type.
Compared with the non-relativistic Brownian motion, this is the most
important difference. Furthermore, we obtain explicit
representations of friction and (momentum)-diffusion coefficients in
terms of expectation values with respect to the heat bath
distribution (see also App. A).
II Non-relativistic Brownian motions
The objective of this section is to recover the well-known
non-relativistic LEs from a simple microscopic collision model for
Brownian motions. As shown by several authors in the past Bogolyubov (1945); Magalinskii (1959); Ford et al. (1965); Zwanzig (1973); Hänggi (1997), non-relativistic LEs can
also be derived by considering a bath of harmonic oscillators
with canonical phase space distribution. Unfortunately,
it is problematic to transfer this approach to the
relativistic case, because any instantaneous linear (or nonlinear)
interactions between Brownian and heat particles would violate the basic
principles of special relativity Wheeler and Feynman (1949); Van Dam and Wigner (1965). To circumvent this problem, we
will pursue a different method here, using only the (non-)relativistic
microscopic conservation laws for energy and momentum, respectively,
known to hold for elastic point-like, binary collisions (contact interactions Van Dam and Wigner (1965)). Conceptually, our
approach is related to that of Pechukas Pechukas (1991) and
Pechukas-Tsonchev Tsonchev and Pechukas (2000), who considered a similar model in
the context of non-relativistic quantum Brownian motion
Hänggi and Ingold (2005). Analogous approaches are also known from unimolecular rate theory, see e.g. Sec. V in Hänggi et al. (1990).
II.1 Microscopic model
For the sake of simplicity only, we will restrict ourselves
throughout to the one-dimensional ($1d)$ case. Generalizations to
higher space dimensions are in principle straightforward, but
certain calculations will become much more cumbersome (cf. comments at the end of App. A).
To start out, consider the following situation in the inertial
laboratory frame $\Sigma_{0}$: A large one-dimensional box volume $\mathcal{V}\equiv[-L/2,L/2]$ contains an ideal non-relativistic gas, consisting of $N$ small
point-like particles with identical masses $m$. The gas particles –
referred to as ‘heat bath’ hereafter – surround a
Brownian particle of mass $M\gg m$. Due to frequent elastic collisions with heat bath particles,
the Brownian particle performs stochastic motions.
II.1.1 Heat bath
The coordinates and momenta of the heat bath particles are denoted
by $x_{r}\in[-L/2,L/2]$ and $p_{r}\in(-\infty;\infty)$, respectively,
where $r=1,\ldots,N$. As usual, we make the following simplifying
assumption concerning the heat bath: The probability density function (PDF)
of the heat bath particles is a spatially homogeneous Maxwell
distribution, i.e., at each time $t>0$, the PDF reads
$$\displaystyle f_{\mathrm{b}}^{N}(x_{1},\ldots,p_{N})$$
$$\displaystyle=$$
$$\displaystyle\left(\frac{\lambda}{L}\right)^{N}\prod_{r=1}^{N}\exp\biggl{(}-%
\frac{p_{r}^{2}}{2mkT}\biggr{)},$$
(1)
where $k$ is the Boltzmann constant, $T$ the temperature,
and $\lambda=(2\pi mkT)^{-1/2}$. Thus, it is implicitly assumed that:
•
the heat bath particles are independently and identically distributed;
•
the collisions with the Brownian particle do not significantly alter the bath distribution.
These assumptions are justified, if the collisions between the gas
particles rapidly reestablish a spatially homogeneous bath distribution.
II.1.2 Kinematics of single collision events
The momentum and energy balance per (elastic) collision reads
$$\displaystyle E+\epsilon=\hat{E}+\hat{\epsilon},\qquad P+p=\hat{P}+\hat{p}.$$
(2)
Here and below, capital letters refer to the Brownian particle and
small letters to particles forming the heat bath; quantities without
(with) hat-symbols refer to the state before (after) the
collision. In the non-relativistic case, we have, e.g., before the collision
$$\displaystyle P=MV,\qquad p=mv,\qquad E=\frac{P^{2}}{2M},\qquad\epsilon=\frac{%
p^{2}}{2m},$$
(3)
where $v$ and $V$ denote the velocities. Taking into account both
conservation of momentum and (kinetic) energy, one finds that the
change $\Delta P\equiv\hat{P}-P$ of the Brownian particle’s
momentum per single collision is given by
$$\displaystyle\Delta P=\frac{-2m}{M+m}\,P+\frac{2M}{M+m}\,p.$$
(4)
II.2 Derivation of the Langevin equation
The total momentum change $\delta P$ of the Brownian particle within the time interval $\tau$ can be written as
$$\displaystyle\delta P(t)\equiv P(t+\tau)-P(t)=\sum_{r=1}^{N}\Delta P_{r}\;I_{r%
}(t,\tau),$$
(5)
where $I_{r}(t,\tau)\in\{0,1\}$ is the indicator function for a collision
with the heat bath particle $r$ during the interval $[t,t+\tau]$; i.e.
$I_{r}(t,\tau)=1$ if a collision has occurred, and, otherwise,
$I_{r}(t,\tau)=0$. In the $1d$ case, the collision indicator can be written explicitly as
$$\displaystyle I_{r}(t,\tau)$$
$$\displaystyle=$$
$$\displaystyle\Theta(X-x_{r})\;\Theta(x^{\prime}_{r}-X^{\prime})\;\Theta(v_{r}-%
V)+$$
(6a)
$$\displaystyle\;\Theta(x_{r}-X)\;\Theta(X^{\prime}-x^{\prime}_{r})\;\Theta(V-v_%
{r}),$$
where $$X=X(t),x_{r}=x_{r}(t)$$, and
$$\displaystyle X^{\prime}=X+V\tau,\qquad x^{\prime}_{r}=x_{r}+v_{r}\tau$$
(6b)
are the projected particle positions at time $t+\tau$. The Heaviside-function is defined by
$$\displaystyle\Theta(x)=\begin{cases}0,&x<0;\\
1/2,&x=0;\\
1,&x>0.\end{cases}$$
The expectation of the collision indicator with respect to the bath distribution, denoted by $\left\langle I_{r}(t,\tau)\right\rangle_{\mathrm{b}}$, gives the
probability that the bath particle $r$ collides with the Brownian
particle between $t$ and $t+\tau$. As shown in
App. A, in the limit $\tau\to 0$, one finds
$$\displaystyle\left\langle I_{r}(t,\tau)\right\rangle_{\mathrm{b}}=\tilde{C}(V)%
\;\frac{\tau}{L}=C(P)\;\frac{\tau}{L},$$
(7a)
with function $$C(P)=\tilde{C}(V(P))$$ given by the
integral formula
$$\displaystyle\tilde{C}(V)$$
$$\displaystyle\equiv$$
$$\displaystyle\frac{1}{2}\int_{V}^{\infty}\!\!\!\mathrm{d}v_{r}\;(v_{r}-V)\;%
\tilde{f}_{\mathrm{b}}^{1}(v_{r})+$$
(7b)
$$\displaystyle\;\frac{1}{2}\int^{V}_{-\infty}\!\!\!\mathrm{d}v_{r}\;(V-v_{r})\;%
\tilde{f}_{\mathrm{b}}^{1}(v_{r}).$$
Here, $\tilde{f}_{\mathrm{b}}^{1}(v_{r})$ is the one-particle velocity PDF of a heat bath particle.
We anticipate that Eqs. (6) and (7) remain valid in
the relativistic case as well, but then one has to insert the relativistic bath distribution in Eq. (7b).
However, in order to recover from Eqs. (5)–(7)
the well-known non-relativistic LE, we still have to make a number of simplifying assumptions:
(i)
The time interval $\tau$ is sufficiently small, so that $|\delta P/P|\ll 1$.
In particular, $\tau$ is supposed to be so small that there occurs at most
only one collision between the Brownian particle and a specific heat bath
particle $r$. One the other hand, the time interval $\tau$ should still
be large enough, so that the total number of collisions within $\tau$ is
larger than $1$. These requirements can be fulfilled simultaneously only if $m/M\ll 1$.
(ii)
Collisions occurring within $[t,t+\tau]$ can be viewed as independent events.
(iii)
Finally, we will (have to) assume that
$$\displaystyle\left\langle[p_{r}\;I_{r}(t,\tau)]^{j}\right\rangle_{\mathrm{b}}$$
$$\displaystyle=$$
$$\displaystyle\left\langle p_{r}^{j}\;I_{r}(t,\tau)\right\rangle_{\mathrm{b}}$$
(8)
$$\displaystyle\simeq$$
$$\displaystyle\left\langle p_{r}^{j}\right\rangle_{\mathrm{b}}\;\left\langle I_%
{r}(t,\tau)\right\rangle_{\mathrm{b}}$$
$$\displaystyle=$$
$$\displaystyle\left\langle p_{r}^{j}\right\rangle_{\mathrm{b}}\;C(P)\;\frac{%
\tau}{L}$$
for $j=1,2,\ldots$. Given the explicit representation of the
indicator function (6a), it is in principle
straightforward to check the quality of the
approximation (8), if a bath distribution has been
specified.
As we shall see immediately, the assumptions (i)–(iii) are
necessary and sufficient for deriving the well-known
non-relativistic LE from Eqs. (5)–(7).
Upon inserting Eq. (4) into (5)
and dividing by $\tau$ we find
$$\displaystyle\frac{\delta P(t)}{\tau}$$
$$\displaystyle\simeq$$
$$\displaystyle-\left[\frac{1}{\tau}\sum_{r=1}^{N}\frac{2m}{m+M}\;I_{r}(t,\tau)%
\right]\,P+$$
(9)
$$\displaystyle\qquad\frac{1}{\tau}\sum_{r=1}^{N}\frac{2M}{M+m}\,p_{r}\;I_{r}(t,%
\tau).$$
The first term on the rhs. in Eq. (9) can be identified as
the ‘friction’ term, whereas the second term represents ‘noise’.
On the rhs. of Eq. (9), it was assumed that for each collision
occurring within $[t,t+\tau]$, the ‘initial’ momentum of the Brownian particle is
approximately equal to some suitably chosen value $P(t^{\prime})$ with $t^{\prime}\in[t,t+\tau]$,
cf. the assumption (i) above and the discussion at the end of this section.
The next step en route to the conventional LE consists in replacing the
square bracket expression in Eq. (9) by the averaged friction coefficient
$$\displaystyle\nu_{0}(P)$$
$$\displaystyle\equiv$$
$$\displaystyle\frac{1}{\tau}\sum_{r=1}^{N}\frac{2m}{m+M}\,\left\langle I_{r}(t,%
\tau)\right\rangle_{\mathrm{b}}.$$
(10a)
Since it was assumed that the heat bath particles are independently and
identically distributed, we can rewrite this as
$$\displaystyle\nu_{0}(P)$$
$$\displaystyle=$$
$$\displaystyle\frac{N}{\tau}\frac{2m}{m+M}\,\left\langle I_{r}(t,\tau)\right%
\rangle_{\mathrm{b}},$$
(10b)
for some $r\in\{1,\ldots,N\}$. The coefficient $\nu_{0}$ can be interpreted
as an average collision rate weighted by some mass ratio. Inserting
Eq. (7a) into Eq. (10b) yields
$$\displaystyle\nu_{0}(P)$$
$$\displaystyle=$$
$$\displaystyle n_{\mathrm{b}}\,\frac{2m}{m+M}\,C(P),$$
(11a)
where $$n_{\mathrm{b}}=N/L$$ is the density of the bath particles. In the case of
the Maxwell distribution, we can evaluate the integral (7b), and find
$$\displaystyle C(P)$$
$$\displaystyle=$$
$$\displaystyle\left(\frac{kT}{2\pi m}\right)^{1/2}\exp\biggl{[}-\frac{m}{2kT}%
\left(\frac{P}{M}\right)^{2}\biggr{]}+$$
(11b)
$$\displaystyle\qquad\frac{P}{2M}\;\mathrm{erf}\biggl{[}\left(\frac{m}{2kT}%
\right)^{1/2}\frac{P}{M}\biggr{]}.$$
In particular, setting (see App. A)
$$\displaystyle C(P)\approx C(0)=\left(\frac{kT}{2\pi m}\right)^{1/2}$$
(12)
corresponds to the commonly used Stokes
approximation.
It then remains to analyze the ‘noise force’
$$\displaystyle\xi(P,t)\equiv\frac{1}{\tau}\sum_{r=1}^{N}\frac{2M}{M+m}\,p_{r}\;%
I_{r}(t,\tau),$$
(13)
corresponding to the last term in Eq. (9). The momentum dependence of the noise enters through the implicit $P$-dependence of the collision indicator functions $I_{r}(t,\tau)$. To keep subsequent formulae as compact as possible, we shall use the abbreviation $\xi(t)\equiv\xi(P,t)$ in the remainder. Then, averaging over the bath distribution $f_{\mathrm{b}}^{N}$ and using Eqs. (8), we
find for the mean value
$$\displaystyle\left\langle\xi(t)\right\rangle_{\mathrm{b}}=0.$$
(14a)
Furthermore, assuming mutual independence of the collisions, the correlation function is obtained as
$$\displaystyle\left\langle\xi(t)\,\xi(s)\right\rangle_{\mathrm{b}}$$
$$\displaystyle=$$
$$\displaystyle\frac{\delta_{ts}}{\tau^{2}}\left(\frac{2M}{M+m}\right)^{2}\sum_{%
r=1}^{N}\,\left\langle p_{r}^{2}\;I_{r}^{2}(t,\tau)\right\rangle_{\mathrm{b}}$$
(14b)
$$\displaystyle\overset{\eqref{e:Stokes-b}}{\simeq}$$
$$\displaystyle\frac{\delta_{ts}}{\tau^{2}}\left(\frac{2M}{M+m}\right)^{2}\sum_{%
r=1}^{N}\,mkT\;\left\langle I_{r}(t,\tau)\right\rangle_{\mathrm{b}}$$
$$\displaystyle\overset{\eqref{e:nu_definition}}{=}$$
$$\displaystyle\frac{\delta_{ts}}{\tau}\left(\frac{2M^{2}}{M+m}\right)\;\nu_{0}kT,$$
with $\delta_{ts}\in\{0,1\}$ denoting the Kronecker-symbol. To
obtain the second line, we have used that $I_{r}^{2}(t,\tau)=I_{r}(t,\tau)$,
and the simplifying assumption (8) that $I_{r}(t,\tau)$ and
$p_{r}$ are (approximately) independent random variables with respect to the bath distribution.
Similar to Eq. (14b), also the higher correlation
functions are determined by the corresponding moments of the Gaussian marginal
bath distribution (1). Thus, under the above
assumptions (i)–(iii), the non-relativistic stochastic force $\xi(t)$ corresponds
to Gaussian white noise (or a Wiener process Wiener (1923), respectively).
Finally, by substituting $\nu_{0}$ from Eq (11) for
the square bracket expression in Eq. (9) and
formally letting $\tau\to 0$, we recover from
Eq. (9) the well-known non-relativistic LE Hänggi and Thomas (1982); Hänggi (1997); Van Kampen (2003)
$$\displaystyle\dot{P}=-\nu_{0}(P)\,P+\xi(t),$$
(15a)
where $$\xi(t)\equiv\xi(P,t)$$ is a momentum-dependent Gaussian white noise force, characterized by
$$\displaystyle\left\langle\xi(t)\right\rangle$$
$$\displaystyle=$$
$$\displaystyle 0,$$
(15b)
$$\displaystyle\left\langle\xi(t)\,\xi(s)\right\rangle$$
$$\displaystyle=$$
$$\displaystyle 2\,D_{0}(P)\;\delta(t-s),$$
(15c)
with (momentum-space) diffusion coefficient
$$\displaystyle D_{0}(P)=\frac{M^{2}}{M+m}\;\nu_{0}(P)\,kT.$$
(15d)
To obtain Eq. (15c), we used that $\delta_{st}/\tau\to\delta(t-s)$ for $\tau\to 0$, where $\delta(t-s)$ is the Dirac-function.
In the limit $m/M\to 0$, Eq. (15d) reduces to the standard
fluctuation-dissipation theorem $D_{0}=M\nu_{0}kT$
Hänggi and Marchesoni (2005); Becker (1967). However, $\nu_{0}$ and $D_{0}$ are constants only if
one adopts the Stokes approximation (12),
cf. App. A. If one goes beyond the Stokes
approximation, then the noise in Eqs. (15) becomes
multiplicative with respect to $P$, and, therefore,
Eqs. (15) must be complemented by a discretization
rule in this
case Ito (1944, 1951); Fisk (1963); Stratonovich (1964, 1966); Hänggi (1978, 1980); Hänggi and Thomas (1982); Klimontovich (1994). As
discussed in Hänggi (1978, 1980); Hänggi and Thomas (1982); Klimontovich (1994), only for the
post-point discretization rule, corresponding to the choice
$\nu_{0}(P)=\nu_{0}(P(t+\tau))$ and $D_{0}(P)=D_{0}(P(t+\tau))$ on the rhs.
of Eq. (15a), one recovers the Maxwellian PDF
$$\displaystyle\Phi_{\infty}(P)=\left(\frac{1}{2\pi MkT}\right)^{1/2}\exp\biggl{%
(}-\frac{P^{2}}{2MkT}\biggr{)}$$
(16)
as the stationary momentum
distribution of the Brownian particle in the limit $t\to\infty$
(assuming that $m/M\to 0$).
III Relativistic Brownian motions
We shall now apply an analogous reasoning to obtain a relativistic LE.
For this purpose we consider an inertial (laboratory) frame $\Sigma_{0}$ with
time coordinate $t$, as e.g. measured by an atomic clock resting in $\Sigma_{0}$.
III.1 Microscopic model
The basic constituents of the microscopic model are the same as
those outlined in Sec. II.1, but in addition we now
have to consider a relativistic heat bath distribution and must
consistently take into account the relativistic collision
kinematics.
III.1.1 Relativistic heat bath
In the relativistic case, we postulate analogous to
Eq. (1) that, with respect to $\Sigma_{0}$,
the heat bath distribution is stationary, spatially homogeneous, and
independent, so that the PDF can be written in the product form
$$\displaystyle f_{\mathrm{b}}^{N}(x_{1},\ldots,p_{N})$$
$$\displaystyle=$$
$$\displaystyle L^{-N}\prod_{r=1}^{N}f_{\mathrm{b}}^{1}(p_{r}).$$
(17a)
As marginal one-particle
momentum PDFs, we will now consider the $$\eta$$-generalized
Jüttner-Maxwell distributions Jüttner (1911); Dunkel and Hänggi (2006), reading:
$$\displaystyle f_{\mathrm{b}}^{1}(p)=\frac{\mathcal{N}_{\eta}}{\epsilon(p)^{%
\eta}}\exp\biggl{[}-\frac{\epsilon(p)}{kT}\biggr{]},\qquad\eta\geq 0,$$
(17b)
where $$p\in(-\infty,+\infty)$$, and
$$\epsilon(p)$$ denotes the relativistic kinetic energy of a heat bath
particle. The normalization constant $$\mathcal{N}_{\eta}$$ is determined
by the condition
$$\displaystyle 1=\int_{-\infty}^{\infty}\mathrm{d}p\;f_{\mathrm{b}}^{1}(p).$$
(17c)
For $\eta=0$, Eq. (17b)
reduces to the standard Jüttner-Maxwell distribution Jüttner (1911).
On the other hand, as discussed recently Dunkel and Hänggi (2006); Marti et al. (2006), the PDF with $\eta=1$ appears to be conserved in relativistic elastic binary collisions.
In general, however, the arguments and results presented below
remain valid for arbitrary one-particle momentum
PDFs $f^{1}_{\mathrm{b}}(p)$, i.e., also for momentum distributions other
than the $\eta$-generalized Jüttner PDFs (17b).
III.1.2 Relativistic collision kinematics
Using natural units such that $c=1$, relativistic kinetic energy, momentum and velocity are related by
$$\displaystyle p$$
$$\displaystyle=mv\,\gamma(v),$$
$$\displaystyle\qquad\epsilon(p)=$$
$$\displaystyle\left(m^{2}+p^{2}\right)^{1/2},$$
(18a)
$$\displaystyle P$$
$$\displaystyle=MV\,\gamma(V),$$
$$\displaystyle\qquad E(P)=$$
$$\displaystyle\left(M^{2}+P^{2}\right)^{1/2},$$
(18b)
where $\gamma(v)\equiv\left(1-v^{2}\right)^{-1/2}$. As before, capital letters
refer to the Brownian particle. Inserting Eqs. (18) into
the conservation laws (2), and solving for $\hat{P}$, one finds Dunkel and Hänggi (2006)
$$\displaystyle\hat{P}$$
$$\displaystyle=$$
$$\displaystyle\frac{2u\,E-(1+u^{2})\,P}{1-u^{2}},$$
(19)
where
$$\displaystyle u(p,P)=\frac{P+p}{E+\epsilon}$$
(20)
is the center-of-mass velocity. Hence, the momentum change $\Delta P=\hat{P}-P$ of
the Brownian particle in a single collision is given by
$$\displaystyle\Delta P=-\frac{2}{1-u^{2}}\;\frac{\epsilon}{E+\epsilon}\;P+\frac%
{2}{1-u^{2}}\;\frac{E}{E+\epsilon}\;p.$$
(21)
In the non-relativistic limit case, where $u^{2}\ll 1$, $E\simeq M$ and $\epsilon\simeq m$,
this reduces to Eq. (4).
III.2 Derivation of the Langevin equation
Inserting Eq. (21) into Eq. (5), one obtains the
relativistic analogon of Eq. (9) as
$$\displaystyle\frac{\delta P(t)}{\tau}$$
$$\displaystyle\simeq$$
$$\displaystyle-\left[\frac{1}{\tau}\sum_{r=1}^{N}\frac{2}{1-{u}_{r}^{2}}\;\frac%
{\epsilon_{r}}{E+\epsilon_{r}}\;I_{r}(t,\tau)\right]P+$$
(22)
$$\displaystyle\qquad\frac{1}{\tau}\sum_{r=1}^{N}\frac{2}{1-{u}_{r}^{2}}\;\frac{%
E}{E+\epsilon_{r}}\,p_{r}\;I_{r}(t,\tau),$$
where ${u}_{r}\equiv u(p_{r},P)$ and $\epsilon_{r}\equiv\epsilon(p_{r})$. Formally, the collision
indicator $I_{r}(t,\tau)$ is still determined by Eqs. (6) and (7),
but differences arise due to the fact that we have to use $V=P/(M^{2}+P^{2})^{1/2}$ and a
relativistic bath distribution now.
Analogous to the non-relativistic case, we can identify the first term on the rhs.
of Eq. (22) as friction, and introduce an averaged friction coefficient by
$$\displaystyle\nu(P)$$
$$\displaystyle\equiv$$
$$\displaystyle\frac{1}{\tau}\sum_{r=1}^{N}\left\langle\frac{2}{1-u_{r}^{2}}\;%
\frac{\epsilon_{r}}{E+\epsilon_{r}}\;I_{r}(t,\tau)\right\rangle_{\mathrm{b}}$$
(23)
$$\displaystyle=$$
$$\displaystyle\frac{N}{\tau}\;\left\langle\frac{2}{1-u_{r}^{2}}\;\frac{\epsilon%
_{r}}{E+\epsilon_{r}}\;I_{r}(t,\tau)\right\rangle_{\mathrm{b}},$$
for some $r\in\{1,\ldots,N\}$. Next, applying a product approximation
similar to (8), we obtain
$$\displaystyle\nu(P)$$
$$\displaystyle\simeq$$
$$\displaystyle\frac{N}{\tau}\left\langle\frac{2}{1-{u}_{r}^{2}}\;\frac{\epsilon%
_{r}}{E+\epsilon_{r}}\right\rangle_{\mathrm{b}}\;\left\langle I_{r}(t,\tau)%
\right\rangle_{\mathrm{b}}$$
(24)
$$\displaystyle\overset{\eqref{e:Stokes}}{=}$$
$$\displaystyle n_{\mathrm{b}}\;C(P)\left\langle\frac{2}{1-{u}_{r}^{2}}\;\frac{%
\epsilon_{r}}{E+\epsilon_{r}}\right\rangle_{\mathrm{b}},$$
where $n_{\mathrm{b}}=N/L$ is the density of the heat bath particles, and
$C(P)$ is determined by Eq. (7b). Figure 1
shows the $P$-dependence of $\nu(P)/[n_{\mathrm{b}}C(P)]$ for the bath
distributions from Eq. (17b). This momentum dependence
is induced by the appearance of ${u}_{r}=u(p_{r},P)$ and $E=E(P)$ in the
expectation value on the rhs. of Eq. (24). Furthermore, the
shape of the one-particle collision coefficient $C(P)=\left\langle I_{r}(t,\tau)\right\rangle_{\mathrm{b}}L/\tau$
is depicted in Fig. 2. As one would intuitively expect, the
friction coefficient grows with the temperature $T$ of the heat bath
(at constant $P$) as well as with the absolute momentum of the Brownian particle (at constant $T$). In the non-relativistic limit case, where $u^{2}\ll 1$, $E\simeq M$ and $\epsilon\simeq m$, the relativistic friction coefficient $\nu(P)$ from Eq. (24) reduces to the non-relativistic result $\nu_{0}(P)$ from Eq. (11).
At this point, it might be worthwhile to emphasize once again
that product approximations of the form
$$\displaystyle\left\langle G(x_{r},p_{r})\;I_{r}(t,\tau)\right\rangle_{\mathrm{%
b}}\simeq\left\langle G(x_{r},p_{r})\right\rangle_{\mathrm{b}}\;\left\langle I%
_{r}(t,\tau)\right\rangle_{\mathrm{b}},$$
(25)
as employed in Eq. (8) and also in the first
line of Eq. (24), can in principle be
omitted by using the explicit representation (6)
of the collision indicator and the Eqs. (39) of the Appendix;
if one opts to avoid such approximations then the accuracy of the Langevin
model increases (note that this statement applies to the non-relativistic case, too).
However, in the following we shall continue to use Eq. (25)
in order to obtain a relativistic LE that is on an equal footing with the
non-relativistic LE (15).
For this purpose, we interpret the second term on the rhs. of
Eq. (22) as ‘noise’, defining
$$\displaystyle\chi(P,t)\equiv\frac{1}{\tau}\sum_{r=1}^{N}\frac{2}{1-u_{r}^{2}}%
\;\frac{E}{E+\epsilon_{r}}\,p_{r}\;I_{r}(t,\tau).$$
(26)
Averaging over the bath distribution $f_{\mathrm{b}}^{N}$, one finds for the mean value
$$\displaystyle\mu(P)$$
$$\displaystyle\equiv$$
$$\displaystyle\left\langle\chi(P,t)\right\rangle_{\mathrm{b}}$$
(27)
$$\displaystyle=$$
$$\displaystyle\frac{N}{\tau}\left\langle\frac{2}{1-u_{r}^{2}}\;\frac{E}{E+%
\epsilon_{r}}\,p_{r}\;I_{r}(t,\tau)\right\rangle_{\mathrm{b}}$$
$$\displaystyle\overset{\eqref{e:product_approx}}{\simeq}$$
$$\displaystyle\frac{N}{\tau}\left\langle\frac{2}{1-u_{r}^{2}}\;\frac{E}{E+%
\epsilon_{r}}\,p_{r}\right\rangle_{\mathrm{b}}\left\langle I_{r}(t,\tau)\right%
\rangle_{\mathrm{b}}$$
$$\displaystyle\simeq$$
$$\displaystyle n_{\mathrm{b}}\;C(P)\;\left\langle\frac{2}{1-u_{r}^{2}}\;\frac{E%
}{E+\epsilon_{r}}\,p_{r}\right\rangle_{\mathrm{b}}.$$
In contrast to the non-relativistic case, the mean value $\mu$ of the
relativistic Langevin force $\chi(t)\equiv\chi(P,t)$ depends on the momentum $P$ of
the Brownian particle. This can be attributed to the appearance
of $u_{r}^{2}=(P+p_{r})^{2}/(E+\epsilon_{r})^{2}$ in Eq. (27).
As shown in Fig. 3, the quantity $\mu(P)/[n_{\mathrm{b}}C(P)]$
is positive for $P>0$ and negative for $P<0$. Thus, on average,
the relativistic stochastic force tends to accelerate particles
in the direction of their motion, but this effect is compensated
by the increase of the friction coefficient $\nu(P)$ at high values of $P$, cf. Fig. 1.
Let us next take a closer look at the covariance function
$$\displaystyle\sigma_{ts}$$
$$\displaystyle\equiv$$
$$\displaystyle\left\langle\,\left[\chi(t)-\left\langle\chi(t)\right\rangle_{%
\mathrm{b}}\right]\,\left[\chi(s)-\left\langle\chi(s)\right\rangle_{\mathrm{b}%
}\right]\,\right\rangle_{\mathrm{b}}.$$
(28)
In the non-relativistic case, the stochastic force possesses a vanishing
mean value, $\left\langle\xi(t)\right\rangle_{\mathrm{b}}=0$. According to Eq. (27),
this is no longer the case for the relativistic noise force, $\left\langle\chi(t)\right\rangle_{\mathrm{b}}\neq 0$.
In order to explicitly calculate $\sigma_{ts}$ for the relativistic case, it is convenient to introduce the abbreviation
$$\kappa_{r}=\frac{2}{1-u_{r}^{2}}\;\frac{E}{E+\epsilon_{r}}\;p_{r}.$$
Assuming, as before, that collisions can be viewed as independent
events, the correlation function (28)
vanishes at non-equal times $t\neq s$, and we thus find
$$\displaystyle\sigma_{ts}$$
$$\displaystyle\overset{\eqref{e:mu}}{\simeq}$$
$$\displaystyle\delta_{ts}\left\langle\left[\frac{1}{\tau}\sum_{r=1}^{N}\kappa_{%
r}I_{r}(t,\tau)\right]^{2}-\mu^{2}(P)\right\rangle_{\mathrm{b}}$$
(29)
$$\displaystyle\overset{\eqref{e:product_approx}}{\simeq}$$
$$\displaystyle\frac{\delta_{ts}}{\tau^{2}}\biggl{\{}\sum_{r=1}^{N}\left\langle%
\kappa_{r}^{2}\right\rangle_{\mathrm{b}}\left\langle I_{r}(t,\tau)\right%
\rangle_{\mathrm{b}}-\tau^{2}\mu^{2}(P)$$
$$\displaystyle\;\sum_{r=1}^{N}\sum_{j\neq r}^{N}\left\langle\kappa_{r}\right%
\rangle_{\mathrm{b}}\left\langle I_{r}(t,\tau)\right\rangle_{\mathrm{b}}\left%
\langle\kappa_{j}\right\rangle_{\mathrm{b}}\left\langle I_{j}(t,\tau)\right%
\rangle_{\mathrm{b}}\biggr{\}}$$
$$\displaystyle=$$
$$\displaystyle\frac{\delta_{ts}}{\tau^{2}}\sum_{r=1}^{N}\biggl{\{}\left\langle%
\kappa_{r}^{2}\right\rangle_{\mathrm{b}}\left\langle I_{r}(t,\tau)\right%
\rangle_{\mathrm{b}}-\frac{\tau^{2}}{N}\mu^{2}(P)+$$
$$\displaystyle\quad\left\langle I_{r}(t,\tau)\right\rangle_{\mathrm{b}}\left%
\langle\kappa_{r}\right\rangle_{\mathrm{b}}\sum_{j\neq r}^{N}\left\langle%
\kappa_{j}\right\rangle_{\mathrm{b}}\left\langle I_{j}(t,\tau)\right\rangle_{%
\mathrm{b}}\biggr{\}}$$
$$\displaystyle\overset{\eqref{e:mu}}{=}$$
$$\displaystyle\frac{\delta_{ts}}{\tau^{2}}\sum_{r=1}^{N}\biggl{\{}\left\langle%
\kappa_{r}^{2}\right\rangle_{\mathrm{b}}\left\langle I_{r}(t,\tau)\right%
\rangle_{\mathrm{b}}-\frac{\tau^{2}}{N}\mu^{2}(P)+$$
$$\displaystyle\qquad\quad\frac{\tau}{N}\,\mu(P)\sum_{j\neq r}^{N}\,\frac{\tau}{%
N}\mu(P)\biggr{\}}.$$
From this, we obtain
$$\displaystyle\sigma_{ts}$$
$$\displaystyle{\simeq}$$
$$\displaystyle\frac{\delta_{ts}}{\tau}\,\frac{N}{\tau}\left\langle\kappa_{r}^{2%
}\right\rangle_{\mathrm{b}}\left\langle I_{r}(t,\tau)\right\rangle_{\mathrm{b}%
}-\frac{\delta_{ts}}{N}\;\mu^{2}(P)$$
$$\displaystyle\overset{\eqref{e:Stokes-a}}{=}$$
$$\displaystyle\frac{\delta_{ts}}{\tau}\,n_{\mathrm{b}}\,C(P)\;\left\langle%
\kappa_{r}^{2}\right\rangle_{\mathrm{b}}-\frac{\delta_{ts}}{N}\;\mu^{2}(P).$$
The last term vanishes, if we consider the thermodynamic limit (TDL)
of an infinite heat bath, i.e., $N,L\to\infty$ such that $n_{\mathrm{b}}=N/L=$constant.
Thus, reinserting the explicit expression for $\kappa_{r}$, we obtain in this limit
$$\displaystyle\sigma_{ts}\to\frac{\delta_{ts}}{\tau}\,n_{\mathrm{b}}\;C(P)\left%
\langle\left(\frac{2}{1-u_{r}^{2}}\;\frac{E}{E+\epsilon_{r}}\;p_{r}\right)^{2}%
\right\rangle_{\mathrm{b}}.$$
(31)
In principle, any higher correlation function can be calculated in the same manner.
It is also evident that the noise force is non-Gaussian,
because the relativistic bath distribution $f_{\mathrm{b}}(p_{r})$ that
determines the averages $\left\langle\;\cdot\;\right\rangle_{\mathrm{b}}$ – and, thus,
the noise correlations – is non-Gaussian.
Finally, by substituting the averaged friction coefficient
$$\displaystyle\nu(P)$$
$$\displaystyle=$$
$$\displaystyle n_{\mathrm{b}}\;C(P)\;\left\langle\frac{2}{1-{u}_{r}^{2}}\;\frac%
{\epsilon_{r}}{E+\epsilon_{r}}\right\rangle_{\mathrm{b}}$$
(32a)
for the square bracket term in
Eq. (22), imposing the TDL for the bath and
letting $$\tau\to 0$$ in Eq. (22), we obtain the
relativistic LE
$$\displaystyle\dot{P}=-\nu(P)\,P+\chi(t),$$
(32b)
where, in view of approximation (25), the non-Gaussian momentum-dependent noise force $$\chi(t)\equiv\chi(P,t)$$ is characterized by the mean
$$\displaystyle\mu(P)$$
$$\displaystyle\equiv$$
$$\displaystyle\left\langle\chi(t)\right\rangle_{\mathrm{b}}$$
(32c)
$$\displaystyle=$$
$$\displaystyle n_{\mathrm{b}}\;C(P)\;\left\langle\frac{2}{1-u_{r}^{2}}\;\frac{E%
}{E+\epsilon_{r}}\,p_{r}\right\rangle_{\mathrm{b}},$$
and the covariance
$$\displaystyle\sigma(t,s)$$
$$\displaystyle\equiv$$
$$\displaystyle\left\langle\,\left[\chi(t)-\left\langle\chi(t)\right\rangle_{%
\mathrm{b}}\right]\,\left[\chi(s)-\left\langle\chi(s)\right\rangle_{\mathrm{b}%
}\right]\,\right\rangle_{\mathrm{b}}$$
(32d)
$$\displaystyle=$$
$$\displaystyle 2\,D(P)\;\delta(t-s),$$
with the (momentum-space) diffusion coefficient given by
$$\displaystyle D(P)=\frac{n_{\mathrm{b}}}{2}\,C(P)\;\left\langle\left(\frac{2}{%
1-u_{r}^{2}}\;\frac{E}{E+\epsilon_{r}}\;p_{r}\right)^{2}\right\rangle_{\mathrm%
{b}}.$$
(32e)
In Fig. 4 the ratio $D(P)/[n_{\mathrm{b}}C(P)]$
is plotted for the same parameters as in Figs. 1 and
3. As it is evident from the diagrams, this quantity
increases with temperature $T$ and absolute momentum $P$ of the
Brownian particle.
IV Resume
We conclude the derivation of the relativistic LE with a set of
general remarks:
(i)
While deriving the relativistic LE (32), we made
use of the stationarity, independence, and homogeneity of the bath
distribution (17a); we did not, however, rely on
the specific properties of the marginal momentum PDF. Hence, the
above results hold true for arbitrary one-particle momentum
distributions $f_{\mathrm{b}}^{1}(p)$.
(ii)
In order to be able to use the LEs (32) derived above, one still needs to
calculate the mean collision rate $C(P)/L$, which is determined by
Eq. (7b); cf. Appendix. We also emphasize once
again that the approximation (25), leading to
the appearance of $C(P)$, can in principle be omitted (in the
non-relativistic as well as in the relativistic case). More precise
results for friction coefficients and noise correlations can then
be extracted from Eq. (39) in the Appendix.
(iii)
The stochastic force $\chi(t)$ in Eq. (32) is
$\delta$-correlated (memory-free), but non-Gaussian; i.e., in order to
completely specify the stochastic process one actually has to
determine all higher order correlation functions. This is
practically unfeasible. Therefore, in numerical studies and/or
practical applications, one could use a Gaussian
approximation (GA) of Eqs. (32), obtained in the following manner:
We rewrite Eq. (32b) equivalently as
$$\displaystyle\dot{P}=-\bar{\nu}(P)\,P+\sqrt{2D(P)}\,\bar{\zeta}(t),$$
(33a)
where
$$\displaystyle\bar{\nu}(P)\equiv\nu(P)-\frac{\mu(P)}{P},\qquad\bar{\zeta}(t)%
\equiv\frac{\chi(t)-\mu(P)}{\sqrt{2D(P)}}.$$
(33b)
Reminiscent of standardized Gaussian white noise, the effective noise force $$\bar{\zeta}(t)$$ is characterized by
$$\displaystyle\left\langle\bar{\zeta}(t)\right\rangle=0,\qquad\left\langle\bar{%
\zeta}(t)\bar{\zeta}(s)\right\rangle=\delta(t-s),$$
(33c)
but the higher moments are non-Gaussian. Accordingly, the GA is achieved by replacing $\bar{\zeta}(t)$ in Eq. (33a) with standardized momentum-independent Gaussian white noise $\zeta(t)$. The resulting stochastic differential equation is a standard LE with multiplicative Gaussian white noise. Hence, after having specified a discretization rule, one can easily write down the corresponding Fokker-Planck equation as well as the corresponding stationary distribution Klimontovich (1994).
The GA obtained this way neglects higher-order cumulants of the noise, so it cannot be expected that the ‘truncated’ LE yields exactly the same relaxation behavior and/or
the same stationary solution as the full relativistic LE (32) Hänggi (1982, 1980). Nevertheless, this approximation should provide useful estimates. In particular, if the stationary momentum distribution of the Brownian particle can be guessed by other arguments Dunkel and Hänggi (2006), then the GA can be made self-consistent with respect to this distribution by fixing a suitably generalized Einstein relation for the friction and noise coefficients. In this case it suffices to calculate, e.g., $\bar{\nu}(P)$, because the corresponding noise amplitude $\bar{D}(P)$ is then uniquely determined by the Einstein relation (i.e., by the stationary distribution).
Due to the multiplicative noise coupling, the results obtained from the GA will also depend on the choice of the discretization
rule Ito (1944, 1951); Fisk (1963); Stratonovich (1964, 1966); Klimontovich (1994); Hänggi (1978, 1980); Hänggi and Thomas (1982).
Loosely speaking, this discretization dilemma is the
price that one has to pay for mapping the large number of collisions
between $t$ and $t+\tau$ onto a single instant of time. Our
experience with the non-relativistic LE (cf. remarks at the end of
Sec. II.2) suggests that the ”transport” or ”kinetic” interpretation, corresponding to the post-point discretization rule Hänggi (1978, 1980); Hänggi and Thomas (1982); Klimontovich (1994), should be preferable in the relativistic case as well.
(iv)
In principle, it should be straightforward to generalize
the above approach to higher space dimensions, by expressing
the momentum vector after the collision, $\hat{\boldsymbol{P}}$,
in terms of the momenta before the collision, $\boldsymbol{P}$
and $\boldsymbol{p}$, analogous to Eq. (21). In
the $2d$ or $3d$-case, complications may arise mostly
due to the fact that one also has take into account the
corresponding collision angles and cross-sections (e.g.,
when determining the collision rates; cf. comments at the
end of the Appendix).
(v)
According to our above results, the previously proposed
‘relativistic’ LEs Debbasch et al. (1997); Zygadlo (2005); Dunkel and Hänggi (2005a, b)
should be viewed as approximations, which can be fruitful for
generating/simulating ensembles of relativistic particles in a
simple manner. It is also evident now why these earlier approaches
were intrinsically limited. Debbasch et al. Debbasch et al. (1997) have
postulated that the relativistic stochastic force in the rest-frame
of the bath is ordinary Gaussian white noise with a constant amplitude $D$,
whereas we in our prior work Dunkel and Hänggi (2005a, b) assumed
Gaussian noise in the (comoving) rest-frame of the Brownian particle.
As follows from the derivation presented here, neither of these assumptions
is accurate if one properly takes into account both the relativistic
conservation laws and the relativistic momentum distributions of the heat bath
particles. However, if we consider suitably chosen momentum-dependent
friction and diffusion coefficients, then the previously proposed
‘relativistic’ LEs Debbasch et al. (1997); Zygadlo (2005); Dunkel and Hänggi (2005a, b) become equivalent
to the Gaussian approximation of the relativistic LE (32).
Acknowledgements.
The authors would like to thank S. Hilbert and P. Talkner for helpful discussions.
References
Hänggi and Marchesoni (2005)
P. Hänggi and
F. Marchesoni,
Chaos 15,
026101 (2005).
Einstein (1905)
A. Einstein,
Ann. Phys. (Leipzig) 17,
549 (1905).
Einstein and von Smoluchowski (1999)
A. Einstein and
M. von Smoluchowski,
Untersuchungen über die Theorie der Brownschen
Bewegung/Abhandlungen über die Brownsche Bewegung und verwandte
Erscheinungen, vol. 199 (Harri
Deutsch, Frankfurt, 1999),
3rd ed.
Uhlenbeck and Ornstein (1930)
G. E. Uhlenbeck
and L. S.
Ornstein, Phys. Rev.
36, 823 (1930).
Chandrasekhar (1943)
S. Chandrasekhar,
Rev. Mod. Phys. 15,
1 (1943).
Wang and Uhlenbeck (1945)
M. C. Wang and
G. E. Uhlenbeck,
Rev. Mod. Phys. 17,
323 (1945).
Hänggi and Jung (1995)
P. Hänggi and
P. Jung,
Adv. Chem. Phys. 89,
239 (1995).
Frey and Kroy (2005)
E. Frey and
K. Kroy,
Ann. Phys. (Leipzig) 14,
20 (2005).
Hänggi et al. (2005)
P. Hänggi,
F. Marchesoni,
and F. Nori,
Ann. Phys. (Leipzig) 14,
51 (2005).
Lämmerzahl (2005)
C. Lämmerzahl,
Ann. Phys. (Leipzig) 14,
71 (2005).
Reimann and Hänggi (2002)
P. Reimann and
P. Hänggi,
Appl. Phys. A 75,
169 (2002).
Bartussek et al. (1996)
R. Bartussek,
P. Reimann, and
P. Hänggi,
Phys. Rev. Lett. 76,
1166 (1996).
Schay (1961)
G. Schay, Ph.D. thesis,
Princeton University (1961),
available through University Microfilms, Ann Arbor, Michigan,
https://wwwlib.umi.com.
Hakim (1965)
R. Hakim, J.
Math. Phys. 6, 1482
(1965).
Dudley (1965)
R. M. Dudley,
Arkiv för Matematik 6,
241 (1965).
Guerra and Ruggiero (1978)
F. Guerra and
P. Ruggiero,
Lett. Nouvo Cimento 23,
529 (1978).
Boyer (1979)
T. H. Boyer,
Phys. Rev. D 19,
1112 (1979).
Ben-Ya’acov (1981)
U. Ben-Ya’acov,
Phys. Rev. D 23,
1441 (1981).
Kostädt and Liu (2000)
P. Kostädt
and M. Liu,
Phys. Rev. D 62,
023003 (2000).
Rapoport (2005)
D. L. Rapoport,
Found. of Physics 35,
1205 (2005).
Dunkel and Hänggi (2005a)
J. Dunkel and
P. Hänggi,
Phys. Rev. E 71,
016124 (2005a).
Dunkel and Hänggi (2006)
J. Dunkel and
P. Hänggi,
Physica A (2006), in
press, doi:10.1016/j.physa.2006.07.013, arXiv:cond-mat/0606487.
Debbasch (2004)
F. Debbasch,
J. Math. Phys. 45,
2744 (2004).
Koide (2005)
T. Koide,
Phys. Rev. E 72,
026135 (2005).
Wolschin (2004)
G. Wolschin,
Phys. Rev. C 69,
024906 (2004).
Zygadlo (2005)
R. Zygadlo,
Phys. Lett. A 345,
323 (2005).
Dunkel and Hänggi (2005b)
J. Dunkel and
P. Hänggi,
Phys. Rev. E 72,
036106 (2005b).
Morato and Viola (1995)
L. M. Morato and
L. Viola, J.
Math. Phys. 36, 4691
(1995), erratum, ibid. 37, 4769
(1996).
Posilicano (1997)
A. Posilicano,
Lett. in Math. Phys. 42,
85 (1997).
Debbasch et al. (1997)
F. Debbasch,
K. Mallick, and
J. P. Rivet,
J. Stat. Phys 88,
945 (1997).
Fingerle (2006)
A. Fingerle
(2006), eprint arXiv:cond-mat/0601303.
Fa (2006)
K. S. Fa,
Braz. J. Phys. 36,
777 (2006).
Abdel-Aziz and Gavin (2004)
M. Abdel-Aziz
and S. Gavin,
Phys. Rev. C 70,
034905 (2004).
Abdel-Aziz and Gavin (2005)
M. Abdel-Aziz
and S. Gavin,
J. Phys. G: Nucl. Phys. 31,
77 (2005).
van Hees
et al. (2006a)
H. van Hees,
V. Greco, and
R. Rapp,
Phys. Rev. C 73,
034913 (2006a).
van Hees
et al. (2006b)
H. van Hees,
V. Greco, and
R. Rapp
(2006b), eprint arXiv:hep-ph/0601166.
Dieckmann et al. (2006)
M. E. Dieckmann,
L. Drury, and
P. K. Shukla,
New J. Phys. 8,
40 (2006).
Marti et al. (2006)
M. Marti,
R. A. Fonseca,
and L. O. Silva
(2006), ”A collision module for OSIRIS”,
P5.015, 32nd European Physical Society Conference on Plasma Physics, Rome.
Duviryuk et al. (2001)
A. Duviryuk,
A. Nazarenko,
and V. Tretyak,
Cond. Mat. Phys. 4,
5 (2001).
Lehmann (2006)
E. Lehmann, J.
Math. Phys. 47, 023303
(2006).
Wheeler and Feynman (1949)
J. A. Wheeler and
R. P. Feynman,
Rev. Mod. Phys. 21,
425 (1949).
Van Dam and Wigner (1965)
H. Van Dam and
E. P. Wigner,
Phys. Rev. 138,
B1576 (1965).
Komar (1978)
A. Komar,
Phys. Rev. D 18,
1887 (1978).
Pechukas (1991)
P. Pechukas, in
Large-Scale Molecular Systems, edited by
W. Gans,
A. Blumen, and
A. Amann
(Plenum, New York,
1991), NATO ASI B258, p. 123.
Tsonchev and Pechukas (2000)
S. Tsonchev and
P. Pechukas,
Phys. Rev. E 61,
6171 (2000).
Wiener (1923)
N. Wiener, J.
Math. Phys. 58, 131
(1923).
Bogolyubov (1945)
N. N. Bogolyubov,
Publ. Acad. Sci. Ukr. SSR Kiev pp.
115–137 (1945), in Russian.
Magalinskii (1959)
V. B. Magalinskii,
Sov. Phys. JETP 9,
1381 (1959).
Ford et al. (1965)
G. W. Ford,
M. Kac, and
P. Mazur, J.
Math. Phys. 6, 505
(1965).
Zwanzig (1973)
R. Zwanzig, J.
Stat. Phys. 9, 215
(1973).
Hänggi (1997)
P. Hänggi, in
Stochastic dynamics
(Springer, Heidelberg,
1997), vol. 484 of
Lecture Notes in Physics, pp.
15–22.
Hänggi and Ingold (2005)
P. Hänggi and
G. L. Ingold,
Chaos 15,
026105 (2005).
Hänggi et al. (1990)
P. Hänggi,
P. Talkner, and
M. Borkovec,
Rev. Mod. Phys. 62,
251 (1990).
Hänggi and Thomas (1982)
P. Hänggi and
H. Thomas,
Phys. Rep. 88,
207 (1982).
Van Kampen (2003)
N. G. Van Kampen,
Stochastic Processes in Physics and Chemistry
(North-Holland Personal Library,
Amsterdam, 2003).
Becker (1967)
R. Becker,
Theory of heat (Springer,
New York, 1967).
Ito (1944)
K. Ito, Proc.
Imp. Acad. Tokyo 20, 519
(1944).
Ito (1951)
K. Ito, Mem.
Amer. Mathem. Soc. 4, 51
(1951).
Fisk (1963)
D. Fisk, Ph.D
thesis, Michigan State University, Dept. of Statistics
(1963).
Stratonovich (1964)
R. L. Stratonovich,
Vestnik Moskov. Univ., Ser. I: Mat., Mekh.
1, 3 (1964).
Stratonovich (1966)
R. L. Stratonovich,
SIAM J. Control 4,
362 (1966).
Hänggi (1978)
P. Hänggi,
Helv. Phys. Acta 51,
183 (1978).
Hänggi (1980)
P. Hänggi,
Helv. Phys. Acta 53,
491 (1980).
Klimontovich (1994)
Y. L. Klimontovich,
Physics-Uspekhi 37,
737 (1994).
Jüttner (1911)
F. Jüttner,
Ann. Phys. (Leipzig) 34,
856 (1911).
Hänggi (1982)
P. Hänggi,
Phys. Rev. A 25,
1130 (1982).
Appendix A Calculation of the collision rate
We aim to derive an explicit expression for the expectation value
$\left\langle I_{r}(t,\tau)\right\rangle_{\mathrm{b}}$ in the limit $\tau\to 0$, as, e.g., required in Eqs. (10).
By definition, the function $I_{r}(t,\tau)\in\{0,1\}$ indicates whether
or not the Brownian particle has collided with the heat bath particle
$r$ during the time interval $[t,t+\tau]$. The positions of the
Brownian and heat bath particle at time $t$ are denoted by $X$ and $x_{r}$,
respectively. Ignoring the possibility of a collision, for small enough $\tau$,
the new positions at time $t+\tau$ would be given by
$$\displaystyle X^{\prime}=X+V\tau,\qquad x^{\prime}_{r}=x_{r}+v_{r}\tau,$$
(34)
where $V$ and $v_{r}$ are the velocities.
Then, the indicator function $I_{r}(t,\tau)$ can be explicitly represented as
$$\displaystyle I_{r}(t,\tau)$$
$$\displaystyle=$$
$$\displaystyle\Theta(X-x_{r})\;\Theta(x^{\prime}_{r}-X^{\prime})\;\Theta(v_{r}-%
V)+$$
(35)
$$\displaystyle\;\Theta(x_{r}-X)\;\Theta(X^{\prime}-x^{\prime}_{r})\;\Theta(V-v_%
{r}),$$
where $\Theta(x)$ is the Heaviside-function, defined by
$$\displaystyle\Theta(x)=\begin{cases}0,&x<0;\\
1/2,&x=0;\\
1,&x>0.\end{cases}$$
(36)
The first (second) summand in Eq. (35) refers
to the initial configuration, where the heat bath particle is
located at the left (right) side of the Brownian particle. Let
us list some properties of the collision indicator $I_{r}(t,\tau)$.
First we note that $I_{r}(t,\tau)$ is idempotent, i.e.,
$$\displaystyle I_{r}^{j}(t,\tau)=I_{r}(t,\tau)$$
(37a)
holds for $$j=1,2\ldots$$. Furthermore, for $$\tau\to 0$$, we have
$$\displaystyle I_{r}(t,0)=0.$$
(37b)
Accordingly, the Taylor-expansion at $$\tau=0$$ gives
$$\displaystyle I_{r}(t,\tau)\simeq\left[\frac{\partial I_{r}}{\partial\tau}(t,0%
)\right]\;\tau.$$
(37c)
In order to determine $$\left\langle\frac{\partial I_{r}}{\partial\tau}(t,0)\right\rangle_{\mathrm{b}}$$, we note that
$$\displaystyle\frac{\partial}{\partial\tau}\Theta(x^{\prime}_{r}-X^{\prime})%
\biggl{|}_{\tau=0}$$
$$\displaystyle=$$
$$\displaystyle\frac{\partial}{\partial\tau}\Theta(x_{r}-X+(v_{r}-V)\tau)\biggl{%
|}_{\tau=0}$$
$$\displaystyle=$$
$$\displaystyle(v_{r}-V)\;\delta(x_{r}-X+(v_{r}-V)\tau)\biggl{|}_{\tau=0}$$
$$\displaystyle=$$
$$\displaystyle(v_{r}-V)\;\delta(x_{r}-X),$$
and, analogously,
$$\displaystyle\frac{\partial}{\partial\tau}\Theta(X^{\prime}-x^{\prime}_{r})%
\biggl{|}_{\tau=0}$$
$$\displaystyle=$$
$$\displaystyle\frac{\partial}{\partial\tau}\Theta(X-x_{r}+(V-v_{r})\tau)\biggl{%
|}_{\tau=0}$$
$$\displaystyle=$$
$$\displaystyle(V-v_{r})\;\delta(X-x_{r}+(V-v_{r})\tau)\biggl{|}_{\tau=0}$$
$$\displaystyle=$$
$$\displaystyle(V-v_{r})\;\delta(X-x_{r}).$$
Hence, we find
$$\displaystyle\frac{\partial I_{r}}{\partial\tau}(t,0)$$
$$\displaystyle=$$
$$\displaystyle(v_{r}-V)\;\Theta(X-x_{r})\;\delta(x_{r}-X)\;\Theta(v_{r}-V)+$$
$$\displaystyle\;(V-v_{r})\;\Theta(x_{r}-X)\;\delta(X-x_{r})\;\Theta(V-v_{r}),$$
$$\displaystyle=$$
$$\displaystyle\Theta(0)\;(v_{r}-V)\;\delta(x_{r}-X)\;\Theta(v_{r}-V)+$$
$$\displaystyle\;\Theta(0)\;(V-v_{r})\;\delta(X-x_{r})\;\Theta(V-v_{r}),$$
and, with $$\Theta(0)=1/2$$, the useful result
$$\displaystyle\frac{\partial I_{r}}{\partial\tau}(t,0)$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{2}(v_{r}-V)\;\;\delta(x_{r}-X)\;\Theta(v_{r}-V)+$$
(37d)
$$\displaystyle\;\frac{1}{2}(V-v_{r})\;\delta(X-x_{r})\;\Theta(V-v_{r}).$$
Now let us consider a spatially homogeneous one-particle bath distribution of the form
$$\displaystyle\tilde{f}_{\mathrm{b}}^{1}(x_{r},v_{r})=\frac{1}{L}\;\tilde{f}_{%
\mathrm{b}}^{1}(v_{r})\begin{cases}1,&x_{r}\in[-L/2,L/2];\\
0,&x_{r}\not\in[-L/2,L/2],\end{cases}$$
(38)
and some function $\tilde{G}(x_{r},v_{r})$ such that the expectation
value $\left\langle\tilde{G}(x_{r},v_{r})\right\rangle_{\mathrm{b}}$ exists. We are
interested in expectations of the form
$$\displaystyle\left\langle\tilde{G}(x_{r},v_{r})\,I_{r}^{j}(t,\tau)\right%
\rangle_{\mathrm{b}}\overset{\eqref{e:idempotent}}{=}\left\langle\tilde{G}(x_{%
r},v_{r})\,I_{r}(t,\tau)\right\rangle_{\mathrm{b}},$$
as required for calculating the mean value of the stochastic
force and its higher correlation functions [e.g., compare first
line of Eq. (29)]. For small $$\tau$$, we may truncate
the Taylor expansion after the linear term, yielding
$$\displaystyle\left\langle\tilde{G}(x_{r},v_{r})\,I_{r}(t,\tau)\right\rangle_{%
\mathrm{b}}\simeq\left\langle\tilde{G}(x_{r},v_{r})\,\frac{\partial I_{r}}{%
\partial\tau}(t,0)\right\rangle_{\mathrm{b}}\;\tau.$$
(39a)
Making use of the result (37d), the mean value on the rhs. is given by
$$\displaystyle\left\langle\tilde{G}(x_{r},v_{r})\,\frac{\partial I_{r}}{%
\partial\tau}(t,0)\right\rangle_{\mathrm{b}}$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{2L}\int_{V}^{\infty}\!\!\!\mathrm{d}v_{r}\;(v_{r}-V)\;\times$$
$$\displaystyle\qquad\quad\tilde{G}(X,v_{r})\;\tilde{f}_{\mathrm{b}}^{1}(v_{r})\;+$$
$$\displaystyle\;\frac{1}{2L}\int^{V}_{-\infty}\!\!\!\mathrm{d}v_{r}\;(V-v_{r})\;\times$$
$$\displaystyle\qquad\quad\tilde{G}(X,v_{r})\;\tilde{f}_{\mathrm{b}}^{1}(v_{r}).$$
In particular, by choosing $\tilde{G}(x_{r},v_{r})\equiv 1$, we find the collision rate
$$\displaystyle\lim_{\tau\to 0}\frac{\left\langle I_{r}(t,\tau)\right\rangle_{%
\mathrm{b}}}{\tau}=\left\langle\frac{\partial I_{r}}{\partial\tau}(t,0)\right%
\rangle_{\mathrm{b}}=\frac{1}{L}\;\tilde{C}(V),$$
(40a)
where
$$\displaystyle\tilde{C}(V)$$
$$\displaystyle\equiv$$
$$\displaystyle\frac{1}{2}\int_{V}^{\infty}\!\!\!\mathrm{d}v_{r}\;(v_{r}-V)\;%
\tilde{f}_{\mathrm{b}}^{1}(v_{r})+$$
(40b)
$$\displaystyle\;\frac{1}{2}\int^{V}_{-\infty}\!\!\!\mathrm{d}v_{r}\;(V-v_{r})\;%
\tilde{f}_{\mathrm{b}}^{1}(v_{r}).$$
The following comments are in order:
(i)
The above derivation is valid for both non-relativistic and relativistic
heat bath distributions $\tilde{f}^{1}_{\mathrm{b}}(v_{r})$. Upon
identifying $C(P)\equiv\tilde{C}(V(P))$, where $P$ is the
momentum of the Brownian particle, we obtain the rigorous
justification for Eq. (7a). However, in the
non-relativistic case we have $V=P/M$, whereas in the
relativistic case $V=P/(M^{2}+P^{2})^{1/2}$. Additionally,
we note that the support interval of the relativistic
velocity distribution $\tilde{f}^{1}_{\mathrm{b}}(v_{r})$ is given by $[-c,c]$,
which determines the effective upper and lower integral boundaries in Eq. (40b).
(ii)
Given a certain bath distribution $\tilde{f}^{1}_{\mathrm{b}}(v_{r})$, the exact
result (39) allows for evaluating the quality of the
product approximations (8) and (25), respectively.
(iii)
The Stokes approximation corresponds to setting $V=0$ in Eq. (40b), yielding
$$\displaystyle\tilde{C}(0)=\frac{1}{2}\int_{-\infty}^{\infty}\mathrm{d}v_{r}\;|%
v_{r}|\;\tilde{f}_{\mathrm{b}}^{1}(v_{r}).$$
(41)
This shows that the Stokes approximation is useful for slow Brownian particles,
but inappropriate at high velocities.
(iv)
It is in principle possible to apply the same procedure to higher space
dimensions, but then the expression (35) for the indicator
unction has to be modified accordingly (e.g., by taking into account the
geometric shape of the Brownian particle). As a consequence, analytic
calculations will become much more difficult. |
Hipparcos parallaxes for $\eta$ Boo and $\kappa^{\scriptscriptstyle 2}$ Boo: two successes for
asteroseismology
Timothy R. Bedding
Hans Kjeldsen and Jørgen Christensen-Dalsgaard
School of Physics, University of Sydney 2006, Australia
Teoretisk Astrofysik Center, Danmarks Grundforskningsfond, and
Institut for Fysik og Astronomi. Aarhus Universitet, DK-8000 Aarhus C,
Denmark
1. Introduction
The release of the Hipparcos catalogue (?) provides an opportunity
to check results from asteroseismology. This has already been done for the
double-mode $\delta$ Scuti star SX Phe: ? (?) found excellent
agreement with the parallax derived from model calculations by
? (?). Here we show that Hipparcos parallaxes for two other
stars are also in good agreement with oscillation results.
2. Solar-like oscillations in $\eta$ Boo
$\eta$ Boo is a bright G0 subgiant and a good target for detecting solar-like
oscillations. We observed this star over six nights with the 2.5-m Nordic
Optical Telescope and, by monitoring equivalent widths of Balmer lines, we
detected oscillations with amplitudes at the expected level (?).
We measured frequencies for thirteen individual modes in the range
750–950 $\mu$Hz and determined the large frequency separation to be
$$\Delta\nu=40.3\pm 0.3\,\mbox{$\mu$Hz}.$$
The measured frequencies were subsequently shown to be consistent with
models by Christensen-Dalsgaard, Bedding & Kjeldsen
(? (?, hereafter CBK95) and also by ? (?). In
the light of a more accurate luminosity, we can revisit these results.
Note that an attempt by ? (?) to confirm oscillations in $\eta$ Boo
using Doppler measurements was not successful. Nevertheless, for the
present we continue to assume the reality of the detection.
The parameters of $\eta$ Boo are summarised in Table 1. Our
adopted luminosity in CBK95 was based on the best available parallax. The
more precise Hipparcos parallax, while being consistent with the
ground-based value, implies a slightly lower luminosity. Also note that in
CBK95 we calculated the luminosity using published estimates of the
effective temperature and angular diameter, which were based on the
infrared flux method. The calculation, also used by ? (?), was
indirect and here we prefer to use $V$-band photometry directly, as
explained in the Table. The new luminosity is accurate to 2.4%, an
improvement by a factor of three over the value adopted in CBK95.
This improved luminosity constrains the expected oscillation frequencies
for $\eta$ Boo. In Figures 1 and 2 we show the
location of $\eta$ Boo in the H-R diagram (these are similar to Figs. 1 and 2
of CBK95). The evolution tracks use the known metallicity of $\eta$ Boo ($Z=0.03$) and the solar value for the ratio of the mixing length to the
pressure scale height. Full details of the calculations are given in
CBK95.
The diagonal lines in Figure 2 join models of constant
$\Delta\nu$. The solid point indicates the 1.6-${M}_{\odot}$ model chosen by
CBK95 to have the frequency separation observed by ? (?). It is clear that the improved luminosity for $\eta$ Boo is in excellent
agreement with the observed frequency separation.
? (?) also computed models for $\eta$ Boo. By matching the observed
oscillation frequencies, they derived a mass of $1.55\pm 0.03\,\mbox{${M}_{\odot}$}$ and
predicted a parallax of $89.5\pm 0.5$ mas. Their parallax agrees with
the Hipparcos measurement, again giving strong support to the reality and
interpretation of the oscillation signal.
3. The $\delta$ Scuti star $\kappa^{\scriptscriptstyle 2}$ Boo
The Aarhus group has also studied the binary system consisting of $\kappa^{\scriptscriptstyle 1}$ Boo (HR 5328; $V=6.69$; F1 V) and $\kappa^{\scriptscriptstyle 2}$ Boo (HR 5329; $V=4.54$; A8 IV). The
brighter component $\kappa^{\scriptscriptstyle 2}$ Boo is a $\delta$ Scuti variable. Based on a model
fit to four observed frequencies, ? (?) derived a distance of
47.9 pc.
The Hipparcos parallax for this system is $21.03\pm 0.83$ mas, which
implies a distance of $47.6\pm 1.9$ pc. This is in excellent agreement
with the distance derived by ? (?. However, we note that their
frequency identifications did not allow for rotational splitting, despite
the fact that $\kappa^{\scriptscriptstyle 2}$ Boo is known to be a rapid rotator. Unless the accuracy
of the predicted parallax is coincidental, we appear to have confirmed
their assumption that the observed modes have $m=0$.
4. Conclusion
The results presented here for $\eta$ Boo and $\kappa^{\scriptscriptstyle 2}$ Boo, together with those for
SX Phe by ? (?), all rely on the same stellar evolution
calculations (?). The fact that asteroseismic analysis has been
successfully performed for three stars covering a range of masses and
evolutionary states is an important validation of the models.
Acknowledgments.
We are indepted to the Hipparcos group for making their wonderful catalogue
available on the Web. This work was supported by the Australian
Research Council, and by the Danish National Research Foundation through
its establishment of the Theoretical Astrophysics Center.
References
1
[ ]
Bell, R. A., Gustafsson, B., 1989, MNRAS 236, 653
2
3
[ ]
Blackwell, D. E., Lynas-Gray, A. E., 1994, A&A 282, 899
4
5
[ ]
Brown, T. M., Kennelly, E. J., Korzennik, S. G., et al., 1997, ApJ 475, 322
6
7
[ ]
Christensen-Dalsgaard, J., 1982, MNRAS 199, 735
8
9
[ ]
Christensen-Dalsgaard, J., Bedding, T. R., Kjeldsen, H., 1995, ApJ 443, L29
(CBK95)
10
11
[ ]
Frandsen, S., Jones, A., Kjeldsen, H., et al., 1995, A&A 301, 123
12
13
[ ]
Guenther, D. B., Demarque, P., 1996, ApJ 456, 798
14
15
[ ]
Harrington, R. S., Dahn, C. C., Kallarakal, V. V., et al., 1993, AJ 105, 1571
16
17
[ ]
Høg, E., Petersen, J. O., 1997, A&A 323, 827
18
19
[ ]
Kjeldsen, H., Bedding, T. R., Viskum, M., Frandsen, S., 1995, AJ 109, 1313
20
21
[ ]
Perryman, M. A. C., Lindegren, L., Kovalevsky, J., et al., 1997, A&A 323, L49
22
23
[ ]
Petersen, J. O., Christensen-Dalsgaard, J., 1996, A&A 312, 463
24 |
Rare $Z$ Decays
††thanks: Talk presented at RADCOR and Loops and Legs in Quantum Field
Theory 2002.
Gad Eilam${}^{\rm a}$
Based on work done in collaboration with D. Atwood,
S. Bar-Shalom and A. Soni [1].
Abstract
Motivated by the well known impact of rare decays of hadrons
and leptons on the evolution
of the Standard Model and on limits for new physics, as well
as by the proposal for Giga-$Z$ option at TESLA, we investigate
the rare decay $Z\to b{\bar{s}}$ in various extensions of the
Standard Model.
Rare $Z$ Decays
††thanks: Talk presented at RADCOR and Loops and Legs in Quantum Field
Theory 2002.
Gad Eilam${}^{\rm b}$
††thanks: Based on work done in collaboration with D. Atwood,
S. Bar-Shalom and A. Soni [1].
${}^{\rm a}$Department of Physics, Technion–Israel
Institute of Technology,
32000, Haifa, Israel
${}^{\rm b}$Department of Physics, Technion–Israel
Institute of Technology,
32000, Haifa, Israel
1 INTRODUCTION
The central role played by rare decays on our understanding of
elementary particle physics, is well known, where “rare”
stands here for Flavor Changing Neutral Currents (FCNC), which are
either small or practically vanishing in the SM. Some highlights:
are:
1. In $K$ physics: The first appearance of charm in loops from
which $m_{c}\approx 1.5$ GeV was predicted [2].
2. In $B$ physics: The importance of $b\to s\gamma$
in the SM and for extracting limits
on Beyond the SM (BSM) scenarios [3].
3. The top quark FCNC provide an excellent tool to
investigate various extensions of the SM [4].
4. The experimental upper limit of the decay $\mu\to e\gamma$
[5], places
severe limits on extensions of the SM.
In view of the above and prompted by the recent discussion of a
Giga-$Z$ option at TESLA [6] in which the center-of-mass energy will
be lowered to $M_{Z}$, producing more than $10^{9}$ $Z$ bosons (i.e.
$\sim 100$ times the number produced at LEP), one should
investigate rare decays of $Z$s. Now since the important subject
of rare leptonic $Z\to\ell_{I}{\bar{\ell}}_{J},~{}\ell_{I}\neq\ell_{J}$ decays,
for which the SM branching ratio is $\leq 10^{-54}$, was covered
by Illana [7], we concentrate here on purely hadronic FCNC
$Z\to d_{I}{\bar{d}}_{J},~{}d_{I}\neq d_{J}$ decays. In fact we only discuss
$Z\to b{\bar{s}}$ which in most models, including the SM, has the
largest branching ratio among hadronic FCNC $Z$ decays. Note however
that experimentally, the latter is practically inseparable from
the $b{\bar{d}}$ mode. Let us also note that when referring
to the $b{\bar{s}}$ mode, we actually mean $Z\to b{\bar{s}}+{\bar{b}}s$.
We note here that in the SM [8]
${\rm BR}({Z\to b\bar{s}})\approx 3\times 10^{-8}$.
In the following sections we will discuss two variants
of 2 Higgs Doublet Models (2HDM) and two of Supersymmetry (SUSY).
Of the latter the first one will be: SUSY with squark mixing,
while in the second one FCNC will result from SUSY with R Parity Violation
(denoted by RPV, or ${R\hskip-6.259606pt/}_{P}$).
As we will see, ${\rm BR}({Z\to b\bar{s}})$
can be either smaller, the same or above the SM with
a maximal value of ${\rm BR}({Z\to b\bar{s}})\approx 10^{-6}$.
Experimentally, the attention devoted to FCNC in hadronic $Z$ decays
at LEP and SLD has been close to nil.
The best upper limit is [9]
$\sum_{q=d,s}\rm{BR}(Z\to b{\bar{q}})\leq 1.8\times 10^{-3}~{}@~{}90\%~{}CL$.
This is a preliminary DELPHI limit (which will probably remain as
such forever…)
based on about $3.5\times 10^{6}$ hadronic decays. Experimentalists
who are privy to LEP and SLD data should be
encouraged to look in their data and improve the above limit.
Due to space limitations, the following discussion of various
BSM models and their predictions for ${\rm Br}({Z\to b\bar{s}})$, will be
sketchy. Many more details and a more complete set of references
can be found in [1]. In fact, almost each reference
should start with: “See e.g.$\dots$” and end with:
“$\dots$ and references therein.”
2 GENERIC CALCULATION
We start with a generic calculation of the diagrams which modify (at
one loop) the $Vd_{I}{\bar{d}}_{J}$ vertex, due to charged or neutral scalar,
as depicted in Fig. 1.
In our case $V=Z$, $d_{I}=b$
and ${\bar{d}}_{J}={\bar{s}}$.
The indices $i,j$ and $\alpha,\beta$ indicate
which fermions and scalars we are considering,
respectively.
The Feynman rules are:
$V_{\mu}f_{I}f_{J}:~{}~{}i\gamma_{\mu}\left(a_{L(Vf)}^{ij}L+a_{R(Vf)}^{ij}R\right)$
$V_{\mu}S_{\alpha}S_{\beta}:~{}~{}ig_{V}^{\alpha\beta}\left(p_{\alpha}-p_{\beta%
}\right)_{\mu}$
$S_{\alpha}d_{J}f_{i}:~{}~{}i\left(b_{L(\alpha)}^{ij}L+b_{R(\alpha)}^{ij}R\right)$,
where $L(R)=[1-(+)\gamma_{5}/]2$.
There are 4 one-loop amplitudes, each corresponding
to one of the 4 one-loop diagrams. Each amplitude is proportional
to $\epsilon_{\mu}(q)$ times
$\bar{u}(p_{b})\left[\gamma^{\mu}\left(A_{L}L+A_{R}R\right)+\left(B_{L}L+B_{R}R%
\right)p_{\mu}\right]v(p_{s}).$ $A_{L,R},~{}B_{L,R}$ are
momentum dependent form factors, calculable from the diagrams.
There are 4 per diagram, thus we have 16 form factors.
$A_{L}$ for diagram (1) is:
$A_{L}=-2\sum_{\alpha,\beta,i}g_{Z}^{\alpha\beta}b_{L(\alpha)}^{iI}b_{L(\beta)}%
^{iJ}C_{24}$,
and similarly for the other 15 form factors.
$C_{24}$ is one of the usual one-loop scalar functions [10] at
$m_{f_{i}}^{2},m_{S_{\alpha}}^{2},m_{S_{\beta}}^{2},m_{d_{I}}^{2},q^{2},m_{d_{J%
}}^{2}.$
Finally:
$$\displaystyle\Gamma(Z\to b\bar{s})=2\frac{N_{C}}{3}\left(\frac{1}{16\pi^{2}}%
\right)^{2}\frac{M_{Z}}{16\pi}\times\left[2\left(\mid A_{L}^{T}\mid^{2}\right.\right.$$
$$\displaystyle\left.\left.+\mid A_{R}^{T}\mid^{2}\right)+\frac{M_{Z}^{2}}{4}%
\left(\mid B_{L}^{T}\mid^{2}+\mid B_{R}^{T}\mid^{2}\right)\right],$$
where $A_{L}^{T}$ is the Total sum of $A_{L}$s from the 4
diagrams, and similarly for $A_{R}^{T}$, $B_{L}^{T}$ and $B_{R}^{T}$.
3 MODELS AND PREDICTIONS
The stage is now ready for identifying, for each
model, the relevant scalars $S_{\alpha}$, fermions $f_{i}$ and the
couplings $a,b~{}\rm{and}~{}g$ (with the appropriate indices),
as expressed in the Feynman rules above. Then, the route
for obtaining $\Gamma({Z\to b\bar{s}})$ using the generic equation in
the previous section is clear.
3.1 Two Higgs doublet models
In 2HDM with flavor diagonal couplings of the neutral Higgs
to down-quarks, the FCNC ${Z\to b\bar{s}}$ go
through the one-loop
diagrams in Fig. 1. The scalars are the charged Higgs bosons,
$S_{\alpha=1}=H^{+}$ and the fermions are $f_{i}=u_{i},~{}i=1,2,3$.
The couplings are:
$Z_{\mu}u_{i}\bar{u}_{j}$ is as in the SM (therefore only $i=j$ survives),
$Z_{\mu}H^{+}H^{-}$ is derived from the kinetic energy part of the
Lagrangian ${\cal L}$ and $H^{+}\bar{u}_{i}d_{j}$ is obtained from the
Yukawa part which, in common notation is [11]:
$$\displaystyle{\cal L}_{Y}$$
$$\displaystyle=$$
$$\displaystyle-\sum_{i,j}\bar{Q}_{L}^{i}\left[\left(U_{ij}^{1}\tilde{\Phi}_{1}+%
U_{ij}^{2}\tilde{\Phi}_{2}\right)u^{j}_{R}\right.$$
$$\displaystyle\left.+\left(D_{ij}^{1}\Phi_{1}+D_{ij}^{2}\Phi_{2}\right)d^{j}_{R%
}\right].$$
A choice of $U$ and $D$, which are
$3\times 3$ matrices in flavor space, leads to
a specific 2HDM. We now study two variants of 2HDM.
3.1.1 Two Higgs doublet model of type II
In this model, called 2HDMII, $U^{1}=D^{1}=0$, ${Z\to b\bar{s}}$ was considered before
[12]. Using realistic values in the $\tan{\beta}-m_{H^{+}}$ plane,
we obtain: $\rm{BR}({Z\to b\bar{s}})\lower 2.15pt\hbox{$\>\buildrel<\over{\sim}\>$}10^{-10}$, two orders of magnitude below
the SM.
3.1.2 Two Higgs doublet model “for top”
In this variant [13], named T2HDM, the top is rewarded for
its “fatness” by having its mass proportional to the large $v_{2}$,
while all other masses are proportional to $v_{1}$. It therefore makes
sense to consider here only $\tan{\beta}>>1$. Using T2HDM parameters
consistent with data we find: $\rm{BR}({Z\to b\bar{s}})\lower 2.15pt\hbox{$\>\buildrel<\over{\sim}\>$}10^{-8}$.
3.2 Supersymmetry with squark mixing
FCNC in SUSY can emanate from squark mixing in:
$$\displaystyle{\cal L}_{\rm{soft}}^{\rm{squark}}$$
$$\displaystyle=-\tilde{Q}_{i}^{\dagger}(M_{Q}^{2})_{ij}\tilde{Q}_{j}-\tilde{U}_%
{i}^{\dagger}(M_{U}^{2})_{ij}\tilde{U}_{j}$$
$$\displaystyle-$$
$$\displaystyle\!\!\!\!\!\!\!\!\tilde{D}_{i}^{\dagger}(M_{D}^{2})_{ij}\tilde{D}_%
{j}+A_{u}^{ij}\tilde{Q}_{i}H_{u}\tilde{U}_{j}+A_{d}^{ij}\tilde{Q}_{i}H_{d}%
\tilde{D}_{j}~{},$$
with the usual notation for the squark fields [1]
and where $i,j$ are generation indices. Furthermore,
$M_{U,D}^{2}=\left(\begin{array}[]{cc}(m_{\tilde{U},\tilde{D}}^{2})_{LL}&(m_{%
\tilde{U},\tilde{D}}^{2})_{LR}\\
(m_{\tilde{U},\tilde{D}}^{2})_{LR}^{\dagger}&(m_{\tilde{U},\tilde{D}}^{2})_{RR%
}\end{array}\right),$
where
$(m_{\tilde{U},\tilde{D}}^{2})_{LL}$ …
are $3\times 3$ matrices.
Under certain assumptions [14] and taking only
$\tilde{b}-\tilde{s}$ or $\tilde{t}-\tilde{c}$ mixing into account:
$$\displaystyle(m_{\tilde{U},\tilde{D}}^{2})_{LL,RR}=\pmatrix{1&0&0\cr 0&1&%
\delta^{U,D(23)}_{LL,RR}\cr 0&\delta^{U,D(32)}_{LL,RR}&1}m_{0}^{2}~{}.$$
The above $\delta$s represent squark mixing from non-diagonal
bilinears in $\cal L$. $m_{0}$ is a common squarks mass scale, obeying:
$m_{0}>>M_{Z}$. Also,
$\delta_{LR}$s will stand for squark mixing from non-diagonal
trilinears in $\cal L$ [1].
For them we adopt the Ansatz of [15],
leading to $\delta_{LR}\propto vA/m_{0}^{2}$,
where $A$ is a common trilinear soft breaking parameter for both up
and down squarks.
$M^{2}_{D,U}$ become $4\times 4$ matrices in the weak bases
$\Phi^{0}_{D,U}=(\tilde{s}_{L},\tilde{s}_{R},\tilde{b}_{L},\tilde{b}_{R}),~{}(%
\tilde{c}_{L},\tilde{c}_{R},\tilde{t}_{L},\tilde{t}_{R}).$
They are diagonalized to obtain the mass eigenstates
$\Phi_{D,U}=(\tilde{s}_{1},\tilde{s}_{2},\tilde{b}_{1},\tilde{b}_{2}),~{}(%
\tilde{c}_{1},\tilde{c}_{2},\tilde{t}_{1},\tilde{t}_{2}),$
with the help of $R_{U,D}$ which rotates
$\Phi$ to $\Phi^{0}$.
We can now describe two cases of squark mixing: $\tilde{b}-\tilde{s}$
and $\tilde{t}-\tilde{c}$ mixing.
3.2.1 ${\tilde{b}}-{\tilde{s}}$ mixing
The scalars here are $S_{\alpha}=\Phi_{D,\alpha},~{}\alpha=1,2,3,4$,
since $\tilde{b}-\tilde{s}$ admixture states run in the loops.
The gluon is the only fermion in the loops, thus $f_{i}={\tilde{g}}$. The $a$
couplings are $0$, since $Z{\tilde{g}}{\tilde{g}}=0$. In other words,
one diagram (out of the four generic diagrams) vanishes.
The $b$ and $g$ couplings are functions of elements of the rotation
matrix $R_{D}$ mentioned above [16]. Since the two $\delta_{LR}\lower 2.15pt\hbox{$\>\buildrel<\over{\sim}\>$}10^{-2}$ [17], we neglect them. For the other four
$\delta$s we assume a common value, i.e.
$\delta^{D(23)}_{LL}=\delta^{D(32)}_{LL}=\delta^{D(23)}_{RR}=\delta^{D(32)}_{RR%
}=\delta^{D}$, and vary
$0<\delta^{D}<1$.
The parameters needed for masses, mixing and
$\Gamma(Z\to b\bar{s})$ are:
$m_{0},~{}\mu,~{}A,~{}\tan\beta,~{}m_{\tilde{g}}$ and $\delta^{D}$.
We vary them subject to $m_{\rm squarks}>150$ GeV and
have plots of practically everything as a function of everything [1].
We find $\rm{BR}({Z\to b\bar{s}})\lower 2.15pt\hbox{$\>\buildrel<\over{\sim}\>$}10^{-6}$, where the
highest value is attained for $m_{\tilde{g}}$ and one $m_{\tilde{d}_{i}}$
$\approx$ the EW scale,
while $m_{\tilde{d}_{j}}$, $j\neq i$ $\approx$ few TeV.
Such splitting
requires “heavy” SUSY mass scale with soft breaking parameters,
which is consistent with the non-observability of SUSY particles so far.
3.2.2 ${\tilde{t}}-{\tilde{c}}$ mixing
In this scenario the
scalars are $S_{\alpha}=\Phi_{U,\alpha},~{}\alpha=1,2,3,4$, similarly
to the previous case, except for $D\to U$. Obviously,
$\tilde{t}-\tilde{c}$ admixture states run in the loops. The loop fermions
are the two charginos $f_{i}=\chi_{i},~{}i=1,2$, and all
four generic diagrams contribute to ${Z\to b\bar{s}}$. The Feynman rules
[16] involve elements of the rotation matrix $R_{U}$
mentioned above and the chargino mixing matrices.
At the end of the day, running with the parameters
over all values consistent with the data, and with $m_{\tilde{q}}>150$ GeV and $m_{\chi}>100$ GeV we obtain:
$\rm{BR}({Z\to b\bar{s}})\lower 2.15pt\hbox{$\>\buildrel<\over{\sim}\>$}10^{-8}$, which we could have anticipated since
BR(through $\tilde{t}-\tilde{c}$ mixing): BR(through $\tilde{b}-\tilde{s}$ mixing)
$\approx\alpha:\alpha_{s}$.
3.3 Supersymmetry with RPV
Since there is no sacred principle which guarantees R-parity
conservation, we assume
in this part of the talk that $R_{P}$ is violated. Then,
${R\hskip-6.259606pt/}_{P}$ terms in the SUSY superpotential $\cal W$
lead to FCNC. $\lambda$ terms (pure $\not{\hbox{\kern-4.0pt$L$}}$) in
$\cal W$ are irrelevant at the 1-loop level. In addition
we assume, for the pure $\not{\hbox{\kern-4.0pt$B$}}$ terms, that
$\lambda^{\prime\prime}<<\lambda^{\prime}$, and also neglect the
bilinear term in the ${R\hskip-6.259606pt/}_{P}$ part of $\cal W$.
Then:
${\cal W}_{RPV}=\epsilon_{ab}\lambda_{ijk}^{\prime}{\hat{L}}^{a}_{i}{\hat{Q}}^{%
b}_{j}{\hat{D}}_{k}^{c}.$
In addition, if
${R\hskip-6.259606pt/}_{P}$ is OK then the $R_{P}$ conserving soft SUSY breaking is
extended. We need only the bilinear:
$V_{RPV}=\epsilon_{ab}b_{i}{\tilde{L}}^{a}_{i}H_{u}^{b}$,
where
$\tilde{L}$, $H_{u}$ are the scalar components of the hatted $L$ and
$H_{u}$, respectively.
We therefore have two types of FCNC:
Type A: Trilinear-trilinear:
$\Gamma({Z\to b\bar{s}})\propto(\lambda^{\prime}\lambda^{\prime})^{2}$.
Type B: Trilinear-bilinear:
$\Gamma({Z\to b\bar{s}})\propto(b\lambda^{\prime})^{2}$.
3.3.1 Type A: Trilinear-trilinear terms
We further sub-divide type A contributions to 6 groups according to the
scalars and fermions running in the loops. For instance, in
type A1 the scalars are $S_{\alpha}={\tilde{e}}_{L,\alpha},~{}\alpha=1,2,3$ and the fermions are $f_{i}=u_{i},~{}i=1,2,3$.
The $a$ couplings are identical to their SM values, $b_{L}=0$ (for
all $i,j$ and $\alpha$), $b_{R(\alpha)}^{i,j}=-\lambda_{\alpha ij}^{\prime*}$ and $g_{Z}^{\alpha\beta}=-e(c^{2}_{W}-s^{2}_{W})\delta_{\alpha\beta}/2s_{W}c_{W}$. Unfortunately, going over all type A groups, taking into
account the available limits on $\lambda{{}^{\prime}}$s and on
the other relevant parameters, we obtain
for the trilinear-trilinear case: $\rm{BR}({Z\to b\bar{s}})\lower 2.15pt\hbox{$\>\buildrel<\over{\sim}\>$}10^{-10}$.
Our results are in agreement with the special cases in [18].
3.3.2 Type B: Trilinear-bilinear terms
In this case, a Higgs exchanged in the loop mixes with a slepton, through
$\epsilon_{ab}b_{3}{\tilde{L}}^{a}_{3}H_{u}^{b}$,
assuming that only $b_{3}\neq 0$.
We choose to work in the “no VEV” basis
$v_{3}=0$ in which:
$H_{u}\equiv\left(h_{u}^{+},~{}(\xi_{u}^{0}+v_{u}+i\phi_{u}^{0})/\sqrt{2}\right),$
$H_{d}\equiv\left((\xi_{d}^{0}+v_{d}+i\phi_{d}^{0})/\sqrt{2},~{}h_{d}^{-}\right),$
$\tilde{L}_{3}\equiv\left((\tilde{\nu}_{+}^{0}+i\tilde{\nu}_{-}^{0})/\sqrt{2}~{%
},\tilde{e}_{3}^{-}\right),$
where $\tilde{\nu}_{+}^{0}$, $\tilde{\nu}_{-}^{0}$,
$\tilde{e}_{3}^{-}$ are
SU(2) CP-even, CP-odd $\tau$-sneutrinos,
${\tilde{\tau}}_{L}$, respectively.
In the basis $\Phi_{C}^{0}=(h_{u}^{+},h_{d}^{+},\tilde{e}_{3}^{+})$
we wrote the mass matrix in the charged scalar sector,
in the basis $\Phi_{E}^{0}=(\xi_{d}^{0},\xi_{u}^{0},{\tilde{\nu}}^{0}_{+})$
we wrote the mass matrix in the CP-even neutral scalar sector, and
in the basis $\Phi_{O}^{0}=(\phi^{0}_{d},\phi^{0}_{u},{\tilde{\nu}}^{0}_{-})$
we wrote the mass matrix in the CP-odd neutral scalar sector.
The new charged scalar and CP-even and CP-odd neutral scalar
mass-eigenstates are obtained
by diagonalizing the above-mentioned matrices. They are:
$\Phi_{C}=\left(H^{+},G^{+},\tilde{\tau}^{+}\right),\Phi_{E}=\left(H,h,\tilde{%
\nu}_{+}^{\tau}\right)$, and
$\Phi_{O}=\left(A,G^{0},\tilde{\nu}_{-}^{\tau}\right)$.
In the limit $b_{3}\to 0$:
$H,h,A,H^{+}$ become the usual ones.
Rotating with the diagonalizing $R_{C,E,O}$ (for Charged, Even-CP,
Odd-CP) matrices, one goes from the $\Phi$s to the $\Phi^{0}$s.
All depends on the four parameters
$A^{0},~{}m_{s\nu}^{0}$ (the masses in the limit $b_{3}\to 0,$),
$b_{3}$ and $\tan\beta$.
Let us sub-divide type B into two types according to the
scalar and fermion in the loop:
Type B1: Here $S_{\alpha}=\Phi_{C,\alpha};~{}f_{i}=u_{i}$
with $\alpha=1,3;~{}i=1,2,3$.
The $a$ couplings are equal to their values in the SM.
The $b$ couplings include elements of the rotation
matrix (for the charged fields) $R_{C}$ and $\lambda^{\prime}$,
and $g_{Z}^{\alpha\beta}=-e\cot{2\theta_{W}}\delta_{\alpha\beta}.$
Type B2: In this case
$S_{\alpha}=\Phi_{E,\alpha}$ and $\Phi_{O,\beta};~{}f_{i}=d_{i}$ with $\alpha=1,2,3$, and $\beta=1,3;~{}i=1,2,3$.
This is the only case for which our generic form is insufficient.
This fact results from the appearance of two new diagrams proportional
to a scalar-vector-vector coupling ($ZZ\Phi_{E}$ in our case). The
other eight diagrams are special cases of the generic ones in Fig. 1.
Inserting parameters consistent with the data we found
that for type B: ${\rm Br}({Z\to b\bar{s}})\lower 2.15pt\hbox{$\>\buildrel<\over{\sim}\>$}10^{-6}$.
4 EXPERIMENTAL FEASIBILITY
Let us briefly comment about the
feasibility of observing
(or limiting) a signal of $BR(Z\to b\bar{s})\sim 10^{-6}$, at a Linear Collider producing $10^{9}$ Z-bosons.
Such a signal leads to
one $b$-jet
and one $q$-jet, where $q$ stands for quarks lighter than $b$.
The main background is from $Z\to b{\bar{b}}$. Using what,
we think, are realistic efficiencies we find that
a new physics signal $Z\to b\bar{s}$, with
a branching ratio of order $10^{-6}$, can reach beyond
the 3-sigma level [1].
We can also get a clue about how low one can go in the value
(or limit) of $BR(Z\to b\bar{s})$ with $10^{9}$ Z-bosons, from the fact that
the DELPHI preliminary result reached [9]
$BR(Z\to b\bar{s})<{\cal O}(10^{-3})$
with ${\cal O}(10^{6})$ $Z$-bosons.
Scaling this limit, especially with the expected advances in $b$-tagging
and identification of non-$b$ jets methods, an ${\cal O}(10^{-6})$ branching
ratios should be easily attained at a Giga-$Z$ factory.
One needs realistic simulations as feasibility studies for this important
rare $Z$ decay mode.
5 SUMMARY AND CONCLUSIONS
Our results are best summarized in Table 1 which shows the best
values for ${\rm Br}({Z\to b\bar{s}})$ in
extensions of the SM discussed in this talk. The
SM result is given for comparison. Note that we have not included
interference with the SM as each of the values “stands alone”.
In some cases such interference may increase the branching ratio
to $\sim 10^{-7}$.
We conclude that Giga-$Z$ experiments will have the opportunity to
place significant limits, or hopefully discover the scenario beyond
the SM, by searching for hadronic (and leptonic [7])
neutral current flavor changing transitions.
Acknowledgements:
I would like to thank my collaborators, especially Shaouly Bar-Shalom,
for teaching me so much. I would also like to express my appreciation to
the organizers of the meeting for a job well done. Thanks also
to members of the theory group in DESY (Hamburg) who gave me the peace
of mind I needed to prepare my talk. This research was supported in
part by the US-Israel Binational Science Foundation, by the Israel
Science Foundation and by the Fund for Promotion of Research at the
Technion.
References
[1]
D. Atwood, S. Bar-Shalom, G. Eilam and A. Soni,
Phys. Rev. D66 (2002) 093005.
[2]
M.K. Gaillard and B.W. Lee,
Phys. Rev. D10 (1974) 897.
[3]
E. Lunghi,
hep-ph/0210379.
[4]
B. Mele, hep-ph/0003064.
[5]
Y. Kuno and Y. Okada,
Rev. Mod. Phys. 73 (2001) 151
[6]
J.A. Aguilar-Saavedra et al.,
ECFA/DESY LC Physics Working Group, hep-ph/0106315.
[7]
J.I Illana, these proceedings.
[8]
G. Mann and T. Riemann,
Annalen Phys. 40 (1984) 334;
M. Clements et al., Phys. Rev. D27 (1983) 570;
V. Ganapathi et al., ibid. D27 1983) 579.
[9]
J. Fuster, F. Martinez-Vidal and P. Tortosa,
preprint DELPHI 99-81 CONF 268, June 1999.
The preprint can be downloaded
from: http://documents.cern.ch/cgi-bin/setlink?base=preprint&categ=cern&id
=open-99-393. SLD plans to improve this limit; see: S. Walston’s talk at
DPF2002.
[10]
G. Passarino and M. Veltman,
Nucl. Phys. B160 (1979) 151. See [1]
for our notation.
[11]
D. Atwood, S. Bar-Shalom, G. Eilam and A.Soni, Phys. Rep. 347 (2001) 1.
[12]
B. Grzadkowski, J.F. Gunion and P. Krawczyk,
Phys. Lett. B268 (1991) 106.
[13]
K. Kiers, A. Soni and G.-H. Wu,
Phys. Rev. D59 (1999) 096001, ibid.
D62 (2000) 116004;
G.-H. Wu and A. Soni, ibid.
D62 056005 (2000).
[14]
M. Misiak, S. Pokorski and J. Rosiek,
Adv. Ser. Direct. High Energy Phys. 15 (1998) 796.
[15]
J.L. Diaz-Cruz, H.-J. He and C.-P. Yuan,
Phys. Lett. 530 (2002) 179.
[16]
J. Rosiek,
Phys. Rev. D41 (1990) 3464, and hep-ph/9511250 (erratum).
See also [1].
[17]
T. Besmer, C. Greub and T. Hurth, Nucl. Phys. B609 (2001) 359.
[18]
M. Chemtob and G. Moreau,
Phys. Rev. D59 (1999) 116012. |
IFT–08–01
k-stabilization in brane models
M. Olechowski
Institute of Theoretical Physics,
University of Warsaw
ul. Hoża 69, PL–00–681 Warsaw, Poland
Abstract
Stabilization of inter–brane distance is analyzed in
5–dimensional models with higher–order scalar kinetic terms.
Equations of motion and boundary conditions for background
and for scalar perturbations are presented.
Conditions sufficient and (with one exception) necessary for
stability are derived and discussed. It is shown that it is possible
to construct stable brane configurations even without
scalar potentials and cosmological constants.
As a byproduct we identify a large class of non–standard
boundary conditions for which the Sturm–Liouville operator
is hermitian.
1 Introduction
Higher dimensional brane models belong to the most interesting
recent developments in the theory of fundamental interactions.
Many models have been proposed in which the space–time consists
of a 5–dimensional (5D) bulk ending at two 4–dimensional (4D)
branes. Usually this space–time has the structure of a warped
product of a maximally symmetric 4D space–time and the one
dimensional orbifold $S^{1}/\mathbb{Z}_{2}$ with the branes located at the
$\mathbb{Z}_{2}$ fixed points. The Standard Model fields may propagate only
on one of the branes called the visible one. Some other fields may
live on the second, hidden, brane. Of course, the gravity fields
can propagate in the whole 5D space–time.
Phenomenological features of such models depend on
fields and interactions other than that of the Standard Model,
on the warping, and on the distance between the branes.
This distance must be fixed in a stable way. Such stabilization
can not be achieved with only gravity propagating in the bulk.
A simple mechanism of fixing the inter–brane distance was
proposed by Goldberger and Wise [1].
The idea is to add a 5D scalar field with some
bulk and brane potentials. If the background value of that field
is not constant in the bulk, then the boundary conditions
(or in another words: equations of motion at the branes)
can be fulfilled only if the branes are located at appropriate
points in the 5th dimension.
It is not enough to have a background solution with some fixed
brane positions. It is necessary also that such a configuration
is stable against all possible small perturbations.
From the 4D point of view, the perturbations can be
describe in terms of Kaluza–Klein (KK) towers of states.
The lightest scalar KK state is usually called the radion
[2].
Tachyonic character of the radion indicates instability
of a given background. The problem of the radion mass, or
of the stability of the inter–brane distance, was investigated
by many authors [3, 4, 5, 6, 7].
Its relation to inflation was discussed in [8].
Quite general criteria for the stability were found in [7].
Generalization of such criteria for models with
the Gauss–Bonnet interactions was presented in [9].
In the present paper we will do the stability analysis for brane
models with non–standard kinetic terms for the scalar field.
Such non–standard kinetic terms appear for example in
string theory due to the $\alpha^{\prime}$– and the loop–corrections.
Very interesting models with generalized scalar kinetic terms
were investigated in the cosmological context. Kinetically driven
inflation, called the k–inflation, was introduced in
[10, 11].
Models of k–essence were proposed as another approach to
the cosmological constant problem [12, 13].
Causality in the context of generalized kinetic terms was
discussed by many authors (see e.g. [14] and
references therein).
There is a simple reason to expect that models with non–standard
scalar kinetic terms may be interesting for the inter–brane
distance stabilization. Their Lagrangians contain terms with
more complicated, than just quadratic,
dependence on the scalar derivatives.
The scalar derivative with respect to the 5th coordinate is
crucial for the stabilization mechanisms similar to that
of Goldberger and Wise. This is analogous to the situations
in cosmological models where the time derivative of the scalar
field is crucial. The problem of radion stabilization in
models with non–standard kinetic terms was addressed in
[15]
but unfortunately the authors used a method which in general
is not correct and obtained incorrect
results111
The authors of [15] integrated a Lagrangian with
fields replaced by their background values. They called the result
“the effective potential” and looked for the minima of such an
object. Of course, in general the potential integrated in a given
background is not equal to the correct effective potential.
It happens to be equal in some simple cases, but this must
be checked case by case by other methods, so the method
of integrating the potential is practically useless.
The authors of [15] claim e.g. that all models with
the standard kinetic terms are unstable, what is in clear conflict
with the results of many previous analyses
[3, 5, 7, 8, 9, 16].
.
In addition to the bulk (non-standard) kinetic terms
we will consider also analogous brane-localized ones.
Many 5D models with (standard) brane kinetic terms
for different bulk fields were proposed. Such localized
kinetic terms were investigated for: pure gravity
[17, 18, 19, 20],
gauge fields
[21, 22, 23, 24, 25, 26],
fermions [22, 24, 25]
and scalars [24, 27]
In section 2 we define our model and derive
background equations of motion and boundary conditions.
Analogous equations for the scalar perturbations are
presented in section 3.
In subsection 3.1 we show that
the spectrum of those perturbations is real.
We identify a large class
of boundary conditions for which the Sturm-Liouville
eigenvalue problem is
self-adjoint.
The stability
conditions are obtained in section 4. They
are discussed and compared to that in models with the standard
kinetic terms in section 5. Finally,
section 6 contains our conclusions.
2 Model and background
We consider 5D models compactified on the $S^{1}/\mathbb{Z}_{2}$ orbifold
with the standard gravitational interactions but with
non–standard kinetic terms for a scalar field $\Phi$.
Two 4D branes are localized at the $\mathbb{Z}_{2}$ fixed points $y=y_{i}$.
The action takes the form
$$\displaystyle S=\int{\rm d}^{4}x\,{\rm d}y\,\sqrt{-G}$$
$$\displaystyle\!\!\!\!\!\!\!\!\left\{\frac{1}{2\kappa^{2}}R-P(\Phi,X)-V(\Phi)\right.$$
(1)
$$\displaystyle\!\!\!\!\!\!\!\!\left.\,\,-\sum_{i=1}^{2}\tilde{\delta}(y-y_{i})%
\big{[}Q^{(i)}(\Phi,X)+U^{(i)}(\Phi)\big{]}\right\}\,,$$
where
$$X=\frac{1}{2}\left(\nabla\Phi\right)^{2}\,,$$
(2)
and $\tilde{\delta}$ is the normalized Dirac delta satisfying
$\int{\rm d}y\,\sqrt{-G}\,\tilde{\delta}(y-y_{i})=\sqrt{-G^{(i)}}$
with $G^{(i)}$ being the determinant of the metric induced
on the brane localized at $y=y_{i}$ (we chose $y_{1}<y_{2}$).
The bulk kinetic term is given by some function $P(\Phi,X)$
depending on the derivatives of $\Phi$ through the combination
$X$ and on the scalar field itself. We choose $P(\Phi,X)$ in
such a way that it vanishes for $X=0$. This way $V(\Phi)$
describes the whole scalar contribution to the action for
constant $\Phi$. In addition to the bulk interactions,
we consider brane localized
contributions to the scalar kinetic term and to the potential:
$Q^{(i)}(\Phi,X)$ and $U^{(i)}(\Phi)$,
respectively222
There are two kinds of brane kinetic terms
considered in the literature.
Some authors assume that such terms
involve derivatives with respect to all 5 coordinates
(e.g. [24]-[26])
while other assume that the derivative in the orbifold
direction is not present
(e.g. [18]-[23]).
We apply the former approach which seems to be natural
when treating thin branes as limits of thick ones.
Generalization of our results to the case of brane kinetic
terms $Q$ which do not depend on $\partial\Phi/\partial y$
is quite straightforward.
.
The terms in the action (1) containing the brane
localized kinetic functions $Q^{(i)}(\Phi,X)$ must
be treated with special care.
Let us discuss now in some detail the meaning of
an integral containing a product of $Q^{(i)}(\Phi,X)$ and
the Dirac delta. Writing explicitly the arguments in one
of such expressions we get
$$\int{\rm d}^{4}x\int{\rm d}y\,\sqrt{G(x,y)}\,\tilde{\delta}(y-y_{i})\,Q^{(i)}%
\!\left(\Phi(x,y),\frac{1}{2}\left(G^{55}(x,y){\Phi^{\prime}}^{2}(x,y)+\ldots%
\right)\right),$$
(3)
where prime denotes differentiation with respect to the
orbifold coordinate $y$ and,
in the second argument of $Q^{(i)}$, the ellipsis stand for
terms in $X$ with derivatives of $\Phi$ in directions
other than $y$.
In brane models the derivatives with respect to the orbifold
coordinate(s) are usually discontinuous at the brane positions.
The derivative $\Phi^{\prime}(x,y)$, being a $\mathbb{Z}_{2}$ odd function,
is exactly zero at $y=y_{i}$.
On the other hand, due to the brane sources, the limits
$\lim_{y\to y_{i}^{\pm}}\Phi^{\prime}(x,y)$ can be different from zero.
The square of the scalar field derivative, ${\Phi^{\prime}}^{2}(x,y)$,
is even under the $\mathbb{Z}_{2}$ symmetry, and can be written as
a product of ${\rm sgn}^{2}(y-y_{i})$ and a smooth function.
Usually ${\Phi^{\prime}}^{2}(x,y)$ is discontinuous at $y=y_{i}$ and,
strictly speaking, its integral with the Dirac delta localized
at $y_{i}$ is not well defined. All expressions of this kind
must be regularized. Physically,
such regularization corresponds to using a thick brane and
taking the limit of its thickness decreasing to zero.
Technically, one replaces
$\delta(y-y_{i})$ and ${\rm sgn}(y-y_{i})$ with some smooth functions
$\delta_{\varepsilon}(y-y_{i})$ and ${\rm sgn}_{\varepsilon}(y-y_{i})$ satisfying
the relation ${\rm sgn}^{\prime}_{\varepsilon}(y-y_{i})=2\delta_{\varepsilon}(y-y_{i})$ and
approaching the Dirac delta and the signum function, respectively,
when $\varepsilon\to 0$.
We calculate the integrals like (3) for regularized expressions
and at the end remove the regulator taking the limit $\varepsilon\to 0$.
Thus, we obtain for example
$$\int{\rm d}^{4}x\int_{y_{a}}^{y_{b}}{\rm d}y\,\sqrt{G}\,\tilde{\delta}(y-y_{i}%
){\Phi^{\prime}}^{2n}(x,y)=\int{\rm d}^{4}x\,\sqrt{G^{(i)}}\lim_{y\to y_{i}}%
\frac{{\Phi^{\prime}}^{2n}(x,y)}{2n+1}\,,$$
(4)
if $y_{a}<y_{i}<y_{b}$.
It is not necessary to specify the direction of the limit
in (4) because
${\Phi^{\prime}}^{2n}(x,y)$
is an even function of $(y-y_{i})$ for any integer $n$.
However, the limit itself is necessary because usually
${\Phi^{\prime}}^{2n}(x,y)$
is discontinuous at $y=y_{i}$.
In the down–stairs approach, one of the limits
of integration is equal to $y_{i}$ and the r.h.s. of the above
equation must be multiplied by $1/2$.
In this work we are interested in warped background solutions
with the flat 4D foliation described by the ansatz
$$\displaystyle{\rm d}s^{2}=a^{2}(y)\left(\eta_{\mu\nu}{\rm d}x^{\mu}{\rm d}x^{%
\nu}+{\rm d}y^{2}\right),$$
(5)
$$\displaystyle\Phi=\phi(y).$$
(6)
The bulk equations of motion for the system described by
action (1) and satisfying ansatz
(5–6)
are given by (we use units $\kappa=1$)
$$\displaystyle\left(P_{X}\phi^{\prime}\right)^{\prime}+3\frac{a^{\prime}}{a}P_{%
X}\phi^{\prime}-a^{2}\left(V_{\Phi}+P_{\Phi}\right)=0\,,$$
(7)
$$\displaystyle\frac{a^{\prime\prime}}{a}-2\left(\frac{a^{\prime}}{a}\right)^{2}%
+\frac{1}{3}P_{X}{\phi^{\prime}}^{2}=0\,,$$
(8)
$$\displaystyle 6\left(\frac{a^{\prime}}{a}\right)^{2}-P_{X}{\phi^{\prime}}^{2}+%
a^{2}\left(V+P\right)=0\,,$$
(9)
where the subscripts $X$ and $\Phi$
denote derivatives with respect to the arguments $X$ and $\Phi$,
respectively333
It is straightforward to generalize the equations of motion
to the case of any non–flat maximally symmetric 4D foliation
of the 5D background. For example, for the 4D dS space–time
characterized by the Hubble constant $H$, the left hand
sides of equations (8) and (9) should
be modified by adding $H^{2}$ and $-6H^{2}$, respectively.
.
The boundary conditions for the background can be obtained
from the full equations of motion resulting from (1)
with the brane terms taken into account. Integrating such
equations over an infinitesimal intervals containing the brane
positions $y_{i}$ one gets
$$\displaystyle\lim_{y\to y_{1}^{+}(y_{2}^{-})}a^{\prime}=$$
$$\displaystyle\!\!\!\!\!\!\!\!\left.\mp\frac{a^{2}}{6}\left(U^{(i)}+\int{\rm d}%
y\,\delta(y-y_{i})Q^{(i)}\right)\right|_{y=y_{1}(y_{2})}\,,$$
(10)
$$\displaystyle\lim_{y\to y_{1}^{+}(y_{2}^{-})}\left(P_{X}\,\phi^{\prime}\right)=$$
$$\displaystyle\!\!\!\!\!\!\!\!\left.\pm\frac{a}{2}\left(U^{(i)}_{\Phi}+\int{\rm
d%
}y\,\delta(y-y_{i})Q^{(i)}_{\Phi}\right)\right|_{y=y_{1}(y_{2})}\,,$$
(11)
where for $y_{1}<y_{2}$ the upper (lower) signs are to be taken
for $y=y_{1}$ ($y=y_{2}$). From now on we use the Dirac delta
distribution with the usual normalization $\int{\rm d}y\delta(y)=1$
(in the upstairs approach).
The above background bulk equations of motion
(7-9) and boundary conditions
(10-11) reduce to the know results
for the standard kinetic terms after substituting
$P=X$ and $Q^{(i)}=0$.
3 Scalar perturbations
Solving the bulk equations of motion (7-9)
and the boundary conditions (10-11) one can
find possible background configurations characterized by the warp
factor $a(y)$ and the scalar field $\phi(y)$. Not all such
background configurations are stable. To check the stability
one has to consider all possible small perturbations around a given
background. Instabilities occur if any of the perturbations
has a tachyonic character. In this paper we concentrate on the
scalar perturbations. From the 4D point of view
they form an infinite Kaluza–Klein tower of scalars.
The state with the lowest (4D) mass squared is called the radion.
The positivity of its mass squared is a necessary condition
for the stabilization of the inter–brane distance.
To find the radion mass we have to investigate the equations of
motion for the scalar perturbations around the background.
Using the generalized longitudinal gauge, the scalar perturbations
can be written in the following way
$$\displaystyle{\rm d}s^{2}=a^{2}\left[\left(1+2F_{1}\right)\left(\eta_{\mu\nu}{%
\rm d}x^{\mu}{\rm d}x^{\nu}\right)+\left(1+2F_{2}\right){\rm d}y^{2}\right],$$
(12)
$$\displaystyle\Phi=\phi+F_{3},$$
(13)
where $a$ and $\phi$ are background solutions depending only
on the 5–th coordinate $y$, while the perturbations
$F_{j}$ are arbitrary (but small) functions of all coordinates.
To find the masses of the KK modes of scalar perturbations it
is enough to consider equations of motion linear in $F_{j}$.
Contrary to the background equations of motion, for the
perturbations we obtain non–trivial off–diagonal
Einstein equations
$$\displaystyle 2F_{1}+F_{2}=0,$$
(14)
$$\displaystyle\left(a^{2}F_{1}\right)^{\prime}+\frac{1}{3}a^{2}P_{X}\phi^{%
\prime}F_{3}=0.$$
(15)
They have to be fulfilled in order to stay in the
longitudinal gauge.
The diagonal Einstein equations, combined with the background
equations of motion (7–9),
give the third equation for the scalar perturbations:
$$\displaystyle\Box F_{1}+4\frac{a^{\prime}}{a}F_{1}^{\prime}-4\left(\frac{a^{%
\prime}}{a}\right)^{2}F_{2}+\frac{a^{\prime}}{a}\left(P_{X}-\frac{2}{3}XP_{XX}%
\right)\phi^{\prime}F_{3}$$
$$\displaystyle+\frac{1}{3}\left(P_{X}+2XP_{XX}\right)\left[{\phi^{\prime}}^{2}F%
_{2}+\phi^{\prime\prime}F_{3}-\phi^{\prime}F_{3}^{\prime}\right]=0\,,$$
(16)
where $\Box$ is the 4–dimensional D’Alembertian. The part of
the boundary conditions linear in the scalar
perturbations are quite complicated and reads
$$\displaystyle\!\!\!\!\!\!\!\!\pm 2\lim_{y\to y_{i}^{\pm}}\left[\left(P_{X}+2XP%
_{XX}\right)F_{3}^{\prime}\right]$$
$$\displaystyle\!\!\!\!\!\!\!\!+\int_{y_{i}}\phi^{\prime\prime}\left[\left(P_{%
\Phi X}+2XP_{\Phi XX}\right)F_{3}-\left(P_{X}+8XP_{XX}+4X^{2}P_{XXX}\right)F_{%
2}\right]$$
$$\displaystyle\!\!\!\!\!\!\!\!\qquad\qquad\qquad\qquad=\left.\left[aF_{3}\left(%
U^{(i)}_{\Phi\Phi}+\int_{y_{i}}\delta_{i}Q^{(i)}_{\Phi\Phi}\right)-\frac{\Box F%
_{3}}{a}\int_{y_{i}}\delta_{i}Q^{(i)}_{X}\right]\right|_{y=y_{i}},$$
(17)
where $\delta_{i}=\delta(y-y_{i})$. The subscript $y_{i}$ at
the integrals indicates that
the range of integration
is an infinitesimal interval containing $y_{i}$.
The off–diagonal Einstein equations (14)
and (15)
can be used to express two of the perturbations introduced
in the ansatz (12-13) in terms
of the third one. It is convenient to eliminate $F_{2}$ and $F_{3}$
and to use the product $a^{2}F_{1}$ as an independent perturbation.
We expand it in the 4D modes as
$$a^{2}(y)F_{1}(t,\vec{x},y)=\sum_{m^{2}}K_{m^{2}}(y)\left[\int{\rm d}^{3}kf_{(m%
^{2},k)}(t)e^{i\vec{k}\vec{x}}\right],$$
(18)
and substitute to eqs. (16) and (17).
Then, the 4D part of the bulk equation (16) takes the
usual form
$$\ddot{f}_{(m^{2},k)}+\left(k^{2}+m^{2}\right)f_{(m^{2},k)}=0.$$
(19)
The equation for the “shape” $K_{m^{2}}(y)$ of the KK mode
with mass squared equal $m^{2}$ can be written as the Sturm–Liouville
equation
$$-\left(pK_{m^{2}}^{\prime}\right)^{\prime}+qK_{m^{2}}=m^{2}rK_{m^{2}}\,,$$
(20)
where $p$, $q$ and $r$ are the following functions depending on the
background
$$p=\frac{3}{2aP_{X}{\phi^{\prime}}^{2}}\,,\qquad q=\frac{1}{a}\,,\qquad r=\frac%
{3}{2a\left(P_{X}+2XP_{XX}\right){\phi^{\prime}}^{2}}=c_{s}^{2}p\,.$$
(21)
In the last equality we have introduced a local ($y$–dependent)
“speed of sound”
$$c_{s}^{2}=\frac{P_{X}}{P_{X}+2XP_{XX}}\,.$$
(22)
The boundary condition (17) in terms of $K_{m^{2}}$
takes the form
$$\left.\left[\left(b_{i}-c_{i}m^{2}\right)\frac{\partial}{\partial n}K_{m^{2}}-%
m^{2}P_{X}K_{m^{2}}\right]\right|_{y_{i}^{\pm}}=0\,,$$
(23)
where from now on $y_{i}^{\pm}$ stands
for $y_{1}^{+}$ or $y_{2}^{-}$. The corresponding limits have
to be taken for quantities discontinuous on the branes.
The $\partial/\partial n$ differentiation is in
the direction of the outer normal at the boundary, i.e.
$(-{\rm d}/{\rm d}y)$ at $y_{1}$ and $(+{\rm d}/{\rm d}y)$ at $y_{2}$.
Quantities $b_{i}$ and $c_{i}$ are the following functions of
the background solution and the brane interactions
$$\displaystyle b_{i}=$$
$$\displaystyle\!\!\!\!\!\!\!\!\frac{1}{2}\left[\left.aU^{(i)}_{\Phi\Phi}\right|%
_{y=y_{i}}+a\int_{y_{i}}\delta_{i}Q^{(i)}_{\Phi\Phi}-\int_{y_{i}}\phi^{\prime%
\prime}\left(P_{\Phi X}+2XP_{\Phi XX}\right)\right]$$
(24)
$$\displaystyle\!\!\!\!\!\!\!\!\mp\lim_{y\to y_{i}^{\pm}}\left(P_{X}+2XP_{XX}%
\right)\left(\frac{\phi^{\prime\prime}}{\phi^{\prime}}-\frac{a^{\prime}}{a}%
\right),$$
$$\displaystyle c_{i}=$$
$$\displaystyle\!\!\!\!\!\!\!\!\frac{1}{2a}\int_{y_{i}}\delta_{i}Q_{X}^{(i)}\,.$$
(25)
All integrals in (24) and (25)
should be calculated with the same
regularization as that used in (4).
The square of the radion mass is given by the lowest eigenvalue
of the equation of motion (20) satisfying the boundary
conditions (23). Of course, in general it is not possible
to find the spectrum of the system (20)-(25)
by solving it explicitly. To get some information about the
smallest eigenvalue we will use methods analogous to those
developed for a similar problem in
[9] (where the corresponding boundary conditions
have a form of (23) with $c_{i}=0$).
But first one has to check whether the differential equation
(20) together with the boundary conditions
(23) constitute a self-adjoint system.
This is a non trivial problem because conditions (23)
are unusual and quite complicated.
The eigenvalue $m^{2}$ of the equation of motion (20)
appears multiplying both $K_{m^{2}}$ and its normal derivative.
In the next subsection we will show that our eigenvalue problem
is self-adjoint with boundary conditions
even more general than (23).
3.1 Self-adjoint eigenvalue problem
Let us consider a differential eigenvalue problem
$${\cal O}v=\lambda v$$
(26)
for the operator ${\cal O}$ of the Sturm-Liouville type
$${\cal O}v=\frac{1}{r}\left[-\left(pv^{\prime}\right)^{\prime}+qv\right]\,.$$
(27)
The boundary conditions on the interval $(y_{1},y_{2})$
have the form
$$\left.\left[\sigma_{1}^{(i)}v+\sigma_{2}^{(i)}v^{\prime}+\sigma_{3}^{(i)}\left%
({\cal O}v\right)+\sigma_{4}^{(i)}\left({\cal O}v\right)^{\prime}\right]\right%
|_{y=y_{i}}=0\,,$$
(28)
where $\sigma^{(i)}_{j}$ are some constants. The spectrum of our
eigenvalue problem is real if ${\cal O}$ is hermitian.
In order to prove this one has to find such a scalar product
$\left(\cdot,\cdot\right)$ for which
$$\left(v,{\cal O}u\right)=\left({\cal O}v,u\right)\,.$$
(29)
The standard boundary conditions discussed in many mathematical
textbooks have the form of (28) with
$\sigma_{3}^{(i)}=\sigma_{4}^{(i)}=0$.
In such a case, ${\cal O}$ is hermitian in the scalar product
$\left(f,g\right)=\int_{y_{1}}^{y_{2}}rfg$ (for simplicity we consider
real functions $f$ and $g$). Let us generalize this scalar product
by adding some boundary
terms444
A simple example of a non-standard scalar product was discussed
for example in [27]. It was calculated for a canonical
kinetic term localized on a brane in a flat background.
In our notation this corresponds to
$p=1$, $q=0$, $r=1$, $\sigma_{1}=0$, $\sigma_{4}=0$.
$$\left(f,g\right)=\int_{y_{1}}^{y_{2}}rfg+\left.\left[\rho_{1}^{(i)}fg+\rho_{2}%
^{(i)}(fg)^{\prime}+\rho_{3}^{(i)}f^{\prime}g^{\prime}\right]\right|_{y_{1}}^{%
y_{2}}\,,$$
(30)
with yet unspecified constants $\rho^{(i)}_{j}$.
For this scalar product we calculate
$$\displaystyle(v,{\cal O}u)$$
$$\displaystyle\!\!\!\!\!\!\!\!-({\cal O}v,u)=\left\{p\left[uv^{\prime}-vu^{%
\prime}\right]+\rho_{1}^{(i)}\left[v({\cal O}u)-({\cal O}v)u\right]\right.$$
(31)
$$\displaystyle+\rho_{2}^{(i)}\left[\left(v({\cal O}u)\right)^{\prime}-\left(({%
\cal O}v)u\right)^{\prime}\right]\left.\left.+\rho_{3}^{(i)}\left[v^{\prime}({%
\cal O}u)^{\prime}-({\cal O}v)^{\prime}u^{\prime}\right]\right\}\right|_{y_{1}%
}^{y_{2}}.$$
Introducing three additional constants
$\rho_{4}^{(i)}$, $\rho_{5}^{(i)}$, $p_{1}^{(i)}$,
at each boundary,
we can rewrite the above equation in the following form
$$\displaystyle(v,{\cal O}u)-({\cal O}v,u)=$$
$$\displaystyle\!\!\!\!\!\!\!\!\left\{v\left[\rho_{4}^{(i)}u-p_{1}^{(i)}u^{%
\prime}+\rho_{1}^{(i)}({\cal O}u)+\rho_{2}^{(i)}({\cal O}u)^{\prime}\right]\right.$$
(32)
$$\displaystyle\!\!\!\!\!\!\!\!-u\left[\rho_{4}^{(i)}v-p_{1}^{(i)}v^{\prime}+%
\rho_{1}^{(i)}({\cal O}v)+\rho_{2}^{(i)}({\cal O}v)^{\prime}\right]$$
$$\displaystyle\!\!\!\!\!\!\!\!+v^{\prime}\left[p_{2}^{(i)}u+\rho_{5}^{(i)}u^{%
\prime}+\rho_{2}^{(i)}({\cal O}u)+\rho_{3}^{(i)}({\cal O}u)^{\prime}\right]$$
$$\displaystyle\!\!\!\!\!\!\!\!-\left.\left.\!\!u^{\prime}\left[p_{2}^{(i)}v+%
\rho_{5}^{(i)}v^{\prime}+\rho_{2}^{(i)}({\cal O}v)+\rho_{3}^{(i)}({\cal O}v)^{%
\prime}\right]\right\}\right|_{y_{1}}^{y_{2}}$$
where $p_{2}^{(i)}=p(y_{i})-p_{1}^{(i)}$.
Our operator ${\cal O}$ is hermitian if the r.h.s. of the above
equation vanishes for all $v$ and $u$ fulfilling
the boundary conditions (28).
This is the case when each square bracket in (32)
is proportional the square bracket in (28):
$$\displaystyle\rho_{4}^{(i)}=n_{1}^{(i)}\sigma_{1}^{(i)}\,,\quad-p_{1}^{(i)}=n_%
{1}^{(i)}\sigma_{2}^{(i)}\,,\quad\rho_{1}^{(i)}=n_{1}^{(i)}\sigma_{3}^{(i)}\,,%
\quad\rho_{2}^{(i)}=n_{1}^{(i)}\sigma_{4}^{(i)}\,,$$
(33)
$$\displaystyle p_{2}^{(i)}=n_{2}^{(i)}\sigma_{1}^{(i)}\,,\quad\rho_{5}^{(i)}=n_%
{2}^{(i)}\sigma_{2}^{(i)}\,,\quad\rho_{2}^{(i)}=n_{2}^{(i)}\sigma_{3}^{(i)}\,,%
\quad\rho_{3}^{(i)}=n_{2}^{(i)}\sigma_{4}^{(i)}\,.$$
(34)
For generic values of $\sigma_{j}^{(i)}$ this set of
linear equations can be easily solved. At each
boundary there are 8 equations and 8 independent constants:
$\rho_{1}^{(i)}$, $\rho_{2}^{(i)}$, $\rho_{3}^{(i)}$, $\rho_{4}^{(i)}$,
$\rho_{5}^{(i)}$, $n_{1}^{(i)}$, $n_{2}^{(i)}$, $p_{1}^{(i)}$.
In fact we are interested only in those three,
$\rho_{1}^{(i)}$, $\rho_{2}^{(i)}$, $\rho_{3}^{(i)}$,
which enter the definition of the scalar product (30).
The solution
reads
$$\displaystyle\left(f,g\right)=\int_{y_{1}}^{y_{2}}rfg\left.+\left[p\,\frac{%
\left(\sigma_{3}^{(i)}\right)^{2}fg+\sigma_{3}^{(i)}\sigma_{4}^{(i)}(fg)^{%
\prime}+\left(\sigma_{4}^{(i)}\right)^{2}f^{\prime}g^{\prime}}{\sigma_{1}^{(i)%
}\sigma_{4}^{(i)}-\sigma_{2}^{(i)}\sigma_{3}^{(i)}}\right]\right|_{y_{1}}^{y_{%
2}}\!\!.$$
(35)
We have shown that the eigenvalue problem (26) with
the boundary conditions (28) is self-adjoint.
Thus, all its eigenvalues $\lambda$ are real and the eigenfunctions
corresponding to different $\lambda$ are orthogonal in the
scalar product (35).
Let us now use the above result for our k-stabilization mechanism.
The boundary conditions (23) have the form of
(28) with
$$\sigma_{1}^{(i)}=0\,,\quad\sigma_{2}^{(i)}=(-1)^{i}b_{i}\,,\quad\sigma_{3}^{(i%
)}=-P_{X}(y_{i})\,,\quad\sigma_{4}^{(i)}=-(-1)^{i}c_{i}\,,$$
(36)
with no summation over $i$.
The factors of $(-1)^{i}$ appear because the outer normal
derivative $\partial/\partial n$ was used in (23).
Substituting (36) into (35) we obtain
the following scalar product appropriate to show that the eigenvalue
problem (20), (23) is self-adjoint:
$$\left(f,g\right)=\int_{y_{1}}^{y_{2}}rfg+{\sum_{i=1,2}}^{\prime}\left.\left[p%
\,\frac{P_{X}^{2}fg+P_{X}c_{i}\frac{\partial}{\partial n}(fg)+c_{i}^{2}\frac{%
\partial}{\partial n}f\frac{\partial}{\partial n}g}{P_{X}b_{i}}\right]\right|_%
{y_{i}}.$$
(37)
The prime at the sum symbol denotes that
the boundary contributions should be taken only for those
boundaries at which $P_{X}b_{i}\neq 0$. The reason is that for
$b_{i}=0$ and/or $P_{X}=0$ the boundary condition (23)
reduces to the standard one for which
$\left(f,g\right)=\int rfg$ without any boundary
terms (at that boundary).
4 Stability conditions
The spectrum of the scalar perturbations in a given background
is given by the eigenvalues of the Strum–Liouville equation
(20) with the boundary conditions (23)
at the branes.
In the previous subsection we have shown that
this spectrum is real.
The most interesting for us is the lowest
eigenvalue which we identify with the square of the radion mass.
The inter–brane distance is stable only if this mass squared
is positive. In this section we will find conditions sufficient
for such stability. We will show also when the radion is massless
and identify some classes of backgrounds which are unstable.
First we check whether there is a massless mode in the KK tower
of the scalar perturbations. In such a case, the
boundary condition (23) at the first brane
reduces, for nonzero $b_{1}$, to $K^{\prime}_{0}(y_{1}^{+})=0$
(the case with vanishing $b_{1}$ will be considered later).
For $m^{2}=0$, the solution of the bulk equation of motion
(20), satisfying
the boundary condition at $y=y_{1}$ and normalized to
$K_{0}(y_{1})=1$, can be written in quite a simple form
$$K_{0}(y)=\frac{a^{2}(y)}{a^{2}(y_{1})}-\frac{2a^{\prime}(y)}{a^{2}(y)a^{2}(y_{%
1})}\int_{y_{1}}^{y}{\rm d}{\tilde{y}}\,a^{3}({\tilde{y}})\,.$$
(38)
Using the background equation of motion (8),
the derivative of the above solution simplifies to
$$K^{\prime}_{0}(y)=\frac{2P_{X}(y){\phi^{\prime}}^{2}(y)}{3a(y)a^{2}(y_{1})}%
\int_{y_{1}}^{y}{\rm d}{\tilde{y}}\,a^{3}({\tilde{y}})\,.$$
(39)
The boundary condition at the second brane reads
$b_{2}K^{\prime}(y_{2}^{-})=0$. The integral in eq.
(39) is strictly positive, so this condition
is fulfilled only when
the product $b_{2}P_{X}(y_{2}^{-})\phi^{\prime}(y_{2}^{-})$ vanishes.
Repeating the same reasoning starting from the second brane,
one gets analogous result for the first brane. Putting
both cases together, we find that for $p$ and $r$ regular
in the bulk, the necessary and sufficient condition
for existence of a massless mode is
$$b_{1}b_{2}P_{X}(y_{1}^{+})P_{X}(y_{2}^{-})\phi^{\prime}(y_{1}^{+})\phi^{\prime%
}(y_{2}^{-})=0\,.$$
(40)
Conditions sufficient for the stability can be
found in the following way.
Multiplying eq. (20) with $K_{m^{2}}$, integrating
over the whole 5th dimension, and using the boundary
conditions (23) we get
$$\displaystyle m^{2}\int_{y_{1}}^{y_{2}}r(K_{m^{2}})^{2}$$
$$\displaystyle\!\!\!\!\!\!\!\!+\left.\sum_{i}\frac{b_{i}}{m^{2}}\frac{p}{P_{X}}%
(K^{\prime}_{m^{2}})^{2}\right|_{y_{i}^{\pm}}$$
(41)
$$\displaystyle\!\!\!\!\!\!\!\!=\int_{y_{1}}^{y_{2}}\left[q(K_{m^{2}})^{2}+p(K^{%
\prime}_{m^{2}})^{2}\right]+\left.\sum_{i}c_{i}\frac{p}{P_{X}}(K^{\prime}_{m^{%
2}})^{2}\right|_{y_{i}^{\pm}}.$$
Let us consider first such models for which
the background dependent
bulk functions $p$, $q$ and $r$ are regular and positive
while the brane parameters $b_{i}$ are positive and $c_{i}$
are non–negative. Then, the r.h.s. of (41) is
positive while the l.h.s. is negative for negative $m^{2}$
and may be divergent for vanishing $m^{2}$. Thus,
the condition (41) can be fulfilled only
for positive $m^{2}$. The function
$q=1/a$ is always positive. Functions $p$ and $r$ have
the same sign as $P_{X}$ and $(P_{X}+2XP_{XX})$, respectively.
They become singular if any of the functions
$P_{X}$, $(P_{X}+2XP_{XX})$ or $\phi^{\prime}$
vanishes for any value of $y$.
Thus, the following conditions
$$\displaystyle b_{i}>0,\qquad\qquad c_{i}\geq 0,$$
(42)
$$\displaystyle\forall_{y\in[y_{1}^{+},\,y_{2}^{-}]}\qquad{\phi^{\prime}}^{2}(y)%
>0,\quad P_{X}(y)>0,\quad P_{X}(y)+2X(y)P_{XX}(y)>0,$$
(43)
are sufficient for the stability of the
inter–brane distance (positivity of the radion mass
squared). By $y\in[y_{1}^{+},y_{2}^{-}]$ we denote the interior
of the bulk, $y_{1}<y<y_{2}$ and the limits $y\to y_{1}^{+}$
and $y\to y_{2}^{-}$.
Showing that the above conditions are sufficient for
stability was quite easy.
It is much more difficult to check which conditions are
necessary.
We will show now that there must be at least one tachyonic mode
if any of the functions, $\phi^{\prime}$, $P_{X}$ or $(P_{X}+2XP_{XX})$,
vanishes anywhere in the bulk.
The arguments are similar to those used in
[7] and [9]. We will compare the properties
of two solutions of the bulk equation of motion (20),
one for $m^{2}=0$ and second for $m^{2}=-M^{2}$ in the limit $M\to\infty$.
Both solutions satisfy the boundary condition at
one brane (let us first choose it to be the first one
located at $y_{1}$).
We start with
solving the bulk equation of motion (20) in the
limit of large negative $m^{2}=-M^{2}$.
In the leading order in $1/M$, equation (20)
has the following approximate solution
$$\displaystyle K_{-M^{2}}(y)\approx$$
$$\displaystyle\!\!\!\!\!\!\!\!\frac{1}{\sqrt{pc_{s}}}\left[C^{+}\exp\left(+M%
\int_{y_{1}}^{y}c_{s}\right)+C^{-}\exp\left(-M\int_{y_{1}}^{y}c_{s}\right)%
\right],$$
(44)
$$\displaystyle K^{\prime}_{-M^{2}}(y)\approx$$
$$\displaystyle\!\!\!\!\!\!\!\!M\sqrt{\frac{c_{s}}{p}}\left[C^{+}\exp\left(+M%
\int_{y_{1}}^{y}c_{s}\right)-C^{-}\exp\left(-M\int_{y_{1}}^{y}c_{s}\right)%
\right].$$
(45)
In the same limit, the boundary condition (23)
at the first brane becomes
$$\left.\left(c_{1}K_{-M^{2}}^{\prime}-P_{X}K_{-M^{2}}\right)\right|_{y=y_{1}}%
\approx 0\,.$$
(46)
Because of the $M$ prefactor in (45), for any $c_{1}\neq 0$
and large enough $M$, the above boundary condition can be fulfilled
when $C^{+}\approx C^{-}$.
We choose $C^{\pm}$ to be positive because later we will
compare this solution with $K_{0}$ normalized to 1 at $y_{1}$.
When $c_{1}$ and $P_{X}(y_{1})$ have the same sign, the boundary condition
(46) can be fulfilled only when $K_{-M^{2}}(y_{1})$
and $K^{\prime}_{-M^{2}}(y_{1})$ have the same sign. Thus,
$C^{+}>C_{-}$ and the square bracket in (44) does
not change its sign in the whole bulk.
For very large $M$ the first term in
(44) starts do dominate over the second one even for
small values of $y-y_{1}$ (it is slightly bigger even at $y_{1}$)
and away from the first brane the solution is approximated by
$$K_{-M^{2}}(y)\approx{C^{+}}\phi^{\prime}\sqrt{\frac{2a}{3}\sqrt{P_{X}\left(P_{%
X}+2XP_{XX}\right)}}\exp\left(M\int\frac{P_{X}}{P_{X}+2XP_{XX}}\right).$$
(47)
Using this solution we can investigate models when
some of the conditions in
(42-43) are not fulfilled.
It is convenient to define the following function of $m^{2}$
$$B_{2}(m^{2})=\left.\left[\left(b_{2}-c_{2}m^{2}\right)\frac{\partial}{\partial
n%
}K_{m^{2}}-m^{2}P_{X}K_{m^{2}}\right]\right|_{y=y_{2}^{-}}.$$
(48)
It is equal to the l.h.s. of the boundary
condition (23)
for $K_{m^{2}}$ satisfying the bulk equation of motion
and the boundary condition at the first brane, and normalized
to 1 at $y_{1}$.
The spectrum of the KK tower of scalar perturbations
consists of those values $m^{2}$ for which $B_{2}(m^{2})=0$.
Now we check whether the positivity of $b_{i}$ and $c_{i}$
are necessary conditions for the stability, assuming that
all the bulk conditions (43) are fulfilled.
For very lage negative $m^{2}$ the boundary function
$B_{2}$ at the second brane is dominated
by the term proportional to $K^{\prime}_{-M^{2}}$. From eq. (47)
and the discussion before it, it follows that
$${\rm sgn}\left[B_{2}(-M^{2})\right]={\rm sgn}\left[M^{2}c_{2}K^{\prime}_{-M^{2%
}}(y_{2}^{-})\right]={\rm sgn}\left[c_{2}\right]\,.$$
(49)
On the other hand, for the solution $K_{0}$ given by
(38) and (39) we get
$${\rm sgn}\left[B_{2}(0)\right]={\rm sgn}\left[b_{2}K^{\prime}_{0}(y_{2}^{-})%
\right]={\rm sgn}[b_{2}]\,,$$
(50)
where we used the fact that $K^{\prime}_{0}$ is always positive
when the inequalities (43) are fulfilled.
Comparing (49) with (50), we conclude
that there must be at least one negative eigenvalue
when the parameters $b_{2}$ and $c_{2}$ have opposite signs.
For $b_{2}c_{2}<0$, the function $B_{2}(m^{2})$ has different
sign for $m^{2}=0$ and for large (enough) negative $m^{2}$.
There must be some negative $m^{2}$ for which $B_{2}$
vanishes because the solutions of (20) change
continuously with $m^{2}$.
Repeating the above reasoning but starting from the brane
at $y_{2}$, we obtain an analogous condition for parameters
$b_{1}$ and $c_{1}$. Thus, the conditions
$$b_{1}c_{1}\geq 0\,,\qquad\qquad b_{2}c_{2}\geq 0\,,$$
(51)
are necessary for the stability.
Now we investigate the stability conditions
for the bulk quantities (43).
The solution (47) for large negative $m^{2}$
vanishes at a point at which $\phi^{\prime}$ or $P_{X}$ vanishes.
It must change sign there because from (20)
it follows that $K$ and $K^{\prime}$ can vanish at the same point
only for trivial solution vanishing everywhere.
Thus, for very large negative $m^{2}$ the function $K(y)$
vanishes close to the point where $P_{X}\phi^{\prime}$ is zero.
On the other hand, from (38) and (39) it follows
that $K_{0}$ is positive for all $y$. So, there must be
some negative $\widetilde{m}^{2}$ for which $K_{\widetilde{m}^{2}}$
has a zero point but is nowhere negative. It is easy to
see that such a zero point must be at the second brane,
$y=y_{2}$, and that the derivative of $K_{\widetilde{m}^{2}}(y_{2}^{-})$
is negative. In such a situation
$${\rm sgn}\left[B_{2}(\widetilde{m}^{2})\right]={\rm sgn}\left[(b_{2}-c_{2}%
\widetilde{m}^{2})K^{\prime}_{\widetilde{m}^{2}}(y_{2}^{-})\right]=-{\rm sgn}[%
b_{2}]\,,$$
(52)
where the last equality follows from the condition
(51).
Comparing (50) and (52) we find that there must be
some negative mode with the eigenvalue ${\widehat{m}}^{2}$
satisfying $\widetilde{m}^{2}<{\widehat{m}}^{2}<0$ for which
$B_{2}({\widehat{m}}^{2})=0$. The radion is tachyonic if
$\phi^{\prime}$ or $P_{X}$ vanishes in the bulk.
The above arguments are rather complicated but the
result is quite intuitive. We consider backgrounds for
which $P_{X}\phi^{\prime}$ vanishes at some $y_{0}<y_{2}$ in the bulk.
For any such background $K_{0}(y)$ defined in (38)
is a zero mode in a model restricted to the interval
$[y_{1},y_{0}]$. It is quite natural that the KK states becomes
lighter when the compact space becomes bigger. So, with
a massless mode on $[y_{1},y_{0}]$ there should be a tachyonic
one on the bigger orbifold $[y_{1},y_{2}]$.
Equation (47) can be used to show that also
$(P_{X}+2XP_{XX})$ should be strictly positive.
If it is not, there are two possibilities
depending on how fast it approaches zero. If the integral
in (47) is finite then $K_{-M^{2}}$ vanishes
because of $(P_{X}+2XP_{XX})$ in the prefactor and a
reasoning similar to that for the case of vanishing
$P_{X}\phi^{\prime}$ may be
applied to prove the existence of at least one tachyonic mode.
On the other hand, a divergent integral in (47)
indicates the breakdown of the perturbativity
assumption. This is not surprising. Vanishing $(P_{X}+2XP_{XX})$
corresponds to infinite speed of sound while negative
$(P_{X}+2XP_{XX})$ gives negative square of the speed of sound
(for positive $P_{X}$, which is anyway necessary for the stability).
In both cases one should expect strong instabilities.
We showed above that the conditions (43)
on the bulk quantities are not only sufficient but
also necessary for the stability.
We were not able to prove the same for the brane conditions
(42). If one of them is fulfilled then the other
has also to be fulfilled. The only possible loophole occurs
when both conditions (42) are violate,
namely when $b_{1}<0$ and $c_{1}<0$ or when $b_{2}<0$ and $c_{2}<0$.
However, these are not very appealing possibilities.
Parameters $c_{i}$ are proportional to the integrals
$\int_{y_{i}}\delta_{i}Q^{(i)}_{X}$ and can be negative only
for localized brane kinetic terms very different from
the standard one.
5 Discussion
With the results presented in the two previous sections
we can investigate how the stabilization of
branes is influenced by the presence of
non–trivial scalar kinetic terms in the bulk and/or
on the branes. Such terms change the background
configurations and the spectrum of the scalar perturbations.
We start the discussion with the background.
Combining eqs. (8)
and (9), the dynamical equation describing
the change of the warp factor can be written as
$$3\frac{a^{\prime\prime}}{a}+a^{2}\left(V+P\right)=0\,.$$
(53)
The source for the change of the warp
factor $a(y)$ is the full “matter” Lagrangian density
$(V+P)$
irrespective of whether the kinetic part is standard
or not. The modification of the scalar equation of motion
given in (7)
$$\left(P_{X}\phi^{\prime}\right)^{\prime}+3\frac{a^{\prime}}{a}\left(P_{X}\phi^%
{\prime}\right)-a^{2}\left(V_{\Phi}+P_{\Phi}\right)=0\,,$$
is twofold. First, similarly as in the case of the
warp factor, the role of the potential in this equation
is played by the full non–gravitational Lagrangian density.
Second, it seems that a natural
variable to describe the change of the scalar
background is the product $P_{X}\phi^{\prime}$
and not $\phi^{\prime}$ itself. The equation of motion
for this generalized variable $P_{X}\phi^{\prime}$ looks formally
the same as that in the standard theory (derivative
of the full Lagrangian as a source and $3a^{\prime}/a$ as “friction”).
Thus, as compared to the standard theory,
for the same local non–gravitational energy density
and the warp factor slope, the scalar
field $\phi$ changes faster (slower) if $P_{X}$ is smaller
(bigger) than 1. Of course this is only a qualitative
feature and in most of the cases any quantitative corrections
can be found only by numerical calculations.
The positions of the branes are determined by
the boundary conditions. The modifications to the boundary
conditions (10) and (11) are
analogous to those in the bulk background equations.
Namely, not only the potentials but the full Lagrangians
localized at the branes determine the jumps of $a^{\prime}$
while their derivatives with respect to $\Phi$ determine
the jumps of $P_{X}\phi^{\prime}$.
Usually in Randall–Sundrum type models, the warp factor changes
monotonically in the bulk, so its derivative has the same
sign for all $y$. Thus, because of opposite overall signs
in the boundary conditions (10) at two branes, one brane
must have positive tension while the second one must have
negative tension. To check the signs of the brane tensions
in the class of models considered in this work we rewrite
eq. (8) in the following form
$$\left(\frac{a^{\prime}}{a^{2}}\right)^{\prime}=-\frac{1}{3a}P_{X}{\phi^{\prime%
}}^{2}\,.$$
(54)
In the previous section we showed that the stability of the
model requires that $P_{X}\phi^{\prime}$ is everywhere non–zero. Thus,
the r.h.s. of the above equation is always negative.
The ratio $a^{\prime}/a^{2}$ always decreases and the warp factor $a(y)$
can not have a minimum in the bulk. Because of that,
it is not possible to construct a stable model with two
positive tension branes. At least
one brane must have a negative tension:
$$\min_{i}\left(\left.U^{(i)}\right|_{y_{i}}+\int_{y_{i}}\delta_{i}Q^{(i)}\right%
)<0\,.$$
(55)
In all stable models $\phi(y)$ must be a monotonic function
($\phi^{\prime}$ can not vanish) and $P_{X}$ can not change sign.
The limit of the product $P_{X}\phi^{\prime}$ has the same sign at
both branes. Thus, it follows from the boundary
condition (11) that
$$\left(\left.U^{(1)}_{\Phi}\right|_{y_{1}}+\int_{y_{1}}\delta_{1}Q^{(1)}_{\Phi}%
\right)\cdot\left(\left.U^{(2)}_{\Phi}\right|_{y_{2}}+\int_{y_{2}}\delta_{2}Q^%
{(2)}_{\Phi}\right)<0\,.$$
(56)
We turn now to the stability conditions. One of them is
the positivity of $b_{i}$ parameters defined in (23).
Using the background equation of motion (7)
the last term in the definition of $b_{i}$
(24) can be rewritten as
$$\displaystyle\!\!\!\!\!\!\!\!\mp\lim_{y\to y_{i}^{\pm}}\left(P_{X}+2XP_{XX}%
\right)\left(\frac{\phi^{\prime\prime}}{\phi^{\prime}}-\frac{a^{\prime}}{a}\right)$$
$$\displaystyle\!\!\!\!\!\!\!\!\qquad\qquad=\lim_{y\to y_{i}^{\pm}}\left[\pm 4P_%
{X}\frac{a^{\prime}}{a}\pm P_{\Phi X}\phi^{\prime}\mp\frac{a^{2}\left(V_{\Phi}%
+P_{\Phi}\right)}{\phi^{\prime}}\right]$$
$$\displaystyle\!\!\!\!\!\!\!\!\qquad\qquad=\lim_{y\to y_{i}^{\pm}}\left[-4P_{X}%
\frac{\frac{\partial}{\partial n}a}{a}-P_{\Phi X}\frac{\partial}{\partial n}%
\phi+\frac{a^{2}\left(V_{\Phi}+P_{\Phi}\right)}{\frac{\partial}{\partial n}%
\phi}\right],$$
(57)
where we used the outer normal derivative introduced in
eq. (23). The first term in the last square bracket
of the above equation gives negative (positive) contribution
to the $b$ parameter on the positive (negative) tension brane.
So, positivity of $b$ at the positive tension brane
is more difficult to achieve. Stability is improved when,
close to the brane(s), $P_{\Phi X}$ has the opposite sign and
$\left(V_{\Phi}+P_{\Phi}\right)$ has the same sign as the
normal derivative of the scalar field $\partial\phi/\partial n$.
There is another term in the definition of $b$ which depends
on the bulk background, namely
$\int\phi^{\prime\prime}(P_{\Phi X}+2XP_{\Phi XX})$. Its sign depends
on the background and on the details of the generalized
bulk kinetic function $P$. Non–trivial $\Phi$–dependence
of $P$ can be, at least in some cases, used to increase
the radion mass. Finally, large enough values of the second
derivatives of the brane kinetic terms $Q^{(i)}$ may be
used to make $b_{i}$ positive.
The second stability condition in (42)
can be quite easily
fulfilled. For example: $c_{i}$ given by eq. (25)
vanishes if there is no kinetic
term localized on the $i$–th brane and it is positive when
such localized term is not much different from the standard
one $Q^{(i)}=X$.
Models with non–standard bulk and/or brane scalar kinetic
terms are quite complicated and usually only performing numerical
calculations one can find the background fields and check
their stability against small perturbations.
Nevertheless, it seems viable that stable
solutions can exist also in models without any scalar
potentials or cosmological constants. The kinetic terms alone
may have structure rich enough for
configurations with stabilized inter–brane distance.
This is similar to the situation in models
proposed in [10, 11] in which inflation was
realized without any scalar potential.
Let us discuss what properties the generalized kinetic
terms should have in order to support stable brane
configurations. Conditions on the bulk kinetic function
$P$ are rather weak. It is enough that eq. (9)
can be fulfilled for some $y_{0}$ and positive values of
${\phi^{\prime}}^{2}$, $P_{X}$ and $(P_{X}+2XP_{XX})$. Then, the dynamical
equations (7) and (8) can be used to
extend the solution to $y\neq y_{0}$. The bulk stability conditions
(43) are fulfilled at $y_{0}$, so they are
fulfilled also in some finite interval in the 5th coordinate.
Any two points in this interval may be used to locate
the branes. Of course, this is possible only when the brane
kinetic terms have appropriate properties.
Restrictions on the brane kinetic functions $Q^{(i)}$
are quite strong
if we want the branes to be stabilized
at given positions in a given background.
First of all, from eq. (56) it is obvious that
without brane potentials it is necessary
that $Q^{(i)}$ have some non–trivial
$\Phi$–dependence. In addition, it follows
from (55) that at least at one of the branes
the kinetic term must give a negative contribution to its
tension.
This does not a priori mean that the system becomes
unstable. Of course, we want the energy to be bounded from
below, so the kinetic term at the second brane
$Q^{(2)}(\Phi,X)$ (we call “second” that brane at which
the expression in (55) is minimized)
should give negative value of $\int\delta_{2}Q^{(2)}$
only for some range of values of its
arguments555
One should remember that in general $Q^{(i)}$ and
$\int\delta_{i}Q^{(i)}$
are not just proportional to
each other and can have negative values for
(slightly) different regions of the parameter space.
This is caused by the regularization procedure discussed
in section 2.
.
More specifically, each $Q^{(i)}$ must satisfy
two equalities (10)
and (11) and two inequalities (42).
The values of $\int\delta_{i}Q^{(i)}$ and
$\int\delta_{i}Q^{(i)}_{\Phi}$ necessary to fulfill the
background boundary
conditions depend on the details of a given background
but their signs are determined by (55) and (56).
All boundary and stability conditions on the brane kinetic
functions may be written in the following form
$$\displaystyle\int_{y_{i}}\delta_{i}Q^{(i)}$$
$$\displaystyle\!\!\!\!\!\!\!\!=\left.\frac{6}{a^{2}}\frac{\partial a}{\partial n%
}\right|_{y_{i}^{\pm}}\,,$$
(58)
$$\displaystyle\int_{y_{i}}\delta_{i}Q^{(i)}_{\Phi}$$
$$\displaystyle\!\!\!\!\!\!\!\!=-\left.\frac{2P_{X}}{a}\frac{\partial\phi}{%
\partial n}\right|_{y_{i}^{\pm}}\,,$$
(59)
$$\displaystyle\int_{y_{i}}\delta_{i}Q^{(i)}_{X}$$
$$\displaystyle\!\!\!\!\!\!\!\!\geq 0\,,$$
(60)
$$\displaystyle\int_{y_{i}}\delta_{i}Q^{(i)}_{\Phi\Phi}$$
$$\displaystyle\!\!\!\!\!\!\!\!>-\frac{2\tilde{b}_{i}}{a(y_{i})}\,,$$
(61)
where $\tilde{b}_{i}$ is the r.h.s. of (24) with
$Q^{(i)}$ set to zero. It is possible to fulfill all the
above conditions for example with the brane kinetic functions
of the form
$$Q^{(i)}=K^{(i)}(\Phi)X+L^{(i)}(\Phi)X^{2}\,.$$
(62)
The most difficult part is to satisfy simultaneously conditions
(58) and (60) at the second (negative tension) brane.
Using eqs. (58), (60) and (4),
one can find a lower bounds on $L^{(i)}$
$$\left.L^{(i)}X^{2}\right|_{y_{i}^{\pm}}\geq\left.-\frac{270}{a^{2}}\,\frac{%
\partial a}{\partial n}\right|_{y_{i}^{\pm}}\,,$$
(63)
which can be translated to an upper bound on $K^{(i)}$
$$\left.K^{(i)}X\right|_{y_{i}^{\pm}}=-\left.\frac{3}{5}L^{(i)}X^{2}\right|_{y_{%
i}^{\pm}}+\left.\frac{18}{a^{2}}\frac{\partial a}{\partial n}\right|_{y_{i}^{%
\pm}}\leq\left.\frac{180}{a^{2}}\,\frac{\partial a}{\partial n}\right|_{y_{i}^%
{\pm}}\,.$$
(64)
At the positive tension brane $L^{(2)}(\phi(y_{2}))$ must be
positive and big enough while $K^{(2)}(\phi(y_{2}))$ must be negative
(with value related to the value of $L^{(2)}$).
Thus, without scalar potentials it is not possible to construct
a stable model with a positive tension brane if
the corresponding $K^{(i)}$ is always positive.
Some $\Phi$–dependence of $K^{(i)}$ and/or $L^{(i)}$ is necessary to
fulfill conditions (59) and (61). The background
boundary condition (59) takes the following form
$$\left.\left[\frac{1}{3}K^{(i)}_{\Phi}X+\frac{1}{5}L^{(i)}_{\Phi}X^{2}\right]%
\right|_{y_{i}^{\pm}}=-\left.\frac{2P_{X}}{a}\frac{\partial\phi}{\partial n}%
\right|_{y_{i}^{\pm}}\,.$$
(65)
In all stable configurations, the r.h.s. of this equation has
opposite signs on the two branes (because $\phi^{\prime}$ can not change
sign). So, there are no consistent brane models without potentials
if all first derivatives of $K^{(i)}$ and $L^{(i)}$ have the same sign.
The stability conditions (61) for the brane kinetic functions
(62) may be rewritten as
$$\displaystyle\lim_{y\to y_{i}^{\pm}}$$
$$\displaystyle\!\!\!\!\!\!\!\!\left[\frac{1}{3}K^{(i)}_{\Phi\Phi}X+\frac{1}{5}L%
^{(i)}_{\Phi\Phi}X^{2}+\frac{P_{\Phi X}}{P_{X}}\left(\frac{1}{3}K^{(i)}_{\Phi}%
X+\frac{1}{5}L^{(i)}_{\Phi}X^{2}\right)\right.$$
(66)
$$\displaystyle\!\!\!\!\!\!\!\!\,\,\,\,-\left.4P_{X}\left(\frac{1}{9}K^{(i)}X+%
\frac{1}{15}L^{(i)}X^{2}+\frac{P_{\Phi}}{\frac{1}{3}K^{(i)}_{\Phi}X+\frac{1}{5%
}L^{(i)}_{\Phi}X^{2}}\right)\right]$$
$$\displaystyle\!\!\!\!\!\!\!\!\qquad\qquad\qquad\qquad\qquad\qquad\qquad>\int_{%
y_{i}}\frac{\phi^{\prime\prime}\left(P_{\Phi X}+2XP_{\Phi XX}\right)}{a}\,.$$
Some of the terms on the l.h.s. of the the above expression
may be negative but they can be compensated
by large enough value of $K_{\Phi\Phi}^{(i)}X/3+L_{\Phi\Phi}^{(i)}X^{2}/5$.
It is clear that it is possible to choose functions
$K^{(i)}$ and $L^{(i)}$ which satisfy all the above boundary
and stability conditions for a given background. So,
models in which the inter–brane distance is fixed in a stable
way can be constructed even without any scalar potentials
or cosmological constants.
The brane induced kinetic terms may have
a relatively simple form $Q=KX+LX^{2}$ if the functions
$K$ and $L$ are generic
enough666Of course, one fine tuning of parameters is necessary
as in all models with flat 4D foliation.
.
It would be interesting
to check whether any higher order kinetic terms predicted
for example by string theories have an appropriate structure.
6 Conclusions
We considered 5D brane models with bulk and
brane scalar kinetic terms generalized to
some functions of $X=(\nabla\phi)^{2}/2$ and the scalar
field itself. The background equations of
motion and boundary conditions have structure similar to
the case with standard kinetic terms. There are two kinds
of modifications. First: the scalar potential is replaced by the
sum of the potential and the kinetic term. Second: derivatives
of the scalar field are multiplied by derivatives of the bulk
kinetic term with respect to $X$.
Stability of background configurations has been checked
by analyzing the spectrum of small scalar perturbations.
A given background with fixed branes positions
is stable only when all
the masses squared in the spectrum are positive.
The bulk equation of motion determining the shape of
the KK modes of such perturbations was written in
the Sturm–Liouville form.
The corresponding boundary conditions have rather
complicated form. They may be expressed in terms of
four parameters
(two for each brane), $b_{i}$ and $c_{i}$,
determined by the background and by the bulk interactions
described effectively by some potentials and generalized
kinetic terms. The boundary conditions depend
also on the eigenvalues and this dependence is
more complicated than in models with standard kinetic terms.
We have shown that our eigenvalue problem is self-adjoint
with those complicated boundary conditions.
We identified even larger class of boundary conditions
for which the Sturm-Liouville operator is hermitian.
The eigenvalue–dependence of the boundary conditions
makes the stability considerations more difficult.
Sufficient conditions for the stability are:
$b_{i}>0$, $c_{i}\geq 0$ at each brane
and the positivity of bulk functions
$P_{X}$, $(P_{X}+2XP_{XX})$ and $\phi^{\prime 2}$
for all values of the 5th
coordinate $y$. If $c_{i}\geq 0$ then the remaining conditions
are not only sufficient but also the necessary ones.
This changes when any of the $c_{i}$ parameters is
negative. It seems that it may be possible to have stable
configurations with negative both $b_{1}$ and $c_{1}$ (or $b_{2}$ and $c_{2}$).
The lowest KK mode, the radion, becomes tachyonic
when any of the quantities $b_{i}c_{i}<0$ or any of the quantities
$\phi^{\prime 2}$, $P_{X}$ or $(P_{X}+2XP_{XX})$ is not strictly positive.
We have shown that stable brane models may be constructed
without bulk and/or brane potentials and cosmological
constants. This may be achieved for example when
the brane localized kinetic terms take the form
$Q^{(i)}=K^{(i)}(\Phi)X+L^{(i)}(\Phi)X^{2}$.
Conditions for the functions $K^{(i)}(\Phi)$ and
$L^{(i)}(\Phi)$ have been found.
Acknowledgments
This work has been supported by a Marie Curie Transfer of Knowledge
Fellowship of the European Community’s Sixth Framework Programme
under contract number MTKD-CT-2005-029466 (2006-2010).
The author would like to thank for the hospitality experienced at
Ludwig Maximilian University and Max Planck Institute in Munich
where this work has been done.
References
[1]
W. D. Goldberger and M. B. Wise,
Phys. Rev. Lett. 83 (1999) 4922
[arXiv:hep-ph/9907447].
[2]
C. Charmousis, R. Gregory and V. A. Rubakov,
Phys. Rev. D 62 (2000) 067505
[arXiv:hep-th/9912160].
[3]
O. DeWolfe, D. Z. Freedman, S. S. Gubser and A. Karch,
Phys. Rev. D 62, 046008 (2000)
[arXiv:hep-th/9909134].
[4]
T. Tanaka and X. Montes,
Nucl. Phys. B 582 (2000) 259
[arXiv:hep-th/0001092].
[5]
C. Csáki, M. L. Graesser and G. D. Kribs,
Phys. Rev. D 63 (2001) 065002
[arXiv:hep-th/0008151].
[6]
S. Mukohyama and L. Kofman,
Phys. Rev. D 65, 124025 (2002)
[arXiv:hep-th/0112115].
[7]
J. Lesgourgues and L. Sorbo,
Phys. Rev. D 69 (2004) 084010
[arXiv:hep-th/0310007].
[8]
A. V. Frolov and L. Kofman,
Phys. Rev. D 69 (2004) 044021
[arXiv:hep-th/0309002].
[9]
D. Konikowska, M. Olechowski and M. G. Schmidt,
Phys. Rev. D 73, 105018 (2006)
[arXiv:hep-th/0603014].
[10]
C. Armendariz-Picon, T. Damour and V. F. Mukhanov,
Phys. Lett. B 458, 209 (1999)
[arXiv:hep-th/9904075].
[11]
J. Garriga and V. F. Mukhanov,
Phys. Lett. B 458, 219 (1999)
[arXiv:hep-th/9904176].
[12]
T. Chiba, T. Okabe and M. Yamaguchi,
Phys. Rev. D 62, 023511 (2000)
[arXiv:astro-ph/9912463].
[13]
C. Armendariz-Picon, V. F. Mukhanov and P. J. Steinhardt,
Phys. Rev. Lett. 85, 4438 (2000)
[arXiv:astro-ph/0004134];
Phys. Rev. D 63, 103510 (2001)
[arXiv:astro-ph/0006373].
[14]
E. Babichev, V. Mukhanov and A. Vikman,
JHEP 0802, 101 (2008)
[arXiv:0708.0561 [hep-th]].
[15]
D. Maity, S. SenGupta and S. Sur,
Phys. Lett. B 643, 348 (2006)
[arXiv:hep-th/0604195];
arXiv:hep-th/0609171;
A. Dey, D. Maity and S. SenGupta,
Phys. Rev. D 75, 107901 (2007)
[arXiv:hep-th/0611262].
[16]
L. Kofman, J. Martin and M. Peloso,
Phys. Rev. D 70, 085015 (2004)
[arXiv:hep-ph/0401189].
[17]
G. R. Dvali, G. Gabadadze and M. Porrati,
Phys. Lett. B 485, 208 (2000)
[arXiv:hep-th/0005016].
[18]
M. S. Carena, A. Delgado, J. D. Lykken, S. Pokorski, M. Quiros
and C. E. M. Wagner,
Nucl. Phys. B 609, 499 (2001)
[arXiv:hep-ph/0102172].
[19]
G. R. Dvali, G. Gabadadze, M. Kolanovic and F. Nitti,
Phys. Rev. D 64, 084004 (2001)
[arXiv:hep-ph/0102216].
[20]
R. Bao, M. S. Carena, J. Lykken, M. Park and J. Santiago,
Phys. Rev. D 73, 064026 (2006)
[arXiv:hep-th/0511266].
[21]
M. S. Carena, T. M. P. Tait and C. E. M. Wagner,
Acta Phys. Polon. B 33, 2355 (2002)
[arXiv:hep-ph/0207056].
[22]
B. s. Kyae,
arXiv:hep-th/0207272.
[23]
H. Davoudiasl, J. L. Hewett and T. G. Rizzo,
Phys. Rev. D 68, 045002 (2003)
[arXiv:hep-ph/0212279].
[24]
F. del Aguila, M. Perez-Victoria and J. Santiago,
JHEP 0302, 051 (2003)
[arXiv:hep-th/0302023].
[25]
F. del Aguila, M. Perez-Victoria and J. Santiago,
Acta Phys. Polon. B 34, 5511 (2003)
[arXiv:hep-ph/0310353].
[26]
A. Mück, L. Nilse, A. Pilaftsis and R. Rückl,
Phys. Rev. D 71, 066004 (2005)
[arXiv:hep-ph/0411258].
[27]
C. Csaki, J. Hubisz and P. Meade,
arXiv:hep-ph/0510275. |
Record dynamics and the observed temperature plateau in
the magnetic creep rate of type II superconductors.
L.P. Oliveira
Henrik Jeldtoft Jensen
h.jensen@ic.ac.uk
http://www.ma.ic.ac.uk/~hjjens/
Department of Mathematics, Imperial College London,
South Kensington campus, London SW7 2AZ, U.K.
Mario Nicodemi
Universitá di Napoli “Federico II”,
Dip. Scienze Fisiche, INFM and INFN, Via Cintia, 80126 Napoli, Italy
Paolo Sibani
Fysisk Institut, Syddansk Universitet, 5230 Odense M, Denmark
(November 20, 2020)
Abstract
We use Monte Carlo simulations of a coarse-grained three dimensional model
to demonstrate that the experimentally observed approximate temperature
independence of the magnetic creep rate for a broad range of temperatures
may be explained in terms of record dynamics, viz. the dynamical
properties of the times at which a stochastic fluctuating signal
establishes records.
pacs: 74.25.Qt, 05.40.-a, 74.40.+k
I Introduction
The magnetization of type II superconductors is determined by the number
of quantized magnetic vortices inside the sample [1].
As an externally imposed magnetic field is increased, vortices penetrate the sample
in a process which at non-zero temperature is driven by thermal activation
over energy barriers produced by the sample surface and by pinning centers
in the bulk. When the external field is lowered, vortices leak out
of the sample. The rate at which vortices move in and
out of the sample determines the magnetic creep rate.
Given that the magnetic relaxation is driven by thermal activation, it
is rather surprising that experiments have found the creep rate
to be essentially temperature independent in a wide temperature range
[2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12].
Several mechanisms have been suggested to explain how competing factors are able to
cancel the typical Arrhenius temperature dependence $\exp(U/T)$ for activation
over an energy barrier $U$. The most prominent theoretical suggestion
so far is probably the description in terms of collective creep [13].
Here we show that the lack of temperature dependence of the
creep rate can be understood quite simply in terms
of record dynamics[14], which
has recently been proposed as a general
mechanism for the irreversible dynamics
following a sudden quench in glassy systems [14, 15].
We base our analysis on the following two experimental and numerical
observations:
A)
For a long time glitches have been observed in the time dependences of the
magnetic relaxation of type II superconductors (see [12]
and references therein). As the external magnetic field is varied, the magnetization
undergoes abrupt jumps whenever vortices suddenly move in or out of
the sample.
B)
A number of studies, experimental as well as theoretical
[16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27],
indicates that in a broad range of not too high temperatures the vortex system
exhibits many characteristic features of glassy dynamics.
Many [28, 29, 30, 31, 32, 33, 34, 35]
but far from all [16, 17, 18, 19, 20] creep experiments
study the response of the external magnetic field to a sweep, including sign
reversal. For simplicity, we will presently concentrate on
a setup in which the external field is initially ramped at a fixed rate up to
a value and then remains constant for the entire duration of the
experiment.
We use a three dimensional version of the Restricted Occupancy Model
(ROM)[21, 22, 23, 24, 25, 26, 27]
to study the response to a fixed applied magnetic field.
Important physical properties of the 3D layered ROM model have a nearly
temperature independent
dynamical evolution, matching in this respect simulation results
previously obtained for the same model
[21, 22, 23, 24, 25, 26, 27]
as well as experimental results on type II superconductors
[3, 4, 5, 6, 11, 12, 16, 17, 18, 19, 20, 28, 29, 30, 31, 32, 33, 34, 35].
As a strong temperature sensitivity is expected for the activated
dynamics of an entirely classical model system, the mechanism behind
the observed temperature independence warrants some further theoretical scrutiny.
A fixed applied magnetic field can be expected to
lead to a non-stationary time dependence of the internal magnetization
which slowly increases from zero to about the value of the applied field.
Interestingly, the low temperature dynamical evolution of the response involves two
types of configuration rearrangements, having widely different time scales.
One type consists of rapid glitches by which the magnetization irreversibly
jumps to higher values, as additional vortices enter the system. We refer to
these glitches as quakes [15, 36, 37] to emphasize
their non-equilibrium nature, their abruptness and their dramatic effect on the
state of the vortex system. The quakes are separated by much longer periods of apparent
quiescence, during which the vortex system is ‘searching’ for a
configuration of larger stability and for a way to accommodate more vortices
inside the sample. Even though the total magnetization does not change,
the internal spatial organization of the vortices undergoes considerable rearrangements.
We argue that the slow vortex creep associated with the quakes
can be analyzed through the statistical properties of record
dynamics, which immediately explains the temperature
independence of the creep rate[14] for the range of temperatures for which this
description is applicable. By record dynamics we mean the following. Consider
a stochastic time signal $\chi(t)$. The corresponding record signal is
$R(t)=\max_{\tau<t}\{\chi(\tau)\}$, i.e. the largest value assumed by
$\chi(t)$ up to the current time $t$. The physical
idea behind record dynamics [14, 36, 38, 39]
is that large irreversible configurational changes in noisy systems with
a macroscopic number of metastable attractors are induced by noise fluctuations
of record size. Similar behavior is observed in other glassy
metastable systems, e.g. gels and spin-glasses, and can be characterized
statistically in a similar way [15, 36, 37].
This highlights the underlying unity of
non-equilibrium glassy dynamics at low temperatures and
supports the possibility of a common theoretical description.
The paper is organized as follows: The next Section summarizes the properties
of the ROM model used in the simulations. Section III
briefly introduces a possible mechanism [38, 39, 15]
by which activated dynamics can become insensitive to the temperature in
a glassy system, and demonstrates its relevance for the ROM model dynamics.
Section IV focuses on creep rates and contains comparison
with experiments.
Finally, Section V presents
a summary and a discussion.
II The 3D ROM model
The two dimensional Random Occupancy Model (ROM)
has been shown to reproduce the essential features of vortex
dynamics at nonzero temperature
[21, 22, 23, 24, 25, 26, 27].
Here we use Monte Carlo (MC) simulations of a generalized
three dimensional layered version of the ROM model
to capture the long time relaxation of interacting vortex matter.
In vortex matter, the length scale of the interactions can be very large
compared with the average distance between (pancake) vortices. At high
densities, this implies that each vortex interacts with many others.
For layered superconductors this situation can roughly be described
by two length scales: the first is the range of the interaction parallel to
the planes, this is the London
penetration depth $\lambda$. The second length
scale is the vortex correlation length, $\xi_{||}$, parallel to the
applied field (which we imagine to be perpendicular to the
copper oxide planes for high temperature superconductors).
The exact identification of this length scale is difficult
and is likely to depend on the anisotropy of the material,
the nature of the pinning, the strength of the magnetic induction
and on the temperature. This length scale may be related to vortex
line
cutting[40, 41, 42, 43, 44, 45, 46, 47].
These length scales respectively give the horizontal
and vertical lattice spacing of our model. The horizontal
coarse-grained length scale $l_{0}$, corresponds to the penetration depth
$l_{0}=\lambda$ of the superconducting material, and the
spacing between the layers in our lattice we consider
$l_{1}\sim\xi_{||}$. Smaller length scales are ignored.
For our purposes this approximation is acceptable because the length
scales smaller than $\lambda$ seem to have little influence on the
long time glassy properties of vortex matter.
Another limitation of the model is that it ignores the
variation of $\lambda$ with the temperature. As will
become clear from our ensuing discussion of record dynamics,
ignoring the temperature dependence of $\lambda$ is not crucial
for our explanation of the observed temperature plateau of
the creep rate.
In a sample of a superconducting material the vortex matter behavior is
determined by the competition of four energy scales [13]: intra
and interlayer vortex-vortex interaction, vortex-pinning interaction and
thermal fluctuations, all of which are schematically included in the ROM
model.
The Hamiltonian of the ROM model is thus the following:
$$H=\sum_{ij}A_{ij}n_{i}n_{j}-\sum_{i}A_{ii}n_{i}+\sum_{i}A_{i}^{p}n_{i}+\sum_{%
\left\langle ij\right\rangle_{z}}A_{2}\left(n_{i}-n_{j}\right)^{2},$$
(1)
where $n_{i}$ is the number of vortices on site $i$ of the lattice.
In a superconducting sample the number of vortex lines per unit area is
restricted by the upper critical field ($B_{c2}$) [1], so in the
model the number of vortices per cell can only assume values smaller than
$N_{c2}=B_{c2}l_{0}^{2}/\phi_{0}$ [48, 25].
Hence the name Restricted Occupancy Model. Moreover,
as we are interested in
a simulation setup that does not require magnetic field inversion and the
vortex-antivortex creation is strongly suppressed,
we simply consider $n_{i}\geq 0$.
The first two terms in Eq. (1)
represent the repulsion energy due to vortex-vortex
interaction in the same layer, and the vortex self energy respectively. Since
the potential that mediates this interaction decays exponentially at
distances longer than our coarse-graining length $\lambda$, interactions
beyond nearest neighbors are neglected. We set
$A_{ii}:=A_{0}=1$, $A_{ij}:=A_{1}$ if $i$ and $j$ are nearest neighbors on
the same layer, and $A_{ij}:=0$ otherwise.
The third term represents the interaction of the vortex pancakes with the pinning
centers. $A_{i}^{p}$ is a random potential and for the purposes of this work we
consider that $A_{i}^{p}$ has the following distribution $P\left(A_{i}^{p}\right)=\left(1-p\right)\delta\left(A_{i}^{p}\right)-p\delta%
\left(A_{i}^{p}-A_{0}^{p}\right)$.
The pinning strength $\left|A_{0}^{p}\right|$ represents the total
action of the pinning centers located on a site. In the present work
we use $\left|A_{0}^{p}\right|=0.3$.
Finally the last term describes the interactions between the vortex sections
in different layers. This term is a nearest neighbor quadratic interaction
along the $z$ axis, so that the number of vortices in neighboring cells along
the $z$ direction tends to be the same.
The parameters of the model are defined in units of $A_{0}$. The
time is measured in units of full MC
sweeps. The relationship between the model parameters and material parameters is
discussed in [48, 25].
Each individual MC update involves the movement to a neighbor site
of a single randomly selected vortex. The movement of the vortex is
automatically accepted if the energy of the system decreases; if the energy of
the system increases, the movement is accepted with probability
$\exp(-\Delta E/T)$ [49].
Given that the movement of pancake vortices is restricted to
the superconducting planes we only allow MC movements parallel
to the planes. We have used periodic boundary conditions along
the $z$ direction.
The external magnetic field is modeled by the edge sites on each
of the planes. The density at the edge is kept at a controlled value.
During a MC sweep vortices may move between the bulk sites and the edge sites.
After each MC sweep the density on the edge sites is brought
to the desired value. Initially the external field is increased to
a desired value ($N_{ext}=10$ vortices per edge site) by a very rapid
increase in the density on the edge sites.
We have here used a sweeping rate $\gamma$ of 0.25
per MC sweep (compared with $\gamma\in[10^{-6},10^{-2}]$ in our
previous studies
[21, 22, 23, 24, 25, 26, 27]).
After this fast initial ramping the external field is kept constant, while
we study how the vortices move into the sample. The age of the system, $t_{w}$,
is taken to be the time since the initial ramping.
We have studied systems of different sizes,
and obtain similar results except for very
small system sizes. Our key
results were obtained in a system consisting of 8 layers
of size $16\times 16$. The model parameters used in our simulations
were:
number of realization
$$\displaystyle=4000$$
$$\displaystyle A_{1}$$
$$\displaystyle=0.28$$
$$\displaystyle A_{2}$$
$$\displaystyle=0.5$$
$$\displaystyle p$$
$$\displaystyle=0.5$$
$$\displaystyle A_{0}^{p}$$
$$\displaystyle=0.3$$
$$\displaystyle N_{c2}$$
$$\displaystyle=27$$
$$\displaystyle\gamma$$
$$\displaystyle=0.25$$
$$\displaystyle N_{ext}$$
$$\displaystyle=10$$
III Record dynamics
We will in this section describe how the observed plateau in the temperature
dependence of the magnetic creep rate can be explained in the framework
of record dynamics. Let us first mention the salient features of what we
mean by record dynamics. Consider a stochastic signal $\chi(t)$ with
no time correlations. Now derive the
record signal $R(t)=\max_{\tau<t}\{\chi(\tau)\}$. We note that $R(t)$ is a
monotonous piecewise constant function which only increases its value
at discrete times $t_{k}$, whenever $\chi(t)$ manages to fluctuate to a
value larger
than any encountered previously. For our present purpose, the most important
property of the statistics of the record times $t_{k}$ is that the probability
that exactly $q$ records occur during
the time interval $[t_{w},t_{w}+t]$ (where $t_{w}$ is the time since the initiation),
is to a good approximation Poisson
distributed on
a logarithmic time scale [14, 36], i.e.,
$$p(q)={\langle q\rangle^{q}\over q!}\exp(-\langle q\rangle)$$
(2)
with average number of quake events proportional to logarithmic time
$$\langle q\rangle=\alpha\log(1+t/t_{w}).$$
(3)
Here $\alpha$ is the logarithmic rate of events.
To get the gist of the mathematics behind Eq. (2)
(for full detail see ref. [14]),
we note that since the largest outcome, i.e. the record, of $t$
independent trials is equally
likely to occur at any of the $t$ instances in the sequence, it occurs at the
first attempt with probability $1/t$. Hence, the probability
$p(1)$ of exactly one record in $t$ trials is $1/t$,
independently of the distribution of the underlying signal $\chi(t)$.
It is important to point out that the same also holds
for the general expression for $p(q)$ in Eq. (2).
The independence of $p(q)$ on the distribution
of random numbers corresponds to the
independence of $\chi(t)$ on the thermal noise
and will translate directly into
the temperature independence of the creep rate.
It has recently been shown [38, 39, 15]
that in (glassy) systems having a large number of dynamically inequivalent
attractors, temperature independence of suitably coarse-grained dynamical
variables can arise from the peculiar way in which the attractors are
selected as the system evolves from a typically rather unstable initial
configuration through gradually more stable ones. A similar noise insensitivity
of stochastic dynamics has been observed with other types of noise,
e.g. in driven dynamical systems [14, 50]
and in evolutionary dynamics[51, 15].
That the ROM model also exemplifies this type of behavior can be gleaned
from Fig. 1, showing the time dependence
of the number of vortices $N(t)$ which during a single run
have entered the system up to time $t$.
Importantly, the length of the
quiescent periods typically increases with time – notice the
logarithmic time axis in Fig. 1. Were this not the case,
the dynamics would appear continuous in terms of a suitably
coarse-grained time scale. Conversely, the lengthening of the
intervals between successive quakes signals the
anticipated gradual entrenching of the dynamics into
dynamically more stable configurations.
Also important is that the overwhelming majority of the
observed glitches lead to states with a higher number of vortices.
This de-facto irreversibility of the dynamics enables one to
meaningfully approximate
the signal $N(t)$ with the monotonically increasing
record signal $R(t)=\max_{[0,t]}\{N(t)\}$.
We stress that this theoretically convenient idealization is only
applicable within the strongly non-equilibrium regime of our
present concern.
Statistical insight into the time evolution
in the number of vortices present within the system is provided
by Fig. 2, where the empirical distributions of $N(t)$
are displayed for three different times, which are equidistantly
placed on a logarithmic time axis.
The insert in Fig. 2 shows the tail of the
probability density function (pdf) of the number of vortices, $p(v)$,
entering during a single quake. To a good approximation, the tail
is exponential and the observed time dependence
and temperature dependence of of $p(v)$ are negligible, except for the
highest temperature $T=0.5$.
The interpretation
in terms of record dynamics suggests that the probability
that exactly $q$ quakes occur during the time interval $[t_{w},t_{w}+t]$
is Poisson distributed on
a logarithmic time scale [14, 36]
according to Eq. (2).
We approximate the pdf for the number of vortices $v$ which enter
during a given quake event (see insert Fig. 2) by an
exponential distribution $p(v)=exp(-v/\bar{v})/\bar{v}$, and assume
that subsequent quakes are statistically
independent. The number of vortices entering during exactly $q$
quakes is then a sum of exponentially distributed
independent variables, and is hence Gamma distributed.
We finally obtain the pdf for the
total number of vortices entering during $[t_{w},t_{w}+t]$ by
averaging the Gamma distribution for $q$ quakes
over the probability Eq. (2) that precisely
$q$ quakes occur within the time interval of interest.
This leads to the following expression for
the pdf of total number of vortices $\Delta N=N(t+t_{w})-N(t_{w})$
which may have entered during the time interval $[t_{w},t_{w}+t]$
$$p(\Delta N,t)=e^{-{\Delta N\over\bar{v}}-\langle q\rangle}\sqrt{{\langle q%
\rangle\over\bar{v}\Delta N}}I_{1}\biggl{(}2\sqrt{{\langle q\rangle\Delta N%
\over\bar{v}}}\biggr{)},$$
(4)
where $I_{1}$ denotes the modified Bessel function of
order $1$[52].
The above theoretical prediction, is compared in Fig. 2
with our simulation results.
To estimate $\langle q\rangle$ according to Eq. (3), we
used $\alpha=22.6$, as obtained from
the logarithmic rate of the quake events. We can determine the
average $\bar{v}$ in two ways. Either directly from the simulated
distributions in the insert of Fig. 2 or
from fitting Eq. (4) to the simulated data in the
main frame of Fig. 2. In both cases we
find $\bar{v}=16$. We also find that $\bar{v}$ is essentially
temperature independent for temperatures below $T\approx 0.1$.
This is also expected from the insert in Fig. 2.
It is important to mention that the MC dynamics does overcome
plenty of positive energy barriers, $\Delta E>0$, through
thermal activation for
temperatures in the range $0.01<T<0.1$. As the temperature is
lowered fewer MC updates correspond to $\Delta E>0$ and
for the lowest temperatures MC steps involve $\Delta E\leq 0$
only[24]. Nevertheless, the record dynamics remains
essentially temperature independent for $T<0.1$.
The agreement is encouraging and suggests that the process of vortex
penetration into the sample can be described in
terms of a Poisson process with logarithmic time argument,
for short the log-Poisson process.
We also note that the log-Poisson statistics
covers the temporal distribution
of the quakes but has nothing to say on the size distribution
of the jumps, i.e. the number $v$ of vortices entering
during a single event. This stochastic quantity could in principle
introduce additional time and temperature dependencies. However, as
mentioned, the insert of Fig. 2 shows that for a
very broad parameter range this is not the case. Accordingly, the creep rate
obtained by convoluting the distribution of the quake sizes and the
log-Poisson distribution of number of quakes
will also be temperature independent.
The link between record statistics and the
stochastic dynamics of a glassy system is discussed
in detail in ref.[36] on the basis of
several idealized physical assumptions.
The first element is the existence of a large number, in principle
a continuum, of attractors. These
are sets of configurations clustered around a local
energy minimum and supporting equilibrium-like reversible thermal
fluctuations. By contrast, attractor changes—our
quakes—are assumed to be irreversible on the time scale
at which they occur. The exact nature of the quakes is
not entirely clear in our system. At intermediate temperatures they are
related to activation over barriers and the jump in $N(t)$ is
associated with a increase in the energy of the interacting vortices.
However, at the lowest temperatures there is not
sufficient thermal energy available for the system to climb
over energy barriers. The thermal fluctuations are only able to
push the vortices along
equipotential trajectories or to lower potential energy configurations.
In this regime the vortex motion is hindered by jamming and the
quakes are of a
mechanical nature [24].
An interpretation in terms of record dynamics implies that the dynamical
bottlenecks overcome by fluctuations are determined by the
actual noise history, and not predetermined in a static fashion.
For other model systems [14, 50],
the validity of the assumed linkage between noise records and barriers
was confirmed by considering white noise perturbations drawn from
a distribution with finite support, e.g. a box distribution,
and by then studying the properties of the selected
attractors as a function of the maximum size of noise.
Record-induced dynamics has thus a number of testable predictions,
the most interesting of which is, for our purposes,
the logarithmic time dependence and the
striking temperature independence of the number of quakes occurring in
the time interval $[0,t]$, which are shown in the following Section.
IV
Creep rates
Let us now turn to the dependence on time and temperature of the
total number of vortices in the sample. At time $t=0$ we rapidly
increases the external field from zero up the value $N_{ext}$
(see section II).
The vortex density of the bulk sites gradually increases
as vortices move in from the boundary.
In Figure 3 we present, for a very broad range of
temperatures, the average density $n(t)$ of the bulk sites
as function of the natural logarithm of time $\log(t)$. As anticipated,
the time dependence is temperature independent for all but the highest
temperatures.
One can identify three different
temporal regimes separated at times $t_{1}\approx 300$ and
$t_{2}\approx 3\times 10^{4}$.
For the remaining of this paper we will focus our
analysis on the intermediate regime $t_{1}<t<t_{2}$ (and choose $t_{w}=t_{1}$).
Our reason for this is that at $t_{1}$
vortex interactions become essential through out the entire system.
The late time regime $t>t_{2}$ is very difficult to resolve
appropriately in simulations and probably equally difficult
to study experimentally.
For times $t_{1}<t<t_{2}$ Fig. 3 demonstrates that $N(t)$
depends linearly on $log(t)$ to a very good approximation.
The linear logarithmic time dependence is of course
entirely consistent with the record dynamics outlined in
the previous section. We consider the total number of
vortices in the system $N(t_{w}+t)$ to be the accumulated effect
of vortices entering during quake events that have occurred prior
to time $t_{w}+t$. Let $t_{k}$ denote the time of occurrence of quake number $k$
and let $v_{k}$ denote the actual number of vortices entering during
this quake. We then have
$$N(t+t_{w})=N(t_{w})+\sum_{t_{w}<t_{k}<t_{w}+t}v_{k},$$
(5)
where the sum is over all quakes that occured during the time
interval $[t_{w},t_{w}+t]$
From Fig. 2 we know that $v_{k}$ is temperature
independent and possesses a well defined average $\bar{v}$.
Since the average number of quakes increases according to Eq. (3),
record dynamics predicts the following (temperature
independent) temporal evolution of the average number of vortices
$$\Delta N\equiv\langle N(t+t_{w})\rangle-\langle N(t_{w})\rangle=\alpha\bar{v}%
\log(1+t/t_{w}).$$
(6)
i.e. for the considered time regime $t/t_{w}\gg 1$ a temperature
independent logarithmic rate given by
$$d\Delta N/d\log(t)=\alpha\bar{v}/(1+t_{w}/t)\approx\alpha\bar{v}.$$
(7)
We extract the rate of the quakes in the simulations from temporal
signals like the one exhibited in Fig. 1.
In Fig. 4 we demonstrate that the quake rate
is indeed approximately independent of temperature
in the broad temperature interval
$10^{-4}<T<2\times 10^{-2}$.
Finally, Fig. 5 shows the near temperature independence
of the actual rate with which $N(t)$ changes. We extract this rate
from the data shown in Fig. 3 in the time region $t_{1}<t<t_{2}$
and plot the normalized creep rate $S=d\log[M(t)]/d\log(t)$ in order
to compare consistently with experiments, here we have used
$M(t)=|N(t)-N_{ext}|$.
V Conclusion
We have presented an analysis of simulated vortex creep data in terms
of record dynamics. This approach allows us to interpret
the observed temperature independence of the creep rate
as a generic property of the dynamics of
records obtained from the underlying fluctuating sequence. To
establish the temperature independence of the creep rate we do not need
to know the detailed nature of the quantity being gradually maximized.
Nor do we need a description of the intermittent vortex quakes which
are responsible for the abrupt changes in the number of vortices.
All we make use of is the assumption, supported by the simulated model,
that the abrupt glitches in the number of vortices inside the sample
can be interpreted as arising from the records of some stochastic
process. We showed that the simulated creep rate behaves in a way
very similar to published experimental data for YBCO.
It is obvious that a better experimental and
theoretical understanding of the nature of the vortex quakes is interesting and
future study of the ROM model will seek to improve our understanding
of the spatial and dynamical properties of these quakes. We can
already conclude that the physical mechanisms involved must be different
at low and high temperatures. At the lowest
temperatures activation over free energy barriers are excluded
and the quakes are related to mechanical rearrangements of the
vortices[24]. At elevated temperatures the quakes are
expected to be triggered by activation over thermal barriers.
The present paper shows that record dynamics can be used to understand
the temperature range from very low temperatures, where no barriers
can be climbed, up through a regime where thermal activation does
take place. For high temperatures (in our case for $T>0.1$) the
description in terms of record dynamics breaks down. This happens
when there is sufficient thermal energy available to make
any trapped metastable configurations short lived.
Let us finally mention that our description in terms of record dynamics may
not exclude aspects of previous descriptions of vortex relaxation
in terms of e.g. correlated collective vortex creep [13].
We would rather think of our approach as contributing to an
understanding of the detailed nature of the dynamics of the correlated
vortex regions.
VI Acknowledgments
We are indebted to Andy Thomas, Dan Moore and
Gunnar Pruessner for their support with the computations. Support
from EPSRC, the Portuguese FCT, a visiting fellowship from EPSRC
and financial support from the Danish SNF are gratefully acknowledged.
References
[1]
M. Tinkham.
Introduction to Superconductivity.
McGraw-Hill, New York, 1996.
[2]
Y. Yeshurun, A.P. Malozemoffand, and A. Shaulov.
Rev. Mod. Phys., 68:911, 1996.
[3]
A.C. Mota, G. Juri, P. Visani, A. Pollini, T. Terizzi, K. Aupke
and B. Hilti.
Physica C, 185-189:343, 1991.
[4]
L. Fruchter, A. P. Malozemoff, I. A. Campbell, J. Sanchez, M.
Konczykowski, R. Griessen, and F. Holtzberg.
Phys. Rev. B, 43:8709, 1991.
[5]
T. Stein, G. A. Levin, C. C. Almasan, D. A. Gajewski and M. B. Maple.
Phys. Rev. Lett., 82:2955, 1999.
[6]
K. Aupke, T. Teruzzi, P. Visani, A. Amann, A. C. Mota and V. N.
Zavaritsky.
Physica C, 209:255, 1993.
[7]
A. C. Mota, G. Juri, A. Pollini, K. Aupke, T. Teruzzi, P. Visani and
B. Hilti.
Physica Scripta, T45:69, 1992.
[8]
A. C. Mota, A. Pollini, G. Juri, P. Visani and B. Hilti.
Physica A, 168:298, 1990.
[9]
A. Pollini, A. C. Mota, P. Visani, G. Juri and J. J. M. Franse.
Physica B, 165-166:365, 1990.
[10]
A. Pollini, A. C. Mota, P. Visani, R. Pittini, G. Juri and T. Teruzzi.
J. Low Temp. Phys., 90:15, 1993.
[11]
A. F. Th Hoekstra.
Phys. Rev. Lett., 80:4293, 1998.
[12]
E.R. Nowak, J. M. Fabijanic, S. Anders and H. M. Jaeger.
Phys. Rev. B, 58:5825, 1998.
[13]
G. Blatter, M. V. Feigelman, V. B. Gheskenbein, A. I. Larkin and V. M.
Vinokur.
Rev. Mod. Phys., 66:1125, 1994.
[14]
P. Sibani and Peter B. Littlewood.
Phys. Rev. Lett., 71:1482, 1993.
[15]
P.E. Anderson, H.J. Jensen, L.P. Oliveria and P. Sibani.
Complexity, 10:49, 2004.
[16]
K. Ghosh, S. Ramakrishnan, A. K. Grover, G. I. Menon, G. Chandra, T. V.
C. Rao, G. Ravikumar, P. K. Mishra, V. C. Sahni, C. V. Tomy,
G. Balakrishnan , D. M. Paul and S. Bhattacharya.
Phys. Rev. Lett., 76:4600, 1996.
[17]
S.B. Roy and P. Chaddah.
J. Phys. Cond. Mat., 9:L625, 1997.
[18]
G. Ravikumar, V. C. Sahni, P. K. Mishra, T. V. C. Rao, S. S. Banerjee,
A. K. Grover, S. Ramakrishnan, S. Bhattacharya, M. J. Higgins, E. Yamamoto
, Y. Haga, M. Hedo, Y. Inada and Y. Onuki.
Phys. Rev. B, 57:R11069, 1998.
[19]
S. Kokkaliaris, P. A. J. de Groot, S. N. Gordeev, A. A. Zhukov, R. Gagnon
and L. Taillefer.
Phys. Rev. Lett., 82:5116, 1999.
[20]
S.S. Banerjee, S. Ramakrishnan, D. Pal, S. Sarkar, A. K. Grover,
G. Ravikumar, P. K. Mishra, T. V. C. Rao, V. C. Sahni, C. V. Tomy,
M. J. Higgins and S. Bhattacharya.
J. Phys. Soc. Jpn, 69:262, 2000.
[21]
D.K. Jackson, M. Nicodemi, G. K. Perkins, N. A. Lindop and H. J. Jensen.
Europhys. Lett., 52:210, 2000.
[22]
M. Nicodemi and H.J. Jensen.
Physica C, 341-348:1065, 2000.
[23]
M. Nicodemi and H.J. Jensen.
J. Phys. A, 34:L11, 2001.
[24]
M. Nicodemi and H.J. Jensen.
Phys. Rev. Lett., 86:4378, 2001.
[25]
H.J Jensen and M. Nicodemi.
Europhys. Lett., 54:566, 2001.
[26]
M. Nicodemi.
Phys. Rev. E, 67:041103, 2003.
[27]
M. Nicodemi.
J. Phys.: Cond Matt., 14:2403, 2002.
[28]
M. Daeumling, J.M. Seuntjens, and D.C. Larbalestier.
Nature, 346:332, 1990.
[29]
M. Chikumoto, M. Konczykowski, N. Motohira and A. P. Malozemoff.
Phys. Rev. Lett., 69:1260, 1992.
[30]
G. Yang, P. Shang, S.D. Sutton, I. P. Jones, J. S. Abell and C. E. Gough.
Phys. Rev. B, 48:4054, 1993.
[31]
Y. Yeshurun, N. Bontemps, L. Burlachkov and A. Kapitulnik.
Phys. Rev. B, 49:1548, 1994.
[32]
X.Y. Cai, A. Gurevich, D. C. Larbalestier, R. J. Kelley, M. Onelion and H.
Berger and G. Maragaritondo.
Phys. Rev. B, 50:16774, 1994.
[33]
A.A. Zhukov, H Kupfer, G. Perkins, L. F. Cohen, A. D. Caplin, S. A.
Klestov, H. Claus, V. I. Voronkova, T. Wolf and H. Wuhl.
Phys. Rev. B, 51:12704, 1995.
[34]
A.K. Pradhan, S. B. Roy, P. Chaddah, D. Kanjilal, C. Chen and B. M.
Wanklin.
Physica C, 264:109, 1996.
[35]
H. Kupfer, A. A. Zhukov, A. Will, W. Jahn, R. MeierHirmer, T. Wolf, V. I.
Voronkova, M Klaser and K. Saito.
Phys. Rev. B, 54:644, 1996.
[36]
P. Sibani and J. Dall.
Europhys. Lett., 64:8, 2003.
[37]
J. Dall and P. Sibani.
Eur. Phys. J B, 34:233, 2003.
[38]
P. Sibani and H.J. Jensen.
cond-mat0403212, 2004.
[39]
P. Sibani and H.J. Jensen.
J. Stat. Mech.: Theor. Exp., P10013, 2004.
[40]
T. Puig and X. Obradors.
Phys. Rev. Lett., 84:1571, 2000.
[41]
M.F. Goffman, J. A. Herbsommer, F. de la Cruz, T. W. Li and P. H. Kes.
Phys. Rev. B, 57:3663, 1998.
[42]
H. Safar, E. Rodriguez, F. de la Cruz, P. L. Gammel,
L. F. Schneemeyer and D. J. Bishop.
Phys. Rev. B, 46:14238, 1992.
[43]
D. Lopez, G. Nieva, F. de la Cruz, Henrik Jeldtoft
Jensen and Dominic O.Kane.
Phys. Rev. B, 50:9684, 1994.
[44]
D. Lopez, E. A. Jagla, E. F. Righi, E. Osquiguil, G.
Nieva, E. Morre, F. de la Cruz and C. A. Balseiro.
Physica C, 260:211, 1996.
[45]
M. B. Gaifullin, Yuji Matsuda, N. Chikumoto, J.
Shimoyama and K. Kishio.
Phys. Rev. Lett., 84:2945, 2000.
[46]
R. Busch, G. Ries, H. Werthner, G. Kreiselmeyer and
G. Saemann-Ischenko.
Phys. Rev. Lett., 69:522, 1992.
[47]
D. T. Fuchs, R. A. Doyle, E. Zeldov, D. Majer, W.
S. Seow, R. J. Drost, T. Tamegai, S. Ooi, M. Konczykowski
and P. H. Kes.
Phys. Rev. B, 55:R6156, 1997.
[48]
M. Nicodemi and H. J. Jensen.
Phys. Rev. B, 65:144517, 2002.
[49]
K. Binder.
Rep. Prog. Phys., 60:487, 1997.
[50]
P. Sibani and C. M. Andersen.
Phys. Rev. E, 64, 021103, 2001.
[51]
P. Sibani and A. Pedersen.
Europhys. Lett., 48:346, 1999.
[52]
M. Abramowitz and I. Stegun.
Handbook of Mathematical Functions.
Dover, New York, 1970.
[53]
L. Civale, A. D. Marwick, M. W. McElfresh, T. K. Worthington, A.
P.
Malozemoff, F. Holtzberg, J. R. Thompson and M. A. Kirk.
Phys. Rev. Lett., 65:1164, 1990.
[54]
D.L. Kaiser, F. Holtzberg, M. F. Chisholm and T. K. Worthington.
J. Cryst. Growth, 85:593, 1987. |
McCulloch-Pitts brains
and pseudorandom functions
Vašek Chvátal 111chvatal@cse.concordia.ca, Mark Goldsmith
222markgoldsmith@gmail.com, and Nan Yang
333nan.yang@me.com
Department of Computer Science and Software Engineering
Concordia University, Montreal
Abstract
In a pioneering classic, Warren McCulloch and Walter Pitts
proposed a model of the central nervous system. Motivated by EEG
recordings of normal brain activity, Chvátal and Goldsmith asked
whether or not these dynamical systems can be engineered to produce trajectories which are
irregular, disorderly, apparently unpredictable. We show that they
cannot build weak pseudorandom functions.
Electroencephalogram recordings of normal brain (or of an epileptic
brain well before a seizure) are usually irregular, disorderly, with
no apparent pattern: see, for instance,
[21, 20, 7, 16, 4, 31, 2]. Chvátal and
Goldsmith [5] asked whether or not the McCulloch-Pitts
model of the brain can be engineered to exhibit similar behaviour.
The same question, although without its physiological interpretation,
was also asked in [10]. Let us begin by briefly describing
the McCulloch-Pitts model.
A linear threshold function is a function $f:{\bf R}^{n}\rightarrow\{0,1\}$ such that, for some real numbers
$w_{1},\ldots,w_{n}$ and $\theta$,
$$\textstyle{f(x_{1},\ldots,x_{n})=H\left(\sum_{j=1}^{n}w_{j}x_{j}-\theta\right)}$$
where $H$ is the Heaviside step function defined by $H(d)=1$ for all
nonnegative $d$ and $H(d)=0$ for all negative $d$. Warren
McCulloch and Walter Pitts [22] proposed a model of the
central nervous system built from linear threshold functions. When
this system has $n$ neurons and no peripheral afferents, its
McCulloch-Pitts model is a mapping $\Phi:\{0,1\}^{n}\rightarrow\{0,1\}^{n}$ defined by
$$\Phi(x)=(f_{1}(x),\ldots,f_{n}(x))$$
for some linear threshold functions $f_{1},\ldots,f_{n}$. We will
refer to such mappings $\Phi$ as McCulloch-Pitts dynamical systems.
Chvátal and Goldsmith [5] asked whether or not these
dynamical systems can produce trajectories which are irregular,
disorderly, apparently unpredictable in the sense of generating random
numbers. In making the meaning of their question precise, they took
the point of view of the practitioners, who mean by a random number
generator any deterministic algorithm that, given a short sequence of
numbers, called a seed, returns a longer sequence of numbers;
such a random number generator is considered to be good if it passes
statistical tests from some commonly agreed on battery. (This point of
view is expounded in [18, Chapter 3].)
In this note, we take the point of view of the theorists: we are going to prove that McCulloch-Pitts dynamical systems cannot produce trajectories which are irregular,
disorderly, apparently unpredictable in the sense of providing weak pseudorandom functions. These
have been introduced in [25] and subsume pseudorandom functions, introduced in [13] under the original name of
‘poly-random collections’. Roughly speaking, a weak pseudorandom function is a probability distribution on a set $F_{n}$ of functions
from $\{0,1\}^{n}$ to $\{0,1\}^{n}$ with the following property: if $x^{1},\ldots,x^{m}$ are chosen independently and uniformly at random from
$\{0,1\}^{n}$,
then no polynomial-time randomized algorithm can distinguish with a non-negligible probability between
(i) a sequence $(x^{1},f(x^{1}),\ldots,x^{m},f(x^{m}))$ where $f$ is chosen at random from $F_{n}$ and (ii) a sequence
$(x^{1},y^{1},\ldots,x^{m},y^{m})$ where
$y^{1},\ldots,y^{m}$ are chosen independently and
uniformly at random from $\{0,1\}^{n}$.
(Distinguishing between (i) and (ii) is a trivial matter when $f$ is known and that is why an unknown $f$ must be drawn from
a probability distribution on $F_{n}$.)
Our result shows that weak pseudorandom functions cannot be built from McCulloch-Pitts dynamical systems:
Theorem 1.
There is a polynomial-time deterministic algorithm that, given
a sequence $(x^{1},y^{1},\ldots,x^{m},y^{m})$ of $n$-bit vectors,
returns either the message McCulloch- Pitts
or the message not McCulloch-Pitts in such a way that
(i)
if $y^{1}=\Phi(x^{1}),\ldots,y^{m}=\Phi(x^{m})$ for some McCulloch-Pitts dynamical system
$\Phi$, then the algorithm returns McCulloch-Pitts,
(ii)
if $x^{1},\ldots,x^{m}$ are chosen independently and
uniformly at random from $\{0,1\}^{n}$, if $y^{1},\ldots,y^{m}$ are chosen independently and
uniformly at random from $\{0,1\}^{n}$, and if $m\geq(2+\varepsilon)n$ for some positive constant
$\varepsilon$, then the algorithm returns
not McCulloch-Pitts with probability at least $1-e^{-\delta n}$, where
$\delta$ is a positive constant depending only on $\varepsilon$.
A dichotomy of a set $X$ is its partition into two disjoint
sets. Unlike Cover [6], for whom a dichotomy is an unordered
pair of sets, we view every dichotomy as an ordered pair of sets. A
dichotomy $(X^{+},X^{-})$ of a subset of $\mathbf{R}^{n}$ is linearly
separable if there are numbers $y_{1},\ldots,y_{n+1}$ such that
$$\begin{split}\displaystyle\textstyle{\sum}_{j=1}^{n}x_{j}y_{j}>y_{n+1}&%
\displaystyle\;\;\text{ whenever }(x_{1},\ldots,x_{n})\in X^{+},\\
\displaystyle\textstyle{\sum}_{j=1}^{n}x_{j}y_{j}<y_{n+1}&\displaystyle\;\;%
\text{ whenever }(x_{1},\ldots,x_{n})\in X^{-}.\end{split}$$
(1)
When $f$ is a function from $\{0,1\}^{n}$ to $\{0,1\}$ and $x^{1},\ldots,x^{m}$ are points in
$\{0,1\}^{n}$, the dichotomy $(\{x^{i}:f(x_{i})=0\},\{x^{i}:f(x_{i})=1\})$ is linearly separable if and only if $f$ is a threshold function. Our proof of Theorem 1 evolves from the propositions that
linearly separable dichotomies are easy to recognize and linearly separable dichotomies are rare:
Lemma 1.
Linearly separable dichotomies of $m$-point subsets of $\{0,1\}^{n}$ can
be recognized in time polynomial in $m$ and $n$.
Lemma 2.
For every positive $\varepsilon$ there is a positive $\gamma$ with the
following property: If $\;X$ is a finite subset of $\mathbf{R}^{n}$ such that
$\lvert X\rvert\geq(2+\varepsilon)n$, then a dichotomy chosen uniformly at random from all
dichotomies of $X$ is linearly separable with probability at most
$e^{-\gamma n}$.
Following the seminal report [34], the subject of
learning a hyperplane that separates, or at least nearly separates, the two parts of a
dichotomy received much attention in the machine learning
community. None of it is relevant to the following standard argument,
implicit in the linear programming proof of Minkowski’s Separating Hyperplane Theorem
for convex polytopes [37].
Proof of Lemma 1.
Deciding whether a prescribed dichotomy of an $m$-point subset of $\{0,1\}^{n}$
is linearly separable amounts to solving system (1) of $m$ strict
linear inequalities in variables $y_{1},\ldots,y_{n+1}$, where each coefficient
$x_{j}$ is $0$ or $1$; the epoch-making result of
Khachiyan [17] guarantees that this can be done in time polynomial in $m$ and $n$.
$\Box$
Proof of Lemma 2.
Without loss of generality, we may assume that
that $0<\varepsilon\leq 1$. Let $m$ denote $\lvert X\rvert$ and
let $p$ denote the probability that a dichotomy chosen uniformly at random from
all dichotomies of $X$ is linearly separable.
Of the $2^{m}$ dichotomies of $X$, at most
$2\sum_{i=0}^{n}\binom{m-1}{i}$ are linearly separable (this is at least implicit in [39] and [6]), and so
$$\textstyle{p\leq 2^{-m+1}\sum_{i=0}^{n}\binom{m-1}{i}\leq 2^{-m+1}\sum_{i=0}^{%
n}\binom{m}{i}.}$$
Since $m\geq(2+\varepsilon)n$ and
$0<\varepsilon\leq 1$, we have $n\leq(0.5-\varepsilon/6)m$; a special case of the
well-known bound on the tail of the binomial distribution (see, for
instance, [15, Theorem 1]) guarantees that for every positive
$\alpha$ smaller than $0.5$ there is a positive $\beta$ such that
$$\textstyle{\sum_{i\leq(0.5-\alpha)m}\binom{m}{i}\;\leq\;2^{m}e^{-\beta m};}$$
setting $\alpha=\varepsilon/6$, we conclude that $p\leq 2e^{-\beta m}$, which
proves the lemma.
$\Box$
An alternative proof of Lemma 2, proposed by one of the reviewers, relies
on the Sauer-Shelah Lemma ([35], [36]):
If a family of subsets of an $m$-point set has Vapnik-Chervonenkis dimension $d$,
then it includes at most $\sum_{i=0}^{d}\binom{m}{i}$ sets. Its other ingredient is
the following corollary of Radon’s theorem [33]: If $\mathcal{H}$ is a family
of half-spaces in $\mathbf{R}^{n}$ and if $X$ is a finite subset of $\mathbf{R}^{n}$, then family
$\{X\cap Y:Y\in\mathcal{H}\}$ has Vapnik-Chervonenkis dimension at most $n+1$. Putting
the two together, we conclude that
$X$ has at most $2\sum_{i=0}^{n+1}\binom{m}{i}$ linearly separable dichotomies.
This upper bound, although weaker than our $2\sum_{i=0}^{n}\binom{m-1}{i}$, also
yields the lemma’s conclusion.
Proof of Theorem 1.
The algorithm goes as follows: Let $\alpha^{i}$ denote the first bit of $y^{i}$ and
define
$$\displaystyle X^{+}$$
$$\displaystyle=\{x^{i}:1\leq i\leq m,\;\alpha^{i}=1\},$$
$$\displaystyle X^{-}$$
$$\displaystyle=\{x^{i}:1\leq i\leq m,\;\alpha^{i}=0\}.$$
If this dichotomy is linearly separable, then return McCulloch-Pitts; else return not McCulloch-Pitts.
Lemma 1 guarantees that the algorithm can be implemented to run in polynomial time.
To prove (i), assume that $y^{1}=\Phi(x^{1}),\ldots,y^{m}=\Phi(x^{m})$ for some McCulloch-Pitts dynamical system
$\Phi:\{0,1\}^{n}\rightarrow\{0,1\}^{n}$ defined
by $\Phi(x)=(f_{1}(x),\ldots,f_{n}(x))$. Now
$\alpha^{i}=f_{1}(x^{i})$ for all $i=1,\ldots,m$, which means that $f_{1}$
takes value $1$ on all points of $X^{+}$ and value $0$ on all points of
$X^{-}$; since $f_{1}$ is a threshold function, the dichotomy $(X^{+},X^{-})$
is linearly separable, and so the algorithm returns McCulloch-Pitts.
To prove (ii), assume that $x^{1},\ldots,x^{m}$ are chosen independently and
uniformly at random from $\{0,1\}^{n}$, that $y^{1},\ldots,y^{m}$ are chosen independently and
uniformly at random from $\{0,1\}^{n}$, and that $m\geq(2+\varepsilon)n$ for some positive constant
$\varepsilon$.
Since the probability that the algorithm returns not McCulloch-Pitts
increases as $m$ increases, we may replace the assumption that $m\geq(2+\varepsilon)n$ by the assumption that $m=\lceil(2+\varepsilon)n\rceil$.
Write $X=X^{+}\cup X^{-}$. Since $x^{1},\ldots,x^{m}$ are chosen independently and
uniformly from $\{0,1\}^{n}$, they are pairwise distinct with probability
$2^{n}(2^{n}-1)\cdots(2^{n}-m+1)/2^{nm}$;
since
$$\frac{2^{n}(2^{n}-1)\cdots(2^{n}-m+1)}{2^{nm}}\geq\left(\frac{2^{n}-m}{2^{n}}%
\right)^{m}=\left(1-\frac{m}{2^{n}}\right)^{m}\geq 1-\frac{m^{2}}{2^{n}},$$
this probability is at least $1-5n^{2}2^{-n}$. When
$\lvert X\rvert=m$, the assumption that $y^{1},\ldots,y^{m}$ are chosen independently and
uniformly from $\{0,1\}^{n}$ implies that the dichotomy $(X^{+},X^{-})$ of $X$ is chosen
uniformly from all dichotomies of $X$, in which case
Lemma 2 guarantees that $(X^{+},X^{-})$ is linearly
separable with probability at most $e^{-\gamma n}$ for
some positive constant $\gamma$ depending only on $\varepsilon$. We conclude that
the algorithm returns not McCulloch-Pitts with probability at least
$1-5n^{2}2^{-n}-e^{-\gamma n}$, which is at least $1-e^{-\delta n}$ for
some positive constant $\delta$ depending only on $\varepsilon$.
$\Box$
There is an obvious refinement of the algorithm used in the proof of Theorem 1:
with $y^{i}_{j}$ standing for the $j$-th bit of $y^{i}$, test each of the $n$ dichotomies
$$\left(\{x^{i}:1\leq i\leq m,\;y^{i}_{j}=1\},\{x^{i}:1\leq i\leq m,\;y^{i}_{j}=%
0\}\right)\phantom{xxx}(j=1,\ldots,n)$$
and return McCulloch-Pitts if and only if all $n$ of them are linearly separable.
In the context of distinguishing McCulloch-Pitts functions from truly random functions, the extra work required in this refinement is pointless. The probability of returning McCulloch-Pitts when $y^{1},\ldots,y^{m}$ are chosen independently and uniformly at random from $\{0,1\}^{n}$ is at most $e^{-\delta n}$ in the original version and that is good enough; reducing it further to $e^{-\delta n^{2}}$ in the refinement is nice, but unnecessary. In addition, the assumption $m\geq(2+\varepsilon)n$ cannot be significantly relaxed even in the refinement: it is at least implicit in [39] and [6] that a dichotomy chosen uniformly at random from
all dichotomies of a set of fewer than $(2-\varepsilon)n$ points in $\mathbf{R}^{n}$ is linearly separable with probability at least $1-e^{-\delta n}$.
Theorem 1 implies that certain simple devices (namely, McCulloch-Pitts dynamical
systems) cannot generate pseudorandomness.
In the opposite direction, it has been proved that certain simple devices can generate pseudorandomness: examples can be found in [24], [19], [27], [26], [3].
The question whether McCulloch-Pitts networks can produce trajectories which are
irregular, disorderly, apparently unpredictable remains open: all
depends on the interpretation of the terms “irregular, disorderly,
apparently unpredictable”. When clinical neurologists visually
inspect an electroencephalogram, their vague criteria for declaring it
random-like are a far cry from the distinguishers that cryptographers use
to separate deterministic sequences from random sequences. As Avi
Wigderson [38, page 6] put it,
“Randomness is in the eye of the beholder, or more precisely, in its
computational capabilities … a phenomenon (be it natural or
artificial) is deemed “random enough,” or pseudorandom, if the class
of observers/applications we care about cannot distinguish it from
random!”
Many examples of generators that appear random
to observers with restricted computational powers are known. In
particular, pseudorandom generators for polynomial size constant depth
circuits have been constructed in [1];
later, this work was greatly simplified and improved in [28]. O’Connor [30] proved that an infinite
binary sequence appears random to all finite-state machines if and
only if it is $\infty$-distributed. Pseudorandom generators for
space-bounded computation have been constructed in [29].
It is conceivable that McCulloch-Pitts dynamical systems could fool
neurologists into finding their trajectories unpredictable just as
they find normal electroencephalograms unpredictable. Proving this in
a formal setting with a suitable definition of ‘neurologists’ is an
interesting challenge.
A variation on our theme comes from the idea that
in a brain of $n$ neurons, only $m$ neurons may be visible to the observer and
the remaining $n-m$ are hidden from view. Formally, given positive integers $m,n$ such that $m\leq n$
and given a McCulloch-Pitts dynamical system $\Phi:\{0,1\}^{n}\rightarrow\{0,1\}^{n}$, we may consider the mapping
${\Phi}_{m}:\{0,1\}^{n}\rightarrow\{0,1\}^{m}$
such that ${\Phi}_{m}(x)$ is the $m$-bit prefix of $\Phi(x)$.
Can such mappings provide pseudorandomness? Our Theorem 1 shows that the
answer is negative when $m=n$; one of the reviewers argued that, under the usual assumption that
one-way functions exist, the answer is close to affirmative when $m=1$. Here is the argument: Every one-way function $f$ (as every boolean function) can be computed by a threshold circuit [32, Chapter 7]. When this circuit has $n$ gates and depth $d$, it can be imbedded in a McCulloch-Pitts dynamical system $\Phi:\{0,1\}^{n}\rightarrow\{0,1\}^{n}$, where the results of its computation show up with the time delay of $d$ units. Now $f$ is represented in the $d$-fold iteration of $\Phi$
and there is an appropriate projection $\pi$, the hard-core bit [11], such
that the sequence $\pi(x)$, $\pi(f(x))$, $\pi(f(f(x)))$, …is pseudorandom.
Statistical properties of ${\Phi}_{1}$ have been studied in [14, Section 4.2].
For instance, there is a McCulloch-Pitts dynamical system $\Phi:\{0,1\}^{37}\rightarrow\{0,1\}^{37}$ such that the restriction of the trajectory of $\Phi$ on the first bit
passes all ten statistical tests of the battery SmallCrush implemented in
the software library TestU01 of [8, 9].
Acknowledgments
This research was undertaken, in part, thanks to funding from the
Canada Research Chairs program and from the Natural Sciences and
Engineering Research Council of Canada. We are grateful to Péter
Gács for helpful comments on a draft of this note and to Avi
Wigderson for telling us about Nisan’s papers [28, 29]. We
also thank the two anonymous reviewers for their thoughtful comments that
made us improve the presentation considerably.
References
[1]
M. Ajtai and A. Wigderson, Deterministic simulation
of probabilistic constant depth circuits, 26th Annual Symposium
on Foundations of Computer Science, pp. 11–19, IEEE, 1985.
[2]
S. Altunay, Z. Telatar, and O. Erogul, Epileptic EEG
detection using the linear prediction error energy, Expert Systems
with Applications 37 (2010), 5661–5665.
[3]
B. Applebaum, Y. Ishai, E. Kushilevitz, Cryptography by cellular
automata or how fast can complexity emerge in nature? in: ICS (pp.
1–19), 2010.
[4]
W.A. Chaovalitwongse, Optimization and Data Mining in
Epilepsy Research: A Review and Prospective, in: Handbook of
Optimization in Medicine (P.M. Pardalos and H.E. Romeijn,
eds.), Springer, 2009, pp. 1–32.
[5]
V. Chvátal and M. Goldsmith, Can brains generate
random numbers?
arXiv:1208.6451 [math.DS].
[6]
T. Cover, Geometrical and statistical properties of
systems of linear inequalities with applications in pattern
recognition, IEEE Transactions on Electronic Computers
(1965), 326–334.
[7]
F.L. Da Silva et al., Epilepsies as dynamical diseases of brain
systems: basic models of the transition between normal and epileptic
activity, Epilepsia 44 (2003), 72–83.
[8]
P. L’Ecuyer, R. Simard, TestU01: A C library for
empirical testing of random number generators, ACM Transactions on
Mathematical Software 33 (2007), Article 22, 40 pages.
[9]
P. L’Ecuyer, R. Simard, TestU01: A software library in ANSI C
for empirical testing of random number generators. UserÂs guide,
compact version.
(Version: August 17, 2009).
http://www.iro.umontreal.ca/~simardr/testu01/guideshorttestu01.pdf
[10]
Y.M. Elyada, D. Horn, Can dynamic neural filters produce pseudo-random sequences?
in: Artiï¬cial neural networks: Biological inspirations - ICANN 2005 (pp. 211–216). Springer, 2005
[11]
O. Goldreich, L.A. Levin, A hard-core predicate for all one-way
functions, Proceedings of the 21st annual ACM symposium on theory
of computing (pp. 25–32), 1989.
[12]
O. Goldreich, Foundations of cryptography. Basic
tools, Cambridge University Press, Cambridge, 2001.
[13]
O. Goldreich, S. Goldwasser, S. Micali, How to construct random
functions, Journal of the ACM 33 (1986), 792–807.
[14]
M. Goldsmith, Neural networks as pseudorandom number generators
(Unpublished doctoral dissertation), Concordia University, 2015.
[15]
W. Hoeffding, Probability inequalities for sums of
bounded random variables, Journal of the American Statistical
Association 58 (1963), 13 – 30.
[16]
L.D. Iasemidis et al., Dynamical resetting of the human brain
at epileptic seizures: application of nonlinear dynamics and global
optimization techniques, IEEE Transactions on Biomedical Engineering
51 (2004), 493–506.
[17]
L.G. Khachiyan, A polynomial algorithm in linear
programming. (Russian), Doklady Akademii Nauk SSSR 244 (1979),
1093–1096.
[18]
D.E. Knuth, The Art of Computer Programming, Volume 2: Seminumerical
Algorithms, Addison-Wesley Professional, 2014.
[19]
M. Krause, S. Lucks, Pseudorandom functions in $TC^{0}$ and cryptographic limitations to proving lower bounds,
Computational complexity 10 (2001), 297–313.
[20]
K. Lehnertz, et al., Nonlinear EEG analysis in epilepsy: Its
possible use for interictal focus localization, seizure anticipation,
and prevention, Journal of Clinical Neurophysiology 18 (2001) 209–222.
[21]
A. Liu, et al., Detection of neonatal seizures through
computerized EEG analysis, Electroencephalography and clinical
neurophysiology 82 (1992) 30– 37.
[22]
W.S. McCulloch, W.Pitts, A logical calculus of the
ideas immanent in nervous activity, Bulletin of Mathematical
Biophysics 5 (1943) 115–133.
[23]
S. Muroga, Threshold logic and its
applications, Wiley-Interscience [John Wiley & Sons], New
York-London-Sydney, 1971.
[24]
M. Naor, B. Pinkas, O. Reingold, Distributed pseudo-random
functions and KDCs, in: Advances in cryptology - EUROCRYPT99 (pp.
327–346), 1999.
[25]
M. Naor, O. Reingold, Synthesizers and their application to the
parallel construction of pseudo-random functions, 36th Annual Symposium
on Foundations of Computer Science, pp. 170–181, IEEE, 1995.
[26]
M. Naor, O. Reingold, Number-theoretic constructions of efficient
pseudo-random functions, Journal of the ACM 51 (2004), 231–262.
[27]
J.B. Nielsen, A threshold pseudorandom function construction and
its applications, in: Advances in cryptology - Crypto 2002 (pp. 401–416),
Springer, 2002.
[28]
N. Nisan, Pseudorandom bits for constant depth circuits, Combinatorica 11 (1991), 63–70.
[29]
N. Nisan, Pseudorandom generators for space-bounded
computation, Combinatorica 12 (1992), 449–461.
[30]
M.G. O’Connor, An unpredictability approach to finite-state
randomness, Journal of Computer and System Sciences 37 (1988),
324–336.
[31]
H. Ocak, Automatic detection of epileptic seizures in EEG using
discrete wavelet transform and approximate entropy, Expert Systems
with Applications 36 (2009), 2027–2036
[32]
I. Parberry, Circuit complexity and neural networks, MIT press, 1994.
[33]
J. Radon, Mengen konvexer körper, die einen gemeinsamen punkt
enthalten, Mathematische Annalen 83 (1921), 113–115.
[34]
F. Rosenblatt, The perceptron — a perceiving and recognizing automation. Cornell Aeronautical Laboratory, Ithaca. Report 85-460-1, 1957.
[35]
N. Sauer, On the density of families of sets, Journal of Combinatorial
Theory Series A 13 (1972), 145–147.
[36]
S. Shelah, A combinatorial problem; stability and order for models
and theories in infinitary languages, Pacific Journal of Mathematics
41 (1972), 247–261.
[37]
A.W. Tucker, Linear inequalities and convex polyhedral sets,
Proc. 2nd symp. linear programming (pp. 569–602), 1955.
[38]
A. Wigderson, Randomness and Pseudorandomness, The Institute Letter, Summer 2009, pp. 1,6,7.
http://www.ias.edu/files/pdfs/publications/letter-2009-summer.pdf
[39]
R.O. Winder, Partitions of $N$-space by hyperplanes,
SIAM Journal on Applied Mathematics 14 (1966), 811–818. |
Momentum-resolved spin splitting in Mn-doped trivial CdTe
and topological HgTe semiconductors
Carmine Autieri
autieri@MagTop.ifpan.edu.pl
International Research Centre MagTop, Institute of Physics, Polish Academy of Sciences, Aleja Lotnikow 32/46, PL-02668 Warsaw, Poland
Cezary Śliwa
Institute of Physics, Polish Academy of Sciences, Aleja Lotnikow 32/46, PL-02668 Warsaw, Poland
Rajibul Islam
International Research Centre MagTop, Institute of Physics, Polish Academy of Sciences, Aleja Lotnikow 32/46, PL-02668 Warsaw, Poland
Giuseppe Cuono
International Research Centre MagTop, Institute of Physics, Polish Academy of Sciences, Aleja Lotnikow 32/46, PL-02668 Warsaw, Poland
Tomasz Dietl
International Research Centre MagTop, Institute of Physics, Polish Academy of Sciences, Aleja Lotnikow 32/46, PL-02668 Warsaw, Poland
WPI-Advanced Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan
Abstract
A series of experiments demonstrated that exchange-induced splitting of magnetooptical spectra of Cd${}_{1-x}$Mn${}_{x}$Te at the $L$ points of the Brillouin zone is, unexpectedly, more than one order of magnitude smaller compared to its magnitude at the zone center and show an unexpected sign of the effective Landé factor. We have determined spin-splitting of the valence and conduction bands in the whole Brillouin zone in Cd${}_{1-x}$Mn${}_{x}$Te and in topologically-nontrivial Hg${}_{1-x}$Mn${}_{x}$Te by means of relativistic first-principles density functional calculations. We find that spin splitting of bands is relatively large at the $L$ points but, in contrast to the $\Gamma$ point, effective exchange integrals have the same sign and similar magnitudes at the $L$ points. This results in comparable energies of the optical transitions for two circular light polarizations leading to small splitting of optical spectra in Cd${}_{1-x}$Mn${}_{x}$Te. Our results substantiate also previous suggestions that the antiferromagnetic sign and a relatively high magnitude of the effective exchange integral in the conduction band away from the $\Gamma$ point results from an admixture of anion $p$-type wave functions and the proximity to upper Hubbard band of Mn $d$ electrons. We use the obtained results to determine parameters of a minimal tight-binding model that describe rather accurately the band structure, including spin-orbit and exchange splittings of bands in the whole Brillouin zone of Cd${}_{1-x}$Mn${}_{x}$Te and Hg${}_{1-x}$Mn${}_{x}$Te.
I Introduction
Dilute magnetic semiconductors (DMSs), such as Cd${}_{1-x}$Mn${}_{x}$Te and Hg${}_{1-x}$Mn${}_{x}$Te, have played a central role in the demonstrating and describing a strong and intricate influence of the $sp$-$d$ exchange interactions upon effective mass states in semiconductors Kos (2010); Dietl (1994); Furdyna (1988), paving the way for the rise of dilute ferromagnetic semiconductors Dietl and Ohno (2014) and magnetic topological insulators Ke et al. (2018); Tokura et al. (2019). One of the key characteristics of DMSs in a giant spin splitting of bands proportional to the field-induced and temperature dependent magnetization of paramagnetic Mn${}^{2+}$ ions, $M(T,H)$. In the case of high electron mobility modulation-doped Cd${}_{1-x}$Mn${}_{x}$Te/Cd${}_{1-y}$Mg${}_{y}$Te heterostructures, the exchange splitting leads to crossings of spin-resolved Landau levels, at which the quantum Hall ferromagnet forms at low temperatures Jaroszyński et al. (2002). It has been recently proposed that magnetic domains of this ferromagnet, if proximitized by a superconductor, can host Majorana modes Kazakov et al. (2016, 2017); Simion et al. (2018). Similarly, Hg${}_{1-x}$Mn${}_{x}$Te/Hg${}_{1-y}$Cd${}_{y}$Te quantum wells of an appropriate thickness and Mn cation concentration $x\lesssim 7$%, which ensures the inverted band structure, show such a lauder of spin sublevels in the magnetic field, which is expected to enable the appearance of the quantum anomalous Hall effect Liu et al. (2008).
According to the present insight there are two exchange mechanisms involved in the interaction ${\cal{H}}_{sp-d}=-J{\bm{s}}\cdot{\bm{S}}$ between effective mass electrons in the vicinity of $\Gamma$ and localized spins residing on the half-filled Mn${}^{2+}$ $d$-shells Kacman (2001). The first on them is the ferromagnetic direct (potential) exchange $J_{sd}$ between band carriers with wave functions derived from Mn $s$ orbitals and electrons localized on the open Mn $d$ shells, usually denoted $N_{0}\alpha$, typically of the order of 0.2 eV. The second one is the antiferromagnetic kinetic exchange between band carriers with anion $p$-type wave functions and $d$ electrons, of the order of $J_{pd}\equiv N_{0}\beta\approx-1$ eV. Incorporation of these interactions into an appropriate multi-band $kp$ Hamiltonian allows one to describe satisfactorily various spectacular magnetotransport and magnetooptical phenomena for carriers near the $\Gamma$ point of the Brillouin zone as a function of $M(T,H)$ Kos (2010); Dietl (1994); Furdyna (1988); Bastard et al. (1978), particularly if effects of strong coupling are taken into account Dietl (2008).
However, in contrast to the $\Gamma$ point, the physics of exchange splittings at the $L$ points of the Brillouin zone is challenging: a series of magnetoreflectivity and magnetic circular dichroism (MCD) studies, notably for Cd${}_{1-x}$Mn${}_{x}$Te Ginter et al. (1983); Coquillat et al. (1986); Ando et al. (2011), has revealed that the magnitudes of spectra splittings for two circular light polarizations at the $L$ points ($E_{1}$ and $E_{1}+\Delta_{1}$ transitions) are smaller by a factor of about seventeen compared to the value at the $\Gamma$ point, the effect not explained by tight-binding modeling Ginter et al. (1983); Bhattacharjee (1990). Furthermore, effective Landé factors corresponding to these transitions show unexpected signs Ando et al. (2011). The situation is also unsettled in Hg${}_{1-x}$Mn${}_{x}$Te, in which a large magnitude of spin-orbit-driven spin-splittings accounts for a controversy concerning the actual magnitudes of the $sp$-$d$ exchange integrals at the $\Gamma$ point Dietl (1994), making their comparison to spin-splitting values at the $L$ points Coquillat et al. (1989) not conclusive. Accordingly, it has been pointed out that the electronic structures of II–VI DMSs have not been as well clarified as we previously believed Ando et al. (2011). Among other issues, this fact may preclude a meaningful evaluation of the role played by interband spin polarization in mediating indirect exchange interactions between magnetic ions. This Bloembergen-Rowland mechanism Bloembergen and Rowland (1955) is known to play a sizable role in p-type dilute ferromagnetic semiconductors, in which it involves virtual transitions between hole valence subbands Dietl et al. (2001); Ferrand et al. (2001). Moreover, this spin-spin exchange is expected to be particularly important in the absence of carriers in the inverted band structure case (such as Hg${}_{1-x}$Mn${}_{x}$Te), in which both the valence and conduction bands are primarily built of anion $p$-type wave functions Bastard and Lewiner (1979); Yu et al. (2010).
In the last years, several ab initio studies of Cd${}_{1-x}$Mn${}_{x}$Te have been carried out Wei and Zunger (1987); Larson et al. (1988); Merad et al. (2006); Liu and Liu (2008); Echeverría-Arrondo et al. (2009); Verma et al. (2011); Wua et al. (2015); Linneweber et al. (2017).
However, these works were not attempted to elucidate the origin of the anomalously exchange-induced splittings of optical spectra corresponding to transitions at the Brillouin zone boundary. In this paper, we present results of our relativistic first-principles density functional computations for both Cd${}_{1-x}$Mn${}_{x}$Te and Hg${}_{1-x}$Mn${}_{x}$Te. Within the model that neglects the spin-orbit coupling (SOC) we discuss the origin of inverted band structure in Hg${}_{1-x}$Mn${}_{x}$Te. Furthermore, we trace the evolution of exchange-induced splittings of the valence and conduction band with the ${\bm{k}}$ vector. These results, together with the determined orbital components of the wave functions, explain qualitatively the origin of a reduction of exchange splittings at $L$ points of the Brillouin zone compared to the $\Gamma$ point as well as substantiate the origin of the change of sign of the conduction band splitting with the ${\bm{k}}$ vector in Cd${}_{1-x}$Mn${}_{x}$Te. We then include SOC and use the obtained ab initio band dispersion to determine parameters of a versatile tight-binding model that describe rather accurately the band structure and $sp$-$d$ exchange splitting of bands in the whole Brillouin zone of Cd${}_{1-x}$Mn${}_{x}$Te and Hg${}_{1-x}$Mn${}_{x}$Te taking SOC into account. This model allows us to determine optical transition energies and magnetic circular dichroism for any point of the Brillouin zone, and for an arbitrary orientation of the magnetization vector in respect to crystal axes. The model is validated by its good agreement with hitherto unexplained magnetic circular dichroism data corresponding to optical transition at the $L$ points of the Brillouin zone in Cd${}_{1-x}$Mn${}_{x}$Te and Hg${}_{1-x}$Mn${}_{x}$Te. An important outcome of our work is the determination, by combining DFT and experimental information, of a tight-binding model that describes quantitatively the electron band structure of CdTe, HgTe, Cd${}_{1-x}$Mn${}_{x}$Te, and Hg${}_{1-x}$Mn${}_{x}$Te in the whole Brillouin zone.
II Computational methodology
We have performed first-principles density functional theory (DFT)
calculations by using the relativistic VASP package based
on plane wave basis set and projector augmented wave method (PAW) Kresse and Furthmüller (1996).
We perform a fully relativistic calculation for the core-electrons while the valence electrons are treated in a scalar relativistic approximation considering the mass-velocity and the Darwin terms.
Spin-orbit coupling of the valence electrons is included using the second-variation method and the scalar-relativistic eigenfunctions of the valence states Hafner (2008).
A plane-wave energy cut-off of 400 eV has been used. For the bulk, we have performed the calculations using 8$\times$8$\times$8 $\bm{k}$-point Monkhorst-Pack grid with 176 $k$-points in the absence of SOC and with 512 $k$-points in presence of SOC
in the irreducible Brillouin zone.
We use the experimental lattice constants corresponding to $a=6.46152$ Å for HgTe and 6.4815 Å for CdTe Skauli and Colin (2001).
For the treatment of exchange-correlation, Perdew-Burke-Ernzerhof (PBE)
generalized gradient approximation (GGA) Perdew et al. (1996) and the the modified Becke-Johnson exchange potential (MBJLDA)Becke and Johnson (2006); Tran and Blaha (2009) have been applied.
According to the computed band structures in GGA, the magnitudes of the band gap $E_{0}=E(\Gamma 6)-E(\Gamma 8)$ are $0.77$ eV and $-0.50$ eV for CdTe and HgTe, to be compared to experimental values at 4.2 K $E_{0}=1.60$ eV and $-0.30$ eV, respectively. These discrepancies reflect the well-known inaccuracies of the GGA in the evaluation of the band-gap. Thus, in order to improve the tight-binding parametrization of CdTe and HgTe band structures, the MBJLDA have been employed for the determination of the hopping parameters.
Our results, obtained within this computationally more demanding approach, confirm that the determined magnitudes of the band gaps Camargo-Martínez and Baquero (2012), as well as of spin-orbit splittings, are close to experimental values.
The effect of Mn doping in Cd${}_{1-x}$Mn${}_{x}$Te and Hg${}_{1-x}$Mn${}_{x}$Te has been studied using a $4\times 4\times 4$ supercell with 64 anions and 64 cations. The SQS calculations have been done using a 2$\times$2$\times$2 $\bm{k}$-point grid. We use the special quasi-random structure (SQS) Zunger et al. (1990) to model the distribution of cation-substitutional Mn atoms in the supercell. To create a large SQS model, we used the mcsqs algorithm Van De Walle et al. (2013) within the framework of alloy theoretic automed toolkit (ATAT) van de Walle et al. (2002).
The mcsqs method is based on the Monte Carlo simulated annealing loop with an objective function that search for a perfectly match maximum number of correlation functions for a fixed shape of the supercell along with the occupation of the atomic site by minimizing the objective function.
The clusters of doublet, triplet and quadruplet are generated using the cordump utility of the ATAT toolkit. We use the parameters -2, - 3, and -4 for which the longest pair, triplet, and quadruplet correlation distance to be matched is 2.0, 1.5, and 1.0 lattice constants, respectively. To create the best SQS structure, we produce all possible structures and choose the one for which the correlation difference in respect to random structure is closest to zero.
In our work, we focused on Mn content $x=2/64,4/64$ and $8/64$. Since we look for magnitudes of $sp$-$d$ exchange splittings, the Mn magnetic moments are always ferromagnetically aligned.
The Hubbard $U$ effects for Mn open $d$ shell have been included. We use the values of $U_{\text{Mn}}=3$ , 5 and 7 eV Paul et al. (2014); Autieri and Sanyal (2014); Keshavarz et al. (2017) and
$J_{\text{H}}=0.15U$ eV for the Mn-$3d$ states.
After obtaining the Bloch wave functions in density functional theory, the maximally localized Wannier functions Marzari and Vanderbilt (1997); Souza et al. (2001) (MLWF)
are constructed using the WANNIER90 code Mostofi et al. (2008). To extract the character of the
electronic bands at low energies, we used the Slater-Koster interpolation scheme based on Wannier functions.
Quantities of interest here are effective exchange energies $J_{c}({\bm{k}})$ and $J_{v}({\bm{k}})$ calculated from ${\bm{k}}$-dependent splittings of the lowest conduction and highest valence bands, generated by exchange interactions with Mn spins $S=5/2$, aligned by an external magnetic field,
$$J_{c}(\bm{k})=\frac{{\Delta}E_{c}}{xS}=\frac{E_{c}^{\downarrow}-E_{c}^{%
\uparrow}}{xS},\quad J_{v}(\bm{k})=\frac{{\Delta}E_{v}}{xS}=\frac{E_{v}^{%
\downarrow}-E_{v}^{\uparrow}}{xS}.$$
(1)
According to this definition, in the weak coupling limit and for the normal band ordering, i.e., for Cd${}_{1-x}$Mn${}_{x}$Te, $J_{c}(k=0)\equiv N_{0}\alpha$ and $J_{v}(k=0)\equiv N_{0}\beta$, where $N_{0}$ is the cation concentration, whereas $\alpha$ and $\beta$ are $s$-$d$ and $p$-$d$ exchange integrals according to the DMS literature Kacman (2001); Wei and Zunger (1987); Larson et al. (1988). The same situation takes place in the case of Hg${}_{1-x}$Mn${}_{x}$Te with $x\gtrsim 0.07$ Furdyna (1988). However, at lower $x$, Hg${}_{1-x}$Mn${}_{x}$Te is a zero-gap semiconductor with an inverted band structure (topological zero-gap semiconductor) for which the $s$-type $\Gamma_{6}$ band is below the $\Gamma_{8}$ $j=3/2$ multiplet forming the conduction and valence bands. In this case, we consider the spin-splitting of the $\Gamma_{6}$ band below the Fermi level as $J_{c}$. We note also that because of antiferromagnetic interactions between Mn spins, an effective Mn concentration $x_{\text{eff}}$ that contributes to the $sp-d$ exchange splitting of bands in a magnetic field is much smaller than $x$, typically $x_{\text{eff}}\lesssim 5$% for any $x$ in relevant magnetic fields $\mu_{0}H\lesssim 6$ T Gaj et al. (1979). For random distribution of Mn over cation sites, these antiferromagnetic interactions result in spin-glass freezing at low temperatures Galazka et al. (1980); Mycielski et al. (1984).
III Results
III.1 DFT band structure for Cd${}_{1-x}$Mn${}_{x}$Te and Hg${}_{1-x}$Mn${}_{x}$Te without spin-orbit coupling
We discuss first the electronic structure of Cd${}_{1-x}$Mn${}_{x}$Te and Hg${}_{1-x}$Mn${}_{x}$Te computed with relativistic corrections in the scalar approximation, i.e., taking into account the Darwin and mass-velocity terms (essential in Hg${}_{1-x}$Mn${}_{x}$Te) but neglecting spin-orbit coupling (SOC). This allows us to extract spin splittings solely due to the exchange interactions between host and Mn spins, i.e., effective exchange integrals $J_{c}$ and $J_{v}$ for relevant bands and arbitrary $\bm{k}$ vectors in the Brillouin zone.
Figure 1 presents the electronic structure of Cd${}_{0.875}$Mn${}_{0.125}$Te for spin up and spin down evaluated assuming $U_{\text{Mn}}=5$ eV. The Mn lower and upper Hubbard $3d$-bands reside around 4.6 eV below and 2.5 eV above the valence band top, respectively.
Hence, in agreement with photoelectron spectroscopy Mizokawa et al. (2002), an effective Hubbard energy of Mn-$3d$ electrons is 7.1 eV for $U_{\text{Mn}}=5$ eV and, of course, would increase with the increasing $U_{\text{Mn}}$. At the same time, experimental data Mizokawa et al. (2002) indicate that the Mn $d$-bands reside by about 1 eV higher in respect to host bands than implied by our DFT results.
In the whole Brillouin zone and for both spin channels, the lowest unoccupied states consist mainly of Cd-$5s$ states, whereas Te-$5p$ states give a dominant contribution to the highest occupied bands. To give an estimation of the orbital contribution in DFT, we evaluate the system
at the $\Gamma$ point, where the $s$-states are decoupled from the $p$ and $d$-states.
For the (Mn,Cd)Te, the conduction band is composed roughly by 70% Cd-$s$ states and 30% Te-$s$ states with a minor contribution from the impurity Mn-$s$ states for the low Mn concentration $x$ in question. The conduction band at the $\Gamma$ point is composed roughly by 80% Te-$p$ states and 20% Cd-$d$ states with a minor contribution from the impurity Mn-d states.
From the electronic structure of Hg${}_{0.875}$Mn${}_{0.125}$Te at $U_{\text{Mn}}=5$ eV without SOC,
the effective Hubbard energy of Mn-$3d$ electrons at the $\Gamma$ point is 7.8 eV for $U_{\text{Mn}}=5$ eV.
As shown in Fig. 2, the $\Gamma_{6}$ bands and the $\Gamma_{8}$ are inverted in Hg${}_{1-x}$Mn${}_{x}$Te resulting in the topological character of the compound. The relativistic Darwin term gives a weak positive contribution to the energy of the for $s$-bands in heavy atoms like Hg, whereas the relativistic mass-velocity term gives a strong negative contribution, and accounts for the band inversion. We can clearly see in Fig. 2 that the $\Gamma_{6}$ band at 0.5 eV below the Fermi level has the spin up component at lower energies indicating the ferromagnetic sign of the exchange interaction with Mn spins. Instead, the carrier spins in the $\Gamma_{8}$ bands are antiferromagnetically coupled to Mn spins.
III.2 Spin splitting along the k path
without spin-orbit coupling
Figure 3 shows exchange energies $J_{c}(\bm{k})$ and $J_{v}(\bm{k})$ computed for Cd${}_{1-x}$Mn${}_{x}$Te with various Mn concentrations $x$ and $U_{\text{Mn}}=5$ eV. In agreement with the experimental results Gaj et al. (1979), the values determined for the $\Gamma$ point do not depend on $x$, and their DFT values, $N_{0}\alpha=0.28$ eV and $N_{0}\beta=-0.63$ eV, describe the sign, and also reasonably well the experimental magnitudes, $N_{0}\alpha=0.22$ eV and $N_{0}\beta=-0.88$ eV Gaj et al. (1979), depicted by horizontal lines in Fig. 3. The exchange splittings at $\Gamma$ means that there is a large energy difference between transitions from the two heavy hole subbands (or for the creation of the heavy hole excitons). This giant Zeeman splitting is described by $\Delta E\simeq xS(N_{0}\alpha-N_{0}\beta)$.
As seen, $J_{v}$ remains negative (antiferromagnetic) for all $\bm{k}$-values, and its magnitude slightly oscillates along the $\bm{k}$-path, it reaches a maximum at $\Gamma$ and a minimum at $L$ for small Mn concentrations,
and between the $\Gamma$ and $X$ points at the highest studied $x=8/64=0.125$.
In contrast to $J_{v}$, $J_{c}$ changes sign and is highly oscillating along the $\bm{k}$-path:
The sign of $J_{c}$ is positive (ferromagnetic potential $s$-$d$ exchange) at the $\Gamma$ point, in which the conduction band wave function has the $s$-type character, but becomes antiferromagnetic away from the $\Gamma$ point. This behavior originates from (i) an admixture of anion $p$ wave functions to the Bloch amplitudes $u_{\bm{k}}$ and, thus, from a significant role of antiferromagnetic $p$-$d$ kinetic exchange and (ii) hybridization of $\exp(i\bm{k}\cdot\bm{r})$ component of the Bloch wave function with $d$ shell resulting in antiferromagnetic exchange, known from the Kondo physics in dilute magnetic metals Schrieffer and Wolff (1966).
In the antiferromagnetic sign region, the absolute value of $J_{c}$ reaches a maximum at the $X$ points and a minimum at the $U$ points at small concentrations and at the $L$ points for $x=0.125$. Such a dependence results from an increase of $p$-$d$ hybridization and, thus, of the kinetic exchange if a given state approaches the $3d$ Mn shell, in the $J_{c}$ case, the upper Hubbard $3d^{6}$ band. In agreement with this interpretation, except for $J_{c}$ at the $\Gamma$ point, the spin splitting gets reduced when we increase $U_{\text{Mn}}$ because the Mn $d$-states are pushed away from bands, and the electronic hybridization between the host bands and the $3d$ shells of Mn-impurities is suppressed.
According to the ab initio results, the exchange splittings $J_{c}$ and $J_{v}$ at the $L$ points have the same sign and similar magnitudes. This qualitatively explains why the experimentally observed splitting of optical spectra is relatively small at $L$ compared to $\Gamma$ Ginter et al. (1983); Coquillat et al. (1986); Ando et al. (2011). In view of our results, the previous attempt to interpret this large reduction of $\Delta E$ at $L$ was quantitatively unsuccessful because a strong dependence of $J_{c}$ on $\bm{k}$ was disregarded Bhattacharjee (1990). At the same time, our data suggest a relatively strong dependence of $J_{c}$ and $J_{v}$ at $L$ on $x$. This is not corroborated by experimental results accumulated so far. In particular, defining a splitting reduction factor $p$ by
$$p=\frac{J_{c}(L)-J_{v}(L)}{J_{c}(\Gamma)-J_{v}(\Gamma)},$$
(2)
the data in Fig. 3 imply $p=-0.082$ and 0.055 for $x=3.125$ and 0.0625, respectively. The experimentally determined ratio of splittings at $L$ and $\Gamma$ was found to be in the range 0.051-0.082, i.e., $p$ is small in accord with our results but stays positive independently of $x$ Ginter et al. (1983); Coquillat et al. (1986); Ando et al. (2011).
Finally, we mention the relevance of our ab initio results to experimental and theoretical studies of $N_{0}\alpha$ as a function of quantum well thickness $t$ in a series of n-Cd${}_{0.98}$Mn${}_{0.02}$Te quantum wells sandwiched between Cd${}_{0.876}$Mn${}_{0.14}$Mg${}_{0.11}$Te barriers Merkulov et al. (1999). Our evaluation indicates that the decrease of $J_{c}$ with $\bm{k}$ (see Fig. 3) together with an increase in the penetration of the electron wave function into barriers can explain an experimentally observed decrease of $N_{0}\alpha$ with decreasing $t$ Merkulov et al. (1999).
Figure 4 shows $J_{c}(\bm{k})$ and $J_{v}(\bm{k})$ extracted from the band structure computations without SOC for Hg${}_{1-x}$Mn${}_{x}$Te with different values of $x$. In the vicinity of the zone center we present single data points corresponding to the exchange energy of the $\Gamma_{6}$ band, i.e., $N_{0}\alpha$. The trends in $\bm{k}$ dependencies are similar to the Mn-doped CdTe. In particular, $J_{v}$ stays negative in the whole Brillouin zone and $J_{c}(\bm{k})$ becomes negative away from the zone center.
On the experimental side, there are two sets of the determined $N_{0}\alpha$ and $N_{0}\beta$ values, differing by more than a factor of two, in the case of Hg${}_{1-x}$Mn${}_{x}$Te Dietl (1994). Our computational results point to the lower values, i.e., $N_{0}\alpha=0.4$ eV and $N_{0}\beta=-0.6$ eV Furdyna (1988); Dobrowolska and Dobrowolski (1981). Furthermore, according to experimental findings,
magnetic circular dichroism at $L$ has the same sign for Hg${}_{1-x}$Mn${}_{x}$Te as found for Cd${}_{1-x}$Mn${}_{x}$Te at $L$ and at $\Gamma$, independently of Mn content $x$ Coquillat et al. (1989). Our data suggest the opposite sign since, according to the results in Fig. 4, $J_{c}(L)-J_{v}(L)<0$ for Hg${}_{1-x}$Mn${}_{x}$Te in the relevant effective Mn concentrations, $x\lesssim 6$%.
III.3 Effects of spin-orbit coupling
The DFT results presented in Figs. 3 and 4, obtained without taking SOC into account, have qualitatively shown how exchange spin-splitting of bands evolves with the ${\bm{k}}$ vector spanning the whole Brillouin zone. This dependence reflects (i) the $\bm{k}$-dependent mixing between cation and anion wave functions, which affects a relative contribution of the potential and kinetic components to the $sp$-$d$ exchange and (ii) the energy position of a given ${\bm{k}}$ state in respect to the open Mn $d$ shells, which controls the magnitude of the $\bm{k}$-dependent kinetic exchange. Quantitatively, however, the position of bands and, thus, their exchange splitting depends significantly on SOC. Moreover, in the presence of SOC, exchange splitting of a given band state changes with the orientation of its ${\bm{k}}$ vector in respect to the direction of ${\bm{M}}(T,{\bm{H}})$. This means that, in general, exchange splitting of particular $L$ valleys differs, depending on the angle between ${\bm{k}}_{L}$ and ${\bm{M}}(T,{\bm{H}})$. Furthermore, under non-zero magnetization $M(T,H)$, degeneracy of states with different projections of the orbital momentum is removed in the presence of SOC. This results in magnetic circular dichroism (MCD), i.e., different transition probabilities for two circular light polarizations $\sigma^{+}$ and $\sigma^{-}$.
We are interested in interpreting magnetooptical results, particularly concerning MCD, for Cd${}_{1-x}$Mn${}_{x}$Te and Hg${}_{1-x}$Mn${}_{x}$Te taken at photon energies corresponding to free excitons at the fundamental gap at the $\Gamma$ point ($E_{0}$ and $E_{0}+\Delta_{0}$ transitions) and at the $L$ points ($E_{1}$ and $E_{1}+\Delta_{1}$ transitions) Ginter et al. (1983); Coquillat et al. (1986); Ando et al. (2011); Coquillat et al. (1989), where $\Delta_{0}$ and $\Delta_{1}$ are the spin-orbit splitting of the valence band at the $\Gamma$ and $L$ points of the Brillouin zone, respectively. Our theoretical approach considering SOC involves four steps. First, we use the DFT calculations with SOC taken into account in order to determine parameters of a tight-binding model for CdTe and HgTe. Second, we consider the Mn-doped case and obtained from DFT on-site and hopping energies for Mn $d$ shell and its coupling to band states in CdTe and HgTe. Third, these parameters are incorporated into $sp$-$d$ exchange hamiltonian that takes into account the presence of $\bm{k}$-dependent kinetic and potential exchange interactions in the molecular-field and virtual-crystal approximations suitable for Cd${}_{1-x}$Mn${}_{x}$Te and Hg${}_{1-x}$Mn${}_{x}$Te.
III.4 DFT with spin-orbit coupling and minimal tight-binding model for CdTe and HgTe
As mention in Sec. II, we use MBJLDA to determine the relativistic band structure of CdTe and HgTe
with experimental lattice constants 6.48 Å for CdTe and 6.46 Å for HgTe.
To extract energy dispersions $E(\bm{k})$ of the electronic bands, the Slater-Koster interpolation scheme is employed. The obtained results are shown in Fig. 5. The extracted values of energy gaps and spin splittings for CdTe and HgTe are summarized in Table 1, and show good agreement with experimental values.
Our aim is to use the ab initio results in order to determine parameters of the one-electron Hamiltonian in the tight-binding approximation (TBA), which will properly describe the band structure and $sp$-$d$ exchange splittings of bands at an arbitrary $\bm{k}$-point of the Brillouin zone with SOC taken into account. Similarly to the previous descriptions of CdTe and HgTe within TBA Tarasenko et al. (2015), we consider $sp^{3}$ orbitals per atom and the nearest neighbor hopping. In particular, from positions of the electronic bands at $\Gamma$ we obtain the TBA on-site energies and the spin-orbit splittings. Then we use as constraints the DFT values of the band energies at the $\Gamma$, $X$ and $L$ points. We create an equation system and search for the values of the hopping energies $V$. If this procedure results in multiple solutions, we select the value of $V$ which has the same sign as the first-neighbor hopping energy among Wannier functions. The TBA parameters obtained in this way are shown in Table 2. Since the atomic radius of Cd is smaller than of Hg, whereas the bond length is greater in CdTe compared to
HgTe, there are no systematic differences in the magnitudes of the hybridizations $V$ between these two compounds.
A comparison of the band structures resulting from MBJLDA and our TBA is shown in Fig. 5.
By construction, the TBA parameters collected in Table 2 lead to similar bandgaps and spin-orbit splittings as determine be DFT, and displayed in Table 1 and Fig. 5. For comparison, we present in Table 1 the magnitudes of bandgaps and spin-orbit splittings computed by using the tight binding parameters determined by Tarasenko et al. Tarasenko et al. (2015) in reference to experimental data. As seen, this empirical tight-binding (ETB) model provides, by designed, the energies in accord with the experimental values.
III.5 Tight-binding parameters from DFT for Cd${}_{1-x}$Mn${}_{x}$Te and Hg${}_{1-x}$Mn${}_{x}$Te
We are interested in evaluating Slater-Koster parameters associated with the presence of open
$3d$ shells of Mn in Cd${}_{1-x}$Mn${}_{x}$Te and Hg${}_{1-x}$Mn${}_{x}$Te, i.e.,
hopping energies between Mn $3d$ orbitals and $5sp^{3}$ states of the nearest neighbor Te anions
as well as energetic positions of Mn $d$ levels. For this purpose we use supercells
with $2\times 2\times 2$ unit cells, each containing 1 Mn atom. The GGA+U technique is employed with $U_{\text{Mn}}=5$ eV
and J${}_{\text{Hund}}=0.75$ eV as well as with the PBE exchange functional. Since in such alloys no $E(\bm{k})$ dependencies
can be derived, we extract the
Slater-Koster parameters $V$ directly from the hopping energies among the relevant Wannier functions, which means that their accuracy
is presumably of the order of 20%. The magnitudes of determined parameters
are shown in Table 3. A lower positions (by about 0.3 eV) of $d$ levels in HgTe compared to CdTe originates
from the valence band offset between these two compounds Kowalczyk et al. (1986); Dietl and Kossut (1988). The spin-up Mn states are more localized
and the hopping energies $V$ related to $d\uparrow$ are smaller. On the other hand, noticeable
dissimilarities in hopping energies of the two compounds are caused by differences in the bond length and in the participation of
cation orbitals to the $s$-like and $p$-like wave functions.
III.6 Tight-binding model with $sp$-$d$ exchange interaction
We now present and discuss the tight-binding Hamiltonian with four $sp^{3}$ orbital per atom containing a term describing giant Zeeman splitting of bands in the presence of spin polarized Mn ions. This splitting is brought about by: (i) direct (potential) exchange coupling of electrons residing on the open Mn $3d$ shell to band carriers visiting Mn $4s$ or $4p$ orbitals (ii) the kinetic exchange resulting from spin-dependent hybridization between Mn $3d$ shells and band states derived from the $5s$ and $5p$ orbitals of the four neighboring Te anions. Our approach is developed within the molecular-field and virtual-crystal approximations, and generalizes previous descriptions of DMSs within TBA Oszwałdowski et al. (2006); Sankowski et al. (2007) by taking into account the $\bm{k}$-dependence of the kinetic exchange according to,
$${\cal{H}}({\bm{k}})={\cal{H}}_{\text{TBA}}({\bm{k}})+{\cal{H}}_{sp-d}({\bm{k}}).$$
(3)
Within our model ${\cal{H}}_{\text{TBA}}(\bm{k})$ is a $16\times 16$ matrix, with on-site energies on the diagonal and $\bm{k}$-dependent hopping $t_{mn}(\bm{k})$ between the orbitals labeled $m,n$,
$$\left<m,s\middle|{\cal{H}}_{\text{TBA}}(\bm{k})\middle|n,s^{\prime}\right>=E_{%
n}\delta_{mn}+t_{mn}(\bm{k})+\frac{\Delta_{a(c)}}{3}\sum_{\alpha}I^{\alpha}_{%
mn}\sigma^{\alpha}_{ss^{\prime}}.$$
(4)
In this equation, if the orbitals labeled $(m,n)$ are located either on the anion and the cation or vice versa,
$t_{mn}(\bm{k})$ is the total of the hoppings to the nearest neighbors in the zinc-blende lattice; momentum $\bm{k}$ enters the phase factors:
$$t_{mn}(\bm{k})=\sum_{\bm{R}_{m}\in\text{n.n.}(\bm{R}_{n})}V_{mn}(\bm{R}_{m}-%
\bm{R}_{n})\exp[i\bm{k}\cdot(\bm{R}_{m}-\bm{R}_{n})].$$
(5)
The Slater-Koster interatomic matrix elements (dependent on the direction cosines of the vector from the location $\bm{R}_{n}$ of the orbital $n$
to the location $\bm{R}_{m}$ of the orbital $m$) are denoted as $V_{mn}(\bm{R}_{m}-\bm{R}_{n})$,
with parameters for the various combinations of orbitals ($ss\sigma$, $sp\sigma$, $pp\sigma$, $pp\pi$) given in Table 2.
The spin-orbit splitting of the anion (cation) $p$ states are denoted by $\Delta_{a(c)}$, respectively,
the spin-1 orbital momentum operator $I^{\alpha}_{\beta\gamma}$ in the Cartesian basis ($\alpha,\beta,\gamma=x,y,z$)
has been written using the Levi-Civita symbol $\epsilon_{\alpha\beta\gamma}$ as
$I^{\alpha}_{\beta\gamma}=-i\epsilon_{\alpha\beta\gamma}$, and $(\sigma^{\alpha})_{\alpha=x,y,z}$ stand for the set of Pauli matrices.
The exchange interaction is taken into account in the molecular-field and virtual-crystal approximations: spin polarization of Mn ions is described by a vector $\mathbf{X}=x_{\text{eff}}\mathbf{S}$, where $x_{\text{eff}}$ is the molar fraction of Mn and $S=5/2$ if all Mn ions are spin polarized. Then, the relevant $sp$-$d$ exchange Hamiltonian assumes the form:
$$\left<m,s\middle|{\cal{H}}_{sp-d}(\bm{k})\middle|n,s^{\prime}\right>=-\frac{1}%
{2}\sum_{\alpha}X^{\alpha}\sigma^{\alpha}_{ss^{\prime}}\left[\frac{1}{S}\sum_{%
d}(\frac{1}{E_{d\uparrow}-E_{\bm{k}}}-\frac{1}{E_{d\downarrow}-E_{\bm{k}}})t_{%
md}(\bm{k})t_{dn}(\bm{k})+\left<m\middle|J_{4s-3d}\hat{P}_{sc}+J_{4p-3d}\hat{P%
}_{pc}\middle|n\right>\right]$$
(6)
The first term in the brackets was given by Schrieffer and Wolff Schrieffer and Wolff (1966), and accounts for the kinetic exchange; this contribution
is $\bm{k}$-dependent and may be non-diagonal. In this term the matrix of hoppings is defined as in (5), with parameters given in Table 3; the $d$ label runs over the $t_{2g}$ and $e_{g}$ orbitals of Mn, and the $d$ orbitals are considered to be located on the cation ($t_{dn}$ may be non-zero only of $n$ is located on the anion).
The second term describes intra-Mn direct (potential) exchange $J_{4s-3d}$ and $J_{4p-3d}$ between electrons residing on $3d$ and $4s$ or $4p$ states, respectively; $\hat{P}_{sc}$ and $\hat{P}_{pc}$ are the projectors on Mn cation $4s$ and $4p$ states. These projectors restrict the exchange interaction to the site on which the Mn ion is located, and the corresponding terms resemble
the frequently encountered expression $J_{sp-d}\,\bm{S}\cdot\bm{s}(\mathbf{r})$.
According to spectroscopic studies, $J_{4s-3d}=0.392$ eV and $J_{4p-3d}=0.196$ eV for free Mn${}^{+1}$ ions Dietl et al. (1994).
IV Conclusions
We have shown, by combining appropriate DFT and TBA approaches, that it is possible to determine, in satisfactory agreement with magnetooptical data, fundamental gaps and exchange splittings of bands at the $\Gamma$ and $L$ points of the Brillouin zone in Mn-doped topologically trivial CdTe and topologically non-trivial HgTe with no adjustable or empirical parameters. In particular, a strong reduction of the splittings at the $L$ points compared to the $\Gamma$ point, hitherto regarded as not understood, originates–according to our results–from (i) $\bm{k}$-dependent hybridization between Bloch states and Mn open $d$ shells, which controls the magnitude of the kinetic exchange; (ii) $\bm{k}$-dependent changes in the orbital components of the Bloch functions, which affects the relative magnitudes of antiferromagnetic kinetic exchange and ferromagnetic potential exchange, and (iii) the important role played by spin-orbit coupling in these systems and magnetooptical phenomena employed to determine exchange splittings. Furthermore, we have discussed the form of the empirical tight-binding model that can serve, by design, to even more accurate description of phenomena relevant to these magnetic and topological compounds.
Acknowledgments
We acknowledge Marcin M. Wysokiński for useful discussions.
The work is supported by the Foundation for Polish Science through the International Research Agendas program co-financed by the
European Union within the Smart Growth Operational Programme.
We acknowledge the access to the computing facilities of the Interdisciplinary
Center of Modeling at the University of Warsaw, Grant No. G73-23 and G75-10.
References
Kos (2010)
in Introduction to the Physics
of Diluted Magnetic Semiconductors, edited by J. Kossut and J. A. Gaj (Springer, Heidelberg, 2010).
Dietl (1994)
T. Dietl, “(Diluted)
Magnetic Semiconductors,” in Handbook of Semiconductors, Vol. 3B, edited by S. Mahajan (North Holland, Amsterdam, 1994) p. 1251.
Furdyna (1988)
J. K. Furdyna, “Diluted
magnetic semiconductors,” J. Appl. Phys. 64, R29–R64 (1988).
Dietl and Ohno (2014)
T. Dietl and H. Ohno, “Dilute ferromagnetic
semiconductors: Physics and spintronic structures,” Rev.
Mod. Phys. 86, 187–251
(2014).
Ke et al. (2018)
He Ke, Yayu Wang, and Qi-Kun Xue, “Topological materials: quantum anomalous
Hall system,” Annu. Rev. Cond. Mat. Phys. 9, 329–3449 (2018).
Tokura et al. (2019)
Y. Tokura, K. Yasuda, and A. Tsukazaki, “Magnetic topological
insulators,” Nat. Rev. Phys. (2019), 110.1038/s42254-018-0011-5.
Jaroszyński et al. (2002)
J. Jaroszyński, T. Andrearczyk, G. Karczewski, J. Wróbel, T. Wojtowicz, E. Papis,
E. Kamińska, A. Piotrowska, Dragana Popovic, and T. Dietl, ‘‘Ising quantum Hall ferromagnet in magnetically doped
quantum wells,” Phys. Rev. Lett. 89, 266802 (2002).
Kazakov et al. (2016)
A. Kazakov, G. Simion,
Y. Lyanda-Geller, V. Kolkovsky, Z. Adamus, G. Karczewski, T. Wojtowicz, and L. P. Rokhinson, “Electrostatic control of quantum Hall ferromagnetic transition: A
step toward reconfigurable network of helical channels,” Phys.
Rev. B 94, 075309
(2016).
Kazakov et al. (2017)
A. Kazakov, G. Simion,
Y. Lyanda-Geller, V. Kolkovsky, Z. Adamus, G. Karczewski, T. Wojtowicz, and L. P. Rokhinson, “Mesoscopic transport in electrostatically defined spin-full channels
in quantum hall ferromagnets,” Phys. Rev. Lett. 119, 046803 (2017).
Simion et al. (2018)
G. Simion, A. Kazakov,
L. P. Rokhinson, T. Wojtowicz, and Y. Lyanda-Geller, “Impurity-generated non-Abelions,” Phys. Rev. B 97, 245107 (2018).
Liu et al. (2008)
Chao-Xing Liu, Xiao-Liang Qi, Xi Dai, Zhong Fang, and Shou-Cheng Zhang, “Quantum anomalous Hall
effect in Hg${}_{1-y}$Mn${}_{y}$Te quantum wells,” Phys. Rev. Lett. 101, 146802 (2008).
Kacman (2001)
P. Kacman, “Spin
interactions in diluted magnetic semiconductors and magnetic semiconductor
structures,” Semicond. Sci. Technol. 16, R25 (2001).
Bastard et al. (1978)
G. Bastard, C. Rigaux,
Y. Guldner, J. Mycielski, and A. Mycielski, “Effect of exchange on interband
magneto-absorption in zero gap Hg${}_{1-k}$Mn${}_{k}$Te mixed crystals,” J. de Physique (Paris) 39, 87–98 (1978).
Dietl (2008)
T. Dietl, “Hole states in
wide band-gap diluted magnetic semiconductors and oxides,” Phys.
Rev. B 77, 085208
(2008).
Ginter et al. (1983)
J. Ginter, J. A. Gaj, and L. S. Dang, “Exchange splittings of
reflectivity maxima E${}_{1}$ and E${}_{1}$ +$\Delta_{1}$ in
Cd${}_{1-x}$Mn${}_{x}$Te,” Solid State Commun. 48, 849 (1983).
Coquillat et al. (1986)
D. Coquillat, J.P. Lascaray, M.C.D. Deruelle, J.A. Gaj, and R. Triboulet, “Magnetoreflectivity of
Cd${}_{1-x}$Mn${}_{x}$Te at L point of the Brillouin zone,” Solid State Commun. 59, 25 – 28 (1986).
Ando et al. (2011)
K. Ando, H. Saito, and V. Zayets, “Anomalous Zeeman splittings of II-VI
diluted magnetic semiconductors at $l$-critical points,” J. Appl.
Phys. 109, 07C304
(2011).
Bhattacharjee (1990)
A. K. Bhattacharjee, “Magneto-optics near the $L$ point of the Brillouin zone in semimagnetic
semiconductors,” Phys. Rev. B 41, 5696–5700 (1990).
Coquillat et al. (1989)
D. Coquillat, J. P. Lascaray, J. A. Gaj,
J. Deportes, and J. K. Furdyna, “Zeeman splittings of optical
transitions at the L point of the Brillouin zone in semimagnetic
semiconductors,” Phys. Rev. B 39, 10088–10093 (1989).
Bloembergen and Rowland (1955)
N. Bloembergen and T. J. Rowland, “Nuclear spin
exchange in solids: Tl${}^{203}$ and Tl${}^{205}$ magnetic resonance in
thallium and thallic oxide,” Phys. Rev. 97, 1679 (1955).
Dietl et al. (2001)
T. Dietl, H. Ohno, and F. Matsukura, “Hole-mediated ferromagnetism in
tetrahedrally coordinated semiconductors,” Phys.
Rev. B 63, 195205
(2001).
Ferrand et al. (2001)
D. Ferrand, J. Cibert,
A. Wasiela, C. Bourgognon, S. Tatarenko, G. Fishman, T. Andrearczyk, J. Jaroszynski, S. Kolesnik, T. Dietl, B. Barbara, and D. Dufeu, “Carrier-induced ferromagnetism in p-Zn${}_{1-x}$Mn${}_{x}$Te,” Phys. Rev. B 63, 085201 (2001).
Bastard and Lewiner (1979)
G. Bastard and C. Lewiner, “Indirect-exchange interactions in zero-gap semiconductors,” Phys. Rev. B 20, 4256–4267 (1979).
Yu et al. (2010)
Rui Yu, Wei Zhang,
Hai-Jun Zhang, Shou-Cheng Zhang, Xi Dai, and Zhong Fang, “Quantized anomalous Hall effect in magnetic topological
insulators,” Science 329, 61–64 (2010), in this paper, the Bloembergen-Rowland mechanism was called the van
Vleck paramagnetism, the notion alredy in use to describe magnetic
susceptibily of magnetic ions having a non-magnetic ground state, see e.g.,
J.M.D. Coey, Magnetism and Magnetic Materials (Cambridge University
Press, 2010), chapter 4.3.
Wei and Zunger (1987)
Su-Huai Wei and A. Zunger, “Total-energy and
band-structure calculations for the semimagnetic Cd${}_{1-x}$Mn${}_{x}$Te
semiconductor alloy and its binary constituents,” Phys.
Rev. B 35, 2340–2365
(1987).
Larson et al. (1988)
B. E. Larson, K. C. Hass,
H. Ehrenreich, and A. E. Carlsson, “Theory of exchange
interactions and chemical trends in diluted magnetic semiconductors,” Phys. Rev. B 37, 4137–4154 (1988).
Merad et al. (2006)
A.E. Merad, M.B. Kanoun, and S. Goumri-Said, “Ab initio study of
electronic structures and magnetism in ZnMnTe and CdMnTe diluted magnetic
semiconductors,” J. Magn. Magn. Mater. 302, 536 – 542 (2006).
Liu and Liu (2008)
Yong Liu and Bang-Gui Liu, “Magnetic
semiconductors in ternary CdMnTe compounds,” phys.
status solidi (b) 245, 973–979 (2008).
Echeverría-Arrondo et al. (2009)
C. Echeverría-Arrondo, J. Pérez-Conde, and A. Ayuela, “First-principles calculations of the magnetic properties of (Cd,Mn)Te
nanocrystals,” Phys. Rev. B 79, 155319 (2009).
Verma et al. (2011)
U. P. Verma, S. Sharma,
N. Devi, P. S. Bisht, and P. Rajaram, “Spin-polarized structural, electronic and magnetic
properties of diluted magnetic semiconductors Cd${}_{1-x}$Mn${}_{x}$Te in zinc
blende phase,” J. Magn. Magn. Mater. 323, 394–399 (2011).
Wua et al. (2015)
Yelong Wua, Guangde Chen,
Youzhang Zhu, Wan-Jian Yin, Yanfa Yan, Mowafak Al-Jassim, and S. J. Pennycook, “LDA+U/GGA+U calculations of structural and electronic properties
of CdTe: Dependence on the effective $U$ parameter,” Comput. Mater. Sci. 98, 18–23 (2015).
Linneweber et al. (2017)
T. Linneweber, J. Bünemann, U. Löw,
F. Gebhard, and F. Anders, “Exchange couplings for Mn ions in
CdTe: Validity of spin models for dilute magnetic II-VI
semiconductors,” Phys. Rev. B 95, 045134 (2017).
Kresse and Furthmüller (1996)
G. Kresse and J. Furthmüller, “Efficiency of ab-initio total energy calculations for metals and
semiconductors using a plane-wave basis set,” Comput. Mat. Sci. 6, 15–50 (1996).
Hafner (2008)
J. Hafner, “Ab-initio
simulations of materials using VASP: Density-functional theory and
beyond,” J. Comput. Chem. 29, 2044–2078 (2008).
Skauli and Colin (2001)
T. Skauli and T. Colin, “Accurate determination of
the lattice constant of molecular beam epitaxial CdHgTe,” J. Crys. Growth 222, 719 (2001).
Perdew et al. (1996)
J. P. Perdew, K. Burke, and M. Ernzerhof, “Generalized gradient
approximation made simple,” Phys. Rev. Lett. 77, 3865–3868 (1996).
Becke and Johnson (2006)
A. D. Becke and E. R. Johnson, “A simple
effective potential for exchange,” J. Chem. Phys. 124, 221101 (2006).
Tran and Blaha (2009)
F. Tran and P. Blaha, “Accurate band gaps of
semiconductors and insulators with a semilocal exchange-correlation
potential,” Phys. Rev. Lett. 102, 226401 (2009).
Camargo-Martínez and Baquero (2012)
J. A. Camargo-Martínez and R. Baquero, “Performance of the modified Becke-Johnson potential for
semiconductors,” Phys. Rev. B 86, 195106 (2012).
Zunger et al. (1990)
A. Zunger, S.-H. Wei,
L. G. Ferreira, and J. E. Bernard, “Special quasirandom
structures,” Phys. Rev. Lett. 65, 353–356 (1990).
Van De Walle et al. (2013)
A. Van
De Walle, P. Tiwary,
M. De Jong, D. L. Olmsted, M. Asta, A. Dick, D. Shin, Yi Wang, Long-qing Chen, and Zi-kui Liu, “Efficient stochastic generation of special
quasirandom structures,” Calphad 42, 13–18 (2013).
van de Walle et al. (2002)
A. van de Walle, M. D. Asta, and G. Ceder, “The alloy theoretic
automated toolkit: A user guide,” Calphad 26, 539–553
(2002).
Paul et al. (2014)
A. Paul, C. Zandalazini,
P. Esquinazi, C. Autieri, B. Sanyal, P. Korelis, and P. Böni, “Structural, electronic and magnetic properties of YMnO${{}_{3}}$/La${{}_{0}.7}$Sr${{}_{0}.3}$MnO${{}_{3}}$ heterostructures,” J. Appl. Crystal. 47, 1054–1064 (2014).
Autieri and Sanyal (2014)
C. Autieri and B. Sanyal, “Unusual
ferromagnetic YMnO${}_{3}$ phase in YMnO${}_{3}$/La${}_{2/3}$Sr${}_{1/3}$MnO${}_{3}$
heterostructures,” New J. Phys. 16, 113031 (2014).
Keshavarz et al. (2017)
S. Keshavarz, Y. O. Kvashnin, D. C. M. Rodrigues, M. Pereiro,
I. Di Marco, C. Autieri, L. Nordström, I. V. Solovyev, B. Sanyal, and O. Eriksson, “Exchange interactions of ${\mathrm{camno}}_{3}$ in the
bulk and at the surface,” Phys. Rev. B 95, 115120 (2017).
Marzari and Vanderbilt (1997)
N. Marzari and D. Vanderbilt, “Maximally
localized generalized Wannier functions for composite energy bands,” Phys. Rev. B 56, 12847–12865 (1997).
Souza et al. (2001)
I. Souza, N. Marzari, and D. Vanderbilt, “Maximally localized
Wannier functions for entangled energy bands,” Phys.
Rev. B 65, 035109
(2001).
Mostofi et al. (2008)
A. A. Mostofi, J. R. Yates,
Y. S. Lee, I. Souza, D. Vanderbilt, and N. Marzari, “Wannier90: A tool for obtaining maximally-localised
Wannier functions,” Comput. Phys. Comm. 178, 685–699 (2008).
Gaj et al. (1979)
J. A. Gaj, R. Planel, and G. Fishman, Solid State Commun. 29, 435 (1979).
Galazka et al. (1980)
R. R. Galazka, S. Nagata, and P. H. Keesom, “Paramagnetic—spin-glass—antiferromagnetic phase transitions in
Cd${}_{1-x}$Mn${}_{x}$Te from specific heat and magnetic susceptibility
measurements,” Phys. Rev. B 22, 3344–3355 (1980).
Mycielski et al. (1984)
A. Mycielski, C. Rigaux,
M. Menant, T. Dietl, and M. Otto, “Spin glass phase transition in Hg${}_{1-k}$Mn${}_{k}$Te
semimagnetic semiconductors,” Solid State Commun. 50, 257–260 (1984).
Mizokawa et al. (2002)
T. Mizokawa, T. Nambu,
A. Fujimori, T. Fukumura, and M. Kawasaki, “Electronic structure of the oxide-diluted
magnetic semiconductor Zn${}_{1-x}$Mn${}_{x}$O,” Phys.
Rev. B 65, 085209
(2002).
Schrieffer and Wolff (1966)
J. R. Schrieffer and P. A. Wolff, “Relation between
the Anderson and Kondo hamiltonians,” Phys.
Rev. 149, 491 (1966).
Merkulov et al. (1999)
I. A. Merkulov, D. R. Yakovlev, A. Keller,
W. Ossau, J. Geurts, A. Waag, G. Landwehr, G. Karczewski, T. Wojtowicz, and J. Kossut, “Kinetic exchange between the conduction band electrons and magnetic ions in
quantum-confined structures,” Phys. Rev. Lett. 83, 1431–1434 (1999).
Dobrowolska and Dobrowolski (1981)
M. Dobrowolska and W. Dobrowolski, “Temperature
study of interband magnetoabsorption in hg${}_{1-x}$mn${}_{x}$te mixed crystals,” J. Phys. C 14, 5689 (1981).
Tarasenko et al. (2015)
S. A. Tarasenko, M. V. Durnev, M. O. Nestoklon, E. L. Ivchenko, Jun-Wei Luo,
and A. Zunger, “Split Dirac cones in
HgTe/CdTe quantum wells due to symmetry-enforced level anticrossing at
interfaces,” Phys. Rev. B 91, 081302 (2015).
Twardowski et al. (1980)
A. Twardowski, E. Pokita,
and J.A. Gaj, “Valence band spin-orbit
splitting in CdTe and Cd${}_{1-x}$Mn${}_{x}$Te and giant Zeeman effect in the
$\gamma_{7}$ band of Cd${}_{1-x}$Mn${}_{x}$Te,” Solid State Comm. 36, 927 – 930 (1980).
Sakuma et al. (2011)
R. Sakuma, C. Friedrich,
T. Miyake, S. Blügel, and F. Aryasetiawan, ‘‘$gw$ calculations including spin-orbit
coupling: Application to Hg chalcogenides,” Phys.
Rev. B 84, 085144
(2011).
Cardona et al. (1967)
M. Cardona, K. L. Shaklee, and F. H. Pollak, “Electroreflectance at a semiconductor-electrolyte interface,” Phys. Rev. 154, 696–720 (1967).
Kowalczyk et al. (1986)
S. P. Kowalczyk, J. T. Cheung, E. A. Kraut, and R. W. Grant, “CdTe-HgTe
$(\bar{1}\bar{1}\bar{1})$ heterojunction valence-band discontinuity: A
common-anion-rule contradiction,” Phys. Rev. Lett. 56, 1605–1608 (1986).
Dietl and Kossut (1988)
T. Dietl and J. Kossut, “Band offsets in HgTe/CdTe
and HgSe/CdSe heterostructures from electron mobility limited by alloy
scattering,” Phys. Rev. B 38, 10941–10942 (1988).
Oszwałdowski et al. (2006)
R. Oszwałdowski, J. A. Majewski, and T. Dietl, ‘‘Influence of band
structure effects on domain-wall resistance in diluted ferromagnetic
semiconductors,” Phys. Rev. B 74, 153310 (2006).
Sankowski et al. (2007)
P. Sankowski, P. Kacman,
J. A. Majewski, and T. Dietl, “Spin-dependent tunneling in modulated
structures of (Ga,Mn)As,” Phys. Rev. B 75, 045306 (2007).
Dietl et al. (1994)
T. Dietl, C. Śliwa,
G. Bauer, and H. Pascher, “Mechanisms of exchange interactions between
carriers and Mn or Eu spins in lead chalcogenides,” Phys.
Rev. B 49, 2230
(1994). |
Equilibrium properties of self-interacting neutrinos in the quasi-particle approach
M. Sirera
Departamento de Astronomía y Astrofísica, Universidad de Valencia,
46100 Burjassot (Valencia) Spain
A. Pérez
Departamento de Física Teórica and IFIC Universidad de Valencia,
46100 Burjassot (Valencia) Spain
Abstract
In this work a neutrino gas in equilibrium is studied both at $T=0$ and at
finite temperature. Neutrinos are treated as massive Dirac quasi-particles
with two generations. We include self-interactions among the neutrinos via
neutral currents, as well as the interaction with a background of matter. To
obtain the equilibrium properties we use Wigner function techniques. To
account for corrections beyond the Hartree approximation, we also introduce
correlation functions. We prove that, under the quasi-particle approximation,
these correlation functions can be expressed as products of Wigner functions.
We analyze the main properties of the neutrino eigenmodes in the medium, such
as effective masses and mixing angle. We show that the formulae describing
these quantities will differ with respect to the case with no self-interactions.
††preprint:
I Introduction.
Propagation in dense media is one of the most interesting issues in present
neutrino physics. The consequences on solar and atmospheric neutrino data, as
well as in baseline neutrino experiments, are crucial for the understanding of
neutrino properties. Moreover, in astrophysical and cosmological scenarios,
they provide an important feedback which modifies the physical evolution of
the system under consideration. This, in fact, is the case for supernova
explosions, where neutrino interactions and oscillations can change the shock
dynamics 1 ; 2 ; 3 .
The nucleosynthesis of heavy elements via r-processes is also largely affected
by neutrino oscillations, since the proton to neutron ratio can be altered by
the oscillations.
In a similar way, they have been shown to play an important role in
establishing the cosmic flavor content when the Big Bang nucleosynthesis
starts, thus influencing the helium production. A crucial ingredient in these
two scenarios is the neutrino self- interaction 4 ; 5 ; 6 ; 3l . As it has
been remarked, such interactions are non-diagonal in flavor space, and give
rise to new phenomena in the oscillatory behavior 7 ; 8 ; 9 .
In this work, we analyze in detail the equilibrium properties of a
self-interacting neutrino gas. Here, we study a system consisting on two
generations of neutrinos in equilibrium with a matter background and
self-interactions taken into account. The formalism, however, can be extended
in a straightforward way to the case of three generations.
We assume that chemical equilibrium has been reached, therefore the chemical
potential is the same for both neutrino species. This is, in fact, the
situation for muon and tau neutrinos inside a supernova, where both kinds of
neutrinos are produced in pairs and, therefore, the chemical potentials are
zero. It seems also the case in the early Universe, with the present values of
mass differences and mixing angles, where chemical equilibrium is achieved (at
least approximately) just before the nucleosynthesis epoch for the three
neutrino families 4 .
Neutrinos will be treated in the quasi-particle approximation, i.e. we
assume that the field corresponding to the mass eigenstates can be described
as a superposition of plane waves with a modified (with respect to the vacuum)
dispersion relation. As we will show, the equilibrium features, such as the
effective masses and mixing angles, are modified when the interactions among
neutrinos are considered, due to the non-diagonal nature of the self-term. To
this end, we use a method based on Wigner Functions, which has been
shown to be appropriate to describe both the equilibrium and kinematics of
many-particle neutrino mixing 10 . To account for corrections beyond the
Hartree approximation, we also introduce correlation functions. We prove that,
under the quasi-particle approximation, these correlation functions can be
expressed as products of Wigner functions. As we show, the results obtained
using these techniques agree with previous calculations SR93 ; R96 ; 3p ; 3s .
This paper is organized as follows. In section II, we derive the equations
satisfied by the Wigner functions of self-interacting massive neutrinos. In
section III we consider the Hartree approximation (i.e., when correlations are
neglected). In this case, the interactions among neutrinos contribute in the
same way as the neutral current interactions of neutrinos with other
particles, such as electrons, protons or neutrons. The addition of
correlations results in the appearance of non-lineal effects due to the
self-interaction. This is described in section IV. In section V we analyze the
resulting dispersion relations of eigenmodes. Our main results are summarized
in section VI. The appendix at the end contains the derivation of the
statistical correlations used in section IV.
II Equations for Wigner functions.
We consider a neutrino gas consisting on two generations of neutrinos which
interact with themselves by means of neutral current interactions. We denote
the two flavors by (e,$\mu$). Of course, the formalism can also be applied to
any two flavors, such as $\mu$ and $\tau$ neutrinos.
Since we deal with two neutrino species, it is convenient to introduce vectors
and matrices in flavor (or mass) space. We therefore define the neutrino and
antineutrino vector fields:
$$\widehat{\nu}(x)\equiv\left(\begin{array}[c]{c}\widehat{\nu}^{e}(x)\\
\widehat{\nu}^{\mu}(x)\end{array}\right),\ \ \widehat{\overline{\nu}}(x)\equiv%
\left(\begin{array}[c]{cc}\widehat{\overline{\nu}}^{e}(x)&\widehat{\overline{%
\nu}}^{\mu}(x)\end{array}\right)$$
(1)
The symbol $\ \char 94$ on top of a magnitude means that we are dealing
with a quantum operator. This will be used to distinguish this magnitudes from
statistical averages. We also introduce the following matrices in flavor space
$$\displaystyle\Lambda^{ab,\mu}$$
$$\displaystyle\equiv\left(\begin{array}[c]{cc}0&0\\
0&\gamma^{\mu}(1-\gamma^{5})\end{array}\right)$$
(2)
$$\displaystyle\quad\Omega^{ab,\mu}$$
$$\displaystyle\equiv\left(\begin{array}[c]{cc}\gamma^{\mu}(1-\gamma^{5})&0\\
0&0\end{array}\right)$$
(3)
With the aid of these notations, the Lagrangian terms which account for the
neutrino self-interactions 7 can be written as
$$\displaystyle\widehat{\mathit{L}}(x)$$
$$\displaystyle=\widehat{\overline{\nu}}(x)i\gamma^{\mu}\partial_{\mu}\widehat{%
\nu}(x)-\widehat{\overline{\nu}}(x)M\widehat{\nu}(x)-\frac{G_{F}}{4\sqrt{2}}%
\widehat{\overline{\nu}}(x)\Omega^{\mu}\widehat{\nu}(x)\widehat{\overline{\nu}%
}(x)\Omega_{\mu}\widehat{\nu}(x)$$
$$\displaystyle-$$
$$\displaystyle\frac{G_{F}}{4\sqrt{2}}\widehat{\overline{\nu}}(x)\Lambda^{\mu}%
\widehat{\nu}(x)\widehat{\overline{\nu}}(x)\Lambda_{\mu}\widehat{\nu}(x)-\frac%
{G_{F}}{2\sqrt{2}}\widehat{\overline{\nu}}(x)\Omega^{\mu}\widehat{\nu}(x)%
\widehat{\overline{\nu}}(x)\Lambda_{\mu}\widehat{\nu}(x),$$
(4)
where $G_{F}$ is the Fermi constant and $M$ is the neutrino mass matrix. The
equation of motion then reads as
$$\displaystyle i\gamma^{\mu}\partial_{\mu}\widehat{\nu}(x)-M\widehat{\nu}(x)-%
\frac{G_{F}}{4\sqrt{2}}\Omega^{\mu}\widehat{\nu}(x)\widehat{\overline{\nu}}(x)%
\Omega_{\mu}\widehat{\nu}(x)$$
$$\displaystyle-$$
$$\displaystyle\frac{G_{F}}{4\sqrt{2}}\widehat{\overline{\nu}}(x)\Omega^{\mu}%
\widehat{\nu}(x)\Omega_{\mu}\widehat{\nu}(x)-\frac{G_{F}}{4\sqrt{2}}\Lambda^{%
\mu}\widehat{\nu}(x)\widehat{\overline{\nu}}(x)\Lambda_{\mu}\widehat{\nu}(x)$$
$$\displaystyle-\frac{G_{F}}{4\sqrt{2}}\widehat{\overline{\nu}}(x)\Lambda^{\mu}%
\widehat{\nu}(x)\Lambda_{\mu}\widehat{\nu}(x)-\frac{G_{F}}{2\sqrt{2}}\Omega^{%
\mu}\widehat{\nu}(x)\widehat{\overline{\nu}}(x)\Lambda_{\mu}\widehat{\nu}(x)$$
$$\displaystyle-$$
$$\displaystyle\frac{G_{F}}{2\sqrt{2}}\widehat{\overline{\nu}}(x)\Omega^{\mu}%
\widehat{\nu}(x)\Lambda_{\mu}\widehat{\nu}(x)=0$$
(5)
(a similar equation can be derived for the adjoint field $\widehat{\overline{\nu}}(x)$ ).
We now introduce the neutrino Wigner operator
$$\widehat{F}_{ij}^{ab}(x,p)=(2\pi)^{-4}\int d^{4}y\;e^{-ipy}\,\widehat{%
\overline{\nu}}_{j}^{b}(x+y/2)\widehat{\nu}_{i}^{a}(x-y/2)$$
(6)
From Eq. (6) one can easily show that the Hermitian conjugate is
given by
$$\widehat{F}_{ij}^{ab\dagger}(x,p)=\gamma_{jk}^{0}\widehat{F}_{kq}^{ba}(x,p)%
\gamma_{qi}^{0}$$
(7)
One obtains the following equation for the neutrino Wigner Operator :
$$\displaystyle\gamma[\partial\widehat{F}(x,p)-2ip\widehat{F}(x,p)]+2iM\widehat{%
F}(x,p)=-(2\pi)^{-4}\frac{iG_{F}}{2\sqrt{2}}\int d^{4}y^{\prime}d^{4}k\,e^{-ik%
(x-y^{\prime})}$$
$$\displaystyle[\Omega\widehat{F}(x,p-k/2)\widehat{\overline{\nu}}(y^{\prime})%
\Psi\widehat{\nu}(y^{\prime})+\widehat{\overline{\nu}}(y^{\prime})\Phi\widehat%
{\nu}(y^{\prime})\Lambda\widehat{F}(x,p-k/2)$$
$$\displaystyle+\widehat{\overline{\nu}}(y^{\prime})\Omega\widehat{\nu}(y^{%
\prime})\Omega\widehat{F}(x,p-k/2)+\Lambda\widehat{F}(x,p-k/2)\widehat{%
\overline{\nu}}(y^{\prime})\Lambda\widehat{\nu}(y^{\prime})]$$
(8)
where we defined the matrices $\Psi=\Omega+2\Lambda$ and $\Phi=2\Omega+\Lambda$.
We are now interested in introducing statistical averages from the quantum
operators defined above. These statistical averages are called Wigner
functions 10 , and are the analogous to the distribution functions we
need to describe many-particles systems. These are, in general, complex
functions, and also contain a Lorentz structure.
The neutrino Wigner functions are defined as:
$$F_{ij}^{ab}(x,p)\equiv<\widehat{F}_{ij}^{ab}(x,p)>=(2\pi)^{-4}\int d^{4}y\,e^{%
-ipy}<\widehat{\bar{\nu}}_{j}^{b}(x+y/2)\widehat{\bar{\nu}}_{i}^{a}(x-y/2)>$$
(9)
Here, the symbol $<\widehat{A}>$ means the average of a given quantum operator
$\widehat{A}$ over a basis of quantum states which are compatible with the
macroscopical knowledge of the system. The latter determines a given density
matrix operator $\widehat{\rho}$ . Thus the averaging is performed according
to
$$<\widehat{A}>\equiv Sp\{\widehat{\rho}\widehat{A}\}$$
(10)
where $Sp$ means the trace performed over the quantum basis. Taking into
account Eq. (7) one immediately obtains :
$$F^{\dagger}(x,p)=\gamma^{0}F(x,p)\gamma^{0}$$
(11)
which implies that $\widehat{F}(x,p)$ is an Hermitian matrix with respect to
generation indices.
Starting from Eq. (8), one can take statistical averages to
obtain the equations of motion for the Wigner Function, which turns out to be
$$\displaystyle[\gamma(\partial-2ip)+2iM]F(x,p)=-(2\pi)^{-4}\frac{iG_{F}}{\sqrt{%
2}}\int d^{4}y^{\prime}d^{4}kd^{4}k^{\prime}\,e^{-ik(x-y^{\prime})}$$
$$\displaystyle[$$
$$\displaystyle<\Omega\widehat{F}(x,p-k/2)Tr\Psi\widehat{F}(y^{\prime},k^{\prime%
})>+<Tr\Psi\widehat{F}(y^{\prime},k^{\prime})\Lambda\widehat{F}(x,p-k/2)>$$
$$\displaystyle+$$
$$\displaystyle<Tr\Omega\widehat{F}(y^{\prime},k^{\prime})\Omega\widehat{F}(x,p-%
k/2)>+<\Lambda\widehat{F}(x,p-k/2)Tr\Lambda\widehat{F}(y^{\prime},k^{\prime})>]$$
(12)
III Neutrino gas in the Hartree approximation
In this section we investigate the neutrino gas under the assumption of global
equilibrium. This means that macroscopic quantities must be invariant under
space-time translations:
$$F(x,p)=F(p)$$
(13)
Moreover, as a first approximation, we neglect the effect of statistical
correlations, which simply translates into replacing the statistical average
of an operator product by the product of their averages:
$$<\widehat{F}_{ij}^{ab}(p)\widehat{F}_{kl}^{cd}(p^{\prime})>=F_{ij}^{ab}(p)F_{%
kl}^{cd}(p^{\prime})$$
(14)
Under these conditions, we obtain that the equation of motion for the Wigner
Function of the neutrinos in equilibrium is 10
$$(\gamma p-M)F(p)=\frac{G_{F}}{2\sqrt{2}}\int d^{4}kTr[\gamma_{\mu}(1-\gamma^{5%
})F(k)]\gamma^{\mu}(1-\gamma^{5})F(p),$$
(15)
In order to obtain a solution to the problem, we simplify our approach to the
neutrino gas by making the following assumptions:
1) Neutrinos are assumed to have a small mass (as compared to typical
energies). Therefore, the neutrino fields consist essentially on
left-chirality projections, i.e. we assume that the right projections are very
small, as compared to the left projections. Moreover, we can consider that the
neutrino gas contains only neutrinos with negative polarization (negative
helicity) and antineutrinos with positive polarization (positive helicity).
Thus we have, to a good approximation, that
$$F(p)\simeq F_{L}(p)\simeq F^{-}(p)+\overline{F}^{+}(p).$$
(16)
2) Quasi-particle approximation. In the equilibrium gas, neutrinos are in
their interacting eigenstates, i.e. a particle with definite momentum and
polarization has also a definite energy. This allows us to treat the neutrinos
of the ensemble as free particles with an effective mass instead of
their own masses in vacuum. In other words, the field can be decomposed in
plane waves, with a dispersion relation which differs from the one in vacuum.
3) The mixing angle, which allows us to write a flavor eigenstate as a linear
combination of effective mass eigenstates, is supposed in principle to depend
on all the degrees of freedom of the particle, in a similar way to what
happens with the mixing angle in matter, when neutrinos are propagating trough
a matter background, but do not interact among them.
With all these hypothesis, we can now rewrite the equation of motion as
$$(\gamma p-M)F_{L}(p)=\sqrt{2}G_{F}\gamma^{\mu}\int d^{4}kTr\gamma_{\mu}F_{L}(k%
)F_{L}(p),$$
(17)
or, in a more simplified way,
$$(\gamma p-\sqrt{2}G_{F}\gamma a-M)F_{L}(p)=0,$$
(18)
where the four-vector $a$ is defined as
$$a_{\mu}=\int d^{4}kTr\gamma_{\mu}F_{L}(k).$$
(19)
In order to obtain an equation which gives the propagation modes of the
neutrinos in the gas, we multiply the above formula by $(\gamma p-\sqrt{2}G_{F}\gamma a+M),$ and neglect the subdominant terms proportional to
$G_{F}^{2}$. We then arrive to the equation:
$$(p^{2}-2\sqrt{2}G_{F}ap-M^{2})F_{L}(p)=0.$$
(20)
Without any loss of generality, we can take $p^{\mu}=(p^{0},|\overrightarrow{p}|,0,0)$ , therefore the interaction term is $ap=a^{0}p^{0}-a^{1}|\vec{p}|$.
On the other hand, using the known properties of the Wigner Functions
10 $\widetilde{F}_{L}^{-}$ and $\widetilde{\overline{F}}_{L}^{+}$
corresponding to mass eigenstates of neutrinos and antineutrinos,
respectively, we obtain
$$\displaystyle Tr\gamma^{\mu}F_{L}(k)$$
$$\displaystyle=Tr\gamma^{\mu}U(k)\widetilde{F}_{L}(k)U^{\dagger}(k)=Tr\gamma^{%
\mu}\widetilde{F}_{L}(k)=$$
$$\displaystyle=Tr\gamma^{\mu}\sum_{a=1,2}[\widetilde{F}_{L}^{-aa}(k)+\widetilde%
{\overline{F}}_{L}^{+aa}(k)],$$
(21)
where $U$ is the unitary transformation that transforms from mass eigenstates
to flavor states. After working out the traces and the integrations, one
obtains that the interaction term is $ap=n_{\nu}p^{0},$ where $n_{\nu}=n_{1}+n_{2}=n_{\widetilde{\nu}_{1}}+$ $n_{\widetilde{\nu}_{2}}-n_{\widetilde{\overline{\nu}}_{1}}-n_{\widetilde{%
\overline{\nu}}_{2}}$ is
the total density of neutrinos (with $p^{0}\simeq|\overrightarrow{p}|$ ) minus
the corresponding antineutrino density. In this way, the equation of motion for neutrinos becomes
$$(p^{2}-2\sqrt{2}G_{F}n_{\nu}|\overrightarrow{p}|-M^{2})F_{L}^{-}(p)=0,$$
(22)
The corresponding equation for antineutrinos (for $\overline{F}^{+}(p)$ ) can
be obtained by replacing the $-$ sign into a $+$ sign in the second term of
this equation.
In order to find the effective masses, we have to diagonalize the
matrix $\widehat{M}^{2}\equiv M^{2}\pm 2\sqrt{2}G_{F}n_{\nu}|\overrightarrow{p}|.$ Obviously, this is made by means of the same rotation angle as in
vacuum. Hence, the effective masses are
$$\displaystyle M_{1}^{2}$$
$$\displaystyle=m_{1}^{2}\pm 2\sqrt{2}G_{F}n_{\nu}|\overrightarrow{p}|,$$
$$\displaystyle M_{2}^{2}$$
$$\displaystyle=m_{2}^{2}\pm 2\sqrt{2}G_{F}n_{\nu}|\overrightarrow{p}|,$$
(23)
for each generation, where $m_{1}$ and $m_{2}$ are the vacuum masses, and the
signs $\pm$ correspond, as above, to neutrinos or antineutrinos, respectively.
Let us now assume that, in addition to the self-interaction among the
neutrinos, we have an electrically neutral background of matter
composed by electrons and nucleons. In this case the effective masses of
neutrinos are 10 ; 11 ; 12
$$M_{1,2}^{2}=1/2(A_{c}+\Sigma)\mp 1/2(A_{c}^{2}+\Delta^{2})^{1/2}+A_{n},$$
(24)
where $\ \ \ $
$$\displaystyle\Sigma$$
$$\displaystyle=m_{1}^{2}+m_{2}^{2}$$
$$\displaystyle\ \Delta$$
$$\displaystyle=m_{1}^{2}-m_{2}^{2}$$
$$\displaystyle\ A_{c}$$
$$\displaystyle=2\sqrt{2}G_{F}|\overrightarrow{p}|n_{e}$$
$$\displaystyle A_{n}$$
$$\displaystyle=2\sqrt{2}G_{F}|\overrightarrow{p}|n_{\nu}-\sqrt{2}G_{F}|%
\overrightarrow{p}|n_{n}$$
(25)
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ %
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ %
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ being
$n_{e}$ the number density of electrons (minus antielectrons) and $n_{n}$ the
number density of neutrons (minus antineutrons).
IV Correction to the Hartree approximation.
We now want to take into account the effect of the statistical correlations in
our treatment. For this purpose, we define the two-body correlation
function for the neutrino fields as
$$D_{ijkl}^{abcd}(x,x^{\prime},p,p^{\prime})\equiv<\widehat{F}_{ij}{}^{ab}(x,p)%
\widehat{F}_{kl}^{cd}(x^{\prime},p^{\prime})>-F_{ij}^{ab}(x,p)F_{kl}^{cd}(x^{%
\prime},p^{\prime}),$$
(26)
where subscripts correspond to spin indices. By inserting this definition in
the general equation of motion Eq. (12), and after some
manipulations, it can be written in the form
$$\displaystyle\text{\ }[\gamma(\partial-2ip)+2iM]F(x,p)=-(2\pi)^{-4}\frac{iG_{F%
}}{2\sqrt{2}}\int d^{4}y^{\prime}d^{4}kd^{4}k^{\prime}e^{-ik(x-y^{\prime})}$$
$$\displaystyle[\Omega Tr\left(D(x,y^{\prime},p-k/2,k^{\prime})\Psi\right)+%
\Omega F(x,p-k/2)Tr\left(\Psi F(y^{\prime},k^{\prime})\right)$$
$$\displaystyle+\Lambda Tr\left(\Phi D(y^{\prime},x,k^{\prime},p-k/2)\right)+Tr%
\left(\Phi F(y^{\prime},k^{\prime})\right)\Lambda F(x,p-k/2)$$
$$\displaystyle+\Omega\left(Tr\Omega D(y^{\prime},x,k^{\prime},p-k/2)\right)+Tr%
\left(\Omega F(y^{\prime},k^{\prime})\right)\Omega F(x,p-k/2)$$
$$\displaystyle+\Lambda Tr\left(D(x,y^{\prime},p-k/2,k^{\prime})\Lambda\right)+%
\Lambda F(x,p-k/2)Tr\left(\Lambda F(y^{\prime},k^{\prime})\right)].$$
(27)
In Eq. (27), the symbol $Tr$ implies summation over both spin
and flavor indexes.
As in the previous section, we will assume a situation of global equilibrium,
where spatial-time invariance is satisfied. For the correlation functions this
implies that
$$D(x,x^{\prime},p,p^{\prime})=D(x-x^{\prime},p,p^{\prime})$$
When this is applied to the above equation we obtain:
$$\displaystyle(\gamma p-M)F(p)=\frac{G_{F}}{2\sqrt{2}}\{\gamma^{\mu}(1-\gamma^{%
5})F(p)\int d^{4}kTr\left(\gamma_{\mu}(1-\gamma^{5})F(k)\right)$$
$$\displaystyle+1/2\int d^{4}k^{\prime}d^{4}k[\Omega Tr\left(\tilde{D}(k,p-k/2,k%
^{\prime})\Psi\right)+\Lambda Tr\left(\tilde{D}(k,p-k/2,k^{\prime})\Lambda\right)$$
$$\displaystyle+\Lambda Tr\left(\Phi\tilde{D}(k,k^{\prime},p+k/2)\right)+\Omega
Tr%
\left(\Omega\tilde{D}(k,k^{\prime},p+k/2)]\right)\},$$
(28)
where the function $\tilde{D}(k,p,p^{\prime})$ is the Fourier transformed of
the correlation function
$$\tilde{D}(k,p,p^{\prime})=\frac{1}{(2\pi)^{-4}}\int d^{4}xe^{-ikx}D(x,p,p^{%
\prime}).$$
(29)
We will impose the same three hypothesis as in the Hartree approximation,
which implies having definite mass eigenstates and a mixing angle, which mixes
the mass eigenstates to produce the flavor eigenstates of neutrinos. Moreover,
under these conditions, we can apply the Wick’s theorem (see, for example
FW ) to calculate correlation functions in terms of the Wigner Function
of the neutrino fields, as derived in the appendix. We obtain:
$$\tilde{D}_{ijkl}^{abcd}(k,p,p^{\prime})=-\delta^{4}(p-p^{\prime})F_{Lkj}^{cb}(%
p^{\prime}+k/2)F_{Lil}^{ad}(p^{\prime}-k/2),$$
(30)
where we are only using the left projections of the neutrino fields in order
to construct the Wigner Functions, and hence the correlation functions. By
inserting the above relation in the equation of motion, this one is finally
left as
$$[\gamma p-\sqrt{2}G_{F}\gamma a+\sqrt{2}G_{F}\int d^{4}q\gamma F_{L}(q)\gamma-%
M]F_{L}(p)=0.$$
(31)
The four-vector $a$ is defined in the same way as in the previous section.
Obviously, the third term of this equation provides us with an additional
correction to the corresponding equation in the Hartree approximation.
V Propagation modes of the neutrinos.
Starting from the latter equation, we can now calculate the propagation modes
of the neutrinos in the gas, by performing the following decomposition,
consistent with the hypothesis made in the previous sections
$$F_{L}(q)=F_{L}^{-}(q)+\bar{F}_{L}^{+}(q)=U^{\ast}(q)F_{L}^{-\ast}(q)U^{{}^{%
\ast}\dagger}(q)+\bar{U}^{\ast}(q)\bar{F}_{L}^{+\ast}(q)\bar{U}^{\ast\dagger}(%
q),$$
(32)
that is, both the corresponding part of neutrinos $F_{L}^{-}(q)$ and the
corresponding part of antineutrinos $\bar{F}_{L}^{+}(q)$ of the Wigner
Function with flavor indices can be expressed in terms of Wigner functions
with effective mass indices $F_{L}^{-\ast}(q)$ and $\bar{F}_{L}^{+\ast}(q)$
for neutrinos and antineutrinos, respectively (whose mass eigenstates will not
be the same as in the Hartree approximation), by means of the unitary
transformation $U^{\ast}(q)$ or $\bar{U}^{\ast}(q)$, each of them defined by
the corresponding rotation angle.
On the other hand, we define the following quantities:
$$\displaystyle n_{\nu_{e}}=\frac{1}{2\pi^{2}}\int_{0}^{\infty}d|\vec{q}||\vec{q%
}|^{2}[c^{2}f_{1}(q)+s^{2}f_{2}(q)],$$
$$\displaystyle n_{\nu\mu}=\frac{1}{2\pi^{2}}\int_{0}^{\infty}d|\vec{q}||\vec{q}%
|^{2}[s^{2}f_{1}(q)+c^{2}f_{2}(q)],$$
$$\displaystyle n_{\nu_{12}}=\frac{1}{2\pi^{2}}\int_{0}^{\infty}d|\vec{q}||\vec{%
q}|^{2}cs[f_{1}(q)-f_{2}(q)],$$
(33)
where $f_{1}(q)$ and $f_{2}(q)$ are the Fermi statistical distribution
functions for each generation, corresponding to quasi-particles with
well-defined effective masses, and $s$ and $c$ are the $sin$ and $cos$ of the
rotation angle $\theta$, which relates the eigenstates of effective masses to
flavor eigenstates. In this way, $n_{\nu_{e}}$ is the number density of
electron neutrinos, $n_{\nu_{\mu}}$is the number density of muon neutrinos and
$n_{\nu_{12}}$ contains interference effects. Analogously, we can define the
number densities for antineutrinos $\bar{n}_{\nu_{e}}$, $\bar{n}_{\nu_{\mu}}$
and $\bar{n}_{\nu_{12}}$.
Carrying out a straightforward calculation, we arrive to
$$\int d^{4}qF_{L}(q)=\frac{1}{2}NP_{L}\gamma^{0},$$
(34)
where $P_{L}$ is the left chirality projector and $N$ is a matrix defined (in
flavor space) as
$$N=\left(\begin{array}[c]{cc}n(e)&n_{12}\\
n_{12}&n(\mu)\end{array}\right)=\frac{1}{2}\left(\begin{array}[c]{cc}n_{\nu}&0%
\\
0&n_{\nu}\end{array}\right)+\frac{1}{2}\left(\begin{array}[c]{cc}\delta&2n_{12%
}\\
2n_{12}&-\delta\end{array}\right),$$
(35)
in which $n(e)=n_{\nu_{e}}-\bar{n}_{\nu_{e}}$ is the electron neutrino (minus
electron antineutrino) number density, $n(\mu)=n_{\nu_{\mu}}-$ $\bar{n}_{\nu_{\mu}}$ is the muon neutrino (minus muon antineutrino) number density,
$n_{\nu}=n(e)+n(\mu)$ is the total number neutrino density, $n_{12}=n_{\nu_{12}}-$ $\bar{n}_{\nu_{12}}$, and $\delta=n(e)-n(\mu)$ is a
statistical parameter of asymmetry between the two flavors. It is important to
note that the above expressions exactly coincide with those obtained in
3p by using a totally different method. We can finally write the
equation of motion in the form
$$(\gamma p-\sqrt{2}G_{F}\gamma^{0}n_{\nu}-\sqrt{2}G_{F}N\gamma^{0}-M)F_{L}(p)=0.$$
(36)
Obviously, if $\delta\neq 0$ and $n_{12}\neq 0$ then the interacting mixing
angle $\theta$ will not be the same as the vacuum mixing angle $\theta_{0}$.
At this point, we can add the contribution from an electrically neutral
background of protons, neutrons and electrons. This amounts to replacing the
matrix $N$ by
$$N^{\ast}=\frac{1}{2}\left(\begin{array}[c]{cc}n_{\nu}+\delta+2n_{e}-n_{n}&2n_{%
12}\\
2n_{12}&n_{\nu}-\delta-n_{n}\end{array}\right)$$
(37)
In this way, Eq. (36) will now become:
$$(\gamma p-\sqrt{2}G_{F}\gamma^{0}n_{\nu}-2\sqrt{2}G_{F}N^{\ast}\gamma^{0}-M)F_%
{L}(p)=0.$$
(38)
To find the propagation modes, we multiply Eq. (38) by
$(\gamma p-\sqrt{2}G_{F}\gamma^{0}n_{\nu}-2\sqrt{2}G_{F}N^{\ast}\gamma^{0}+M)$
, and neglect the terms of order $G_{F}^{2}$, as in the Hartree approximation.
Then, one obtains:
$$(p^{2}-\sqrt{2}G_{F}n_{\nu}p^{0}-\sqrt{2}G_{F}N^{\ast}p^{0}-M^{2})F_{L}(p)=0$$
(39)
In order to find the neutrino effective masses we have to diagonalize the
matrix
$$\hat{M}^{2}=M^{2}+\sqrt{2}G_{F}n_{\nu}p^{0}+2\sqrt{2}G_{F}N^{\ast}p^{0},$$
(40)
and to find the mixing angle in matter we have to obtain the unitary
transformation, which is determined by the corresponding rotation. After some
algebra, we arrive to the following expression for the effective masses in the
medium:
$$M_{1,2}^{\ast 2}\left(p^{0}\right)=\frac{1}{2}\left[\Sigma+2\sqrt{2}G_{F}\left%
(3n_{\nu}+n_{e}-n_{n}\right)p^{0}\right]\mp\frac{1}{2}\Delta^{\ast},$$
(41)
where
$$\Delta^{\ast}=\left[\left(2\sqrt{2}G_{F}p^{0}\left(n_{e}+\delta\right)-\Delta%
\cos 2\theta_{0}\right)^{2}+\left(4\sqrt{2}G_{F}p^{0}n_{12}-\Delta\sin 2\theta%
_{0}\right)^{2}\right]^{1/2}$$
(42)
is the effective mass difference. In Eq. (41) the upper
(lower) sign corresponds to $M_{1}^{\ast}$ ($M_{2}^{\ast}$) ,where $\Sigma$
and $\Delta$ have been defined in Eq. (25). The mixing angle is
given by
$$\sin 2\theta=\frac{\Delta\sin 2\theta_{0}-4\sqrt{2}G_{F}p^{0}n_{12}}{\Delta^{%
\ast}}.$$
(43)
Notice that Eqs. (41-43) depend on $n_{\nu}$
and $\delta$ which, in turn, have to be evaluated using the above equations.
In other words, the whole set of equations has to be solved
self-consistently.
Eq. (39) gives the dispersion relation for neutrinos and
antineutrino mass eigenstates, which can be written as:
$$p^{2}-M_{1,2}^{\ast 2}\left(p^{0}\right)=0$$
(44)
and provides (as an implicit equation) the energy $p^{0}$ as a function of the
momentum $|\vec{p}|$. As a first approximation, one can use the fact that,
under most situations of interest, neutrinos are extremely relativistic
particles. Thus, for neutrinos one can replace $p^{0}$ by $|\vec{p}|$. In this
way, the above dispersion equation can be approximately solved as:
$$p_{0}=\sqrt{|\vec{p}|^{2}+M_{1,2}^{\ast 2}\left(|\vec{p}|\right)}$$
(45)
To obtain the corresponding formulae for antineutrinos we only have to change
$|\vec{p}|$ to $-|\vec{p}|$ in the previous equation.
A consequence of Eq. (43) is that the condition for the MSW
resonance is modified with respect to the situation where there is not a
neutrino background. In fact, the condition for the resonance is now:
$$p^{0}=\frac{\Delta\left[\left(\delta+n_{e}\right)\cos 2\theta_{0}+2n_{12}\sin 2%
\theta_{0}\right]}{2\sqrt{2}G_{F}\left[\left(\delta+n_{e}\right)^{2}+4n_{12}^{%
2}\right]}$$
(46)
This new condition can be of interest if $\sin 2\theta_{0}\simeq 1$, as
suggested for both solar and atmospheric neutrino oscillation values. In this
case, the MSW resonance might be dominated by the neutrino background. In
order to investigate this possibility, we need to calculate $n_{12}$ and
$\delta$. Using Eqs. (33) and (43) we arrive to the
formulae
$$\displaystyle n_{12}$$
$$\displaystyle=\frac{\Delta\sin(2\theta_{0})F_{1}}{1+4\sqrt{2}G_{F}F_{2}}$$
$$\displaystyle\delta+n_{e}$$
$$\displaystyle=\frac{n_{e}+2\Delta\cos(2\theta_{0})F_{1}}{1+4\sqrt{2}G_{F}F_{2}}$$
(47)
where
$$\displaystyle F_{1}$$
$$\displaystyle=\frac{1}{4\pi^{2}}\int_{0}^{\infty}dpp^{2}\frac{f_{1}(p)-f_{2}(p%
)}{\Delta^{\ast}(p)}$$
$$\displaystyle F_{2}$$
$$\displaystyle=\frac{1}{4\pi^{2}}\int_{0}^{\infty}dpp^{3}\frac{f_{1}(p)-f_{2}(p%
)}{\Delta^{\ast}(p)}$$
(48)
The numerator in the integrand of the above equations is a small quantity, due
to the small mass difference $\Delta^{\ast}(p)$, therefore it is convenient to
expand the numerator using $\Delta^{\ast}(p)$ as a parameter. We then obtain:
$$f_{1}(p)-f_{2}(p)\simeq-\frac{\Delta^{\ast}(p)}{2p}\frac{\partial f_{1}(p)}{%
\partial p}$$
Moreover, in the extremely relativistic limit, we can write
$$f_{1}(p)\simeq\frac{1}{1+\exp\left[\left(p-\mu\right)/T\right]}$$
In this way, the integrals in Eq. (48) can be performed, giving
$$\displaystyle F_{1}$$
$$\displaystyle=\frac{T}{8\pi^{2}}\ln\left[1+\exp\left(\mu/T\right)\right]$$
$$\displaystyle F_{1}$$
$$\displaystyle=-\frac{T^{2}}{4\pi^{2}}L_{i2}\left[-\exp\left(\mu/T\right)\right]$$
(49)
with $L_{i2}$ the dilogarithmic function.
The correction to the MSW condition, as given in Eq. (46), is then
of the order
$$\frac{n_{12}}{n_{e}}\tan 2\theta_{0}$$
Let us consider the conditions in the Early Universe, just before the
nucleosynthesis. We then have a temperature $T\sim 1$ MeV , $\mu\sim 0.1$ MeV
and $n_{e}\sim 0.1MeV^{3}$. By substituting into the above expression, one
finds that the correction to the MSW condition is only meaningful if
$$\tan 2\theta_{0}>10^{15}$$
which implies $\theta_{0}=\pi/4$. Such values seem to be disfavored (although
not excluded) for $\nu_{e}\rightarrow(\nu_{\mu},\nu_{\tau})$ oscillations
14 . Only if this value is allowed, the neutrino background can play a
role in establishing the MSW condition. On the other hand, one can check that
the corrections of the neutrino background to the effective masses, as given
by Eq. (41), are negligible.
Another possibility consists in the adiabatic $\nu_{\mu}\rightarrow\nu_{\tau}$
conversion in the presence of a neutrino background. The corresponding
resonance formula can be found to be
$$p^{0}=\frac{\Delta\left[\delta\cos 2\theta_{0}+2n_{12}\sin 2\theta_{0}\right]}%
{2\sqrt{2}G_{F}\left[\delta^{2}+4n_{12}^{2}\right]}$$
(50)
As an example, we consider the proto-neutron star interior with $T\sim 40$ MeV
and $\mu\sim 100$ MeV. We then find $p^{0}\sim 10^{9}$ MeV which is, of course,
too high for the neutrinos produced in such a scenario.
VI Conclusions.
In this paper, we have investigated the equilibrium properties of a system of
two generations of mixed massive Dirac neutrinos in equilibrium, when
self-interactions are taken into account. To this end, we have used techniques
based on Wigner functions. We assume that well-defined quasi-particle states
exist for the neutrinos, i.e. the fermion fields for states with a definite
mass can be expanded in plane-wave states, similarly to the non-interacting
case, but with a different dispersion relation.
The equilibrium state is characterized by a single chemical potential $\mu$
and a temperature $T$. First, we analyzed the Hartree approximation (when
correlations are neglected). In this case, self-interactions are diagonal in
flavor space and do not modify the mixing angle, although they change the
effective masses.
The inclusion of correlations can be done, under the conditions assumed above,
using the derivations made in appendix. Our results for these corrections
agree with previous calculations 3s ; 3p , using completely different
techniques. These corrections give a non-diagonal term in the effective
mass matrix. Therefore, in addition to a modification in the effective masses
of eigenstates, there is a change in the in-medium mixing angle, as compared
to the Hartree result. Also, the condition for the MSW resonance (when a
matter background is included) differs from the usual MSW condition. We have
shown, however, that, for typical astrophysical and cosmological scenarios,
the corrections are small, although they can be of some interest in cosmology
if the $\nu_{e}$-($\nu_{\mu},\nu_{\tau}$) mixing angle is exactly $\pi/4$, as
still allowed by neutrino oscillation experiments and theoretical models
14 ; 15 .
Acknowledgements.
This work has been supported by the Spanish grants FPA2002-00612 and
AYA2001-3490-C02. We are indebted to Sergio Pastor for a fruitful discussion
and comments.
Appendix A Calculation of correlations.
In this appendix we calculate the statistical correlations defined by Eq.
(26) under the quasi-particle hypothesis. Let us consider, for
simplicity, that we have a neutrino field in equilibrium consisting on one
generation of massive neutrinos. Using the corresponding unitary
transformation to mass eigenstates, the following procedure can be easily
generalized to more than one generation. The Wigner function for this field
is
$$\widehat{F}_{ij}(x,p)=(2\pi)^{-4}\int d^{4}y\,e^{-ipy}<\widehat{\overline{\Psi%
}}_{j}(x+y/2)\widehat{\Psi}_{i}(x-y/2)>,$$
(51)
and the correlation functions can be expressed as
$$D_{ijkl}(x,x^{\prime},p,p^{\prime})=<\widehat{F}_{ij}(x,p)\widehat{F}_{kl}(x^{%
\prime},p^{\prime})>-F_{ij}(x,p)F_{kl}(x^{\prime},p^{\prime})$$
(52)
We can now evaluate the first term on the right hand using Wick’s theorem
FW . In this way, we arrive to:
$$\displaystyle<\widehat{F}_{ij}(x,p)\widehat{F}_{kl}(x^{\prime},p^{\prime})>=(2%
\pi)^{-8}\int d^{4}yd^{4}y^{\prime}\,e^{-ipy}e^{-ip^{\prime}y^{\prime}}$$
(53)
$$\displaystyle<\widehat{\overline{\Psi}}_{j}(x+y/2)\widehat{\Psi}_{i}(x-y/2)%
\widehat{\overline{\Psi}}_{l}(x^{\prime}+y^{\prime}/2)\widehat{\Psi}_{k}(x^{%
\prime}-y^{\prime}/2)>$$
$$\displaystyle=$$
$$\displaystyle(2\pi)^{-8}\int d^{4}yd^{4}y^{\prime}\,e^{-ipy}e^{-ip^{\prime}y^{%
\prime}}[$$
$$\displaystyle<\widehat{\overline{\Psi}}_{j}(x+y/2)\widehat{\Psi}_{i}(x-y/2)><%
\widehat{\overline{\Psi}}_{l}(x^{\prime}+y^{\prime}/2)\widehat{\Psi}_{k}(x^{%
\prime}-y^{\prime}/2)$$
$$\displaystyle-$$
$$\displaystyle<\widehat{\overline{\Psi}}_{j}(x+y/2)\widehat{\overline{\Psi}}_{l%
}(x^{\prime}+y^{\prime}/2)><\widehat{\Psi}_{i}(x-y/2)\widehat{\Psi}_{k}(x^{%
\prime}-y^{\prime}/2)$$
$$\displaystyle+$$
$$\displaystyle<\widehat{\overline{\Psi}}_{j}(x+y/2)\widehat{\Psi}_{k}(x^{\prime%
}-y^{\prime}/2)><\widehat{\Psi}_{i}(x-y/2)\widehat{\overline{\Psi}}_{l}(x^{%
\prime}+y^{\prime}/2)>].$$
(54)
The first term in this expression leads to a product of Wigner functions which
is canceled by the second term in Eq. (52). The second term
vanishes. Finally, the third term can be expressed as:
$$\displaystyle(2\pi)^{-8}$$
$$\displaystyle\int d^{4}yd^{4}y^{\prime}\,e^{-ipy}e^{-ip^{\prime}y^{\prime}}<%
\widehat{\overline{\Psi}}_{j}(x+y/2)\widehat{\Psi}_{k}(x^{\prime}-y^{\prime}/2)>$$
$$\displaystyle\times$$
$$\displaystyle\left[-<\widehat{\overline{\Psi}}_{l}(x^{\prime}+y^{\prime}/2)%
\widehat{\Psi}_{i}(x-y/2)>-iS_{il}(x-y/2-x^{\prime}-y^{\prime}/2)\right]$$
(55)
In this formula $S_{il}()$ is the propagator of the neutrino field. It appears
because normal ordering of the operators has no been considered. We now impose
normal ordering, which amounts to neglecting the last term. In this way, we
find:
$$\displaystyle D_{ijkl}(x,x^{\prime},p,p^{\prime})=-(2\pi)^{-8}\int d^{4}yd^{4}%
y^{\prime}\,e^{-ipy}e^{-ip^{\prime}y^{\prime}}$$
$$\displaystyle<\widehat{\overline{\Psi}}_{j}(x+y/2)\widehat{\Psi}_{k}(x^{\prime%
}-y^{\prime}/2)><\widehat{\overline{\Psi}}_{l}(x^{\prime}+y^{\prime}/2)%
\widehat{\Psi}_{i}(x-y/2)>$$
(56)
By using the equality
$$\widehat{\overline{\Psi}}_{i}(x+z/2)\widehat{\Psi}_{j}(x-z/2)=\int d^{4}p\,e^{%
ipz}\widehat{F}_{ji}(x,p),$$
(57)
and the conditions of equilibrium, which imply that $F(x,p)=F(p),$ we obtain
$$D_{ijkl}(x-x^{\prime},p,p^{\prime})=-\int d^{4}q\delta(p-p^{\prime})e^{-i2(p^{%
\prime}-q)(x-x^{\prime})}F_{kj}(q)F_{il}(2p^{\prime}-q).$$
(58)
If we take the Fourier transformation of the correlations, we have that
$$\widetilde{D}_{ijkl}(k,p,p^{\prime})=(2\pi)^{-4}\int d^{4}(x-x^{\prime})D_{%
ijkl}(x-x^{\prime},p,p^{\prime})e^{-ik(x-x^{\prime})}$$
(59)
and we end up with the following result:
$$\widetilde{D}_{ijkl}(k,p,p^{\prime})=-\delta(p-p^{\prime})F_{kj}(p^{\prime}+k/%
2)F_{il}(p^{\prime}-k/2).$$
(60)
References
(1)
G. Raffelt. Nucl. Phys B (Proc. Suppl.) 95 (2001) 183; G.
Raffelt. New Astron. Rev. 46 (2002) 699-708.
(2)
S. Pastor and G. Raffelt. Phys. Rev. Lett. 89 (2002) 191101.
(3)
Yong-Zhong Qian and G.M. Fuller. Phys. Rev. Lett. 71 (1993)
1965 ; ; G. Sigl Phys. Rev. D 51 (1995) 4035; G.L. Fogli, E. Lisi, D.
Montanino and A. Palazzo. Phys. Rev D 65 (2002) 073008.
(4)
M.J. Thomson and B. McKellar. Phys. Lett. B 259 (1991)
113; V.A. Kostelecky and M.J. Thomson. Phys. Rev. D 49 (1994) 1740. B.
McKellar and M.J. Thomson. Phys. Rev. D 49 (1994) 2710.
(5)
K.N. Abazajian, J.F. Beacom and N.F. Bell Phys.Rev D 66
(2002) 013008.
(6)
S. Pastor, G. Raffelt and D.V. Semikoz Phys. Rev D 65
(2002) 053011 ; A.D. Dolgov, S.H. Hansen, S. Pastor, S.T. Petcov, G. Raffelt
and D.V. Semikoz. Nucl. Phys B 632 (2002) 363.
(7)
Y.Y.Y. Wong. Phys.Rev D 66 (2002) 025015.
(8)
J. Pantaleone. Phys. Lett. B 287 (1992) 128.
(9)
J. Pantaleone. astro-ph/9405008.
(10)
A. Friedland and C. Lunardini. hep-ph/0304055.
(11)
M. Sirera and A. Pérez. Phys. Rev. D 59 (1999) 125011.
(12)
G. Sigl and G. Raffelt. Nucl. Phys. B 406 (1993) 423.
(13)
G. Raffelt. Stars as Laboratories for Fundamental
Physics. The University of Chicago Press (1996).
(14)
Yong-Zhong Qian and G.M. Fuller. Phys. Rev. D 51 (1995) 1479.
(15)
S. Samuel. Phys. Rev. D 48 (1993) 1462.
(16)
D. Nötzold and G. Raffelt. Nucl. Phys B 307 (1988) 924.
(17)
C.W. Kim and A. Pevsner. Neutrinos in Physics and
Astrophysics. Ed. Harwood Academic Publishers (1993).
(18)
A. L. Fetter and J.D. Walecka, Quantum Theory of
Many-Particle Physics. McGraw-Hill Inc. (1971).
(19)
M. Maltoni, T. Schwetz, M.A. Tórtola and J.W.F. Valle. hep-ph/0309130.
(20)
N.F. Bell, A.A. Rawlinson and R.F. Sawyer. hep-ph/0304082.
(21)
S. Pakvasa and J.W.F. Valle. hep-ph/0301061.
(22)
Mad. Prakash, I. Bombaci, Man. Prakash, P.J. Ellis, J.M. Lattimer
and R. Knorren. Phys.Rep 280 (1997) 1-77.
(23)
E. Kh. Akhmedov, M. Frigerio and A. Yu. Smirnov. JHEP 0309
(2003) 021; We-Fu Chang, J. N. Ng. JHEP 0310 (2003) 036. D. Falcone.
Phys.Rev. D68 (2003) 033002. |
Independence, Relative Randomness, and PA Degrees
Adam R. Day
Adam R. Day
Department of Mathematic
University of California, Berkeley, CA
USA
adam.day@math.berkeley.edu
and
Jan Reimann
Jan Reimann
Department of Mathematics
Pennsylvania State University
University Park, PA, USA
reimann@math.psu.edu
(Date:: December 4, 2020)
Abstract.
We study pairs of reals that are mutually Martin-Löf random with respect to a common, not necessarily computable probability measure. We show that a generalized version of van Lambalgen’s Theorem holds for non-computable probability measures, too. We study, for a given real $A$, the independence spectrum of $A$, the set of all $B$ so that there exists a probability measure $\mu$ so that $\mu\{A,B\}=0$ and $(A,B)$ is $\mu\times\mu$-random. We prove that if $A$ is r.e., then no $\Delta^{0}_{2}$ set is in the independence spectrum of $A$. We obtain applications of this fact to PA degrees. In particular, we show that if $A$ is r.e. and $P$ is of PA degree so that $P\not\geq_{\operatorname{T}}A$, then $A\oplus P\geq_{\operatorname{T}}\emptyset^{\prime}$.
Reimann was partially supported by NSF grants DMS-0801270 and DMS-1201263.
1. Independence and relative randomness
The property of independence is central to probability theory. Given a probability space with measure $\mu$, we call two measurable sets $\mathcal{A}$ and $\mathcal{B}$ independent if
$$\mu\mathcal{A}=\frac{\mu(\mathcal{A}\cap\mathcal{B})}{\mu\mathcal{B}}.$$
The idea behind this definition is that if event $\mathcal{B}$ occurs, it does not make event $\mathcal{A}$ any more or less likely. This paper considers a similar notion, that of relative randomness. The theory of algorithmic randomness provides a means of defining which elements of Cantor space ($2^{\omega}$) are random. We call $A\in 2^{\omega}$ Martin-Löf random if $A$ is not an element of any effective null set. We denote the class of all Martin-Löf random reals by $\operatorname{MLR}$.111For a comprehensive presentation of the theory of Martin-Löf randomness, see the monographs by Downey and Hirschfeldt (2010) and Nies (2009).
We say that $A$ is Martin-Löf random relative to $B$, or $A\in\operatorname{MLR}(B)$ if $A$ is not an element of any null set effective in $B$. Relative randomness is analogous to independence because if $A\in\operatorname{MLR}(B)$, then not only is $A$ a random real but even given the information in $B$, we cannot capture $A$ in an effective null set.
If we start with the assumption that $A$ and $B$ are both Martin-Löf random, then the following theorem of van Lambalgen establishes that relative randomness is symmetrical.
Theorem 1.1 (Van Lambalgen (1987)).
If $A,B\in\operatorname{MLR}$ then
$A\in\operatorname{MLR}(B)$ if and only if $B\in\operatorname{MLR}(A)$ if and only if
$A\oplus B\in\operatorname{MLR}$.
We can extend the notion of relative randomness to any probability measure. We take $\mathcal{P}(2^{\omega})$ to be the set of all Borel probability measures on Cantor space. Endowed with the weak-$*$ topology, $\mathcal{P}(2^{\omega})$ becomes a compact metrizable space. The measures that are a finite, rational-valued, linear combination of Dirac measures form a countable dense subset, and one can choose a metric on $\mathcal{P}(2^{\omega})$ that is compatible with the weak-$*$ topology so that the distance between the those basic measures is a computable function, and with respect to which $\mathcal{P}(2^{\omega})$ is complete. In other words, $\mathcal{P}(2^{\omega})$ can be given the structure of an effective Polish space. We can represent measures via Cauchy sequences of basic measures. This allows for coding measures as reals, and one can show that there exists a continuous mapping $\rho:{2^{\omega}}\to\mathcal{P}(2^{\omega})$ so that for any $X\in{2^{\omega}}$,
$$\rho^{-1}(\{\rho(X)\})\text{ is a $\Pi^{0}_{1}(X)$ class.}$$
For details of this argument, see Day and Miller (ta). If $\mu\in\mathcal{P}(2^{\omega})$, any real $R$ with $\rho(R)=\mu$ is called a representation of $\mu$.
We want to define randomness relative to a parameter with respect to a probability measure $\mu$. Martin-Löf’s framework easily generalizes to tests that have access to an oracle. However, our test should have access to two sources: the parameter of relative randomness and the measure (in form of a representation).
Definition 1.2.
Let $R_{\mu}$ be a representation of a measure $\mu$, and let $A\in{2^{\omega}}$.
(a)
A $(R_{\mu},A)$-test is given by a sequence $(\mathcal{V}_{n}\colon n\in\mathbb{N})$ of uniformly $\Sigma^{0}_{1}(R_{\mu}\oplus A)$-classes $\mathcal{V}_{n}\subseteq{2^{\omega}}$ such that for all $n$, $\mu(\mathcal{V}_{n})\leq 2^{-n}$.
(b)
A real $X\in{2^{\omega}}$ passes an $(R_{\mu},A)$-test $(\mathcal{V}_{n})$ if $X\not\in\bigcap_{n}\mathcal{V}_{n}$.
(c)
A real $X\in{2^{\omega}}$ is $(R_{\mu},A)$-random if it passes all $(R_{\mu},A)$-tests.
If, in the previous definition, $A=\emptyset$, we simply speak of an $R_{\mu}$-test and of $X$ being $R_{\mu}$-random.
The previous definition defines randomness with respect to a specific representation. If $X$ is random for one representation, it is not necessarily random for other representations.
On the other hand, we can ask whether a real exhibits randomness with respect to some representation, so the following definition makes sense.
Definition 1.3.
A real $X\in{2^{\omega}}$ is $\mu$-random relative to $A\in{2^{\omega}}$, or simply $\mu$-$A$-random if there exists a representation $R_{\mu}$ of $\mu$ so that $X$ is $(R_{\mu},A)$-random. We denote by $\operatorname{MLR}_{\mu}(A)$ the set of all $\mu$-$A$-random reals.
For Lebesgue measure $\lambda$, we sometimes suppress the measure. Hence, in accordance with established notation, $\operatorname{MLR}(A)$ denotes the set of all Martin-Löf random reals.
A most useful property of the theory of Martin-Löf randomness is the existence of universal tests. Universal tests subsume all other tests. Furthermore, they can be defined uniformly with respect to any parameter. The construction can be extended to tests with respect to a measure $\mu$. More precisely, there exists a uniformly c.e. sequence $(U_{n}\colon n\in\mathbb{N})$ of sets $U_{n}\subseteq{2^{<\omega}}$ such that, if we set for $R,A\in{2^{\omega}}$
$$\mathcal{U}^{R,A}_{n}=\{[\sigma]\colon\langle\sigma,\tau_{0},\tau_{1}\rangle%
\in U_{n},\;\tau_{0}\prec R,\tau_{1}\prec A\},$$
then $(\mathcal{U}^{R,A}_{n})$ is an $(R,A)$-test and $X\in{2^{\omega}}$ is $(R,A)$-random if and only if $X\not\in\bigcap_{n}\mathcal{U}^{R,A}_{n}$. We call $(U_{n})$ a universal oracle test.
Since for any $R\in{2^{\omega}}$, $\rho^{-1}(\rho(R))$ is $\Pi^{0}_{1}(R)$, we can eliminate the representation of a measure in a test for randomness by defining, for any $A\in{2^{\omega}}$,
$$\widetilde{\mathcal{U}}^{R,A}_{n}=\bigcap_{S\in\rho^{-1}(\{\rho(R)\})}\mathcal%
{U}^{S,A}_{n}.$$
The resulting class $\widetilde{\mathcal{U}}^{A}_{n}$ is still $\Sigma^{0}_{1}(R)$, since $\rho^{-1}(\{\rho(R)\})$ is $\Pi^{0}_{1}(R)$ and hence compact.
Proposition 1.4.
For any $R,A\in{2^{\omega}}$ with $\rho(R)=\mu$, a real $X$ is $\mu$-$A$-random if and only if
$$X\not\in\bigcap_{n}\widetilde{\mathcal{U}}^{R,A}_{n}.$$
Proof.
If $X$ is $\mu$-$A$-random, then it passes every $(R_{\mu},A)$-test for some representation $R_{\mu}$ of $\mu$, in particular the instance $(\mathcal{U}^{R_{\mu},A}_{n})$ of the universal oracle test. Since $R_{\mu}\in\rho^{-1}(\{\rho(R)\})$, it follows that $X$ passes $\widetilde{\mathcal{U}}^{R,A}_{n}$.
On the other hand, if for every representation $R_{\mu}$ of $\mu$, $X$ fails the test $(\mathcal{U}^{R,A}_{n})$, then $X\in\bigcap_{n}\widetilde{\mathcal{U}}^{R,A}_{n}.$
∎
The previous proposition shows that the test $\widetilde{\mathcal{U}}^{R,A}_{n}$ is related to the concept of a uniform test, originally introduced by Levin (1976), and further developed by Gács (2005) and Hoyrup and Rojas (2009). Hence we call it a uniform oracle test.
Note that if $R,S$ are both representations of a measure $\mu$, then the uniform oracle tests $(\widetilde{\mathcal{U}}^{R,A}_{n})_{n}$ and $(\widetilde{\mathcal{U}}^{S,A}_{n})_{n}$ are identical.
Definition 1.5.
Take $A,B\in 2^{\omega}$ and $\mu\in\mathcal{P}(2^{\omega})$. We say that $A$ and $B$ are relatively random with respect to $\mu$ if $A\in\operatorname{MLR}_{\mu}(B)$ and $B\in\operatorname{MLR}_{\mu}(A)$.
Note that the representations of $\mu$ witnessing randomness for $A$ and $B$, respectively, do not have to be identical.
If $A$ and $B$ are relatively random with respect to some measure $\mu$, then $\mu$ might offer some information about the relationship between $A$ and $B$. For example, we know that if $A$ and $B$ are relatively random with respect to Lebesegue measure, then any real they both compute must be K-trivial. If $A$ and $B$ are both atoms of $\mu$ then clearly $A$ and $B$ are relatively random with respect to $\mu$. Given this, perhaps the most obvious question to ask about relative randomness is the following.
Question 1.6.
For which $A,B\in 2^{\omega}$ does there exist a measure $\mu$ such that $A$ and $B$ are relatively random with respect to $\mu$ and neither $A$ nor $B$ is an atom of $\mu$?
This question is closely related to a theorem of Reimann and Slaman (2008).
They proved that an element $X$ of Cantor space is non-recursive if and only if there exists a measure $\mu$ such that $X$ is $\mu$-random and $X$ is not an atom of $\mu$.
Van Lambalgen’s theorem shows that $A$ and $B$ are relatively random if and only if $A\oplus B\in\operatorname{MLR}$. If we take $\lambda$ to be the uniform measure, then $A\oplus B\in\operatorname{MLR}$ if and only if the the pair $(A,B)\in 2^{\omega}\times 2^{\omega}$ is Martin-Löf random with respect to the product measure $\lambda\times\lambda$ i.e. $(A,B)\in\operatorname{MLR}_{\lambda\times\lambda}$. We begin our investigation into relative randomness by showing that van Lambalgen’s theorem holds for any Borel probability measure on Cantor space.
Theorem 1.7.
Let $\mu\in\mathcal{P}(2^{\omega})$ and let $A,B\in{2^{\omega}}$ then
$(A,B)\in\operatorname{MLR}_{\mu\times\mu}$ if and only if $A\in\operatorname{MLR}_{\mu}$ and $B\in\operatorname{MLR}_{\mu}(A)$.
Proof.
Let $R$ be any representation of $\mu$.
First let us consider if $B\not\in\operatorname{MLR}_{\mu}(A)$. In this case we have that
$B\in\bigcap_{n}\mathcal{U}_{n}^{R,A}$. We define an $(R,\emptyset)$-test for ${2^{\omega}}\times{2^{\omega}}$ by
$\mathcal{V}_{n}^{R}=\{[\tau]\times[\sigma]:\exists\eta\prec R\;(\langle\sigma,%
\eta,\tau\rangle\in U_{n})\}$. This ensures that $(A,B)\in\bigcap_{n}\mathcal{V}^{R}_{n}$.
By applying Fubini’s theorem we can establish that:
$$\displaystyle(\mu\times\mu)(\mathcal{V}^{R}_{n})$$
$$\displaystyle=\int_{{2^{\omega}}\times{2^{\omega}}}\chi_{\mathcal{V}^{R}_{n}}(%
X,Y)d\mu\times d\mu$$
$$\displaystyle=\int_{2^{\omega}}\left(\int_{{2^{\omega}}}\chi_{\mathcal{U}_{n}^%
{R,X}}(Y)d\mu(Y)\right)d\mu(X)$$
$$\displaystyle\leq\int_{2^{\omega}}2^{-n}d\mu(X)=2^{-n}$$
Hence $(A,B)$ is not $(R,\emptyset)$-random. As this is true for any representation $R$ of $\mu$
we have that $(A,B)\not\in\operatorname{MLR}_{\mu\times\mu}$. The same argument shows a fortiori that if $A\not\in\operatorname{MLR}_{\mu}$ then $(A,B)\not\in\operatorname{MLR}_{\mu\times\mu}$.
To establish the other direction assume that
$(A,B)\not\in\operatorname{MLR}_{\mu\times\mu}$. Again let $R$ be any
representation of $\mu$.
Hence $(A,B)\in\bigcap_{n}\mathcal{V}^{R}_{n}$, where $(\mathcal{V}^{R}_{n})$ is a universal $R$-test for ${2^{\omega}}\times{2^{\omega}}$. Let
$$\mathcal{W}_{n}^{R,X}=\{Y:(X,Y)\in\mathcal{V}^{R}_{n}\}$$
We have that
$\mathcal{W}_{n}^{R,X}$ is a $\Sigma^{0}_{1}(R\oplus X)$ class and this is uniform in $n$. However, given any $X$, we do not know whether or not $\mu(\mathcal{W}_{n}^{R,X})\leq f(n)$ for some decreasing recursive function $f$ such that $\lim_{n}f(n)=0$. Hence we cannot necessarily turn this into a Martin-Löf test relative to $X$. In fact it is not even necessarily true that $\liminf_{n}\mu(\mathcal{W}_{n}^{R,X})=0$.
We will show that the failure to turn this into a Martin-Löf test for some $X\in{2^{\omega}}$ implies that $X\not\in\operatorname{MLR}_{\mu}$.
This is a slight strengthening of the result that van Lambalgen obtained in his thesis. Van Lambalgen showed that if $\liminf_{n}\mu(\mathcal{W}_{n}^{R,X})\neq 0$ then $X\not\in\operatorname{MLR}_{\mu}$.
However, we can generalize the proof of van Lambalgen’s theorem given in Nies (2009). We define another $R$-test by letting
$\mathcal{T}_{n}^{R}=\{X\in{2^{\omega}}:\mu(\mathcal{W}_{2n}^{R,X})>2^{-n}\}$.
To see that $\mathcal{T}_{n}^{R}\leq 2^{-n}$, note that
$$\displaystyle(\mu\times\mu)\mathcal{V}^{R}_{2n}$$
$$\displaystyle\geq\int_{\mathcal{T}_{n}^{R}\times{2^{\omega}}}\chi_{\mathcal{V}%
^{R}_{2n}}(X,Y)d\mu\times d\mu$$
$$\displaystyle=\int_{\mathcal{T}_{n}^{R}}\int_{{2^{\omega}}}\chi_{\mathcal{W}_{%
2n}^{R,X}}(Y)d\mu(Y)d\mu(X)$$
$$\displaystyle\geq\int_{\mathcal{T}_{n}^{R}}2^{-n}d\mu(X)=2^{-n}\mu(\mathcal{T}%
_{n}^{R})$$
Now as $2^{-2n}\geq(\mu\times\mu)\mathcal{V}_{2n}^{R}$, we have that $\mu(\mathcal{T}_{n}^{R})\leq 2^{-n}$. Hence $\cap_{n}\mathcal{T}_{n}^{R}$ is an $R$-test.
Assume that $A\not\in\operatorname{MLR}_{\mu}(R)$. Then $A$ avoids all but finitely many of the sets $\mathcal{T}_{n}^{R}$. Hence for all but finitely many $n$ we have that
$\mu\,\mathcal{W}^{R,A}_{2n}\leq 2^{-n}$ and so by modifying finitely many $\mathcal{W}^{R,A}_{2n}$ we can obtain an $(R,A)$-test that covers $B$. Therefore $B$ is not $R$-random relative to $A$.
For all representations $R$ of $\mu$, we have shown that either $A$ is not $R$-random or $B$ is not $R$-random relative to $A$. However, to prove the theorem, it is essential that we get the same outcome for all representations
i.e. if $(A,B)\not\in\operatorname{MLR}_{\mu\mu}$ then either for all representations $R$ of $\mu$, $A$ is not $R$-random or for all representations $R$ of $\mu$, $B$ is not $R$-random relative to $A$.
We can resolve this problem by taking our test $(\mathcal{V}^{R}_{n})$ on the product space to be a uniform test. In this case we always obtain the same “projection tests” $(\mathcal{W}_{n}^{R,X})$ (independent of $R$) and hence the same outcome for any representation of $\mu$.
∎
Corollary 1.8.
If $A,B\in{2^{\omega}}$ and $\mu\in\mathcal{P}(2^{\omega})$, then $A$ and $B$ are relatively random with respect to $\mu$ if and only if $(A,B)\in\operatorname{MLR}_{\mu\times\mu}$.
Corollary 1.9.
If $A\geq_{\operatorname{T}}B$ and $(A,B)\in\operatorname{MLR}_{\mu\times\mu}$ then $B$ must be an atom of $\mu$.
Proof.
This holds because $B\in\operatorname{MLR}_{\mu}(A)$ if and only if $B$ is an atom of $\mu$.
∎
We note that we cannot extend one direction of van Lambalgen’s theorem to product measures of the form $\mu\times\nu$. In particular it is not true that if $(A,B)\in\operatorname{MLR}_{\mu\times\nu}$ then $A\in\operatorname{MLR}_{\mu}$ and $B\in\operatorname{MLR}_{\nu}(A)$. For example we can code $B$ into $\mu$ and obtain $A\in\operatorname{MLR}_{\mu}$, $B\in\operatorname{MLR}_{\nu}(A)$, $(A,B)\not\in\operatorname{MLR}_{\mu\times\nu}$.
Given any $X\in 2^{\omega}$, we will use $\mathcal{R}(X)$ to denote the set of reals $Y$ such that $X$ and $Y$ are relatively random with respect to some measure $\mu$ and neither $X$ nor $Y$ are atoms of $\mu$ i.e.
$$\mathcal{R}(A)=\{B\in 2^{\omega}:(\exists\mu\in\mathcal{P}(2^{\omega}))[(A,B)%
\in\operatorname{MLR}_{\mu\times\mu}\text{ and }\mu\{A,B\}=0]\}.$$
We call $\mathcal{R}(A)$ the independence spectrum of $A$.
The following proposition lists some basic properties of the independence spectrum.
Proposition 1.10.
For all $A,B\in 2^{\omega}$ the following hold:
(1)
$A\in\mathcal{R}(B)$ if and only if $B\in\mathcal{R}(A)$.
(2)
$B\in\mathcal{R}(A)$ implies that $A\mid_{\operatorname{T}}B$.
(3)
If $A$ is non-recursive and $\nu$ is a computable, non-atomic measure (i.e. a measure with a computable representation and $\nu\{X\}=0$ for all $X\in{2^{\omega}}$), then $\mathcal{R}(A)$ has $\nu$-measure $1$.
(4)
If $A\in\operatorname{MLR}$ then $\operatorname{MLR}(A)\subsetneq\mathcal{R}(A)$.
Proof.
(1) is by definition and (2) is by Corollary 1.9.
(3) Suppose $A$ is non-recursive and $\nu$ is a computable measure with $\nu\{A\}=0$. There is a measure $\mu$ such that $A$ is not an atom of $\mu$ and $A\in\operatorname{MLR}_{\mu}$, say via a representation $R_{\mu}$. Let $\kappa=(\mu+\nu)/2$. There exists a representation $R_{\kappa}\leq_{\operatorname{T}}R_{\mu}$, as $\nu$ is computable.
We claim that $A$ is $R_{\kappa}$-random. For if not, then $A$ fails some $R_{\kappa}$-test $(\mathcal{W}^{R_{\kappa}}_{n})$. We have
$$\mu\mathcal{W}^{R_{\kappa}}_{n}=2\kappa\mathcal{W}^{R_{\kappa}}_{n}-\nu%
\mathcal{W}^{R_{\kappa}}_{n}\leq 2\kappa\mathcal{W}^{R_{\kappa}}_{n}\leq 2^{n-%
1}.$$
Since $R_{\kappa}\leq_{\operatorname{T}}R_{\mu}$, $(\mathcal{W}^{R_{\kappa}}_{n+1})$ would define an $R_{\mu}$-test that covers $A$, contradicting the assumption that $A$ is $R_{\mu}$-random. Furthermore, by assumption on $\mu$ and $\nu$, $\kappa\{A\}=0$. Hence
$$(\operatorname{MLR}_{\kappa}(A)\setminus\{B:\kappa\{B\}\neq 0\})\subseteq%
\mathcal{R}(A),$$
by van Lambalgen’s Theorem.
Now $\nu(\operatorname{MLR}_{\kappa}(A))=1$ because the complement of $\operatorname{MLR}_{\nu}(A)$ is a $\kappa$ null set and hence a $\nu$ null set ($\nu$ is absolutely continuous with respect to $\kappa$ by definition). Moreover, the
set of atoms of $\kappa$ is countable and so has $\nu$-measure $0$ by the assumption that $\nu$ is non-atomic. This gives us that
$$\nu(\operatorname{MLR}_{\kappa}(A)\setminus\{B:\kappa\{B\}\neq 0\})=1$$
and thus $\nu\mathcal{R}(A)=1$.
(4) Suppose $A$ is Martin-Löf random. By the definition of $\mathcal{R}(A)$ and Theorem 1.7 we have that $\operatorname{MLR}(A)\subseteq\mathcal{R}(A)$.
On the other hand, $A$ is not recursive and hence by (3),
$\mathcal{R}(A)$ has measure $1$ for any computable, non-atomic measure.
Let $\nu$ be a computable, non-atomic measure orthogonal to Lebesgue measure (e.g. the $(1/3,2/3)$-Bernoulli measure). Since $\nu\mathcal{R}(A)=1$, $\mathcal{R}(A)$ has to contain a $\nu$-random element $X$. But $X$ cannot be relatively Martin-Löf random. Therefore, $\operatorname{MLR}(A)\subsetneq\mathcal{R}(A)$.
∎
The proposition shows that, outside the upper and lower cone of a real $A$, the complement of $\mathcal{R}(A)$ is rather small measure wise. On the other hand,
the above properties leave open the possibility that $\mathcal{R}(A)$ is just the set of reals that are Turing incomparable with $A$. We will now establish that this is not necessarily the case.
Proposition 1.11.
Let $R$ be a representation of a measure $\mu$. If $A\in{2^{\omega}}$ is such that
(1)
$A$ is r.e.222We mean here, of course, that $A$ is recursively enumerable viewed as a subset of $\mathbb{N}$, by identifying a subset of $\mathbb{N}$ with the real given by its characteristic sequence.,
(2)
$A$ is $R$-random, and
(3)
$A$ is not an atom of $\mu$,
then $R\oplus A\geq_{\operatorname{T}}R^{\prime}$.
Proof.
Given such an $R$ and $A$, let $A_{s}$ be a recursive approximation to $A$. We define the function $f\leq_{\operatorname{T}}A\oplus R$ by:
$$f(x)=\min\{s:(\exists m\leq s)(A_{s}\upharpoonright m=A\upharpoonright m\wedge%
\mu_{s}[A\upharpoonright m]<2^{-x})\}.$$
In this definition we take $\mu_{s}[\sigma]$ to be an $R$-recursive approximation to $\mu[\sigma]$ from above.
Note that $f$ is well defined because $A$ is not an atom of $\mu$. We claim that if $g$ is any partial function recursive in $R$, then for all but finitely many $x\in\mbox{dom}(g)$, we have that $f(x)>g(x)$.
To establish this claim, let $g$ be an $R$-recursive partial function. We will build an $R$-test $\{U_{n}\}_{n\in\omega}$ by defining $U_{n}$ to be:
$$\{X\in{2^{\omega}}:(\exists x>n)(\exists m)(g(x)\downarrow\wedge\,\mu[A_{g(x)}%
\upharpoonright m]<2^{-x}\wedge X\succ(A_{g(x)}\upharpoonright m))\}.$$
Because any $x\in\mbox{dom}(g)$ adds a single open set ($[A_{g(x)}\upharpoonright m]$ for some $m$) of measure less than $2^{-x}$ to those $U_{n}$ with $n<x$, we have constructed a valid test.
Now if $g(x)\downarrow\geq f(x)$, then by definition of $f$, there is some $m\leq f(x)$ such that
$\mu[A\upharpoonright m]<2^{-x}$ and $A\upharpoonright m=A_{f(x)}\upharpoonright m=A_{g(x)}\upharpoonright m$. Thus for all $n<x$, $A\in U_{n}$. Because $A\in\operatorname{MLR}_{\mu}(R)$ we have that $f(x)>g(x)$ for all but finitely many $x$ in $\mbox{dom}(g)$.
Let $g(x)$ be the $R$-recursive partial function with domain $R^{\prime}$ such that $g(x)$ is the unique $s$ such that $x\in R^{\prime}_{s+1}\setminus R^{\prime}_{s}$. For almost all $x$, we have that $x\in R^{\prime}$ if and only if $x\in R^{\prime}_{f(x)}$ and so $R^{\prime}\leq_{\operatorname{T}}A\oplus R$.
∎
Theorem 1.12.
Let $R$ be a representation of a measure $\mu$. If
(1)
$A$ is r.e.,
(2)
$A$ is $\mu$-random, and
(3)
$A$ is not an atom of $\mu$,
then $R\oplus A\geq_{\operatorname{T}}\emptyset^{\prime}$.
Proof.
Note the following characteristics of the previous proof. First the totality of $f$ does not depend on the fact that $A$ is $R$-random, it only depends on the fact that $A$ is not an atom of $\mu$. The construction is uniform so there is a single index $e$ such that $\Phi_{e}(A\oplus\hat{R})$ is total if
$\hat{R}$ is any representation of $\mu$. Additionally if $A$ is $\hat{R}$-random then for all but finitely many $x$, $\Phi_{e}(A\oplus\hat{R};x)\geq g(x)$ where $g$ is any $\hat{R}$-recursive partial computable function.
Let $R$ be any representation of $\mu$. The set
$\{A\oplus\hat{R}:\hat{R}$ is a representation of $\mu\}$ is a
$\Pi^{0}_{1}(A\oplus R)$ class and $\Phi_{e}$ is total on this class. From $A\oplus R$ we can compute a function $f$ that dominates $\Phi_{e}(A\oplus\hat{R})$ where $A$ is $\hat{R}$-random. As $f$ dominates any $\hat{R}$-recursive partial function we have that $A\oplus R\geq_{\operatorname{T}}\emptyset^{\prime}$.
∎
Corollary 1.13.
If $A$ is r.e. and $B\leq_{\operatorname{T}}\emptyset^{\prime}$ then $B\not\in\mathcal{R}(A)$.
The question remains, however, how big the independence of a real can be outside its upper and lower cones.
Question 1.14.
Is the set of all $X$ so that $X\mid_{\operatorname{T}}A$ and $X\not\in\mathcal{R}(A)$ countable?
2. Recursively enumerable sets and $\mathrm{PA}$ degrees
We will now give two (somehow unexpected) applications of Theorem 1.12 to the interaction between recursively enumerable sets and sets of $\mathrm{PA}$ degree. Recall that a set $A\subseteq\mathbb{N}$ is of PA degree if it is Turing equivalent to a set coding a complete extension of Peano arithmetic (PA). PA degrees have many interesting computability theoretic properties. For instance, a set is of PA degree if and only if it computes a path through every non-empty $\Pi^{0}_{1}$ class. However, a complete degree-theoretic characterization of the PA degrees is still not known. If $A\geq_{\operatorname{T}}\emptyset^{\prime}$, then $A$ is of PA degree. On the other hand, Gödel’s First Incompleteness Theorem implies that no r.e. set can be a complete extension of PA. Jockusch and Soare (1972) showed moreover that if a set is of incomplete r.e. degree, it cannot be of PA degree.
It seems therefore worthwile to gain a complete understanding how r.e. sets and PA degrees are related. The crucial fact that links Theorem 1.12 to PA degrees is a result by Day and Miller (ta). They showed that every set of PA degree computes a representation of a neutral measure.
Such a measure has the property that every real is random with respect to it, i.e. ${2^{\omega}}=\operatorname{MLR}_{\mu}$. The existence of neutral measures was first established by Levin (1976).
Our first result shows that below $\emptyset^{\prime}$, r.e. sets and PA degrees behave quite complementary
333After the authors announced the result presented in Corollary 2.1, proofs not involving measure theoretic arguments have been found independently by A. Kučera and J. Miller..
Corollary 2.1 (to Theorem 1.12).
If $A$ is an r.e. set and $P$ a set of $\mathrm{PA}$ degree such that $P\not\geq_{\operatorname{T}}A$ then $P\oplus A\geq_{\operatorname{T}}\emptyset^{\prime}$.
Proof.
By the result of Day and Miller (ta) mentioned above, $P$ computes a representation $R_{\mu}$ of a neutral measure $\mu$ and $A\in\operatorname{MLR}_{\mu}$. Day and Miller (ta) also showed that a real $X$ is an atom of a neutral measure if every representation of the measure computes $X$. Now because $P\not\geq_{\operatorname{T}}A$, we have that $A$ is not an atom of $\mu$. Thus all hypotheses of Theorem 1.12 are satisfied and we have $P\oplus A\geq_{\operatorname{T}}\emptyset^{\prime}$.
∎
Corollary 2.1 strengthens a result due to Kučera and Slaman (unpublished). Recall that a function $f:\mathbb{N}\to\mathbb{N}$ is diagonally non-recursive if $f(n)\neq\varphi_{n}(n)$ for all $n$, where $\varphi_{n}$ denotes, as usual, the $n$th partial recursive function.
Kučera and Slaman constructed a a $\operatorname{low}_{2}$ r.e. set so that $A\oplus f\equiv_{\operatorname{T}}\emptyset^{\prime}$ for any diagonally non-recursive function $f\leq_{\operatorname{T}}\emptyset^{\prime}$.
It is well-known that every PA degree computes a $\{0,1\}$-valued diagonally non-recursive function. Hence the set constructed by Kučera and Slaman joins any PA degree below $\emptyset^{\prime}$ to $\emptyset^{\prime}$. Corollary 2.1 yields that this is in fact true for any r.e. set.
One can now ask which kind of incomplete r.e. sets can be bounded by PA degrees below $\emptyset^{\prime}$.
This question was first raised by Kučera (2004)
For which incomplete r.e. sets $A$ does there exist set $P$ of PA degree such that $A<_{\operatorname{T}}P<_{\operatorname{T}}\emptyset^{\prime}$?
We can use Corollary 2.1 to completely answer this question.
We say a set $B$ is of PA degree relative to a set $A$, written $B\gg A$ (see (Simpson, 1977)), if $B$ computes a path through every $\Pi^{0}_{1}(A)$ class. One well-known fact we will make use of is the following. If $P$ is of PA degree, then there exists a set $Q$ of PA degree such that $P\gg Q$. One way to prove this fact is to observe that the $\Pi^{0}_{1}$ class
$$\{(A,B)\in 2^{\omega}\times 2^{\omega}\colon A\in\mathrm{DNR}_{2}\wedge B\in%
\mbox{DNR}_{2}(A)\}$$
is non-empty, where $\mathrm{DNR}_{2}$ and $\mathrm{DNR}_{2}(A)$ are the classes of $\{0,1\}$-valued diagonally non-recursive functions and $\{0,1\}$-valued diagonally non-recursive functions relative to $A$, respectively.
Theorem 2.2.
If $A$ is an r.e. set then the following are equivalent:
(1)
$A$ is low.
(2)
There exists $P$, $P\gg A$ and $P$ is low.
(3)
There exists $P$ of PA degree such that $\emptyset^{\prime}>_{\operatorname{T}}P>_{\operatorname{T}}A$.
Proof.
(1) $\Rightarrow$ (2): There is a (non-empty) $\Pi^{0}_{1}(A)$ class of sets $B\gg A$. Relativize the low basis theorem to find $P\gg A$ and $P^{\prime}\equiv_{\operatorname{T}}A^{\prime}$. As $A$ is low so is $P$.
(2) $\Rightarrow$ (3): This is clear.
(3) $\Rightarrow$ (1): Take any $Q$ of $\mathrm{PA}$ degree such that $P\gg Q$.
Now $Q\geq_{\operatorname{T}}A$ because otherwise $Q\oplus A\geq\emptyset^{\prime}$ but this is impossible because $P\geq_{\operatorname{T}}Q\oplus A$ and
$P\not\geq_{\operatorname{T}}\emptyset^{\prime}$. Hence $P\gg A$. But now we have that $\emptyset^{\prime}$ is r.e. in $A$ and also $\emptyset^{\prime}$ computes a DNR function relative to $A$. Hence by relativizing Arslanov’s completeness criterion we have that $A^{\prime}\equiv_{\operatorname{T}}\emptyset^{\prime}$.
∎
Observe that in the proof of (3) $\Rightarrow$ (1), showing $P\gg A$ only used the facts that $P\not\geq_{T}\emptyset^{\prime}$ and $P\geq_{T}A$. Hence we get a final corollary.
Corollary 2.3.
If $P$ is a set of PA degree and $A$ is an r.e. set such that $P\geq_{T}A$ and $P\not\geq_{T}\emptyset^{\prime}$, then
$P\gg A$.
3. Acknowledgments
The authors would like to thank Steve Simpson for stimulating and insightful discussions.
References
Day and Miller [ta]
A. Day and J. Miller.
Randomness for non-computable measures.
Trans. Amer. Math. Soc., to appear.
Downey and Hirschfeldt [2010]
R. G. Downey and D. R. Hirschfeldt.
Algorithmic randomness and complexity.
Springer, 2010.
Gács [2005]
P. Gács.
Uniform test of algorithmic randomness over a general space.
Theoretical Computer Science, 341:91–137, 2005.
Hoyrup and Rojas [2009]
M. Hoyrup and C. Rojas.
Computability of probability measures and Martin-Löf
randomness over metric spaces.
Information and Computation, 207(7):830 –
847, 2009.
Jockusch and Soare [1972]
C. G. Jockusch, Jr. and R. I. Soare.
Degrees of members of $\Pi^{0}_{1}$ classes.
Pacific J. Math., 40:605–616, 1972.
Kučera [2004]
A. Kučera.
Remarks on randomness and PA sets.
Presentation at the Conference on Logic, Computability and Randomness 2004. Cordoba, Argentina, 2004.
Levin [1976]
L. A. Levin.
Uniform tests for randomness.
Dokl. Akad. Nauk SSSR, 227(1):33–35,
1976.
Nies [2009]
A. Nies.
Computability and randomness, volume 51 of Oxford Logic
Guides.
Oxford University Press, Oxford, 2009.
Reimann and Slaman [2008]
J. Reimann and T. A. Slaman.
Measures and their random reals.
arXiv.org, math.LO, Feb. 2008.
Simpson [1977]
S. G. Simpson.
Degrees of unsolvability: A survey of results.
In Handbook of Mathematical Logic (J. Barwise, ed.), pp. 631–652, North-Holland Publishing Co., 1977.
Van Lambalgen [1987]
M. van Lambalgen.
Random Sequences.
PhD Thesis, University of Amsterdam, 1987. |
Fine-Grained Representation Learning and Recognition by
Exploiting Hierarchical Semantic Embedding
Tianshui Chen
Sun Yat-sen University
tianshuichen@gmail.com
,
Wenxi Wu
Sun Yat-sen University
ngmanhei@foxmail.com
,
Yuefang Gao
South China Agricultural University
gaoyuefang@scau.edu.cn
,
Le Dong
University of Electronic Science and Technology of China
ledong@uestc.edu.cn
,
Xiaonan Luo
Guilin University of Electronic Technology
luoxn@guet.edu.cn
and
Liang Lin
Sun Yat-sen University
linliang@ieee.org
(2018)
Abstract.
Object categories inherently form a hierarchy with different levels of concept abstraction, especially for fine-grained categories. For example, birds (Aves) can be categorized according to a four-level hierarchy of order, family, genus, and species. This hierarchy encodes rich correlations among various categories across different levels, which can effectively regularize the semantic space and thus make prediction less ambiguous. However, previous studies of fine-grained image recognition primarily focus on categories of one certain level and usually overlook this correlation information. In this work, we investigate simultaneously predicting categories of different levels in the hierarchy and integrating this structured correlation information into the deep neural network by developing a novel Hierarchical Semantic Embedding (HSE) framework. Specifically, the HSE framework sequentially predicts the category score vector of each level in the hierarchy, from highest to lowest. At each level, it incorporates the predicted score vector of the higher level as prior knowledge to learn finer-grained feature representation. During training, the predicted score vector of the higher level is also employed to regularize label prediction by using it as soft targets of corresponding sub-categories. To evaluate the proposed framework, we organize the 200 bird species of the Caltech-UCSD birds dataset with the four-level category hierarchy and construct a large-scale butterfly dataset that also covers four level categories. Extensive experiments on these two and the newly-released VegFru datasets demonstrate the superiority of our HSE framework over the baseline methods and existing competitors.
Semantic Embedding, Fine-Grained Image Recognition, Category Hierarchy
††journalyear: 2018††copyright: acmcopyright††conference: 2018 ACM Multimedia Conference; October 22–26, 2018; Seoul, Republic of Korea††price: 15.00††doi: 10.1145/3240508.3240523††isbn: 978-1-4503-5665-7/18/10
1. Introduction
Object categories inherently form a hierarchy with different levels of concept abstraction, in which nodes closer to the root of the hierarchy refer to more abstract concepts while nodes closer to the leaves refer to finer-grained concepts. This hierarchy organization is especially important and obvious for fine-grained categories. For example, the fine-grained categories of birds (Aves) can be organized with a four-level hierarchy of order, family, genus and species, where an order consists of several families while a family consists of several genera, and so on. This category hierarchy provides very rich semantic correlations among categories across different levels, which can effectively regularize semantic space and provide extra guidance to attend more subtle regions for better recognition. For example, to recognize the fine-grained category of a given object (e.g., the species of a bird), we might first recognize its superclass (e.g., genus). Then, we prefer to concentrate on the fine-grained categories that are subject to this superclass and fixate on object parts that are more distinguishable among these fine-grained categories.
Existing methods on fine-grained image recognition (FGIR) primarily focus on classifying categories of one particular level, e.g., categorizing 200 species of birds (Lin
et al., 2015a; Zheng
et al., 2017) or 431 models of cars (Hu
et al., 2017), and usually overlook this correlation information. In this work, we simultaneously predict categories of all levels in the hierarchy, and integrate this structured correlation information into the deep neural network to progressively regularize label prediction and guide representation learning. To this, we formulate a novel Hierarchical Semantic Embedding (HSE) framework that orderly predicts the score vector of each level, from highest to lowest. At each level, it incorporates the predicted score vector of the higher level as prior knowledge to learn finer-grained feature representation. This is implemented by a semantic guided attentional mechanism that learns to fixate on more discriminative regions for better distinguishing. During training, we also utilize the predicted score vector of the higher level as soft targets to regularize the label prediction, thus that the predicted result at this level finely accords with that predicted at the higher level.
Caltech-UCSD birds dataset (Wah et al., 2011) is the most widely used benchmark for evaluating the FGIR task. To evaluate our proposed HSE framework on this benchmark, we organize the 200 bird categories with a four-level hierarchy of 13 orders, 37 families, 122 genera, and 200 species according to the ornithological systematics (Salvador et al., 2017; Remsen Jr et al., 2016). In addition, we also create a new large-scale butterfly (namely Butterfly-200) dataset that also covers four-level categories for multi-granularity image recognition. Currently, this dataset consists of 200 prevalent species of butterflies, which are grouped into 116 genera, 23 sub-families, and 5 families according to the insect taxonomy (Verovnik and
Popović, 2013; SAMBHU and
NANKISHORE, 2018). It contains 25,279 images in total and at least 30 images per species. It’s worth noting that these category hierarchies can be obtained from the literature of taxonomy (Salvador et al., 2017; Verovnik and
Popović, 2013) or directly retrieved from Wikipedia conveniently, thus the methods of embedding this structured information can be easily adapted to various domains.
The major contributions of this work are concluded to three folds: 1) We formulate a novel Hierarchical Semantic Embedding (HSE) framework that integrates semantically structured information of category hierarchy into the deep neural network for FGIR. To our knowledge, this is the first work that explicitly incorporates this structured information to aid FGIR. 2) We introduce a four-level category hierarchy for the Caltech-UCSD birds dataset (Wah et al., 2011) and construct a new large-scale butterfly dataset that also covers four-level categories for evaluation. To our knowledge, these two datasets are the first that involves in four-level categories in FGIR and they may benefit research on multi-granularity image recognition. 3) We conduct experiments on the two and the VegFru (Hou
et al., 2017) datasets, and demonstrate the effectiveness of our proposed HSE framework over the baseline and existing state-of-the-art methods. Moreover, we also conduct ablative studies to carefully evaluate and analyze the contribution of each component of the proposed framework. The code, trained models, and dataset are available online: https://github.com/HCPLab-SYSU/HSE.
2. Related Work
2.1. Fine-grained image recognition
Recent progress on image classification mainly benefited from the advancement of deep Convolutional Neural Networks (CNNs) (LeCun
et al., 1998; Krizhevsky
et al., 2012; Simonyan and
Zisserman, 2014; He
et al., 2016; Chen
et al., 2018b, 2016) that learned powerful feature representation via stacking multiple nonlinear transformations. To adapt the deep CNNs for handling the FGIR task, a bilinear model (Lin
et al., 2015a) was proposed to compute high-order image representation that captured local pairwise interactions between features generated by two independent sub-networks, but the bilinear feature is extremely high-dimensional, making it impractical for subsequent analysis. To reduce the feature dimension while keeping comparable performance on FGIR task, Gao et al. (Gao
et al., 2016) developed a compact model that approximates bilinear feature with the polynomial kernels. Kong et al. (Kong and Fowlkes, 2016) proposed classifier co-decomposition to further compress the bilinear model.
To better capture subtle visual difference among sub-ordinate categories, a series works (Zhang
et al., 2014; Huang
et al., 2016; Zhang et al., 2016b) were also proposed to leverage extra supervision of bounding boxes and parts to locate discriminative regions. However, the heavy involvement of manual annotations prevents these methods from application to large-scale real-world problems. Recently, visual attention models (Mnih
et al., 2014; Chen
et al., 2018c; Wang
et al., 2017; Liu
et al., 2018) were intensively proposed to automatically search the informative regions and various works successfully applied this technique to FGIR (Liu
et al., 2016; Fu et al., 2017; Zheng
et al., 2017; Jaderberg et al., 2015). Liu et al. (Liu
et al., 2016) formulated a reinforcement learning framework to adaptively glimpse local regions regarding discriminative object parts and trained the framework using a greedy reward strategy with image-level labels. Zheng et al. (Zheng
et al., 2017) introduced a multi-attention convolutional neural network that learned channel grouping for parts localization, and aggregated features from the located regions as well as the global object for classification. These works learned to locate informative regions merely based on image content by the self-attention mechanism. In contrast, some works also introduced extra guidance to learn more meaningful and semantic-related regions to aid FGIR. For example, Liu et al., (Liu
et al., 2017; Chen
et al., 2018a) introduce part-based attribute to guide learning more discriminative features for fine-grained bird recognition. Similarly, He et al. (He and Peng, 2017) further utilized more detailed language descriptions to help mine discriminative parts or characteristics.
Our framework is also related to some existing works that exploit category hierarchy. For example, Srivastava et al. (Srivastava and
Salakhutdinov, 2013) exploited class hierarchy prior to transfer knowledge among similar lower-level classes for transfer learning. Jia et al. (Deng et al., 2014) proposed a probabilistic classification model based on a hierarchy and exclusion graph to capture label relations of mutual exclusion, overlap, and subsumption for object classification. Works (Wang
et al., 2016; Chen
et al., 2018c) utilized an RNN to model label co-occurrence dependencies for multi-label recognition. In contrast to these methods that merely model dependencies on label space, our HSE framework introduces the hierarchical information to progressively regularize label prediction and simultaneously guide learning finer-grained feature representation. Besides, using predicted results of the higher level as soft targets for label regularization can distill knowledge learned from the high level to lower level, which is also original compared with these methods.
2.2. Fine-grained image datasets
In the past decade, datasets of FGIR have intensively emerged across various domains ranging from man-made objects to natural plants or animals, including FGVC-Aircraft (Maji et al., 2013), Stanford Cars (Krause
et al., 2013), Caltech-UCSD birds (Wah et al., 2011), Stanford Dogs (Khosla
et al., 2011), Oxford Flowers (Nilsback and
Zisserman, 2008), to name a few. As a representative dataset that was widely used in previous FGIR works (Liu
et al., 2017; Gao
et al., 2016; He and Peng, 2017), Caltech-UCSD birds dataset contained 11,788 images and covered 200 species of birds. These datasets significantly evolved the research of FGIR, but they primarily focus on categories of one certain level, e.g., Caltech-UCSD birds with 200 species of birds and Stanford Dogs with 120 breeds of dogs. More recently, there also released some datasets that involved categories of multiple levels, like CompCars (Yang
et al., 2015), Boxcars (Sochor
et al., 2016), Cars-333 (Xie
et al., 2015) with three-level car categories of make, model, and year, and VegFru (Hou
et al., 2017) with 25 upper-level categories and 292 sub-ordinate classes of vegetables and fruits. These datasets mainly include man-made vehicles (Yang
et al., 2015; Sochor
et al., 2016; Xie
et al., 2015) and domestic food materials (Hou
et al., 2017). To better evaluate our proposed frameworks and increase the diversity of dataset with categories of multiple levels, we further organize the 200 bird species with four-level category hierarchy and construct a new butterfly dataset that also covers four-level categories. Besides the research on FGIR with categories of multiple levels, these two datasets have potential to benefit practical applications of wildlife recognition, protection, and discovery.
3. HSE Framework
In this section, we describe the proposed HSE framework in detail. Given an image, the framework first utilizes a trunk network to extract image feature maps $\mathbf{f}_{I}\in\mathcal{R}^{W^{\prime}\times H^{\prime}\times C^{\prime}}$, where $W^{\prime}$, $H^{\prime}$ and $C^{\prime}$ denote the width, height and channel number of the feature maps, respectively. Then, it orderly utilizes a small branch network to predict the score vectors of all levels, from highest to lowest. At each level, the branch network incorporates the predicted score vector of higher level as prior guidance to learn finer-grained representation via a soft attention mechanism and aggregates this representation with features learned without guidance to predict the score vector of this level. During training, we further use the predicted score vector of higher level as soft targets to regularize the label prediction, such that the predicted result at this level tends to accord with that predicted at the higher level. Since there is no guidance at the first level, we merely use the representation learned without guidance to make prediction and no label regularization is involved either. Fig. 1 gives an overall illustration of the HSE framework.
Before delving deep into the formulation, we first present some notations associated with our task that will be used throughout this article. Without loss of generality, we consider the FGIR task with a category hierarchy of $L$ levels. We utilize $l_{1}$, $l_{2}$, $\dots$, $l_{L}$ to denote each level and $\mathbf{s}_{1}$, $\mathbf{s}_{2}$, $\dots$, $\mathbf{s}_{L}$ to denote the predicted score vectors correspondingly. $n_{1}$, $n_{2}$, $\dots$, $n_{L}$ are used to represent the category number for each level, respectively.
3.1. Semantic embedding representation learning
As we orderly predict the score vector of each level, $\mathbf{s}_{i-1}$ is given when making prediction at level $l_{i}$. Generally, $\mathbf{s}_{i-1}$ encodes the category that the object of the given image belongs to with a high probability at level $l_{i-1}$, and make prediction at level $l_{i}$ may tend to distinguish the sub-ordinate categories of this category. As discussed above, some certain parts play key roles to distinguish the sub-ordinate categories of a superclass. In this work, we take full advantage of this information by incorporating $\mathbf{s}_{i-1}$ to guide learning finer-grained feature representation at level $l_{i}$. Naturally, this can be implemented by a soft mechanism that learns to fixate on the discriminative regions under the guidance of $\mathbf{s}_{i-1}$.
At level $l_{i}$, we first map the image feature maps $\mathbf{f}_{I}$ to higher-level features $\hat{\mathbf{f}}_{i}\in\mathcal{R}^{W\times H\times C}$ via
(1)
$$\hat{\mathbf{f}}_{i}=\phi_{i}(\mathbf{f}_{I}),$$
where $\phi_{i}(\cdot)$ is a transformation that is implemented by a small network. Then, at each location $(w,h)$, we introduce a shared attentional mechanism $a_{i}(\cdot)$ to compute the attention coefficient vector under the guidance of $\mathbf{s}_{i-1}$ by
(2)
$$\hat{\mathbf{e}}_{iwh}=a_{i}([\hat{\mathbf{f}}_{iwh},\varphi_{i}(\mathbf{s}_{{%
i-1}})]),$$
where $\hat{\mathbf{e}}_{iwh}=\{\hat{e}_{iwh1},\hat{e}_{iwh2},\dots,\hat{e}_{iwhC}\}$ denote the importance of each neuron of feature vector $\mathbf{f}_{iwh}$. In the equation, $\varphi_{i}(\cdot)$ is a linear transformation that transforms $s_{i-1}$ to a semantic feature vector. To make the coefficients easily comparable across different channels, we normalize the coefficients across all the locations of each channels $c$ using a softmax function
(3)
$$e_{iwhc}=\frac{\exp(\hat{e}_{iwhc})}{\sum_{w^{\prime},h^{\prime}}{\exp(\hat{e}%
_{iw^{\prime}h^{\prime}c})}}.$$
In this way, we can obtain $\mathbf{e}_{iwh}=\{e_{iwh1},e_{iwh2},\dots,e_{iwhC}\}$ denoting the normalized weight of each neuron of feature vector $\mathbf{f}_{iwh}$. Finally, we perform weighted average across all locations of each channel to produce the final finer-grained features
(4)
$$\mathbf{f}_{i}=\sum_{w,h}{\mathbf{e}_{iwh}\odot\hat{\mathbf{f}}_{iwh}},$$
where $\odot$ denotes the element-wise multiplication operation.
As the feature vector $\mathbf{f}_{i}$ pays much attention to the local discriminative regions that may tend to capture subtle difference for distinguishing sub-ordinate categories of a superclass. It may ignore the overall description of the object and some background information that may provide contextual cues. Thus, we further extract a feature vector directly from the image feature maps $\mathbf{f}_{I}$ without guidance for complementary. Similarly, we also adopt a simple transformation $\psi_{i}(\cdot)$ on $\mathbf{f}_{I}$ by
(5)
$$\hat{\mathbf{f}}^{\prime}_{i}=\psi_{i}(\mathbf{f}_{I}),$$
where $\hat{\mathbf{f}}^{\prime}_{i}\in\mathcal{R}^{W\times H\times C}$. Similar to (He
et al., 2016), we simply perform average pooling to obtain the feature vector
(6)
$$\mathbf{f}^{\prime}_{i}=\frac{1}{WH}\sum_{w,h}{\hat{\mathbf{f}}^{\prime}_{iwh}}.$$
The obtained feature vectors $\mathbf{f}^{\prime}_{i}$, $\mathbf{f}_{i}$ and the concatenation of them $[\mathbf{f}_{i},\mathbf{f}^{\prime}_{i}]$ are fed to three classifiers to predict the score vectors independently, which are then averaged to produce the final score vector $\mathbf{s}_{i}$.
Network details. Similar to recent FGIR works (Liu
et al., 2016, 2017), we implement our framework based on the widely used ResNet-50 (He
et al., 2016). Specifically, we implement the trunk network with the preceding 41 convolutional layers of the ResNet-50, and the transformations of $\phi_{i}(\cdot)$, $\psi_{i}(\cdot)$ with the following 9 layers of the ResNet-50. We make the trunk network be shared across different levels to better balance prediction accuracy and computational efficiency. $\varphi_{i}(\cdot)$ is simply implemented by a single fully connected layer that map the $c$-dim score vector to a 1,024-dimemsion features and the attention mechanism $a_{i}(\cdot)$ is implemented by two stacked fully connected layers, in which the first one is $c$+1,024 to 1,024 followed by the tanh non-linear function and the second one is 1,024 to $c$. As we use the identical architecture with ResNet-50, $c$ is 2,048 in this paper.
3.2. Semantic guided label regularization
The hierarchy encodes rich semantic correlations among categories across different levels. For example, the ground truth category at level $l_{i}$ is the child sub-category of the ground truth category at level $l_{i-1}$. This correlation information can effectively regularize semantic space and thus make prediction less ambiguous. These correlations should also be maintained among predicted categories of different levels. To this, we incorporate $\mathbf{s}_{i-1}$ as soft targets to regularize label prediction at level $l_{i}$.
Given the predicted score vector $\mathbf{s}_{i-1}=\{s_{i-1,1},s_{i-1,2},\dots,s_{i-1,n_{i-1}}\}$, a high value $s_{i-1,c}$ denotes high confidence that the object in given image belongs to category $c$ at level $l_{i-1}$, and the predicted scores for the corresponding child sub-categories at level $l_{i}$ should also be assigned with high values. To this, we first extend $\mathbf{s}_{i-1}$ to $\mathbf{s}^{\prime}_{i-1}$ according to the structured correlations thus that $\mathbf{s}^{\prime}_{i-1}$ has the same dimension as $\mathbf{s}_{i}$ and pull $\mathbf{s}_{i}$ close to $\mathbf{s}^{\prime}_{i-1}$, as shown in Fig. 2. Concretely, if category $c$ at level $l_{i-1}$ has $k$ child sub-categories at level $l_{i}$, we duplicate the score $s_{i-1,c}$ by $k$ times. Then we orderly get these duplicated scores together and re-arrange their subscripts to obtain the extended score vector $\mathbf{s}^{\prime}_{i-1}=\{s^{\prime}_{i-1,1},s^{\prime}_{i-1,2},\dots,s^{%
\prime}_{i-1,n_{i}}\}$. To make these two vectors easily comparable, we normalize them into probability distribution using the softmax function with temperature $T$
(7)
$${p}^{\prime T}_{i-1,c}=\frac{\exp(\frac{s^{\prime}_{i-1,c}}{T})}{\sum_{c^{%
\prime}}{\exp(\frac{s^{\prime}_{i-1,c^{\prime}}}{T})}},p^{T}_{i,c}=\frac{\exp(%
\frac{s_{i,c}}{T})}{\sum_{c^{\prime}}{\exp(\frac{s_{i,c^{\prime}}}{T})}},$$
where $T$ is normally set to 1, and we use a high temperature to produce softer probability distribution over classes in our experiment. In this way, we can obtain two normalized probability distributions, i.e., $\mathbf{p}^{\prime T}_{i-1}=\{{p}^{\prime T}_{i-1,1},{p}^{\prime T}_{i-1,2},%
\dots,{p}^{\prime T}_{i-1,n_{i}}\}$ and $\mathbf{p}^{T}_{i}=\{p^{T}_{i,1},p^{T}_{i,2},\dots,p^{T}_{i,n_{i}}\}$, and define a regularization term as the Kullback-Leibler divergence from $\mathbf{p}^{T}_{i}$ to $\mathbf{p}^{\prime T}_{i-1}$
(8)
$$\ell_{i}^{r}=D_{KL}(\mathbf{p}^{\prime T}_{i-1}||\mathbf{p}^{T}_{i})=-\sum_{c}%
{p}^{\prime T}_{i-1,c}\log{\frac{p^{T}_{i,c}}{{p}^{\prime T}_{i-1,c}}}.$$
As $\ell_{i}^{r}$ is defined on a single sample, we simply sum up $\ell_{i}^{r}$ over the training set to define the regularization loss term $\mathcal{L}_{i}^{r}$. As suggested in (Hinton
et al., 2015), when using soft targets that have high entropy, more information can be provided than hard target per training sample, and the gradient between training samples enjoy less variance. Thus, it can be trained more steadily and using much less training samples. In our experiments, $T$ is set as 4 to produce a sufficiently soft target.
3.3. Optimization
Besides the regularization term, we also employ the cross-entropy loss with the correct labels as the objective function. We first normalize the predicted score vector using exactly the same logits in softmax function but at a normal temperature of 1, expressed as
(9)
$$p_{i,c}=\frac{\exp({s_{i,c}})}{\sum_{c^{\prime}}{\exp({s_{i,c^{\prime}}})}}.$$
Then suppose the ground truth label at level $l_{i}$ is $c_{i}$, its loss can be defined as
(10)
$$\ell_{i}^{c}=-\sum_{c}\mathbf{1}(c=c_{i})\log{p_{i,c}},$$
where $\mathbf{1}(\cdot)$ is the indication function that is assigned as 1 if the expression is true, and assigned as 0 otherwise. We have define the same loss for the score vectors predicted by the three classifier, respectively. Thus, each sample has four losses, and we sum up the four losses over the training set to define the classification loss $\mathcal{L}_{i}^{c}$.
The proposed framework consists of a trunk network and $L$ branch network, and it is trained using a weighted combination of the classification and regularization losses. The training process is empirically divided into two stages, i.e. level-wise training followed by joint fine tuning.
Stage 1: Level-wise training.
When training the branch network of level $l_{i}$, it needs the predicted score vector of level $l_{i-1}$ to define the regularization loss. Thus, we first train the branch networks in a level-wise manner, from level $l_{1}$ to $l_{L}$. As our framework is implemented based on the ResNet-50 (He
et al., 2016), we initialize the parameters with those of the corresponding layers of ResNet-50 pre-trained on the ImageNet dataset (Deng
et al., 2009). Concretely, the parameters of the trunk network are initialized by those of the corresponding 41 convolutional layers and the parameters of the transformation $\phi_{i}(\cdot)$ and $\psi_{i}(\cdot)$ are initialized with those of the 9 corresponding layers. The parameters of other modules, including the attentional mechanism $a_{i}(\cdot)$, semantic mapper $\varphi_{i}(\cdot)$ and the three classifiers, are automatically initialized with the Xavier algorithm (Glorot and Bengio, 2010). As the trunk network is shared by all branch networks, its parameters are kept fixed at this stage. We train the branch network of level $l_{i}$ with a weighted combination of the classification and regularization losses
(11)
$$\mathcal{L}_{i}=\mathcal{L}_{i}^{c}+\gamma\mathcal{L}_{i}^{r},$$
where $\gamma$ is a balance parameter. As discussed in (Hinton
et al., 2015), the magnitudes of the gradients produced by $\mathcal{L}_{i}^{r}$ are scaled by $\frac{1}{T^{2}}$, thus it is important to multiply them by a scale of $T^{2}$. Thus, we set $\gamma$ as $T^{2}$, i.e., 16 in our experiments. Note that we merely use the classification loss $\mathcal{L}_{1}^{c}$ to train the branch network of level $l_{1}$, as there is no guidance to define the regularization loss term at this level. Similar to previous works (Liu
et al., 2016; Lin
et al., 2015a) on FGIR task, we resize the input images to $512\times 512$ and perform randomly cropping with a size of $448\times 448$ and their horizontal reflections for data augmentation. Then, we train the branch network using the stochastic gradient descent (SGD) algorithm with a batch size of 8, a momentum of 0.9 and a weight decay of 0.00005. The initial learning rate is set as 0.001, and it is divided by 10 when the error plateaus.
Stage 2: Joint fine tuning. After all branch networks are trained, we jointly fine tune the entire framework by combining the loss terms over all granularities
(12)
$$\mathcal{L}=\mathcal{L}_{1}^{c}+\sum_{i=2}^{L}{\mathcal{L}_{i}}.$$
We adopt the same strategies for data augmentation and hyper-parameter setting as Stage 1 except using a smaller initial learning rate 0.0001.
4. Datasets
We construct a new large-scale butterfly (Butterfly-200) dataset with four-level categories and organize the 200 bird species of the Caltech-UCSD Birds (CUB) dataset also with four-level categories. We evaluate our proposed framework, the baseline methods and the existing competitors on these two and the VegFru (Hou
et al., 2017) datasets. In this section, we first introduce these three datasets.
4.1. Butterfly-200 dataset construction
We select 200 common species of butterflies and build the hierarchical structure with 116 genera, 23 subfamilies, and 5 families according to the insect taxonomy. The butterfly images are collected from two scenarios, natural images with the butterfly in their natural living environment and standard images with the butterfly in the form of specimens, as both are widely used in the real-world applications. The natural images are collected by searching the keywords of butterfly species names on the internet including Google, Flicker, Bing, Baidu, etc. The standard images are collected by capturing the samples in Lab. In this way, a large number of candidate images for each species are collected. To ensure the dataset highly reliable, the candidate images are carefully identified by four experts on butterflies. Currently, we have collected 25,279 butterfly images of the 200 species, with each species containing 30 images at least, which are divided into training, validation, and test set for evaluation. For each species, we randomly select 20% for training, 20% for validation and the rest 60% for test, resulting in a training of 5,135 images, a validation set of 5,135 images, and a test set of 15,009 images, respectively. Figure 3 shows some samples from the family of ”Pieridae” and their corresponding hierarchical labels.
4.2. Caltech-UCSD birds dataset extenstion
The CUB dataset (Wah et al., 2011) is the most widely used benchmark for FGIR task. It covers 200 species of birds and contains 11,788 bird images that are divided into a training set of 5,994 images and a test set of 5,794 images. In this work, we build a bird taxonomy hierarchy according to the ornithological systematics, which groups the 200 species into 122 genera, 37 families, and 13 orders. We follow the standard train/test split as (Wah et al., 2011) for evaluation. Figure 3 also shows some samples from the order of ”Passeriformes” and their corresponding hierarchical labels.
4.3. VegFru dataset introduction
VegFru (Hou
et al., 2017) is a newly released large-scale dataset for fine-grained vegetables and fruits recognition. It covers two-level categories of 25 upper-level categories and 292 subordinate classes. The dataset contains 160,731 images in total, including a training set of 29,200 images, a validation set of 14,600 images, and a test set of 116,931 images. Similarly, we follow this standard train/val/test splits as (Hou
et al., 2017) to evaluate our HSE framework and the existing methods for fair comparison.
5. Experiment
5.1. Significance of semantic embedding
We first implement two baseline methods that use network architecture similar to ours but do not consider the structured correlations to demonstrate the effectiveness of the proposed HSE framework.
Baseline. Similar to our framework, we utilize a trunk network to extract image features and then utilize four small networks to predict the category of all levels, separately. For fair comparison, we also implement the trunk network with the preceding 41 convolutional layers of the ResNet-50 and the small network with the following 9 layers.
Baseline+backtrack. We utilize the baseline methods to predict the category of the finest level, and backtrack through the hierarchy to obtain the categories of the other levels.
We compare the HSE with these two baseline methods on the CUB and Butterfly-200 datasets in Table 1. Here, we present the accuracies of all levels for comprehensive comparisons. At level $l_{1}$, we find the HSE achieves comparable accuracies with those of the two baseline methods, as there is no semantic guidance at this level. However, at level $l_{2}$ to $l_{4}$, the HSE performs consistently better than the baseline methods on both datasets. For example on the CUB dataset, the HSE achieves accuracies of 95.7%, 92.7%, and 88.1%, outperforming the baseline methods by 0.6%, 1.2%, and 2.9%, respectively. It is noteworthy that the improvement is more obvious for predicting categories of finer levels, e.g., 1.2% accuracy improvement at level $l_{3}$ while 2.9% at level $l_{4}$ on the CUB dataset. This phenomenon suggests that incorporating semantic correction information benefits more to challenging tasks.
To delve deep into the effect of semantic embedding on network learning, we further present the curve of loss v.s. training epoch on the training set and the curve of accuracy v.s. training epoch on the test set in Fig 4. These experiments are conducted on recognizing the category of $l_{4}$ on the CUB dataset. Compared with the baseline, the HSE can be trained more stably and converged faster.
The foregoing comparisons with the baseline methods demonstrate the effectiveness of the HSE as a whole. Actually, the HSE incorporates the semantic correlation information from two aspects, i.e., semantic embedding representation learning (SERL) and semantic guided label regularization (SGLR). Here, we further conduct ablative studies to assess the actual contributions of these two components.
Contribution of semantic guided label regularization (SGLR). We first evaluate the contribution of SGLR by comparing the performance with and without regularization loss. Specifically, we simply remove the regularization loss terms of each level with others keep fixed and re-train the model in an identical way. As shown in Table 1, removing this term leads to an obvious drop in performance over all levels on both datasets.
We further analyze how SGLR improves the performance. When the category of an image is wrongly predicted, we denote it as an inter-superclass error if the wrongly predicted category and ground truth category do not belong to the same superclass, and denote it as an intra-superclass error if they belong to the same superclass. As discussed before, SGLR regularizes label prediction thus that the predicted category at level $l_{i}$ tends to be the child sub-category of the predicted category at level $l_{i-1}$. Thus, this tends to help correct the inter-superclass error. To validate this, we present the sample number of inter-superclass and the intra-superclass errors at level $l_{4}$ of our HSE with and without SGLR on both datasets. As shown in Fig. 5, introducing SGLR mainly reduces the sample number of inter-superclass error (17.5% relative reduction on the CUB dataset and 13.5% on the Butterfly-200 dataset), finely in accordance with our motivation.
Contribution of semantic embedding representation learning (SERL). Here, we evaluate the benefit of SERL. To this, we remove the feature embedding module (i.e., $\phi_{i}$ and $a_{i}$) and simply use the feature without guidance for recognition. To ensure fair comparisons, we also re-train the model with both of the classification and regularization losses. Similarly, the performance at each level suffers from an evident drop on both datasets.
As discussed before, SERL helps to attend regions that help to distinguish sub-ordinate categories of the predicted superclass of the higher level. Here, we visualize the attentional regions learned by our HSE framework in Fig. 6. At each row, we present some samples of a specific species, and the first two species belong to the same genus while the last two belong to another genus. For the samples from different species of the same genus, our framework actually attends discriminative regions to better distinguish these species. For example, to differentiate the species of “Bohemian Waxwing” and “Cedar Waxwing” that belong to the genus of “Phoebastria”, the HSE pay much attention to the throat and wing tail regions that provide most discriminative information.
5.2. Comparison with state-of-the-art methods
In this subsection, we compare the HSE framework with existing state-of-the-art methods on the CUB (Wah et al., 2011) and VegFru (Hou
et al., 2017) datasets. Here, we evaluate on recognizing the categories of the finest level (200 species on CUB and 292 subcategories on VegFru) as existing methods primarily report their results of this level.
Comparison on Caltech-UCSD birds dataset. CUB dataset is the most widely used benchmark for FGIR task, and most works have reported their results on this dataset. We compare our HSE framework with 17 state-of-the-art methods, including Deep Localization, Alignment and Classification (DeepLAC) (Lin
et al., 2015b), Semantic Part Detection and Abstraction (SPDA-CNN) (Zhang et al., 2016b), Part-RCNN (Zhang
et al., 2014), Part Alignment-based (PA-CNN) (Krause
et al., 2015), Pose Normalized CNN (PN-CNN) (Branson et al., 2014), Picking Deep Filter Responses (PDFR) (Zhang
et al., 2016a), Multiple Granularity (MG-CNN) (Wang
et al., 2015), Spatial Transformer (ST-CNN) (Jaderberg et al., 2015), Bilinear-CNN (B-CNN) (Lin
et al., 2015a), Compact Bilinear CNN (CB-CNN) (Gao
et al., 2016), Two-Level Attention Network (TLAN) (Xiao
et al., 2015), Diverse Attention Network (DAN) (Zhao
et al., 2017), Fully Convolutional Attentional Network (FCAN) (Liu
et al., 2016), Recurrent Attention (RA-CNN) (Fu et al., 2017), Combine Vision and Language (CVL) (He and Peng, 2017), Attribute-Guided Attention Localization (AGAL) (Liu
et al., 2017), Multi-Attentional CNN (MA-CNN) (Zheng
et al., 2017). Among these methods, some use merely image-level labels (i.e., image-level setting), and some also use bounding box/parts annotations (i.e., box-level setting); thus we also present these information for fair and direct comparisons.
Under the box-level setting, the previous well-performing methods include PN-CNN and B-CNN that achieve accuracies of 85.4% and 85.1%. However, PN-CNN requires strong supervision of both human-defined bounding box and ground truth parts while B-CNN relies on a very high-dimension feature representation (250k dimensions). Under the image-level setting, most works resort to attentional model that automatically search the discriminative regions and aggregate deep features of these regions for classification. For example, MA-CNN learns to attend multiple discriminative regions, and adopt a CNN to extract the global feature from the whole and multiple part-CNNs to extract the local feature from each attentional regions. It achieves an accuracy of 86.5%, which is the best among existing methods. Different from these methods, our HSE framework requires no bounding box and part annotations and does not use multiple CNN to extract local and global features. Instead, it embeds structure information of category hierarchy to learn fine-grained feature representation and regularize label prediction, leading to obvious performance improvement, i.e., 88.1% in accuracy.
Note that our HSE introduces extra guidance of the category hierarchy. However, this hierarchy can be easily obtained from the literature of taxonomy or retrieved from the Wikipedia. Besides, we also compare with existing methods that also rely on extra supervisions, like AGAL requiring attribute annotations and CVL depending on sentence description. Our HSE achieves an accuracy of 88.1%, much better than theirs, i.e., 85.5% and 85.6%, respectively.
Comparison on VegFru dataset. VegFru is a newly released large-scale dataset for fine-grained vegetables and fruits recognition, and some works also report their results on this dataset. Here, we also present comparisons with the baseline and existing methods on this dataset in Table 3. As shown, the HSE also significantly outperforms all these methods.
6. Conclusion
Fine-grained categories naturally form a hierarchy with different levels of concept abstraction, and this hierarchy encodes rich correlations among categories across different levels. In this work, we investigate simultaneously predicting categories of all levels in the hierarchy and integrating this structured correlation information into the deep neural network by developing a novel Hierarchical Semantic Embedding (HSE) framework. Specifically, the HSE orderly predicts the score vector for each level, and at each level, it incorporates the predicted score vector of the higher level to guide learning finer-grained feature representation and simultaneously regularize label prediction during training. To evaluate the HSE framework, we extend the Caltech-UCSD birds with four-level categories and construct a butterfly dataset also with four-level categories. Extensive experiments and thorough analysis on these two and the VegFru datasets demonstrate the superiority of the proposed HSE framework over the baseline methods and existing competitors.
Acknowledgement
We would like to thank Prof. Min Wang, Associate Prof. XiaoLing Fan, Dr. Haiming Xu, and Dr. Hailing Zhuang with Department of Entomology, College of Agriculture, South China Agricultural University for their assistance in butterfly image annotations. This work was supported in part by the Chinese National Science Foundation (NSFC No. 61702196), and Science and Technology Planning Project of Guangdong Province, China (No. 2017A020208041). This work is jointly supported by State Key Development Program under Grant 2018YFC0830103.
References
(1)
Branson et al. (2014)
Steve Branson, Grant
Van Horn, Serge Belongie, and Pietro
Perona. 2014.
Bird species categorization using pose normalized
deep convolutional nets.
British Machine Vision Conference
(2014).
Chen
et al. (2018a)
Tianshui Chen, Liang Lin,
Riquan Chen, Yang Wu, and
Xiaonan Luo. 2018a.
Knowledge-Embedded Representation Learning for
Fine-Grained Image Recognition. In Proc. of
International Joint Conference on Artificial Intelligence.
627–634.
Chen
et al. (2016)
Tianshui Chen, Liang Lin,
Lingbo Liu, Xiaonan Luo, and
Xuelong Li. 2016.
DISC: Deep Image Saliency Computing via Progressive
Representation Learning.
IEEE Trans. Neural Netw. Learning Syst.
27, 6 (2016),
1135–1149.
Chen
et al. (2018b)
Tianshui Chen, Liang Lin,
Wangmeng Zuo, Xiaonan Luo, and
Lei Zhang. 2018b.
Learning a Wavelet-like Auto-Encoder to Accelerate
Deep Neural Networks. In Proc. of AAAI Conference
on Artificial Intelligence. 6722–6729.
Chen
et al. (2018c)
Tianshui Chen, Zhouxia
Wang, Guanbin Li, and Liang Lin.
2018c.
Recurrent Attentional Reinforcement Learning for
Multi-label Image Recognition. In Proc. of AAAI
Conference on Artificial Intelligence. 6730–6737.
Deng et al. (2014)
Jia Deng, Nan Ding,
Yangqing Jia, Andrea Frome,
Kevin Murphy, Samy Bengio,
Yuan Li, Hartmut Neven, and
Hartwig Adam. 2014.
Large-scale object classification using label
relation graphs. In European conference on
computer vision. Springer, 48–64.
Deng
et al. (2009)
Jia Deng, Wei Dong,
Richard Socher, Li-Jia Li,
Kai Li, and Li Fei-Fei.
2009.
Imagenet: A large-scale hierarchical image
database. In Computer Vision and Pattern
Recognition, 2009. CVPR 2009. IEEE Conference on. IEEE,
248–255.
Fu et al. (2017)
Jianlong Fu, Heliang
Zheng, and Tao Mei. 2017.
Look closer to see better: recurrent attention
convolutional neural network for fine-grained image recognition. In
IEEE Conference on Computer Vision and Pattern
Recognition (CVPR).
Gao
et al. (2016)
Yang Gao, Oscar Beijbom,
Ning Zhang, and Trevor Darrell.
2016.
Compact bilinear pooling. In
Proceedings of the IEEE Conference on Computer
Vision and Pattern Recognition. 317–326.
Glorot and Bengio (2010)
Xavier Glorot and Yoshua
Bengio. 2010.
Understanding the difficulty of training deep
feedforward neural networks. In Proceedings of the
thirteenth international conference on artificial intelligence and
statistics. 249–256.
He
et al. (2016)
Kaiming He, Xiangyu
Zhang, Shaoqing Ren, and Jian Sun.
2016.
Deep residual learning for image recognition. In
Proceedings of the IEEE conference on computer
vision and pattern recognition. 770–778.
He and Peng (2017)
Xiangteng He and Yuxin
Peng. 2017.
Fine-graind Image Classification via Combining
Vision and Language.
Proceedings of the IEEE Conference on
Computer Vision and Pattern Recognitions (2017).
Hinton
et al. (2015)
Geoffrey Hinton, Oriol
Vinyals, and Jeff Dean.
2015.
Distilling the knowledge in a neural network.
arXiv preprint arXiv:1503.02531
(2015).
Hou
et al. (2017)
Saihui Hou, Yushan Feng,
and Zilei Wang. 2017.
VegFru: A Domain-Specific Dataset for Fine-grained
Visual Categorization. In 2017 IEEE International
Conference on Computer Vision (ICCV). IEEE, 541–549.
Hu
et al. (2017)
Qichang Hu, Huibing Wang,
Teng Li, and Chunhua Shen.
2017.
Deep CNNs With Spatially Weighted Pooling for
Fine-Grained Car Recognition.
IEEE Transactions on Intelligent
Transportation Systems 18, 11
(2017), 3147–3156.
Huang
et al. (2016)
Shaoli Huang, Zhe Xu,
Dacheng Tao, and Ya Zhang.
2016.
Part-stacked CNN for fine-grained visual
categorization. In Proceedings of the IEEE
Conference on Computer Vision and Pattern Recognition.
1173–1182.
Jaderberg et al. (2015)
Max Jaderberg, Karen
Simonyan, Andrew Zisserman, et al.
2015.
Spatial transformer networks. In
Advances in Neural Information Processing
Systems. 2017–2025.
Khosla
et al. (2011)
Aditya Khosla, Nityananda
Jayadevaprakash, Bangpeng Yao, and
Fei-Fei Li. 2011.
Novel dataset for fine-grained image
categorization: Stanford dogs. In Proc. CVPR
Workshop on Fine-Grained Visual Categorization (FGVC),
Vol. 2. 1.
Kong and Fowlkes (2016)
Shu Kong and Charless
Fowlkes. 2016.
Low-rank Bilinear Pooling for Fine-Grained
Classification.
arXiv preprint arXiv:1611.05109
(2016).
Krause
et al. (2015)
Jonathan Krause, Hailin
Jin, Jianchao Yang, and Li Fei-Fei.
2015.
Fine-grained recognition without part annotations.
In Proceedings of the IEEE Conference on Computer
Vision and Pattern Recognition. 5546–5555.
Krause
et al. (2013)
Jonathan Krause, Michael
Stark, Jia Deng, and Li Fei-Fei.
2013.
3d object representations for fine-grained
categorization. In Computer Vision Workshops
(ICCVW), 2013 IEEE International Conference on. IEEE,
554–561.
Krizhevsky
et al. (2012)
Alex Krizhevsky, Ilya
Sutskever, and Geoffrey E Hinton.
2012.
Imagenet classification with deep convolutional
neural networks. In Advances in neural information
processing systems. 1097–1105.
LeCun
et al. (1998)
Yann LeCun, Léon
Bottou, Yoshua Bengio, and Patrick
Haffner. 1998.
Gradient-based learning applied to document
recognition.
Proc. IEEE 86,
11 (1998), 2278–2324.
Lin
et al. (2015b)
Di Lin, Xiaoyong Shen,
Cewu Lu, and Jiaya Jia.
2015b.
Deep lac: Deep localization, alignment and
classification for fine-grained recognition. In
Proceedings of the IEEE Conference on Computer
Vision and Pattern Recognition. 1666–1674.
Lin
et al. (2015a)
Tsung-Yu Lin, Aruni
RoyChowdhury, and Subhransu Maji.
2015a.
Bilinear cnn models for fine-grained visual
recognition. In Proceedings of the IEEE
International Conference on Computer Vision. 1449–1457.
Liu
et al. (2018)
Lingbo Liu, Hongjun Wang,
Guanbin Li, Wanli Ouyang, and
Liang Lin. 2018.
Crowd Counting using Deep Recurrent Spatial-Aware
Network. In Proc. of International Joint
Conference on Artificial Intelligence.
Liu
et al. (2017)
Xiao Liu, Jiang Wang,
Shilei Wen, Errui Ding, and
Yuanqing Lin. 2017.
Localizing by Describing: Attribute-Guided
Attention Localization for Fine-Grained Recognition.. In
AAAI. 4190–4196.
Liu
et al. (2016)
Xiao Liu, Tian Xia,
Jiang Wang, and Yuanqing Lin.
2016.
Fully convolutional attention localization
networks: Efficient attention localization for fine-grained recognition.
arXiv preprint arXiv:1603.06765
(2016).
Maji et al. (2013)
Subhransu Maji, Esa
Rahtu, Juho Kannala, Matthew Blaschko,
and Andrea Vedaldi. 2013.
Fine-grained visual classification of aircraft.
arXiv preprint arXiv:1306.5151
(2013).
Mnih
et al. (2014)
Volodymyr Mnih, Nicolas
Heess, Alex Graves, et al.
2014.
Recurrent models of visual attention. In
Advances in neural information processing
systems. 2204–2212.
Nilsback and
Zisserman (2008)
Maria-Elena Nilsback and
Andrew Zisserman. 2008.
Automated flower classification over a large number
of classes. In Computer Vision, Graphics & Image
Processing, 2008. ICVGIP’08. Sixth Indian Conference on. IEEE,
722–729.
Remsen Jr et al. (2016)
JV Remsen Jr, Alexis FLA
Powell, Richard Schodde, F Keith Barker,
and Scott M Lanyon. 2016.
A revised classification of the Icteridae (Aves)
based on DNA sequence data.
Zootaxa 4093,
2 (2016), 285–292.
Salvador et al. (2017)
Rodrigo B Salvador, Henk
Van der Jeugd, and Barbara M Tomotani.
2017.
Taxonomy of the European Pied Flycatcher Ficedula
hypoleuca (Aves: Muscicapidae).
Zootaxa 4291,
1 (2017), 171–182.
SAMBHU and
NANKISHORE (2018)
HEMCHANDRANAUTH SAMBHU and
ALLIEA NANKISHORE. 2018.
Butterflies (Lepidoptera) of Guyana: A compilation
of records.
Zootaxa 4371,
1 (2018), 1–187.
Simonyan and
Zisserman (2014)
Karen Simonyan and
Andrew Zisserman. 2014.
Very deep convolutional networks for large-scale
image recognition.
arXiv preprint arXiv:1409.1556
(2014).
Sochor
et al. (2016)
Jakub Sochor, Adam
Herout, and Jiri Havel.
2016.
Boxcars: 3d boxes as cnn input for improved
fine-grained vehicle recognition. In Proceedings
of the IEEE Conference on Computer Vision and Pattern Recognition.
3006–3015.
Srivastava and
Salakhutdinov (2013)
Nitish Srivastava and
Ruslan R Salakhutdinov. 2013.
Discriminative transfer learning with tree-based
priors. In Advances in Neural Information
Processing Systems. 2094–2102.
Verovnik and
Popović (2013)
Rudi Verovnik and
Miloš Popović. 2013.
Annotated checklist of Albanian butterflies
(Lepidoptera, Papilionoidea and Hesperioidea).
ZooKeys 323
(2013), 75.
Wah et al. (2011)
Catherine Wah, Steve
Branson, Peter Welinder, Pietro Perona,
and Serge Belongie. 2011.
The caltech-ucsd birds-200-2011 dataset.
(2011).
Wang
et al. (2015)
Dequan Wang, Zhiqiang
Shen, Jie Shao, Wei Zhang,
Xiangyang Xue, and Zheng Zhang.
2015.
Multiple granularity descriptors for fine-grained
categorization. In Proceedings of the IEEE
International Conference on Computer Vision. 2399–2406.
Wang
et al. (2016)
Jiang Wang, Yi Yang,
Junhua Mao, Zhiheng Huang,
Chang Huang, and Wei Xu.
2016.
Cnn-rnn: A unified framework for multi-label image
classification. In Computer Vision and Pattern
Recognition (CVPR), 2016 IEEE Conference on. IEEE,
2285–2294.
Wang
et al. (2017)
Zhouxia Wang, Tianshui
Chen, Guanbin Li, Ruijia Xu, and
Liang Lin. 2017.
Multi-label Image Recognition by Recurrently
Discovering Attentional Regions. In IEEE
International Conference on Computer Vision. IEEE,
464–472.
Xiao
et al. (2015)
Tianjun Xiao, Yichong Xu,
Kuiyuan Yang, Jiaxing Zhang,
Yuxin Peng, and Zheng Zhang.
2015.
The application of two-level attention models in
deep convolutional neural network for fine-grained image classification. In
Proceedings of the IEEE Conference on Computer
Vision and Pattern Recognition. 842–850.
Xie
et al. (2015)
Saining Xie, Tianbao
Yang, Xiaoyu Wang, and Yuanqing Lin.
2015.
Hyper-class augmented and regularized deep learning
for fine-grained image classification.
Proceedings of the IEEE Conference on
Computer Vision and Pattern Recognition (2015).
Yang
et al. (2015)
Linjie Yang, Ping Luo,
Chen Change Loy, and Xiaoou Tang.
2015.
A large-scale car dataset for fine-grained
categorization and verification. In Proceedings of
the IEEE Conference on Computer Vision and Pattern Recognition.
3973–3981.
Zhang et al. (2016b)
Han Zhang, Tao Xu,
Mohamed Elhoseiny, Xiaolei Huang,
Shaoting Zhang, Ahmed Elgammal, and
Dimitris Metaxas. 2016b.
Spda-cnn: Unifying semantic part detection and
abstraction for fine-grained recognition. In
Proceedings of the IEEE Conference on Computer
Vision and Pattern Recognition. 1143–1152.
Zhang
et al. (2014)
Ning Zhang, Jeff Donahue,
Ross Girshick, and Trevor Darrell.
2014.
Part-based R-CNNs for fine-grained category
detection. In European conference on computer
vision. Springer, 834–849.
Zhang
et al. (2016a)
Xiaopeng Zhang, Hongkai
Xiong, Wengang Zhou, Weiyao Lin, and
Qi Tian. 2016a.
Picking deep filter responses for fine-grained
image recognition. In Proceedings of the IEEE
Conference on Computer Vision and Pattern Recognition.
1134–1142.
Zhao
et al. (2017)
Bo Zhao, Xiao Wu,
Jiashi Feng, Qiang Peng, and
Shuicheng Yan. 2017.
Diversified Visual Attention Networks for
Fine-Grained Object Classification.
IEEE Transactions on Multimedia
19, 6 (2017),
1245–1256.
Zheng
et al. (2017)
Heliang Zheng, Jianlong
Fu, Tao Mei, and Jiebo Luo.
2017.
Learning multi-attention convolutional neural
network for fine-grained image recognition. In
Proceedings of the IEEE Conference on Computer
Vision and Pattern Recognition. 5209–5217. |
Correlations and the ridge in the Color Glass Condensate beyond the glasma graph approximation
Tolga Altinoluk,
b
Néstor Armesto
b
and Douglas E. Wertepny
tolga.altinoluk@ncbj.gov.pl
nestor.armesto@usc.es
douglas.wertepny@usc.es
National Centre for Nuclear Research, 00-681 Warsaw, PolandDepartamento de Física de Partículas and IGFAE,
Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Galicia-Spain
Abstract
We consider two-gluon production in dilute-dense collisions within the Color Glass Condensate framework, applicable to both proton-nucleus and heavy-light ion collisions. We go beyond the glasma graph approximation which is valid in the dilute-dilute limit and show the correspondence between the glasma graphs and the $k_{T}$-factorized approach that we use in our calculation. We then identify the classical uncorrelated, and the Hanbury-Brown-Twiss (HBT) and Bose enhancement correlated contributions, with the Bose enhancement contribution being suppressed by the number of degrees of freedom with respect to the uncorrelated piece. We show that both the HBT and the Bose enhancement pieces survive the inclusion of higher order contributions in density and that they stem from the quadrupole piece of the two-gluon inclusive cross section. Finally, we illustrate the results using a toy model that allows a simple numerical implementation.
1 Introduction
The ridge phenomenon - the existence of long range pseudorapidity two-particle correlations peaking for identical or opposite azimuthal directions - is one of the main findings of the Relativistic Heavy Ion Collider (RHIC) at BNL and the Large Hadron Collider (LHC) at CERN concerning Quantum Chromodynamics (QCD). First discovered in AuAu collisions at RHIC Alver:2009id ; Abelev:2009af , it was later found in pp collisions at the LHC Khachatryan:2010gv ; Khachatryan:2015lva ; Aad:2015gqa and subsequently observed in all collisional systems, including the small ones (pp, pPb, dAu, ${}^{3}$HeAu) for high multiplicity events CMS:2012qk ; Abelev:2012ola ; Aad:2012gla ; Aaij:2015qcq ; Khachatryan:2016ibd ; Adare:2014keg ; Adamczyk:2015xjc ; Adare:2015ctn . More recently, sizeable azimuthal anisotropies have also been observed in small systems for events with multiplicities much closer to average Khachatryan:2016txc ; Aaboud:2016yar ; Aaboud:2017acw ; Aaboud:2017blb .
The standard explanation of the ridge in heavy-ion collisions comes through the coupling of an initial long range pseudorapidity correlation to an expanding medium. This medium is accurately described by viscous relativistic hydrodynamics Bozek:2012gr ; Shuryak:2013ke ; Bzdak:2013zma ; Werner:2010ss ; Gavin:2008ev . Nevertheless, the validity of the assumptions underlying the hydrodynamical explanation becomes tenuous for small systems where isotropization and smallness of the mean free path are difficult to justify. One alternative currently under exploration is that hydrodynamics seems to be applicable for out-of-equilibrium systems. This has been argued in both weak and strong coupling approaches Chesler:2009cy ; Heller:2011ju ; Kurkela:2015qoa . The ridge phenomenon is thus a key observable for our understanding of the emergence of a macroscopic description in hadronic and nuclear collisions from the underlying QCD microscopic dynamics Romatschke:2016hle .
On the other hand, explanations alternative to hydrodynamics exist that may shed light on the emergence problem and, in any case, should be used to provide the initial conditions for and characterise the dynamics prior to hydrodynamic evolution. In this work we focus on those given by the weak coupling but non-perturbative realisation of QCD at high energies provided by the Color Glass Condensate (CGC) effective field theory Gelis:2010nm , but other approaches exist based on different non-perturbative ideas, see e.g. Hwa:2008um ; Bjorken:2013boa ; Shuryak:2013sra ; Andres:2014bia .
In the CGC framework, the ridge was phenomenologically addressed through an approximation valid when both projectile and target are dilute, called the “glasma graphs" Armesto:2006bv ; Dumitru:2008wn ; Dumitru:2010iy , see Kovchegov:2012nd ; Kovchegov:2013ewa for an analogous calculation in slightly different language. Such an approximation for gluon-gluon correlations, assuming that it can be translated to the final particles, was used to successfully describe the measurements in pp Dusling:2012iga ; Dusling:2012cg , and was later extended to pA collisions in various phenomenological ways Dusling:2012wy ; Dusling:2013qoz ; Dusling:2017dqg ; Dusling:2017aot . The dilute-dilute results have been argued recently to stem from coherence in parton radiation Blok:2017pui without any reliance on the CGC formalism. The extension to high densities has been explored recently to quantify the validity of the glasma graph approach and to analyze the odd Fourier harmonics in the azimuthal particle correlations Skokov:2014tka ; Schenke:2015aqa ; Lappi:2015vta ; McLerran:2016snu ; Kovner:2016jfp ; Kovchegov:2018jun . It has also been extended to partons other than gluons Altinoluk:2016vax ; Kovner:2017gab ; Martinez:2018ygo and to the forward region linking with the multiple parton scattering language Kovner:2017vro ; Kovner:2017ssr ; Kovner:2018vec . On the other hand, other ideas within the CGC framework exist, considering the existence of domains of oriented chromoelectric fields in the hadron or nucleus Kovner:2010xk ; Kovner:2011pe ; Kovner:2012jm ; Dumitru:2014vka , or justifying the azimuthal correlations through the density profile of the hadron Levin:2011fb .
In Altinoluk:2015uaa , the origin of the ridge azimuthal correlations in the glasma graph approach was identified to come from the Bose enhancement of gluons in the wave function of the incoming hadrons. Similar calculations also showed the existence of Hanbury-Brown-Twiss (HBT) correlations of the produced particles Kovchegov:2012nd ; Kovchegov:2013ewa ; Altinoluk:2015eka . The aim of the present work is to establish whether the Bose enhancement contribution found in the glasma graph approach survives the density corrections that appear in the dilute-dense situation and, if so, to identify which contributions from the color ensembles in the target are dominant. We anticipate that our answer is positive and that, somewhat unexpectedly (although previously claimed in e.g. Kovner:2017ssr ), we find that the contribution to the Bose enhancement terms (also to the HBT ones) comes from the quadrupole distribution of Wilson lines in the target. The contribution from the target average of two dipoles turns out to be suppressed by a relative factor of $\frac{1}{N_{c}^{2}-1}$.
The plan of the paper is as follows: In Section 2 we present the setup and describe the previous results on Bose-enhanced contributions to the ridge. Section 3 contains our main results. In Section 4 we illustrate them using a toy model. Finally, in Section 5 we summarise and present our conclusions.
2 Setup and previous results
2.1 Bose Enhancement and the ridge
As mentioned in the Introduction, in the Color Glass Condensate particle correlations have been studied for phenomenological purposes using the glasma graph approach, which has produced successful comparisons with experimental data. In this approach, it was shown in Kovchegov:2012nd ; Kovchegov:2013ewa ; Altinoluk:2015uaa ; Altinoluk:2015eka that the ridge receives two contributions:
•
One contribution comes from the Bose enhancement of the gluons in the projectile wave function, that results in a form of the normalised two-particle correlation of gluons in the projectile wave function
$$D(q_{1},q_{2})\propto 1+\frac{1}{S_{\perp}(N_{c}^{2}-1)}\left[\delta^{(2)}(q_{%
1}-q_{2})+\delta^{(2)}(q_{1}+q_{2})\right],$$
(1)
with $q_{1},q_{2}$ the transverse momentum of the gluons, see Fig. 1. In this equation, 1 stands for the classical, uncorrelated term, and the $\delta$-functions come from the Bose statistics of the partons in the adjoint (real) representation, the gluons. Bose enhancement can be clearly seen by the fact that these latter are suppressed by the number of degrees of freedom: transverse area $S_{\perp}$ times number of gluons. The wave function of the projectile in the CGC is boost invariant up to rapidities $y\sim 1/\alpha_{s}$, large in the weak coupling regime.
This is the correlation in the projectile wave function; in order to get the corresponding cross section, one convolutes with the probability density for scattering of the two gluons in the target, which amounts to multipole distributions that would smear the correlations by momenta of order the saturation scale of the target.
•
Another contribution comes from the HBT correlations in the final state (with gluon momenta $k_{1},k_{2}$), so after rescattering with the target. These correlations are sensitive to the size of the projectile, therefore they appear at smaller distance (in $|k_{1}\pm k_{2}|$) from the peak than the previous ones, and are enhanced with respect to them by the number of particle sources ($S_{\perp}$ times the saturation scale of the target squared).
In Fig. 1 we show the relevant graphs for the computation of correlations in the CGC. Two gluons are emitted from two different color sources (grey blobs) from the projectile, with momentum $q_{1}$ and $q_{2}$. The curved blue lines indicate the color contractions of the projectile sources. The diagram on the right, type 3, does not produce any correlations (apart from those that may stem from geometry, see below). Note that we have not yet performed the rescattering with the target. In the glasma graph approach, such rescattering is done at the lowest order in the target density (two gluon exchange leading to dilute-dilute scattering), with the final state gluons coming from Lipatov vertices (black blobs). Both Bose enhancement and HBT correlations result from the two leftmost diagrams, type 1 and 2.
The aim of this paper is the extension of these calculations to the dilute-dense situation, i.e., beyond the lowest order in target density. For that we will use the language in Kovchegov:2012nd ; Kovchegov:2013ewa . It was proved there that, after several manipulations, a $k_{T}$-factorized form can be written for two-particle correlations. In this formalism, analogous results were obtained with geometrical (classical), HBT and Bose contributions. We present the $k_{T}$-factorized formalism in the next section and its correspondence to the glasma graph approach that we have just discussed.
2.2 Two-particle correlations in the $k_{T}$-factorized form
The two-gluon correlation function in the dilute-dense (or heavy-light) limit, where the projectile is defined by two color sources as in the previous Subsection and all rescatterings in the target are taken into account, was originally derived in Kovchegov:2012nd ; Kovchegov:2013ewa in a $k_{T}$-factorized form.
Three distinct processes contributed, see Fig. 2:
•
The ones exemplified on the right, type 3, where each projectile source emitted the same gluon in the amplitude and in the complex conjugate amplitude. These diagrams, referred to as squared diagrams, correspond exactly to the type 3 ones in Fig. 1 and, at large $N_{c}$, do not provide any correlations apart from trivial, geometrical ones.
•
The type 2 ones in which each projectile source is attached to one gluon in the amplitude and to the other gluon in the complex conjugate amplitude. These are part of the connected diagrams and correspond to type 2 diagrams in Fig. 1.
•
The type 1 ones where one projectile source emitted two gluons in the amplitude that are attached to the other projectile source in the complex conjugate amplitude. They provide the remaining part of the connected diagrams and correspond to type 1 diagrams in Fig. 1.
Diagrams of type 3 contain the double trace of Wilson lines while type 1 and 2 contain the quadrupole contribution.
One should note that, since we have two indistinguishable gluons, this process is invariant under interchange of $\vec{k}_{1}$ and $\vec{k}_{2}$.
The separated diagrams’ contribution is explicitly invariant when $\vec{k}_{2}\rightarrow-\vec{k}_{2}$ and diagrams of type 1 become diagrams of type 2 under this interchange. Thus, the total expression is invariant under the interchange of $\vec{k}_{1}$ and $\vec{k}_{2}$ and $\vec{k}_{2}\rightarrow-\vec{k}_{2}$. This means the two-gluon production cross section is explicitly even and will generate a symmetric, in azimuthal angle, correlation function, a fact that in the previous Subsection resulted from the reality of the gluon color representation. This can be seen explicitly in the full results presented in the original work and in the equations presented later in this one.
While the form presented in the original work Kovchegov:2012nd contains all the information required to see if the Bose correlation terms survive the inclusion of higher order rescatterings in the target, it is rather difficult to extract the terms that correspond to the various correlations.
In a follow up paper Kovchegov:2013ewa , a $k_{T}$-factorized form was derived for the two-gluon production cross section.
Here we will explicitly see the way in which the gluon distribution functions associated with the projectile and target nuclei conspire to produce the correlations.
This form ends up being much easier to manipulate in order to isolate the Bose-enhanced contribution.
Before delving into the equation itself, it is first necessary to define the distribution functions that compose the correlation function.
The $k_{T}$-factorized form of the single inclusive gluon production cross section involves the usual unintegrated gluon distributions,
$$\displaystyle\phi_{A_{1}}({\vec{q}};y)=\frac{C_{F}}{\alpha_{s}(2\pi)^{3}}\int d%
^{2}b\,d^{2}r\;e^{-i{\vec{q}}\cdot{\vec{r}}}\;\nabla_{{\vec{r}}}^{2}\;n_{G}({%
\vec{b}}+{\vec{r}},{\vec{b}};y)$$
(2)
for the projectile nucleus and
$$\displaystyle\phi_{A_{2}}({\vec{q}};y)=\frac{C_{F}}{\alpha_{s}(2\pi)^{3}}\int d%
^{2}b\,d^{2}r\;e^{-i{\vec{q}}\cdot{\vec{r}}}\;\nabla_{{\vec{r}}}^{2}\;N({\vec{%
b}}+{\vec{r}},{\vec{b}};y)$$
(3)
for the target.
Note that our wave function is rapidity independent and that in the semiclassical calculation that we are presenting, no quantum evolution is allowed neither between the two gluons not between them and any of the projectile or target hadrons or nuclei. Therefore $y$ is simply a label for the rapidity distance between projectile and target, i.e., for the collision energy, that would correspond to the rapidity at which the target multipole distributions are evaluated and, thus, it is not well determined without computing the quantum evolution.
Here $n_{G}({\vec{b}}+{\vec{r}},{\vec{b}};y)$ is the distribution associated with the proton (or light nucleus or, in general, dilute projectile) which in our semiclassical calculation is 111In the toy model that we will present later, the saturation scale does not depend on impact parameter $\vec{b}$.
$$n_{G}({\vec{b}}+{\vec{r}},{\vec{b}};y=0)=\frac{1}{4}Q_{s,1}^{2}(\vec{b})\,r^{2%
}\ln{\left(\frac{1}{r\,\Lambda}\right)},$$
(4)
where $\Lambda$ is an infrared (IR) cutoff and $y$ is the rapidity.
Note that usually this is notated as $n_{G}(\vec{x},\vec{y};y)$ where the $\vec{r}=\vec{x}-\vec{y}$ and the saturation scale, $Q_{s,1}^{2}(\vec{b})$, is evaluated at $\frac{1}{2}\left(\vec{x}+\vec{y}\right)$.
In our case we assume that the saturation scale is slowly varying such that $Q_{s,1}^{2}(\vec{b})\approx Q_{s,1}^{2}(\vec{b}+\vec{r})$ so the difference between these two notations is negligible. $N({\vec{b}}+{\vec{r}},{\vec{b}};y)$ is the gluon dipole scattering amplitude on the target nucleus, where we have also assumed that $Q_{s,2}^{2}(\vec{b})\approx Q_{s,2}^{2}(\vec{b}+\vec{r})$.
In order to write a $k_{T}$-factorized form for the two-gluon production cross section one must introduce new distribution functions that come with an extra transverse coordinate dependence and various Wilson line objects.
We have the unintegrated gluon distribution functions with extra coordinate dependence associated with the projectile
$$\displaystyle\left\langle\frac{d\phi_{A_{1}}({\vec{q}};y)}{d^{2}b}\right%
\rangle_{A_{1}}=\frac{C_{F}}{\alpha_{s}(2\pi)^{3}}\int d^{2}r\;e^{-i{\vec{q}}%
\cdot{\vec{r}}}\;\nabla_{{\vec{r}}}^{2}\;n_{G}({\vec{b}}+{\vec{r}},{\vec{b}};y)$$
(5)
and target
$$\displaystyle\left\langle\frac{d\phi_{A_{2}}({\vec{q}};y)}{d^{2}b}\right%
\rangle_{A_{2}}=\frac{C_{F}}{\alpha_{s}(2\pi)^{3}}\int d^{2}r\;e^{-i{\vec{q}}%
\cdot{\vec{r}}}\;\nabla_{{\vec{r}}}^{2}\;N({\vec{b}}+{\vec{r}},{\vec{b}};y).$$
(6)
Here the coordinate dependence of the distribution functions only affects where the saturation scale is measured, so for a translationally invariant nucleus the saturation scale becomes a constant and this coordinate dependence is absent.
Then, one finds that the distribution function above can be written in terms of unintegrated gluon distribution functions divided by the transverse area of the corresponding hadron or ion, $S_{\perp,1}$ for the projectile and $S_{\perp,2}$ for the target,
$$\displaystyle\left\langle\frac{d\phi_{A_{i}}({\vec{q}};y)}{d^{2}b}\right%
\rangle_{A_{i}}=\frac{1}{S_{\perp,i}}\phi_{A_{i}}({\vec{q}};y).$$
(7)
In the $k_{T}$-factorized form, the interaction with the target can be described through two different distribution functions. We have the double trace distribution
$$\displaystyle\left\langle\frac{d\phi_{A_{2}}^{D}({\vec{q}}_{1},{\vec{q}}_{2};y%
)}{d^{2}b_{1}\;d^{2}b_{2}}\right\rangle_{A_{2}}$$
$$\displaystyle=\left(\frac{C_{F}}{\alpha_{s}(2\pi)^{3}}\right)^{2}\int d^{2}r_{%
1}\;d^{2}r_{2}\,e^{-i{\vec{q}}_{1}\cdot{\vec{r}}_{1}-i{\vec{q}}_{2}\cdot{\vec{%
r}}_{2}}$$
(8)
$$\displaystyle\times\nabla_{{\vec{r}}_{1}}^{2}\;\nabla_{{\vec{r}}_{2}}^{2}\;N_{%
D}({\vec{b}}_{1}+{\vec{r}}_{1},{\vec{b}}_{1},{\vec{b}}_{2}+{\vec{r}}_{2},{\vec%
{b}}_{2};y),$$
where
$$\displaystyle N_{D}({\vec{x}},{\vec{y}},{\vec{z}},{\vec{w}};y)=\;\frac{1}{(N_{%
c}^{2}-1)^{2}}\;\left\langle\mbox{Tr}\left[\mathds{1}-U_{{\vec{x}}}U_{{\vec{y}%
}}^{\dagger}\right]\mbox{Tr}\left[\mathds{1}-U_{{\vec{z}}}U_{{\vec{w}}}^{%
\dagger}\right]\right\rangle_{A_{2}}(y).$$
(9)
$U_{\vec{x}}$ is the Wilson line in the adjoint representation,
$$U_{\vec{x}}={\cal P}\exp\left[ig\int_{-\infty}^{+\infty}dx^{+}{\cal A}^{-}(x^{%
+},x^{-}=0,\vec{x})\right],$$
(10)
where ${\cal A}$ is the gauge field of the hadron or nucleus in the adjoint representation, $x^{-}=0$ is the position of the infinitely contracted target nucleus (the shockwave), the light-cone gauge ${\cal A}^{+}=0$ is used and quantities with arrows ($\vec{x},\vec{y},\dots$) are two-dimensional vectors that denote the transverse position.
Type 1 diagrams also contain the quadrupole distribution
$$\displaystyle\left\langle\frac{d\phi_{A_{2}}^{Q}({\vec{q}}_{1},{\vec{q}}_{2};y%
)}{d^{2}b_{1}\;d^{2}b_{2}}\right\rangle_{A_{2}}$$
$$\displaystyle=\left(\frac{C_{F}}{\alpha_{s}(2\pi)^{3}}\right)^{2}\int d^{2}r_{%
1}\;d^{2}r_{2}\,e^{-i{\vec{q}}_{1}\cdot{\vec{r}}_{1}-i{\vec{q}}_{2}\cdot{\vec{%
r}}_{2}}$$
(11)
$$\displaystyle\times\nabla_{{\vec{r}}_{1}}^{2}\;\nabla_{{\vec{r}}_{2}}^{2}\;N_{%
Q}({\vec{b}}_{1}+{\vec{r}}_{1},{\vec{b}}_{1},{\vec{b}}_{2}+{\vec{r}}_{2},{\vec%
{b}}_{2};y),$$
with
$$\displaystyle N_{Q}({\vec{x}},{\vec{y}},{\vec{z}},{\vec{w}};y)=\;\frac{1}{N_{c%
}^{2}-1}\;\left\langle\mbox{Tr}\left[\left(\mathds{1}-U_{{\vec{x}}}U_{{\vec{y}%
}}^{\dagger}\right)\left(\mathds{1}-U_{{\vec{z}}}U_{{\vec{w}}}^{\dagger}\right%
)\right]\right\rangle_{A_{2}}(y).$$
(12)
Using these distribution functions we can write the cross section for two gluon production in dilute-dense collisions as the convolution of these various distribution functions and a kinematic kernel (see Kovchegov:2013ewa for details),
$$\displaystyle\frac{d\sigma}{d^{2}k_{1}dy_{1}d^{2}k_{2}dy_{2}}=\left(\frac{2\;%
\alpha_{s}}{C_{F}}\right)^{2}\frac{1}{k_{1}^{2}\;k_{2}^{2}}\int d^{2}B\;d^{2}b%
_{1}\;d^{2}b_{2}\int d^{2}q_{1}\;d^{2}q_{2}$$
(13)
$$\displaystyle\times\left\langle\frac{d\phi_{A_{1}}({\vec{q}}_{1};y=0)}{d^{2}({%
\vec{B}}-{\vec{b}}_{1})}\right\rangle_{\!\!A_{1}}\left\langle\frac{d\phi_{A_{1%
}}({\vec{q}}_{2};y=0)}{d^{2}({\vec{B}}-{\vec{b}}_{2})}\right\rangle_{\!\!A_{1}%
}\left\{\left\langle\frac{d\phi_{A_{2}}^{D}({\vec{q}}_{1}-{\vec{k}}_{1},{\vec{%
q}}_{2}-{\vec{k}}_{2},;y)}{d^{2}b_{1}\;d^{2}b_{2}}\right\rangle_{\!\!A_{2}}\!%
\!\right.$$
$$\displaystyle\left.+\,e^{-i\,({\vec{k}}_{1}-{\vec{k}}_{2})\cdot({\vec{b}}_{1}-%
{\vec{b}}_{2})}\;\frac{\mathcal{K}({\vec{k}}_{1},{\vec{k}}_{2},{\vec{q}}_{1},{%
\vec{q}}_{2})}{N_{c}^{2}-1}\left\langle\frac{d\phi_{A_{2}}^{Q}({\vec{q}}_{1}-{%
\vec{k}}_{1},{\vec{q}}_{2}-{\vec{k}}_{2};y)}{d^{2}b_{1}\;d^{2}b_{2}}\right%
\rangle_{\!\!A_{2}}\right\}+({\vec{k}}_{2}\rightarrow-{\vec{k}}_{2}),$$
with $\vec{B}$ the relative impact parameter between the target and the projectile (which is considered to be homogeneous and small compared with the target, consistent with the fact that gluons in the projectile should lie close enough to be correlated), and the kernel
$$\displaystyle\mathcal{K}({\vec{k}}_{1},{\vec{k}}_{2},{\vec{q}}_{1},{\vec{q}}_{%
2})$$
$$\displaystyle=\frac{1}{q_{1}^{2}\;q_{2}^{2}\;({\vec{k}}_{1}-{\vec{q}}_{1})^{2}%
({\vec{k}}_{2}-{\vec{q}}_{2})^{2}}\;\left\{k_{1}^{2}\;k_{2}^{2}({\vec{q}}_{1}%
\cdot{\vec{q}}_{2})^{2}\right.$$
$$\displaystyle-\;k_{1}^{2}\;({\vec{q}}_{1}\cdot{\vec{q}}_{2})\left[({\vec{k}}_{%
2}\cdot{\vec{q}}_{1})\;q_{2}^{2}\;+\;({\vec{k}}_{2}\cdot{\vec{q}}_{2})\;q_{1}^%
{2}\;-\;q_{1}^{2}\;q_{2}^{2}\right]$$
$$\displaystyle-\;k_{2}^{2}\;({\vec{q}}_{1}\cdot{\vec{q}}_{2})\left[({\vec{k}}_{%
1}\cdot{\vec{q}}_{1})\;q_{2}^{2}\;+\;({\vec{k}}_{1}\cdot{\vec{q}}_{2})\;q_{1}^%
{2}\;-\;q_{1}^{2}\;q_{2}^{2}\right]$$
$$\displaystyle\left.+\;q_{1}^{2}\;q_{2}^{2}\;\left[({\vec{k}}_{1}\cdot{\vec{q}}%
_{1})({\vec{k}}_{2}\cdot{\vec{q}}_{2})\;+\;({\vec{k}}_{1}\cdot{\vec{q}}_{2})({%
\vec{k}}_{2}\cdot{\vec{q}}_{1})\right]\right\}.$$
(14)
In the next Section we will examine this expression and isolate the various contributions.
3 Isolating the Various Contributions
The first contribution to the two gluon production cross section beyond the dilute-dilute limit (the glasma graph approximation) that one can isolate trivially is the classical contribution which stems from treating the produced gluons as two distinguishable particles. Effectively, this corresponds to ignoring all the interference diagrams and considering only the diagrams of type 3. Moreover, in this contribution the emissions of the two gluons are completely independent of each other.
The correlations encoded in the classical contribution only depends on the geometry of the collision. Thus, it corresponds to uncorrelated production and it is the leading contribution in the large-$N_{c}$ limit (as discussed in Altinoluk:2015uaa ). Therefore, one can isolate this contribution by taking the large-$N_{c}$ limit of the two gluon production cross section, Eq.(13). In this limit, the double dipole operator given in Eq.(9) can be approximated as
$$\displaystyle N_{D}({\vec{x}},{\vec{y}},{\vec{z}},{\vec{w}};y)$$
$$\displaystyle\simeq$$
$$\displaystyle\;\frac{1}{N_{c}^{2}-1}\;\left\langle\mbox{Tr}\left[\mathds{1}-U_%
{{\vec{x}}}U_{{\vec{y}}}^{\dagger}\right]\right\rangle_{A_{2}}(y)\;\frac{1}{N_%
{c}^{2}-1}\left\langle\mbox{Tr}\left[\mathds{1}-U_{{\vec{z}}}U_{{\vec{w}}}^{%
\dagger}\right]\right\rangle_{A_{2}}(y)$$
(15)
$$\displaystyle=$$
$$\displaystyle N_{D}({\vec{x}},{\vec{y}};y)\;N_{D}({\vec{z}},{\vec{w}};y).$$
We would like to emphasize that this is not the only contribution containing the product of two target averaged dipoles but it is the only one that gives the uncorrelated piece and, moreover, the others are suppressed by factors $\frac{1}{(N_{c}^{2}-1)^{2}}$.
This corresponds to approximating the double trace distribution in the two gluon production cross section as
$$\left\langle\frac{d\phi_{A_{2}}^{D}({\vec{q}}_{1}-{\vec{k}}_{1},{\vec{q}}_{2}-%
{\vec{k}_{2}};y)}{d^{2}b_{1}\;d^{2}b_{2}}\right\rangle_{A_{2}}\simeq\left%
\langle\frac{d\phi_{A_{2}}({\vec{q}}_{1}-{\vec{k}_{1}};y)}{d^{2}b_{1}}\right%
\rangle_{A_{2}}\;\left\langle\frac{d\phi_{A_{2}}({\vec{q}}_{2}-{\vec{k}_{2}};y%
)}{d^{2}b_{2}}\right\rangle_{A_{2}}\;.$$
(16)
Finally, the classical contribution to the two-gluon production cross section reads
$$\displaystyle\frac{d\sigma_{classical}}{d^{2}k_{1}dy_{1}d^{2}k_{2}dy_{2}}=%
\left(\frac{2\;\alpha_{s}}{C_{F}}\right)^{2}\frac{1}{k_{1}^{2}\;k_{2}^{2}}\int
d%
^{2}B\;d^{2}b_{1}\;d^{2}b_{2}\int d^{2}q_{1}\;d^{2}q_{2}$$
(17)
$$\displaystyle\times\left\langle\frac{d\phi_{A_{1}}({\vec{q}}_{1};y=0)}{d^{2}({%
\vec{B}}-{\vec{b}}_{1})}\right\rangle_{\!\!A_{1}}\left\langle\frac{d\phi_{A_{1%
}}({\vec{q}}_{2};y=0)}{d^{2}({\vec{B}}-{\vec{b}}_{2})}\right\rangle_{\!\!A_{1}%
}\left\langle\frac{d\phi_{A_{2}}({\vec{q}}_{1}-{\vec{k}}_{1};y)}{d^{2}b_{1}}%
\right\rangle_{\!\!A_{2}}\left\langle\frac{d\phi_{A_{2}}({\vec{q}}_{2}-{\vec{k%
}}_{2};y)}{d^{2}b_{2}}\right\rangle_{\!\!A_{2}}.$$
As it was mentioned previously, the only correlations contained in the classical contribution are the ones that are related with the geometry of the collision. However, these correlations (that are not of particular interest for this study) can be neglected assuming a translationally invariant target. In this case, integrations over various impact parameters can be performed trivially and the classical contribution to the two gluon production cross section reads
$$\displaystyle\frac{d\sigma_{classical}}{d^{2}k_{1}dy_{1}d^{2}k_{2}dy_{2}}=%
\frac{1}{S_{\perp,2}}\left(\frac{2\;\alpha_{s}}{C_{F}}\right)^{2}\frac{1}{k_{1%
}^{2}\;k_{2}^{2}}\int d^{2}q_{1}\;d^{2}q_{2}$$
$$\displaystyle\times\phi_{A_{1}}({\vec{q}}_{1};y=0)\;\phi_{A_{1}}({\vec{q}}_{2}%
;y=0)\;\phi_{A_{2}}({\vec{q}}_{1}-{\vec{k}}_{1};y)\;\phi_{A_{2}}({\vec{q}}_{2}%
-{\vec{k}}_{2};y)$$
(18)
which is just the square of the single-gluon production cross section divided by the transverse area, i.e.,
$$\frac{d\sigma_{classical}}{d^{2}k_{1}dy_{1}d^{2}k_{2}dy_{2}}=\frac{1}{S_{\perp%
,2}}\frac{d\sigma_{g}}{d^{2}k_{1}dy_{1}}\frac{d\sigma_{g}}{d^{2}k_{2}dy_{2}}.$$
(19)
Our next order of business is to isolate the HBT and Bose enhancement contributions. As it was discussed in detail in Altinoluk:2015uaa and Altinoluk:2015eka , these contributions are suppressed by a factor of $\frac{1}{N_{c}^{2}-1}$ when compared to the classical contribution and they can originate either from the quadrupole distribution or the double dipole distribution terms in the two gluon production cross section, Eq.(13).
First, we consider the quadrupole distribution and factorize the target averaging of the four Wilson lines into averaging over the pairs as it was shown in Kovchegov:2013ewa 222In principle, a pair of gluons can be found in all of the seven irreducible representations of $SU(N_{c})$ that result from the product of two adjoints. However, we have approximated the quadrupole operator by only considering the singlet projector which gives the factorized double dipole operator as it was argued in Kovchegov:2013ewa .. Then, the quadrupole term reads
$$\displaystyle\left\langle\mbox{Tr}\left[U_{{\vec{r}}_{1}+{\vec{b}}_{1}}U^{%
\dagger}_{{\vec{b}}_{1}}U_{{\vec{r}}_{2}+{\vec{b}}_{2}}U^{\dagger}_{{\vec{b}}_%
{2}}\right]\right\rangle_{A_{2}}(y)$$
(20)
$$\displaystyle\quad\quad\quad=\frac{1}{N_{c}^{2}-1}\left\langle\mbox{Tr}\left[U%
_{{\vec{r}}_{1}+{\vec{b}}_{1}}U^{\dagger}_{{\vec{b}}_{1}}\right]\right\rangle_%
{A_{2}}(y)\left\langle\mbox{Tr}\left[U_{{\vec{r}}_{2}+{\vec{b}}_{2}}U^{\dagger%
}_{{\vec{b}}_{2}}\right]\right\rangle_{A_{2}}(y)$$
$$\displaystyle\quad\quad\quad+\frac{1}{N_{c}^{2}-1}\left\langle\mbox{Tr}\left[U%
_{{\vec{r}}_{1}+{\vec{b}}_{1}}U^{\dagger}_{{\vec{b}}_{2}}\right]\right\rangle_%
{A_{2}}(y)\left\langle\mbox{Tr}\left[U_{{\vec{r}}_{2}+{\vec{b}}_{2}}U^{\dagger%
}_{{\vec{b}}_{1}}\right]\right\rangle_{A_{2}}(y)$$
$$\displaystyle\quad\quad\quad+\frac{1}{(N_{c}^{2}-1)^{2}}\left\langle\mbox{Tr}%
\left[U_{{\vec{r}}_{1}+{\vec{b}}_{1}}U^{\dagger}_{{\vec{r}}_{2}+{\vec{b}}_{2}}%
\right]\right\rangle_{A_{2}}(y)\left\langle\mbox{Tr}\left[U_{{\vec{b}}_{1}}U^{%
\dagger}_{{\vec{b}}_{2}}\right]\right\rangle_{A_{2}}(y)$$
$$\displaystyle\quad\quad\quad+\cdots\ .$$
Using this factorization, the gradients act on related dipoles and the quadrupole operator can be written in terms of the dipole operators as
$$\displaystyle\nabla^{2}_{{\vec{r}}_{1}}\nabla^{2}_{{\vec{r}}_{2}}N_{Q}({\vec{r%
}}_{1}+{\vec{b}}_{1},{\vec{b}}_{1},{\vec{r}}_{2}+{\vec{b}}_{2},{\vec{b}}_{2};y)$$
$$\displaystyle=$$
$$\displaystyle\nabla^{2}_{{\vec{r}}_{1}}N({\vec{r}}_{1}+{\vec{b}}_{1},{\vec{b}}%
_{1};y)\nabla^{2}_{{\vec{r}}_{2}}N({\vec{r}}_{2}+{\vec{b}}_{2},{\vec{b}}_{2};y)$$
$$\displaystyle+$$
$$\displaystyle\nabla^{2}_{{\vec{r}}_{1}}N({\vec{r}}_{1}+{\vec{b}}_{1},{\vec{b}}%
_{2};y)\nabla^{2}_{{\vec{r}}_{2}}N({\vec{r}}_{2}+{\vec{b}}_{2},{\vec{b}}_{1};y)$$
$$\displaystyle+$$
$$\displaystyle\frac{1}{N_{c}^{2}-1}\nabla^{2}_{{\vec{r}}_{1}}\nabla^{2}_{{\vec{%
r}}_{2}}N({\vec{r}}_{1}+{\vec{b}}_{1},{\vec{r}}_{2}+{\vec{b}}_{2};y)N({\vec{b}%
}_{1},{\vec{b}}_{2};y)$$
$$\displaystyle+$$
$$\displaystyle\cdots\ .$$
One can use this factorized form of the quadrupole operator, Eq. (3), to get the explicit expressions for the quadrupole distributions of the target in terms of the dipole distributions which then will be used to identify various contributions to the two gluon production cross section.
Once the first term of Eq. (3) is substituted into the quadrupole distribution, one gets
$$\displaystyle\left\langle\frac{\phi_{term\,1}({\vec{q}}_{1}-{\vec{k}}_{1},{%
\vec{q}}_{2}-{\vec{k}}_{2};y)}{d^{2}b_{1}d^{2}b_{2}}\right\rangle_{A_{2}}=%
\left(\frac{C_{F}}{\alpha_{s}(2\pi)^{3}}\right)^{2}\int d^{2}r_{1}d^{2}r_{2}e^%
{-i{\vec{r}}_{1}\cdot({\vec{q}}_{1}-{\vec{k}}_{1})-i{\vec{r}}_{2}\cdot({\vec{q%
}}_{2}-{\vec{k}}_{2})}$$
$$\displaystyle\times\nabla^{2}_{{\vec{r}}_{1}}N({\vec{r}}_{1}+{\vec{b}}_{1},{%
\vec{b}}_{1};y)\nabla^{2}_{{\vec{r}}_{2}}N({\vec{r}}_{1}+{\vec{b}}_{2},{\vec{b%
}}_{2};y)$$
$$\displaystyle=\left\langle\frac{d\phi_{A_{2}}({\vec{q}}_{1}-{\vec{k}}_{1};y)}{%
d^{2}b_{1}}\right\rangle_{A_{2}}\left\langle\frac{d\phi_{A_{2}}({\vec{q}}_{2}-%
{\vec{k}}_{2};y)}{d^{2}b_{2}}\right\rangle_{A_{2}}\,.$$
(22)
Now, this distribution can be plugged into the two gluon production cross section, Eq. (13), to get the contribution of the first term of Eq. (3) which simply reads
$$\displaystyle\frac{d\sigma_{term\,1}}{d^{2}k_{1}dy_{1}d^{2}k_{2}dy_{2}}=\left(%
\frac{2\;\alpha_{s}}{C_{F}}\right)^{2}\frac{1}{k_{1}^{2}\;k_{2}^{2}}\int d^{2}%
B\;d^{2}b_{1}\;d^{2}b_{2}\int d^{2}q_{1}\;d^{2}q_{2}$$
$$\displaystyle\times\left\langle\frac{d\phi_{A_{1}}({\vec{q}}_{1};y=0)}{d^{2}({%
\vec{B}}-{\vec{b}}_{1})}\right\rangle_{\!\!A_{1}}\left\langle\frac{d\phi_{A_{1%
}}({\vec{q}}_{2};y=0)}{d^{2}({\vec{B}}-{\vec{b}}_{2})}\right\rangle_{\!\!A_{1}%
}\left\langle\frac{d\phi_{A_{2}}({\vec{q}_{1}}-{\vec{k}_{1}};y)}{d^{2}b_{1}}%
\right\rangle_{A_{2}}\left\langle\frac{d\phi_{A_{2}}({\vec{q}_{2}}-{\vec{k}_{2%
}};y)}{d^{2}b_{2}}\right\rangle_{A_{2}}$$
$$\displaystyle\times\left[e^{-i\,({\vec{k}}_{1}-{\vec{k}}_{2})\cdot({\vec{b}}_{%
1}-{\vec{b}}_{2})}\;\frac{\mathcal{K}({\vec{k}}_{1},{\vec{k}}_{2},{\vec{q}}_{1%
},{\vec{q}}_{2})}{N_{c}^{2}-1}\right]+({\vec{k}}_{2}\rightarrow-{\vec{k}}_{2})\,.$$
(23)
This term was identified as the HBT contribution in Kovchegov:2013ewa and in Altinoluk:2015eka . It is easier to understand this identification in the case of a translationally invariant target:
$$\displaystyle\frac{d\sigma_{HBT}}{d^{2}k_{1}dy_{1}d^{2}k_{2}dy_{2}}=\frac{1}{S%
_{\perp,1}S_{\perp,2}}\left(\frac{2\;\alpha_{s}}{C_{F}}\right)^{2}\frac{1}{k_{%
1}^{2}\;k_{2}^{2}}\int d^{2}q_{1}\;d^{2}q_{2}$$
$$\displaystyle\times\phi_{A_{1}}({\vec{q}}_{1};y=0)\;\phi_{A_{1}}({\vec{q}}_{2}%
;y=0)\;\phi_{A_{2}}({\vec{q}}_{1}-\vec{k}_{1};y)\;\phi_{A_{2}}({\vec{q}}_{2}-%
\vec{k}_{2};y)$$
$$\displaystyle\times\frac{2\pi}{N_{c}^{2}-1}\left(\delta(\vec{k}_{1}-\vec{k}_{2%
})+\delta(\vec{k}_{1}+\vec{k}_{2})\right)\mathcal{K}({\vec{k}}_{1},{\vec{k}}_{%
2},{\vec{q}}_{1},{\vec{q}}_{2})$$
(24)
which clearly gives a peak at ${\vec{k}}_{1}={\vec{k}}_{2}$ and at ${\vec{k}}_{1}=-{\vec{k}}_{2}$ (as expected from the HBT contribution), with ${\vec{k}}_{1}$ and ${\vec{k}}_{2}$ being the transverse momenta of the produced gluons. In order to observe the relative enhancement of the HBT contribution with respect to the Bose-enhanced one, we should consider the origin of the $\delta$-functions in (3). They come from an integral in $\vec{b}_{1}-\vec{b_{2}}$ and thus, in the non-translational invariant case, provide a factor $S_{\perp,1}$ that enhance this contribution by the number of sources $S_{\perp,1}Q^{2}_{s,1}$ with respect to the Bose-enhanced term that we discuss next.
In a similar manner, we can consider the second term of Eq. (3). When substituted in the quadrupole distribution, it reads
$$\displaystyle\left\langle\frac{\phi_{term\,2}({\vec{q}}_{1}-{\vec{k}}_{1},{%
\vec{q}}_{2}-{\vec{k}}_{2};y)}{d^{2}b_{1}d^{2}b_{2}}\right\rangle_{A_{2}}=%
\left(\frac{C_{F}}{\alpha_{s}(2\pi)^{3}}\right)^{2}\int d^{2}r_{1}d^{2}r_{2}e^%
{-i{\vec{r}}_{1}\cdot({\vec{q}}_{1}-{\vec{k}}_{1})-i{\vec{r}}_{2}\cdot({\vec{q%
}}_{2}-{\vec{k}}_{2})}$$
$$\displaystyle\times\nabla^{2}_{{\vec{r}}_{1}}N({\vec{r}}_{1}+\Delta{\vec{b}}+{%
\vec{b}}_{2},{\vec{b}}_{2};y)\nabla^{2}_{{\vec{r}}_{2}}N({\vec{r}}_{1}-\Delta{%
\vec{b}}+{\vec{b}}_{1},{\vec{b}}_{1};y),$$
(25)
where we have introduced $\Delta{\vec{b}}={\vec{b}}_{1}-{\vec{b}}_{2}$. It is convenient to define the shifted variables
$$\displaystyle{\vec{r}}\,^{\prime}_{1}$$
$$\displaystyle=$$
$$\displaystyle{\vec{r}}_{1}+\Delta{\vec{b}},$$
$$\displaystyle{\vec{r}}\,^{\prime}_{2}$$
$$\displaystyle=$$
$$\displaystyle{\vec{r}}_{2}-\Delta{\vec{b}},$$
(26)
in order to write the quadrupole distribution of the second term of Eq. (3) in factorized form as
$$\displaystyle\left\langle\frac{\phi_{term\,2}({\vec{q}}_{1}-{\vec{k}}_{1},{%
\vec{q}}_{2}-{\vec{k}}_{2};y)}{d^{2}b_{1}d^{2}b_{2}}\right\rangle_{A_{2}}=%
\left(\frac{C_{F}}{\alpha_{s}(2\pi)^{3}}\right)^{2}e^{-i(\Delta{\vec{b}})\cdot%
({\vec{q}}_{2}-{\vec{k}}_{2}-{\vec{q}}_{1}+{\vec{k}}_{1})}$$
$$\displaystyle\times\int d^{2}r^{\prime}_{1}d^{2}r^{\prime}_{2}e^{-i{\vec{r}}\,%
^{\prime}_{1}\cdot({\vec{q}}_{1}-{\vec{k}}_{1})-i{\vec{r}}\,^{\prime}_{2}\cdot%
({\vec{q}}_{2}-{\vec{k}}_{2})}\nabla^{2}_{{\vec{r}}\,^{\prime}_{1}}N({\vec{r}}%
\,^{\prime}_{1}+{\vec{b}}_{2},{\vec{b}}_{2};y)\nabla^{2}_{{\vec{r}}\,^{\prime}%
_{2}}N({\vec{r}}\,^{\prime}_{2}+{\vec{b}}_{1},{\vec{b}}_{1};y)$$
$$\displaystyle=\,e^{-i(\Delta{\vec{b}})\cdot({\vec{q}}_{2}-{\vec{k}}_{2}-{\vec{%
q}}_{1}+{\vec{k}}_{1})}\left\langle\frac{d\phi_{A_{2}}({\vec{q}_{1}}-{\vec{k}_%
{1}};y)}{d^{2}b_{2}}\right\rangle_{A_{2}}\left\langle\frac{d\phi_{A_{2}}({\vec%
{q}_{2}}-{\vec{k}_{2}};y)}{d^{2}b_{1}}\right\rangle_{A_{2}}\ \!\!\!.$$
(27)
We plug this factorized form of the second term of the quadrupole distribution into the two gluon production cross section and the result reads
$$\displaystyle\frac{d\sigma_{term\,2}}{d^{2}k_{1}dy_{1}d^{2}k_{2}dy_{2}}=\left(%
\frac{2\;\alpha_{s}}{C_{F}}\right)^{2}\frac{1}{k_{1}^{2}\;k_{2}^{2}}\int d^{2}%
B\;d^{2}b_{1}\;d^{2}b_{2}\int d^{2}q_{1}\;d^{2}q_{2}$$
$$\displaystyle\times\left\langle\frac{d\phi_{A_{1}}({\vec{q}}_{1};y=0)}{d^{2}({%
\vec{B}}-{\vec{b}}_{1})}\right\rangle_{\!\!A_{1}}\left\langle\frac{d\phi_{A_{1%
}}({\vec{q}}_{2};y=0)}{d^{2}({\vec{B}}-{\vec{b}}_{2})}\right\rangle_{\!\!A_{1}%
}\left\langle\frac{d\phi_{A_{2}}({\vec{q}_{1}}-{\vec{k}_{1}};y)}{d^{2}b_{2}}%
\right\rangle_{A_{2}}\left\langle\frac{d\phi_{A_{2}}({\vec{q}_{2}}-{\vec{k}_{2%
}};y)}{d^{2}b_{1}}\right\rangle_{A_{2}}$$
$$\displaystyle\times\frac{1}{N_{c}^{2}-1}\,e^{-i{\Delta{\vec{b}}}\cdot(2{\vec{k%
}}_{1}-2{\vec{k}}_{2}-{\vec{q}}_{1}+{\vec{q}}_{2})}\;\mathcal{K}({\vec{k}}_{1}%
,{\vec{k}}_{2},{\vec{q}}_{1},{\vec{q}}_{2})\;+\;({\vec{k}}_{2}\rightarrow-{%
\vec{k}}_{2}).$$
(28)
For convenience, let us define the average transverse momentum ${\vec{q}}$ and shifted transverse momenta difference $\Delta{\vec{q}}$ as
$$\displaystyle{\vec{q}}$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{2}({\vec{q}}_{1}+{\vec{q}}_{2}),$$
(29)
$$\displaystyle\Delta{\vec{q}}$$
$$\displaystyle=$$
$$\displaystyle{\vec{q}}_{1}-{\vec{q}}_{2}-2\Delta{\vec{k}},$$
where we have defined the transverse momenta difference of the produced gluons as $\Delta{\vec{k}}={\vec{k}}_{1}-{\vec{k}}_{2}$. After these change of variables the contribution of the second term of the quadrupole distribution to the two gluon production cross section can be written as
$$\displaystyle\frac{d\sigma_{term\,2}}{d^{2}k_{1}dy_{1}d^{2}k_{2}dy_{2}}=\left(%
\frac{2\;\alpha_{s}}{C_{F}}\right)^{2}\frac{1}{k_{1}^{2}\;k_{2}^{2}}\int d^{2}%
B\;d^{2}b_{1}\;d^{2}b_{2}\int d^{2}q\;d^{2}\Delta q\;\frac{1}{N_{c}^{2}-1}\,e^%
{i{\Delta{\vec{b}}}\cdot\Delta{\vec{q}}}$$
$$\displaystyle\times\left\langle\frac{d\phi_{A_{1}}({\vec{q}}+\Delta{\vec{q}}/2%
-\Delta{\vec{k}};y=0)}{d^{2}({\vec{B}}-{\vec{b}}_{1})}\right\rangle_{\!\!A_{1}%
}\left\langle\frac{d\phi_{A_{1}}({\vec{q}}-\Delta{\vec{q}}/2-\Delta{\vec{k}};y%
=0)}{d^{2}({\vec{B}}-{\vec{b}}_{2})}\right\rangle_{\!\!A_{1}}$$
$$\displaystyle\times\left\langle\frac{d\phi_{A_{2}}({\vec{q}}-\Delta{\vec{q}}/2%
-{\vec{k}}_{1};y)}{d^{2}b_{1}}\right\rangle_{A_{2}}\left\langle\frac{d\phi_{A_%
{2}}({\vec{q}}+\Delta{\vec{q}}/2-{\vec{k}}_{2};y)}{d^{2}b_{2}}\right\rangle_{A%
_{2}}$$
$$\displaystyle\times\;\mathcal{K}\left({\vec{k}}_{1},{\vec{k}}_{2},{\vec{q}}+%
\frac{\Delta{\vec{q}}}{2}+\Delta{\vec{k}},{\vec{q}}-\frac{\Delta{\vec{q}}}{2}-%
\Delta{\vec{k}}\right)\;+\;({\vec{k}}_{2}\rightarrow-{\vec{k}}_{2})\;.$$
(30)
In the case of a translationally invariant target, this contribution reads
$$\displaystyle\frac{d\sigma_{Bose}}{d^{2}k_{1}dy_{1}d^{2}k_{2}dy_{2}}=\left(%
\frac{2\;\alpha_{s}}{C_{F}}\right)^{2}\frac{1}{k_{1}^{2}\;k_{2}^{2}}\;\frac{1}%
{S_{\perp,1}S_{\perp,2}}\;\frac{1}{N_{c}^{2}-1}\int d^{2}q\;\mathcal{K}\left({%
\vec{k}}_{1},{\vec{k}}_{2},{\vec{q}}+\Delta{\vec{k}},{\vec{q}}-\Delta{\vec{k}}\right)$$
$$\displaystyle\times\phi_{A_{1}}({\vec{q}}+\Delta{\vec{k}};y=0)\,\phi_{A_{1}}({%
\vec{q}}-\Delta{\vec{k}};y=0)\,\phi_{A_{2}}({\vec{q}}-{\vec{k}}_{1};y)\,\phi_{%
A_{2}}({\vec{q}}-{\vec{k}}_{2};y)$$
$$\displaystyle+\;({\vec{k}}_{2}\rightarrow-{\vec{k}}_{2})\;.$$
(31)
We have identified this term as Bose enhancement of the projectile even though the projectile gluon distributions seem to have different momenta. However, as discussed in detail in Altinoluk:2015uaa , the Bose enhancement contribution to the correlated production is peaked when the ${\vec{k}}_{1}={\vec{k}}_{2}$ for the nearside and when ${\vec{k}}_{1}=-{\vec{k}}_{2}$ for the away side ridge. Thus, for the near side ridge, $\Delta{\vec{k}}\to 0$ and clearly we get the peak in the first term of Eq. (3). Note that, the same argument holds for the away side ridge and the second term of Eq. (3). In conclusion, the Bose enhancement contribution for the near side ridge reads
$$\displaystyle\frac{d\sigma_{Bose}}{d^{2}k_{1}dy_{1}d^{2}k_{2}dy_{2}}\bigg{|}_{%
{\vec{k}}_{1}={\vec{k}}_{2}={\vec{k}}}=\left(\frac{2\;\alpha_{s}}{C_{F}}\right%
)^{2}\frac{1}{k^{4}}\;\frac{1}{S_{\perp,1}S_{\perp,2}}\;\frac{1}{N_{c}^{2}-1}%
\int d^{2}q\;\mathcal{K}\left({\vec{k}},{\vec{k}},{\vec{q}},{\vec{q}}\right)$$
(32)
$$\displaystyle \times\phi_{A_{1}}({\vec{q}};y%
=0)\,\phi_{A_{1}}({\vec{q}};y=0)\,\phi_{A_{2}}({\vec{q}}-{\vec{k}};y)\,\phi_{A%
_{2}}({\vec{q}}-{\vec{k}};y),$$
with the kernel $\mathcal{K}\left({\vec{k}},{\vec{k}},{\vec{q}},{\vec{q}}\right)$ being
$$\mathcal{K}\left({\vec{k}},{\vec{k}},{\vec{q}},{\vec{q}}\right)=\bigg{\{}1+%
\frac{1}{(|{\vec{k}}-{\vec{q}}|^{2})^{2}}\Big{[}4q^{2}({\vec{k}}\cdot{\vec{q}}%
)-|q^{2}|^{2}+2({\vec{k}}\cdot{\vec{q}})^{2}\Big{]}\bigg{\}}\;.$$
(33)
We have identified the HBT, Eq. (3), and the Bose enhancement, Eq. (3), contributions to the correlated production that stem respectively from the first and second terms of the quadrupole distributions given in Eq. (3). Besides, we have also identified the classical contribution, Eq. (3) that originates from the double dipole term, Eq. (16). It contributes to uncorrelated production and it is the leading term in the large $N_{c}$ limit. These are the main results of the paper. However, we would like to comment about the remaining terms of the quadrupole and the double dipole distributions. It is clear from the last line of Eq. (3) that the third contribution of the quadrupole distribution is suppressed by an extra power of $\frac{1}{N_{c}^{2}-1}$ with respect to the first two contributions that are responsible for HBT and Bose enhancement contributions. On the other hand, the remaining two contributions of the double dipole distribution are suppressed an extra power of $\frac{1}{N_{c}^{2}-1}$ as well 333This can be easily understood from the fact that two color projectors, one on the right and one on the left, are required to express the double dipole term with a combination of coordinates that would result in Bose enhancement, see Kovchegov:2013ewa ..
Therefore, the complete results for the uncorrelated, HBT and Bose-enhanced pieces of the two-gluon production cross section in the dilute-dense limit correspond to Eqs. (3), (3) and (3) for the translational invariant case.
4 A toy model
In this section, we perform the numerical analysis of the main results of our study, namely the HBT and the Bose enhancement contributions to the correlated two-gluon production, by adopting a toy model for both the projectile and the target distributions.
The unintegrated gluon distributions of the projectile are defined in Eq. (2) with $n_{G}({\vec{b}}+{\vec{r}},{\vec{b}};y)$ being the distribution associated with the dilute projectile, whose expression is given in Eq. (4). In our toy model, we assume translational invariance of the dilute projectile. Effectively, this is equivalent to approximate the saturation scale of the projectile that depends on the impact parameter $b$ by a constant which serves as an infrared cut off, i.e. $Q^{2}_{s,1}({\vec{b}})\approx Q^{2}_{1}$. Within the limits of this approximation, the unintegrated gluon distribution of the projectile can be written as
$$\phi_{A_{1}}({\vec{q}})\approx\frac{C_{F}\,S_{\perp,1}\,Q_{1}^{2}}{\alpha_{s}%
\,(2\pi)^{3}}\,\frac{1}{4}\int d^{2}r\,e^{-i{\vec{q}}\cdot{\vec{r}}}\,\nabla^{%
2}_{\vec{r}}\left[r^{2}\mbox{ln}\left(\frac{1}{r\,\Lambda}\right)\right]=\frac%
{C_{F}\,S_{\perp,1}\,Q_{1}^{2}}{\alpha_{s}\,(2\pi)^{3}}\,\frac{2\pi}{q^{2}}\ .$$
(34)
On the other hand, we adopt the Golec-Biernat–Wüsthoff (GBW) model GolecBiernat:1998js for the dipole distribution of the target:
$$N_{G}({\vec{r}}+{\vec{b}},{\vec{b}};y)=1-e^{-\frac{Q_{2}^{2}}{4}r^{2}},$$
(35)
with $Q_{2}$ being the saturation scale of the target. Then, the target distribution reads
$$\displaystyle\phi_{A_{2}}({\vec{q}})=\frac{C_{F}}{\alpha_{s}(2\pi)^{3}}\int d^%
{2}rd^{2}b\,e^{-i{\vec{q}}\cdot{\vec{r}}}\,\nabla^{2}_{{\vec{r}}}\left(1-e^{-%
\frac{Q_{2}^{2}}{4}r^{2}}\right)=\frac{C_{F}}{\alpha_{s}(2\pi)^{3}}\,S_{\perp,%
2}\,\frac{q^{2}}{Q_{2}^{2}}\,4\pi\,e^{-\frac{q^{2}}{Q_{2}^{2}}}.$$
(36)
Using the projectile distribution, Eq.(34), and the target distribution, Eq. (36), the HBT contribution to the two gluon production cross section, Eq. (3), can be written as
$$\displaystyle\frac{d\sigma_{HBT}}{d^{2}k_{1}dy_{1}d^{2}k_{2}dy_{2}}$$
$$\displaystyle=$$
$$\displaystyle\left(\frac{C_{F}}{\alpha_{s}}\right)^{2}16\,\frac{1}{(2\pi)^{8}}%
\,S_{\perp,1}S_{\perp,2}\,\left(\frac{Q_{1}^{2}}{Q_{2}^{2}}\right)^{2}\,\frac{%
1}{k_{1}^{2}k_{2}^{2}}\,\int d^{2}q_{1}d^{2}q_{2}\frac{(\vec{q}_{1}-\vec{k}_{1%
})^{2}(\vec{q}_{2}-\vec{k}_{2})^{2}}{q_{1}^{2}q_{2}^{2}}$$
(37)
$$\displaystyle\times\frac{2\pi}{N_{c}^{2}-1}e^{-\frac{1}{Q_{2}^{2}}\left[(\vec{%
q}_{1}-\vec{k}_{1})^{2}+(\vec{q}_{2}-\vec{k}_{2})^{2}\right]}\,\mathcal{K}(%
\vec{k}_{1},\vec{k}_{2},\vec{q}_{1},\vec{q}_{2})\,\left[\delta^{(2)}(\vec{k}_{%
1}-\vec{k}_{2})+\delta^{(2)}(\vec{k}_{1}+\vec{k}_{2})\right]\,.$$
Similarly, the Bose enhancement contribution to the two gluon production cross section reads
$$\displaystyle\frac{d\sigma_{Bose}}{d^{2}k_{1}dy_{1}d^{2}k_{2}dy_{2}}$$
$$\displaystyle=$$
$$\displaystyle\left(\frac{C_{F}}{\alpha_{s}}\right)^{2}16\,\frac{1}{(2\pi)^{8}}%
\,S_{\perp,1}S_{\perp,2}\,\left(\frac{Q_{1}^{2}}{Q_{2}^{2}}\right)\,\frac{1}{k%
_{1}^{2}k_{2}^{2}}$$
(38)
$$\displaystyle\times\int d^{2}q\frac{\left(\vec{q}-\bar{k}-\frac{\Delta\vec{k}}%
{2}\right)^{2}\left(\vec{q}-\bar{k}+\frac{\Delta\vec{k}}{2}\right)^{2}}{(\vec{%
q}+\Delta\vec{k})^{2}(\vec{q}-\Delta\vec{k})^{2}}\frac{2\pi}{N_{c}^{2}-1}\,%
\exp{\left\{-\frac{2}{Q_{2}^{2}}(\vec{q}-\bar{k})^{2}-\frac{1}{Q_{2}^{2}}(%
\Delta\vec{k})^{2}\right\}}$$
$$\displaystyle\times\ \mathcal{K}(\vec{k}_{1},\vec{k}_{2},\vec{q}+\Delta\vec{k}%
,\vec{q}-\Delta\vec{k})+({\vec{k}}_{2}\rightarrow-{\vec{k}}_{2}),$$
where $\bar{k}=(\vec{k}_{1}+\vec{k}_{2})/2$.
In Figs. 3 and 4 we plot the result of Eq. (38) divided by $\left(\frac{C_{F}}{\alpha_{s}}\right)^{2}16\,\frac{1}{(2\pi)^{8}}\,S_{\perp,1}%
S_{\perp,2}\,\frac{1}{k_{1}^{2}k_{2}^{2}}$ for different choices of $k_{1},k_{2}$ and the angle $\phi$ between them, and $Q_{1}=0.2$ GeV and $Q_{2}=1$ GeV. The denominators in (38) resulting in divergent contributions have been regulated by adding them $Q_{1}^{2}$ or $Q_{2}^{2}$ if they stem from the projectile or target dipole distributions respectively.
While the model that we have used cannot be considered realistic, it illustrates several of the features of the result. First, in Fig. 3 the ridge structure can be seen, symmetric for the near and away side peaks in this calculation. The dip observed at $\phi\simeq 0.2,\pi-0.2$ comes from the double Gaussian structure in Eq. (38) in this model. Second, a fast degradation of the ridge for $k_{1}\neq k_{2}$ can also be observed, a clear signal of the effect of Bose statistics. Finally, in Fig. 4 the height of the peak for $k_{1}=k_{2}$ has a maximum for $k_{1}$ slightly above $Q_{2}$ and is seen to decrease very fast for larger $k_{1}$.
5 Conclusions
In conclusion, in this paper we have explicitly considered the two-gluon production cross section in the CGC for the collision of a dilute projectile on a dense target, going beyond the glasma graph approximation that has been used in the dilute-dilute limit. In this dilute-dense limit, applicable for proton-nucleus or heavy-light ion collisions, we have identified the HBT and the Bose enhancement contributions to the correlated production that are given in Eqs. (3) and (3) respectively. The latter comes suppressed by the number of particles sources with respect to the HBT one, and shows the characteristic suppression by the number of degress of freedom with respect to the uncorrelated contribution.
We have shown that both contributions survive the inclusion of higher order density corrections and that they stem from the quadrupole distribution of the target. We have established the correspondence between the glasma graph approximation and the $k_{T}$-factorized approach, showing that these contributions come from type 1 and type 2 diagrams in the glasma graph approach that correspond to the interference diagrams in the $k_{T}$-factorized formulation.
On the other hand, we have also identified the classical contribution and have shown that it contributes to the uncorrelated production in the case of a translationally invariant target. We have identified that the origin of this contribution is the type 3 diagrams.
Finally, we have developed a toy model that allows a simple numerical implementation and whose results illustrate some of the features of the approach.
The findings in this study are coherent with the results of the previous works Kovchegov:2012nd ; Kovchegov:2013ewa ; Altinoluk:2015uaa ; Altinoluk:2015eka on two-gluon production, and also with similar works Altinoluk:2016vax ; Martinez:2018ygo ; Kovner:2017vro ; Kovner:2017ssr ; Kovner:2018vec on double and multi-quark production.
Acknowledgements.TA expresses his gratitude to the Departamento de Física de Partículas at Universidade de Santiago de Compostela, for support and hospitality when part of this work was done. The work of TA is supported by Grant No. 2017/26/M/ST2/01074 of the National Science Centre, Poland. The work of NA and DEW were supported by the European Research Council grant HotLHC ERC-2011-StG-279579, Ministerio de Ciencia e Innovación of Spain under project FPA2014-58293-C2-1-P and Unidad de Excelencia María de Maetzu under project MDM-2016-0692, Xunta de Galicia (Consellería de Educación) within the Strategic Unit AGRUP2015/11, and FEDER.
This work has been performed in the framework of COST Action CA15213 "Theory of hot matter and relativistic heavy-ion collisions" (THOR).
References
(1)
B. Alver et al. [PHOBOS Collaboration],
Phys. Rev. Lett. 104 (2010) 062301
[arXiv:0903.2811 [nucl-ex]].
(2)
B. I. Abelev et al. [STAR Collaboration],
Phys. Rev. C 80 (2009) 064912
[arXiv:0909.0191 [nucl-ex]].
(3)
V. Khachatryan et al. [CMS Collaboration],
JHEP 1009 (2010) 091
[arXiv:1009.4122 [hep-ex]].
(4)
Phys. Rev. Lett. 116 (2016) 172302
[arXiv:1510.03068 [nucl-ex]].
(5)
G. Aad et al. [ATLAS Collaboration],
Phys. Rev. Lett. 116 (2016) 172301
[arXiv:1509.04776 [hep-ex]].
(6)
S. Chatrchyan et al. [CMS Collaboration],
Phys. Lett. B 718 (2013) 795
[arXiv:1210.5482 [nucl-ex]].
(7)
B. Abelev et al. [ALICE Collaboration],
Phys. Lett. B 719 (2013) 29
[arXiv:1212.2001 [nucl-ex]].
(8)
G. Aad et al. [ATLAS Collaboration],
Phys. Rev. Lett. 110 (2013) 182302
[arXiv:1212.5198 [hep-ex]].
(9)
R. Aaij et al. [LHCb Collaboration],
Phys. Lett. B 762 (2016) 473
[arXiv:1512.00439 [nucl-ex]].
(10)
V. Khachatryan et al. [CMS Collaboration],
Phys. Rev. C 96 (2017) no.1, 014915
[arXiv:1604.05347 [nucl-ex]].
(11)
A. Adare et al. [PHENIX Collaboration],
Phys. Rev. Lett. 114 (2015) no.19, 192301
[arXiv:1404.7461 [nucl-ex]].
(12)
L. Adamczyk et al. [STAR Collaboration],
Phys. Lett. B 747 (2015) 265
[arXiv:1502.07652 [nucl-ex]].
(13)
A. Adare et al. [PHENIX Collaboration],
Phys. Rev. Lett. 115 (2015) no.14, 142301
[arXiv:1507.06273 [nucl-ex]].
(14)
V. Khachatryan et al. [CMS Collaboration],
Phys. Lett. B 765 (2017) 193
[arXiv:1606.06198 [nucl-ex]].
(15)
M. Aaboud et al. [ATLAS Collaboration],
Phys. Rev. C 96 (2017) no.2, 024908
[arXiv:1609.06213 [nucl-ex]].
(16)
M. Aaboud et al. [ATLAS Collaboration],
Eur. Phys. J. C 77 (2017) no.6, 428
[arXiv:1705.04176 [hep-ex]].
(17)
M. Aaboud et al. [ATLAS Collaboration],
Phys. Rev. C 97 (2018) no.2, 024904
[arXiv:1708.03559 [hep-ex]].
(18)
P. Bozek and W. Broniowski,
Phys. Lett. B 718 (2013) 1557
[arXiv:1211.0845 [nucl-th]].
(19)
E. Shuryak and I. Zahed,
Phys. Rev. C 88 (2013) 044915
[arXiv:1301.4470 [hep-ph]].
(20)
A. Bzdak, B. Schenke, P. Tribedy and R. Venugopalan,
Phys. Rev. C 87 (2013) 064906
[arXiv:1304.3403 [nucl-th]].
(21)
K. Werner, I. Karpenko and T. Pierog,
Phys. Rev. Lett. 106 (2011) 122004
[arXiv:1011.0375 [hep-ph]].
(22)
S. Gavin, L. McLerran and G. Moschelli,
Phys. Rev. C 79 (2009) 051902
[arXiv:0806.4718 [nucl-th]].
(23)
P. M. Chesler and L. G. Yaffe,
Phys. Rev. D 82 (2010) 026006
[arXiv:0906.4426 [hep-th]].
(24)
M. P. Heller, R. A. Janik and P. Witaszczyk,
Phys. Rev. Lett. 108 (2012) 201602
[arXiv:1103.3452 [hep-th]].
(25)
A. Kurkela and Y. Zhu,
Phys. Rev. Lett. 115 (2015) no.18, 182301
[arXiv:1506.06647 [hep-ph]].
(26)
P. Romatschke,
Eur. Phys. J. C 77 (2017) no.1, 21
[arXiv:1609.02820 [nucl-th]].
(27)
F. Gelis, E. Iancu, J. Jalilian-Marian and R. Venugopalan,
Ann. Rev. Nucl. Part. Sci. 60 (2010) 463
[arXiv:1002.0333 [hep-ph]].
(28)
C. B. Chiu, R. C. Hwa and C. B. Yang,
Phys. Rev. C 78 (2008) 044903
[arXiv:0801.2183 [nucl-th]].
(29)
J. D. Bjorken, S. J. Brodsky and A. Scharff Goldhaber,
Phys. Lett. B 726 (2013) 344
[arXiv:1308.1435 [hep-ph]].
(30)
E. Shuryak and I. Zahed,
Phys. Rev. D 89 (2014) 094001
[arXiv:1311.0836 [hep-ph]].
(31)
C. Andres, A. Moscoso and C. Pajares,
Phys. Rev. C 90 (2014) 054902
[arXiv:1405.3632 [hep-ph]].
(32)
N. Armesto, L. McLerran and C. Pajares,
Nucl. Phys. A 781 (2007) 201
[hep-ph/0607345].
(33)
A. Dumitru, F. Gelis, L. McLerran and R. Venugopalan,
Nucl. Phys. A 810 (2008) 91
[arXiv:0804.3858 [hep-ph]].
(34)
A. Dumitru, K. Dusling, F. Gelis, J. Jalilian-Marian, T. Lappi and R. Venugopalan,
Phys. Lett. B 697 (2011) 21
[arXiv:1009.5295 [hep-ph]].
(35)
Y. V. Kovchegov and D. E. Wertepny,
Nucl. Phys. A 906 (2013) 50
[arXiv:1212.1195 [hep-ph]].
(36)
Y. V. Kovchegov and D. E. Wertepny,
Nucl. Phys. A 925 (2014) 254
[arXiv:1310.6701 [hep-ph]].
(37)
K. Dusling and R. Venugopalan,
Phys. Rev. Lett. 108 (2012) 262001
[arXiv:1201.2658 [hep-ph]].
(38)
K. Dusling and R. Venugopalan,
Phys. Rev. D 87 (2013) no.5, 051502
[arXiv:1210.3890 [hep-ph]].
(39)
K. Dusling and R. Venugopalan,
Phys. Rev. D 87 (2013) no.5, 054014
[arXiv:1211.3701 [hep-ph]].
(40)
K. Dusling and R. Venugopalan,
Phys. Rev. D 87 (2013) no.9, 094034
[arXiv:1302.7018 [hep-ph]].
(41)
K. Dusling, M. Mace and R. Venugopalan,
Phys. Rev. Lett. 120 (2018) no.4, 042002
[arXiv:1705.00745 [hep-ph]].
(42)
K. Dusling, M. Mace and R. Venugopalan,
Phys. Rev. D 97 (2018) no.1, 016014
[arXiv:1706.06260 [hep-ph]].
(43)
B. Blok, C. D. Jäkel, M. Strikman and U. A. Wiedemann,
JHEP 1712 (2017) 074
[arXiv:1708.08241 [hep-ph]].
(44)
V. Skokov,
Phys. Rev. D 91 (2015) no.5, 054014
[arXiv:1412.5191 [hep-ph]].
(45)
B. Schenke, S. Schlichting and R. Venugopalan,
Phys. Lett. B 747 (2015) 76
[arXiv:1502.01331 [hep-ph]].
(46)
T. Lappi, B. Schenke, S. Schlichting and R. Venugopalan,
JHEP 1601 (2016) 061
[arXiv:1509.03499 [hep-ph]].
(47)
L. McLerran and V. Skokov,
Nucl. Phys. A 959 (2017) 83
[arXiv:1611.09870 [hep-ph]].
(48)
A. Kovner, M. Lublinsky and V. Skokov,
Phys. Rev. D 96 (2017) no.1, 016010
[arXiv:1612.07790 [hep-ph]].
(49)
Y. V. Kovchegov and V. V. Skokov,
arXiv:1802.08166 [hep-ph].
(50)
T. Altinoluk, N. Armesto, G. Beuf, A. Kovner and M. Lublinsky,
Phys. Rev. D 95 (2017) no.3, 034025
[arXiv:1610.03020 [hep-ph]].
(51)
A. Kovner, M. Lublinsky and V. Skokov,
Phys. Rev. D 96 (2017) no.9, 096003
[arXiv:1706.02330 [hep-ph]].
(52)
M. Martinez, M. D. Sievert and D. E. Wertepny,
arXiv:1801.08986 [hep-ph].
(53)
A. Kovner and A. H. Rezaeian,
Phys. Rev. D 95 (2017) no.11, 114028
[arXiv:1701.00494 [hep-ph]].
(54)
A. Kovner and A. H. Rezaeian,
Phys. Rev. D 96 (2017) no.7, 074018
[arXiv:1707.06985 [hep-ph]].
(55)
A. Kovner and A. H. Rezaeian,
arXiv:1801.04875 [hep-ph].
(56)
A. Kovner and M. Lublinsky,
Phys. Rev. D 83 (2011) 034017
[arXiv:1012.3398 [hep-ph]].
(57)
A. Kovner and M. Lublinsky,
Phys. Rev. D 84 (2011) 094011
[arXiv:1109.0347 [hep-ph]].
(58)
A. Kovner and M. Lublinsky,
Int. J. Mod. Phys. E 22 (2013) 1330001
[arXiv:1211.1928 [hep-ph]].
(59)
A. Dumitru and V. Skokov,
Phys. Rev. D 91 (2015) no.7, 074006
[arXiv:1411.6630 [hep-ph]].
(60)
E. Levin and A. H. Rezaeian,
Phys. Rev. D 84 (2011) 034031
[arXiv:1105.3275 [hep-ph]].
(61)
T. Altinoluk, N. Armesto, G. Beuf, A. Kovner and M. Lublinsky,
Phys. Lett. B 751 (2015) 448
[arXiv:1503.07126 [hep-ph]].
(62)
T. Altinoluk, N. Armesto, G. Beuf, A. Kovner and M. Lublinsky,
Phys. Lett. B 752 (2016) 113
[arXiv:1509.03223 [hep-ph]].
(63)
K. J. Golec-Biernat and M. Wusthoff,
Phys. Rev. D 59 (1998) 014017
[hep-ph/9807513]. |
Shaping the distribution of vertical velocities of antihydrogen in GBAR
G. Dufour
Laboratoire Kastler-Brossel, CNRS, ENS, UPMC, Campus Jussieu, F-75252 Paris, France
P. Debu
Institut de Recherche sur les lois Fondamentales de l’Univers, CEA-Saclay, F-91191 Gif sur Yvette, France
A. Lambrecht
Laboratoire Kastler-Brossel, CNRS, ENS, UPMC, Campus Jussieu, F-75252 Paris, France
V.V. Nesvizhevsky
Institut Max von Laue - Paul Langevin, 6 rue Jules Horowitz, F-38042, Grenoble, France
S. Reynaud
Laboratoire Kastler-Brossel, CNRS, ENS, UPMC, Campus Jussieu, F-75252 Paris, France
A.Yu. Voronin
P.N. Lebedev Physical Institute, 53 Leninsky prospect, Ru-117924 Moscow, Russia
()
Abstract
GBAR is a project aiming at measuring the free fall acceleration of
gravity for antimatter, namely antihydrogen atoms ($\overline{\mathrm{H}}$). Precision
of this timing experiment depends crucially on the dispersion of
initial vertical velocities of the atoms as well as on the reliable
control of their distribution. We propose to use a new method for
shaping the distribution of vertical velocities of $\overline{\mathrm{H}}$, which
improves these factors simultaneously. The method is based on
quantum reflection of elastically and specularly bouncing $\overline{\mathrm{H}}$ with
small initial vertical velocity on a bottom mirror disk, and
absorption of atoms with large initial vertical velocities on a top
rough disk. We estimate statistical and systematic uncertainties,
and show that the accuracy for measuring the free fall acceleration
$\overline{g}$ of $\overline{\mathrm{H}}$ could be pushed below $10^{-3}$ under realistic
experimental conditions.
Keywords :Antihydrogen, Gravitation, Quantum reflection
PACS : 04.80.Cc, 06.30.Ft, 34.35.+a, 36.10.Gv
1 Introduction
Gravitational properties of antimatter have never been measured
directly. A promising experimental method to do so consists in
producing sufficiently cold antihydrogen atoms ($\overline{\mathrm{H}}$) and timing
their free fall in the Earth’s gravity field. This approach is being
pursued by AEGIS [1], ATHENA-ALPHA [2],
ATRAP [3] and GBAR [4] collaborations.
In order to get the highest accuracy for measuring the free fall
acceleration $\overline{g}$ of $\overline{\mathrm{H}}$, one has to cool atoms down to low
temperatures and to measure, or at least to deduce from design and
calculations, the initial velocity distribution. We discuss here the
method proposed by Walz and Hänsch [5] which is used
in the GBAR project to reach very low temperatures : ${\overline{\mathrm{H}}}^{+}$ ions
are trapped and cooled down to the lowest quantum state in a Paul
trap, and $\overline{\mathrm{H}}$ is then produced by photo-detaching the excess
positron. The photo-detachment pulse is the START signal for the
free fall timing measurement, while the STOP signal is provided by
the annihilation of $\overline{\mathrm{H}}$ atoms on a detection plate placed at a
height $H$ below the center of the ion trap.
Precision of this measurement depends crucially on the dispersion of vertical
velocities before the free fall, which corresponds to the residual kinetic energy of the atoms after the cooling process.
The aim of the present paper is to propose a new filtering method to further reduce
the initial distribution of vertical velocities and
thus improve the accuracy in the measurement of $\overline{g}$.
In section 2 we justify our choice of characteristic
values for the spatial localization of the initial atomic cloud by
considering the spreading of the freely-falling wave-packet of $\overline{\mathrm{H}}$
in the gravitational field. We describe in section 3 the
new method for shaping vertical velocities of $\overline{\mathrm{H}}$ in the
quasi-classical approximation, and show in section 4
that the improvement of accuracy due to the velocity selection
overcomes the degradation associated with the decrease of the
statistics. We then present in section 5 a
quantum-mechanical description of the experiment in order to
validate the quasi-classical estimations of the preceding sections.
In section 6 we list possible systematic effects and
show that they scale down compared to those in the case of
unrestricted free fall of $\overline{\mathrm{H}}$. We then deduce the accuracy which
could be reached on the measurement of $\overline{g}$ under realistic
experimental conditions.
We neglect throughout this paper systematic effects
related to the energy-dependent probability of quantum reflection of
$\overline{\mathrm{H}}$ from the detection plate [6].
The atomic recoil in the photo-detachment process induces
an additional velocity dispersion which is discussed in the last section on systematic effects.
2 Spreading of a freely-falling wave-packet
In the simplified description presented in the introduction, the
initial distribution at time $t=0$ is the lowest quantum state in
the Paul trap. This corresponds to a Gaussian wave-packet with
vertical velocity dispersion $\upsilon$ and vertical position
dispersion $\zeta$ reaching the minimum in the Heisenberg
uncertainty relation:
$$\displaystyle m\upsilon\zeta=\frac{\hbar}{2}$$
(2.1)
where $\hbar$ is the reduced Planck constant and $m$ the inertial
mass of $\overline{\mathrm{H}}$.
After their release from the trap at time $t=0$, atoms start
falling freely in the Earth’s gravity field until they reach the
detection plate placed at a height $H$ below the center of the trap.
The time of fall is measured as the delay $t$ from their release to
their annihilation on the detection plate. The acceleration of
gravity $\overline{g}$ for antihydrogen is then deduced from the distribution
of these fall times. This acceleration $\overline{g}$ for antihydrogen is
related to the analog quantity $g$ defined for hydrogen by
$\overline{g}=Mg/m$, where $M$ is the gravitational mass of $\overline{\mathrm{H}}$.
We now discuss the distribution of free fall times, assuming for
simplicity that this distribution is determined by initial
dispersions of vertical velocity and position (other sources of
uncertainty negligible). If the initial quantum state is poorly
localized (large values of $\zeta$) then the spread of the fall
times is too large because of the initial dispersion of height. In
the opposite case where the wave-packet is too localized (small
values of $\zeta$) then the spread of the fall times is too large
because of initial dispersion of vertical velocity. An optimum
localization of the initial quantum state should be found as a
compromise between these two limiting cases.
As the variations of position and velocity are uncorrelated in the
initial wave-packet, a classical calculation gives the relative
spread $\Delta t$ of the free fall times arising from both effects:
$$\displaystyle\frac{\Delta t}{t_{H}}=$$
$$\displaystyle\sqrt{{\left(\frac{\zeta}{2H}\right)}^{2}+{\left(\frac{\upsilon}{%
v_{H}}\right)}^{2}}$$
(2.2)
$$\displaystyle=$$
$$\displaystyle\sqrt{{\left(\frac{\zeta}{2H}\right)}^{2}+{\left(\frac{\hbar}{2mv%
_{H}\zeta}\right)}^{2}}~{}.$$
(2.3)
The second of these relations uses (2.1) while the
first one is valid even when $\upsilon$ and $\zeta$ do not reach the
minimum in Heisenberg uncertainty relation. We have defined
$t_{H}=\sqrt{2H/\overline{g}}$ and $v_{H}=\sqrt{2\overline{g}H}$ as the free fall time
and velocity for a free fall height $H$ with zero initial velocity.
The optimum size of the initial state, which minimizes $\Delta t$ in
(2.3), is:
$$\displaystyle{\zeta}_{{opt}}=\sqrt{\frac{{\hbar H}}{mv_{H}}}~{}.$$
(2.4)
It leads to an optimum resolution for the free fall measurement:
$$\displaystyle{\left(\frac{\Delta t}{t_{H}}\right)}_{{opt}}=\sqrt{\frac{\hbar}{%
2mv_{H}H}}~{}.$$
(2.5)
The larger the product $mv_{H}H$ with respect to $\hbar/2$, the
better this optimal resolution is.
Better precisions are also obtained by increasing the fall height
with the characteristics of the trap kept fixed. However, the setup
size is limited by practical arguments involving price and space
considerations. Note that equation (2.5) is
translated in an uncertainty twice larger on the acceleration of
gravity
$$\displaystyle\frac{\Delta\overline{g}}{\overline{g}}=2\frac{\Delta t}{t}$$
(2.6)
in the simple derivation presented here (a detailed analysis based
on Monte-Carlo simulations is given in [4]).
With the typical numbers used for the design [4] of the GBAR
experiment ($H=0.3$ m so that $v_{H}\approx 2.4$ m/s if $\overline{g}\approx g$), one obtains $\zeta_{opt}\approx 88$ $\mu$m and $\left(\Delta t/t_{H}\right)_{opt}\approx 2.1{\times}{10}^{-4}$. If this optimum
operation could be experimentally realized, the accuracy would reach
$\left(\Delta\overline{g}/\overline{g}\right)_{opt}\approx 4.2{\times}{10}^{-4}$ for
each detection of an annihilation event. With a total number of
events $N_{\mathrm{tot}}\approx 2.6{\times}{10}^{4}$, calculated for a
typical measuring time of 1 month and an average production rate of
1 ultracold $\overline{\mathrm{H}}$ atoms per period of 100 s, this would lead to the
resolution after one month:
$$\displaystyle\left(\frac{\Delta\overline{g}}{\overline{g}\sqrt{N_{\mathrm{tot}%
}}}\right)_{opt}=\sqrt{\frac{2\hbar}{mv_{H}HN_{\mathrm{tot}}}}\approx 2.6%
\times 10^{-6}~{}.$$
(2.7)
We have assumed there were no large systematic effect.
However, the size of the initial cloud used in the design of the
GBAR experiment is far from this optimum.
The Paul trap is characterized by its oscillation frequency $\omega$ which fixes the velocity and position dispersions in the ground state:
$$\displaystyle\zeta=\sqrt{\hbar/2m\omega}\quad,\quad\upsilon=\sqrt{\hbar\omega/%
2m}~{}.$$
(2.8)
The mean kinetic energy in the ground state is then $m\upsilon^{2}/2=\hbar\omega/4$.
Therefore the range of trap frequencies that can be used is limited by the residual kinetic energy of the atoms after cooling.
In GBAR, the considered frequency range is 0.1 MHz $<\omega/2\pi<$ 1 MHz, so that one gets
0.22 $\mu$m $>\zeta>$ 0.07 $\mu$m and 0.14 m/s $<\upsilon<$ 0.44 m/s. This means that
the initial cloud is smaller than the optimum by about 3 orders of
magnitude. The resolution is thus limited by the dispersion of
initial velocity:
$$\displaystyle\frac{\Delta\overline{g}}{\overline{g}\sqrt{N_{\mathrm{tot}}}}%
\approx\frac{2\upsilon}{v_{H}\sqrt{N_{\mathrm{tot}}}}~{}.$$
(2.9)
As it is not experimentally feasible to further cool down the ions to reach the
optimum size of the initial cloud, we propose in this paper to
select the initial vertical velocity of the atoms.
This will improve
the resolution after each annihilation event by a factor scaling as
the reduced velocity range $\Delta v/\upsilon$. The
statistics is reduced by a factor scaling as $\sqrt{N/N_{\mathrm{tot}}}{\propto}\sqrt{\Delta v/\upsilon}$ (see equation (3.1)) so that an overall
improvement is expected. Also systematic uncertainties will decrease
dramatically. The description of the shaping device and the
evaluation of its performance are discussed in more details in the
next sections.
3 Shaping the distribution of vertical velocities of $\overline{\mathrm{H}}$ in GBAR
The current design for GBAR is a classical free-fall experiment
which aims at an accuracy of the order of 1% [4]. With a
quantum detection technique, one could get significantly higher
precision, in analogy to spectroscopy [7] or
interferometry [8] of near-surface quantum
states [9] of ultracold neutrons (UCNs)
[10, 11]. However these techniques require
high energy resolution and sufficient statistics
[12, 13]. The method that we propose in this
paper is an intermediate step in this direction which is less
precise than the full quantum detection technique but allows for
better statistics and simpler design.
This method is analogous to the one used in the experiment on the
observation of gravitational quantum states of ultra-cold neutrons
[7, 14, 15, 16].
The distribution of initial vertical velocities is shaped by
selecting the atoms passing through a shaping device consisting of
two disks. A scheme of principle of the shaping device where all useful quantities are defined is shown in
figure 1. In the sequel of this section, a simple
analysis of the problem is presented in terms of quasi-classical
arguments, to be confirmed in the next sections. A more complete
quantum-mechanical description is also available in papers devoted
to ultra-cold neutrons
[15, 17, 18, 19, 20].
In the zone between the two disks, atoms with sufficiently small
vertical velocities bounce on the bottom mirror disk due to the high
efficiency of quantum reflection in the Casimir-Polder potential
[6]. If the top surface of the mirror disk is flat,
smooth and horizontal, the horizontal velocity component as well as
the total energy of the vertical motion do not change and atoms thus
pass through the shaping device with high probability. This last
statement would be precisely valid for ideal quantum reflection from
the mirror surface; otherwise corresponding corrections have to be
taken into account (more discussions below). On the other hand,
atoms with large vertical velocities rise in the Earth’s gravity
field to the height of the rough surface of the top disk and scatter
non-specularly on this surface. As this scattering mixes horizontal
and vertical velocity components, it leads to rapid loss of
scattered $\overline{\mathrm{H}}$ through annihilation on the top or bottom disk.
A few remarks are useful at this point: 1) the shaping device has to
be coupled with the Paul trap (not shown on the figure); this point
is not discussed in this paper except for the role of the openings
left in the center of the disks for operating the Paul trap; note
that the disks may consist of several sectors not covering the
complete $2\pi$ horizontal angle in order to include the Paul trap
in the overall design; 2) annihilation events are supposed to be
detected with position-sensitive and time-resolving detectors; this
will allow one to account for the time spent in the shaping device
(see below); 3) due to the cylindrical symmetry of the device, all
atoms with small enough vertical velocity components and any value
and direction of the horizontal velocity component can pass through
it with high probability.
In order to describe the operation of the shaping device, we follow
possible classical trajectories of atoms from the initial point
where they are released to the points where they annihilate. As the
size of the initial spot (discussed in section 2) is
much smaller than any other characteristic size of the shaping
device, it plays no role in the following. We suppose the initial
spot of atoms to be placed at the height $H$ of the top surface of
the bottom disk (origin for altitude placed at the detection plate).
In a first step, we let the radius $r$ of the central opening tend
to zero and the radius $R$ of the disk tend to infinity.
Disregarding the losses due to imperfect quantum reflection on the
mirror disk, we obtain the fraction of atoms going through the
angular acceptance of the shaping device as:
$$\displaystyle\frac{N}{N_{\mathrm{tot}}}\approx\frac{\Delta v}{\upsilon}\sqrt{%
\frac{1}{2\pi}}$$
(3.1)
where $\upsilon$ is the standard deviation of the Gaussian
distribution of vertical velocities and $\Delta v$ the range of
vertical velocities fitting the aperture of the shaping device. With
the geometry sketched in figure 1, the latter corresponds to
atoms with vertical velocities $0<v<\Delta v$ with $\Delta v$
deduced from the energy needed to rise the height from $H$ to $H+h$
in the gravity field:
$$\displaystyle\Delta v=\sqrt{2\overline{g}h}~{}.$$
(3.2)
Note that the fraction of atoms going through the angular acceptance
of the shaping device changes as a function of the height of the
initial spot above the mirror disk as well as a function of the
radius $r$; therefore equation (3.1) has to be modified for
other positions of the spot. Also equation (3.1) has been
written in the limit of a good velocity selection $\Delta v<\upsilon$, which entails through (3.2) that $h$ has a
maximum value $h_{\max}$:
$$\displaystyle h<h_{\max}=\frac{{\upsilon}^{2}}{2\overline{g}}~{}.$$
(3.3)
With the GBAR numbers considered above, the maximum value $h_{\max}$
lies in the interval 1-10 mm. If this condition is not obeyed,
equation (3.1) has to be replaced by the appropriate
integral.
We now take into account the finite values of the radii of the
central openings $r$ and of the disks $R$. In order to do it
properly, we have to consider the shape of the angular distribution
of initial velocities. The operation of the Paul trap may indeed
require anisotropy to be introduced between horizontal and vertical
directions. This can be described by a ratio $\varepsilon$ between
frequencies of operation in horizontal and vertical directions
${\omega}_{\mathrm{hor}}={\varepsilon\omega}$ ($\omega$ is the frequency
already introduced for operation of the vertical trap). This ratio
should be in the interval $2<\varepsilon<4$ for a proper operation
of the Paul trap [21]. Using the same reasoning as in
the preceding section, we deduce that the horizontal dispersions are
$$\displaystyle\upsilon_{\mathrm{hor}}=\upsilon\sqrt{\varepsilon}~{},\quad\zeta_%
{\mathrm{hor}}=\zeta/\sqrt{\varepsilon}~{}.$$
(3.4)
where $\upsilon$ and $\zeta$ are the dispersions already introduced
for operation of the vertical trap.
We can now discuss the role of the finite radius $r$ of the central
openings. We want to avoid extra loss of statistics at the entrance
of the device, and thus choose $r$ small enough so that the angular
divergence there fits the angular acceptance of the shaping device:
$$\frac{h}{r}>\frac{\Delta v}{\upsilon\sqrt{\varepsilon}}~{},\quad r<r_{\max}=%
\frac{\upsilon\sqrt{\varepsilon h}}{\sqrt{2\overline{g}}}=\sqrt{\varepsilon hh%
_{\max}}~{}.$$
(3.5)
To write these relations, we have neglected the effect of gravity on
the short distance $r$ and used the value in (3.4) of
the root-mean-square (rms) dispersion of horizontal velocity.
We then consider the role of the finite radius $R$ of the disk,
using the following classical arguments. We want to produce an
efficient loss of atoms having too large velocities with respect to
the designed angular acceptance of the shaping velocity, and thus
choose $R$ large enough so those atoms efficiently touch the top
disk. Saying that they touch it at least once, this implies that the
time $T$ they spend in the zone between the disks is about two times
larger than the time $t_{h}=\sqrt{2h/\overline{g}}$ corresponding to a free
fall on a height $h$:
$$\displaystyle T=\frac{R}{\upsilon\sqrt{\varepsilon}}>2t_{h}=2\sqrt{\frac{2h}{%
\overline{g}}}$$
$$\displaystyle\quad\Rightarrow\quad R>{R}_{\min}=\frac{4\upsilon\sqrt{{%
\varepsilon h}}}{\sqrt{2\overline{g}}}=4r_{\max}~{}.$$
(3.6)
Again, we have used the dispersion (3.4) of horizontal
velocity to calculate the time $T$ spent in the shaping device for
an atom with the rms velocity. Of course, the time $T$ depends on
the actual horizontal velocity (not its rms value) so that a value
larger than that calculated in (3.6) is required to produce
an effective shaping for the whole distribution.
We want also to stress that the time $T$ appears as a systematical
delay in the free fall timing experiment so that its knowledge is
crucial for accuracy. Here the fact that annihilation event
detectors are position-sensitive is important. Measuring the
horizontal distance $L$ between the initial spot and the detection
point indeed gives the actual horizontal velocity of the atom
$L/T_{\mathrm{tot}}$ with $T_{\mathrm{tot}}$ the time between escape from the trap
and annihilation on the detector and allows one to correct the
timing measurement for the time spent in the shaping device $T=RT_{\mathrm{tot}}/L$.
At the exit of the shaping device, the height lies in the interval
$\left[H,H+h\right]$ while the vertical velocity lies in the
interval $\left[-\Delta v,+\Delta v\right]$. As discussed in the
next section, this affects the resolution of the timing measurement
in the same manner as the dispersion of velocities did affect the
free fall measurement discussed in section 2. In order
to optimize the various parameters, in particular the value of the
radius $R$, we have to simulate the whole experiment, that is the photo-detachment, the
passage through the shaping device, the free fall from its output
slit to the detection plate, the timing of annihilation events, and
the correction from the time spent in the device. In the present
paper, we use simpler arguments to estimate the resulting accuracy
of the measurement.
4 Estimation of statistical uncertainty
At this point, we have all information needed to give a simple
estimation of the statistical accuracy in this experiment. To this
aim, we use the analogy with the free fall timing measurement to
write the relative spread of the free fall times as (compare with
(2.2)):
$$\displaystyle\frac{\Delta t}{t_{H}}=\sqrt{\alpha{\left(\frac{h}{2H}\right)}^{2%
}+\beta{\left(\frac{\Delta v}{v_{H}}\right)}^{2}}$$
(4.1)
$\alpha$ and $\beta$ are dimensionless numbers smaller than unity
describing the shapes of position and velocity distributions at the
output slit of the shaping device. For simplicity, we have supposed
that these distributions are uncorrelated and we have considered
that the correction for the time $T$ spent in the shaper has been
done. As $\Delta v=\sqrt{2\overline{g}h}$ and $v_{H}=\sqrt{2\overline{g}H}$ with
$h{\ll}H$, it follows that the relative spread $\left(\Delta t/t_{H}\right)$ is dominated by the effect of velocity dispersion and
can be written as:
$$\displaystyle\frac{\Delta t}{t_{H}}\approx\sqrt{\frac{{\beta h}}{H}}~{}.$$
(4.2)
This corresponds to an accuracy $\left(\Delta\overline{g}/\overline{g}\right)\approx 2\sqrt{{\beta h}/H}$ for each detection of an annihilation event. We
then obtain the resolution after one month of measurement, taking
into account that the number of events is reduced by the velocity
selection (compare with (3.1)):
$$\displaystyle\frac{\Delta\overline{g}}{\overline{g}\sqrt{N}}=2\sqrt{\frac{%
\beta h}{H}}\sqrt{\frac{\upsilon\sqrt{2\pi}}{N_{\mathrm{tot}}\Delta v}}=2\!%
\left(\frac{{\pi h}{\beta}^{2}{\upsilon}^{2}}{\overline{g}{H}^{2}N_{\mathrm{%
tot}}^{2}}\right)^{1/4}.$$
(4.3)
It is instructive to compare this resolution with the analogous
result obtained without the velocity selection mechanism. The
improvement is described by the ratio of (4.3) to
(2.9):
$$\displaystyle 2\left(\frac{{\pi h}{\beta}^{2}{\upsilon}^{2}}{\overline{g}{H}^{%
2}N_{\mathrm{tot}}^{2}}\right)^{1/4}\left(\frac{2\upsilon}{v_{H}\sqrt{N_{%
\mathrm{tot}}}}\right)^{-1}=\left(\frac{2{\pi h}{\beta}^{2}}{h_{\max}}\right)^%
{1/4}~{}.$$
(4.4)
The best accuracy is therefore achieved for smaller slit sizes. We
take the value $H=0.3$ m chosen in the current design of GBAR, the
worst case of $\beta=1$ and a velocity dispersion
$\upsilon=$ 0.44 m/s and discuss three cases corresponding to
decreasing values of $h$:
1.
Equation (4.4) shows that $h$ should be smaller
than $\approx h_{\max}/2\pi$ for the shaping device to improve the
resolution of the experiment. We choose as an example $h=1$ mm, so
that the statistics is $N\approx 3.3\times 10^{3}$. The opening radius
has to be smaller than $r_{\max}\simeq 3.2\sqrt{\varepsilon}$ mm
and the disk radius should be larger than $R_{\min}\simeq 13\sqrt{\varepsilon}$ mm. The statistical accuracy is then
$\Delta\overline{g}/\overline{g}\sqrt{N}\approx 2.0\times 10^{-3}$. Note that for a
conducting mirror and a maximal vertical velocity $\sqrt{2gh}\approx 0.14$ m/s, the reflection probability for an atom is $78\%$
[6]. To simultaneously improve the resolution and
reduce losses from annihilation on the bottom mirror, we move to
smaller values of $h$.
2.
For $h<50$ $\mu$m, the atom flux through the slit can no longer
be evaluated from classical arguments and the quantum behavior of
$\overline{\mathrm{H}}$ in the slit between the disks has to be taken into account
[7, 14, 15]. At
the boundary $h=50$ $\mu$m, the statistics is $N\approx 7.3\times 10^{2}$ and the statistical accuracy is $\Delta\overline{g}/\overline{g}\sqrt{N}\approx 1.0{\times}{10}^{-3}$. The opening radius
has to be smaller than $r_{\max}\simeq 0.7\sqrt{\varepsilon}$ mm.
Note that the reflection probability for an atom with the maximal
velocity $\sqrt{2gh}=3.1\times 10^{-2}$ m/s is 94% for a perfect
mirror.
3.
For $h<20~{}\mu$m, only atoms in the lowest quantum state
can pass through the slit. The reflection probability approaches
unity in this case which also corresponds to the highest accuracy
for the free fall timing measurement. This quantum limit is analyzed
in sections 5.2 and 5.3.
The first and second cases provide more comfortable conditions for
merging the proposed shaping device and the Paul trap, as well as
better statistics. In this discussion, we have disregarded several
factors which may decrease statistics (annihilation of $\overline{\mathrm{H}}$ in the
bottom disk, non-perfect merging of the angular acceptance of the
optical device and the incoming beam of $\overline{\mathrm{H}}$, quantum reflection
of $\overline{\mathrm{H}}$ from the reference plate, etc.). These factors have to be
evaluated at a later stage.
5 Quantum mechanical description
We now perform a quantum-mechanical description of the experiment,
which will turn out to reproduce the main features and estimations
of the quasi-classical treatment given above.
5.1 Free fall of a wave-packet
We consider the free fall of a pre-formed quantum wave-packet of
$\overline{\mathrm{H}}$ in the Earth’s gravity field, and estimate the accuracy of the
corresponding time-of-fall measurement. We know that the initial
state ${\Psi}_{0}\left(z\right)$ of the wave-packet is a Gaussian
function centered in the vertical direction $z$ around the height
$H$ of the center of the trap, with the vertical position dispersion given by
(2.8):
$$\displaystyle\Psi_{0}(z)=\left(\frac{m\omega}{\hbar\pi}\right)^{1/4}\exp\left(%
-\frac{m\omega}{2\hbar}\left(z-H\right)^{2}\right)~{}.$$
(5.1)
This wave-function is calculated prior to the release, at a time
where the gravity is compensated by the trap. After the
photo-detachment event, the atom is suddenly released and its state
is modified by the free fall in the gravity field.
This evolution is given by the propagation equation:
$$\displaystyle\Psi\left(z,t\right)=\int_{-\infty}^{\infty}G\left(z,z^{\prime},t%
\right)\Psi_{0}\left(z^{\prime}\right)\mathrm{d}z^{\prime}$$
(5.2)
where $t$ is the free fall time and $G$ the propagator:
$$\displaystyle G\left(z,z^{\prime},t\right)=$$
$$\displaystyle\sqrt{\frac{m}{2i\pi\hbar t}}\exp\left[\frac{im}{2\hbar t}\left(z%
-z^{\prime}+\frac{\overline{g}t^{2}}{2}\right)^{2}\right]$$
$$\displaystyle\quad\times\exp\left[\frac{m\overline{g}zt+\frac{1}{6}m\overline{%
g}^{2}t^{3}}{i\hbar}\right]~{}.$$
(5.3)
Integrating (5.2) for the initially Gaussian
wave-packet (5.1), one gets :
$$\displaystyle\Psi\left(z,t\right)$$
$$\displaystyle=\left(\frac{m\omega}{\hbar\pi(1+i\omega t)^{2}}\right)^{1/4}\!\!%
\!\exp\left[\frac{m\overline{g}zt+\frac{1}{6}m\overline{g}^{2}t^{3}}{i\hbar}\right]$$
$$\displaystyle\times\exp\left[-\frac{m\omega}{2\hbar(1+i\omega t)}\left(z-H+%
\frac{\overline{g}t^{2}}{2}\right)^{2}\right]~{}.$$
(5.4)
Assuming that all atoms annihilate instantaneously when they
touch the detection plate at $z=0$, we deduce that the distribution
for annihilation times is given by the flux $\mathcal{F}(t)$ of
atoms passing through the plane at height $z=0$, that is also the
opposite of the current (downward velocities have negative values):
$$\displaystyle\mathcal{F}(t)$$
$$\displaystyle=-j(0,t)=-\frac{\hbar}{m}\text{Im}\left(\Psi(0,t)^{*}\frac{%
\partial}{\partial z}\Psi(0,t)\right)~{},$$
$$\displaystyle=\sqrt{\frac{m\omega^{5}t^{2}}{\hbar\pi(1+\omega^{2}t^{2})^{3}}}%
\left(H+\frac{\overline{g}t^{2}}{2}+\frac{\overline{g}}{\omega^{2}}\right)$$
$$\displaystyle\times\exp\left[-\frac{m\omega}{\hbar(1+\omega^{2}t^{2})}\left(%
\frac{\overline{g}t^{2}}{2}-H\right)^{2}\right]~{}.$$
(5.5)
This probability distribution is shown in figure
2 for an initially Gaussian wavepacket dropped from
30 cm, in the two cases of an initial size typically expected for
the GBAR expected (upper plot) and the optimal size discussed above
(lower plot).
The optimal case (lower plot) leads to an extremely narrow time
distribution, with a peak having the Gaussian shape deduced by
expanding at lowest order in $(t-t_{H})$ the distribution
(5.5) :
$$\displaystyle\mathcal{F}(t)\underset{t\approx t_{H}}{\simeq}C\exp\left[-\frac{%
(t-t_{H})^{2}}{2\Delta t^{2}}\right]~{}.$$
(5.6)
The width $\Delta t$ of the distribution agrees with the classical
result (2.3) :
$$\displaystyle\Delta t=\sqrt{\frac{\hbar(1+\omega^{2}t_{H}^{2})}{2m\omega%
\overline{g}^{2}t_{H}^{2}}}=t_{H}\sqrt{{\left(\frac{\zeta}{2H}\right)}^{2}+{%
\left(\frac{\hbar}{2mv_{H}\zeta}\right)}^{2}}~{}.$$
(5.7)
The upper plot in figure 2, which corresponds to the
typical numbers of the GBAR design, leads to a much broader
distribution and shows a deformed shape with respect to a Gaussian
distribution. As already discussed, this is a consequence of the
large dispersion of initial vertical velocities.
5.2 Gravitational quantum states in the shaping device
We come now to the discussion of the shaping device in the regime
where quantum gravitational states play an important role. The
wave-function of the atoms can thus be developed over the basis of
eigenstates $\Psi_{n}$ with energies $E_{n}$ in the gravity field, here
calculated above a perfectly reflecting mirror [17],
$$\displaystyle\Psi_{n}(z)=\frac{1}{\sqrt{l}}\frac{\text{Ai}(z/l-\lambda_{n})}{%
\text{Ai}^{\prime}(-\lambda_{n})}\quad,\quad E_{n}=mgl\lambda_{n}~{}.$$
(5.8)
The typical scale $l$ of gravitational quantum states is:
$$\displaystyle l=\left(\frac{\hbar^{2}}{2m^{2}\overline{g}}\right)^{1/3}\approx
5%
.9~{}\mu\mathrm{m}~{}.$$
(5.9)
and the quantized energy levels are determined by the zeros of the
Airy function Ai:
$$\displaystyle\text{Ai}(-\lambda_{n})=0$$
(5.10)
$$\displaystyle\lambda_{1}\approx 2.34\;,\;\lambda_{2}\approx 4.09\;,\;\lambda_{%
3}\approx 5.52\;,\;\ldots$$
The high-$n$ states are given by the asymptotic law
$$\displaystyle\lambda_{n}\underset{n\to\infty}{\approx}\left(\frac{3\pi}{2}%
\left(n-\frac{1}{4}\right)\right)^{2/3}$$
(5.11)
Selectivity of the shaping device is based on the sharp dependence
of the transmission of eigenstates $\Psi_{n}$ versus the height $h$ of
the slit. The detailed formalism in [17] leads to a
propagation through the device described by the following
propagator:
$$\displaystyle K(z,z^{\prime},t)=\sum_{n}\Psi_{n}(z)\Psi_{n}(z^{\prime})\exp%
\left[\frac{(E_{n}-i\Gamma_{n})t}{i\hbar}\right]~{}.$$
(5.12)
The width $\Gamma_{n}$ of level $n$ becomes large for high values of
$n$ [17], as explained by the following qualitative
interpretation. When the spatial dispersion $l\lambda_{n}$ of the
state $\Psi_{n}$ is smaller than the slit size $h$, the overlap with
the absorber is small and the atom has a high probability to pass
through the device ($\Gamma_{n}$ small). On the other hand, when
$l\lambda_{n}$ is larger than $h$, the overlap of the wave-function
with the absorber is significant and atoms have a high probability
to be absorbed ($\Gamma_{n}$ large).
As a quantitative illustration, figure 3 shows the
probability of transmission for atoms in the two lowest
gravitational states $\Psi_{1}$ and $\Psi_{2}$ when the length of the
shaping device is $R-r=5$ cm and the roughness amplitude of the top
absorber is 1 $\mu$m. A slit size $h=24$ $\mu$m provides 72%
transmission probability for the first state but only 0.3% for the
second state. This implies that a nearly pure ground state or a
superposition of a few lowest gravitational states can be prepared
by a suitable choice of the parameters of the shaping device.
5.3 Free fall experiment after the velocity shaping
The output of the velocity shaping device is a superposition of
gravitational quantum states $\Psi_{n}$, determined by the propagator
(5.12) calculated for a time $t=(R-r)/v_{\mathrm{hor}}$ for
an atomic horizontal velocity $v_{\mathrm{hor}}$. This shaped superposition
then falls freely to the detection plate so that the time
distribution of annihilation events depends on the properties of the
shaped state. We stress again at this point that this supposes that
the time $R/v_{\mathrm{hor}}$ spent in the shaping device, and before its
entrance, is corrected in the data analysis, $v_{\mathrm{hor}}$ being deduced
from the position of the annihilation event.
The spatial and velocity dispersions of the state $\Psi_{n}$ can be
expressed in terms of $\lambda_{n}$ [22]:
$$\displaystyle\Delta z_{n}=\frac{2l\lambda_{n}}{3\sqrt{5}}$$
$$\displaystyle\Delta v_{n}=\frac{\hbar}{ml}\sqrt{\frac{\lambda_{n}}{3}}$$
(5.13)
In contrast with the case of Gaussian wave-packets discussed above,
these dispersions do not reach the minimum in the Heisenberg
inequality. Furthermore, $\Delta v_{n}$ and $\Delta z_{n}$ increase
simultaneously as functions of $n$. The dispersion of the
annihilation time (after correction of the time spent in the device)
is thus given for the state $\Psi_{n}$ by (2.2) with
$\zeta,\upsilon$ replaced by $\Delta z_{n},\Delta v_{n}$ :
$$\displaystyle\frac{\Delta t}{t_{H}}=\sqrt{\frac{l^{2}\lambda_{n}^{2}}{45H^{2}}%
+\frac{l\lambda_{n}}{3H}}{\approx}\sqrt{\frac{\lambda_{n}l}{3H}}$$
(5.14)
As $l\lambda_{n}\sim h\ll H$, the initial velocity spread still
dominates the uncertainty on the annihilation time. It follows that
the dispersion of these times is determined by $\Delta v_{n}$ and
scales as $\sqrt{\lambda_{n}}$.
In order to get an estimate of the dispersions, we suppose
that the state in the shaper is an incoherent superposition of the
quantum states which fit in the slit. It follows from the arguments
in the preceding section that the quantum states which fit in the
slit correspond to
$$n\leq n_{\max}~{},\quad l\lambda_{n_{\max}}\approx h~{}.$$
(5.15)
We then deduce the dispersion of annihilation times as
$$\frac{\Delta t}{t_{H}}=\sqrt{\sum_{n}\pi_{n}\frac{l\lambda_{n}}{3H}}~{},$$
(5.16)
where $\pi_{n}$ is the population in the state $\Psi_{n}$. As the slit
size is small compared with the incoming wave-function size, we
expect that the states are equally populated among the fitting
gravitational quantum states, so that $\pi_{n}\approx 1/{n_{\max}}$ for
$n\leq n_{\max}$, $\pi_{n}\approx 0$ otherwise. In the quasi-classical
limit where $n_{\max}\gg 1$, we can use the asymptotic expression
(5.11) for $\lambda_{n}$ and replace the sum by an
integral to find:
$$\frac{\Delta t}{t_{H}}\approx\sqrt{\frac{h}{5H}}~{}.$$
(5.17)
This expression scales like the classical result (4.2)
with $\beta$ now specified to be $1/5$.
The preceding argument disregards the coherence between the
components $\Psi_{n}$ in the superposition prepared by the shaping
device. This approximation can be justified qualitatively by
considering that the effects of coherence are washed out in the
averaging associated with free fall propagation as well as
horizontal velocity dispersion. However it cannot be considered as
exact, and it will have to be confirmed by more precise simulations,
to be published in forthcoming papers.
Exact quantum calculations can be performed for the special
case of an initial state for free fall prepared by the shaper as the
ground gravitational state $\Psi_{1}$. The initial velocity
distribution, shown in figure 4, has a width $\Delta v\approx 9.5$ mm/s. This is 30 times larger than the optimal
velocity spread $\upsilon_{opt}\approx 0.36$ mm/s, but two orders of
magnitude smaller than the initial velocity spread in the GBAR
experiment. The exact quantum evolution of this initial wave-packet
is then obtained by integrating the propagation equations
(5.2-5.3). The annihilation time
distribution calculated in this manner is shown in figure
5. Its spread is in excellent agreement with
the prediction $\Delta t=t_{H}\sqrt{{l\lambda_{1}}/{3H}}\simeq 0.97$ ms deduced from (5.14). As a comparison, this
spread was of the order of 45 ms for the free fall measurement
performed without velocity shaping. The improvement reflects the
velocity selection by the shaping device, which is only partly
balanced by the degradation of the statistics (as discussed above).
6 Estimation of systematic effects
For our proposal to be useful as an improved option of the GBAR
measurement, one must ensure that there are no large systematic
uncertainties which could contribute at a level comparable to the
estimated statistical uncertainty of ${\ 10}^{-3}$.
We first examine the additional velocity dispersion caused by the photo-detachment recoil.
As discussed in [23], the vertical velocity dispersion due to the absorption of the photon and the positron emission can be kept small ($\sim 0.5$ m/s) by using a horizontal polarized laser beam with an energy tuned at around $\Delta E\approx 10$ $\mu$eV $\approx 0.1$ cm${}^{-1}$ above threshold. The photo-detachment cross section near threshold follows the Wigner law and can be estimated by using the available information in the literature to be $\sigma=6.8\times 10^{-26}(\Delta E/1\text{ cm}^{-1})^{3/2}\approx 2\times 10^{%
-27}$ m${}^{2}$ [24, 25, 26, 27]. With a $P=1$ W laser beam tuned close the threshold energy $E_{T}=6083$ cm${}^{-1}=0.76$ eV focused on an area $A=10$ $\mu$m $\times 10$ $\mu$m covering the Paul trap center, the photo-detachment rate is $R=\sigma P/AE_{T}=130$ s${}^{-1}$.
In GBAR, antihydrogen ions can be produced only every 110 s, the ejection period of the antiproton decelerator at CERN. This time is sufficient to photo-detach the excess positron with high efficiency. The method is to illuminate the ion during a short enough time so as to define the start time with high precision, at a low enough repetition rate so that in case of successful photo-detachment, the free fall is completed before the next laser shot.
For example, since the free fall time on 30 cm is only 250 ms, laser shots of 100 $\mu$s duration at a repetition rate of 2 Hz during 100 s allows the start time to be known with enough precision ($4\times 10^{-4}$), it also avoids ambiguity on identifying the successful shot, and leads to a photo-detachment efficiency larger than of 90 %.
Since the velocity dispersion induced by the atomic recoil is of the same order as that from the confinement in the Paul trap, one would not gain by trying to get closer to the optimal cloud size. Finally, this effect is equivalent to a slightly warmer antihydrogen cloud, which changes the effective value of the frequency $\omega$ to be used in the calculations, without affecting the principle of the method.
A careful analysis of other systematic effects has to be performed in the
future, in particular for the following list of possibilities:
1)
Uncertainty of shaping/measuring the distribution of vertical
velocity components of $\overline{\mathrm{H}}$ within the range of acceptance of the
two-disk system;
2)
Finite positioning and timing resolution for the detection
of annihilation events;
3)
Accuracy and reliability for the correction for the time spent
in the shaping device;
4)
Diffraction of atoms on the mirror edges;
5)
Residual electromagnetic effects, and in particular patch effect
on mirror surfaces;
6)
Defects of mechanical alignments, such as inclinations of the
disks and detection plate;
7)
Finite precision of production and adjustment of optical elements;
8)
Vibrations able to cause parasitic transitions between gravitational
quantum states.
Monte-Carlo simulations of the experiment are underway; they take into account
photo-detachment, coupling of the shaping device with the Paul trap and detector vessel, as well as points 1) and 2).
For most of these systematic effects, one may also rely on the experience
accumulated in experiments with UCNs
[7, 9, 14, 15].
We note that the main systematic uncertainties (in particular 1) are
proportional to the ratio $h/H$, and thus decrease strongly when
slit heights are decreased. We therefore think that the control of
these systematic effects will be improved at small slit heights.
7 Conclusion
In this paper, we have proposed a new method for shaping vertical
velocities of antihydrogen atoms in the timing experiment to be
performed by the GBAR collaboration [4]. We have given first
estimations of the corresponding statistic uncertainties and listed
possible systematic effects. The conclusion of these preliminary
estimations, to be confirmed by further analysis, is that the
accuracy in the measurement of the free fall acceleration $\overline{g}$ of
$\overline{\mathrm{H}}$ atoms could be pushed below 10${}^{-3}$ in realistic
experimental conditions.
Statistical uncertainties in the experiment are improved for smaller
slit heights, which lead to better defined vertical velocities of
$\overline{\mathrm{H}}$. This means that a better selection of the range of vertical
velocities overweighs the loss in statistics. Systematical
uncertainties are expected to decrease even more dramatically for
smaller heights of the slit between the two disks in the proposed
experimental design. In the optimum experiment where atomic
wave-packet is shaped to the lowest quantum state, the effective
temperature corresponding to the vertical motion of $\overline{\mathrm{H}}$ is as low
as 10 nK.
These preliminary estimations have to be confirmed by more
complete simulations. We are currently working to develop a fully
quantum treatment of the shaping device as well as a complete
Monte-Carlo simulations.
Let us also mention that an even better accuracy could in principle
be obtained by studying interference effects in the time-of-arrival
distribution of a coherent superposition of a few lowest-lying
gravitational quantum states [12, 13].
Acknowledgements
The authors thank the ESF Research Networking Programme CASIMIR
(casimir-network.org), the GRANIT collaboration and the GBAR collaboration (gbar.in2p3.fr) for
providing excellent possibilities for discussions and exchange.
References
[1]
A. Kellerbauer, M. Amoretti, A.S. Belov, G. Bonomi, I. Boscolo, R.S. Brusa,
M. Büchner, V.M. Byakov, L. Cabaret, C. Canali, C. Carraro,
F. Castelli, S. Cialdi, M. de Combarieu, D. Comparat, G. Consolati,
N. Djourelov, M. Doser, G. Drobychev, A. Dupasquier, G. Ferrari, P. Forget,
L. Formaro, A. Gervasini, M.G. Giammarchi, S.N. Gninenko, G. Gribakin,
S.D. Hogan, M. Jacquey, V. Lagomarsino, G. Manuzio, S. Mariazzi, V.A.
Matveev, J.O. Meier, F. Merkt, P. Nedelec, M.K. Oberthaler, P. Pari,
M. Prevedelli, F. Quasso, A. Rotondi, D. Sillou, S.V. Stepanov, H.H.
Stroke, G. Testera, G.M. Tino, G. Trénec, A. Vairo, J. Vigué,
H. Walters, U. Warring, S. Zavatarelli, and D.S. Zvezhinskij.
Proposed antimatter gravity measurement with an antihydrogen beam.
Nuclear Instruments and Methods in Physics Research Section B:
Beam Interactions with Materials and Atoms, 266(3):351–356, February 2008.
[2]
The ALPHA Collaboration and A. E. Charman.
Description and first application of a new technique to measure the
gravitational mass of antihydrogen.
Nature Communications, 4:1785, April 2013.
[3]
G Gabrielse.
The production and study of cold antihydrogen.
Technical Report CERN-SPSC-2010-006.SPSC-SR-057, CERN, Geneva, Jan
2010.
[4]
G Chardin, P Grandemange, D Lunney, V Manea, A Badertscher, P Crivelli,
A Curioni, A Marchionni, B Rossi, A Rubbia, V Nesvizhevsky, P-A Hervieux,
G Manfredi, P Comini, P Debu, P Dupré, L Liszkay, B Mansoulié,
P Pérez, J-M Rey, N Ruiz, Y Sacquin, A Voronin, F Biraben, P Cladé,
A Douillet, A Gérardin, S Guellati, L Hilico, P Indelicato, A Lambrecht,
R Guérout, J-P Karr, F Nez, S Reynaud, V-Q Tran, A Mohri, Y Yamazaki,
M Charlton, S Eriksson, N Madsen, D-P van der Werf, N Kuroda, H Torii, and
Y Nagashima.
Proposal to measure the gravitational behaviour of antihydrogen at
rest.
Technical Report CERN-SPSC-2011-029.SPSC-P-342, CERN, Geneva, Sep
2011.
[5]
Jochen Walz and Theodor W. Hänsch.
A proposal to measure antimatter gravity using ultracold antihydrogen
atoms.
General Relativity and Gravitation, 36(3):561–570, March 2004.
[6]
G. Dufour, A. Gérardin, R. Guérout, A. Lambrecht, V. V. Nesvizhevsky,
S. Reynaud, and A. Yu. Voronin.
Quantum reflection of antihydrogen from the casimir potential above
matter slabs.
Physical Review A, 87(1):012901, January 2013.
[7]
Valery V. Nesvizhevsky, Hans G. Börner, Alexander K. Petukhov, Hartmut
Abele, Stefan Baeßler, Frank J. Rueß, Thilo Stöferle, Alexander
Westphal, Alexei M. Gagarski, Guennady A. Petrov, and Alexander V. Strelkov.
Quantum states of neutrons in the earth’s gravitational field.
Nature, 415(6869):297–299, January 2002.
[8]
Valery V. Nesvizhevsky, Alexei Yu Voronin, Robert Cubitt, and Konstantin V.
Protasov.
Neutron whispering gallery.
Nature Physics, 6(2):114–117, February 2010.
[9]
Valerii V. Nesvizhevsky.
Near-surface quantum states of neutrons in the gravitational and
centrifugal potentials.
Physics-Uspekhi, 53(7):645, October 2010.
[10]
Vladimir Kazimirovich Ignatovich.
The physics of ultracold neutrons.
Oxford series on neutron scattering in condensed matter ; Oxford
science publications , Oxford University Press. Clarendon Press, Oxford,
Royaume-Uni, 1990.
[11]
Robert Golub, David J Richardson, and Lamoreaux .
Ultra-cold neutrons.
Adam Hilger, Bristol; Philadelphia, 1991.
[12]
A. Yu Voronin, P. Froelich, and V. V. Nesvizhevsky.
Gravitational quantum states of antihydrogen.
Physical Review A, 83(3):032903, 2011.
[13]
A. Yu Voronin, V. V. Nesvizhevsky, and S. Reynaud.
Interference of the whispering gallery states of antihydrogen.
J. Physics B, 45:165007, 2012.
[14]
V. V. Nesvizhevsky, H. G. Börner, A. M. Gagarski, A. K. Petoukhov, G. A.
Petrov, H. Abele, S. Baeßler, G. Divkovic, F. J. Rueß, Th.
Stöferle, A. Westphal, A. V. Strelkov, K. V. Protasov, and A. Yu.
Voronin.
Measurement of quantum states of neutrons in the
earth’s gravitational field.
Physical Review D, 67(10):102002, May 2003.
[15]
V. V. Nesvizhevsky, A. K. Petukhov, H. G. Borner, T. A. Baranova, A. M.
Gagarski, G. A. Petrov, K. V. Protasov, A. Y. Voronin, S. Baessler, H. Abele,
A. Westphal, and L. Lucovac.
Study of the neutron quantum states in the gravity field.
European Physical Journal C, 40(4):479–491, April 2005.
[16]
V.V Nesvizhevsky, H Börner, A.M Gagarski, G.A Petrov, A.K Petukhov,
H Abele, S Bäßler, T Stöferle, and S.M Soloviev.
Search for quantum states of the neutron in a gravitational field:
gravitational levels.
Nuclear Instruments and Methods in Physics Research Section A:
Accelerators, Spectrometers, Detectors and Associated Equipment,
440(3):754–759, February 2000.
[17]
A. Yu. Voronin, H. Abele, S. Baeßler, V. V. Nesvizhevsky, A. K. Petukhov,
K. V. Protasov, and A. Westphal.
Quantum motion of a neutron in a waveguide in the gravitational
field.
Physical Review D, 73(4):044029, February 2006.
[18]
A. E. Meyerovich and V. V. Nesvizhevsky.
Gravitational quantum states of neutrons in a rough waveguide.
Physical Review A, 73(6):063616, June 2006.
[19]
R. Adhikari, Y. Cheng, A. E. Meyerovich, and V. V. Nesvizhevsky.
Quantum size effect and biased diffusion of gravitationally bound
neutrons in a rough waveguide.
Physical Review A, 75(6):063613, June 2007.
[20]
M. Escobar and A. E. Meyerovich.
Beams of gravitationally bound ultracold neutrons in rough
waveguides.
Physical Review A, 83(3):033618, March 2011.
[21]
A. Walther, F. Ziesel, T. Ruster, S. T. Dawkins, K. Ott, M. Hettrich,
K. Singer, F. Schmidt-Kaler, and U. Poschinger.
Controlling fast transport of cold trapped ions.
Physical Review Letters, 109(8):080501, August 2012.
[22]
R. W. Robinett.
The stark effect in linear potentials.
European Journal of Physics, 31(1):1, January 2010.
[23]
Pascal Debu.
GBAR.
Hyperfine Interactions, 212(1-3):51–59, December 2012.
[24]
Oliver Harms, Michael Zehnpfennig, Victor Gomer, and Dieter Meschede.
Photodetachment spectroscopy of stored ions.
Journal of Physics B: Atomic, Molecular and Optical Physics,
30(17):3781, September 1997.
[25]
John T. Broad and William P. Reinhardt.
One- and two-electron photoejection from H-: A
multichannel j-matrix calculation.
Physical Review A, 14(6):2159–2173, December 1976.
[26]
K. R. Lykke, K. K. Murray, and W. C. Lineberger.
Threshold photodetachment of H-.
Physical Review A, 43(11):6104–6107, June 1991.
[27]
The formula given is a result of a compilation made by C. Blondel, private
communication (2012). |
Generation of monocycle squeezed light in chirped quasi-phase-matched nonlinear crystals
D. B. Horoshko
Dmitri.Horoshko@univ-lille1.fr
Univ. Lille, CNRS, UMR 8523 - PhLAM - Physique des Lasers Atomes et Molécules, F-59000 Lille, France
B. I. Stepanov Institute of Physics, NASB, Nezavisimosti Ave. 68, Minsk 220072 Belarus
M. I. Kolobov
Univ. Lille, CNRS, UMR 8523 - PhLAM - Physique des Lasers Atomes et Molécules, F-59000 Lille, France
(December 2, 2020)
Abstract
We present a quantum theory of parametric down-conversion of light in chirped quasi-phase-matched second-order nonlinear crystals with undepleted quasi-monochromatic pump. This theory allows us to consider generation of ultrabroadband squeezed states of light and is valid for arbitrary, sufficiently slowly-varying nonlinear poling profiles. Using a first-order approximate quantum solution for the down-converted light field, we calculate the squeezing spectra and the characteristic squeezing angles. We compare the approximate solutions with the exact and numerical ones and find a very good agreement. This comparison validates our approximate solution in the regime of moderate gain, where the existing approaches are not applicable. Our results demonstrate that aperiodically poled crystals are very good candidates for generating ultrabroadband squeezed light with the squeezing bandwidth covering almost all the optical spectrum and the correlation time approaching a single optical cycle.
pacs: 42.50.Dv, 42.65.Lm
I Introduction
Squeezed light is a non-classical electromagnetic field at optical frequency with the fluctuations of one quadrature component below the level of the vacuum fluctuations within certain frequency bandwidth. Squeezed light is one of the central objects of study in modern quantum optics, being, on the one hand, a macroscopic object with substantially quantum properties, and, on the other hand, a valuable resource for metrology, quantum communication and quantum information processing Kilin01 . Both the degree of squeezing and the squeezing bandwidth are important for potential applications of squeezed light. To date, successful generation has been reported of continuous-wave optical beams with 15 dB squeezing in a band of about 100 MHz Vahlbruch16 and 2 dB in a band of 1.2 GHz Ast13 . Experiments with pulsed light reach the bandwidth of several THz Spalter98 ; Wenger04 ; Agafonov10 ; Pinel12 and even tens of THz Iskhakov09 .
In our recent paper Horoshko13 we gave a theoretical description of a method allowing for generation of squeezed light with the squeezing bandwidth comprising the whole optical spectrum, i.e., hundreds of THz. After a proper compensation of the phase, such light would demonstrate a monocycle two-mode squeezing with the sideband-frequency quadrature components quantum-correlated at the time scale of a single optical period. The proposed method is based on parametric down-conversion (PDC) of light in an aperiodically poled nonlinear crystal with quasi-phase-matching (QPM) in a broad band of frequencies, resulting from a linear chirp of the spatial frequency of the poling. Such crystals are widely used for parametric amplification of ultrashort pulses of light Baker03 ; Fejer05 ; Fejer08a ; Fejer08b ; Heese10 ; Phillips14 and also for generation of photon pairs with a correlation time of the order of one optical cycle Harris07 ; Nasr08 ; Brida09 ; Sensarn10 ; Tanaka12 .
In the present article we develop a general quantum theory of generation of ultrabroadband squeezed light by PDC of light in aperiodically poled crystals. Our purpose is twofold. First, we provide a detailed description for the analytic solution of the case of linear chirp, presented in Ref. Horoshko13 , where many details were omitted. Second, we present an approximate solution for the quantum field in a QPM nonlinear crystal, which is important for qualitative understanding of the underlying physical processes and for crystal design in practical applications. Our theory is valid for arbitrary nonlinear poling profiles, which should be sufficiently slowly varying, and for both low and high parametric gains. We compare the approximate analytical solution with exact and numerical ones for linear and quadratic-hyperbolic QPM poling profiles and find a very good agreement within the amplification band.
Our approach is conceptually close to the classical description of optical parametric amplification in QPM media developed in Refs. Fejer08a ; Fejer08b . We use a similar perturbation approach for obtaining an approximate solution of the wave equation for the slowly-varying field amplitudes. We restrict our consideration to the first-order approximation; however our results can be easily generalized to the second-order solution. The main difference between our approach and that of Ref. Fejer08a is that our solution is for the slowly-varying Heisenberg field operators and, therefore, can be applied to arbitrary quantum states of light such as squeezed or entangled states. The solution of Ref. Fejer08a is for the classical slowly-varying field amplitudes and is not suitable for evaluation of the squeezing spectra and squeezing angles of the ultrabroadband squeezed light, which is the main objective of our work.
It should be understood also that the classical and the quantum theories of PDC in aperiodically poled crystals are oriented at different values of the parametric gain and put different meaning into the term “high-gain regime”. For the classical theory of parametric amplification the gain is “high” if it provides a practically important increase of the signal peak power, above 10 dB, sometimes even above 60 dB Jovanovic05 . In the quantum theory of PDC the “low-gain regime” corresponds to spontaneous emission of the downconverted photon pairs, so-called biphotons, while the “high-gain regime” corresponds to stimulated emission of photons, when the mean photon number per mode well surpasses unity. The latter regime is characterized by squeezing of one field quadrature and can be observed at the values of the power gain, which are not practical for the pulse amplification. Indeed, for the power gain $G$ the variance of the squeezed quadrature is reduced $[G^{\frac{1}{2}}+(G-1)^{\frac{1}{2}}]^{2}$ times below the vacuum level. Thus, the widely available values of squeezing from 3 to 12 dB correspond to the power gain from 0.5 to 7 dB, which is of relatively little interest for the purpose of amplification of light pulses. We note in this connection, that our quantum solution gives an adequate description of the field evolution in the high (above 0.5 dB) and very high (above 10 dB) gain regimes, and in the latter case is in good agreement with the classical formulas obtained in Ref. Fejer08a .
The article is organized as follows. In Sec. II we derive a differential equation for the slowly-varying Heisenberg field operators of the electromagnetic field in an aperiodically poled nonlinear crystal. This equation is solved exactly for a linear poling profile in Sec. III and approximately for an arbitrary sufficiently slowly varying poling profile in Sec. IV. An example of a crystal with more than octave-wide QPM is considered in Sec. V, where we compare the exact analytical solution for a linear poling profile with the approximate one. In the same section a similar comparison is presented for numerical and approximate analytical solutions for a nonlinear, quadratic-hyperbolic poling profile. Here we discuss also the limits of applicability of our analytical approximation. In Sec. VI we summarize the results and discuss their importance for the experiments with ultrabroadband squeezed light.
II Parametric down-conversion in an aperiodically poled nonlinear crystal
II.1 Differential equation for the field
We consider the process of collinear PDC in a nonlinear crystal, where after annihilation of one photon of the pump wave with the frequency $\omega_{\mathrm{p}}$ two photons are created with the same polarization and frequencies $\omega_{\mathrm{0}}+\Omega$ and $\omega_{\mathrm{0}}-\Omega$, where $\omega_{\mathrm{0}}=\omega_{\mathrm{p}}/2$. The phase mismatch for this process has the form $\Delta(\Omega)=k_{\mathrm{p}}-k(\Omega)-k(-\Omega)$, where $k_{\mathrm{p}}$ is the wave vector of the pump wave, accepted to be an undepleted monochromatic plane wave, and $k(\Omega)$ is the wave vector of the down-converted wave at the frequency $\omega_{\mathrm{0}}+\Omega$. In general there is no phase matching at degeneracy, $k_{p}\neq 2k_{0}$, where $k_{0}=k(0)$. Let us direct the $z$ axis along the propagation of the waves, placing the origin on the front edge of the crystal. For the description of the field we use two operators: the photon annihilation operator at the frequency $\omega_{\mathrm{0}}+\Omega$ and position $z$, which we denote $b\left(\Omega,z\right)$, and the sideband photon annihilation operator at detuning $\Omega$, which is given by $a(\Omega,z)=b(\Omega,z)e^{i\left(k(\Omega)-k_{0}\right)z}$. The field operator $E^{(+)}(t,z)$ of the down-converted light is expressed (in photon flux units) through these operators as follows
$$\displaystyle E^{(+)}(t,z)$$
$$\displaystyle=$$
$$\displaystyle\int a(\Omega,z)e^{i\left(k_{0}z-(\omega_{0}+\Omega)t\right)}d\Omega$$
$$\displaystyle=$$
$$\displaystyle\int b(\Omega,z)e^{i\left(k(\Omega)z-(\omega_{0}+\Omega)t\right)}%
d\Omega.$$
The operator $b\left(\Omega,z\right)$ corresponds to the modal function, which is a solution of the wave equation in the absence of nonlinear interaction; therefore in its presence $b\left(\Omega,z\right)$ is the slowly-varying amplitude. In terms of this operator the equation for the down-converted waves at frequencies $\omega_{\mathrm{0}}+\Omega$ and $\omega_{\mathrm{0}}-\Omega$ takes the well-known form Bloembergen65 ; Armstrong62 ,
$$\frac{\partial b(\Omega,z)}{\partial z}=\chi^{\left(2\right)}b_{p}b^{\dagger}(%
-\Omega,z)e^{i\Delta(\Omega)z},$$
(2)
where $\chi^{\mathrm{(2)}}$ is the appropriately scaled element of the nonlinear susceptibility tensor of the second order, responsible for the nonlinear interaction, while $b_{\mathrm{p}}$ is the pump-wave amplitude in units of photon flux. An effective interaction of the tree waves is possible only for such frequencies $\Omega$ where the phase-matching condition $\Delta(\Omega)\approx 0$ is approximately satisfied. Usually, in an experiment the phase matching is realized for a narrow frequency band by selecting an angle of propagation with respect to the optical axis of the crystal Bloembergen65 , birefringence being taken into consideration.
If reaching the phase matching is impossible at the desired frequency $\omega_{\mathrm{0}}+\Omega$, one can apply the method of QPM, which consists of the following Armstrong62 . An artificial periodic layered structure is produced out of the original crystal, where the width of each layer is $\Lambda/2$, and each subsequent layer is different from the previous one by inversion of the crystal structure. As a result of such an inversion the second-order nonlinear susceptibility tensor changes its sign, though the linear properties of the crystal remain unchanged. The spatial modulation of the second-order nonlinear susceptibility in such a layered structure has the form of a meander
$$\chi^{(2)}(z)=\chi_{0}\operatorname{sgn}\left(\sin{Kz}\right)=\frac{-i\chi_{0}%
}{\pi}\sum\limits_{n=-\infty}^{+\infty}{\frac{1-\left(-1\right)^{n}}{n}e^{inKz%
}},$$
(3)
where $K=2\pi/\Lambda$ is the spatial frequency of the created grating, $\chi_{\mathrm{0}}$ is the second-order nonlinear susceptibility of the first layer, and the Fourier series decomposition of the meander function has been used, containing only odd values of $n$ (the term with $n=0$ is implied to be zero). Quasi-phase-matching of the first order for frequencies $\omega_{\mathrm{0}}+\Omega$ and $\omega_{\mathrm{0}}-\Omega$ consists of choosing the grating vector such that $K=\Delta(\Omega)$. In this case the additional phase factor, corresponding to $n=-1$, will compensate the phase mismatch at the desired frequency, when Eq. (3) is substituted into Eq. (2). All other terms in Eq. (3) can be disregarded under typical conditions Powers11 .
In practice such periodically oriented crystals are created by a number of different methods Powers11 ; Houe95 . The most widely used of them is the method based on the property of a ferroelectric crystal to change its crystal structure under the action of an external electric field and then to maintain this structure when the external field is removed. Applying a spatially-periodic constant electric field to a ferroelectric with a significant second-order nonlinear susceptibility, such as lithium niobate, allows one to create artificial structures with QPM for practically any combination of wavelengths in various nonlinear optical processes. Such crystals are generally known as periodically poled and represent today a versatile tool in nonlinear optics.
In the past decades much interest has been concentrated on the development of the above-described method, based on a slow change of the spatial frequency $K(z)$ along the crystal, allowing one to reach QPM at different frequencies in different parts of the crystal (Fig. 1). Such crystals received the name of aperiodically poled crystals and are widely used for parametric amplification of ultrashort optical pulses Baker03 ; Fejer05 ; Fejer08a ; Fejer08b ; Heese10 ; Phillips14 and generation of broadband entangled photon pairs Harris07 ; Nasr08 ; Brida09 ; Sensarn10 ; Tanaka12 .
When the spatial frequency modulation is weak, the local period of the grating, $\Lambda(z)=2\pi/K(z)$, is a slowly varying function of the coordinate $z$ and under the condition $|\Lambda^{\prime}(z)|\ll 1$ Eq. (3) can be rewritten as
$$\displaystyle\chi^{(2)}(z)$$
$$\displaystyle=$$
$$\displaystyle\chi_{0}\operatorname{sgn}\left(\sin\int_{0}^{z}{K(z^{\prime})dz^%
{\prime}}\right)$$
$$\displaystyle\approx$$
$$\displaystyle\frac{-i\chi_{0}}{\pi}\sum\limits_{n=-\infty}^{+\infty}{\frac{1-%
\left(-1\right)^{n}}{n}e^{in\int_{0}^{z}{K\left(z^{\prime}\right)dz^{\prime}}}}.$$
Leaving only the term with $n=-1$ and substituting Eq. (II.1) into Eq.(2), we obtain
$$\displaystyle\frac{\partial b(\Omega,z)}{\partial z}$$
$$\displaystyle=$$
$$\displaystyle i\gamma b^{\dagger}(-\Omega,z)e^{i\Delta(\Omega)z-i\int_{0}^{z}{%
K(z^{\prime})dz^{\prime}}},$$
(5)
$$\displaystyle\frac{\partial b^{\dagger}(-\Omega,z)}{\partial z}$$
$$\displaystyle=$$
$$\displaystyle-i\gamma^{\ast}b(\Omega,z)e^{-i\Delta(\Omega)z+i\int_{0}^{z}{K(z^%
{\prime})dz^{\prime}}},$$
where $\gamma=2\chi_{\mathrm{0}}b_{\mathrm{p}}/\pi$ is the coefficient of nonlinear coupling, and the second equation is obtained from the first one by a Hermitian conjugation and a sign inversion for $\Omega$. We note, that the function $\Delta(\Omega)$ is even by definition for the considered case of type-I phase matching. Equations (5) represent a closed system, having a unique solution for given boundary conditions. For finding this solution we introduce a new field operator $\tilde{b}(\Omega,z)$ by the following relation:
$$\displaystyle b(\Omega,z)=\tilde{b}(\Omega,z)e^{\frac{i}{2}\left(\Delta(\Omega%
)z-\int_{0}^{z}{K(z)dz}+\varphi_{0}\right)},$$
(6)
where $\varphi_{0}=\arg(i\gamma)$ combines the phases of the pump wave and $\chi_{\mathrm{0}}$. Now the system of Eqs. (5) takes the form
$$\displaystyle\frac{\partial\tilde{b}(\Omega,z)}{\partial z}$$
$$\displaystyle+$$
$$\displaystyle\frac{i}{2}\left(\Delta(\Omega)-K(z)\right)\tilde{b}(\Omega,z)$$
$$\displaystyle=$$
$$\displaystyle|\gamma|\tilde{b}^{\dagger}(-\Omega,z)$$
$$\displaystyle\frac{\partial\tilde{b}^{\dagger}(-\Omega,z)}{\partial z}$$
$$\displaystyle-$$
$$\displaystyle\frac{i}{2}\left(\Delta(\Omega)-K(z)\right)\tilde{b}^{\dagger}(-%
\Omega,z)$$
$$\displaystyle=$$
$$\displaystyle|\gamma|\tilde{b}(\Omega,z).$$
Solution of this system with the boundary conditions
$\tilde{b}(\Omega,0)$, $\tilde{b}^{\dagger}(-\Omega,0)$ will give a
transformation of the field operators in the nonlinear crystal. Practical
interest is represented by their values at the output of the crystal, at
the point $z=L$, where $L$ is the length of the crystal, i.e.,
$\tilde{b}(\Omega,L)$ and $\tilde{b}(-\Omega,L)$.
Excluding the operator $\tilde{b}^{\dagger}(-\Omega,z)$ from the system of Eqs. (II.1), we obtain one equation of second order:
$$\displaystyle\frac{\partial^{2}\tilde{b}(\Omega,z)}{{\partial z}^{2}}$$
$$\displaystyle+$$
$$\displaystyle\left(\frac{1}{4}\left(\Delta(\Omega)-K(z)\right)^{2}-|\gamma|^{2%
}\right.$$
$$\displaystyle-$$
$$\displaystyle\left.\frac{i}{2}K^{\prime}(z)\right)\tilde{b}(\Omega,z)=0.$$
In the next section we discuss the general structure of the solution of this equation.
II.2 The general structure of solution
The system of Eqs. (II.1) with the boundary conditions at $z=0$ has a unique solution in the form of a Bogoliubov transformation for the field operators:
$$\tilde{b}(\Omega,L)=A(\Omega)\tilde{b}(\Omega,0)+B(\Omega)\tilde{b}^{\dagger}(%
-\Omega,0),$$
(9)
where $A(\Omega)$ and $B(\Omega)$ are some complex functions. Note, that the frequency detuning enters Eqs. (II.1) only through $\Delta(\Omega)$ which is an even function. Therefore, the functions $A(\Omega)$ and $B(\Omega)$ are also even. Equation (9) can be rewritten in terms of the sideband photons annihilation and creation operators as
$$a(\Omega,L)=U(\Omega)a(\Omega,0)+V(\Omega)a^{\dagger}(-\Omega,0),$$
(10)
where
$$\displaystyle U(\Omega)$$
$$\displaystyle=$$
$$\displaystyle A(\Omega)e^{i[k(\Omega)-k_{0}+\frac{1}{2}\Delta(\Omega)]L-\frac{%
i}{2}\int_{0}^{L}{K(z)dz}},$$
(11)
$$\displaystyle V(\Omega)$$
$$\displaystyle=$$
$$\displaystyle B(\Omega)e^{i[k(\Omega)-k_{0}+\frac{1}{2}\Delta(\Omega)]L-\frac{%
i}{2}\int_{0}^{L}{K(z)dz}+i\varphi_{0}},$$
and these functions are not even in general because of their dependence on $k(\Omega)$.
The transformation (10) at a frequency where $V(\Omega)\neq 0$ corresponds to generation of a two-mode squeezed field state Kolobov99 . As any Bogoliubov transformation, it is fully characterized by four real parameters. Indeed, Eq. (10) together with its Hermitian conjugate with opposite detuning represents a closed linear transformation for a pair of operators $\{a(\Omega,z),a^{\dagger}(-\Omega,z)\}$ from $z=0$ to $z=L$. This transformation for a fixed $\Omega$ is fully characterized by four complex numbers $U(\pm\Omega)$, $V(\pm\Omega)$. Unitarity of the Bogoliubov transformation imposes four real conditions $|U(\pm\Omega)|^{2}-|V(\pm\Omega)|^{2}=1$, and $U(\Omega)/V(\Omega)=U(-\Omega)/V(-\Omega)$ (note that the latter complex equation is equivalent to two real conditions), so only four real parameters remain. They can be defined as one squeezing parameter and three characteristic angles Kolobov99 by the following expressions:
$$\displaystyle r(\Omega)$$
$$\displaystyle=$$
$$\displaystyle\ln\left(|U(\Omega)|+|V(\Omega)|\right),$$
(12)
$$\displaystyle\psi_{L}(\Omega)$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{2}\arg\left[U(\Omega)V(-\Omega)\right],$$
(13)
$$\displaystyle\psi_{0}(\Omega)$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{2}\arg\left[U^{-1}(\Omega)V(\Omega)\right],$$
(14)
$$\displaystyle\kappa(\Omega)$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{2}\arg\left[U(\Omega)U^{-1}(-\Omega)\right],$$
(15)
where the first three parameters are even functions of $\Omega$, while the fourth one is odd.
The physical meaning of these parameters becomes clear from the definition of the squeezed quadrature. For each pair of modes with opposite detunings we construct two quadrature operators as Kolobov99
$$\displaystyle X_{1}(\Omega,z)$$
$$\displaystyle=$$
$$\displaystyle a(\Omega,z)e^{-i\psi_{z}(\Omega)}+a^{\dagger}(-\Omega,z)e^{i\psi%
_{z}(\Omega)},$$
(16)
$$\displaystyle X_{2}(\Omega,z)$$
$$\displaystyle=$$
$$\displaystyle-i\left[a(\Omega,z)e^{-i\psi_{z}(\Omega)}-a^{\dagger}(-\Omega,z)e%
^{i\psi_{z}(\Omega)}\right],$$
where for the aims of the present discussion $z$ is equal only to zero and $L$. In terms of these quadratures the transformation Eq. (10) can be rewritten in a simple form,
$$X_{j}(\Omega,L)=e^{\pm r(\Omega)+i\kappa(\Omega)}X_{j}(\Omega,0),$$
(17)
where the upper (lower) sign corresponds to $j=1$ ($j=2$). It follows from Eq. (17) that the quadrature $X_{2}(\Omega,L)$ is squeezed below the standard quantum limit, while the conjugate quadrature $X_{1}(\Omega,L)$ is stretched above that limit. The squeezing parameter $r(\Omega)$ determines the degree of this effect, while the angle of squeezing, $\psi_{L}(\Omega)$, determines the quadrature at which the squeezing is to be observed at the output of the nonlinear crystal.
The angle $\psi_{0}(\Omega)$ determines the quadrature at the input, which is subject to the squeezing operation. For an unseeded PDC this angle is irrelevant, since all quadratures of the input field are in the vacuum state. However, for a seeded PDC this angle is to be taken into account, as discussed in Sec. IV.5.
The last parameter $\kappa(\Omega)$ in our case of even $A(\Omega)$ and $B(\Omega)$ is independent of the nonlinear properties of the crystal and is given by
$$\kappa(\Omega)=\frac{1}{2}\left[(k(\Omega)-k(-\Omega)\right]L\approx\tau_{g}\Omega,$$
(18)
where $\tau_{g}=k^{\prime}(0)L$ is the time of light propagation through the crystal at the group velocity of the central wavelength of the downconverted light.
Below we are interested in finding the functions $r(\Omega)$, $\psi_{L}(\Omega)$, and $\psi_{0}(\Omega)$, characterizing the nonlinear transformation of the field in QPM crystals. In the next section we present an exact solution for a linear poling profile, while in Sec. IV we discuss in detail an approximate solution for a sufficiently slowly varying, but otherwise arbitrary, poling profile $K(z)$.
III Exact solution for a linear poling profile
In this section we present an exact solution of the system of Eqs. (II.1) in the case of linear chirp of the grating vector $K(z)=K_{0}-\zeta z$, where $\zeta>0$ is the chirp rate.
For a fixed $\Omega$ we introduce a new variable $x=\sqrt{\zeta}z+\left(\Delta(\Omega)-K_{0}\right)/\sqrt{\zeta}$. In the variables $(\Omega,x)$, Eqs. (II.1) for a linear chirp take the following form:
$$\displaystyle\frac{\partial\tilde{b}(\Omega,x)}{\partial x}+\frac{i}{2}x\tilde%
{b}(\Omega,x)$$
$$\displaystyle=$$
$$\displaystyle\sigma\tilde{b}^{\dagger}(-\Omega,x),$$
(19)
$$\displaystyle\frac{\partial\tilde{b}^{\dagger}(-\Omega,x)}{\partial x}-\frac{i%
}{2}x\tilde{b}^{\dagger}(-\Omega,x)$$
$$\displaystyle=$$
$$\displaystyle\sigma\tilde{b}(\Omega,x),$$
where $\sigma=|\gamma|/\sqrt{\zeta}$ is a new coupling
coefficient. The second-order equation (II.1) in the new variables is
$$\frac{\partial^{2}\tilde{b}(\Omega,x)}{{\partial x}^{2}}+\left(\frac{1}{4}x^{2%
}-\sigma^{2}+\frac{i}{2}\right)\tilde{b}(\Omega,x)=0,$$
(20)
and the corresponding equation for the operator $\tilde{b}^{\dagger}(-\Omega,x)$ is
$$\frac{\partial^{2}\tilde{b}^{\dagger}(-\Omega,x)}{{\partial x}^{2}}+\left(%
\frac{1}{4}x^{2}-\sigma^{2}-\frac{i}{2}\right)\tilde{b}^{\dagger}(-\Omega,x)=0.$$
(21)
Equations (20) and (21) have solutions in the class of parabolic cylinder functions Abramowitz72 . Let us denote two linearly independent solutions of Eq. (20) with a constant Wronskian $W$ as $\phi_{1}(x)$ and $\phi_{2}(x)$. For these two functions we introduce “reciprocal” functions $\tilde{\phi}_{i}(x)$, $i=1,2$, by the relation
$$\frac{1}{\sigma}\left(\frac{\partial}{\partial x}+\frac{i}{2}x\right)\phi_{i}(%
x)=\tilde{\phi}_{i}(x).$$
(22)
By construction, pairs $\left(\phi_{i}(x),\tilde{\phi}_{i}(x)\right)$ are solutions of the system of Eqs. (19). Let us prove that the functions $\tilde{\phi}_{1}(x)$ and $\tilde{\phi}_{2}(x)$ represent solutions of Eq. (21). This can be easily seen from writing Eqs. (20) and (21) in an operator form, $T^{\ast}T\tilde{b}(\Omega,x)=\tilde{b}(\Omega,x)$ and $TT^{\ast}\tilde{b}^{\dagger}(-\Omega,x)=\tilde{b}^{\dagger}(-\Omega,x)$, respectively, where we have introduced a differential operator
$$T=\frac{1}{\sigma}\left(\frac{\partial}{\partial x}+\frac{i}{2}x\right),$$
(23)
having, after Eq. (22), a meaning of mapping onto the “reciprocal”
function: $T\phi_{i}(x)=\tilde{\phi}_{i}(x)$, and asterisk stands for complex conjugation. Substituting the last expression into the operator form of Eq. (21), and using the associative property of differential operators, we obtain $TT^{\ast}\tilde{\phi}_{i}(x)=(TT^{\ast})T\phi_{i}(x)=T(T^{\ast}T)\phi_{i}(x)=T%
\phi_{i}(x)=\tilde{\phi}_{i}(x)$, which had to be
proven. Also we easily obtain $T^{\ast}\tilde{\phi}_{i}(x)={T^{\ast}T\phi}_{i}(x)=\phi_{i}(x)$,
which is a complex conjugate operator $T^{\ast}$ that maps back the reciprocal
function onto the original one. Let us denote by $\tilde{W}$ the Wronskian
of functions $\tilde{\phi}_{1}(x)$ and $\tilde{\phi}_{2}(x)$. Then
$$\displaystyle\tilde{W}$$
$$\displaystyle=$$
$$\displaystyle\left|\begin{array}[]{*{20}c}\tilde{\phi}_{1}(x)&\tilde{\phi}_{2}%
(x)\\
\tilde{\phi}_{1}^{{}^{\prime}}(x)&\tilde{\phi}_{2}^{{}^{\prime}}(x)\\
\end{array}\right|=\sigma\left|\begin{array}[]{*{20}c}\tilde{\phi}_{1}(x)&%
\tilde{\phi}_{2}(x)\\
T^{\ast}\tilde{\phi}_{1}(x)&T^{\ast}\tilde{\phi}_{2}(x)\\
\end{array}\right|$$
$$\displaystyle=$$
$$\displaystyle\sigma\left|\begin{array}[]{*{20}c}T\phi_{1}(x)&T\phi_{2}(x)\\
\phi_{1}(x)&\phi_{2}(x)\\
\end{array}\right|=\left|\begin{array}[]{*{20}c}\phi_{1}^{{}^{\prime}}(x)&\phi%
_{2}^{{}^{\prime}}(x)\\
\phi_{1}(x)&\phi_{2}(x)\\
\end{array}\right|=-W,$$
where we have used the property of invariance of the determinant under addition to one of its rows of another row, multiplied by an arbitrary factor. Equation (III) shows that the reciprocal functions $\tilde{\phi}_{1}(x)$ and $\tilde{\phi}_{2}(x)$ are linearly independent if their original functions are.
Taking the complex conjugate of Eq. (20) in the operator form, we obtain $TT^{\ast}\phi_{i}^{\ast}(x)=\phi_{i}^{\ast}(x)$; i.e., the functions $\phi_{1}^{\ast}(x)$ and $\phi_{2}^{\ast}(x)$ are solutions of Eq. (21) and, therefore, are linear combinations of the functions $\tilde{\phi}_{1}(x)$ and $\tilde{\phi}_{2}(x)$. Let us write this dependence in the matrix form
$$\left[\begin{array}[]{*{20}c}\phi_{1}^{\ast}(x)\\
\phi_{2}^{\ast}(x)\\
\end{array}\right]=\left[\begin{array}[]{*{20}c}m_{11}&m_{12}\\
m_{21}&m_{22}\\
\end{array}\right]\left[\begin{array}[]{*{20}c}\tilde{\phi}_{1}(x)\\
\tilde{\phi}_{2}(x)\\
\end{array}\right],$$
(25)
where $m_{ij}$ are complex numbers. Now the Wronskian of $\phi_{1}^{\ast}(x)$ and $\phi_{2}^{\ast}(x)$ can be written as
$$\displaystyle W^{\ast}$$
$$\displaystyle=$$
$$\displaystyle\det\left[\begin{array}[]{*{20}c}\phi_{1}^{\ast}(x)&\phi_{2}^{%
\ast}(x)\\
\phi_{1}^{\ast^{\prime}}(x)&\phi_{2}^{\ast^{\prime}}(x)\\
\end{array}\right]$$
$$\displaystyle=$$
$$\displaystyle\det\left\{\left[\begin{array}[]{*{20}c}\tilde{\phi}_{1}(x)&%
\tilde{\phi}_{2}(x)\\
\tilde{\phi}_{1}^{{}^{\prime}}(x)&\tilde{\phi}_{2}^{{}^{\prime}}(x)\\
\end{array}\right]M^{T}\right\}=\tilde{W}\det M,$$
where $M$ is a matrix with the coefficients $m_{ij}$ from Eq. (25), and the superscript $T$ stands for transposition. Applying to Eq. (25) first the operator $T^{\ast}$, and then the complex conjugation, we obtain the property $M^{-1}=M^{\ast}$.
A general solution of the system of Eqs. (19) with the boundary conditions at the point $x_{0}=(\Delta(\Omega)-K_{0})/\sqrt{\zeta}$ can be written in the form
$$\displaystyle\tilde{b}(\Omega,x)$$
$$\displaystyle=$$
$$\displaystyle A(x,x_{0})\tilde{b}(\Omega,x_{0})+B(x,x_{0})\tilde{b}^{\dagger}(%
-\Omega,x_{0}),$$
(27)
$$\displaystyle\tilde{b}^{\dagger}(-\Omega,x)$$
$$\displaystyle=$$
$$\displaystyle\tilde{A}(x,x_{0})\tilde{b}^{\dagger}(-\Omega,x_{0})+\tilde{B}(x,%
x_{0})\tilde{b}(\Omega,x_{0}),$$
where
$$\displaystyle A(x,x_{0})$$
$$\displaystyle=$$
$$\displaystyle\frac{\sigma}{W}\left|\begin{array}[]{*{20}c}\phi_{1}(x)&\phi_{2}%
(x)\\
\tilde{\phi}_{1}(x_{0})&\tilde{\phi}_{2}(x_{0})\\
\end{array}\right|,$$
(28)
$$\displaystyle B(x,x_{0})$$
$$\displaystyle=$$
$$\displaystyle-\frac{\sigma}{W}\left|\begin{array}[]{*{20}c}\phi_{1}(x)&\phi_{2%
}(x)\\
\phi_{1}(x_{0})&\phi_{2}(x_{0})\\
\end{array}\right|,$$
(29)
$$\displaystyle\tilde{A}(x,x_{0})$$
$$\displaystyle=$$
$$\displaystyle-\frac{\sigma}{W}\left|\begin{array}[]{*{20}c}\tilde{\phi}_{1}(x)%
&\tilde{\phi}_{2}(x)\\
\phi_{1}(x_{0})&\phi_{2}(x_{0})\\
\end{array}\right|,$$
(30)
$$\displaystyle\tilde{B}(x,x_{0})$$
$$\displaystyle=$$
$$\displaystyle\frac{\sigma}{W}\left|\begin{array}[]{*{20}c}\tilde{\phi}_{1}(x)&%
\tilde{\phi}_{2}(x)\\
\tilde{\phi}_{1}(x_{0})&\tilde{\phi}_{2}(x_{0})\\
\end{array}\right|.$$
(31)
The structure of the solution, Eqs. (27), becomes more clear if we notice that both expressions are linear combinations of solutions of Eqs. (20) and (21), respectively, and, therefore, are also their solutions. Moreover, $TA(x,x_{0})=\tilde{B}(x,x_{0})$ and $TB(x,x_{0})=\tilde{A}(x,x_{0})$, and therefore the pair of functions defined by Eqs. (27) satisfies the system of Eqs. (19). Correspondence to the boundary conditions is seen from the following considerations. It is easy to see that $B(x_{0},x_{0})=\tilde{B}(x_{0},x_{0})=0$ because of the presence of two identical rows in both determinants. In addition,
$$\displaystyle A(x_{0},x_{0})$$
$$\displaystyle=$$
$$\displaystyle\frac{\sigma}{W}\left|\begin{array}[]{*{20}c}\phi_{1}(x_{0})&\phi%
_{2}(x_{0})\\
\tilde{\phi}_{1}(x_{0})&\tilde{\phi}_{2}(x_{0})\\
\end{array}\right|$$
$$\displaystyle=$$
$$\displaystyle\frac{\sigma}{W}\left|\begin{array}[]{*{20}c}\phi_{1}(x_{0})&\phi%
_{2}(x_{0})\\
T\phi_{1}(x_{0})&T\phi_{2}(x_{0})\\
\end{array}\right|=1,$$
and similarly $\tilde{A}(x_{0},x_{0})=1$. Also, using Eqs. (III)-(III), we find that
$$\displaystyle A^{\ast}(x,x_{0})$$
$$\displaystyle=$$
$$\displaystyle\frac{\sigma}{W^{\ast}}\det\left\{\left[\begin{array}[]{*{20}c}%
\tilde{\phi}_{1}(x)&\tilde{\phi}_{2}(x)\\
\phi_{1}(x_{0})&\phi_{2}(x_{0})\\
\end{array}\right]M^{T}\right\}$$
$$\displaystyle=$$
$$\displaystyle-\frac{W}{W^{\ast}}\tilde{A}(x,x_{0})\det M=\tilde{A}(x,x_{0}),$$
and similarly $B^{\ast}(x,x_{0})=\tilde{B}(x,x_{0})$; i.e., for the coefficients in Eqs. (28)-(31) an exchange of the original and the reciprocal functions is equivalent to complex conjugation, though in general $\tilde{\phi}_{i}(x)\neq\phi_{i}^{\ast}(x)$. In particular, it follows that the second of Eqs. (27) can be obtained from the first one by taking a Hermitian conjugation, as expected. It should be noted, that by definition $x$ is an even function of the frequency detuning $\Omega$, since it is determined by $\Delta(\Omega)$.
It is left to prove that Eqs. (27), as required for a Bogoliubov transform, preserve the commutator of the field operators. To this end we need to show the fulfillment of two conditions Kolobov99 : evenness of $A(x,x_{0})/B(x,x_{0})$ as a function of $\Omega$ and the relation $\left|A(x,x_{0})\right|^{2}-\left|B(x,x_{0})\right|^{2}=1$. Fulfillment of the first condition follows from the evenness of both coefficients $A(x,x_{0})$ and $B(x,x_{0})$ as functions of $\Omega$. Let us show that the second condition is always satisfied by these coefficients:
$$\displaystyle\left|A(x,x_{0})\right|^{2}-\left|B(x,x_{0})\right|^{2}$$
(34)
$$\displaystyle=A(x,x_{0})\tilde{A}(x,x_{0})-B(x,x_{0})\tilde{B}(x,x_{0})$$
$$\displaystyle=-\frac{\sigma^{2}}{W^{2}}\det\left\{\left[\begin{array}[]{*{20}c%
}\phi_{1}(x)&\phi_{2}(x)\\
\tilde{\phi}_{1}(x_{0})&\tilde{\phi}_{2}(x_{0})\\
\end{array}\right]\left[\begin{array}[]{*{20}c}\tilde{\phi}_{1}(x)&\phi_{1}(x_%
{0})\\
\tilde{\phi}_{2}(x)&\phi_{2}(x_{0})\\
\end{array}\right]\right\}$$
$$\displaystyle+\frac{\sigma^{2}}{W^{2}}\det\left\{\left[\begin{array}[]{*{20}c}%
\phi_{1}(x)&\phi_{2}(x)\\
\phi_{1}(x_{0})&\phi_{2}(x_{0})\\
\end{array}\right]\left[\begin{array}[]{*{20}c}\tilde{\phi}_{1}(x)&\tilde{\phi%
}_{1}(x_{0})\\
\tilde{\phi}_{2}(x)&\tilde{\phi}_{2}(x_{0})\\
\end{array}\right]\right\}$$
$$\displaystyle=\frac{\sigma^{2}}{W^{2}}\left|\begin{array}[]{*{20}c}\phi_{1}(x_%
{0})&\phi_{2}(x_{0})\\
\tilde{\phi}_{1}(x_{0})&\tilde{\phi}_{2}(x_{0})\\
\end{array}\right|\left|\begin{array}[]{*{20}c}\phi_{1}(x)&\tilde{\phi}_{1}(x)%
\\
\phi_{2}(x)&\tilde{\phi}_{2}(x)\\
\end{array}\right|=1,$$
where we have used the invariance of the matrix determinant to the operation of transposition.
Thus, Eqs. (27) together with their Hermitian conjugation give a correct
description of the evolution of quantum field operators in the nonlinear
medium, in full correspondence to Eq. (9), with $A(\Omega)=A(x,x_{0})$ and $B(\Omega)=B(x,x_{0})$.
Let us consider Eq. (10) for the sideband operator. The functions $U(\Omega)$ and $V(\Omega)$ are given by Eqs. (11), where the right-hand side is defined by the exact solution obtained above. We see easily that the transformation of Eq. (10) is unitary. The ratio $U(\Omega)/V(\Omega)$ is even as a function of $\Omega$ because it is proportional to $A(\Omega)/B(\Omega)$, which is even. The relation $\left|U(\Omega)\right|^{2}-\left|V(\Omega)\right|^{2}=1$ follows from the corresponding properties of the functions $A(\Omega)$ and $B(\Omega)$, since these functions differ only by phase.
For practical calculations in the rest of this article we choose the
functions $\phi_{1}(x)$ and $\phi_{2}(x)$ from the family of Whittaker
functions (see §19.3.7 in Ref. Abramowitz72 ), which are represented in the system of computer algebra
Mathematica 10. Thus, for a fixed parameter $\nu=\sigma^{2}$ we let
$$\displaystyle\phi_{1}(x)$$
$$\displaystyle=$$
$$\displaystyle D_{i\nu}(xe^{i\pi/4}),$$
(35)
$$\displaystyle\phi_{2}(x)$$
$$\displaystyle=$$
$$\displaystyle D_{-1-i\nu}(-xe^{-i\pi/4}),$$
with the corresponding reciprocal functions
$$\displaystyle\tilde{\phi}_{1}(x)$$
$$\displaystyle=$$
$$\displaystyle{\nu^{1/2}e}^{i3\pi/4}D_{i\nu-1}(xe^{i\pi/4}),$$
(36)
$$\displaystyle\tilde{\phi}_{2}(x)$$
$$\displaystyle=$$
$$\displaystyle{\nu^{-1/2}e}^{-i\pi/4}D_{-i\nu}(-xe^{-i\pi/4}).$$
The Wronskian of the functions, defined by Eqs. (36), is equal to $W=e^{-i\pi/4}e^{\pi\nu/2}$. The spectra of PDC calculated with these functions are presented in Sec. V.1.
Thus, we have seen that in the case of a linear poling profile an analytic solution for the Heisenberg equations of motion for the field exists in the class of special functions. Unfortunately, in the general case of the nonlinear profile it is not so, and the solution can be computed numerically only. In the next section we show how an approximate solution can be obtained in a more general case.
IV Approximate solution for a nonlinear poling profile
IV.1 Formulation of the equivalent “potential barrier” problem
In this section we derive an approximate solution for the field transformation in an aperiodically poled crystal in a very simple analytic form. Our approach is based on the similitude of the field evolution to that of a quantum particle in a given potential and is similar to the approach of Refs. Baker03 ; Fejer08a with the main difference that we consider the evolution of a quantum field and are interested in a unitary transformation of the field operators. In addition, we obtain the approximate solution directly in the first-order approximation, without deriving a second-order solution and then simplifying it, as in Ref. Fejer08a .
Equation (II.1) is similar to the Schrödinger equation for a particle of mass $1/2$ in a given potential,
$$\displaystyle\frac{\partial^{2}}{\partial z^{2}}\Psi(z)+\left(E-\mathcal{U}(z)%
\right)\Psi(z)=0,$$
(37)
where $\Psi(z)$ is the particle wavefunction, $E=-|\gamma|^{2}$ is the energy of the particle, and the potential is defined as
$$\mathcal{U}(z)=-\frac{1}{4}\left(\Delta(\Omega)-K(z)\right)^{2}.$$
(38)
Rewriting Eq. (II.1) in the form of Eq. (37), we have omitted the term $K^{\prime}(z)$, which is justified for a sufficiently slowly varying profile Baker03 ; Fejer08a .
An approximate solution of Eq. (37) can be obtained in the first-order approximation Landau65 . In this approximation the solution is oscillating in the regions where $\mathcal{U}(z)<E$ and exponentially growing or decaying in the regions where $\mathcal{U}(z)>E$. For the sake of simplicity we limit ourselves to monotonous profiles $K(z)$, which, for definiteness, we consider to be decreasing functions of $z$.
In a crystal with a monotonous profile $K(z)$ for every pair of frequencies $\omega_{0}+\Omega$ and $\omega_{0}-\Omega$ from the parametric amplification band, there is a perfect phase-matching point $0\leq z_{pm}(\Omega)\leq L$, defined by the relation
$$K\left(z_{pm}(\Omega)\right)=\Delta(\Omega).$$
(39)
At this point the potential $\mathcal{U}(z)$ is maximal and equal to zero. To the left and right of this point there are the so-called “turning points”, where $\mathcal{U}(z)=E$. These points are defined by the relation
$$K\left(z_{1,2}(\Omega)\right)=\Delta\left(\Omega\right)\pm 2|\gamma|$$
(40)
and represent the borders of the region of the exponential solution (see Fig. 2). Note that in our case the oscillatory solutions exist in the regions $(-\infty,z_{1}]$ and $[z_{2},\infty)$, while the exponentially growing and decaying solutions, corresponding to the parametric amplification and attenuation, exist in the region $[z_{1},z_{2}]$. Therefore, our equivalent quantum particle problem corresponds to passing through a “potential barrier” and not to oscillating in a “potential well.”
Using the approach described above, the evolution of the signal field in the crystal can be represented by the following simplified picture. In the region $[0,z_{1}(\Omega)]$ the field remains in its vacuum state. A major part of photons at a given pair of frequencies $\omega_{0}+\Omega$ and $\omega_{0}-\Omega$ is generated in a narrow layer of crystal between $z_{1}(\Omega)$ and $z_{2}(\Omega)$, which we call the amplification layer. Afterwards both waves propagate through the crystal with practically unchanging amplitudes, acquiring a phase difference due to the crystal dispersion. The field operator at the crystal output $\tilde{b}(\Omega,L)=\Psi(L)$ is given by the solution of Eq. (37) with the conditions at some point $z=z_{0}$,
$$\displaystyle\Psi(z_{0})$$
$$\displaystyle=$$
$$\displaystyle\tilde{b}(\Omega,z_{0}),$$
(41)
$$\displaystyle\Psi^{\prime}(z_{0})$$
$$\displaystyle=$$
$$\displaystyle\pm i\sqrt{-\mathcal{U}(z_{0})}\tilde{b}(\Omega,z_{0})+\sqrt{-E}%
\tilde{b}^{\dagger}(-\Omega,z_{0}).$$
Here the second condition is derived from Eq. (II.1) and for a decreasing poling profile the upper (lower) sign should be taken for $z_{0}<z_{pm}$ ($z_{0}>z_{pm}$). Note that in the equivalent “potential barrier” problem $\Psi(z)$ is considered as a c-number. However, after a solution of Eq. (37) is obtained in the form of a linear combination of initial values $\Psi(0)$ and $\Psi^{\prime}(0)$, the latter can be substituted by operator-valued expressions, Eqs. (41) with $z_{0}=0$, corresponding to the original physical problem.
In what follows we derive a solution of Eq. (37) together with the conditions in Eqs. (41) in the first-order approximation. A second-order, Wentzel-Kramers-Brillouin (WKB) solution, was obtained for a classical field in the high-gain regime in Refs. Baker03 ; Fejer08a . However, below we demonstrate that even the first-order solution describes very well the average shapes of optical and squeezing spectra and with a very high precision the characteristic squeezing angles. In the calculations which follow we often omit the frequency detuning, which is always equal to $\Omega$.
IV.2 Oscillating solution before amplification
In the first-order approximation the solution of Eq. (37) in the region $[0,z_{tp1}]$ is given by Landau65
$$\Psi(z)=C_{1}e^{+i\int_{0}^{z}{\sqrt{E-\mathcal{U}(z)}dz}}+C_{2}e^{-i\int_{0}^%
{z}{\sqrt{E-\mathcal{U}(z)}dz}},$$
(42)
where $C_{1}$ and $C_{2}$ are constants, to be determined from the initial conditions. In the considered region almost everywhere we have $|E|\ll|\mathcal{U}(z)|$. Thus, disregarding $E$ compared to $\mathcal{U}(z)$ in Eqs. (41) and (42), we obtain $C_{2}=0$ and write the solution in the form of a phase shift $\tilde{b}(\Omega,z_{1})=\tilde{b}(\Omega,0)e^{i\varphi(\Omega)}$, with
$$\varphi(\Omega)=-\frac{1}{2}\int_{0}^{z_{pm}\left(\Omega\right)}{\left(\Delta(%
\Omega)-K(z)\right)dz},$$
(43)
where we have replaced the upper integration limit $z_{1}(\Omega)$ by $z_{pm}(\Omega)$, which is a good approximation for a sufficiently thin amplification layer.
IV.3 Exponential solution inside the amplification layer
In the first-order approximation the solution of Eq. (37) in the region $[z_{1},z_{2}]$ is given by Landau65 ,
$$\Psi(z)=\tilde{C}_{1}e^{+\int_{z_{1}}^{z}{\sqrt{\mathcal{U}(z)-E}dz}}+\tilde{C%
}_{2}e^{-\int_{z_{1}}^{z}{\sqrt{\mathcal{U}(z)-E}dz}},$$
(44)
where $\tilde{C}_{1}$ and $\tilde{C}_{2}$ are some constants. Unfortunately, these constants cannot be obtained from the initial condition at $z=z_{1}$ since it is a turning point, where $\mathcal{U}(z_{1})=E$ and therefore $\Psi^{\prime}(z_{1})=0$. The problem of tailoring the solutions at the turning points is well known for both the first-order and the WKB approximations Landau65 . Below we show how this problem can be circumvented in our case.
Accepting that the amplification layer is very thin compared to the distance at which the poling profile $K(z)$ is substantially nonlinear, we can approximate the profile inside the amplification layer by its Taylor expansion around $z=z_{pm}$ up to the linear term: $K(z)\approx K(z_{pm})+K^{\prime}(z_{pm})(z-z_{pm})$. Substituting such a linearized profile into Eq. (44) and performing the integration, we obtain
$$\Psi(z)=\tilde{C}_{1}e^{\nu(\xi_{z}+\frac{1}{2}\sin{2\xi_{z}}+\frac{\pi}{2})}+%
\tilde{C}_{2}e^{-\nu(\xi_{z}+\frac{1}{2}\sin{2\xi_{z}}+\frac{\pi}{2})},$$
(45)
where $\xi_{z}=\arcsin{s_{z}}$, and $s_{z}=|K^{\prime}(z_{pm})|(z-z_{pm})/(2|\gamma|)$ is the normalized coordinate inside the amplification layer varying from $-1$ to $1$. Note that the parameter $\nu=|\gamma^{2}/K^{\prime}(z_{pm})|$ is defined exactly as in Sec. III, if we replace $\zeta$ by the local chirp rate $|K^{\prime}(z_{pm})|$.
Substituting $z=z_{2}$ into Eq. (45) gives the following expression for the field at the crystal output,
$$\tilde{b}(\Omega,z_{2})=\tilde{C}_{1}e^{\pi\nu}+\tilde{C}_{2}e^{-\pi\nu}.$$
(46)
Taking the condition in Eqs. (41) at $z_{0}=z_{1}$ we obtain $\tilde{C}_{1}+\tilde{C}_{2}=\tilde{b}(\Omega,z_{1})$. In the absence of the second initial condition, Eq. (46) represents a family of solutions, from which one member should be selected with some considerations. Let us parametrize properly this family. The solution should be a linear combination of $\tilde{b}(\Omega,z_{1})$ and $\tilde{b}^{\dagger}(-\Omega,z_{1})$. Let us write
$$\displaystyle\tilde{C}_{1}$$
$$\displaystyle=$$
$$\displaystyle\frac{1+\mu}{2}\tilde{b}(\Omega,z_{1})+\frac{\tilde{\mu}}{2}%
\tilde{b}^{\dagger}(-\Omega,z_{1}),$$
(47)
$$\displaystyle\tilde{C}_{2}$$
$$\displaystyle=$$
$$\displaystyle\frac{1-\mu}{2}\tilde{b}(\Omega,z_{1})-\frac{\tilde{\mu}}{2}%
\tilde{b}^{\dagger}(-\Omega,z_{1}),$$
where $\mu$ and $\tilde{\mu}$ are some complex coefficients. Such a parametrization is the most general one satisfying the relation for the sum of $\tilde{C}_{1}$ and $\tilde{C}_{2}$. Now Eq. (46) has the form
$$\displaystyle\tilde{b}(\Omega,z_{2})$$
$$\displaystyle=$$
$$\displaystyle\left[\cosh{(\pi\nu)}+\mu\sinh{(\pi\nu)}\right]\tilde{b}(\Omega,z%
_{1})$$
$$\displaystyle+$$
$$\displaystyle\tilde{\mu}\sinh{(\pi\nu)}\tilde{b}^{\dagger}(-\Omega,z_{1}).$$
Unitarity of this transformation demands
$$|\tilde{\mu}|=\frac{\sqrt{\left|\cosh{(\pi\nu)}+\mu\sinh{(\pi\nu)}\right|^{2}-%
1}}{\sinh{(\pi\nu)}}.$$
(49)
There are two candidates for $\mu$, met in similar physical problems. First, we notice that the case $\mu=0$, $|\tilde{\mu}|=1$ resembles the field transformation in a medium with perfect phase matching Kolobov99 . Indeed, in this case the signal field is multiplied (up to a phase) by $\cosh(gl)$, where $g$ is proportional to the pump amplitude and $l$ is the length of the medium. In our case the width of the medium (amplification layer) is also proportional to the pump amplitude [see Eq. (40)] and, as consequence, $gl$ is proportional to the pump intensity, exactly as the parameter $\nu$. However, such a choice is related to neglecting the phase mismatch close to the edges of the amplification layer, which may become significant with growing pump power, leading to widening of the amplification layer. Second, the case of $\mu=1$ resembles a solution obtained by Rosenbluth for a similar problem in plasma physics. In Ref. Rosenbluth72 a parametric interaction of three waves in plasma is considered, which is governed by Eq. (II.1), written for c-numbers. In our case c-numbers appear when one considers the mean field, $\langle\tilde{b}(\Omega,z)\rangle$, which, of course, satisfies the same Eq. (II.1) because of its linearity. The initial conditions of Ref. Rosenbluth72 correspond to the presence of the mean field at the input at the signal frequency, but not at the idler one, which is also a typical scenario of parametric amplification in classical optics Baker03 ; Fejer05 ; Fejer08a ; Fejer08b ; Heese10 . Rosenbluth’s solution Rosenbluth72 can be written for the mean field as $\langle\tilde{b}(\Omega,z_{2})\rangle=\langle\tilde{b}(\Omega,z_{1})\rangle e^%
{\pi\nu}$, which obviously corresponds to $\mu=1$ in Eq. (IV.3). Unfortunately, there is no clear intuitive reason for giving a preference to $\mu=0$ or $\mu=1$, or maybe some other value of $\mu$.
Fortunately, the field transformation for a linearized poling profile can be written in an analytic form by the approach of the previous section where the constant chirp rate $\zeta$ is to be substituted by the local chirp rate $|K^{\prime}(z_{pm})|$ and $K_{0}$ by $K(z_{pm})-K^{\prime}(z_{pm})z_{pm}$. Thus, the exact form of the field transformation in the amplification layer for a linearized profile, but without first-order approximation for an equivalent problem, is given by Eq. (27) with $x=\sqrt{\zeta}(z-z_{pm})$. It is easy to find that the turning points correspond to the values $x_{1,2}=\pm 2\sqrt{\nu}$. Equation (28) in our case takes the form
$$\displaystyle A(x_{2},x_{1})$$
$$\displaystyle=$$
$$\displaystyle e^{-\pi\nu/2}\left(\left|D_{i\nu}(2\sqrt{\nu}e^{i\pi/4})\right|^%
{2}\right.$$
$$\displaystyle+$$
$$\displaystyle\left.\nu\left|D_{i\nu-1}(-2\sqrt{\nu}e^{i\pi/4})\right|^{2}%
\right),$$
where we have used the expression of the elementary solutions through the Whittaker functions given by Eqs. (35) and (36). We see that $A(x_{2},x_{1})$ is real, which corresponds to the choice of a real $\mu$. In Fig. 3 we show the amplitude gain as a function of $\nu$ for the exact solution and the approximate solutions corresponding to different values of $\mu$. We see that in the region of significant, but not too high squeezing, $0.5<\nu<2$, the case of $\mu=1$, i.e., the Rosenbluth formula, complies very well with the exact solution. A similar analysis of $B(x_{2},x_{1})$ shows that its phase is a slow function of $\nu$ and for $\nu<2$ can be approximated as $\varphi_{1}=\arg[B(x_{2},x_{1})]\approx-\nu+\nu^{2}/4$.
Thus, the total transformation in the amplification layer can be written as
$$\tilde{b}(\Omega,z_{2})=e^{\pi\nu(\Omega)}\tilde{b}(\Omega,z_{1})+e^{i\varphi_%
{1}}\sqrt{e^{2\pi\nu(\Omega)}-1}\tilde{b}^{\dagger}(-\Omega,z_{1}),$$
(51)
which can be viewed as a quantum extension of the Rosenbluth formula.
IV.4 Oscillating solution after amplification
In the region $[z_{2},L]$, analogously to Sec. IV.2, the solution of Eq. (37) in the first-order approximation is given by
$$\Psi(z)=\bar{C}_{1}e^{+i\int_{z_{2}}^{z}{\sqrt{E-\mathcal{U}(z)}dz}}+\bar{C}_{%
2}e^{-i\int_{z_{2}}^{z}{\sqrt{E-\mathcal{U}(z)}dz}},$$
(52)
where $\bar{C}_{1}$ and $\bar{C}_{2}$ are constants, to be determined from the initial conditions. In the considered region again, almost everywhere we have $|E|\ll|\mathcal{U}(z)|$, and disregarding $E$ compared to $\mathcal{U}(z)$ in Eqs. (41) and (52), we obtain $\bar{C}_{2}=0$ and write the solution in the form of a phase shift $\tilde{b}(\Omega,L)=\tilde{b}(\Omega,z_{2})e^{i\theta(\Omega)}$, with
$$\theta(\Omega)=-\frac{1}{2}\int_{z_{pm}\left(\Omega\right)}^{L}{\left(\Delta(%
\Omega)-K(z)\right)dz},$$
(53)
where we have replaced the integration limit $z_{2}(\Omega)$ by $z_{pm}(\Omega)$.
IV.5 Total trasformation and the characteristic angles
Combining the results of the previous three sections and coming back to the sideband operator $a(\Omega,z)$, we write the total field transformation in the crystal in the following form:
$$a(\Omega,L)=U_{1}(\Omega)a(\Omega,0)+V_{1}(\Omega)a^{\dagger}(-\Omega,0),$$
(54)
where
$$\displaystyle U_{1}(\Omega)$$
$$\displaystyle=$$
$$\displaystyle e^{\pi\nu(\Omega)}e^{i(k(\Omega)-k_{0})L},$$
(55)
$$\displaystyle V_{1}(\Omega)$$
$$\displaystyle=$$
$$\displaystyle\sqrt{e^{2\pi\nu(\Omega)}-1}e^{-2i\varphi(\Omega)+i(k(\Omega)-k_{%
0})L+i\varphi_{A}}.$$
Here index $1$ stands for the first-order approximation, and $\varphi_{A}=\varphi_{0}+\varphi_{1}$ is the phase added in the amplification layer. The properties $|U_{1}(\pm\Omega)|^{2}-|V_{1}(\pm\Omega)|^{2}=1$ and
$U_{1}(\Omega)/V_{1}(\Omega)=U_{1}(-\Omega)/V_{1}(-\Omega)$, required for the unitarity of this transformation, are straightforward to verify.
We show in the next section that the first-order approximate solution, given by Eq. (54) is very close to the analytical and numerical solutions of the initial Heisenberg equation for both small and considerable values of the pump power. In this way it differs from the solution, called the “Rosenbluth formula” in Ref. Fejer08a , which provides the same value $e^{\pi\nu}$ for moduli of $U_{1}(\Omega)$ and $V_{1}(\Omega)$ inside the amplification band and is valid only at $\pi\nu\gg 1$. In this limit our Eqs. (55) give the same result. At the same time, they give also a very good approximation at a moderate pump power, where $\pi\nu$ is less than or comparable to unity.
The three parameters in Eqs. (12)-(14), characterizing the nonlinear transformation of the field in the first-order approximation, are equal to
$$\displaystyle r(\Omega)$$
$$\displaystyle=$$
$$\displaystyle\ln\left(e^{\pi\nu(\Omega)}+\sqrt{e^{2\pi\nu(\Omega)}-1}\right),$$
(56)
$$\displaystyle\psi_{L}(\Omega)$$
$$\displaystyle=$$
$$\displaystyle-\varphi(\Omega)-\frac{1}{2}\Delta(\Omega)L+\frac{\varphi_{A}}{2},$$
(57)
$$\displaystyle\psi_{0}(\Omega)$$
$$\displaystyle=$$
$$\displaystyle-\varphi(\Omega)+\frac{\varphi_{A}}{2}.$$
(58)
As mentioned in Sec. II, the angle $\psi_{0}(\Omega)$ determines the quadrature at the input which is subject to the squeezing effect and is important in the case of seeded PDC. For example, to obtain an amplitude squeezing at all frequencies, one needs to shape the signal seed pulse so that $\langle X_{1}(\Omega,0)\rangle=0$, $\langle X_{2}(\Omega,0)\rangle\neq 0$, where at each frequency the quadratures are determined by $\psi_{0}(\Omega)$. For this the idler component should be equal to the conjugated and phase-shifted signal component, $\langle a(-\Omega,0)\rangle=-e^{i2\psi_{0}(\Omega)}\langle a(\Omega,0)\rangle^%
{*}$, where we have assumed $\Omega>0$. Equation (14) shows that this phase shift is exactly compensated by the relative phase difference $-2\varphi(\Omega)$ acquired by the idler field with respect to the signal field before the amplification layer, and by the phase $\varphi_{A}$, added during the amplification. As a result, the squeezing is in-phase with the coherent component of the field, as required for the amplitude squeezing.
Now we proceed to explain the physical meaning of the expression for the angle of squeezing, Eq. (13). Differentiating this equation and taking into account Eq. (43), we obtain
$$\frac{d\psi_{L}(\Omega)}{d\Omega}=-\frac{1}{2}\Delta^{\prime}(\Omega)\left[L-z%
_{pm}(\Omega)\right].$$
(59)
Equation (59) has a simple physical meaning in terms of the classical notion of the relative group delay between the signal and the idler waves. In the classical treatment of PDC the field operators $a(\pm\Omega,z)$ are replaced by classical complex amplitudes $\langle a(\pm\Omega,z)\rangle$ ($\Omega$ is assumed to be positive). Equation (54) with the corresponding replacement gives the field amplitudes at the crystal output. As mentioned above, in the scenario of parametric amplification, treated classically, it is typically assumed that at the crystal input only the signal wave is present, i.e., $\langle a(\Omega,0)\rangle=1$, $\langle a(-\Omega,0)\rangle=0$. In this case the signal wave at the output is equal to $\langle a(\Omega,L)\rangle=U(\Omega)$ and the idler wave to $\langle a(-\Omega,L)\rangle=V(-\Omega)$. Then, the group delay of the signal wave is given by $\tau_{s}(\Omega)=-\frac{d}{d\Omega}\arg\{U(\Omega)\}$ and that of the corresponding idler wave by $\tau_{i}(\Omega)=-\frac{d}{d(-\Omega)}\arg\{V(-\Omega)\}$. The relative delay is
$$\displaystyle\tau(\Omega)$$
$$\displaystyle=$$
$$\displaystyle\tau_{s}(\Omega)-\tau_{i}(\Omega)$$
$$\displaystyle=$$
$$\displaystyle-\frac{d}{d\Omega}\arg\{U(\Omega)V(-\Omega)\}=-2\frac{d\psi_{L}(%
\Omega)}{d\Omega}.$$
Now Eq. (59) can be rewritten as
$$\tau(\Omega)=\frac{L-z_{pm}(\Omega)}{v_{g}(\Omega)}-\frac{L-z_{pm}(\Omega)}{v_%
{g}(-\Omega)},$$
(61)
where $v_{g}(\Omega)=[k^{\prime}(\Omega)]^{-1}$ is the group velocity at the corresponding frequency. Equation (61) shows that the relative delay is equal to the difference of propagation times of two waves from the perfect phase-matching point to the end of the crystal with the corresponding group velocities, which is quite a natural result. This result is well known in the classical consideration of parametric amplification; see, e.g., Ref. Fejer05 . In our quantum treatment we have shown that the relative group delay is related to the angle of squeezing via Eq. (IV.5).
Before finishing this section, let us estimate the condition of “slow” variation for the poling profile. In order that its linearization inside the amplification layer be valid, it is necessary that the second-order term in the Taylor expansion is much smaller than the first-order term, i.e., the parameter
$$\epsilon=\frac{1}{2}\frac{|K^{\prime\prime}(z_{pm})(z_{tp}-z_{pm})|}{|K^{%
\prime}(z_{pm})|}$$
(62)
should be much smaller than unity, where $z_{tp}$ is the most distant of two turning-points with respect to $z_{pm}$. We may obtain a more compact expression for the smallness parameter. If $\epsilon\ll 1$ and the linearization is valid, then $|z_{tp}-z_{pm}|\approx 2|\gamma/K^{\prime}(z_{pm})|$ and the parameter
$$\epsilon^{\prime}=\frac{|\gamma K^{\prime\prime}(z_{pm})|}{(K^{\prime}(z_{pm})%
)^{2}}$$
(63)
is also much smaller than unity. Thus, $\epsilon^{\prime}\ll 1$ is a necessary condition for the applicability of the first-order approximation of this section. Note that, since $|\gamma|$ is proportional to $|b_{\mathrm{p}}|$, this condition imposes a limitation on the pump power.
V Spectra of parametric down-conversion for a model crystal
V.1 Linearly chirped crystal: comparison of exact and approximate solutions
For making a comparison of the exact and the approximate solutions of the
wave equation we consider PDC in a nonlinear crystal of 5% MgO-doped congruent aperiodically poled LiNbO${}_{3}$ of length $L=$4.5 mm, continuously pumped at the wavelength $\lambda_{p}=532$ nm. The pump frequency is $\omega_{\mathrm{p}}=2\pi c/\lambda_{p}$, and the central frequency of the downconverted light is $\omega_{0}=\omega_{\mathrm{p}}/2$. The signal band is chosen to be from $1.1\omega_{0}$ to $1.5\omega_{0}$, and the idler band from $0.5\omega_{0}$ to $0.9\omega_{0}$. This corresponds to signal wavelengths of 709–967 nm, and to idler wavelengths of 1182–2128 nm. To obtain the desired frequency band of the downconverted light we need to vary the spatial frequency of aperiodical poling $K(z)$ from $K_{0}=\Delta(0.1\omega_{0})=894$ rad/mm to $K_{1}=\Delta(0.5\omega_{0})=720$ rad/mm. In this subsection we consider a linear dependence $K_{L}(z)=K_{0}-\zeta z$, where $\zeta=(K_{0}-K_{1})/L=38.5$ rad/mm${}^{2}$. The phase mismatch for such a crystal, obtained from its Sellmeier equation Gayer08 , is shown in Fig. 4. In the same figure we show the quadratic approximation of the phase mismatch
$$\Delta_{q}(\Omega)=-\alpha\left(\frac{\Omega}{\omega_{0}}\right)^{2}+\beta,$$
(64)
where $\alpha=-\Delta^{\prime\prime}(0)\omega_{0}^{2}/2=735$ rad/mm, and $\beta=\Delta(0)=901$ rad/mm.
The optical spectrum of PDC, $S(\omega)$, is defined by the relation
$\langle a^{\dagger}(\Omega,L)a(\Omega^{\prime},L)\rangle=S(\omega_{0}+\Omega)%
\delta(\Omega-\Omega^{\prime})$, wherefrom, taking into account the commutation relations $\left[a(\Omega,z),a^{\dagger}(\Omega^{\prime},z)\right]=\frac{1}{2\pi}\delta%
\left(\Omega-\Omega^{\prime}\right)$ and Eq. (10), we obtain
$$S(\omega_{0}+\Omega)=\frac{1}{2\pi}\left|V(\Omega)\right|^{2}.$$
(65)
In Fig. 5 we show the PDC spectra, calculated with the help of the exact
solution, defined by Eqs. (10), (11), (28), and (29), and its approximation, Eq. (54), for various values of the parameter $\nu=4\left|\chi_{0}b_{p}\right|^{2}/(\pi^{2}\zeta)$.
We see in Fig. 5 that for both small and comparable-with-unity values of parameter $\nu$ the approximation in Eq. (54) gives the true average value for the spectrum in the generation band, though it does not follow the rapid oscillations around this average value. Validity of the approximate formula at small values of $\nu$ is a consequence of unitarity of the field transformation in the amplification layer, Eq. (51). This is the key difference from the approach of Ref. Fejer08a , where a purely classical consideration was undertaken, valid for a sufficiently strong pump, $\pi\nu\gg 1$. It is demonstrated in the same reference that the rapid oscillations of the spectrum can be fairly well described by the second-order (WKB) approximation. For the purposes of design of aperiodically poled crystals these oscillations may be secondary and a general, “averaged,” shape of the spectrum provided by Eq. (54) may be sufficient.
The squeezing spectrum of the field at the crystal output $S_{2}(\Omega)$ for an unseeded PDC is defined by the relation $\langle X_{2}(\Omega,L)X_{2}^{\dagger}(\Omega^{\prime},L)\rangle=S_{2}(\Omega)%
\langle X_{2}(\Omega,0)X_{2}^{\dagger}(\Omega^{\prime},0)\rangle$; i.e., it shows the change of variance of the quadrature $X_{2}$. It is given by Kolobov99 ,
$$S_{2}(\Omega)=\left(\left|U(\Omega)\right|-\left|V(\Omega)\right|\right)^{2}.$$
(66)
In Fig. 6 we show the squeezing spectra, calculated on the basis of the exact
solution, defined by Eqs. (10), (11), (28), and (29), and its approximation, Eq. (54), for various values of the parameter $\nu$.
We see in Fig. 6 that for both small and comparable-with-unity values of parameter $\nu$ the approximation in Eq. (54) gives a value for the spectrum very close to the exact value in the generation band. Outside this band there is a significant difference between the two solutions: the exact solution shows some squeezing, though the approximate one predicts no squeezing at all.
The first-order approximate angle of squeezing can be easily obtained for a linear chirp by integrating Eq. (43) and substituting the result into Eq. (13), which gives
$$\psi_{L}^{(lin)}(\Omega)=\frac{1}{2}\left[(k(\Omega)+k(-\Omega))L-\frac{\left(%
\Delta(\Omega)-K_{0}\right)^{2}}{2\zeta}+\bar{\varphi}_{A}\right],$$
(67)
where $\bar{\varphi}_{A}=\varphi_{A}-2k_{0}L$. A similar expression (with a factor of $-2$) was obtained in Ref. Harris07 in the low-gain regime for the phase of the compensating optical element, necessary to provide simultaneous arrival of the signal and idler photons at the distant detector or summing crystal. Thus the angle of squeezing is related to this phase as $\psi_{L}(\Omega)=-\frac{1}{2}\arg\{H(\Omega)\}+\frac{1}{2}\bar{\varphi}_{A}$, where $H(\Omega)$ is the transfer function of the compensating element. The additional term $2k_{0}L$ in the definition of $\bar{\varphi}_{A}$ reflects the difference between the phase of a sideband operator and that of the full field operator. The phase $\varphi_{1}$, which is part of $\varphi_{A}$, is very small in the low-gain regime and can be disregarded. In the high-gain regime it is not small, but for a linear chirp it is constant and does not affect the simultaneity of photon arrivals. Thus, even in the high-gain regime we can understand the dispersion of the squeezing angle as caused by a relative delay of the photons at the conjugated frequencies.
In a similar way we obtain the second characteristic angle,
$$\psi_{0}^{(lin)}(\Omega)=-\frac{\left(\Delta(\Omega)-K_{0}\right)^{2}}{4\zeta}%
+\varphi_{A}.$$
(68)
We see that in the first-order approximation both angles at all frequencies are independent of the pump power up to an additive constant, which is quite a general result.
We show in Fig. 7 the exact and the approximate values for both angles, calculated at $\nu=0.01$. We see that the difference between the two solutions is much smaller than $\pi/2$ everywhere. With the growing $\nu$ this difference increases, and a numerical study shows that up to $\nu=1$ it remains less than or comparable to $\pi/2$.
We can conclude that the approximate formula, Eq. (54), looks very promising for designing aperiodically poled crystals in the case of a linear chirp profile. It provides a good qualitative description of the squeezing spectrum and almost exact values of the characteristic angles for a sufficiently low pump power, $\nu$ well below 1.
V.2 Nonlinearly chirped crystal: comparison of approximate and numerical solutions
In this section we analyze a nonlinear profile of aperiodic poling. Several specific shapes of nonlinear profiles have been studied to date: a $z^{n}$ profile Baker03 , and sinusoidal and tapered profiles Fejer08a . For demonstrating the efficiency of the results of Sec. IV we need to consider a rather slowly varying profile, which is selected from the following physical considerations. In the previous section we have seen that a linear profile produces an almost flat spectrum of the down-converted light, but the angle of squeezing, $\psi_{L}(\Omega)$, is a complicated function of frequency detuning, including non-negligible third- and fourth-order components [see Eq. (67)]. Any observation of the ultrabroadband character of squeezing requires compensation of this angle in a wide range of frequencies. For a bandwidth of the order of 10 THz the quadratic term is dominant and compensation can be performed by a passive optical element (optical fibre Brida09 , a glass block Sensarn10 , or a pair of prisms Tanaka12 ), but for a 100-THz-wide PDC spectrum an active compensation is required Peer05 ; Lukens13 , which is the state of art of modern quantum optics. When looking for a nonlinear spatial frequency profile we could demand that it is a monotonous function $K(z)$ such, that the corresponding angle of squeezing $\psi(\Omega)$ is a second-order polynomial, which can be compensated by passive optical elements.
The relative delay of the signal with respect to the idler in the first-order approximation is given by Eq. (59). Our aim is to obtain $\tau(\Omega)=a\Omega+b$, where $a$ and $b$ are some real parameters. From Eqs. (39), (59), and (IV.5) we obtain the following equation, which should be satisfied by the profile function:
$$K\left(L-\frac{a\Omega+b}{\Delta^{\prime}(\Omega)}\right)=\Delta(\Omega).$$
(69)
This equation can be easily solved in the approximation of quadratic dispersion, Eq. (64), where the inverse group velocity difference has a simple form: $\Delta^{\prime}(\Omega)=-2\alpha\Omega/\omega_{0}^{2}$.
For a quadratic phase (linear delay) we need $z_{pm}(\Omega)=L+d+db/(a\Omega)$, where $d=a\omega_{0}^{2}/(2\alpha)$. The inverse of this function is
$$\Omega_{pm}(z)=-\frac{db/a}{L+d-z},$$
(70)
and has a meaning of the frequency, for which perfect phase matching is reached at the point $z$. The sought profile is found in the form
$$K(z)=\Delta(\Omega_{pm}(z))=-\frac{\alpha}{\omega_{0}^{2}}\left(\frac{db/a}{L+%
d-z}\right)^{2}+\beta.$$
(71)
Let us determine the possible values of the coefficients $a$ and $b$. From Eq. (70) we obtain the phase-matched frequencies at the edges of the crystal,
$$\Omega_{pm}(0)=-\frac{b}{a}\frac{d}{L+d},\quad\Omega_{pm}(L)=-\frac{b}{a}.$$
(72)
Since both frequencies should be positive, we have two possibilities:
•
$a>0$, $b<0$, $d>0$, and $K(z)$ is decreasing, and
•
$a<0$, $b>0$, $d<-L$, and $K(z)$ is increasing.
In this section we limit the lower frequency of the signal amplification band to $0.25\omega_{0}$, because otherwise the profile does not satisfy the requirement of slow variation. In the first of the cases listed above, substituting $\Omega_{pm}(0)=0.25\omega_{0}$, $\Omega_{pm}(L)=0.5\omega_{0}$, we obtain $d=L$ and
$$K(z)=-\frac{\alpha}{4\left(2-z/L\right)^{2}}+\beta.$$
(73)
In the second case, substituting $\Omega_{pm}(0)=0.5\omega_{0}$, $\Omega_{pm}(L)=0.25\omega_{0}$, we obtain $d=-2L$ and
$$K(z)=-\frac{\alpha}{4\left(1+z/L\right)^{2}}+\beta.$$
(74)
A decreasing profile is more interesting from the practical point of view, because it generates a negatively chirped field ($a>0$), where lower signal frequencies are more delayed than the higher ones, which requires a compensating medium with normal dispersion, e.g., an optical fiber Brida09 or a glass block Sensarn10 . In the rest of this section we compare the first-order approximate and numerical solutions for the case of decreasing quadratic-hyperbolic poling profile, given by Eq. (73). Substituting Eq. (73) and Eq. (64) into Eq. (II.1) and solving numerically this second-order differential equation, we calculate optical spectra and spectra of squeezing. These spectra are presented in Figs. 8 and 9, where they are compared with the first-order values, predicted by Eq. (54). For better comparison with Figs. 5 and 6, we introduce normalized pump intensity,
$$\nu_{0}=\frac{|\gamma|^{2}L}{|K(0)-K(L)|},$$
(75)
which has a physical meaning of the Rosenbluth parameter for the linear profile, providing a quasi-phase-matching in the same frequency band for the given crystal length.
From the same numerical solution we can calculate the angle of squeezing. Its first-order value is given by integrating Eq. (43) with the profile defined by Eq. (73) and substituting the result into Eq. (13),
$$\psi_{L}^{(qh)}(\Omega)=-\frac{\alpha L}{2}\left(\frac{\Omega-0.5\omega_{0}}{%
\omega_{0}}\right)^{2}+\psi_{\mathrm{c}},$$
(76)
where $\psi_{\mathrm{c}}=(\varphi_{A}+\alpha L/8-\beta L)/2$ is a constant. In a similar way we obtain the input angle
$$\psi_{0}^{(qh)}(\Omega)=-\alpha L\left(\frac{\Omega-0.25\omega_{0}}{\omega_{0}%
}\right)^{2}+\frac{\varphi_{A}}{2},$$
(77)
where the superscript $qh$ denotes the quadratic-hyperbolic profile. The approximate angles are plotted in Fig. 10 together with their numerical solutions. We see that the agreement of both solutions is very good. As in the previous section, the numerical study shows that the two solutions start to differ significantly when $\nu_{0}$ approaches unity.
Finally, let us evaluate two parameters, characterizing the slow variation of the profile, given by Eq. (73). We find easily that $\max|\Lambda^{\prime}(z)|=|\Lambda^{\prime}(L)|=0.001$, so the poling period is changing sufficiently slowly for applying Eq. (II.1). We also find that $\max(\epsilon^{\prime})=0.18\sqrt{\nu_{0}}$, and, therefore, the linearization of the poling profile is justified for values $\nu_{0}<0.31$, where $\epsilon^{\prime}<0.1$. We see in Fig. 8(d) that at $\nu_{0}=0.3$ the optical spectrum starts to deflect from the prediction of the first-order approximation not only in rapid oscillations but also in the average value. With growing $\nu_{0}$ this deflection becomes greater, meaning that the linearized solution is not valid anymore.
To ascertain that the good correspondence of the approximate solution to the exact analytic and the numerical ones is not a particular property of the chosen crystal settings, we applied the analysis of the current section to four other crystal designs. First, we considered MgO:LiNbO${}_{3}$ crystals of different lengths, quasi-phase-matched for the same bandwidth: a 2-mm-long crystal with the chirp rate $\zeta=87$ rad/mm${}^{2}$, and a 20-mm-long crystal with the chirp rate $\zeta=8.7$ rad/mm${}^{2}$. In addition, we considered a 20-mm-long crystal of undoped LiNbO${}_{3}$ pumped at 420 nm and quasi-phase-matched from 464 to 750 nm, as in Ref. Harris07 , and a 22-mm-long crystal of stoichiometric LiTaO${}_{3}$ pumped at 532 nm and quasi-phase-matched from 680 to 800 nm, as in Ref. Fejer05 . In all these cases we obtained the correspondence of the solutions similar to that of Figs. 5–10.
VI Conclusions
We have considered the process of ultrabroadband collinear PDC in an aperiodically poled crystal, designed to produce QPM in a wide range of wavelengths (hundreds on nanometers). In the case of the high-gain regime with an undepleted pump such a process generates an ultrabroadband squeezed-light wave at the output of the crystal. The components of such a light wave at the frequencies symmetric with respect to the central frequency $\omega_{0}$ are highly quantum correlated, and their correlation time may be made as small as one optical period. This ultrabroadband squeezing can be observed, for example, in second-harmonic generation as described in Ref. Horoshko13 , after the compensation of the angle of squeezing at all frequencies. For a sufficiently broadband squeezed light the correlation time can be as short as a single optical period.
We can estimate the number of squeezed modes in the considered ultrabroadband source of squeezed light. In our model of the monochromatic pump the number of squeezed modes is formally infinite. When the spectral width $\delta\omega$ of the pump is taken into account, the number of such modes in the low-gain regime is given roughly by the ratio $\Delta\omega/\delta\omega$, where $\Delta\omega$ is the amplification bandwidth Horoshko12 . In the high-gain regime the modes are expected to be approximately the same, but each pair of modes will be characterized by a high degree of squeezing. Thus, we can estimate the number of entangled modes for a nanosecond pump pulse as $200\mathrm{THz}/1\mathrm{GHz}=2\times 10^{5}$ modes, where the amplification bandwidth of 200 THz corresponds to the example analyzed in Sec. V.1. Such a highly multimode field can be used in various applications of quantum information, from metrology to cluster state quantum computation.
Let us summarize the results obtained in this article. First, we have analyzed in detail the exact solution of the differential equation for PDC with an undepleted quasi-monochromatic pump in an aperiodically poled nonlinear crystal with a linear poling profile. The solution is expressed through parabolic cylinder special functions. We have analyzed the properties of this solution and proven the conservation of the commutation relations for the field operators. Second, we have obtained a unitary approximate solution, a “quantum Rosenbluth formula,” in the first-order approximation and have demonstrated that it is in good agreement with the exact solution within the amplification band for various values of the pump power. We have shown that, taking into consideration the quantum conditions, one arrives at a solution, applicable in the high-gain regime of PDC with the gain, corresponding to practical values of squeezing from 3 to 12 dB. We have also shown a good correspondence of the approximate solution to the numerical one for the case of a nonlinear (quadratic-hyperbolic) profile. Third, we have shown that a quadratic-hyperbolic profile of aperiodic poling results in a negatively chirped output field, compressible by a passive dispersive element with normal quadratic dispersion. These results will help to design aperiodically poled crystals for generation of squeezed light with monocycle squeezing, which is important for applications requiring ultra-short correlations in the temporal domain or an ultra-high number of entangled modes in the spectral domain.
Acknowledgements.The authors are grateful to Maria Chekhova and Chris Phillips for fruitful discussions. This work was supported in part by the European Union’s Horizon 2020 research and innovation programme under grant agreement No 665148 (QCUMbER) and in part by the Belarusian Republican Foundation for Fundamental Research.
References
(1)
S. Ya. Kilin, Prog. Opt. 42, 1 (2001).
(2)
H. Vahlbruch, M. Mehmet, K. Danzmann, and R. Schnabel, Phys. Rev. Lett. 117, 110801 (2016).
(3)
S. Ast, M. Mehmet, and R. Schnabel, Opt. Expr. 21, 13572 (2013).
(4)
S. Spalter, N. Korolkova, F. Konig, A. Sizmann, and G. Leuchs, Phys. Rev.
Lett. 81, 786 (1998).
(5)
J. Wenger, R. Tualle-Brouri, and P. Grangier, Opt. Lett. 29, 1267
(2004).
(6)
I. N. Agafonov, M. V. Chekhova, and G. Leuchs, Phys. Rev. A 82,
011801 (2010).
(7)
O. Pinel et al., Phys. Rev. Lett. 108, 083601 (2012).
(8)
T. Iskhakov, M. V. Chekhova, and G. Leuchs, Phys. Rev. Lett.
102, 183602 (2009).
(9)
D. B. Horoshko and M. I. Kolobov, Phys. Rev. A 88, 033806
(2013).
(10)
K. L. Baker, Appl. Phys. Lett. 82, 3841 (2003).
(11)
M. Charbonneau-Lefort, M. M. Fejer, and B. Afeyan, Opt. Lett. 30, 634 (2005).
(12)
M. Charbonneau-Lefort, B. Afeyan, and M. M. Fejer, J. Opt. Soc. Am. B
25, 463 (2008).
(13)
M. Charbonneau-Lefort, B. Afeyan, and M. M. Fejer, J. Opt. Soc. Am. B
25, 680 (2008).
(14)
C. Heese, C. R. Phillips, L. Gallmann, M. M. Fejer, and U. Keller, Opt.
Lett. 35, 2340 (2010).
(15)
C. R. Phillips, B. W. Mayer, L. Gallmann, M. M. Fejer, and U. Keller, Opt. Expr. 22, 9627 (2014).
(16)
S. E. Harris, Phys. Rev. Lett. 98, 063602 (2007).
(17)
M. B. Nasr et al., Phys. Rev. Lett. 100, 183601 (2008).
(18)
G. Brida et al., Phys. Rev. Lett. 103, 193602 (2009).
(19)
S. Sensarn, G. Y. Yin, and S. E. Harris, Phys. Rev. Lett. 104,
253602 (2010).
(20)
A. Tanaka et al., Opt. Expr. 20, 25228 (2012).
(21)
I. Jovanovic, C. G. Brown, C. A. Ebbers, C. P. J. Barty, N. Forget and C. Le Blanc, Opt. Lett. 30, 1036 (2005).
(22)
N. Bloembergen, Nonlinear Optics (W.A. Benjamin Inc., 1965).
(23)
J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, Phys.
Rev. 127, 1918 (1962).
(24)
P. E. Powers, Fundamentals of Nonlinear Optics (CRC Press, 2011).
(25)
M. Houé, P. D. Townsend, J. Phys. D 28, 1747 (1995).
(26)
L. D. Landau, E. M. Lifshitz, Quantum Mechanics (Pergamon Press, 1965)
(27)
M. I. Kolobov, Rev. Mod. Phys. 71, 1539 (1999).
(28)
M. N. Rosenbluth, Phys. Rev. Lett. 29, 565 (1972).
(29)
M. Abramowitz and I. A. Stegun, Eds. Handbook of Mathematical Functions (National Bureau of Standards, 1972).
(30)
O. Gayer et al., Appl. Phys. B 91, 343-348 (2008).
(31)
A. Pe’er, B. Dayan, A. A. Friesem, and Y. Silberberg, Phys. Rev. Lett. 94, 073601 (2005).
(32)
J. M. Lukens, A. Dezfooliyan, C. Langrock, M. M. Fejer, D. E. Leaird, and A. M. Weiner, Phys. Rev. Lett. 111, 193603 (2013).
(33)
D. B. Horoshko, G. Patera, A. Gatti and M. I. Kolobov, Eur. Phys. J. D. 66, 239 (2012). |
$\widetilde{\tau}$ searches at the ILC
M.T. Núñez Pardo de Vera
Corresponding author: maria-teresa.nunez-pardo-de-vera@desy,de
DESY
M. Berggren
DESY
J. List
DESY
(May 2021)
Abstract
The direct pair-production of the superpartner of the $\tau$-lepton, the $\widetilde{\tau}$, is one
of the most interesting channels to search for SUSY in. First of all, the $\widetilde{\tau}$ is
likely to be the lightest of the scalar leptons. Secondly the
signature of $\widetilde{\tau}$ pair production signal events is one of the experimentally most difficult
ones, thereby constituting the “worst” possible scenario for SUSY searches.
The current model-independent $\widetilde{\tau}$ limits comes from analyses performed at
LEP but they suffer from the limited energy of this facility. Limits obtained at the LHC do
extend to higher masses, but they are only valid under strong assumptions.
ILC, a future electron-positron collider with
energy up to 500 GeV and upgrade capability111The initial ILC energy is planned to be 250 GeV., is a promising facility for SUSY
searches. The capability of the ILC for determining exclusion/discovery limits
for the $\widetilde{\tau}$ in a model-independent way is shown in this paper, together
with an overview of the current state-of-the-art.
Results of the last studies of $\widetilde{\tau}$ pair-production at the ILC are presented,
showing the improvements with respect to previous results222Talk presented at the International Workshop on Future Linear Colliders (LCWS2021), 15-18 March 2021. C21-03-15.1..
1 Introduction
Supersymmetry (SUSY) [1][2][3][4][5]
is one of the most promising candidates for new physics. It could explain or
at least give some hint at solutions to current problems of the Standard Model (SM), such as the gauge hierarchy
problem, the nature of Dark Matter or the possible theory-experiment discrepancy of the muon magnetic moment. SUSY is a symmetry of
spacetime relating fermions and bosons. For every SM particle it introduces a superpartner with the
same quantum numbers except for the spin. The spin differs by half a unit from the value of its SM partner.
A new parity, R-parity, is commonly introduced in SUSY, which has a crucial impact in SUSY phenomenology.
R-parity takes an even value for SM particles and odd value for the SUSY ones.
Multiplicative R-parity conservation 333The introduction and conservation of
this symmetry is inspired by flavour physics constraints since the most general SUSY Lagrangian induces
flavour-changing neutral interactions that are avoided imposing R-parity conservation., assumed in
most of the SUSY models, implies that the SUSY particles are always created in
pairs and that the lightest SUSY particle (the LSP) is stable and, when cosmological constraints are taken into account,
also neutral.
An important point in this kind of studies is the fundamental SUSY principle stating that each
SUSY particle couples as its corresponding SM particle. This allows to know the cross sections
for SUSY pair production solely from the knowledge of initial centre-of-mass energy of the collider and the masses of the involved SUSY
particles.
2 SUSY searches
Considerable efforts have been and are being devoted to the search of SUSY at different facilities.
Searches at hadron colliders, such as the LHC, are mainly sensitive to the production of coloured particles,
gluino and squarks. They are most probably the heaviest ones. The search of the lighter colour-neutral SUSY states, such as sleptons, charginos or neutralinos, at
hadron colliders is challenged by the much smaller cross sections,
and high backgrounds. Mass limits have been obtained at the LHC, but they
are only valid if many constraints on the model parameters are fulfilled.
Lepton colliders, like LEP, have a higher sensitivity to the production of colour-neutral SUSY states, but
the searches up to now were limited by the beam energy. Limits computed at these facilities are however
valid for any value of the model parameters not shown in the exclusion plots.
The future International Linear Collider (ILC) [6], an electron-positron collider operating at centre-of-mass
energies of $250-500$ GeV and with upgrade capability to $1$ TeV, is seen as an ideal environment for SUSY studies.
SUSY searches at the ILC would profit from the high electron and positron beam polarisations,
80$\%$ and 30$\%$ respectively, a well defined initial state (in 4-momentum and spin configuration),
a clean and reconstructable final state, near absence of pile-up, hermetic detectors (almost
4$\pi$ coverage) and trigger-less operation, which is a huge advantage for precision measurements and
unexpected signatures.
3 Motivation for $\widetilde{\tau}$ searches
For evaluating the power of SUSY searches at future facilities,
it is beneficial to focus on the lightest particle in the SUSY spectrum that could be accessible. Since
the cosmological constraints requires a neutral and colourles LSP, the next-to-lightest SUSY
particle, the NLSP, would be the first one to be detected. The NLSP can only decay to the LSP and
the SM partner of the NLSP (or via it’s SM partner, if the LSP-NLSP mass-difference is smaller than the mass
of the SM partner). This already makes the NLSP production special: heavier states might well decay in
cascades, and thus have signatures that depend strongly on the model.
Furthermore, there is only a finite set of sparticles that could be the NLSP, so a systematic search
for each possible case is feasible.
This also means that one can a priori estimate which will be the most difficult case, namely the
NLSP that combines small production cross-section with a difficult experimental signature.
The $\widetilde{\tau}$ satisfies both these conditions. Therefore, studies of $\widetilde{\tau}$ production
might be seen as the way to determine the guaranteed discovery or exclusion reach for SUSY: any other NLSP
would be easier to find.
The $\widetilde{\tau}$ is the super-partner of the ${\tau}$. Like for any other fermion or sfermion, there
are two weak hyper-charge states, $\widetilde{\tau}_{R}$ and $\widetilde{\tau}_{L}$. For the fermions the
chiral symmetry assures that both weak hyper-charge states are degenerate in mass. However this symmetry
does not apply to sfermions, since they are scalars, rather than fermions. Hence there is no reason to
expect that $\widetilde{\tau}_{R}$ and $\widetilde{\tau}_{L}$ would have the same mass.
Furthermore, mixing between the weak hyper-charge states
yields the physical states. The strength of the couplings involved in the mixing of states depend on the fermion mass and hence only
the third generation of the sleptons, $\widetilde{\tau}$, will mix 444This is also the case for the
squarks, where the third generation, the stop the and the sbottom, will also mix.. As a consequence of the mixing the lightest
$\widetilde{\tau}$, $\widetilde{\tau}_{1}$, would most likely be the lightest slepton, due to the seesaw
mechanism. The mass of the lightest physical (mixed) state would be smaller than the mass of
any un-mixed weak hyper-charge state. The cross section of the $\widetilde{\tau}$ also differs from the
one of the ${\tau}$, not only due to phase-space limitations - the $\widetilde{\tau}$ being more massive
than the ${\tau}$ - but also due to the mixing.
In $e^{+}e^{-}$ colliders, assuming R-parity conservation, the $\widetilde{\tau}$ will be pair-produced, with
contribution of the s-channel only, via $Z^{0}$/$\gamma$ exchange. The strength of the $Z^{0}$/$\gamma$ $\widetilde{\tau}$ $\widetilde{\tau}$ coupling depends on the $\widetilde{\tau}$ mixing, reaching its minimum value when
the coupling $\widetilde{\tau}_{1}$ $\widetilde{\tau}_{1}$ $Z^{0}$ vanishes, which will lead to the worst possible
scenario in $\widetilde{\tau}$ and, in general, in slepton searches.
Another property making the search of $\widetilde{\tau}$ the worst case, is the fact that its SM partner
is unstable. It decays before it can be detected, and, as a further complication, some of its decay products are
undetectable neutrinos.
This on one hand makes the identification
more difficult than the direct decay to electrons or muons, and on the other hand,
since the decay products are only partially detectable, that blurs kinematic signatures.
The search of a light $\widetilde{\tau}$ is also theoretically motivated: SUSY models with a light
$\widetilde{\tau}$ can accommodate the observed relic density, by enhancing the $\widetilde{\tau}$-neutralino coannihilation process.
4 Limits at LEP, LHC and previous ILC studies
The most model-independent limit on the $\widetilde{\tau}$ mass comes from the LEP experiments [7].
They set a minimum value that ranges from 87 to 93 GeV depending on the mass difference between the $\widetilde{\tau}$ and
the neutralino, not smaller than 7 GeV. These limits, shown in figure 1, are valid for any mixing and any value of the model-parameters, other than the two masses explicitly shown in the plot.
An analysis by the DELPHI experiment, targeted at low mass differences, excludes a $\widetilde{\tau}$ with mass below 26.3 GeV, for any mixing, and any mass difference larger than the $\tau$ mass [8].
At the LHC, ATLAS and CMS have determined limits on the $\widetilde{\tau}$ mass, analysing data from Run 1 and
Run 2 [9] [10].
These limits, however, are only valid under certain assumptions. Both experiments assume $\widetilde{\tau}_{R}$
and $\widetilde{\tau}_{L}$ to be mass-degenerate. This is a very unlikely scenario, the running of the $\widetilde{\tau}_{R}$
and $\widetilde{\tau}_{L}$ masses from the GUT scale to the weak scale follows renormalisation group equations with $\beta$-functions
that are inevitably different for the two weak hyper-charge states.
They also assume that there is no mixing between the weak hyper-charge eigenstates, which is again very improbable.
The mixing will yield to cross section of the lightest physical state smaller than that of any unmixed state.
Putting together $\widetilde{\tau}_{R}$ and $\widetilde{\tau}_{L}$ by adding the cross sections, ATLAS excludes
$\widetilde{\tau}$ masses between approximately 120 and 390 GeV for a nearly massless neutralino555100$\%$
decay to $\widetilde{\tau}$ and neutralino is assumed, as it is in the analysis presented in this paper.
Under the same conditions, CMS extends the lower limit to 90 GeV closing the gap with the LEP limit. Analysis of a pure
$\widetilde{\tau}_{L}$ pair production set limits between 150 and 310 GeV from ATLAS data and up to 125 GeV from CMS; both limits again
assume a nearly massless neutralino.
The future HL-LHC should provide an improvement on the $\widetilde{\tau}$ limits provided by ATLAS and
CMS, not only because of an increase of the luminosity but also because of an expected gain in sensitivity
to direct $\widetilde{\tau}$ production due to the use of different analysis methods.
Simulation studies have already been performed in both
experiments [11] [12]. Upper limits for $\widetilde{\tau}$ masses
are indeed increased by about 300 GeV, but they suffer from the same constraints as the previous
studies. ATLAS adds limits for pure $\widetilde{\tau}_{R}$ pair production, that could be considered
the closest case to the physical lightest $\widetilde{\tau}$ since it is likely to be the lightest of the two
weak hyper-charge states and is the one with the lowest cross section. These limits, presented in figure 2,
show that no discovery
potential is expected in this case, only exclusion potential. They do not have exclusion potential for $\widetilde{\tau}$ co-annihilation
scenarios, a highly motivated scenario if SUSY is to provide a viable DM candidate:
Such a scenario requires that the $\widetilde{\tau}$-LSP mass difference is small, $\lesssim$ 10 GeV.
$\widetilde{\tau}$ searches at the ILC have been also performed in previous studies [13].
They assume an integrated luminosity of 500 fb${}^{-1}$ at $\sqrt{s}=500$ GeV and average
beam polarisations of $P(e^{-},e^{+})=(+80\%,-30\%)$. The same beam polarisations are used in the
current studies, since the signal to background ratio is favoured, but the luminosity is increased
to the one corresponding the the foreseen running scenario, 1.6 ab${}^{-1}$.
The limits presented in that study do not have a dedicated
analysis for low mass differences between the $\widetilde{\tau}$ and the LSP, $\Delta M$, and are only
valid down to $\Delta M$ 3-4 GeV. The exclusion limit goes up to 240 GeV with a discovery potential
up to 230 GeV for large mass differences.
5 Conditions and tools
The study was done assuming an integrated luminosity of 1.6 ab${}^{-1}$ at $\sqrt{s}=500$ GeV
with beam polarisations $P(e^{-},e^{+})=(+80\%,-30\%)$, according to the H-20
running scenario in the ILC500 benchmark [14] 666$\sqrt{s}$=500 GeV, total integrated
luminosity 4 ab${}^{-1}$ with 1.6 ab${}^{-1}$ for $P(e^{-},e^{+})=(-80\%,+30\%)$ and
$P(e^{-},e^{+})=(+80\%,-30\%)$, 0.4 ab${}^{-1}$ for $P(e^{-},e^{+})=(+80\%,+30\%)$ and
$P(e^{-},e^{+})=(-80\%,-30\%)$. The polarisation was selected due to the
increase of the signal to background ratio, as will be shown in the description of the
analysis.
The study assumes R-parity conservation and a 100$\%$ decay of the $\widetilde{\tau}$ to ${\tau}$
and the lightest neutralino, the LSP in this case.
In order to select the worst scenario, the $\widetilde{\tau}$ mixing angle was set to 53 degrees,
corresponding to the lowest cross section due to the suppression of the s-channel with Z exchange
in the $\widetilde{\tau}$ pair production.
The SGV fast detector simulation [15], adapted to the ILD concept [16] at ILC, was used for detector simulation and event
reconstruction. Signal events were generated inside SGV using Pythia 6.422 [17].
The generated background event samples were those of the standard “DBD” production [18]. They were generated
with Whizard 1.95 [19], and were written in stdhep format. These files were read by SGV, and passed through the same detector simulation and reconstruction as the signal samples.
The relevant information of the reconstructed events were written to Root files.
6 Signal characterisation
Assuming R-parity conservation and assuming that the $\widetilde{\tau}$ is the NLSP, $\widetilde{\tau}$’s
will be produced in pairs via $Z^{0}$/$\gamma$ exchange in the s-channel and they will decay to
a ${\tau}$ and an LSP (assuming mass differences above the mass of the ${\tau}$, as is done
in this study).
The LSP, as already mentioned, is stable and weakly interacting, hence it will leave the detector
without being detected. The ${\tau}$, with a lifetime of the order of 2.9 x 10${}^{-13}$ s, will decay
before leaving any signal in the detectors. The only detectable activity in the signal events is therefore
the decay products of the two ${\tau}$’s.
Signal events are then characterised by a large missing energy and momentum, not only due to the
invisible LSPs but also to the neutrinos from both ${\tau}$-decays. Since the $\widetilde{\tau}$’s are scalars
and hence isotropically produced, these events have a large fraction of the detected activity in the
central region of the detector. The $\widetilde{\tau}$’s must also be rather heavy, so they will not have a large boost in the lab-frame,
and since the LSP is also quite heavy, the direction of the $\widetilde{\tau}$ does not strongly correlate to that of
the visible $\tau$ after the decay. As a consequence the two $\tau$-leptons are expected to go in directions quite independent
of each other resulting in events with un-balanced transverse momentum, large angles between the two $\tau$-lepton directions and absence of
forward-backward asymmetry.
These properties are however not necessarily present in any event - the two $\tau$’s could accidentally happen
to be back to back, for example.
7 Main background sources
The main sources of background, given the generic signal topology, i.e. two $\tau$’s and an anseen
recoil system, are SM processes with real or fake missing energy. They can be classified into
“irreducible” and “almost irreducible” sources. The first are events with two $\tau$’s
and real missing energy, i.e. neutrinos. The main contribution to this group are ZZ events
with one Z decaying to two neutrinos and the other to two $\tau$’s, and fully leptonic WW events,
where at most one of the W’s decays to $\tau$ and neutrino.
ZWW and ZZZ events decaying to two $\tau$’s and four neutrinos
are not an issue due to their low cross sections.
The second group of events are those which are not really two $\tau$’s and neutrinos, but after reconstruction looks very similar.
They are events with two soft $\tau$-jets, with two other leptons plus true
missing energy or two $\tau$’s plus fake missing energy.
The main sources for events with true missing energy in this group are $\tau$
pair production, with the $\tau$’s decaying such that most energy goes to the neutrinos, ZZ events where one of the Z’s decays to an electron or a muon
pair and the other one to neutrinos, and WW events
with each W decaying to an electron or muon and a neutrino.
The background with fake missing energy comes mainly from $\tau$ pair
production with Initial State Radiation (ISR) at very low angles, events with two $\tau$’s and
two very low angle electrons (below the acceptance of the BeamCAL) in the final state and events where two $\tau$’s are produced
by a $\gamma\gamma$ interaction and not from an $e^{+}e^{-}$ one; in that case there
is not really missing energy but an initial state with much less energy
than that of the electron-positron interactions.
8 General cuts
Taking into account the signal signature and the main background sources,
different cuts have been designed in order to separate the signal from
the background.
Since the study was focused on small differences between the $\widetilde{\tau}$ and LSP
masses777Larger mass differences were also analysed in order
to cross-check and try to improve the limits from the previous studies.,
$M_{\tau}$ $<$ $\Delta M$ $<$ 11 GeV, the absence of signal in the calorimeter closest to the beam pipe
(the BeamCAL) was required as a pre-selection step before applying the following cuts.
The first group of cuts are those in properties that the
$\widetilde{\tau}$-events must have. Since the two LSP’s from the
$\widetilde{\tau}$-decays are invisible to the detector, signal events
have to have a missing energy greater than two times the mass of the
LSP and the visible mass can not be bigger than this quantity. Also a
cut in the maximum total momentum, smaller than 70$\%$ the beam momentum
is applied for the same reason. The multiplicity of the event can also
be constrained taking into account that the visible part comes only from
the decays of the two $\tau$’s and maybe an ISR photon. For that reason
the number of charged particles is asked to be between 2 and 6, with
only 2 or 3 clusters identified as $\tau$’s and a total charge between
-1 and 1. An specific algorithm for $\tau$-identification was also applied.
This algorithm consists in a first set of conditions requiring to have a pattern
of charged tracks typical for $\tau$-decay, viz. exactly two jets
(obtained with the DELPHI tau-finder [8]) with
charged particles, 1 or 3 charged particles in each charged jet,
jet-charge $\pm$1, and opposite charge between both jets. A set of conditions
is devoted to the reduction of background from sources with
leptons not from $\tau$-decays. To reduce the background of single W
production in e$\gamma$ events, with W decaying to $\tau$ and neutrino,
none of the jets should consist of a single positron (this cut takes
into account the polarisation selected for the study). This background
together with the background from $WW\rightarrow e\nu_{e}\mu\nu_{\mu}$ and
from $\gamma\gamma$ events with a beam-remnant deflected to larger
angles is further reduced by rejecting those events in which the most energetic
jet consists of a single electron. The two charged jets were also required to
neither be made by single leptons with the same flavour nor to have one hadronic
jet and one leptonic.
This algorithm reduces the signal efficiency by 38$\%$ but with
a reduction of the background of the order of 94$\%$, depending on the
region of the SUSY parameter space one is working with.
The last cut in this first group of cuts is on the maximum momentum of the jets. Since
the $\widetilde{\tau}$-decay is a two body decay, it is possible to
determine the maximum and minimum momentum of each of the decay products as
a function of the $\widetilde{\tau}$ mass, the mass of LSP
and the centre-of-mass energy of the collider. The cut in the minimum momentum
can not be applied due to the presence of neutrinos in the $\tau$ decay, with
the corresponding decrease of observable momentum. The maximum value can be used even if
it is smeared by the missed neutrinos. The expression for the maximum
jet momentum is given by:
$$P_{max}=\frac{\sqrt{s}}{4}(1-\left(\frac{M_{LSP}}{M_{\widetilde{\tau}}}\right)^{2})(1+\sqrt{(1-\frac{4M_{\widetilde{\tau}}~{}^{2}}{s}})$$
(1)
Excluding the cut applied by the $\tau$-identification algorithm, the signal
efficiency for each of the cuts is at least 95$\%$ at all model points.
A second group of cuts is based on those properties that the
$\widetilde{\tau}$-events might have, but will rarely be
present in background events. As already pointed out, the $\widetilde{\tau}$’s are
scalars, and therefore isotropically produced, while the backgrounds are
either fermions or vector bosons, and tend to be produced at small angles to the beam
axis. This allows to set cuts requiring events with high missing transverse
momentum, large acoplanarity, high angles to the beam and high
values of the variable $\rho$.
The latter is calculated by first projecting the event on the x-y
(transverse) plane, and calculating the thrust axis in that plane. $\rho$
is then the transverse momentum (in the plane) with respect to the
thrust axis.
The cut in $\rho$ helps to reject events with two $\tau$’s back-to-back in the transverse
projection with the visible part of the decay of one of the $\tau$’s in the direction of its parent,
while the other $\tau$ decays with the invisible $\nu$ closely aligned with the direction of its parent.
These events fake the signal topology,
having a large missing transverse momentum and high acoplanarity, but would have
a small value of $\rho$. The values at which the cut in
these properties is set depends on the $\widetilde{\tau}$ mass and the mass difference
between the $\widetilde{\tau}$ and the LSP. Cutting in these properties has a certain
cost in efficiency but improves the signal-to-background ratio.
The third group of cuts uses properties of some of the “almost irreducible” sources of
background. WW events with each of the W’s decaying to a lepton (not $\tau$) and a neutrino are
highly forward-backward asymmetric; they can be almost entirely removed by requiring the sum of the product of the
charge and the cosine of the polar angle of the two most energetic jets to be above -1.
ZZ events with one Z decaying to two neutrinos and the second one to a electron or muon
pair are highly suppressed demanding a visible mass more than 4 GeV from the Z mass,
since the visible mass in those events equals the Z mass quite precisely.
A last cut is based on a property that the signal often does not have, viz. sizeable energy
detected at low angles to the beam. Events with more than 2 GeV detected at angles lower than 20
degrees to the beam axis are therefore rejected.
This cut is however not useful for small mass differences.
After applying these cuts the main sources of remaining background are WW events with each
W decaying to $\tau\nu$ and events with four fermions in the final state coming from
$\gamma\gamma$ interactions, mostly $\tau\tau$ events.
The selected polarisation plays an important role in the capability of excluding/discovering
the different regions of the SUSY space.
Table 1 shows the number of signal and background events for a specific spectrum point for
the two main ILC running polarisations and for unpolarised beams.
Since the polarisation of the $\tau$ coming from the $\widetilde{\tau}$ decays was not considered in
this study, the difference in the number of signal events comes only from the dependence of the
cross section on the polarisation. This is also the main factor for the difference in WW events, ee $\rightarrow\tau\nu\tau\nu$.
One can see that the signal-to-background ratio is clearly enhanced in the selected polarisation.
Taking the definition of exclusion at 95$\%$ CL as $S>2\sqrt{S+B}$, with S and B the number of signal and background
events respectively, it is also shown that unpolarised beams would allow neither exclusion nor discovery.
Polarisation is not only important in the enhancement of the signal over background but also
plays an important role in the parameter determination.
The cuts described above are mainly suited for mass differences up to 3 GeV.
When the mass difference is between 3 GeV and the mass of the $\tau$ 888For mass differences
below the mass of the $\tau$ the lifetime of the $\widetilde{\tau}$ increases exponentially and
the study has to be done based on a signature of long-lived particles travelling through the
detector. the kinematic of the signal events is very close to that of the $\gamma\gamma$ background
events and the described cuts are not enough for discovering/excluding the signal.
An additional cut was done based on the Initial State Radiation photons (ISR). Events with isolated photons
with high transverse momentum were selected, allowing to extension of the limits into the region under study.
This cut is effective against the remaining $\gamma\gamma$ background because these events become candidates due
to fake missing transverse momentum. If the presence of an ISR is requested, the incoming electron or positron that
emitted the ISR must have recoiled against the ISR.
Since this is a
scattering process, not an annihilation one, the electron (positron) is still present in the final state.
Therefore, if it is required to see a high transverse momentum ISR, the final state
electron (positron) will have acquired a transverse momentum big enough to be deflected into the BeamCAL, and
thus to have been rejected already at the pre-selection stage. On the other hand, if the ISR was emitted from
an electron or positron that was subsequently annihilated into a Z, as is the case for the signal process,
the transverse momentum of the ISR is included
in the decay products of the Z, and no signal is expected in the BeamCAL.
9 Exclusion/discovery limits
The exclusion and discovery limits extracted from this study are shown in figure 3.
They assume the lightest $\widetilde{\tau}$ to be the NLSP and the lightest neutralino the LSP,
and are valid for any $\widetilde{\tau}$ mixing angle. Results from previous ILC studies,
computed for 500 fb${}^{-1}$ total integrated luminosity, are also shown for comparison, as well
as an extrapolation of the current results from 1.6 ab${}^{-1}$ to 500 fb${}^{-1}$. The
comparison of these two curves shows that the extension of the limits is not only due to an increase of the
total integrated luminosity but also to an improvement of the analysis. The main reason of this improvement
is the application of individual limits depending on the $\widetilde{\tau}$ mass and the mass difference.
The previous studies were only making a difference for mass differences above or below 10 GeV and
were not optimised for the low mass difference region. Another difference in the analysis is
a change in the ${\tau}$-identification algorithm, excluding events with two jets consisting of single leptons of
the same flavour or one jet being hadronic and other leptonic, which was found to be necessary for the exclusion/discovery
of some points.
It is also relevant to compare these results with the current $\widetilde{\tau}$ limits coming from
LEP. Figure 4 shows this comparison. LEP limits are also valid for any value of the not shown
model parameters.
The projection of the limits in the M${}_{\widetilde{\tau}}$-$\Delta$M plane is shown in figure 5. The
region for mass differences below the mass of the $\tau$, not included in the current study, is shown for
completeness. In the region with $\Delta$M larger than M${}_{\tau}$ exclusion and discovery ILC limits
are compared to the ones from LEP. Since the LHC limits are highly model-dependent, the comparison in
this case have to be taken with care. Limits considering only the $\widetilde{\tau}_{R}$-pair production
are shown, as, while still being optimistic, they are the closest to the ones expected for the lightest
$\widetilde{\tau}$ at minimal cross-section.
For the region with $\Delta$M smaller than M${}_{\tau}$ results from LEP and LHC are shown.
LEP studies cover not only the region where the $\widetilde{\tau}_{1}$ travels through the
detector without decaying but also the region with decays at a certain distance from the production
vertex. In those regions acoplanar leptons, tracks with large impact parameters and kinked tracks
are looked for, depending on the $\widetilde{\tau}_{1}$ lifetime [20] [21].
Figure 6 extends the previous one adding the extrapolation of the ILC limits for the scenarios
with centre-of-mass energy 250 GeV and 1 TeV.
10 Outlook and conclusions
The capability of the ILC for excluding/discovering $\widetilde{\tau}$-pair production up to a
few GeV below the kinematic limit, without model dependencies and even in the worst scenario,
has been shown.
The study has been done assuming the $\widetilde{\tau}$ mixing angle to be the one corresponding to
the lowest cross section for unpolarised beams. This is also the mixing angle that
gives the smallest number of signal events when simply combining the samples with
polarisations $P(e^{-},e^{+})=(+80\%,-30\%)$ and $P(e^{-},e^{+})=(-80\%,-30\%)$ with equal
integrated luminosity, as it is planned in the ILC running scenarios. However, due to the
clear enhancement of the signal-to-background ratio with the polarisation $P(e^{-},e^{+})=(+80\%,-30\%)$,
as shown in table 1, only this dataset was used for the calculation of
the limits. The study will be extended taking into account the contribution of both polarisations. In this extension we will
consider different $\widetilde{\tau}$ mixing angles
for confirming the one corresponding to the worst scenario.
Without considering the polarisation of the ${\tau}$ coming from the $\widetilde{\tau}$ decay, as it
is done in the present study, the number of detected signal events
for each mixing angle and each beam polarisation depends only on the cross section for
$\widetilde{\tau}$-pair production in those conditions. However the signal efficiency is
affected by the ${\tau}$ polarisation due to the effect on the momentum distribution of the ${\tau}$-decays,
being softer or harder depending on the neutralino mixing angle. This effect will be also
considered in the extension of the study, being an important point in the determination of the
worst scenario.
The calculation of the exclusion/discovery limits in the region with mass differences below the
${\tau}$ mass, meaning an exponential increase of the $\widetilde{\tau}$ lifetime and consequently
a study of long-lived particles going through or decaying in different parts of the detector, is
also foreseen.
11 Acknowledgements
We would like to thank the LCC generator working group for producing the Monte Carlo samples used in this study.
We also thankfully acknowledge the support by the the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy EXC 2121 “Quantum Universe” 390833306.
This work has benefited from computing services provided by the German National Analysis Facility (NAF) [22].
References
[1]
S. P. Martin,
“A Supersymmetry primer,”
Adv. Ser. Direct. High Energy Phys. 18 (1998), 1-98
[arXiv:hep-ph/9709356 [hep-ph]].
[2]
J. Wess and B. Zumino,
“Supergauge Transformations in Four-Dimensions,”
Nucl. Phys. B 70 (1974), 39-50
[3]
H. P. Nilles,
“Supersymmetry, Supergravity and Particle Physics,”
Phys. Rept. 110 (1984), 1-162
[4]
H. E. Haber and G. L. Kane,
“The Search for Supersymmetry: Probing Physics Beyond the Standard Model,”
Phys. Rept. 117 (1985), 75-263
[5]
R. Barbieri, S. Ferrara and C. A. Savoy,
“Gauge Models with Spontaneously Broken Local Supersymmetry,”
Phys. Lett. B 119 (1982), 343
[6]
T. Behnke, J. E. Brau, B. Foster, J. Fuster, M. Harrison, J. M. Paterson, M. Peskin, M. Stanitzki, N. Walker and H. Yamamoto,
[arXiv:1306.6327 [physics.acc-ph]].
[7]
LEP SUSY Working Group, ALEPH, DELPHI, L3 and OPAL Collaborations,
“Combined LEP Selectron/Smuon/Stau Results, $183$-$208$ GeV,”
LEPSUSYWG/04-01.1,
http://lepsusy.web.cern.ch/lepsusy/www/sleptons_summer04/slep_final.html
[8]
J. Abdallah et al. [DELPHI],
“Searches for supersymmetric particles in e+ e- collisions up to 208-GeV and interpretation of the results within the MSSM,”
Eur. Phys. J. C 31 (2003), 421-479
[arXiv:hep-ex/0311019 [hep-ex]].
[9]
G. Aad et al. [ATLAS],
“Search for direct stau production in events with two hadronic $\tau$-leptons in $\sqrt{s}=13$ TeV $pp$ collisions with the ATLAS detector,”
Phys. Rev. D 101 (2020) no.3, 032009
[arXiv:1911.06660 [hep-ex]].
[10]
A. M. Sirunyan et al. [CMS],
“Search for direct pair production of supersymmetric partners to the $\tau$ lepton in proton-proton collisions at $\sqrt{s}=$ 13 TeV,”
Eur. Phys. J. C 80 (2020) no.3, 189
[arXiv:1907.13179 [hep-ex]].
[11]
[ATLAS],
“Prospects for searches for staus, charginos and neutralinos at the high luminosity LHC with the ATLAS Detector,”
ATL-PHYS-PUB-2018-048.
[12]
[CMS],
“Search for supersymmetry with direct stau production at the HL-LHC with the CMS Phase-2 detector,”
CMS-PAS-FTR-18-010.
[13]
M. Berggren,
“Simplified SUSY at the ILC,”
[arXiv:1308.1461 [hep-ph]].
[14]
T. Barklow, J. Brau, K. Fujii, J. Gao, J. List, N. Walker and K. Yokoya,
“ILC Operating Scenarios,”
[arXiv:1506.07830 [hep-ex]].
[15]
M. Berggren,
“SGV 3.0 - a fast detector simulation,”
[arXiv:1203.0217 [physics.ins-det]].
[16]
H. Abramowicz et al. [ILD Concept Group],
“International Large Detector: Interim Design Report,”
[arXiv:2003.01116 [physics.ins-det]].
[17]
T. Sjostrand, S. Mrenna and P. Z. Skands,
“PYTHIA 6.4 Physics and Manual,”
JHEP 05 (2006), 026
[arXiv:hep-ph/0603175 [hep-ph]].
[18]
T. Behnke, J. E. Brau, P. N. Burrows, J. Fuster, M. Peskin, M. Stanitzki, Y. Sugimoto, S. Yamada, H. Yamamoto, H. Abramowicz, et al.
“The International Linear Collider Technical Design Report - Volume 4: Detectors,”
[arXiv:1306.6329 [physics.ins-det]].
[19]
W. Kilian, T. Ohl and J. Reuter,
“WHIZARD: Simulating Multi-Particle Processes at LHC and ILC,”
Eur. Phys. J. C 71 (2011), 1742
[arXiv:0708.4233 [hep-ph]].
[20]
LEP SUSY Working Group, ALEPH, DELPHI, L3 and OPAL Collaborations,
“Stable Heavy Charged Particles”
LEPSUSYWG/02-05.1,
http://lepsusy.web.cern.ch/lepsusy/www/stable_summer02/stable_208.html
[21]
LEP SUSY Working Group, ALEPH, DELPHI, L3 and OPAL Collaborations,
“Combined LEP GMSB Stau//Smuon/Selectron Results, 189-208 GeV”
LEPSUSYWG/02-09.2,
http://lepsusy.web.cern.ch/lepsusy/www/gmsb_summer02/lepgmsb.html
[22]
A. Haupt and Y. Kemp
“The NAF: National analysis facility at DESY,”
J. Phys.: Conf. Ser. 219 (2010), 052007 |
N
C. Pich and F. Schwabl
Abstract
We calculate the Néel temperature $T_{N}$ for two-dimensional isotropic
dipolar Heisenberg antiferromagnets via linear spin-wave theory and a high
temperature expansion, employing the method of Callen. The theoretical
predictions for $T_{N}$ for K${}_{2}$MnF${}_{4}$, Rb${}_{2}$MnF${}_{4}$, Rb${}_{2}$MnCl${}_{4}$ and
(CH${}_{3}$NH${}_{3}$)${}_{2}$MnCl${}_{4}$ are in good agreement with the measured values.
pacs: PACS numbers: 75.10 J, 75.30 D, 75.30 G, 75.50 E
éel temperature for quasi two-dimensional dipolar antiferromagnets
{instit}
Institut für Theoretische Physik,
Physik-Department der Technischen Universität München,
D-85747 Garching, Federal Republic of Germany
Recently[1] it has been shown that long-range order is possible in
two-dimensional isotropic Heisenberg antiferromagnets due to the anisotropy
of the dipole-dipole interaction. The occurence of a finite energy gap goes in
hand with a nonvanishing order parameter and a finite Néel-temperature.
In the present paper a quantitative improvement of the theory is achieved by
means of Callen’s extension of the Tyablikov decoupling scheme [2].
In[1] we have used linear spin wave theory based on the
Holstein-Primakoff transformation to evaluate the magnon dispersion relation.
The evaluation of the Néel temperature by means of the temperature
independent dispersion relation leads to an overestimate of $T_{N}$. In reality
the magnon frequency softens with increasing temperature and thus the actual
transition temperature is lower. This feature is accounted for by an extension
of the Tyablikov decoupling scheme due to Callen[2]. In essence the
dependence on the magnitude of the spin $S$ is replaced by $\sigma$, the
temperature dependent order parameter. Even at $T=0$ the zero point
fluctuations lead to a reduction of $\sigma$ as compared to $S$.
The resulting transition temperature is lowered in comparison to the estimate
of Ref.[1] such that a satisfactory agreement between theory and
experiment is achieved.
The Hamiltonian of a dipolar antiferromagnet reads
$$H=-\sum_{l\neq l^{\prime}}\sum_{\alpha\beta}\left(J_{ll^{\prime}}\delta_{%
\alpha\beta}+A^{\alpha\beta}_{ll^{\prime}}\right)S_{l}^{\alpha}S_{l^{\prime}}^%
{\beta}~{},$$
(1)
with spins ${\bf S}_{l}$ at lattice sites ${\bf x}_{l}$. The first term in brackets
is the exchange interaction $J_{ll^{\prime}}$ and the second the usual dipole-dipole
interaction.
We consider a square lattice in the $xy$ plane with lattice constant $a$ and
the spins orientated alternatingly along the $z$ axis. The out-of-plane
orientation is the classical ground state for the isotropic dipolar
antiferromagnet with a nearest-neighbor exchange energy $|J|$ much larger than
the dipole energy [1].
Let us introduce the retarded double-time Green functions according to Callen
[2]
$$\displaystyle G^{1}({\bf R}_{k}-{\bf R}_{0},t)$$
$$\displaystyle=$$
$$\displaystyle-i\Theta(t)\langle[S^{+}_{k}(t),e^{bS^{z}_{0}}S^{-}_{0}(0)]%
\rangle\equiv\langle\langle S^{+}_{k}|e^{bS^{z}_{0}}S^{-}_{0}\rangle\rangle,$$
(2a)
$$\displaystyle G^{2}({\bf R}_{k}-{\bf R}_{0},t)$$
$$\displaystyle=$$
$$\displaystyle-i\Theta(t)\langle[S^{-}_{k}(t),e^{bS^{z}_{0}}S^{-}_{0}(0)]%
\rangle\equiv\langle\langle S^{-}_{k}|e^{bS^{z}_{0}}S^{-}_{0}\rangle\rangle,$$
(2b)
which obey the following Fourier transformed equation of motion[3, 4]
$$\displaystyle\omega\langle\langle S^{+}_{k}|e^{bS^{z}_{0}}S^{-}_{0}\rangle%
\rangle_{\omega}$$
$$\displaystyle=$$
$$\displaystyle\Theta_{b}{}~{}+~{}\langle\langle[S^{+}_{k},H]|e^{bS^{z}_{0}}S^{-%
}_{0}\rangle\rangle_{\omega}~{},$$
(3a)
$$\displaystyle\omega\langle\langle S^{-}_{k}|e^{bS^{z}_{0}}S^{-}_{0}\rangle%
\rangle_{\omega}$$
$$\displaystyle=$$
$$\displaystyle\langle\langle[S^{-}_{k},H]|e^{bS^{z}_{0}}S^{-}_{0}\rangle\rangle%
_{\omega}~{},$$
(3b)
with the equal time commutator
$$\displaystyle\Theta_{b}$$
$$\displaystyle=$$
$$\displaystyle\langle[S^{+},e^{bS^{z}}S^{-}]\rangle.$$
Here ${\bf R}_{0}$ is a lattice point of sublattice one and ${\bf R}_{k}$ is a
general lattice vector. The higher order Green functions generated by the
commutator in Eqs. (3a) and (3b) are approximated by the Tyablikov decoupling
scheme
$$\langle\langle S^{z}_{l}S^{+}_{m}|e^{bS^{z}_{0}}S^{-}_{0}\rangle\rangle~{}~{}%
\to~{}~{}\langle S^{z}_{l}\rangle\langle\langle S^{+}_{m}|e^{bS^{z}_{0}}S^{-}_%
{0}\rangle\rangle.$$
(4)
As a consequence of translational symmetry the mean spin value is independent
of the lattice site for each sublattice: $\langle S^{z}_{l_{1}}\rangle=-\langle S^{z}_{l_{2}}\rangle=\sigma$ where $l_{1}\in L_{1}$ ($l_{2}\in L_{2}$) refers to
sublattice one (two). Using this approximation we obtain from Eqs. (3a) and Eq.
(3b) a set of four equations for the Green functions $G^{1}_{\bf q}$, $G^{1}_{{\bf q}+{\bf q}_{0}}$, $G^{2}_{\bf q}$ and $G^{2}_{{\bf q}+{\bf q}_{0}}$. The evaluation of
the order parameter and the magnon dispersion relation requires only two of
them:
$$\displaystyle\tilde{G}^{1}_{\bf q}(\omega)\equiv{1\over 2}(G^{1}_{\bf q}(%
\omega)+G^{1}_{{\bf q}+{\bf q}_{0}}(\omega))$$
$$\displaystyle=$$
$$\displaystyle\Theta_{b}{A^{(1)}+B^{(1)}\omega+C^{(1)}\omega^{2}+\omega^{3}%
\over(\omega^{2}-\epsilon_{1}^{2})(\omega^{2}-\epsilon_{2}^{2})},$$
(5)
with
$$\displaystyle-A^{(1)}$$
$$\displaystyle=$$
$$\displaystyle{1\over 2}A_{\bf q}(A_{{\bf q}+{\bf q}_{0}}^{2}-B_{{\bf q}+{\bf q%
}_{0}}^{2})+{1\over 2}A_{{\bf q}+{\bf q}_{0}}(A_{\bf q}^{2}-B_{\bf q}^{2})$$
$$\displaystyle\quad+(C_{\bf q}^{2}+C_{{\bf q}+{\bf q}_{0}}^{2})(A_{\bf q}+A_{{%
\bf q}+{\bf q}_{0}})+(C_{\bf q}^{2}-C_{{\bf q}+{\bf q}_{0}}^{2})(B_{{\bf q}+{%
\bf q}_{0}}-B_{\bf q}),$$
$$\displaystyle-B^{(1)}$$
$$\displaystyle=$$
$$\displaystyle{1\over 2}(A_{\bf q}^{2}-B_{\bf q}^{2}+A_{{\bf q}+{\bf q}_{0}}^{2%
}-B_{{\bf q}+{\bf q}_{0}}^{2}+8C_{\bf q}C_{{\bf q}+{\bf q}_{0}}),$$
$$\displaystyle C^{(1)}$$
$$\displaystyle=$$
$$\displaystyle{1\over 2}(A_{\bf q}+A_{{\bf q}+{\bf q}_{0}}).$$
Here ${\bf q}$ denotes the wave-vector of the chemical Brillouin zone and
${\bf q}_{0}={\pi\over a}(1,1,0)$. In the magnetic Brillouin zone which is
half the chemical there exist two distinct spin-wave branches with frequencies
$\epsilon_{i}$ ($i=1,2$) which read
$$\epsilon_{i}^{2}={1\over 2}(\Omega_{1}\pm\Omega_{2}),$$
(6)
$$\Omega_{1}=A_{\bf q}^{2}-B_{\bf q}^{2}+A_{{\bf q}+{\bf q}_{0}}^{2}-B_{{\bf q}+%
{\bf q}_{0}}^{2}+8C_{\bf q}~{}C_{{\bf q}+{\bf q}_{0}}~{},$$
$$\Omega_{2}^{2}=(A_{\bf q}^{2}-B_{\bf q}^{2}-A_{{\bf q}+{\bf q}_{0}}^{2}+B_{{%
\bf q}+{\bf q}_{0}}^{2})^{2}+16[C_{{\bf q}+{\bf q}_{0}}(A_{{\bf q}+{\bf q}_{0}%
}-B_{{\bf q}+{\bf q}_{0}})-C_{\bf q}~{}(A_{\bf q}-B_{\bf q})]$$
$$\times[C_{\bf q}~{}(A_{{\bf q}+{\bf q}_{0}}+B_{{\bf q}+{\bf q}_{0}})-C_{{\bf q%
}+{\bf q}_{0}}(A_{\bf q}+B_{\bf q})].$$
In Eq. (6) the coefficients
$$\displaystyle A_{\bf q}$$
$$\displaystyle=$$
$$\displaystyle\sigma(2J_{\bf q_{0}}-J_{\bf q}-J_{{\bf q}+{\bf q}_{0}})+\sigma(2%
A^{zz}_{{\bf q}_{0}}-A^{xx}_{\bf q}-A^{yy}_{{\bf q}+{\bf q}_{0}}),$$
(7a)
$$\displaystyle B_{\bf q}$$
$$\displaystyle=$$
$$\displaystyle\sigma(J_{{\bf q}+{\bf q}_{0}}-J_{\bf q})+\sigma(A^{yy}_{{\bf q}+%
{\bf q}_{0}}-A^{xx}_{\bf q}),$$
(7b)
$$\displaystyle C_{\bf q}$$
$$\displaystyle=$$
$$\displaystyle i\sigma A^{xy}_{\bf q},$$
(7c)
have been introduced. This result, Eq. (6), coincides with the magnon
frequencies derived by the Holstein-Primakoff transformation when $\sigma$ is
replaced by $S$ and the external magnetic field is set to zero[1]. But
there is a difference even at absolute zero
temperature, because the ground state of the antiferromagnet is not the Néel
state of fully aligned spins, i.e. $\sigma(T=0)<S$ as will be seen later.
Note that the magnon frequency scales with the order parameter $\sigma$, i.e.
the whole spectrum softens with increasing temperatures and vanishes at the
phase transition.
Now we turn to the evaluation of $\sigma$. For arbitrary spin $S$ the spin
expectation value is given by the well known relation
$$\displaystyle\langle S^{z}\rangle=S(S+1)-\langle(S^{z})^{2}\rangle-\langle S^{%
-}S^{+}\rangle.$$
(8)
For $S=1/2$ the order parameter can be calculated directly via the Green
functions of Eq. (2a) and (2b) with $b\equiv 0$ and $\Theta_{0}=2\sigma$. For
higher spin quantum numbers the above formula is not so helpful. Then
a convenient starting point is the following gerneralized thermal
average[2]
$$\displaystyle\psi(b)\equiv\langle e^{bS_{0}^{z}}S_{0}^{-}S_{0}^{+}\rangle,$$
(9)
by means of which a self-consistent system of equations can be derived by the
method of Callen. The above thermal average can be represented by the spectral
theorem[3, 4]:
$$\displaystyle\langle e^{bS_{0}^{z}}S_{0}^{-}S_{0}^{+}\rangle$$
$$\displaystyle=$$
$$\displaystyle\lim_{\delta\to 0}-{i\over 2\pi}\int_{-\infty}^{\infty}d\omega\ n%
(\omega){1\over N}\sum_{\bf q}\{\tilde{G}^{1}_{\bf q}(\omega+i\delta)-\tilde{G%
}^{1}_{\bf q}(\omega-i\delta)\}$$
(10)
$$\displaystyle=$$
$$\displaystyle\Theta_{b}(n-1/2),$$
where Eq. (5) has been used in the last step and $N$ denotes the total number
of spins. Here we have introduced the Bose occupation number
$$\displaystyle n(\omega)=\left(e^{\omega\over k_{B}T}-1\right)^{-1}~{},$$
and
$$\displaystyle n$$
$$\displaystyle=$$
$$\displaystyle{1\over N}\sum_{\bf q}{-A^{(1)}+C^{(1)}\epsilon_{1}\epsilon_{2}%
\over 2\epsilon_{1}\epsilon_{2}(\epsilon_{1}+\epsilon_{2})}+{A^{(1)}+C^{(1)}%
\epsilon_{1}^{2}\over\epsilon_{1}(\epsilon_{1}^{2}-\epsilon_{2}^{2})}n(%
\epsilon_{1})+{A^{(1)}+C^{(1)}\epsilon_{2}^{2}\over\epsilon_{2}(\epsilon_{2}^{%
2}-\epsilon_{1}^{2})}n(\epsilon_{2})~{}.$$
(11)
The right hand side of Eq. (11) depends on $\sigma$ only via the occupation
numbers $n(\epsilon_{i})$. For spin $S=1/2$ and vanishing parameter $b$ the
thermal average [Eq. (9)] represents the number of spin wave excitations,
which reduce the staggered magnetization from the totally ordered Néel state,
not only for finite but also for zero temperature.
For arbitrary spin one has to express $\sigma$ in terms of $n$ which can be
achieved by the method of Callen with the result[2, 5]:
$$\sigma=(S+{1\over 2}){(n+1/2)^{2S+1}+(n-1/2)^{2S+1}\over(n+1/2)^{2S+1}-(n-1/2)%
^{2S+1}}-n.$$
(12)
Eqs. (11) and (12) constitute a self-consistent system of equations for $n$ and
the spin expectation value $\sigma$.
Let us now discuss the dispersion relation for $T=0$. In this limit Eq. (11)
reduces to
$$\displaystyle n_{0}\equiv n(T=0)={1\over N}\sum_{\bf q}{-A^{(1)}+C^{(1)}%
\epsilon_{1}\epsilon_{2}\over 2\epsilon_{1}\epsilon_{2}(\epsilon_{1}+\epsilon_%
{2})},$$
(13)
where the right-hand side is independent of $\sigma$. Let us denote the spin
expectation value at zero temperature by $\sigma_{0}=\sigma(T=0)$ which is found
by inserting Eq. (13) into (12). Knowing $\sigma_{0}$ one obtains for the
staggered magnetization $N(0)$ at $T=0$:
$$\displaystyle N(0)=g\mu_{B}N\sigma_{0}.$$
(14)
One can convince oneself that Eq. (14) for large $S$ coincides with the
expression derived by the Holstein-Primakoff transformation[1] (see
also [5]). This must be so because the latter is an expansion
in $1/S$. In addition we derive an energy gap (q $=0$) from Eq. (6)
$$E_{0}^{\sigma}\equiv\epsilon_{1}(T=0)=\epsilon_{2}(T=0)=2\sigma_{0}\sqrt{A^{zz%
}_{\bf q_{0}}-A^{\rho\rho}_{{\bf q}_{0}}}\sqrt{(J_{{\bf q}_{0}}-J_{0})-(A^{%
\rho\rho}_{0}-A^{zz}_{{\bf q}_{0}})}~{},$$
(15)
with
$$\displaystyle A^{\rho\rho}_{0}=A^{xx}_{0}=A^{yy}_{0},\qquad A^{\rho\rho}_{{\bf
q%
}_{0}}=A^{xx}_{{\bf q}_{0}}=A^{yy}_{{\bf q}_{0}}.$$
This is of the same form as the result from the Holstein-Primakoff
transformation $E_{0}$[1] except for the prefactor which is smaller
by the ratio $\sigma_{0}/S$.
Now we turn to the evaluation of the transition temperature $T_{N}$, i.e.
consider the limit $\sigma\to 0$.
Since the spin-wave energy (Eq. 6) is proportional to $\sigma$ the Bose
occupation numbers can be replaced by their classical limit:
$$\displaystyle n(\epsilon_{i})\to{k_{B}T\over\epsilon_{i}}~{}~{}.$$
If this is inserted into Eq. (11) together with $\sigma\to 0$ one obtains
$$\displaystyle n={k_{B}T_{N}\over N\sigma}\sum_{\bf q}{-\tilde{A}^{(1)}\over(%
\tilde{\epsilon}_{1}\tilde{\epsilon}_{2})^{2}}~{}~{}.$$
(16)
To keep track of the $\sigma$ dependence we have introduced the
$\sigma$-independent quantities $\tilde{A}^{(1)}=A^{(1)}/\sigma^{3}$ and
$\tilde{\epsilon}_{i}=\epsilon_{i}/\sigma$. According to Eq. (16) $n$ increases
indefinitly with $\sigma\to 0$ and thus the second relation between $\sigma$
and $n$, Eq. (12), becomes[2]
$$\sigma={S(S+1)\over 3}{1\over n}.$$
(17)
Combining Eqs. (16) and (17) we obtain an explicit expression for the Néel
temperature:
$$\displaystyle T_{N}$$
$$\displaystyle=$$
$$\displaystyle{S(S+1)\over 3k_{B}}F^{-1}$$
(18)
with
$$\displaystyle F$$
$$\displaystyle=$$
$$\displaystyle{1\over N}\sum_{\bf q}{-\tilde{A}^{(1)}\over(\tilde{\epsilon}_{1}%
\tilde{\epsilon}_{2})^{2}}~{}~{}.$$
For purely isotropic antiferromagnets the coefficient $F$ diverges,
excluding long-range order at finite temperature in two dimensions in accord
with the Hohenberg-Mermin-Wagner theorem[6, 7].
In the presence of the dipolar interaction there is an energy gap and $T_{N}$
becomes finite. If the dipolar interaction is weak in comparison with the
exchange energy $(g\mu_{B})^{2}/a^{3}\ll|J|$ and if the argument of the summation is
approximated by its small q limit one obtains
$$\displaystyle T_{N}\sim{|J|\over ln{|J|\over E_{0}^{\sigma}}}~{},$$
(19)
which coincides with the analogous formula derived by the Holstein-Primakoff
transformation (Eq. (18) in Ref.[1]). In the general case the
above sum [Eq. (18)] is evaluated with the full dispersion relation Eq. (6)
and by computing $100\times 100$ points in the 2D Brillouin zone and
determining the other points by linear extrapolation. The dipole sums have been
calculated via Ewald summation[8].
Now we apply our theory to real quasi two-dimensional antiferromagnets.
Prominent examples of almost two-dimensional antiferromagnets are the
tetragonal antiferromagnetic halides K${}_{2}$MnF${}_{4}$, Rb${}_{2}$MnF${}_{4}$,
Rb${}_{2}$MnCl${}_{4}$ and (CH${}_{3}$NH${}_{3}$)${}_{2}$MnCl${}_{4}$. In these quadratic layer
structures the out-of-plane exchange interaction is neglegible in comparison to
the in-plane exchange interaction (about $10^{-4}$) [9, 10] whereas
the dipole energy is larger by an order of magnitude. This two-dimensional
character has been shown experimentally by the absence of any dispersion along
the $z$-direction[10]. For these halides the measured exchange energy
$|J|$, the lattice constant $a$, the energy gap $E^{\rm exp}_{0}$, the spin-flop
field $H^{\rm exp}_{\rm sf}$ and the transition temperature
$T_{N}^{\rm exp}$[10, 11, 12] are listed in table I. The spin-flop field
$H_{\bf sf}$ is the critical magnetic field at which the antiferromagnetic
Néel ground state changes to the spin-flop ground state. It can be calculated
by adding to the Hamiltonian, Eq. (1), the Zeeman energy[13].
From the full dispersion-relation the spin-flop field $H_{\bf sf}^{\sigma}$ is
defined by that field for which the magnon energy vanishes
$$H_{\bf sf}^{\sigma}={1\over g\mu_{B}}E_{0}^{\sigma},$$
(20)
in close analogy to the formula obtained by the Holstein-Primakoff method (Eq.
(11) in Ref.[1]). The table also
contains the theoretical energy gaps $E^{\sigma}_{0}$ and $E_{0}$ calculated via Eq.
(15) and Eq. (9) of Ref.[1], the resulting spin-flop fields
$H^{\sigma}_{\rm sf}$ and $H_{\bf sf}$ via Eq. (20) and Eq. (11) of
Ref.[1] and the theoretical transition temperature $T_{N}$ [Eq. (18)].
All these substances have spin $S=5/2$, which yields from Eq. (13) and (12)
$\sigma_{0}=2.30$ for pure isotropic antiferromagnets. This value is increased
only neglegibly by the dipolar interaction as can be seen from table I.
We find a good agreement with the measured the Néel temperature although our
theory accounts only for the dipolar interaction and no other anisotropy.
Corrections due to dipolar interactions between different planes
are neglegible because of the large lattice constant in $z$-direction; e.g.
the energy gap for K${}_{2}$MnO${}_{4}$ [Eq. (6)] is altered by
$$\displaystyle E^{\sigma}_{0}(3D)=E^{\sigma}_{0}(2D)[1+{\cal O}(10^{-5})],$$
(21)
which justifies the application of the two-dimensional model. Note that the
nearest-neighbor exchange energy $J$ is the only parameter entering in our
theory. Experimentally this parameter has been derived by fitting the measured
spin wave spectrum with a dispersion relation which is different of ours [Eq.
(6)].
For the halides listed in table I, the energy gap obtained from Eq. (15) and
the transition temperature are lower than the experimental values. The
following reasons may be responsible for that: (i) In the Holstein-Primakoff
approximation the softening of the magnons is neglected entirely. This leads to
an overestimate of $T_{N}$. In the Callen method the magnons soften in the
entire Brillouin zone, thus particularly near the phase transition the
softening is overestimated and leads to a $T_{N}$ which is somewhat too low.
(ii) A small readjustment of $J$ could be necessary if our dispersion relation,
Eq. (6), is used to fit the data.
(iii) A small additional anisotropy from the crystal field might be present as
suggested by[14].
Acknowledgements.This work has been supported by the German Federal Ministry
for Research and Technology (BMFT) under the contract number 03-SC2TUM.
References
[1]
C. Pich and F. Schwabl, Phys. Rev. B 47, 7957 (1993).
[2]
H.B. Callen, Phys. Rev 130, 890 (1963).
[3]
K. Elk and W. Gasser, Die Methode der Greenschen Funktionen
in der Festkörperphysik, (Akademie-Verlag, Berlin, 1979).
[4]
D.N. Zubarev, Usp. Fiz. Nauk. 71, 71 (1960) [Sov. Phys.
Usp. 3, 320 (1960)].
[5]
D.A. Yablonskiy, Phys. Rev. B 44, 4467 (1991).
[6]
P.C. Hohenberg, Phys. Rev. 158, 383 (1967).
[7]
N.D. Mermin and H.Wagner, Phys. Rev. Lett. 17, 1133
(1966).
[8]
L. Bonsal and A.A. Maradudin, Phys. Rev. B 15, 1959
(1977).
[9]
H.W. De Wijn, L.R. Walker, and R.E. Walstedt, Phys. Rev. 8, 285 (1973).
[10]
J.R. Birgeneau, H.J. Guggenheim, and G. Shirane, Phys. Rev. B
8, 304 (1973).
[11]
A.F.M. Arts and H.W. De Wijn, Magnetic Properties Of Layered
Transition Metal Compounds ed. by L.J. De Jongh, (Kluwer Academic Puplishers,
1990), p. 191ff.
[12]
B. Schröder, V. Wagner, N. Lehner, K.M. Kesharwani, and R.
Geick, Phys. stat. sol. (b) 97, 501 (1980).
[13]
In the presence of the Zeeman term $-g\mu_{B}H_{0}\sum_{l}S_{l}^{z}$
the quantities $\Omega_{1}$ and $\Omega_{2}^{2}$ in the spin-wave energy of Eq. (6)
contain the additional terms $2(g\mu_{B}H_{0})^{2}$ and $4(g\mu_{B}H_{0})^{2}[(A_{\bf q}+A_{{\bf q}+{\bf q}_{0}})^{2}-(B_{\bf q}+B_{{%
\bf q}+{\bf q}_{0}})^{2}]$
respectively.
[14]
C.M.J. van Uijen and H.W. De Wijn, Phys. Rev. B 30, 5265
(1984).
[15]
H.W. De Wijn, L.R. Walker, S. Geschwind, and H.J. Guggenheim,
Phys. Rev. 8, 299 (1973).
[16]
Landolt-Börnstein, Numerical Data and Functional
Relationship in Science and Technology, Vol. 12 (Springer-Verlag, 1978).
[17]
J.R. Birgeneau, H.J. Guggenheim, G. Shirane, Phys. Rev. B 1, 2211 (1970). |
Sample variance, source clustering and their influence on the counts of faint radio sources
Ian Heywood${}^{1,2}$, Matt J. Jarvis${}^{1,3}$ and James J. Condon${}^{4}$
${}^{1}$Astrophysics, Department of Physics, University of Oxford, Keble Road, Oxford, OX1 3RH
${}^{2}$Department of Physics and Electronics, Rhodes University, PO Box 94, Grahamstown, 6140, South Africa
${}^{3}$Physics Department, University of the Western Cape, Private Bag X17, Bellville 7535, South Africa
${}^{4}$National Radio Astronomy Observatory, 520 Edgemont Road, Charlottesville, VA 22903, USA
ianh@astro.ox.ac.uk
(Accepted 2013 February 08. Received 2013 February 07; in original form 2013 January 28)
Abstract
The shape of the curves defined by the counts of radio sources per unit area as a function of their flux density was one of the earliest cosmological probes. Radio source counts continue to be an area of astrophysical interest as they can be used to study the relative populations of galaxy types in the Universe (as well as investigate any cosmological evolution in their respective luminosity functions). They are also a vital consideration for determining how source confusion may limit the depth of a radio interferometer observation, and are essential for characterising the extragalactic foregrounds in Cosmic Microwave Background experiments. There is currently no consensus as to the relative populations of the faintest (sub-mJy) source types, where the counts show a turn-up. Most of the source count data in this regime are gathered from multiple observations that each use a deep, single pointing with an interferometric radio telescope. These independent count measurements exhibit large amounts of scatter (factors of order a few) that significantly exceeds their respective stated uncertainties. In this article we use a simulation of the extragalactic radio continuum emission to assess the level at which sample variance may be the cause of the scatter. We find that the scatter induced by sample variance in the simulated counts decreases towards lower flux density bins as the raw source counts increase. The field-to-field variations make significant contributions to the scatter in the measurements of counts derived from deep observations that consist of a single pointing, and could even be the sole cause at $>$100 $\mu$Jy. We present a method for evaluating the flux density limit that a radio survey must reach in order to reduce the count uncertainty induced by sample variance to a specific value. We also derive a method for correcting Poisson errors on source counts from existing and future deep radio surveys in order to include the uncertainties due to the cosmological clustering of sources. A conclusive empirical constraint on the effect of sample variance at these low luminosities is unlikely to arise until the completion of future large-scale radio surveys with next-generation radio telescopes.
keywords:
galaxies: general – galaxies: statistics – radio continuum: galaxies
††pagerange: Sample variance, source clustering and their influence on the counts of faint radio sources–Sample variance, source clustering and their influence on the counts of faint radio sources††pubyear: 2013
1 Introduction
Astrophysical radio emission, at least that which we observe away from the plane of the Milky Way, tends to originate from extragalactic objects at great distances. The differential counts111i.e. the number of sources per unit area on the sky with flux densities in the interval $S$ $\rightarrow$ $S+dS$. of these distant radio sources formed one of the earliest cosmological probes (e.g. Longair, 1966). In a non-expanding Euclidean universe222A Euclidean universe filled with sources of luminosity $L$ with number density $n$ contains $N$ = 4$\pi$$n$$d^{3}$/3 such sources out to distance $d$. Since the flux $S$ = $L$$/$4$\pi$$d^{2}$ it is trivial to show that $N(S)$ $\propto$ $S^{-3/2}$. populated with non-evolving sources we would see the integrated source counts $n(S)$ scaling with source flux density $S$ according to the relationship $n(S)~{}\propto~{}S^{-3/2}$. Observed departures from this relationship thus inform on the geometry of the Universe, and radio source counts were being invoked as early as the 1950s as one of the key evidential cruxes in the Big Bang versus Steady State debate (Ryle & Clarke, 1961), a cosmological contention that was eventually effectively ended by the discovery of the Cosmic Microwave Background (CMB) radiation (see e.g. Longair, 2004).
Source counts are thus an area of study that is almost as old as the science of radio astronomy itself. Today the primary interest in source counts (across the whole electromagnetic spectrum) stems from the need to determine the contributions that different galaxy populations make to the total number of objects in the Universe, in particular the relative numbers of star-forming galaxies and those harbouring active-galactic nuclei (AGN), and how the luminosity functions of these populations evolve over cosmic time. Radio source counts are essential for foreground subtraction in CMB experiments, and are also vital for determining where confusion becomes a fundamental limitation in a radio synthesis image. This may occur either due to classical confusion imposed by the sources at the faint end of the distribution that lie within the target field (Condon, 2009), or due to the presence of point spread function sidelobes associated with the brighter sources that lie in distant regions of the array primary beam (Smirnov et al, in prep.). This is particularly relevant at present as we await the arrival of the next-generation of radio instruments. These have been designed to deliver ultra-deep radio imaging and fast survey speeds by virtue of their extreme sensitivities and novel detector technologies, eventually culminating in the deployment of the Square Kilometre Array (SKA; Dewdney et al., 2009).
The faint end of the source count distribution is of particular interest, and there are many publications on the nature of the sub-mJy source population. The 1.4 GHz counts exhibit a turn up at $<$1 mJy (e.g. Windhorst et al., 1984; Hopkins et al., 1999), that persists at higher frequencies (e.g. Fomalont et al., 2002; Heywood et al. 2013). Many publications assert the nature of the source population at these levels and it is generally accepted that this is due to the increasing dominance of star-forming galaxies over AGN at these low luminosities (e.g. Seymour et al., 2008; Padovani et al., 2009), although radio-weak AGN and FR-I type galaxies may make still make significant contributions (Jarvis & Rawlings, 2004; Simpson et al., 2006; Gendre & Wall, 2008; Smolčić et al., 2009). There is however no clear consensus as to the relative fractions that these objects occupy.
Additional interest in the faintest end of the radio source counts was recently stimulated due to the balloon-borne Absolute Radiometer for Cosmology, Astrophysics and Diffuse Emission (ARCADE2; Fixsen et al., 2011) experiment which detected a significant excess in the sky brightness temperature at 3 GHz (Seiffert et al., 2011). These data suggest that if the result is genuine there must be a significant population of hitherto unknown faint radio emitters responsible for the excess (Vernstrom et al, 2011). Condon et al. (2012) performed a probability of deflection ($P(D)$; Scheuer, 1957) analysis of a confusion-limited Very Large Array (VLA) image at 3 GHz with a depth of 1 $\mu$Jy. Their results suggest that if the ARCADE2 result is indeed produced by a population of discrete radio sources then they are exceptionally numerous, not associated with known galaxies and must have 1.4 GHz flux densities of $<$0.03 $\mu$Jy.
Clearly there remains much to learn from surveys of extragalactic radio sources in the $\mu$Jy regime. Examination of the differential source counts from multiple surveys immediately highlights an issue that blights the current data: interpretation of the measured source counts at flux densities $<$1 mJy proves difficult when the derived source counts from survey to survey do not agree to within their respective errors. Possible explanations for the scatter include different calibration accuracies, uncertainties in the method of correcting for the array primary beam and smearing effects (e.g. Section 2.4, Fomalont et al., 2006), correction of detection thresholds due to resolved sources (e.g. Section 3.2, Bondi et al., 2008), as well as non-instrumental considerations such as the clustering bias of the sources in the field, i.e. due to sample variance.
Avoiding sample variance in faint source counts requires a large-area sky survey down to sub-mJy depths, which would require multiple, deep pointings on existing radio telescopes. Condon (2007) investigated the effect of sample variance by measuring the count fluctuations in 17 non-overlapping VLA pointings from the Spitzer First Look Survey and determined that the scatter due to sample variance is (1.07 $\pm$ 0.26) times the fluctuations expected in the absence of source clustering, concluding that the field-to-field variations are likely to be non-cosmic in origin.
We take a different approach to quantifying the effect of sample variance by exploiting an existing extragalactic sky simulation in order to present a simple measurement of the scatter induced in the measured counts. For an in-depth review of the subject of radio source counts we refer the reader to de Zotti et al. (2010).
2 Method and results
We investigate the effect of sample variance on the scatter in the measured source counts by comparing observationally-derived measurements with matched samples drawn from an existing simulation of the extragalactic radio sky.
The data points and associated error bars on Figure 1 show the Euclidean-normalized differential source counts from a variety of radio surveys at 1.4 GHz. The observational source count data that we use for comparison is drawn from fourteen individual studies, most of which are conveniently tabulated by de Zotti et al. (2010). The solid angle sky coverage of the individual surveys are partitioned into three bins: those that resulted from a single, deep pointing with the VLA, resulting in a nominal survey area of approximately 0.196 deg${}^{2}$ (hereafter known as the ‘deep’ bin; Mitchell & Condon, 1985; Biggs & Ivison, 2006; Fomalont et al., 2006; Kellermann et al., 2008; Owen & Morrison, 2008; Seymour et al., 2008; Ibar et al., 2009), surveys covering approximately 4-4.5 deg${}^{2}$ (hereafter referred to as the ‘broad’ bin; Ciliegi et al., 1999; Gruppioni et al., 1999; Hopkins et al., 2003), and finally surveys that were in general conducted over sky areas that exceeded the footprint of the simulation described below, and thus cannot be compared. These include the source counts derived from the FIRST survey (White et al., 1997), as well as those from the targeted surveys of Bridle et al. (1972). Also plotted on the figure are the counts from the 2 deg${}^{2}$ radio survey of the COSMOS field (Schinnerer et al., 2004; Bondi et al., 2008).
Counts from surveys in the deep and broad bins are plotted on Figure 1 as the blue and red points respectively. The smaller black points correspond to all other surveys. Immediately apparent from this selection and colour-coding alone is the large spread in source counts derived from the deep sample. This is the issue we aim to address with the simulation.
Our next step is to compare these measured values to matched samples of simulated source counts. For this we make use of the semi-empirical extragalactic sky simulation (hereafter referred to as ‘the simulation’) of Wilman et al. (2008; 2010)333The simulation database can be accessed online via http://s-cubed.physics.ox.ac.uk.. Briefly, the simulation uses observed and extrapolated luminosity functions to populate an evolving dark matter skeleton with various galaxy types. The 20 $\times$ 20 deg${}^{2}$ sky area of the simulation contains $\sim$260 million sources down to a flux density limit of approximately 10 nJy.
We extract multiple sky patches with areas of 0.196 and 4.5 deg${}^{2}$ from the simulation for comparison to the measured counts in both the deep and broad bins. This process results in 1936 and 64 unique source catalogues for the deep and broad samples respectively. For each of these simulated source subsets we compute the Euclidean-normalized differential source counts in 58 logarithmically-spaced flux density bins from 10 nJy to 100 Jy. For each bin the maximum and minimum value of the counts delineate a region on the left hand panel of Figure 1 that corresponds to the possible range of field-to-field fluctuations in the source counts of a survey of matched area. This is plotted for both the deep bin (blue area) and the broad bin (red area).
We stress that this process is not blighted by the biases inherent in deriving accurate counts from observations, such as those mentioned briefly in Section 1, and the scatter will be induced purely by the source clustering, itself governed by the underlying model dark matter density field upon which the simulated galaxy population is placed. Our chosen bin widths are well matched to those used in the observations: for every flux density bin used in the set of observations we calculate the ratio of that bin width to that of the simulated bin that is closest to it in terms of central flux, and the mean value of these ratios over all bins considered is 0.96 with a median value of 0.83.
The right hand panel of Figure 1 shows a zoomed-in region covering the 10 $\mu$Jy to 1 mJy flux density region. Again the blue points show the observed source counts for single pointing experiments. The mean value of the simulated counts from the 1936 independent distributions in each bin is shown by the black line. The shaded regions surrounding this correspond to 1, 2, 3 and 5 times the standard deviation of the count measurements.
Figure 1 clearly shows that the scatter induced in the source counts by the clustering of radio sources across the sky for a survey of fixed area is thus strongly dependent on the depth of the survey, due to the unmodified surface density of radio sources rising with decreasing flux density. Observational challenges notwithstanding, larger areas are required to accurately quantify the counts of faint radio sources. Count fluctuations induced by sample variance are significant enough to dominate the observed scatter at flux densities above $\sim$100 $\mu$Jy, and contribute significantly below this. Notable outliers on Figure 1 are the anomalously high and rising count values from Owen & Morrison (2008). The $P(D)$ analysis of Condon et al. (2012) was conducted over the same field as the Owen & Morrison (2008) observations, partially motivated by the prospect of confirming the high counts previously seen in that region. Condon et al. (2012) determine new counts with their 8” resolution VLA C-array observations that are a factor of $\sim$4 lower than the existing ones derived from the multi-configuration, 1.6” resolution observations of Owen & Morrison (2008), and speculate that overestimation of the resolution corrections are responsible for the discrepancy.
There is a deviation of the measured broad area counts from the corresponding simulated samples in the left hand panel of Figure 1 below approximately 150 $\mu$Jy. At this depth the broad area counts are drawn solely from the Phoenix Deep Survey (Hopkins et al., 2003). This survey includes a deeper tier that has an effective area that is notably less ($\sim$1–1.5 deg${}^{2}$) than the 4.5 deg${}^{2}$ probed by the multiple samples of the simulation, and it is from this smaller, deeper region that these counts originate. The deviation illustrates that even on scales of $\sim$1 deg${}^{2}$ the sampling variation in the counts is not negligible.
As noted by Wilman et al. (Section 4, 2008), in order to predict the behaviour of the radio sky at levels that are beyond present observation requires extrapolation of the known luminosity functions. We naturally cannot rule out departures of the simulation from reality below the limits of the observationally measured source counts. Our results are also sensitive to the accuracy of the clustering model in the simulation. Wilman et al. (2008) test the validity of the source clustering by comparing the simulated and measured angular two point correlation functions, and find good agreement. For further details, including potential (less significant) limitations of the simulation we refer the reader to Wilman et al. (2008).
Note also that the brightest end of the source counts also have uncertainties in the measurements comparable to those associated with the faintest counts. The effect that causes the large scatter is analogous at both ends of the scale: in the case of the bright sources it is a combination of small effective survey volumes for nearby sources and the intrinsic rarity of extremely bright sources at large distances, resulting in low number counts in both scenarios.
The following two subsections broaden the utility of the above results by presenting a pair of tools for observers who wish to carry out deep radio surveys in order to investigate the faint radio source population.
2.1 Optimisation of survey area according to flux density detection threshold
Here we present a method for approximately evaluating the area that a survey of a given detection threshold must cover in order to limit the uncertainty in the counts induced by sample variance to a certain level. The standard deviation derived from the multiple count samples per flux density bin ($\sigma$) is expressed as a fraction of the mean count value ($\mu$) in that bin, and these data are plotted in log space as solid lines on Figure 2. These calculations are performed for a representative group of nine survey areas ranging from 0.1 to 4.9 deg${}^{2}$, as listed on the legend of Figure 2. Testing sky areas larger than this becomes problematic as the number of independent catalogues that can be extracted from the simulation decreases with sky area. This is reflected in the increasing ripple levels of the curves on Figure 2 as the sky area increases.
A good approximation to the measured curves is provided by a least-squares fitted polynomial of the form
$$\mathrm{log}(\mu/\sigma)~{}=~{}p_{1}+p_{2}\mathrm{log}(S)+p_{3}\mathrm{log}(S)%
^{2}+p_{4}\mathrm{log}(S)^{3}.$$
(1)
The fitted curves are shown by the dashed lines on Figure 2. The coefficients $p_{n}$ are provided in Table 1 for the nine survey areas, allowing the approximate uncertainties to be calculated for arbitrary surveys. As this is a polynomial fit it should not be used to extrapolate outside the range of the data to which it was fitted, however the lower limit of 10 nJy is the formal flux-density limit of the simulation, and the source counts are generally well constrained observationally beyond the 10 mJy upper limit and up to the rare $>$1 Jy population.
Table 1 also lists the survey limits required to reduce the scatter in the source counts to 1, 5, 10 and 25% of the mean values (shown by the horizontal lines on Figure 2) for the nine hypothetical surveys. To illustrate how these limits are determined the 5% case is presented as an example by the colour coded vertical lines on Figure 2. Note that the four smallest sky areas do not provide the accuracy to ever reach a 1% uncertainty within the limits of the simulation, hence the missing values in Table 1.
2.2 Corrections for Poisson uncertainties in order to include the effects of source clustering
The sample variance is equivalent to the variance of the counts in the cells into which the simulation is divided, and consists of two components, namely the Poisson variance and a second contribution caused purely by the cosmological clustering of the sources. In this section we provide an estimate of the contribution to the sample variance that is solely due to source clustering as a function of flux density and survey solid angle. This allows existing and future experiments that measure the counts of faint radio sources to correct their Poisson errors in order to include clustering effects.
The 1$\sigma$ percentage errors due to both Poisson scatter ($\sigma^{\%}_{P}$) and sample variance ($\sigma^{\%}_{S}$) can be calculated for the simulated Euclidean-normalized differential source counts for each flux density bin. An estimate of pure Poisson errors that does not include the effects of source clustering is derived by randomising the position of each source in the simulation and measuring the variance of the counts in each cell. This procedure is carried out 100 times and the mean variance is used to calculate the 1$\sigma$ Poisson percentage error $\sigma^{\%}_{P}$. The sample variance uncertainty is taken as the standard deviation of the individual count values in each cell of the unperturbed simulation, as per the 1$\sigma$ limits presented in Section 2.1. These calculations are performed in flux density bins with a logarithmic width of 0.2 Jy over the full flux-density range of the simulation.
How can the contribution to the sample variance that is purely due to cosmological source clustering be distilled? We assume that the source clustering multiplies the number of galaxies in each independent cell by a factor $f$ that has a mean value of 1. The rms percentage scatter in this factor is denoted by $\sigma^{\%}_{CL}$, and is independent of the raw source counts in any given bin (and thus independent of the Poisson errors). Furthermore, the factor $f$ is assumed to be a function of flux density that varies slowly enough such that $f$ can be treated as constant across each flux density bin in which sources are counted.
If the distribution of radio sources were devoid of any clustering then the Poisson variance ($\sigma^{\%}_{P}$) would be the sole cause of the scatter in the Euclidean-normalized counts ($N_{bin}$) in any given flux-density bin. We assume that the source clustering contributes to the measured variance ($\sigma^{\%}_{S}$) from the simulation in a way that conforms to the behaviour of the $f$ parameter described above, i.e. the clustering adjusts the measured counts to a value of $f$ $\times$ $N_{bin}$. The sample variance (i.e. the variance of the counts in each cell of the simulation, $\sigma^{\%}_{S}$) is thus the quadratic sum of the Poisson variance ($\sigma^{\%}_{P}$) and the additional variance due to cosmological clustering ($\sigma^{\%}_{CL}$). It does not drop to zero even in the absence of any source clustering. We can extract the rms percentage scatter in $f$ using error propagation rules:
$$\sigma^{\%}_{CL}~{}=~{}\sqrt{(\sigma^{\%}_{S})^{2}~{}-~{}(\sigma^{\%}_{P})^{2}}$$
(2)
since in the absence of clustering the Poisson variance is the sole contributor to the sample variance. The parameter $\sigma_{CL}$ is independent of the choice of bin width, and its values derived from our simulation can be used in conjunction with an observationally-derived value of $\sigma_{P}^{\%}$ to determine
$$\sigma^{\%obs}_{S}~{}=~{}\sqrt{(\sigma^{\%obs}_{P})^{2}~{}+~{}(\sigma^{\%}_{CL%
})^{2}},$$
(3)
i.e.
$$\sigma^{\%obs}_{S}~{}=~{}\sqrt{\frac{100^{2}}{N^{obs}_{bin}}~{}+~{}(\sigma^{\%%
}_{CL})^{2}}$$
(4)
where $N^{obs}_{bin}$ is the number of sources in that flux density bin.
Figure 3 shows the values of $\sigma_{CL}^{\%}$ derived from the simulation that are applicable to faint flux density bins (10.0 nJy $<$ $S_{centre}$ $<$ 0.3 mJy) for a range of effective survey solid angles. For a given measurement of the Euclidean-normalized differential source counts, Figure 3 can be used in conjunction with Equation 4 in order to correct the percentage error estimate ($\sigma^{\%obs}_{S}$) in the observed counts ($N_{bin}^{obs}$) to include clustering effects. We impose the condition that for the derived value of $\sigma_{CL}^{\%}$ to be trustworthy, it must exceed 5$\sigma_{P}^{\%}$. This is to account for the fact that the Poisson errors derived from flux density bins containing average counts of $<$1 cannot be reliably used. These conditions lead to the cut-offs in the lines on Figure 3. The cut-offs manifest themselves at fainter flux densities with smaller survey solid angles as the raw source counts per bin decrease with sky coverage.
The seven sky survey areas in Figure 3 cover the range 0.003 to 3.0 deg${}^{2}$. The smallest areas are chosen to make the figure relevant for the current deepest observations, where the faintest sources are detected in effective areas much smaller than the primary beam size. The broader areas make the plot relevant for future radio continuum surveys with MeerKAT (13.5 m dishes) and the SKA (15 m dishes)444The Australian SKA Pathfinder (ASKAP) is a special case as it has been designed to deliver an instantaneous field of view at 1.4 GHz of $\sim$30 deg${}^{2}$. The sample variance contribution due to the clustering of cosmological sources is not likely to be an issue for the surveys that are planned for it..
We can compare our predictions for the effects of source clustering to the measurement of Condon (2007). Seventeen independent pointings of approximately 0.2 deg${}^{2}$ each were extracted from the Spitzer First Look Survey, and with approximately 100 sources per field with a flux density limit of 150 $\mu$Jy, our simulation predicts a $\sigma_{CL}$ value of approximately 12.5%, as shown by the intersecting dashed lines on Figure 3. Applying these values to Equation 4 results in a percentage error in the observed counts of $\sigma_{S}^{\%obs}$ = 16%. This is slightly higher than but still broadly consistent with the observed value of (10.7 $\pm$ 2.6)%.
The shapes of the $\sigma^{\%}_{CL}$ curves on Figure 3 are worthy of comment as they say something about the clustering strength of radio sources as a function of their flux densities. The plot shows the area-dependent trend that one would instinctively expect. The effect of source clustering rises with flux density although this is not a smooth change over the plotted range. This is likely due to the brighter end of the source counts likely being dominated by more massive elliptical galaxies that are more strongly clustered than the faint sources, the less clustered star-forming spiral galaxies.
Finally, we compare the trend that these lines exhibit to existing theory. Clustering will increase the variance of the source counts in each individual cell. If each cell contains $N$ sources in a solid angle $\Omega$ and a (fairly narrow) flux-density range $\Delta S$, then the mean number of sources per cell is
$$\bar{N}=n(S)\Delta S\,\Omega~{}.$$
(5)
The sample variance can be written as the sum of the Poisson variance $\bar{N}$ and the variance caused solely by clustering. Peebles (1980) expresses this in terms of $w(\theta)$, the two-point correlation as a function of angular separation $\theta$:
$$\langle(N-\bar{N})^{2}\rangle=\bar{N}+{\bar{N}^{2}\over\Omega^{2}}\int w(%
\theta)d\Omega_{1}d\Omega_{2}~{}.$$
(6)
The function $w(\theta)$ is usually approximated by a power-law of the form
$$w(\theta)=A\biggl{(}{\theta\over{\rm deg}}\biggr{)}^{-\alpha}~{}.$$
(7)
Blake & Wall (2002a,b) measured $w(\theta)$ in the range $0.1<\theta{\rm\,(deg)}<10$ for NRAO VLA Sky Survey (NVSS; Condon et al., 1998) sources stronger than about 10 mJy at 1.4 GHz and found $A\approx 1.0\times 10^{-3}$, $\alpha\approx 0.8$. Blake et al. (2004) combined Sydney University Molonglo Sky Survey (SUMSS; Bock, Large & Sadler, 1999), NVSS, and Westerbork Northern Sky Survey (WENSS; Rengelink et al., 1997) data to estimate a slightly larger $A\approx 1.6\times 10^{-3}$ and
a slightly steeper $\alpha\approx 1.1$.
Following de Zotti et al. (2010), we note that the fractional variance
$${\langle(N-\bar{N})^{2}\rangle\over\bar{N}^{2}}={1\over\bar{N}}+{1\over\Omega^%
{2}}\int w(\theta)d\Omega_{1}d\Omega_{2}$$
(8)
has the advantage that the clustering term does not explicitly depend on $\bar{N}$ or $\Delta S$. Using our notation
$${\langle(N-\bar{N})^{2}\rangle\over\bar{N}^{2}}={1\over\bar{N}}+\sigma_{CL}^{2%
}~{},$$
(9)
where
$$\sigma_{{CL}}^{2}={1\over\Omega^{2}}\int w(\theta)d\Omega_{1}d\Omega_{2}%
\approx 2.36A\biggl{(}{\Omega\over{\rm deg^{2}}}\biggr{)}^{-\alpha/2}$$
(10)
is the fractional variance contributed by clustering alone. Thus
$$\sigma_{CL}^{\%}\approx 5\biggl{(}{\Omega\over{\rm deg^{2}}}\biggr{)}^{-\alpha%
/4}$$
(11)
declines more slowly with $\Omega$ than the Poisson scatter, which
is proportional to $\Omega^{-1/2}$, as is reflected in our results in Figure 3, in which the fainter dotted lines show the mean fractional percentage Poisson errors.
3 Concluding remarks
Observationally-derived values of the counts of faint radio sources exhibit levels of scatter that can be up to a factor of several greater than the quoted uncertainties in the counts. We have provided an estimate of the scatter induced in the counts of faint radio sources due to the sample variance induced by cosmological source clustering by using many independent samples of an extragalactic sky simulation, and comparing these results to matched observations. The deepest observations to date have been carried out using single deep pointings with the VLA. The fluctuations induced by sample variance in the counts derived from such an observation may be large enough to completely explain the observed scatter at flux densities above approximately 100 $\mu$Jy, and we have quantified their contribution as a function of survey area below this level.
We have presented a method for estimating the count uncertainty induced by sample variance for an arbitrary radio survey, or reciprocally, for determining the depth that a radio survey of fixed solid angle coverage must reach in order to limit the count uncertainty. We have also derived a method for correcting Poisson errors in order to include the effects of source clustering. This method is applicable to the deepest surveys that exist today and should remain applicable for future deep continuum surveys with the VLA, MeerKAT and the SKA, down to survey flux density limits of 0.1 $\mu$Jy. We stress again the distinction between survey flux density limits and the rms sensitivity of the corresponding radio images when applying these methods.
The amount that cosmological clustering affects the counts is as one would expect strongly dependent on survey area but also on flux density limits, likely due to the preferential clustering of massive elliptical galaxies at the brighter end, with the less clustered star-forming spiral galaxies dominating the fainter counts.
The method for correcting Poisson uncertainties is broadly consistent with the observationally-derived measurement of the count fluctuations presented by Condon (2007), who concluded that human-induced instrumental calibration and interpretative differences are likely to dominate the scatter. Such effects are certainly contributing factors to the difference in published counts in cells between different authors; the potential overestimation of the resolution correction resulting in the very high counts of Owen & Morrison (2008) being a prime example that is not explained by our results. The sample variance in the case of the deepest surveys such as this is only marginally larger than the actual Poisson variance due to the source counts per bin being very low, counted over effective areas much smaller than the primary beam size.
Current facilities are not suited to deriving a low-uncertainty measurement of the faint radio source counts without an unfeasibly large investment of telescope time. It is likely that the issue will lack an empirical resolution until the completion of the next-generation of legacy radio surveys with future instruments such as ASKAP, MeerKAT and eventually the SKA.
Acknowledgments
We thank Andrew Hopkins for refereeing this paper, and suggesting numerous significant improvements to its content. I.H. thanks the South-East Physics Network (SEPnet) and the University of Oxford. This research has made use of NASA’s Astrophysics Data System.
References
Biggs & Ivison (2006)
Biggs, A. D., & Ivison, R. J. 2006, MNRAS, 371, 963
Blake et al. (2004)
Blake, C., Mauch, T., & Sadler, E. M. 2004, MNRAS, 347, 787
Blake & Wall (2002a)
Blake, C., & Wall, J. 2002a, MNRAS, 329, L37
Blake & Wall (2002b)
Blake, C., & Wall, J. 2002b, MNRAS, 337, 993
Bock et al. (1999)
Bock, D. C.-J., Large, M. I., & Sadler, E. M. 1999, AJ, 117, 1578
Bondi et al. (2008)
Bondi, M., Ciliegi, P.,
Schinnerer, E., et al. 2008, ApJ, 681, 1129
Bridle et al. (1972)
Bridle, A. H., Davis, M. M., Fomalont, E. B., & Lequeux, J. 1972, AJ, 77, 405
Ciliegi et al. (1999)
Ciliegi, P., McMahon,
R. G., Miley, G., et al. 1999, MNRAS, 302, 222
Condon et al. (1998)
Condon, J. J., Cotton,
W. D., Greisen, E. W., et al. 1998, AJ, 115, 1693
Condon (2007)
Condon, J. J. 2007, in “At the Edge of the Universe: Latest Results from the Deepest Astronomical Surveys”, ASP Conference Series 380 (Edited by José Afonso, Henry C. Ferguson, Bahram Mobasher, and Ray Norris), 189
Condon (2009)
Condon, J. J., 2009, SKA Memo 114, http://www.skatelescope.org/publications
Condon et al. (2012)
Condon, J. J., Cotton, W. D., Fomalont, E. B., et al. 2012, ApJ, 758, 23
de Zotti et al. (2010)
de Zotti, G., Massardi, M., Negrello, M., & Wall, J. 2010, A&A Rev., 18, 1
Dewdney et al. (2009)
Dewdney, P. E., Hall, P. J., Schilizzi, R. T., & Lazio, T. J. L. W. 2009, IEEE Proceedings, 97, 1482
Fixsen et al. (2011)
Fixsen, D. J., Kogut, A., Levin, S., et al. 2011, ApJ, 734, 5
Fomalont et al. (2002)
Fomalont, E. B., Kellermann, K. I., Partridge, R. B., Windhorst, R. A., & Richards, E. A. 2002, AJ, 123, 2402
Fomalont et al. (2006)
Fomalont, E. B., Kellermann, K. I., Cowie, L. L., et al. 2006, ApJS, 167, 103
Gendre & Wall (2008)
Gendre, M. A., & Wall, J. V. 2008, MNRAS, 390, 819
Gruppioni et al. (1999)
Gruppioni, C.,
Ciliegi, P., Rowan-Robinson, M., et al. 1999, MNRAS, 305, 297
Heywood et al. (2013)
Heywood, I., Bielby, R. M., Hill, M. D., et al. 2013, MNRAS, 428, 935
Hopkins et al. (1999)
Hopkins, A., Afonso,
J., Cram, L., & Mobasher, B. 1999, ApJL, 519, L59
Hopkins et al. (2003)
Hopkins, A. M., Afonso,
J., Chan, B., et al. 2003, AJ, 125, 465
Ibar et al. (2009)
Ibar, E., Ivison, R. J.,
Biggs, A. D., et al. 2009, MNRAS, 397, 281
Jarvis & Rawlings (2004)
Jarvis, M. J., Rawlings S., 2004, NewAR, 48, 1173
Kellermann et al. (2008)
Kellermann, K. I.,
Fomalont, E. B., Mainieri, V., et al. 2008, ApJS, 179, 71
Longair (1966)
Longair, M. S. 1966, MNRAS,
133, 421
Longair (2004)
Longair, M. S. 2004, in Measuring and Modeling the Universe, Carnegie Observatories Astrophysics Series, Vol. 2
Mitchell
& Condon (1985)
Mitchell, K. J., & Condon, J. J. 1985, AJ, 90, 1957
Owen & Morrison (2008)
Owen, F. N., & Morrison, G. E. 2008, AJ, 136, 1889
Padovani et al. (2009)
Padovani, P.,
Mainieri, V., Tozzi, P., et al. 2009, ApJ, 694, 235
Peebles (1980)
Peebles, P. J. E. 1980, “The Large Scale Structure of the Universe,” Princeton University Press
Rengelink et al. (1997)
Rengelink, R. B., Tang, Y., de Bruyn, A. G., et al. 1997, A&A Suppl., 124, 259
Ryle & Clarke (1961)
Ryle, M., & Clarke, R. W. 1961, MNRAS, 122, 349
Scheuer (1957)
Scheuer, P. A. G. 1957, Proceedings of the Cambridge Philosophical Society, 53, 764
Schinnerer et al. (2004)
Schinnerer, E.,
Carilli, C. L., Scoville, N. Z., et al. 2004, AJ, 128, 1974
Seiffert et al. (2011)
Seiffert, M., Fixsen, D. J., Kogut, A., et al. 2011, ApJ, 734, 6
Seymour et al. (2008)
Seymour, N., Dwelly,
T., Moss, D., et al. 2008, MNRAS, 386, 1695
Simpson et al. (2006)
Simpson, C., Martínez-Sansigre, A., Rawlings, S., et al. 2006, MNRAS, 372, 741
Smolčić et al. (2009)
Smolčić V., et al., 2009, ApJ, 696, 24
Vernstrom et al. (2011)
Vernstrom, T., Scott, D., & Wall, J. V. 2011, MNRAS, 415, 3641
White et al. (1997)
White, R. L., Becker,
R. H., Helfand, D. J., & Gregg, M. D. 1997, ApJ, 475, 479
Wilman et al. (2008)
Wilman, R. J., Miller, L., Jarvis, M. J., et al. 2008, MNRAS, 388, 1335
Wilman et al. (2010)
Wilman, R. J., Jarvis,
M. J., Mauch, T., Rawlings, S., & Hickey, S. 2010, MNRAS, 405, 447
Windhorst et al. (1984)
Windhorst, R. A., van Heerde, G. M., & Katgert, P. 1984, A&AS, 58, 1 |
The nature of the present
Gustavo E. Romero
Abstract
The feeling of a moving present or ‘now’ seems to form part of our most basic perceptions about reality. Such a present, however, is not reflected in any of our theories of the physical world. In this short note I argue for a tenseless view of time, where what we call ‘the present’ is just an emergent secondary quality arising from the interaction of perceiving self-conscious individuals with their environment. I maintain that there is no flow of time, but just an ordered system of events.
Keywords:Philosophy of science, general relativity, space-time
: 01.70.+w, 04.20.-q, 04.20.Gz
1 Introduction
Time has always puzzle philosophers and scientists alike. Traditionally, there are two broad views about the nature of time. These views are usually called the “tensed” and the “tenseless” views, or, for simplicity, the A and B views of time. For an A-oriented person, only present things exist. There are many varieties of this ontological position: presentism, becoming theory, primitive tenses, branching universe theory, and so on. All of them distinguish the present in some way. In particular, presentism is the doctrine that it is always the case that, for every $x$, $x$ is present. The quantification in this definition is unrestricted, it ranges over all existents. In order to make this definition meaningful, the presentist must provide a specification of the term “present”. A standard definition is:
Present: The mereological sum of all objects with null temporal distance Crisp (2003).
Since the mereological sum of objects is always an object, we can infer that for a presentist the present is an object, i.e. and individual with some properties.
A B-oriented person will consider all this as pure nonsense. She will maintain that past, present and future ‘equally’ exist. For her, the fundamental temporal properties are relations of ‘earlier than’, ‘later than’ and ‘simultaneous with’. These are relations between events. There is no distinguished present in any absolute sense. The present is not an object. Then, it cannot move, since only objects can move with respect to each other. There is no objective ‘flow’ or passage of time.
What is, then, the present in this view? My aim, in this short note, is to answer this question from a B-perspective.
2 Shoot the presentist
The Englishman John McTaggart Ellis McTaggart (1866-1925) presented a disproof of presentism in his famous paper Unreality of Time mcTaggart (1908). He reasoned as follows.
1.
There is no time without change.
2.
If time passes, events should change respect to the properties of pastness, presentness, and futureness.
3.
A given event, then, should be able to be in absolute sense, past, present and future.
4.
These properties exclude each other.
Then: Events do not pass, just are.
There is no passage of time. There is no moving present. The mere idea of a flowing time simply does not make any sense. An additional problem is that if time flows, it should move with respect to something. If we say that there is a super-time with respect to time flows, then we shall need a super-super-time for this super-time, and we shall have an infinite regress. In addition, there is no flow without a rate of flow. At what rate does time go by? The answer 1 sec per sec is meaningless. It is like saying that a road extends along a distance of one km per each km that it extends!
On the physical side, the theory of special relativity seems not to be friendly to the idea of an absolute present, at least in its usual Minkowskian 4-dimensional interpretation (for arguments against presentism from general relativity see the paper by Romero and Pérez in this volume). Special relativity is the theory of moving bodies formulated by Albert Einstein in 1905 Einstein (1905). It postulates the Lorentz-invariance of all physical law statements that hold in a special type of reference systems, called inertial frames. Hence the ‘restricted’ or ‘special’ character of the theory. The equations of Maxwell electrodynamics are Lorentz-invariant, but those of classical mechanics are not. When classical mechanics is revised to accommodate invariance under Lorentz transformations between inertial reference frames, several modifications appear. The most notorious is the impossibility of defining an absolute simultaneity relation between events. The simultaneity relation results to be frame-dependent. Then, some events can be future events in some reference system, and present or past in another system. Since what exists cannot depend on the reference frame adopted for the description of nature, it is concluded that past, present, and future events exist. Then, presentism, the doctrine that only present exists, is false.
The presentist or A-theorist of time might find a way around this argument adopting a different (purely Lorentzian) interpretation of the theory Crisp (2007). The problems of this approach has been discussed at length by Saunders Sau (2002), and I shall not insist on the topic here.
Said all that, yet, we all have a kind of feeling that “our time is running out”. Where does this feeling come from?
3 When is ‘now’?
If we assume that the present is an instant of time instead of a thing, then the question of “which instant is present?” follows. One possible answer is “now”. But…when is ‘now’?
‘Now’, like ‘here’, is an indexical word. To say that I exist now gives no information on when I exist. Similarly, to say that I am here, gives no information on where I am. There is no particular moment of time defined as an absolute now.
I maintain that ‘nowness’ and ‘hereness’ emerge from the existence of perceiving self-conscious beings in a certain environment. What these beings perceive is not time, but changes in things. Similarly, they do not perceive space, but spatial relations among things. In particular, we do not perceive the passage of time. We perceive how our brain changes. I submit that there is no present per se, in the same way that there is no smell, no pain, no joy, no beauty, no noise, no secondary qualities at all without sentient beings. What we call “the present” is not in the world. It emerges from our interaction with the world.
We group various experienced inputs together as present; we are tempted to think that this grouping is done by the world, not by us. But this is just delusional.
I maintain that tenses aren’t needed and in fact aren’t wanted by the natural sciences. This idea is clearly expressed by Poeppel on the basis of neurological research Poeppel (1978):
our brain furnishes an integrative mechanism that shapes sequences of events to unitary forms…that which is integrated is the unique content of consciousness which seems to us present. The integration, which itself objectively extends over time, is thus the basis of our experiencing a thing as present.
The now, the subjective present, is nothing independently; rather it is an attribute of the content of consciousness. Every object of consciousness is necessarily always now - hence the feeling of nowness.
The perception of motion gives an additional argument against the idea that the present is an instant of time. According to Le Poidevin Le Poidevin (2009):
1.
What we perceive, we perceive as present.
2.
We perceive motion.
3.
Motion occurs over an interval.
Therefore: What we perceive as present occurs over an interval.
Any tentative definition of ‘present’ compatible with modern neuro-biological science must take into account the role of the perceiving and sentient individual. In the next section I shall offer some provisional definitions that meet this requirement and distinguish among the different meanings in which the word ‘present’ is used.
4 Defining the present
Physical events are ordered by the relation ‘earlier than’ or ‘later than’, and ‘simultaneous with’ Gru (1973). There is no ‘now’ or ‘present’ in the representation of the physical laws.
What we call ‘present’ is not an intrinsic property of the events nor an instant of time, much less a moving thing. ‘Present’ is a relation between a certain number of events and a self-conscious individual.
Present: Class of all events simultaneous to a given brain event.
To every brain event there is a corresponding present. The individual, however, needs not to be aware of all events that form the present. The present, being a class of events, is an abstract object without any causal power.
Psychological present: Class of local events that are causally connected to a given brain event.
Notice that from a biological point of view only local events are relevant. These events are those that directly trigger neuro-chemical reactions in the brain. Such events are located in the immediate causal past of the brain event. The psychological present is a conceptual construction of the brain, based on abstraction from events belonging to an equivalence class. The present, then, is not a thing or a change in a thing (an event).
William James introduced the concept of ‘specious present’ as “the short duration of which we are immediately and incessantly sensible” Jam (1893). We re-elaborate this to get the following definition:
Specious present: Length of the time-history of brain processes necessary to integrate all local events that are physically (causally) related to given brain event.
The specious present, then, being related to a brain process, can be different for different individuals equipped with different brains. The integration of the specious present can be performed in different ways, depending on the structure of the brain. It is even possible to imagine integration systems that can produce more than one specious present or even systems that might ‘recall’ the future Hartle (2005). If biological evolution has not produced such systems seems to be a consequence of the existence of space-time asymmetric boundary conditions that introduce a preferred direction for the occurrence of processes Romero-Perez (2011).
Finally, we can introduce a physical present.
Physical present: Class of events that belong to a space-like hypersurface in a smooth and continuous foliation of a time-orientable space-time.
Since in the manifold model of space-time every event is represented by an element of the manifold, the introduction of this class does not signal a special time identified with ‘now’. Every space-like hypersurface corresponds to a different time and none of them is an absolute present “moving” into the future. Actually, naming ‘the future’ to a set of surfaces in the direction opposite to the so-called Bing Bang is purely conventional.
5 Final remarks
We have distinguished three different types of present: psychological, physical, and specious. The former two are classes of events, hence they are concepts. The latter is not an instant of time but an interval in space-time associated with the world history of a sentient individual.
In any case, the present does not flow or move. Only material individuals (and their brains, if they have one) can change. Becoming is not a property of physical events, but of the consciousness of the events. We call ‘becoming’ to the series of states of consciousness associated with a certain string of physical events. Events do not become. Events just are.
I thank Daniela Pérez and Felipe Tovar Falciano for insightful comments. This work was supported by CONICET (PIP 0078).
References
Crisp (2003)
Crisp, T. (2003).
Presentism. (In M. J. Loux & D. W. Zimmerman (Eds.), The Oxford Handbook of Methaphysics (pp. 211-245). Oxford: Oxford University Press).
mcTaggart (1908)
McTaggart, J.M.E. (1908).
Unreality of time.
Mind, 17, 456-473.
Einstein (1905)
Einstein, A. (1905).
Zur Elektrodynamik bewegter Körper. Annalen der Physik, 17 (10), 891921.
Crisp (2007)
Crisp, T. (2007).
Presentism, Eternalism and Relativity Physics. (In W. L. Craig & Q. Smith (Eds.), Einstein, Relativity and Absolute Simultaneity (pp. 262-278). London: Routledge)
Sau (2002)
Saunders, S. (2002).
How relativity contradicts presentism. (In C. Callender (Ed.), Time, Reality $\&$ Experience, Royal Institute of Philosophy, Supplement (pp. 277-292). Cambridge, New York: Cambridge University Press)
Poeppel (1978)
Poeppel, E. (1978).
Time perception.
(In Richard Held et al. (eds.), Handbook of Sensory Physiology, Vol. VIII: Perception (pp. 713-729). Berlin: Springer-Verlag).
Le Poidevin (2009)
Le Poidevin, R. (2009).
The experience and perception of time.
The Standfrod Encyclopedia of Philosophy.
(http://plato.stanford.edu/entries/time-experience/).
Gru (1973)
Grünbaum, A. (1973).
Philosophical Problems of Space and Time.
(Dordrecht: Reidel).
Jam (1893)
James, W. (1893).
The Principles of Psychology.
(New York: H. Holt and Company).
Hartle (2005)
Hartle, J.B. (2005).
The physics of now.
American Journal of Physics, 73, 101-109.
Romero-Perez (2011)
Romero, G.E. and Pérez, D. (2011).
Time and irreversibility in an accelerating universe.
International Journal of Modern Physics D, 20, 1-8. |
††thanks:
Present address: MITRE Corporation, 7515 Colshire Dr.
McLean, VA 22102, USA
††thanks:
Present address: Raytheon Intelligence and Space, 2000 E El Segundo Blvd, El Segundo, CA 90245
Probing the Optical Dynamics of Quantum Emitters in Hexagonal Boron Nitride
Raj N. Patel
Quantum Engineering Laboratory, Department of Electrical and Systems Engineering, University of Pennsylvania, Philadelphia, PA 19104, United States
David A. Hopper
Quantum Engineering Laboratory, Department of Electrical and Systems Engineering, University of Pennsylvania, Philadelphia, PA 19104, United States
Jordan A. Gusdorff
Quantum Engineering Laboratory, Department of Electrical and Systems Engineering, University of Pennsylvania, Philadelphia, PA 19104, United States
Department of Materials Science and Engineering, University of Pennsylvania, Philadelphia, PA 19104, USA
Mark E. Turiansky
Department of Physics, University of California, Santa Barbara, CA 93106, United States
Tzu-Yung Huang
Quantum Engineering Laboratory, Department of Electrical and Systems Engineering, University of Pennsylvania, Philadelphia, PA 19104, United States
Rebecca E. K. Fishman
Quantum Engineering Laboratory, Department of Electrical and Systems Engineering, University of Pennsylvania, Philadelphia, PA 19104, United States
Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104, USA
Benjamin Porat
Quantum Engineering Laboratory, Department of Electrical and Systems Engineering, University of Pennsylvania, Philadelphia, PA 19104, United States
Chris G. Van de Walle
Materials Department, University of California, Santa Barbara, CA 93106, United States
Lee C. Bassett
lbassett@seas.upenn.edu
Quantum Engineering Laboratory, Department of Electrical and Systems Engineering, University of Pennsylvania, Philadelphia, PA 19104, United States
Abstract
Hexagonal boron nitride is a van der Waals material that hosts visible-wavelength quantum emitters at room temperature.
However, experimental identification of the quantum emitters’ electronic structure is lacking, and key details of their charge and spin properties remain unknown.
Here, we probe the optical dynamics of quantum emitters in hexagonal boron nitride using photon emission correlation spectroscopy.
Several quantum emitters exhibit ideal single-photon emission with noise-limited photon antibunching, $\bm{g^{(2)}{(0)}=0}$.
The photoluminescence emission lineshapes are consistent with individual vibronic transitions.
However, polarization-resolved excitation and emission suggests the role of multiple optical transitions, and photon emission correlation spectroscopy reveals complicated optical dynamics associated with excitation and relaxation through multiple electronic excited states.
We compare the experimental results to quantitative optical dynamics simulations, develop electronic structure models that are consistent with the observations, and discuss the results in the context of ab initio theoretical calculations.
††preprint: APS/123-QED
I Introduction
Hexagonal boron nitride (h-BN) is a wide-bandgap ($\sim$6 eV) van der Waals material that hosts fluorescent quantum emitters (QEs) at room temperature [1, 2, 3, 4, 5, 6, 7].
The QEs in h-BN are bright and photostable with narrow emission linewidths and high single-photon purity, as required for quantum technologies [6, 8, 9, 10].
Recent observations of room-temperature magnetic field dependence and spin resonance of QEs in h-BN make them attractive for spin-based quantum sensing and computation [11, 12, 13, 14, 15].
Despite intense interest in h-BN’s QEs, their chemical and electronic structures remain uncertain, as do key details regarding their optical, spin, and charge dynamics.
The pronounced heterogeneity of observations suggests that QEs originate from multiple distinct defect structures [16, 17, 6, 18, 19].
Ultraviolet emission around 4.1 eV has been attributed to the carbon dimer $C_{B}C_{N}$ [20], whereas near-infrared emission around 1.7 eV and an associated optically detected magnetic resonance signal is attributed to the negatively-charged boron vacancy, $V_{B}^{-}$ [12].
For QEs in the visible spectrum, experiments utilizing various forms of electron and optical microscopy, spectroscopy, and materials growth and treatments have generated a detailed, yet complicated, empirical understanding of the QEs’ creation, stabilization, and principal optical signatures [21, 22, 23, 3, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39].
Theoretical work suggests that vacancies and their complexes, along with substitutional carbon atoms and dangling bonds, are likely candidates, although consensus is still lacking [40, 41, 42, 43, 44, 45].
Specific candidates include $V_{\rm N}N_{\rm B}$, $V_{\rm N}C_{\rm B}$, $V_{\rm B}C_{\rm N}$, and the boron dangling bond.
Even less is known about the visible QEs’ optical dynamics.
Optical dynamics arise from a QE’s electronic structure together with radiative and nonradiative transitions between electronic states.
State transitions can involve multiple processes including electron-phonon interactions, intersystem crossings between different spin manifolds, and ionization or recombination events.
For QEs in h-BN, previous studies have reported photon bunching associated with metastable dark states [3, 6, 11], and yet the nature of these states and the transitions between them remains unclear.
Some QEs exhibit magnetic-field-dependent modulation of their photoluminescence (PL) signal, consistent with a spin-dependent intersystem crossing, whereas others do not [11, 14].
An optically detected magnetic resonance signal was observed for a particular QE under excitation at 633 nm but not at 532 nm [13].
Such observations present a complicated picture of the visible QEs, likely involving multiple defect classes (e.g., different chemical structures or charge states), strong local perturbations, and complex excitation and relaxation pathways.
Improved understanding of the QEs’ optical dynamics can resolve these mysteries.
Such understanding is also a prerequisite to designing quantum control protocols that would facilitate their use in quantum technologies.
Here, we use quantitative spectral, spatial, and temporal PL spectroscopy to investigate the optical dynamics of h-BN’s QEs.
Photons emitted by a QE carry a wealth of information about its electronic structure and optical dynamics.
For vibronic optical transitions, the photon energy and polarization distributions reflect the details of electron-phonon coupling and optical dipole selection rules, respectively.
The QEs in h-BN generally exhibit linearly-polarized PL and strong electron-phonon coupling associated with a single vibronic transition [6], and yet other experimental and theoretical evidence points to the involvement of multiple excited states in the optical dynamics [28, 43, 10, 46].
Time-dependent measurements provide complementary information.
The second-order photon autocorrelation function, $g^{(2)}(\tau)$, is widely used to identify single-photon emitters.
As a more general analytical tool, photon emission correlation spectroscopy (PECS) yields quantitative information about a QE’s optical dynamics [47].
Qualitatively, we distinguish between photon antibunching ($g^{(2)}(\tau)<1$) as a signature of non-classical light, with single-photon emission as a special case when $g^{(2)}(0)=0$, and photon bunching ($g^{(2)}(\tau)>1$ for $\tau\neq 0$) as a signature of dark, metastable states accessed via nonradiative transitions.
Quantitative measurements of $g^{(2)}(\tau)$ as a function of optical excitation power or wavelength can elucidate a QE’s excitation and emission pathways as well as bunching mechanisms.
Prior observations of h-BN’s visible QEs feature both bunching and antibunching signatures, although with some unusual, conflicting patterns.
Some QEs respond to applied dc and ac magnetic fields in a manner consistent with spin-mediated intersystem crossing transitions, whereas others do not [11, 13, 14].
A lack of evidence for pure single-photon emission motivated a proposal that h-BN’s QEs occur in pairs as “double defects” [48].
In this work, we compare quantitative PL spectroscopy and PECS measurements of h-BN’s QEs with theoretical simulations.
We show that QEs in room-temperature h-BN can exhibit pure single-photon emission, with $g^{(2)}(0)=0$ within experimental uncertainty.
Furthermore, we find evidence for multiple electronic states connected by radiative and nonradiative transitions, with associated timescales spanning over five orders of magnitude.
Comparing the experiments to theoretical proposals, we find that the boron dangling bond model provides a consistent, quantitative understanding of the observations for individual QEs as well as their heterogeneity.
II Results
The Results and Discussion are organized as follows.
First, we report the basic optical characteristics of five well-isolated QEs across three samples, specifically including their PL spectra, PL saturation as a function of excitation power, polarization properties, and single-photon purity.
Next, we investigate the QEs’ optical dynamics using PECS as a function of excitation power and wavelength.
We consider different models for the electronic level structure and simulate the corresponding optical dynamics.
Finally, we compare the theoretical simulations to experimental observations and discuss the implications for understanding the QEs’ electronic and chemical structure along with their spin and charge dynamics.
II.1 Photoluminescence characterization
We used a custom-built confocal microscope to study individual QEs in h-BN under ambient conditions.
The h-BN bulk crystals were mechanically exfoliated into thin flakes using a dry transfer process [49] and transferred to a SiO${}_{2}$/Si substrate patterned with circular trenches [6].
Prior to the optical studies, the samples were annealed in a tube furnace at $850\text{\,}\mathrm{\SIUnitSymbolCelsius}\text{/}$ in Ar atmosphere for 2 hours.
This annealing process has been shown to brighten the emitters [35].
The QEs are illuminated with either of two continuous-wave lasers operating at 532 nm and 592 nm wavelengths, where excitation power and polarization are controlled.
To differentiate between excitation wavelengths in this work, data recorded under 532 nm (592 nm) excitation are plotted in green (orange) in the relevant figures.
Some QEs disappeared during experiments, hence the set of measurements is not identical for each QE.
See Materials and Methods for additional details on sample preparation, data acquisition, and analysis.
Figure 1 summarizes the PL characterization measurements.
Each row corresponds to a particular QE (labeled A-E), and each column corresponds to a different experiment.
The first column includes $\micro$-PL images of each QE, acquired by scanning a fast steering mirror and recording the accumulated counts at each pixel.
A two-dimensional Gaussian fit to each $\micro$-PL image yields the background and signal levels for subsequent studies.
The second column displays $g^{(2)}(\tau)$ measurements over short delay times, showing characteristic antibunching dips fit by an empirical model for a multi-level system (see Materials and Methods).
The third column displays the steady-state PL signal as a function of excitation power.
These data are fit using an empirical saturation model,
$$\displaystyle C(P)=\frac{C_{\mathrm{sat}}^{\lambda}P}{P+P_{\mathrm{sat}}^{\lambda}}$$
(1)
where
$C$ is the background-subtracted, steady-state PL count rate,
$P$ is the optical excitation power,
$C_{\mathrm{sat}}^{\lambda}$ is the saturation count rate for a particular excitation wavelength, $\lambda$,
and $P_{\mathrm{sat}}^{\lambda}$ is the corresponding saturation power.
The best-fit results are reported in Supplementary Table S2.
The fourth column of Fig. 1 presents PL emission spectra and polarization measurements.
In each PL spectrum, the long pass filter cut-on wavelength is indicated as a vertical dotted line, and the excitation wavelength is a solid line.
The inset to each PL spectra panel presents measurements of the QE’s excitation and emission polarization properties.
These data are acquired by varying the linear polarization of the excitation laser (colored circles) or by passing the PL through a linear polarizer placed in the collection path (black squares).
For excitation polarization measurements, the linear polarizer in the collection path is removed.
For emission polarization measurement, the excitation polarization is set to maximize the PL.
At each polarization setting, we record the steady-state PL intensity as well as a background intensity from a spatial location offset $\sim 1\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$ from the QE, which is subtracted to yield the PL signal.
The order of the polarization angles is set randomly to minimize effects of drift and hysteresis.
Solid curves are fits to the data using the model function
$$I_{s}^{\lambda}(\theta)=A_{s}^{\lambda}\cos^{2}(\theta-\theta_{s}^{\lambda})+B_{s}^{\lambda}$$
(2)
where $\lambda$ indicates the excitation wavelength, $s$ indicates excitation (ex) or emission (em), $A_{s}^{\lambda}$ is the amplitude, $\theta_{s}^{\lambda}$ is the polarization angle of maximum intensity, and $B_{s}^{\lambda}$ is the offset.
From the fit results, the visibility is calculated as
$$V_{s}^{\lambda}=\frac{I^{\mathrm{max}}_{s}-I^{\mathrm{min}}_{s}}{I^{\mathrm{max}}_{s}+I^{\mathrm{min}}_{s}}=\frac{A_{s}^{\lambda}}{A_{s}^{\lambda}+2B_{s}^{\lambda}}$$
(3)
where $I^{\mathrm{max}}_{s}$ and $I^{\mathrm{min}}_{s}$ are the maximum and minimum intensities, respectively.
The misalignment between the excitation and emission polarization angles is
$$\Delta\theta^{\lambda}=\theta_{\mathrm{ex}}^{\lambda}-\theta_{\mathrm{em}}^{\lambda}$$
(4)
The best-fit parameters are reported in Supplementary Table S2.
II.2 Spectral emission lineshapes
For spectra that are not cut off by the excitation filter (namely, the 532 nm excitation spectra for QEs A, C, and E, and the 592 nm spectrum for QE B), we find that the lineshapes are consistent with a Huang-Rhys model for a vibronic transition associated with a single zero-phonon line (ZPL).
We use the analysis method described by Ref. [6] to fit the observed PL spectra using an empirical model in which the ZPL energy, ZPL width, Huang-Rhys factor, and vibronic coupling lineshape are free parameters; see Materials and Methods for additional details.
The results are shown in Fig. 2.
In the left column, we plot the normalized observed emission lineshape, $L(\Delta E)\propto S(\Delta E)/E^{3}$, where $S(\Delta E)$ is the spectral intensity distribution as a function of the relative energy $\Delta E=E_{\mathrm{ZPL}}-E$, with $E$ denoting the photon emission energy and $E_{\mathrm{ZPL}}$ denoting the ZPL energy.
The factor $1/E^{3}$ accounts for the photon energy dependence in spontaneous emission.
Each solid curve is the result of a weighted least-squares fit of the model to the experimental lineshapes.
The right column of Fig. 2 shows the corresponding 1-phonon vibronic coupling lineshape for each fit.
Best-fit parameters are reported in Supplementary Table S2.
II.3 Photon emission correlation spectroscopy
Temporal correlations between fluorescence photons reveal information about a QE’s excitation and emission dynamics.
In this work, we use PECS for two purposes: to verify the single-photon purity of the QEs and to probe their optical dynamics as a function of optical excitation rate.
We calculate $g^{(2)}(\tau)$ from the photon arrival times acquired from two detectors in a Hanbury Brown and Twiss interferometer using a time-correlated single-photon counting module.
For QEs in h-BN, the timescales over which antibunching and bunching occur can vary over at least 6 orders of magnitude [6, 11, 16].
For this reason, we initially calculate and analyze $g^{(2)}(\tau)$ over a logarithmic scale spanning from 100 ps to 1 s.
We fit the background-corrected data using a general empirical model for a QE’s optical dynamics with a varying number of levels:
$$g^{(2)}(\tau)=1-C_{1}e^{-\gamma_{1}|\tau|}+\sum_{i=2}^{n}C_{i}e^{-\gamma_{i}|\tau|}$$
(5)
Here, $\gamma_{1}$ is the antibunching rate, $C_{1}$ is the antibunching amplitude, $\gamma_{i}$ for $i\geq 2$ are bunching rates, and $C_{i}$ for $i\geq 2$ are the corresponding bunching amplitudes.
We determine the number of resolvable timescales, $n$, by calculating and comparing the Akaike Information Criterion (AIC) and the reduced chi-squared statistic for each best-fit model.
In optical dynamics models, an $N$-level system is characterized by $N-1$ rates, corresponding to the nonzero eigenvalues of the generator matrix (see, e.g., Eq. (16) in Materials and Methods).
Therefore, the inferred value of $n$ places a lower limit on the number of electronic levels required to describe the observations, $N\geq n+1$.
We extract the rates, amplitudes, and their corresponding uncertainty from these fits for comparisons with theoretical simulations.
In order to assess the single-photon purity associated with the value of $g^{(2)}(0)$, we perform a subsequent analysis of $g^{(2)}(\tau)$ calculated over a linear scale of delay times, $\tau\in[-20,20]$ ns.
Examples of such data are shown in Fig. 1, along with constrained fits in which only the antibunching parameters $\gamma_{1}$ and $C_{1}$ are allowed to vary, and which account for the instrument response function associated with detector timing jitter.
See Materials and Methods for further details.
II.4 Verifying single-photon emission
Any observation of sub-Poissonian statistics, $g^{(2)}(0)<1$, indicates the presence of quantized photon emission.
The threshold $g^{(2)}(0)<0.5$ is often used to indicate single-photon emission; however, a more precise interpretation is that a PL signal is dominated by a single-photon emitter when $g^{(2)}(0)<0.5$ [47].
An observation of $g^{(2)}(0)>0$ implies a non-zero probability of observing two detection events simultaneously, either due to background fluorescence, detection timing jitter, or the presence of multiple QEs.
Studies of h-BN’s QEs routinely report $g^{(2)}(0)<0.5$, however we are unaware of any prior room-temperature observations of pure single-photon emission with $g^{(2)}(0)=0$.
Partially on the basis of such observations, Ref. [48] proposed that h-BN’s QEs actually occur in pairs as double defects with parallel emission pathways.
We find that QEs in h-BN can indeed exhibit pure single-photon emission at room temperature.
Figure 3 shows $g^{(2)}(0)$ for each QE as a function of excitation power.
These data are corrected for background fluorescence and detector timing jitter, as described in Materials and Methods.
For QEs C, D and E, we observe $g^{(2)}(0)=0$ within the experimental uncertainty, particularly at low excitation powers.
For QEs A and B, we observe $g^{(2)}(0)\sim 0.1-0.2$.
The offset from zero could reflect a contribution from additional dim emitters, however we believe it is more likely to result from incomplete estimation of the background.
For instance, QE B sits on an extended background feature whose contribution is not captured by our standard analysis method.
For QE C, we attribute the increase in $g^{(2)}(0)$ as a function of excitation power to limitations in the instrument-response-function correction as the antibunching rate exceeds the detector timing resolution.
II.5 Probing the optical dynamics
Figure 4 summarizes the results of fitting the empirical model of Eq. (5) to PECS measurements as a function of optical excitation power.
The figure includes the best-fit antibunching rate (top row) as well as the first two bunching rates and amplitudes (lower rows).
The PECS data for QEs B, D, and E are best described by a three-timescale model ($n=3$), whereas QE A exhibits four resolvable timescales ($n=4$).
For QE C, we resolve two or three timescales depending on the excitation power and wavelength.
The best-fit results for QE A’s third bunching component ($\gamma_{4}$ and $C_{4}$), as well as the antibunching amplitude ($C_{1}$) for all emitters are shown in Supplementary Figures S6 and S7, respectively.
As in previous figures, colors indicate the optical excitation wavelength.
The PL decay rate of QE A was directly measured to be $355\text{\,}\mathrm{MHz}\text{/}$ using a picosecond pulsed laser (see Section S6 and Fig. S4 in the Supplementary Materials); this measurement is shown in Fig. 4(a) as a dashed black line.
The PL lifetime measurement was only performed for QE A given the susceptibility of h-BN’s QEs to disappear under pulsed excitation.
To fit these metadata, we consider the following empirical models:
$$\displaystyle\text{Model I (Linear)}:$$
$$\displaystyle\qquad R(P)=R_{0}+m_{0}P$$
(6a)
$$\displaystyle\text{Model II (First-Order Saturation)}:$$
$$\displaystyle\qquad R(P)=R_{0}+\frac{R_{sat}P}{P+P_{sat}}$$
(6b)
$$\displaystyle\text{Model III (Second-Order Saturation)}:$$
$$\displaystyle\qquad R(P)=R_{0}+\frac{(m_{0}P_{sat}P+m_{1}P^{2})}{P+P_{sat}}$$
(6c)
$$\displaystyle\text{Model IV (Quadratic)}:$$
$$\displaystyle\qquad R(P)=R_{0}+m_{0}P+m_{1}P^{2}$$
(6d)
where $P$ is the excitation power and the other variables are free parameters representing zero-power offset, $R_{0}$, low-power slope, $m_{0}$, high power slope, $m_{1}$, saturation value, $R_{sat}$, and saturation power, $P_{sat}$.
Dotted curves in Fig. 4 show the best-fit results for the model listed in each panel;
in each case, we select the model with the fewest free parameters that qualitatively fits the data.
Best-fit parameters and uncertainties for each fit are reported in Supplementary Tables S5 and S6.
The antibunching rate, $\gamma_{1}$, exhibits a markedly nonlinear power dependence for QEs A, B, and C whereas the dependence appears to be linear for QEs D and E.
However, we note that the power range in the data for QEs D and E might not be large enough for nonlinearities to emerge.
For comparison, QEs B and C are excited with up to $\sim 4P_{\mathrm{sat}}^{\lambda}$ whereas QE D is excited with up to $\sim 2P_{\mathrm{sat}}^{\lambda}$ (see Supplementary Table S2).
The zero-power antibunching-rate offset ($R_{0}$) for QEs B-E is clearly nonzero, whereas the fits using Model II for QE A are poorly constrained, yielding $R_{0}=0\pm 261$ MHz and $R_{0}=0\pm 167$ MHz for green and orange excitation, respectively.
The antibunching amplitudes (see Supplementary Figure S7) for all QEs show a nonlinear saturation dependence on excitation power with an expected convergence to $C_{1}\sim 1$ at zero excitation power.
The bunching dynamics exhibit significant quantitative and qualitative variations across emitters.
The fastest bunching rate, $\gamma_{2}$, scales linearly with excitation power and has a non-zero offset for QEs A, D and E, whereas it exhibits saturation behavior and zero offset for QEs B and C.
The magnitudes of $\gamma_{2}$ range from several kilohertz (QEs A, B, and D) up to several megahertz or faster (QEs C and E).
The slower bunching rate, $\gamma_{3}$, exhibits the largest qualitative variation across emitters, including linear (QE D), quadratic (QEs A and E), and saturation models (QEs B and C).
Only QE D exhibits clear evidence for a non-zero offset for $\gamma_{3}$.
The magnitudes of $\gamma_{3}$ are typically in the kilohertz range, with the exception of QE C, whose $\gamma_{3}$ increases beyond 1 MHz at high powers.
The bunching amplitudes primarily depend nonlinearly on excitation power, except for $C_{2}$ of QEs B, C (green excitation) and E, which scale linearly with excitation power.
All of the bunching-amplitude fits are consistent with zero offset, except for QE E, where small residual offsets ($R_{0}<0.1$) likely reflect minor systematic errors in the analysis or inaccuracies of the empirical models.
For QE A, we restrict the meta-analysis of bunching parameters to the orange-excitation data, which extend to higher excitation power.
However, we note that the green-excitation bunching parameters generally track the data for orange excitation.
II.6 Electronic model and optical dynamics simulations
We find that the key features observed in Fig. 4 can be understood using the four-level electronic model shown in Fig. 5(a).
Figure 5 summarizes the results of optical dynamics simulations for this model.
Given a set of transition rates for the model, we simulate $g^{(2)}(\tau)$ including the effects of timing resolution and shot noise (e.g., Fig. 5(b)), and we subsequently fit the simulated data using the empirical model of Eq. (5) with $n=2$ to extract the antibunching and bunching parameters, as shown in Figs. 5(c)-(e).
For reasons explained later in this section, the simulated data were best described by an $n=2$ model despite having three eigenvalues.
See Materials and Methods for more information regarding the simulations.
The four-level model consists of a ground state (level 1), an excited radiative state (level 2), a higher-lying excited state (level 3) and a nonradiative metastable state (level 4).
We consider two optical excitation pathways from the ground state to excited states 2 or 3, represented by the rates $\Gamma_{12}$ and $\Gamma_{13}$, respectively.
The magnitudes of these two rates depend on the corresponding optical cross-sections for absorption at the excitation wavelength.
A difference in cross section can result from the difference in electric dipole matrix elements between the different electronic states, the atomic configuration coordinate overlap for vibronic transitions, or both of these factors.
For the simulations in Fig. 5, we set $\Gamma_{12}=0$, since we are particularly interested in the situation where $\Gamma_{12}/\Gamma_{13}\ll 1$, such that the dynamics feature indirect excitation of the radiative state 2 via nonradiative relaxation from excited state 3, at a rate $\kappa_{32}$.
This was informed by the nonlinear power-scaling of $\gamma_{1}$ for QEs A, B and C.
In the Supplementary Materials, we report simulations over a range of settings where $\Gamma_{12}/\Gamma_{13}\in[0,2]$, with qualitatively similar results (see Supplementary Figure S8).
In addition to the indirect excitation pathway formed by states 1, 2, and 3, optical excitation results in population and relaxation of metastable state 4 via nonradiative transitions with rates $\kappa_{24}$ and $\kappa_{41}$.
We consider two types of nonradiative transition mechanism for the metastable state: spontaneous and optically pumped.
Spontaneous transition rates are independent of the optical excitation rate (in this case, $\Gamma_{13}$), whereas optically pumped transition rates scale linearly with $\Gamma_{13}$.
In this model, the optically pumped transition rates $\kappa_{24}$ and $\kappa_{41}$ can approximate more complicated processes; for example, they could involve re-pumping from levels $2\rightarrow 3$ or from levels $4\rightarrow 3$ with subsequent nonradiative relaxation (see Supplementary Figure S10), or they could involve transient population of additional levels.
Their approximation as individual pumped transitions remains accurate as long as optical pumping remains the rate-limiting step.
The key observable difference between spontaneous and optically pumped transitions manifests in the excitation power dependence of the corresponding bunching rate (Fig. 5d); the bunching rate for spontaneous transitions features a non-zero zero-power offset and saturates at high power, whereas the bunching rate for optically pumped transitions has zero offset and scales nearly linearly with power, even as the corresponding bunching amplitude (Fig. 5e) saturates.
For both bunching mechanisms, the simulated data were best described by only a single bunching level ($n=2$ in Eq. 5) despite the fact that there should be 3 eigenvalues that describe this system.
The reason for this is that the indirect excitation and emission process through levels 1, 2, and 3 can lead to two of the eigenvalues being complex.
These eigenvalues have the largest real values and are responsible for the antibunching dynamics.
When we include practical limitations on timing resolution and signal-to-noise ratio at short delay times, the fit cannot distinguish these two values, and the goodness-of-fit analysis prefers a single real rate that approximates the true model.
The result is an effective antibunching rate that scales nonlinearly with increasing excitation rate.
This effect persists even when a direct transition from state $1\rightarrow 2$ is included.
We performed simulations varying the ratio $\Gamma_{12}/\Gamma_{13}$, and observed qualitatively similar results (see Supplementary Figure S8).
III Discussion
III.1 Photoluminescence, spectral, and polarization properties
Our experiments provide clear evidence that visible QEs in h-BN occur as isolated point defects with emission originating from a single, dominant optical transition.
The QEs are spatially resolved in high signal-to-background $\micro$-PL images.
They exhibit PL saturation, high polarization visibility in emission, and emission spectra consistent with individual vibronic transitions.
Most convincingly, several emitters exhibit pure single-photon emission, with $g^{(2)}(0)=0$ within small experimental uncertainty, as shown in Fig. 3.
This finding contrasts with previous suggestions that h-BN’s QEs occur in pairs [48].
We do not contend, however, that such pairing cannot occur.
On the contrary, in the course of our experiments we observed multiple instances of spatially isolated emitters with high polarization visibility, and yet $g^{(2)}(0)$ is substantially larger than zero.
We have focused here on emitters showing the highest likelihood of being single defects.
Qualitatively, the QEs’ room-temperature PL spectra are similar to those reported elsewhere in the literature [6, 19, 28, 27].
The analysis shown in Fig. 2 indicates that the PL spectra are consistent with individual vibronic transitions between two optical-dipole-coupled electronic states.
The one-phonon lineshapes for QEs A, B, C, and E are all qualitatively similar despite the fact that QEs A and B feature ZPL energies near 1.9 eV, compared to 2.1 eV for QEs C and E.
All four emitters exhibit strong coupling to low-energy phonons ($\lesssim$50 meV) as well as to higher-energy phonons (150-200 meV) that are typically associated with longitudinal optical modes in bulk h-BN [4].
Coupling to low-energy phonons is a key feature in determining the asymmetric shape of the dominant emission peak [50, 27].
The ZPL corresponds to the transition from the lowest vibrational level of the initial (excited) state to the lowest vibrational level of the final (ground) state.
When low-energy phonons are involved, transitions can occur from the lowest vibrational level of the initial state to the first vibrational level of the final state, showing up on the low-energy side of the ZPL and leading to the asymmetric shape.
Failure to account for low-energy phonons in interpreting experimental spectra leads to an underestimation of the Huang-Rhys factor, $S_{\mathrm{HR}}$, which quantifies the strength of the vibronic coupling and is a crucial parameter for comparing with theoretical calculations.
Our model captures the asymmetric spectral shape.
However, the precise details of the low-energy phonon-coupling lineshape become correlated with the ZPL width (assumed to be Lorentzian) and $S_{\mathrm{HR}}$ when fitting the model to experimental data.
We account for these correlations by performing the fits using varied constraints on the low-energy phonon coupling motivated by scaling considerations.
We follow the method described in Ref. [6], in order to estimate uncertainties on $S_{\mathrm{HR}}$ and the ZPL linewidth.
Overall, we find that the best-fit ZPL linewidths are narrower or comparable to those reported in the literature for off-resonant excitation of h-BN’s QEs at room temperature [51, 10, 52], and the values of $S_{\mathrm{HR}}$ are somewhat higher.
We consider comparisons to theoretical proposals in detail later; we note here that the ZPL energies and $S_{\mathrm{HR}}$ values closely match the calculated properties of boron dangling bonds [43, 53].
The QE’s polarization-resolved PL excitation and emission characteristics (Fig. 1) begin to reveal more complicated features of their optical dynamics.
Both QEs A and C exhibit linearly polarized emission with nearly complete visibility, again consistent with emission through a single optical dipole transition.
For QE A, the PL intensity varies as a function of excitation polarization angle in a manner consistent with excitation through a single optical dipole, with high visibility and an angle aligned with the emission dipole, independent of excitation wavelength (532 nm or 592 nm).
In the case of QE C, the emission polarization visibility and dipole angle is similarly independent of the excitation wavelength.
However, QE C’s excitation polarization dependence varies dramatically as a function of excitation wavelength; the excitation dipole is aligned with the emission under 592 nm excitation, but misaligned under 532 nm excitation with substantially reduced visibility.
Emission polarization data are not available for QEs B, D, and E, but the excitation polarization measurements are qualitatively similar to those for QEs A and C.
All three QEs show polarized absorption with varying degrees of visibility.
The heterogeneous polarization responses are consistent with previous observations for QEs in h-BN [27, 6, 28, 18].
In particular, Ref. [28] studied the variation of polarization visibility and alignment between excitation and emission as a function of the energy difference between the excitation photon energy and the ZPL photon energy, $\Delta E$.
They observed that the excitation and emission dipoles are aligned ($\Delta\theta=0$) when $\Delta E\lesssim$200\text{\,}\mathrm{meV}\text{/}$$, whereas if $\Delta E\gtrsim$200\text{\,}\mathrm{meV}\text{/}$$, $\Delta\theta$ can take any value.
Our observations are consistent with this empirical finding.
For QE A, the excitation and emission angles are aligned despite relatively large energy differences, $\Delta E^{592}=$169\text{\,}\mathrm{meV}\text{/}$$ and $\Delta E^{532}=$405\text{\,}\mathrm{meV}\text{/}$$, for 592 nm and 532 nm excitation, respectively.
For QE C, the dipoles are aligned for excitation at 592 nm ($\Delta E^{592}=$10\text{\,}\mathrm{meV}\text{/}$$; $\Delta\theta^{592}={0.0\pm 1.4}{}$) but misaligned at 532 nm ($\Delta E^{532}=$247\text{\,}\mathrm{meV}\text{/}$$; $\Delta\theta^{532}={46.5\pm 1.6}{}$).
Misalignment between absorption and emission dipoles is expected if the optical dynamics involve multiple excited states.
Whereas the invariance of the PL polarization, visibility, and spectral shape to exctiation energy implies that PL emission occurs through a single optical transition, off-resonant optical pumping can involve transient excitation of higher-lying excited states through transitions with different optical dipole orientations, which subsequently relax to the radiative state as shown in Fig. 5(a).
Depending on energy level arrangement and the vibronic copuling strengths, a single excitation laser can drive both transitions between states $1\rightarrow 2$ and $1\rightarrow 3$.
The excitation polarization dependence will then reflect a superposition of two optical dipole transitions, with an orientation and visibility determined by the underlying dipole transition orientations and their relative optical cross section.
To test this hypothesis, we performed a simultaneous fit of QE C’s emission and excitation polarization data under 532 nm excitation assuming a single shared dipole for excitation and emission via states $1\leftrightarrow 2$ together with a second dipole for excitation via $1\rightarrow 3$ (see Section S4 and Fig. S2 in the Supplementary Materials).
We find that the data are consistent with such a model, in which the dipole projection for transition $1\rightarrow 3$ is misaligned from that of transition $1\leftrightarrow 2$ by $63\pm 1{}$, and the ratio of excitation cross sections is $\Gamma_{12}/\Gamma_{13}\sim 0.5$.
In interpreting these results, we note that the observation of highly polarized emission implies the presence of at least one symmetry axis for the underlying electronic states.
For most defect models under consideration, symmetry allows for optical dipole transitions aligned perpendicular to the h-BN plane (along $z$) or within the plane either parallel or perpendicular to the defect’s symmetry axis (along $x$ or $y$).
Hence, the observation of dipoles misaligned by $\sim$60 seems surprising.
However, since our polarization-resolved experiments are primarily sensitive to the projection of the dipole perpendicular to the microscope’s optical axis, it is possible that sample misalignment or local distortions of the defect that tend to tilt the $z$ axis could explain the observations.
Alternatively, our model of two superposed excitation dipoles might not capture all salient features of the excitation process; more than two transitions might be involved, and yet the superposition of any number of dipole absorption patterns will ultimately yield a polarization dependence consistent with Eq. (2).
III.2 Optical dynamics
PECS experiments reveal key details regarding the nature of the QEs’ excited states and optical dynamics.
The PECS results summarized in Fig. 4 resolve individual dynamical processes, their associated timescales, and their dependence on optical excitation power.
All QEs feature three or more timescales in their autocorrelation spectra, which implies that the optical dynamics involve at least four electronic levels.
In addition to antibunching on nanosecond timescales, all QEs exhibit bunching with two or more resolvable timescales that are orders of magnitude longer (typically microseconds to milliseconds).
These bunching timescales are broadly consistent with past observations [6, 3, 4, 54, 26, 14], and they indicate the role of metastable dark states in the optical dynamics.
Here, we emphasize and discuss two key features of the PECS measurements in Fig. 4: the nonlinear power dependence of the antibunching rate, $\gamma_{1}$, that is clearly observed for QEs A, B, and C; and the heterogeneous behavior of the bunching rates and amplitudes, which feature qualitatively diverse power-dependent variations.
For a QE featuring a direct optical transition between a ground state and a radiative excited state, the antibunching rate scales linearly as a function of optical excitation power, with a zero-power offset corresponding to the QE’s spontaneous emission rate.
This is the case even for QEs that also feature metastable charge and spin states, such as the NV center in diamond [55].
In Supplementary Figure S5, we present measurements of the antibunching rate of single NV centers in nanodiamonds as a function of excitation power; the results show clear linear scaling and a zero-power offset for $\gamma_{1}$ consistent with the expected optical lifetime.
The PECS observations of h-BN’s QEs in Fig. 4 defy this expectation.
The power-scaling of $\gamma_{1}$ for QEs A, B, and C is clearly sublinear, with a saturation behavior (Model II or Model III) characterized by a steep slope at low power tapering off to a shallow slope at high power.
Moreover, the $\gamma_{1}$ measurements for QE A are all less or equal to the measured spontaneous decay rate (dashed line in the upper-left panel of Fig. 4), whereas $\gamma_{1}$ always exceeds the spontaneous rate for a direct optical transition.
The zero-power offset for $\gamma_{1}$ in QEs A and B is consistent with zero but poorly constrained due to the steep low-power slope; the offset is non-zero for QEs C, D, and E.
QEs D and E exhibit linear power-scaling of $\gamma_{1}$, however the range of available powers is smaller than for the other emitters, and we cannot rule out a saturation behavior at higher power.
Previous studies of QEs in h-BN have revealed hints of power-independent antibunching rates [54] and nonlinear power scaling [3, 26], however these observations were never satisfactorily explained.
The antibunching rate’s nonlinear power dependence can be understood in the context of an indirect excitation mechanism, as illustrated in Fig. 5(a), where optical excitation leads to the population of multiple states: levels 2 and 3, with competing rates $\Gamma_{12}$ and $\Gamma_{13}$.
Indirect population of the radiative state (level 2) through such a mechanism creates a rate-limiting step ($3\rightarrow 2$) to the optical emission pathway ($2\rightarrow 1$) that leads to nonlinear scaling of the observed antibunching rate, as shown in Fig. 5(c).
The rate-limiting nature of this process is intuitively obvious in the limit where $\Gamma_{12}/\Gamma_{13}\ll 1$.
However, we find that the nonlinear saturation behavior remains qualitatively consistent independent of the pumping-rate ratio across a wide range of simulation settings where $\Gamma_{12}/\Gamma_{13}\in[0,2]$ (see Supplementary Figure S8).
In the regime $\Gamma_{12}/\Gamma_{13}\ll 1$, the population of level 2 is still mostly determined by the indirect excitation pathway through level 3, with rate $\kappa_{32}$, and the dominant antibunching rate saturates to a value close to $\kappa_{32}+\Gamma_{21}$.
In the regime where $\Gamma_{12}/\Gamma_{13}>1$, two underlying rates in the dynamical system are associated with the antibunching dip.
As discussed previously, the eigenvalues associated with these rates can be real or complex depending on the relative magnitudes of transitions in the system.
However, the fast rate associated with the direct population of level 2 and the subtle signatures of complex eigenvalues on the shape of the antibunching dip turn out not to be detectable when we include realistic assumptions for the experimental limits on timing resolution and shot noise.
Instead, we observe a single effective antibunching rate $\gamma_{1}$ that exhibits nonlinear saturation similar to the slow rate.
The bunching dynamics observed in Fig. 4 can also be understood within our optical dynamics models by including metastable shelving states.
In Fig. 5, the key qualitative difference between spontaneous population of the metastable state(s) (e.g., spin-dependent intersystem crossings) and optically pumped transitions (e.g., ionization/recombination) appears in the power-scaling and zero-power offset of the associated bunching rate.
Spontaneous transitions are characterized by a rate with a non-zero offset that tends to saturate with increasing pumping power, whereas optically pumped transitions have zero offset and increase quasi-linearly.
Previous studies considering the power-scaling of bunching rates for QEs in h-BN nano-flakes and exfoliated flakes have proposed similar optically-pumped models [26, 3].
Similar behavior has also been observed in color centers such as the silicon-vacancy center in diamond, attributed to power-dependent de-shelving from higher lying states to the metastable state [56].
However, the heterogeneity and complexity of these processes for QEs in h-BN, both regarding the number of levels and the type of transitions, have not been considered before.
We observe both qualitative bunching behaviors in the data of Fig. 4, with several QEs exhibiting multiple bunching levels that apparently have different transition mechanisms.
In some cases, individual bunching rates exhibit power scalings with features of both phenomena; for example, $\gamma_{2}$ for QEs A, B, D, and E appears to have a non-zero offset and yet increase linearly with power.
This could indicate that the associated state can be populated both spontaneously and through an optically pumped pathway.
Our simulations support this intuitive reasoning.
For example, Supplementary Figure S9 shows the results of simulations of the same four-level system as in Fig. 5(a), but with rates chosen to reproduce the observations for QE A from Fig. 4.
We indeed find that a combination of spontaneous and optically pumped transitions to the metastable state ($\kappa_{24}$ and $\kappa_{41}$) yields a bunching rate $\gamma_{2}$ with a non-zero offset that scales linearly with pumping power.
Moreover, setting $\kappa_{32}<\Gamma_{21}$ creates a situation where the spontaneous emission rate exceeds the observed antibunching rate, $\Gamma_{21}>\gamma_{1}$, over a wide range of pumping power, in agreement with our observations.
The quantitative magnitudes of $\gamma_{1}$, $\gamma_{2}$, and the bunching amplitude, $C_{2}$, are also reproduced by the simulations.
This highlights the versatility of optical dynamics simulations as a valuable tool to recreate or predict optical dynamics based on complex combinations of radiative and nonradiative processes.
To fully capture the observed dynamics of any particular QE, including the additional bunching rates $\gamma_{3}$ and $\gamma_{4}$ (where applicable), more metastable states are required in the simulations.
We further note that the number of observed bunching timescales represents a lower limit on the number of metastable states, and hence some states could actually represent multiplets associated with different spin manifolds.
Even with those caveats, these observations present the opportunity for quantitative comparisons with theoretical predictions.
III.3 Consistency with theoretical proposals
Several defect structures have been proposed as the origin of visible-wavelength single-photon emission in h-BN, including the boron dangling bond (DB) [43], $V_{\rm N}N_{\rm B}$ [2], $V_{\rm N}C_{\rm B}$ [46], and $V_{\rm B}C_{\rm N}$ [34].
The negatively-charged boron vacancy, $V_{\rm B}^{-}$, has been suggested to give rise to an optically detected magnetic resonance signal observed for emitter ensembles [12], however $V_{\rm B}^{-}$ has a ZPL of $\sim$1.7 eV and couples more strongly to phonons ($S_{\mathrm{HR}}\sim 3.5$) [57], producing a PL band between 800-900 nm that does not overlap with the emitters considered here.
Early studies highlighted $V_{\rm N}N_{\rm B}$ as the potential origin of visible QEs [2], but recent calculations show that the coupling to phonons is substantially larger than observations [46].
More recently, $V_{\rm B}C_{\rm N}$ has been proposed based on the observation that carbon is correlated with the emission signal, but the calculated PL spectrum [34] does not match our observations.
The $V_{\rm B}C_{\rm N}$ calculations also predict a single, linearly-polarized absorption dipole, which is inconsistent with our measurements.
The calculated PL spectrum and strain dependence of $V_{\rm N}C_{\rm B}$ [46] are in reasonable agreement with the our observations.
However, the optical transition for $V_{\rm N}C_{\rm B}$ occurs in the triplet channel, while the calculated ground state is a singlet;
the authors did not propose a mechanism through which the triplet channel is populated quickly enough to give rise to the optical emission they considered.
The boron DB is predicted to possess an optical transition at 2.06 eV with a Huang-Rhys factor of 2.3 [43], which is in close agreement with the values observed in this study.
In addition, the variations in ZPL and $S_{\mathrm{HR}}$ for the observed emitters can be explained by out-of-plane distortions [53].
The ground state of the boron DB is a singlet, and the predicted existence of a triplet excited state can explain the presence of level 4 in Fig. 5(a).
Another important feature of the boron DB model is the proximity of the states to h-BN’s conduction band [43];
this allows electrons to be optically excited directly into the conduction band, depending on the excitation energy, explaining the misalignment of the absorptive and emissive dipole when the excitation energy is increased.
Other proposed models do not provide an explanation for the misalignment.
For instance, in the case of $V_{\rm N}C_{\rm B}$ the optical transition occurs in the neutral charge state, and for the excitation energies considered here, photoionization will not occur [58].
Within the boron DB model, we would interpret level 3 in Fig. 5(a) as the conduction band and $\kappa_{32}$ as the nonradiative capture rate.
To support this interpretation, we have estimated the relevant capture rate $\kappa_{32}$ of a photoionized electron from the conduction-band minimum into the DB excited state [level 2 in Fig. 5(a)].
This capture rate is a product of a capture coefficient and the density of electrons in the conduction band.
A first-principles calculation yields a capture coefficient of $4\times 10^{-7}$ cm${}^{3}$ s${}^{-1}$ (see Supplementary Materials, Section S9).
The density of electrons is estimated based on the thermal velocity of the photoionized electron ($\sim 10^{5}$ m s${}^{-1}$) and a typical electron energy relaxation time of $\sim 1$ ps [59].
In the time it takes the electron to relax to the conduction-band minimum, it can thus travel $\sim 100$ nm;
this distance corresponds to an effective electron density of $2.4\times 10^{14}$ cm${}^{-3}$.
Multiplying this value with the calculated capture coefficient gives a rate of $\kappa_{32}\sim 100$ MHz, in compelling agreement with the observed saturation antibunching rates of $\gamma_{1}\sim 300$-800 MHz for QEs A, B, and C.
Our calculations also show that capture into the excited state is favored over capture into the ground state by more than 5 orders of magnitude, justifying the neglect of $\kappa_{31}$ in the general model of Fig. 5(a).
The inclusion of photoionization allows us to further rationalize the heterogeneity in bunching behavior of the observed emitters:
the photoionized electron is not necessarily re-captured at the same QE, but may instead be captured by a neighboring defect, leaving the QE in a nonfluorescent, ionized configuration that likely requires optical excitation of additional free electrons to restore emission via subsequent electron capture.
This process would be represented in Fig. 5(a) by an optically pumped transition, where level 4 represents an ionized state of the QE.
The emitters may therefore be highly sensitive to the local defect environment.
Unlike other proposed defect models, we conclude that the boron DB model is thus capable of explaining numerous aspects of the experimental observations, lending support to this proposed microscopic structure.
IV Conclusion
The observations in this work reveal that h-BN’s QEs have intricate electronic level structures and complex optical dynamics including multiple charge or spin manifolds.
Our proposed electronic-structure models complement previous reports [54, 3, 26] and explain the quantitative features of our observations.
In particular, the models explain the observation of nonlinear power-scaling of the antibunching rate as well as heterogeneous magnitudes and power-scaling behavior of multiple bunching rates.
Whereas past reports have lacked consensus on mechanisms to explain the observed optical dynamics of h-BN’s QEs, and many posited chemical and electronic structure models have failed to adequately explain the heterogeneous observations, we show that the boron dangling bond model is remarkably consistent with experiments, especially accounting for the role of local distortions, photoionization, electron capture, and the QEs’ heterogeneous local defect environment.
Future experiments should be designed to investigate these details, for example time-domain studies of transients associated with charge and spin dynamics, and temperature- and excitation-energy-dependent variations of the PL lineshape, vibronic spectrum, and polarization-dependent excitation cross section.
Combined with theoretical models, such experiments can resolve the underlying transition rates and resolve the disparate influences of the QEs’ intrinsic properties with those of their local environments.
The observation of pure single-photon emission with $g^{(2)}(0)=0$ resolves earlier questions about h-BN’s QEs [48], affirming their potential for use in photonic quantum technologies.
More generally, we hope that the approach and techniques presented in this work — especially the quantitative use of PECS — present a model to formulate optical dynamics models for QEs in any material platform [47, 60].
Our models can be adapted to account for recent observations of magnetic-field-dependent optical dynamics [11] and optically detected magnetic resonance [13, 14] in h-BN.
Subsequently, they can be used to design protocols for initialization, control, and readout of quantum-coherent spin states for quantum information processing and quantum sensing.
V Materials and Methods
V.1 Sample preparation
We used a confocal microscope (see Supplementary Materials) to isolate individual QEs in h-BN under ambient conditions.
The h-BN samples were sourced from two batches (purchased $\sim$2 years apart) of bulk, single crystals from HQ Graphene.
Each batch consisted of roughly 20 different individual crystals.
The bulk crystals were mechanically exfoliated using a dry transfer process [49] resulting in thin ($\leq 100\text{\,}\mathrm{nm}\text{/}$) and large area ($\sim 10\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$) flakes of h-BN.
The exfoliated flakes were transferred to a SiO${}_{2}$/Si substrate with micro-fabricated circular trenches $4\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}8\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$ in diameter and $5\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$ deep [6] using a dry transfer process.
Table S1 in Supplementary Materials highlights the crystal from which the h-BN flake under study came.
Prior to the optical studies, the exfoliated h-BN samples were cleaned with a soft O${}_{2}$ plasma (Anatech SCE 106 Barrel Asher, $50\text{\,}\mathrm{W}\text{/}$ of power, $50\text{\,}\mathrm{sccm}\text{/}$ O${}_{2}$ flow rate) for 5 minutes to remove polymer residues resulting from the transfer process.
The samples were then annealed in a tube furnace at $850\text{\,}\mathrm{\SIUnitSymbolCelsius}\text{/}$ in low flow Ar atmosphere for 2 hours.
Annealing h-BN has been found to brighten the emitters [35].
While not the focus of study here, annealing for longer time (2 hours vs commonly used 30 minutes) appears to improve emitter stability.
One sample was also exposed to a focused ion beam chamber operated in scanning electron mode (FEI Strata DB235 FIB SEM) but was not directly exposed to the electron beam.
Supplementary Table S1 summarizes the three samples investigated, the QEs studied in each sample and the annealing treatment received by each sample.
V.2 Experimental details
Supplementary Figure S1 depicts a simplified schematic of the room-temperature confocal microscope used to measure the QEs.
There are two available excitation sources: a $532\text{\,}\mathrm{nm}\text{/}$ (green) cw laser (Coherent, Compass 315M-150) and a $592\text{\,}\mathrm{nm}\text{/}$ (orange) cw laser (MPB Communications, VF-P-200-592).
The power and polarization of each excitation path can be independently selected.
Reported power values are measured just prior to the objective.
In addition, a shutter completely blanks the excitation source when imaging is not in use to mitigate unnecessary light exposure.
The excitation paths are combined with the collection path using a long pass (LP) dichroic mirror (Semrock, BrightLine FF560-FDi01 for green and Semrock, BrightLine FF640-FDi01 for orange).
The LP dichroic cut-off is 560 nm for green excitation and 640 nm for orange excitation.
A fixed half-wave plate in each of the excitation paths corrects for the birefringence induced by the dichroic mirrors.
The co-aligned excitation and collection paths are sent through a $4f$ lens system with a fast steering mirror (Optics in Motion, OIM101) and a 0.9 NA 100x objective (Olympus, MPI Plan Fluor) at the image planes.
This allows for the collection of wide-field, rastered, micro-photoluminescence ($\micro$-PL) images.
The objective is mounted on a stage system for changing the field of view.
The collection path consists of a linear polarizer (Thorlabs, WP25M-VIS) for measuring the emission polarization as well as a wide-band variable retarder (Meadowlark, LRC-100) which compensates for the birefringence induced by the dichroic.
A LP filter specific to the excitation color fully extinguishes any scattered excitation light and the Raman signal.
The cut-on wavelengths are $578\text{\,}\mathrm{nm}\text{/}$ (Semrock, BLP01-568R-25) and $650\text{\,}\mathrm{nm}\text{/}$ (Semrock, BLP01-635R-25) for green and orange, respectively.
The filtered light is focused onto the core of a $50\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$ core multi-mode fiber (Thorlabs, M42L01) acting as a pinhole.
The output of the fiber is connected to a fiber switch (DiCon, MEMS 1x2 Switch Module) which can switch the collected emission to either a 50:50 visible fiber splitter (Thorlabs, FCMM50-50A-FC) or a spectrometer (Princeton Instruments, IsoPlane160 and Pixis 100 CCD).
The outputs of the fiber splitter are sent to two identical single-photon counting modules (SPCM, Laser Components, Count T-100) resulting in a Hanbury Brown Twiss interferometer.
The outputs of the SPCMs are either measured by a data acquisition card (National Instruments, DAQ6323) for general-purpose counting or a time-correlated single-photon counting (TCSPC) module (PicoQuant, PicoHarp 300) for recording the photon time-of-arrival information with a full system resolution of $\sim 350\text{\,}\mathrm{ps}\text{/}$.
V.3 Spectra analysis
The PL spectra are collected as multiple exposures and averaged after correcting for dark counts, cosmic rays and wavelength-dependent photon collection efficiency.
The PL spectra are measured as a function of wavelength, $\lambda$ and binned to determine spectral distribution function, $S(\lambda)$.
To analyze the vibronic coupling, the measured spectra must be converted to a form suitable for analysis with the general theory of electron-phonon coupling in three dimensional crystals [61, 62].
To do this, the spectral probability distribution function is obtained through
$$S(E)=S(\lambda)\frac{hc}{E^{2}}$$
(7)
where $h$ is Planck’s constant, $c$ is the speed of light, and $E$ is the photon energy.
The emission lineshape, $L(E)$ is derived from $S(E)$ as
$$L(E)=\frac{S(E)}{E^{3}}$$
(8)
which accounts for the photon-energy dependence of spontaneous emission.
The emission lineshape is then fit following the method described in [6].
From the fit, the following free parameters are determined: the ZPL energy, $E_{\mathrm{ZPL}}$, the ZPL Lortentzian linewidth, $\Gamma_{\mathrm{ZPL}}$, the Huang-Rhys factor, $S_{\mathrm{HR}}$, and the one-phonon vibronic coupling lineshape, approximated as an interpolated vector of values spanning the phonon spectrum in h-BN.
The Debye-Waller factor, $w_{\mathrm{DW}}$, can be calculated from $w_{\mathrm{DW}}=e^{-S_{\mathrm{HR}}}$.
V.4 Second-order photon autocorrelation function
For a given QE, all autocorrelation measurements are performed with the excitation polarization set at the angle of maximum excitation and the collection path has the polarizer removed.
Due to the varying QE brightness, which affects the signal-to-noise ratio of the antibunching signal, measurements are integrated for 10 s to 140 min with repositioning occurring every 2 min.
All errors from the fitting denote one standard deviation.
Due to timing jitter in the single photon counting modules (SPCMs) introducing systematic artifacts at short delay times, the data are analyzed in two stages: logarithmic and linear scales.
First, the the autocorrelation data are binned over a log scale for visualizing the dynamics over 9 orders of magnitude in time (0.1 $ns$ to 1 $s$) corrected for background [63] (see Supplementary Materials), and then fit by multiple instances of Eq. 5 with $n=[2,5]$.
The best fit, and corresponding $n$, is then determined by calculating the AIC and comparing the reduced chi-squared statistic.
This method determines the number of bunching levels and their rates and amplitudes that best explain the observations.
The QE’s autocorrelation data are then binned over a linear scale that contains the antibunching features ($\tau\leq$30\text{\,}\mathrm{ns}\text{/}$$).
To account for the timing jitter in the SPCMs, the instrument response function (IRF) is found by measuring the autocorrelation signal of an attenuated picosecond pulsed laser sent through the HBT interferometer and binned over the same linear scale as the QE.
A convolution of the IRF with a modified Eq. 5 is fit to the background-corrected data, given by
$$\tilde{g}^{(2)}(\tau)=\mathrm{IRF}*(1-C_{1}e^{-\gamma_{1}|\tau|}+C_{\mathrm{B}}(\tau))$$
(9)
where $C_{\mathrm{B}}(\tau)$ is the total bunching contribution found from the log scale analysis (first step) and only $C_{1}$ and $\gamma_{1}$ are allowed to vary.
The autocorrelation at zero delay is then given by
$$\tilde{g}^{(2)}(0)=1-C_{1}+\sum_{i=2}^{n}C_{i}$$
(10)
which is used to determine the purity of single-photon emission from the QE.
V.5 Electronic level structure simulations
A four-level optical rate equation is used to model aspects of the observed autocorrelation data.
The model is defined as
$$\displaystyle\mathbf{\dot{P}}=\mathbf{G}\mathbf{P}$$
(11)
where $\mathbf{P}$ is a vector of state populations $P_{i}$ and $\mathbf{G}$ is a generator matrix describing the transition rates and is given by
$$\displaystyle\bf{G}=\ \begin{pmatrix}-\Gamma_{13}&\Gamma_{21}&0&\kappa_{41}\\
0&-\Gamma_{21}-\kappa_{24}&\kappa_{32}&0\\
\Gamma_{13}&0&-\kappa_{32}&0\\
0&\kappa_{24}&0&-\kappa_{41}\end{pmatrix}$$
(16)
where $\Gamma_{13}$ is the excitation rate, $\Gamma_{21}$ is the radiative emission rate, and $\kappa_{ij}$ are nonradiative rates that are either fixed or a proportion of the excitation rate.
The $g^{(2)}(\tau)$ can be found by calculating the time evolution of the radiative state, $P_{2}$ given the system started in state $P_{1}$ and normalizing by the steady state population of $P_{2}$.
This is given by
$$\displaystyle g^{(2)}(\tau)=\frac{P_{2}(t_{2}|P(t_{1})=(1,0,0,0))}{P_{2}(\infty)}$$
(17)
where $\tau=t_{2}-t_{1}$.
The differential equation (Eq. 11) given the initial state is solved in MATLAB using the function ode15s.
Timing resolution limitations and shot noise are added to the simulated autocorrelation function to best recreate the measurements.
To model timing resolution, the simulated data are only analyzed for $t_{0}\geq$0.5\text{\,}\mathrm{ns}\text{/}$$.
To include shot noise, a standard deviation, $\sigma_{0}$, is set for the first delay time.
Assuming shot noise, this standard deviation is converted to mean number of photons as
$$\displaystyle\braket{N_{0}}=\sigma_{0}^{-2}$$
(18)
The log-scale processing results in the average number of photon correlations detected in each bin increasing linearly with the delay time,
$$\displaystyle\braket{N(\tau)}=\braket{N_{0}}\frac{\tau}{\tau_{0}}$$
(19)
From this, a simulated, noisy $g^{(2)}(\tau)$ is calculated as
$$\displaystyle g^{(2)}(\tau)_{\mathrm{Noisy}}=\frac{\mathrm{Poiss}(g^{(2)}(\tau)\braket{N(\tau)})}{\braket{N(\tau)}}$$
(20)
where Poiss is a Poission distribution.
The simulated autocorrelation data are analyzed with the same fitting framework as the measured data.
The general model parameters are as follows:
$\Gamma_{21}=$300\text{\,}\mathrm{MHz}\text{/}$$,
$\Gamma_{13}=a\Gamma_{21}$ where $a=[.01,10]$,
$\Gamma_{12}=x\Gamma_{13}$ where $x=[0,2]$,
$\kappa_{32}=$600\text{\,}\mathrm{MHz}\text{/}$$.
For the spontaneous bunching,
$\kappa_{24}=$60\text{\,}\mathrm{kHz}\text{/}$$ and $\kappa_{41}=$30\text{\,}\mathrm{kHz}\text{/}$$.
For the pumped bunching,
$\kappa_{24}=$6\text{\,}\mathrm{kHz}\text{/}\mathrm{MHz}$\times\Gamma_{13}$ and $\kappa_{41}=$3\text{\,}\mathrm{kHz}\text{/}\mathrm{MHz}$\times\Gamma_{13}$.
V.6 Theoretical calculations
We perform first-principles density-functional theory calculations as implemented in the VASP code [64, 65].
We utilize the hybrid functional of Heyd, Scuseria, and Ernzerhof [66, 67] to ensure accurate energetics, electronic structure, and atomic geometries.
The fraction of Hartree-Fock exchange is set to 40%, consistent with previous studies [43, 53].
A plane-wave basis with projector augmented-wave potentials [68] is used, and the energy cutoff for the basis is set to 520 eV.
The boron dangling bond is modeled in a 240-atom supercell with volume 2110 Å${}^{3}$ within periodic boundary conditions [69].
A single, special k-point (0.25, 0.25, 0.25) is used to sample the Brillouin zone.
Lattice vectors are held fixed while the atomic forces are relaxed to below 0.01 eV/Å.
To calculate the nonradiative capture coefficient, we utilize the formalism of Ref. [70] implemented in the Nonrad code [71].
VI Acknowledgements
This work was primarily supported by the National Science Foundation (NSF) award DMR-1922278.
J.A.G. was supported by an NSF Graduate Research Fellowship (DGE-1845298).
The authors gratefully acknowledge use of facilities and instrumentation in the Singh Center for Nanotechnology at the University of Pennsylvania, supported by NSF through the National Nanotechnology Coordinated Infrastructure (NNCI; Grant ECCS-1542153) and the University of Pennsylvania Materials Research Science and Engineering Center (MRSEC; DMR-1720530).
M.E.T. was supported by the NSF through Enabling Quantum Leap: Convergent Accelerated Discovery Foundries for Quantum Materials Science, Engineering and Information (Q-AMASE-i; DMR-1906325).
Computational resources were provided by the Extreme Science and Engineering Discovery Environment (XSEDE), supported by the NSF (ACI-1548562).
We gratefully acknowledge fruitful discussions with A. Alkauskas.
Author Contributions
R.N.P. and L.C.B. designed the study.
R.N.P. carried out the experiments.
R.N.P., D.A.H., and L.C.B. analyzed and interpreted the experimental data and performed optical dynamics simulations.
J.A.G. and L.C.B. carried out the PL spectra analysis and vibronic coupling calculations.
R.E.K.F. contributed to the analysis and interpretation of photon emission correlation spectroscopy experiments.
M.E.T. and C.G.V.W. performed theoretical calculations of defect models.
R.N.P. and B.P. prepared the samples.
T.-Y.H. and D.A.H. contributed to designing the experiment setup and data acquisition software.
R.N.P., D.A.H., M.E.T., and L.C.B. wrote the manuscript.
All authors contributed to discussions and commented on the manuscript.
References
Meuret et al. [2015]
S. Meuret, L. H. G. Tizei, T. Cazimajou,
R. Bourrellier, H. C. Chang, F. Treussart, and M. Kociak, Photon Bunching in Cathodoluminescence, Phys. Rev. Lett. 114, 197401 (2015).
Tran et al. [2016a]
T. T. Tran, K. Bray, M. J. Ford, M. Toth, and I. Aharonovich, Quantum emission from hexagonal boron nitride
monolayers, Nat. Nanotechnol. 11, 37 (2016a).
Chejanovsky et al. [2016]
N. Chejanovsky, M. Rezai,
F. Paolucci, Y. Kim, T. Rendler, W. Rouabeh, F. F. de Oliveira, P. Herlinger, A. Denisenko, S. Yang, I. Gerhardt, A. Finkler,
J. H. Smet, and J. Rg Wrachtrup, Structural Attributes and
Photodynamics of Visible Spectrum Quantum Emitters in Hexagonal Boron
Nitride, Nano Lett. 16, 7037 (2016).
Martinez et al. [2016]
L. J. Martinez, T. Pelini,
V. Waselowski, J. R. Maze, B. Gil, G. Cassabois, and V. Jacques, Efficient single photon emission from a high-purity hexagonal boron nitride
crystal, Phys. Rev. B 94, 121405(R) (2016).
Shotan et al. [2016]
Z. Shotan, H. Jayakumar,
C. R. Considine, E. Mackoit, H. Fedder, R. Wrachtrup, A. Alkauskas, M. W. Doherty, V. M. Menon, and C. A. Meriles, Photoinduced Modification of Single-Photon Emitters in Hexagonal Boron
Nitride, ACS Photonics 3, 2490 (2016).
Exarhos et al. [2017]
A. L. Exarhos, D. A. Hopper,
R. R. Grote, A. Alkauskas, and L. C. Bassett, Optical Signatures of Quantum Emitters in
Suspended Hexagonal Boron Nitride, ACS Nano 11, 3328 (2017).
Noh et al. [2018]
G. Noh, D. Choi, J.-H. Kim, D.-G. Im, Y.-H. Kim, H. Seo, and J. Lee, Stark
Tuning of Single-Photon Emitters in Hexagonal Boron Nitride, Nano Lett. 18, 4710 (2018).
Chakraborty et al. [2019]
C. Chakraborty, N. Vamivakas, and D. Englund, Advances in quantum
light emission from 2D materials, Nanophotonics 8, 2017 (2019).
Kim et al. [2019]
S. Kim, N. M. H. Duong,
M. Nguyen, T. J. Lu, M. Kianinia, N. Mendelson, A. Solntsev, C. Bradac, D. R. Englund, and I. Aharonovich, Integrated on Chip Platform with Quantum Emitters in Layered Materials, Adv. Opt. Mater. 7 (2019).
Dietrich et al. [2020]
A. Dietrich, M. W. Doherty, I. Aharonovich, and A. Kubanek, Solid-state single
photon source with Fourier transform limited lines at room temperature, Phys. Rev. B 101, 081401(R) (2020).
Exarhos et al. [2019]
A. L. Exarhos, D. A. Hopper,
R. N. Patel, M. W. Doherty, and L. C. Bassett, Magnetic-field-dependent quantum emission in
hexagonal boron nitride at room temperature, Nat. Commun. 10, 222 (2019).
Gottscholl et al. [2020]
A. Gottscholl, M. Kianinia, V. Soltamov,
S. Orlinskii, G. Mamin, C. Bradac, C. Kasper, K. Krambrock, A. Sperlich, M. Toth, I. Aharonovich, and V. Dyakonov, Initialization and
read-out of intrinsic spin defects in a van der Waals crystal at room
temperature, Nat. Mater. 19, 540 (2020).
Chejanovsky et al. [2021]
N. Chejanovsky, A. Mukherjee, J. Geng,
Y.-C. Chen, Y. Kim, A. Denisenko, A. Finkler, T. Taniguchi, K. Watanabe, D. Bhaktavatsala Rao Dasari, P. Auburger, A. Gali, J. H. Smet, and J. Wrachtrup, Single-spin resonance in a van der Waals embedded paramagnetic defect, Nat. Mater. 20, 1079 (2021).
Stern et al. [2021]
H. L. Stern, J. Jarman,
Q. Gu, S. Eizagirre Barker, N. Mendelson, D. Chugh, S. Schott, H. H. Tan, H. Sirringhaus, I. Aharonovich, and M. Atatüre, Room-temperature
optically detected magnetic resonance of single defects in hexagonal boron
nitride, arxiv arXiv:2103 (2021).
Gottscholl et al. [2021]
A. Gottscholl, M. Diez,
V. Soltamov, C. Kasper, A. Sperlich, M. Kianinia, C. Bradac, I. Aharonovich, and V. Dyakonov, Room
temperature coherent control of spin defects in hexagonal boron nitride, Sci. Adv. 7, eabf3630 (2021).
Tran et al. [2016b]
T. T. Tran, C. Elbadawi,
D. Totonjian, C. J. Lobo, G. Grosso, H. Moon, D. R. Englund, M. J. Ford, I. Aharonovich, and M. Toth, Robust Multicolor Single
Photon Emission from Point Defects in Hexagonal Boron Nitride, ACS Nano 10, 7331 (2016b).
Bourrellier et al. [2016]
R. Bourrellier, S. Meuret,
A. Tararan, O. Stehan, M. Kociak, L. H. G. Tizei, and A. Zobelli, Bright
UV Single Photon Emission at Point Defects in h‑BN, Nano Lett. 16, 4317 (2016).
Ziegler et al. [2018]
J. Ziegler, A. Blaikie,
A. Fathalizadeh, D. Miller, F. S. Yasin, K. Williams, J. Mohrhardt, B. J. McMorran, A. Zettl, and B. Alema, Single-Photon Emitters in Boron Nitride Nanococoons, Nano Lett. 18, 2683 (2018).
Stern et al. [2019]
H. L. Stern, R. Wang,
Y. Fan, R. Mizuta, J. C. Stewart, L.-M. Needham, T. D. Roberts, R. Wai, N. S. Ginsberg,
D. Klenerman, S. Hofmann, and S. F. Lee, Spectrally Resolved Photodynamics of Individual Emitters in
Large-Area Monolayers of Hexagonal Boron Nitride, ACS Nano 13, 4538 (2019).
Mackoit-Sinkeviciene et al. [2019]
M. Mackoit-Sinkeviciene, M. Maciaszek, C. G. Van de
Walle, and A. Alkauskas, Carbon dimer defect as
a source of the 4.1 eV luminescence in hexagonal boron nitride, Appl. Phy. Lett. 115, 212101 (2019).
Hayee et al. [2020]
F. Hayee, L. Yu, J. L. Zhang, C. J. Ciccarino, M. Nguyen, A. F. Marshall, I. Aharonovich, J. Vučković, P. Narang, T. F. Heinz, and J. A. Dionne, Revealing
multiple classes of stable quantum emitters in hexagonal boron nitride with
correlated optical and electron microscopy, Nat. Mater. 19, 534 (2020).
Cretu et al. [2021]
O. Cretu, A. Ishizuka,
K. Yanagisawa, K. Ishizuka, and K. Kimoto, Atomic-Scale Electrical Field Mapping of Hexagonal Boron
Nitride Defects, ACS Nano 15, 5316 (2021).
Feng et al. [2018]
J. Feng, H. Deschout,
S. Caneva, S. Hofmann, I. Lončarić, P. Lazić, and A. Radenovic, Imaging of Optically Active Defects with Nanometer Resolution, Nano Lett. 18, 1739 (2018).
Hernández-Mínguez et al. [2018]
A. Hernández-Mínguez, J. Lähnemann, S. Nakhaie, J. Lopes, and P. V. Santos, Luminescent
Defects in a Few-Layer h -BN Film Grown by Molecular Beam Epitaxy, Phys. Rev. Appl. 10, 044031 (2018).
Hou et al. [2018]
S. Hou, M. D. Birowosuto,
S. Umar, M. A. Anicet, R. Y. Tay, P. Coquet, B. K. Tay, H. Wang, and E. H. T. Teo, Localized emission from
laser-irradiated defects in 2D hexagonal boron nitride, 2D Mater. 5, 015010 (2018).
Boll et al. [2020]
M. K. Boll, I. P. Radko,
A. Huck, and U. L. Andersen, Photophysics of quantum emitters in hexagonal
boron-nitride nano-flakes, Opt. Express 28, 7475 (2020).
Jungwirth et al. [2016]
N. R. Jungwirth, B. Calderon,
Y. Ji, M. G. Spencer, M. E. Flatte, and G. D. Fuchs, Temperature Dependence of Wavelength Selectable Zero-Phonon
Emission from Single Defects in Hexagonal Boron Nitride, Nano Lett. 16, 6052 (2016).
Jungwirth and Fuchs [2017]
N. R. Jungwirth and G. D. Fuchs, Optical Absorption and
Emission Mechanisms of Single Defects in Hexagonal Boron Nitride, Phys. Rev. Lett. 119, 057401 (2017).
Toledo et al. [2018]
J. R. Toledo, D. B. de Jesus, M. Kianinia,
A. S. Leal, C. Fantini, L. A. Cury, G. A. M. Sáfar, I. Aharonovich, and K. Krambrock, Electron paramagnetic resonance signature of point defects in
neutron-irradiated hexagonal boron nitride, Phys. Rev. B 98, 155203 (2018).
Comtet et al. [2019]
J. Comtet, E. Glushkov,
V. Navikas, J. Feng, V. Babenko, S. Hofmann, K. Watanabe, T. Taniguchi, and A. Radenovic, Wide-Field Spectral Super-Resolution Mapping of Optically Active Defects in
Hexagonal Boron Nitride, Nano Lett. 19, 2516 (2019).
Grosso et al. [2020]
G. Grosso, H. Moon,
C. J. Ciccarino, J. Flick, N. Mendelson, L. Mennel, M. Toth, I. Aharonovich, P. Narang, and D. R. Englund, Low-Temperature
Electron-Phonon Interaction of Quantum Emitters in Hexagonal Boron
Nitride, ACS Photonics 7, 1410 (2020).
Duong et al. [2018]
N. M. H. Duong, M. Anh Phan Nguyen, M. Kianinia, T. Ohshima,
H. Abe, K. Watanabe, T. Taniguchi, J. H. Edgar, I. Aharonovich, and M. Toth, Effects of High-Energy Electron Irradiation on Quantum Emitters in
Hexagonal Boron Nitride, ACS Appl. Mater. &
Interfaces 10, 24886
(2018).
Duong et al. [2019]
N. M. H. Duong, E. Glushkov, A. Chernev,
V. Navikas, J. Comtet, M. Anh, P. Nguyen, M. Toth, A. Radenovic, T. T. Tran, and I. Aharonovich, Facile Production of
Hexagonal Boron Nitride Nanoparticles by Cryogenic Exfoliation, Nano Lett. 19, 5417 (2019).
Mendelson et al. [2020]
N. Mendelson, D. Chugh,
J. R. Reimers, T. S. Cheng, A. Gottscholl, H. Long, C. J. Mellor, A. Zettl, V. Dyakonov, P. H. Beton, S. V. Novikov, C. Jagadish,
H. H. Tan, M. J. Ford, M. Toth, C. Bradac, and I. Aharonovich, Identifying carbon as the source of visible single-photon emission from
hexagonal boron nitride, Nat. Mater. 20, 321 (2020).
Breitweiser et al. [2020]
S. A. Breitweiser, A. L. Exarhos, R. N. Patel,
J. Saouaf, B. Porat, D. A. Hopper, and L. C. Bassett, Efficient Optical Quantification of Heterogeneous Emitter
Ensembles, ACS Photonics 7, 288 (2020).
Fournier et al. [2021]
C. Fournier, A. Plaud,
S. Roux, A. Pierret, M. Rosticher, K. Watanabe, T. Taniguchi, S. Buil, X. Quélin, J. Barjon, J.-P. Hermier, and A. Delteil, Position-controlled
quantum emitters with reproducible emission wavelength in hexagonal boron
nitride, Nat. Commun. 12, 3779 (2021).
Xu et al. [2021]
X. Xu, Z. O. Martin,
D. Sychev, A. S. Lagutchev, Y. Chen, T. Taniguchi, K. Watanabe, V. M. Shalaev, and A. Boltasseva, Creating Quantum Emitters in Hexagonal Boron Nitride Deterministically on
Chip-Compatible Substrates, Nano Lett. 21, 8182 (2021).
Stewart et al. [2021]
J. C. Stewart, Y. Fan,
J. S. Danial, A. Goetz, A. S. Prasad, O. J. Burton, J. A. Alexander-Webber, S. F. Lee, S. M. Skoff, V. Babenko, and S. Hofmann, Quantum Emitter Localization in Layer-Engineered Hexagonal Boron
Nitride, ACS Nano 15, 13591 (2021).
Glushkov et al. [2021]
E. Glushkov, N. Mendelson,
A. Chernev, R. Ritika, M. Lihter, R. R. Zamani, J. Comtet, V. Navikas, I. Aharonovich, and A. Radenovic, Direct Growth of Hexagonal Boron Nitride on Photonic Chips for
High-Throughput Characterization, ACS Photonics 8, 2033 (2021).
Tawfik et al. [2017]
S. A. Tawfik, S. Ali,
M. Fronzi, M. Kianinia, T. T. Tran, C. Stampfl, I. Aharonovich, M. Toth, and M. J. Ford, First-principles investigation of quantum emission from hBN defects, Nanoscale 9, 13575 (2017).
Weston et al. [2018]
L. Weston, D. Wickramaratne, M. Mackoit, A. Alkauskas, and C. G. Van de Walle, Native point defects and impurities
in hexagonal boron nitride, Phys. Rev. B 97, 214104 (2018).
Sajid et al. [2018]
A. Sajid, J. R. Reimers, and M. J. Ford, Defect states in hexagonal boron
nitride: Assignments of observed properties and prediction of properties
relevant to quantum computation, Phys. Rev. B 97, 064101 (2018).
Turiansky et al. [2019]
M. E. Turiansky, A. Alkauskas, L. C. Bassett, and C. G. Van de Walle, Dangling Bonds in
Hexagonal Boron Nitride as Single-Photon Emitters, Phys. Rev. Lett. 123, 127401 (2019).
Korona and Chojecki [2019]
T. Korona and M. Chojecki, Exploring point defects
in hexagonal boron-nitrogen monolayers, Int. J. Quantum Chem. 119, e25925 (2019).
Dev [2020]
P. Dev, Fingerprinting quantum
emitters in hexagonal boron nitride using strain, Phys. Rev. Res. 2, 022050(R) (2020).
Sajid and Thygesen [2020]
A. Sajid and K. S. Thygesen, VNCB defect as source
of single photon emission from hexagonal boron nitride, 2D Mater. 7, 031007 (2020).
Fishman et al. [2021]
R. E. K. Fishman, R. N. Patel, D. A. Hopper, T.-Y. Huang, and L. C. Bassett, Photon emission correlation
spectroscopy as an analytical tool for quantum defects, arxiv arXiv:2111 (2021).
Bommer and Becher [2019]
A. Bommer and C. Becher, New insights into
nonclassical light emission from defects in multi-layer hexagonal boron
nitride, Nanophotonics 8, 2041
(2019).
Huang et al. [2015]
Y. Huang, E. Sutter,
N. N. Shi, J. Zheng, T. Yang, D. Englund, H.-J. Gao, and P. Sutter, Reliable Exfoliation of
Large-Area High-Quality Flakes of Graphene and Other Two-Dimensional
Materials, ACS Nano 9, 10612 (2015).
Wigger et al. [2019]
D. Wigger, R. Schmidt,
O. Del Pozo-Zamudio,
J. A. Preuß, P. Tonndorf, R. Schneider, P. Steeger, J. Kern, Y. Khodaei, J. Sperling,
S. M. De Vasconcellos,
R. Bratschitsch, and T. Kuhn, Phonon-assisted emission and absorption of
individual color centers in hexagonal boron nitride, 2D Mater. 6, 035006 (2019).
Dietrich et al. [2018]
A. Dietrich, M. Bürk, E. S. Steiger, L. Antoniuk,
T. T. Tran, M. Nguyen, I. Aharonovich, F. Jelezko, and A. Kubanek, Observation of Fourier transform limited lines in hexagonal boron
nitride, Phys. Rev. B 98, 081414(R) (2018).
Akbari et al. [2021]
H. Akbari, W.-H. Lin,
B. Vest, P. K. Jha, and H. A. Atwater, Temperature-dependent Spectral Emission of Hexagonal
Boron Nitride Quantum Emitters on Conductive and Dielectric Substrates, Phys. Rev. Appl. 15, 014036 (2021).
Turiansky and Van de
Walle [2021]
M. E. Turiansky and C. G. Van de Walle, Impact of dangling
bonds on properties of h-BN, 2D Mater. 8, 024002 (2021).
Sontheimer et al. [2017]
B. Sontheimer, M. Braun,
N. Nikolay, N. Sadzak, I. Aharonovich, and O. Benson, Photodynamics of quantum emitters in hexagonal boron nitride
revealed by low-temperature spectroscopy, Phys. Rev. B 96, 121202(R) (2017).
Doherty et al. [2013]
M. W. Doherty, N. B. Manson,
P. Delaney, F. Jelezko, J. Wrachtrup, and L. C. L. Hollenberg, The nitrogen-vacancy colour centre in diamond, Phys. Rep. 528, 1 (2013).
Neu et al. [2012]
E. Neu, M. Agio, and C. Becher, Photophysics of single silicon vacancycentersin
diamond: implications forsingle photon emission, Opt. Express 20 (2012).
Ivády et al. [2020]
V. Ivády, G. Barcza,
G. Thiering, S. Li, H. Hamdi, J.-P. Chou, O. Legeza, and A. Gali, Ab initio
theory of the negatively charged boron vacancy qubit in hexagonal boron
nitride, npj Comput. Mater. 6, 1 (2020).
Wu et al. [2017]
F. Wu, A. Galatas,
R. Sundararaman, D. Rocca, and Y. Ping, First-principles engineering of charged defects for two-dimensional
quantum technologies, Phys. Rev. Mater. 1, 071001(R) (2017).
Yan et al. [2014]
Q. Yan, E. Kioupakis,
D. Jena, and C. G. Van de Walle, First-principles study of high-field-related
electronic behavior of group-III nitrides, Phys. Rev. B 90, 121201(R) (2014).
Bassett et al. [2019]
L. C. Bassett, A. Alkauskas,
A. L. Exarhos, and K.-M. C. Fu, Quantum defects by design, Nanophotonics 8, 1867 (2019).
Maradudin [1966]
A. Maradudin, Theoretical and
experimental aspects of the effects of point defects and disorder on the
vibrations of crystals (Academic Press, 1966) pp. 273–420.
Davies [1974]
G. Davies, Vibronic spectra in
diamond, J.
Phys. C: Solid State Phys. 7, 3797 (1974).
Brouri et al. [2000]
R. Brouri, A. Beveratos,
J.-P. Poizat, and P. Grangier, Single-photon generation by pulsed excitation of
a single dipole, Phys. Rev. A 62, 063817
(2000).
Kresse and Furthmüller [1996a]
G. Kresse and J. Furthmüller, Efficient
iterative schemes for ab initio total-energy calculations using a plane-wave
basis set, Phys. Rev. B 54, 11169 (1996a).
Kresse and Furthmüller [1996b]
G. Kresse and J. Furthmüller, Efficiency of
ab-initio total energy calculations for metals and semiconductors using a
plane-wave basis set, Comput. Mater. Sci. 6, 15 (1996b).
Heyd et al. [2003]
J. Heyd, G. E. Scuseria, and M. Ernzerhof, Hybrid functionals based on a
screened Coulomb potential, J. Chem. Phys. 118, 8207 (2003).
Heyd et al. [2006]
J. Heyd, G. E. Scuseria, and M. Ernzerhof, Erratum: “Hybrid functionals based
on a screened Coulomb potential” [J. Chem. Phys. 118, 8207 (2003)], J. Chem. Phys. 124, 219906 (2006).
Blöchl [1994]
P. E. Blöchl, Projector
augmented-wave method, Phys. Rev. B 50, 17953 (1994).
Freysoldt et al. [2014]
C. Freysoldt, B. Grabowski, T. Hickel,
J. Neugebauer, G. Kresse, A. Janotti, and C. G. Van de Walle, First-principles calculations for point defects in
solids, Rev. Mod. Phys. 86, 253 (2014).
Alkauskas et al. [2014]
A. Alkauskas, Q. Yan, and C. G. Van de Walle, First-principles theory of
nonradiative carrier capture via multiphonon emission, Phys. Rev. B 90, 075202 (2014).
Turiansky et al. [2021]
M. E. Turiansky, A. Alkauskas, M. Engel,
G. Kresse, D. Wickramaratne, J.-X. Shen, C. E. Dreyer, and C. G. Van de Walle, Nonrad: Computing nonradiative capture coefficients from
first principles, Comput. Phy. Commun. 267, 108056 (2021). |
Collision-model-based approach to non-Markovian quantum dynamics
F. Ciccarello${}^{1}$, G. M. Palma${}^{2}$, and V. Giovannetti${}^{1}$
${}^{1}$NEST, Scuola Normale Superiore and Istituto Nanoscienze-CNR, Piazza dei Cavalieri 7, I-56126 Pisa, Italy
${}^{2}$NEST, Istituto Nanoscienze-CNR and Dipartimento di Fisica e Chimica, Universit$\grave{a}$ degli Studi di Palermo, via Archirafi 36, I-90123 Palermo, Italy
(December 7, 2020)
Abstract
We present a theoretical framework to tackle quantum non-Markovian dynamics based on a microscopic collision model (CM), where the bath consists of a large collection of initially uncorrelated ancillas.
Unlike standard memoryless CMs, we endow the bath with memory by introducing inter-ancillary collisions between next system-ancilla interactions. Our model interpolates between a fully Markovian dynamics and the continuous interaction of the system with a single ancilla, i.e., a strongly non-Markovian process. We show that in the continuos limit one can derive a general master equation, which while keeping such features is guaranteed to describe an unconditionally completely positive and trace-preserving dynamics. We apply our theory to an atom in a dissipative cavity for a Lorentzian spectral density of bath modes, a dynamics which can be exactly solved. The predicted evolution shows a significant improvement in approaching the exact solution with respect to two well-known memory-kernel master equations.
pacs: 03.65.Yz, 03.67.-a, 42.50.Lc
In open system dynamics the focus is on a system “$S$” in contact with an external environment. Typically, the goal is to seek
a master equation (ME) where the degrees of freedom (DOFs) of $S$ are the only explicit variables. Hence, the environmental interactions should be accounted for through an effective but reliable description. When it comes to quantum objects, this problem turns out to be especially thorny petruccione ; weiss ; huelga ; breuer . Within this context, “reliable” means that the ME to be worked out should give rise to a completely positive and trace-preserving (CPT) dynamics. It is well-assessed that Markovian, i.e., memoryless, environments are described by MEs in the so called Lindblad form petruccione entailing unconditionally CPT dynamics. Markovianity is in most cases only an approximation, though: in general, the environment is not forgetful and there is indeed a broad variety of actual phenomena featuring strong non-Markovian (NM) effects nm-actual . Yet, a general systematic framework for describing these has not been developed to date, which is a topical issue of concern to a manifold of variegated research areas. Rather, many different approaches have been proposed nota-citaz . Typically, they are to some extent regularly underpinned by phenomenological assumptions and/or approximations (testifying the formidable hurdles to cope with in such problems). As a consequence, non-CPT – namely unphysical – dynamics can turn out in certain parameter ranges vacchini1 . Among these descriptive tools are the so called memory-kernel MEs, e.g. those in Refs. barnett ; lidar . These are integro-differential MEs featuring a history integral, where past states of $S$ are weighted through a certain memory-kernel function. There exist regimes in which such MEs can fail to be CPT budini ; daffer ; sabrina ; vacchini2 ; wilkie ; laura ; kossa . What is more, it was recently tested laura-breuer whether MEs in Refs. barnett ; lidar are non-Markovian according to a non-Markovianity indicator proposed by Breuer et al. measure . It turned out that this is null laura-breuer , which suggests that such MEs should rather be regarded as time-dependent Markovian. This means that when they entail a CPT dynamics this is anyway very close to the purely Markovian regime (weak non-Markovianity).
In this work we tackle the problem to derive a non-Markovian ME by employing a suitably defined collision model (CM) rau ; alicki ; scarani ; buzek ; pelle1 ; pelle2 ; vittorio ; indivisible
of the system-bath interactions. This allows us to identify a new class of MEs featuring two attractive properties that rarely hold simultaneously. First, they unconditionally fulfill the CPT condition. Second, they nicely allow to interpolate between the purely Markovian regime and the strongly non-Markovian situation where $S$ is continuously interacting with a
low dimensional, hence non-forgetful, environment. Also, the model applies regardless of the dimensionality of $S$ and the form of the system-ancilla coupling.
We recall that in a conventional CM approach to open dynamics, the bath is modeled as a large collection of non-interacting identical ancillas (each can be thought as a low-dimensional system even though this is not necessary). By hypothesis, $S$ “collides” with each of these one at a time and, importantly, is not allowed to interact more than once with a given ancilla. Demonstrably, such a process gives rise to an irreversible dynamics for $S$
corresponding to a Lindblad-type petruccione , i.e., Markovian, ME buzek . This can be expected since, as stressed, at each step $S$ comes into contact with a fresh ancilla which is still in its initial state. Hence, there is no way for the bath to keep track of the system’s past history.
Although they are somewhat fictitious, latest research is unveiling the potential of CMs as effective theoretical tools for tackling open system dynamics vittorio ; indivisible . First, they are conceptually intuitive, hence potentially easier to cope with: a complex coupling to a large environment is decomposed as a succession of elementary interactions with its subparts. A key feature is that CMs lead to Lindblad-type MEs without demanding any approximation: in fact, only the passage to the continuous limit is needed buzek . This is in contrast to standard microscopic system-reservoir models petruccione , where Markovianity must be somehow enforced through drastic assumptions such as the requirement of small coupling and short enough bath correlation time (Born-Markov approximation). Should such a feature be maintained in a NM generalization of a CM, this would be quite appealing: as stressed above, approximations and phenomenological assumptions can lead to unphysical predictions. First progress along this line has been made very recently pelle1 ; vittorio ; indivisible . In particular, Rybar et al. indivisible introduced a CM able to simulate any indivisible channel petruccione (thus highly NM) when $S$ is a single qubit NC , i.e., a two-level system. Memory was introduced by taking the bath ancillas initially in a nontrivial quantum-correlated state, whose form depends on the specific simulated channel.
In the present work we tackle the problem from a completely different perspective: in line with physical intuition, we describe the memory effects as arising directly from the internal dynamics of the bath itself.
Specifically, in the very spirit of standard CMs, a natural memory mechanism to devise is adding inter-ancillary (AA) collisions between next system-ancilla (SA) ones.
This way, quantum information received from $S$ can be conveyed across the bath and returned to $S$ in next SA collisions (information backflow).
Here, we introduce a CM with memory precisely built upon this idea.
In the beginning (see Fig. 1) $S$ collides with ancilla 1. In standard (Markovian) CMs, $S$-2 collision would then follow, then $S$-3 and so on
[see Fig. 1(a)]. This way, each ancilla would still be in the initial state before colliding with $S$, thus fully “unaware” of previous collisions. In contrast, as sketched in Fig. 1(b), we assume that an extra AA collision between 1 and 2 occurs after $S$-1 but before $S$-2. Thereby, prior to its interaction with $S$, ancilla 2 will now be in a perturbed state in which information over past history of $S$ is imprinted. The process proceeds by mere iteration: once $S$-2 collision is over, a 2-3 interaction follows, then $S$-3, 3-4 etc.
Each collision, either SA or AA, is described by a CPT quantum map affecting the DOFs of the two involved particles.
Specifically, without loss of generality the SA collision involving the $i$th ancilla is defined as the mapping
$\sigma\rightarrow\mathcal{U}_{Si}[\sigma]\!=\!\hat{U}_{Si}\sigma\hat{U}_{Si}^{\dagger}$, with $\hat{U}_{Si}\!=\!e^{-i\hat{H}_{Si}\tau}$ being a unitary operator which depends upon the collision time $\tau$ and the interaction Hamiltonian $\hat{H}_{Si}$ (we set $\hbar\!=\!1$ throughout).
Instead, the AA collision involving the $i$-th and the $(i+1)$-th ancillas is defined
in terms of a stochastic process $\mathcal{S}_{i+1,i}$ which, with probability $p$, exchanges their states or leaves the system unaffected.
Formally, this is described by the transformation
$$\displaystyle\sigma\rightarrow\mathcal{S}_{i+1,i}[\sigma]$$
$$\displaystyle\!=$$
$$\displaystyle(1-p)\;\sigma\!+\!p\;\hat{S}_{i+1,i}\sigma\hat{S}_{i+1,i}\,,$$
(1)
where $\hat{S}_{i+1,i}$ is the swap operator NC on ancillas $i$ and $i+1$. As discussed in the last part of the manuscript, different AA collisional mechanisms can be
selected: the one in Eq. (1) however has the advantage that it allows for a simple analytical treatment
while perfectly capturing the idea of information backflow mediated by the environment. Here, the parameter $p$ plays the role of a knob for tuning the bath memory.
The overall state of the system at the $n$th step of the evolution is therefore given by
$$\sigma_{n}=\left(\mathcal{U}_{Sn}\!\circ\!\mathcal{S}_{n,n\!-\!1}\!\circ\ldots%
\circ\!\mathcal{U}_{S2}\!\circ\!\mathcal{S}_{2,1}\!\circ\!\mathcal{U}_{S1}%
\right)[\sigma_{0}]\;,$$
(2)
where “$\circ$” represents the super-operator composition and will be henceforth omitted and
$\sigma_{0}\!=\!\rho_{0}\left|{\bf 0}\right\rangle_{B}\!\left\langle{\bf 0}\right|$
is the system-bath initial state notanotaz with
$\rho_{0}$ being the input density matrix of $S$ and $\left|{\bf 0}\right\rangle_{B}\!=\!\left|0\right\rangle_{1}\!\left|0\right%
\rangle_{2}\cdots$ the initial ancillary state purification . Exploiting the properties of the swap operator and the translational symmetry of the environmental initial state, we find it useful to cast Eq. (2) in a recursive form, where $\sigma_{n}$ is expressed as a sum of terms involving
states $\{\sigma_{m<n}\}$. Specifically, for $n\!\geq\!2$ we straightforwardly obtain sm
$$\sigma_{n}\!=\!(1-p)\sum_{j=1}^{n\!-\!1}p^{j-1}{\cal U}_{Sn}^{j}\,[\sigma_{n\!%
-\!j}]\;\!+\!\;p^{n-1}{\cal U}^{n}_{Sn}[\sigma_{0}]\,,$$
(3)
where now ${\cal U}^{j}_{Sn}$ represents $j$ consecutive applications of the unitary gate ${\cal U}_{Sn}$, i.e., ${\cal U}_{Sn}^{j}[\sigma]\!=\!e^{-i\hat{H}_{Si}j\tau}\sigma e^{i\hat{H}_{Si}j\tau}$. Note that this corresponds to a coherent interaction process between $S$ and the $n$th ancilla only, which continued for a time $j\tau$.
This and the fact that in Eq. (3) each ${\cal U}_{Sn}^{j}$ is applied to $\sigma_{n-j}$ (with $n$ still in $\left|0\right\rangle_{n}$) entail the attractive property that
an expansion for $\rho_{n}\!=\!{\rm Tr}_{B}\sigma_{n}$ similar to Eq. (3) holds. Tracing this over $B$ indeed yields
$$\rho_{n}\!=\!(1-p)\sum_{j=1}^{n\!-\!1}p^{j-1}\mathcal{E}_{j}[\rho_{n\!-\!j}]\!%
+\!p^{n-1}\mathcal{E}_{n}[\rho_{0}]\,,$$
(4)
where a transformation $\mathcal{E}_{j}$ is a CPT map on $S$ only defined in terms of the unitary map ${\cal U}_{Sn}^{j}$ and the initial bath state as
$$\displaystyle\mathcal{E}_{j}[\rho]\!=\!\mbox{Tr}_{B}\left\{{\cal U}_{Sn}^{j}[%
\rho\otimes|\bf{0}\rangle_{B}\langle\bf{0}|]\right\}\;.$$
(5)
Interestingly, the structure of Eq. (4) shares features with the discrete model used by Shabani and Lidar lidar to derive their ME (there, in particular, $\mathcal{E}_{j}$ is the dynamical map in absence of measurements performed on the bath).
Two major differences occur, though. First, Eq. (4) cannot be written as a single sum due to the missing $(1\!-\!p)$ factor in the last term, which in fact means that here we deal with a time-inhomogeneous memory-kernel function (MKF). Second, map $\mathcal{E}_{j}$ is in general strongly NM: it describes the reduced dynamics of $S$ for a continuous coherent interaction between $S$ and a single ancilla (e.g. once can think of two coupled spins periodically exchanging an excitation). Indeed, as anticipated earlier, our model interpolates between two extreme regimes depending on the value of the probability $p$. When $p\!=\!0$, AA collisions are absent [cf. Eq. (1)]: Eq. (4) reduces to $\rho_{n}\!=\!\mathcal{E}_{1}[\rho_{n-1}]$ and we retrieve a standard Markovian CM rau ; alicki ; scarani ; buzek .
Quite differently, for $p\!=\!1$, Eq. (4) yields $\rho_{n}\!=\!\mathcal{E}_{n}[\rho_{0}]$, i.e., $S$ behaves as if it interacts with a single ancilla all the time. This can be seen by noting that for $p\!=\!1$ Eq. (1) reduces to a perfect swap: once $S$ has undergone a $\tau$-long interaction with $i$, the final state of $i$ is fully transferred to $i\!+\!1$ (with $i$ returning to $\left|0\right\rangle_{i}$).
Our next goal is to work out the ME corresponding to Eq. (4) in the continuous limit and then prove
that (i) the resulting equation is still capable to interpolate between the two opposite limits depicted above and (ii) it unconditionally satisfies the CPT condition.
For this aim, we first subtract from Eq. (4) the analogous identity for $n\!-\!1$. This gives rise to an equation for the variation of $\rho_{n}$ between two next steps $\Delta\rho_{n}\!=\!\rho_{n}\!-\!\rho_{n\!-\!1}$, which reads
$$\displaystyle\!\Delta\rho_{n}$$
$$\displaystyle\!=$$
$$\displaystyle(1\!-\!p)\!\sum_{j=1}^{n\!-\!2}p^{j-1}\mathcal{E}_{j}[\Delta\rho_%
{n\!-\!j}]\!+\!(1\!-\!p)p^{n\!-\!1}\mathcal{E}_{n\!-\!1}[\rho_{1}]$$
(6)
$$\displaystyle\!+\!\;\;\Delta\left(p^{n\!-\!1}\mathcal{E}_{n}\right)[\rho_{0}].$$
This can now be transformed into a differential equation for the continuous time evolution of the system density matrix $\rho(t)$
by taking the limit of infinite collisions [$n,j\rightarrow\infty$] while sending the
collision time to zero [i.e. $\tau\rightarrow 0$] in such a way that the elapsed times $t=n\tau$ and $t^{\prime}=j\tau$ remain finite.
Also, when $j$ becomes very large the probability $p^{j}$ of multiple AA collisions clearly must not vanish.
We thus set $p=\exp[-\Gamma\tau]$, where $\Gamma\!=\!-(\log p)/\tau$ is interpreted as the memory rate. We require that, when $\tau\!\rightarrow\!0$, $p$ approaches 1 in such a way that $\Gamma$ remains finite. This allows to express each power of $p$ as a decaying exponential $p^{j}\!=\!({p^{\frac{1}{\tau}}})^{j\tau}\!=\!e^{-\Gamma t^{\prime}}$. Note that in the continuous limit $\tau$ should be far shorter than any characteristic time, in particular $\Gamma^{-1}$. This gives $\Gamma\tau\!\ll\!1$ and thus $1\!-\!p\!=\!1\!-\!e^{-\Gamma\tau}\!\simeq\!\Gamma\tau$. Using this, the sum over $j$ in Eq. (6) becomes a time integral as $\tau\!\rightarrow\!0$. By identifying $\Delta\rho_{n}/\tau\!\rightarrow\!\dot{\rho}(t)=\!d\rho(t)/dt$, after a few straightforward steps sm we end up with the ME
$$\dot{\rho}(t)=\Gamma\!\int_{0}^{t}\!\!dt^{\prime}e^{-\Gamma t^{\prime}}\;%
\mathcal{E}({t^{\prime}})\left[\dot{\rho}(t-t^{\prime})\right]+e^{-\Gamma t}\;%
\dot{\mathcal{E}}(t)[\rho_{0}]\;,$$
(7)
where the CPT map $\mathcal{E}(t)$ is the continuous analogue of Eq. (5) and the dot stands for the total derivative. This is an integro-differential equation in $\rho(t)$ featuring a history integral term with an associated MKF $\Gamma e^{-\Gamma t^{\prime}}$ and, notably, a term $\sim\!\!\rho_{0}$. The latter, which stems from the formerly discussed inhomogeneity of the discrete MKF in Eq. (4), is a strong signature of NM behavior. Indeed, in the limit where memory effects persist indefinitely, i.e., $\Gamma\!\rightarrow\!0$, it is the only term surviving in Eq. (7) yielding $\dot{\rho}(t)\!\rightarrow\!\dot{\mathcal{E}}(t)[\rho_{0}]$, i.e., $\rho(t)\!\rightarrow\!\mathcal{E}(t)[\rho_{0}]$ in full analogy with the discrete model (we address the opposite limit $\Gamma\!\rightarrow\!\infty$ later on). Next, we derive the solution of Eq. (7) $\rho(t)\!=\!\Lambda(t)[\rho_{0}]$ and prove that the dynamical map petruccione $\Lambda(t)$ is always CPT [$\Lambda(0)\!=\!\mathcal{I}$ with $\mathcal{I}$ the identity superoperator]. Evidently, $\Lambda(t)$ obeys Eq. (7) under the formal replacement $\rho\!\rightarrow\!\Lambda$. By taking the Laplace transform (LT) of such equation, this is easily solved as sm
$$\displaystyle\tilde{{\Lambda}}(s)=\frac{\tilde{\mathcal{E}}(s+\Gamma)}{{%
\mathcal{I}}-\Gamma\;\tilde{\mathcal{E}}(s+\Gamma)}\;,$$
(8)
where $\tilde{\Lambda}(s)$ and $\tilde{\mathcal{E}}(s)$ are the LTs of $\Lambda(t)$ and $\mathcal{E}(t)$, respectively [Eq. (8) is well-defined since the numerator and denominator commute].
Expanding Eq. (8) in powers of $\Gamma$ gives $\tilde{{\Lambda}}(s)\!=\!\sum_{k=1}^{\infty}\left[\tilde{\mathcal{E}}(s\!+\!%
\Gamma)\right]^{k}\!\Gamma^{k\!-\!1}$, whose inverse LT is
$$\displaystyle\Lambda(t)\!=\!\mathcal{L}^{-1}[\tilde{\Lambda}(s)](t)\,\!=\!\sum%
_{k=1}^{\infty}\Gamma^{k\!-\!1}\;\mathcal{L}^{-1}\\;.$$
(9)
Basic properties of LT allow to immediately calculate the inverse LT within braces as sm
$$\displaystyle\!\mathcal{L}^{\!-\!1}\![\tilde{\mathcal{E}}^{k}\!(\!s\!+\!\Gamma%
\!)]$$
$$\displaystyle\!=$$
$$\displaystyle e^{-\Gamma t}\!\!\!\int_{0}^{t}\!\!\!dt_{1}\!\!\!\int_{0}^{t_{1}%
}\!\!\!\!\!\!dt_{2}\!\cdot\!\cdot\!\cdot\!\!\!\!\int_{0}^{t_{k\!-\!2}}\!\!\!\!%
\!\!\!dt_{k\!-\!1}$$
(10)
$$\displaystyle\times\;\mathcal{E}(t_{k\!-\!1}\!)\mathcal{E}(t_{k\!-\!2}\!\!-\!%
\!t_{k\!-\!1}\!)\cdot\!\cdot\!\cdot\!\mathcal{E}(t\!\!-\!\!t_{1}\!).$$
We have thus expressed $\Lambda(t)$ as a weighted series of multiple auto-convolutions of the CPT map $\mathcal{E}(t)$. The integrand in Eq. (10) is evidently a composition of CPT $\mathcal{E}$ maps, hence it is CPT itself. Therefore, we see that the dynamical map Eq. (9) is in fact a combination of CPT maps with positive weights [factors $\Gamma^{k-1}$ and $e^{-\Gamma t}$ in Eqs. (9) and (10) are all positive]. This proves the complete positivity of map $\Lambda(t)$. Moreover, the state obtained by applying the integrand in Eq. (10) (a CPT map as discussed) to $\rho_{0}$ has evidently unitary trace. As is easily checked sm , this entails ${\rm Tr}\left\{\Lambda(t)[\rho_{0}]\right\}\!=\!1$. We conclude that, since $\mathcal{E}(t)$ is CPT,
$\Lambda(t)$ is CPT. The last remaining task is to prove that, in line with Eq. (4) for $p\!=\!0$, the Markovian behavior arises from Eq. (7) for $\Gamma\!\rightarrow\!\infty$. Indeed, Eq. (7) is such that for $\Gamma$ large enough we can approximate ${\mathcal{E}}(t)\!\simeq\!{\mathcal{I}}\!+\!{\mathcal{F}}t$, where $\mathcal{F}\!=\!\dot{\mathcal{E}}(0)$. Under LT, this becomes $\tilde{\mathcal{E}}(s)\!=\!{1}/{(s\!+\!\Gamma)}\!+\!{{\mathcal{F}}}/{(s\!+\!%
\Gamma)^{2}}$, which once plugged into Eq. (8) and in the limit of $\Gamma\!\rightarrow\!\infty$ yields $\tilde{{\Lambda}}(s)\!=\!\left.{(s\!+\!\Gamma\!+\!{\mathcal{F}})}/{[s^{2}\!+\!%
\Gamma(s\!-\!{\mathcal{F}})]}\right|_{\Gamma\rightarrow\infty}\!=\!{1}/{(s\!-%
\!{\mathcal{F}})}$. By transforming back, we end up with $\Lambda(t)\!=\!e^{\mathcal{F}t}$ entailing that the semigroup property is fulfilled and thus, necessarily, $\mathcal{F}$ is a Lindbladian superoperator petruccione with Eq. (7) reducing to the Lindblad-form $\dot{\rho}(t)\!=\!\mathcal{F}[\rho]$.
It is worth noticing that a Markovian dynamics can also be generated from (7) for a finite $\Gamma$, by properly choosing the integral generator map ${\mathcal{E}}(t)$. Indeed, if one assume $\mathcal{E}(t)\!=\!e^{\mathcal{F}t}$, then $\Lambda(t)\!=\!e^{\mathcal{F}t}$ is the exact solution of Eq. (7) for any $\Gamma$:
in other words, the mapping $\mathcal{E}(t)\rightarrow\Lambda(t)$ induced by (8) is transparent for dynamical semigroups. This seems reasonable: no information can propagate through AA collisions if each SA collision is already forgetful.
This further marks the difference from the Shabani-Lidar ME lidar where the fact that $\Lambda(t)\!=\!e^{\mathcal{F}t}$ is not a solution in that context is exploited in a perturbative way to deliver the conclusions, and strengthens the importance of the last term in Eq. (7). Also note that our ME agrees with the general form predicted by Nakajima and Zwanzig petruccione provided that the corresponding $t\!-\!t^{\prime}$-dependent memory-kernel superoperator exhibits a discontinuity at $t\!=\!t^{\prime}$ [through part integration method, Eq. (7) can be easily expressed in a form where the time integral involves $\rho(t\!-\!t^{\prime})$ instead of $\dot{\rho}(t\!-\!t^{\prime})$].
To test the predictive power of our approach, we consider the dynamics of a two-level atom [whose ground (excited) state is denoted by $\left|0\right\rangle_{S}$ ($\left|1\right\rangle_{S}$)] coupled to a continuum of electromagnetic modes in the rotating-wave approximation petruccione . The case in which the field spectral density $J(\omega)$ is a Lorentzian centered on the atomic frequency can be solved exactly garraway , which makes it a useful benchmark to assess the effectiveness of a ME sabrina .
This solution can be expressed in terms of an amplitude damping channel (ADC) NC as $\rho(t)\!=\!\mathcal{A}_{G(t)}[\rho_{0}]$, where $\mathcal{A}_{\eta}[\rho_{0}]\!=\!1\!-\!p|\eta|^{2}\left|0\right\rangle_{S}\!%
\left\langle 0\right|\!+\!p|\eta|^{2}\left|1\right\rangle_{S}\!\left\langle 1%
\right|\!+\!\left\{r\,\eta\left|0\right\rangle_{S}\!\left\langle 1\right|\!+\!%
{\rm H.c.}\right\}$ is the general form of an ADC ($p$ and $r$ are the atom’s initial populations and coherences). Specifically petruccione , $G(t)\!=\!e^{-\lambda/2t}[\cosh(dt/2)+\lambda/d\sinh(dt/2)]$ with $d\!=\!\sqrt{\lambda^{2}\!-\!2\gamma_{0}\lambda}$. Here, $\lambda$ measures the width of $J(\omega$), while $\gamma_{0}$ is related to the strength of the coupling petruccione . For $\lambda\!\gg\!\gamma_{0}$, $J(\omega)$ becomes about flat and $G(t)\!\rightarrow\!e^{-\gamma_{0}/2t}$: the atom undergoes standard spontaneous emission at a rate $\gamma_{0}$ and $\dot{\rho}\!\rightarrow\!\mathcal{L}[\rho]$, namely the Markovian regime occurs ($\mathcal{L}$ is the usual zero-temperature atomic Lindbladian petruccione with associated rate $\gamma_{0}$). For $\lambda\!<\!\gamma_{0}/2$, instead, damped oscillations take place as a signature of non-Markovianity. In particular, in the regime $\lambda\!\ll\!\gamma_{0}$, $G(t)\!\simeq\!e^{-\lambda t}\!\cos(\Omega t)$ with $\Omega\!=\!\sqrt{\gamma_{0}\lambda/2}$ showing that the atom undergoes damped Rabi oscillations at a rate $\Omega$ due to its coupling to the cavity protected mode. For vanishing $\lambda$ (ideal cavity with infinite quality factor) we would thus obtain $\mathcal{A}_{G(t)}[\rho_{0}]\simeq\mathcal{A}_{\cos(\Omega t)}[\rho_{0}]$. This strongly suggests to regard the cavity protected mode as a generic ancilla in our CM framework and thus set $\mathcal{E}(t)\!\equiv\!\mathcal{A}_{\cos(\Omega t)}$ and, additionally, $\Gamma\!\equiv\!\lambda$. Indeed, we have shown earlier that if $\Gamma$, namely $\lambda$, vanishes, the system behaves as if interacting all the time with a single ancilla, namely the protected mode. On the other hand, we have seen that when $\Gamma$ is very large (Markovian limit) at each collision the system interacts with a fresh ancilla still in the initial state. Note that even this case can be viewed as an effective single-ancilla process if one supposes such ancilla to be reset to its initial state between two next collisions with $S$. Correspondingly, in the atom-field model, for very large $\lambda$ the cavity quality factor is very low: the leakage of the protected mode is so effective that at any time – not only at the beginning – the atom in fact “sees” such mode in its vacuum state. With the above settings [$\mathcal{E}(t)\!\equiv\!\mathcal{A}_{\cos(\Omega t)}$ and $\Gamma\!\equiv\!\lambda$] the dynamical map $\Lambda(t)$ can be calculated exactly through Eqs. (8)–(10). In Fig. 2, we display the time behavior of the atomic excitation, i.e., the excited-state population, and coherences (normalized to the respective initial values) as given by the exact solution and our CM. For comparison, we also report the corresponding functions predicted by the memory-kernel MEs $\dot{\rho}\!=\!\mathcal{L}\int\!dt^{\prime}k(t^{\prime})\rho(t\!-\!t^{\prime})$ barnett and $\dot{\rho}\!=\!\mathcal{L}\int\!dt^{\prime}k(t^{\prime})e^{\mathcal{L}t^{%
\prime}}\rho(t\!-\!t^{\prime})$ lidar for $k(t^{\prime})\!=\!\lambda e^{-\lambda t^{\prime}}$, to which we will refer as phenomenological and Shabani-Lidar MEs, respectively (a similar comparison was carried out in Ref. sabrina ).
For large $\lambda$ (compared to $\gamma_{0}$) the Markovian regime occurs: all the models basically yield the same purely exponential behavior [see Figs. 2(a) and (d)]. As $\lambda$ becomes low, significant deviations arise. The Shabani-Lidar model keeps predicting exponential decays [cf. Figs. 2(b),(c),(e) and (f)] in contrast to the damped oscillations predicted by the exact solution. The phenomenological ME predicts coherences matching the exact solution [cf. Figs. 2(e) and (f)], however positivity is drastically violated barnett ; sabrina [see Figs. 2(b) and (c)]. Our ME Eq. (7) yields a substantial improvement on both the above models. As the phenomenological ME, it accurately reproduces the exact coherences [see Figs. 2(e) and (f)]. Quite differently, though, it does not break positivity [see Figs. 2(b) and (c)] in line with our general proof. The latter feature is shared with the Shabani-Lidar ME. Yet, unlike this, the CM captures the physics of the process far better: damped oscillations for populations rather close to the exact ones are predicted (the discrepancy decreases as $\lambda/\gamma_{0}\!\rightarrow\!0$). Note that, while in Figs. 2(b) and (c) the minima of the exact solution are zero, the corresponding CM minima are small but strictly positive. This is most likely to stem from the incoherent mixture of the identity and swap operator entering Eq. (1). If this is replaced by a coherent sum (in which case every AA collision becomes unitary) a regime featuring damped oscillations with zero minima indeed occurs inprep .
To summarize, we have introduced a NM microscopic CM, where the bath memory is added dynamically through simple inclusion of inter-ancillary collisions each modeled as a CPT swapping operation. The model interpolates between two extreme situations: a fully Markovian regime (retrieving a standard CM) and a strongly NM one (corresponding to the continuous interaction of the system with a single ancilla). The continuous limit gives rise to a ME which was proven to be unconditionally CPT. To test the effectiveness of our approach, we have applied it to an atom coupled to a bath of modes featuring a Lorentzian spectral density and compared the outcomes with the analytical solution and two popular memory-kernel MEs. While all the advantageous features of such MEs simultaneously occur in ours, this in addition succeeds to capture distinctive traits of the NM dynamics.
We thank R. Fazio for comments and discussions and acknowledge support from the MIUR through the FIRB-IDEAS project RBID08B3FM.
Supplementary Material
In this Supplementary Material, we supply some technical details related to the derivation of some properties discussed the paper’s main text.
I Derivation of Eq. (3)
Map $\mathcal{S}_{i+1,i}$ is given by Eq. (1). By definition, $\hat{S}_{i+1,i}$ swaps the states of $i$ and $(i\!+\!1)$, hence it transforms each $\hat{U}_{Si}$ as $\hat{S}_{i+1,i}\hat{U}_{Si}\,\hat{S}_{i+1,i}\!=\!\hat{U}_{S,i+1}$ (we recall that $\hat{S}_{ij}^{\dagger}\!\equiv\!\hat{S}_{ij}$). Equivalently, $\hat{S}_{i+1,i}\hat{U}_{Si}\!=\!\hat{U}_{S,i+1}\,\hat{S}_{i+1,i}$: a swap operator on the left of a $\hat{U}$-type one can jump to the right side provided that the ancillary index of $\hat{U}$ is increased by one unity. By construction,
$\sigma_{2}\!=\!\mathcal{U}_{S2}\mathcal{S}_{21}[\sigma_{1}$], i.e.,
$\sigma_{2}\!=\!(1\!-\!p)\hat{U}_{S2}\,\sigma_{1}\hat{U}_{S2}^{\dagger}\!+\!p%
\hat{U}_{S2}\hat{S}_{21}(\hat{U}_{S1}\,\sigma_{0}\hat{U}_{S1}^{\dagger})\hat{S%
}_{21}\hat{U}_{S2}^{\dagger}$, where we have replaced (only in the latter term) $\sigma_{1}\!=\!\hat{U}_{S1}\,\sigma_{0}\hat{U}_{S1}^{\dagger}$ [see Fig. 1(b)]. Operator $\hat{U}_{S1}$ can be eliminated as follows. We use $\hat{S}_{21}\hat{U}_{S1}\!=\!\hat{U}_{S,2}\,\hat{S}_{21}$ alongside $\hat{S}_{21}\sigma_{0}\!=\!\rho_{0}\hat{S}_{21}\!\left|{\bf 0}\right\rangle_{B%
}\!\left\langle{\bf 0}\right|\!\equiv\!\sigma_{0}$. This yields
$\sigma_{2}\!=\!(1-p)\hat{U}_{S2}\,\sigma_{1}\hat{U}_{S2}^{\dagger}\!+\!p\hat{U%
}_{S2}^{2}\,\sigma_{0}\,\!\left(\hat{U}_{S2}^{\dagger}\right)^{2}$, which indeed corresponds to Eq. (3) in the main text for $n\!=\!2$.
The arbitrary-$n$ case can be proven by induction as follows. By construction, $\sigma_{n+1}\!=\!\mathcal{U}_{S,n+1}\mathcal{S}_{n+1,n}[\sigma_{n}]\!=\!(1\!-%
\!p)\hat{U}_{S,n+1}\,\sigma_{n}\hat{U}_{S,n+1}^{\dagger}\!+\!p\hat{U}_{S,n+1}%
\hat{S}_{n+1,n}\sigma_{n}\hat{S}_{n+1,n}\,\hat{U}_{S,n+1}^{\dagger}$. By replacing Eq. (3) in the second term and using $\hat{S}_{n+1,n}\hat{U}_{sn}\!=\!\hat{U}_{s,n+1}\hat{S}_{n+1,n}$ along with $\hat{S}_{n+1,n}\sigma_{n-j}\!=\!\sigma_{n-j}$ for $j\!\geq\!1$, we end up with Eq. (3) for $n\!\rightarrow\!n\!+\!1$.
II Derivation of Eq. (7)
When Eq. (6) in MT is divided by $\tau$ and by using the limiting expressions discussed in the main text, the terms on the right-hand side in the continuous limit take the form
$$\displaystyle\frac{c^{2}\sum_{j=1}^{n\!-\!2}s^{2(j-1)}\mathcal{E}_{j}\left[{%
\rho_{n\!-\!j}\!-\!\rho_{n\!-\!1\!-\!j}}\right]}{\tau}\!\simeq\!\Gamma\!\int_{%
0}^{t}\!dt^{\prime}e^{-\Gamma t^{\prime}}\mathcal{E}(t^{\prime})\left[\frac{d%
\rho(t\!-\!t^{\prime})}{d(t\!-\!t^{\prime})}\right]$$
$$\displaystyle\frac{\Delta(s^{2(n\!-\!1)}\mathcal{E}_{n})}{\tau}\,[\rho_{0}]\!=%
\!\frac{(s^{2(n\!-\!1)}\mathcal{E}_{n}\!-\!s^{2(n\!-\!2)}\mathcal{E}_{n\!-\!1}%
)}{\tau}\,[\rho_{0}]$$
$$\displaystyle\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,%
\,\,\,\,\!\!\!\!\!\!\!\!\!\!\!\!\!\!\simeq\!\frac{e^{-\Gamma(t+2\tau)}\mathcal%
{E}_{t+\tau}\!-\!e^{-\Gamma(t+\tau)}\mathcal{E}(t)}{\tau}\,[\rho_{0}]\!=\!%
\frac{d}{dt}\!\left(e^{-\Gamma t}\mathcal{E}(t)\right)[\rho_{0}]$$
$$\displaystyle\frac{c^{2}s^{2(n\!-\!1)}\mathcal{E}_{n\!-\!1}}{\tau}\,[\rho_{1}]%
\!\simeq\!\Gamma e^{-\Gamma t}\mathcal{E}(t)\,[\rho_{0}]$$
whereas the left-hand side of Eq. (6) in MT clearly reduces to the time derivative of $\rho(t)$. Using these, Eq. (7) is immediately obtained.
III Derivation of $\tilde{\Lambda}(s)$
By replacing $\rho(t)=\Lambda(t)[\rho_{0}]$ in MT’s Eq. (7) and using that $\rho_{0}$ is arbitrary, the equation obeyed by $\Lambda(t)$ tuns out to be
$$\displaystyle\dot{\Lambda}(t)=\Gamma\!\int_{0}^{t}dt^{\prime}\exp[-\Gamma t^{%
\prime}]\;{\mathcal{E}}(t^{\prime})\dot{\Lambda}(t-t^{\prime})+\exp[-\Gamma t]%
\;\dot{{\mathcal{E}}}(t)$$
(11)
in addition to the requirement ${\Lambda}(0)\!=\!{\mathcal{I}}$.
Upon Laplace transform (LT), the equation becomes
$$\displaystyle s\tilde{{\Lambda}}(s)\!-\!{\mathcal{I}}\!=\!\Gamma\tilde{%
\mathcal{E}}(s+\Gamma)[s\tilde{{\Lambda}}(s)\!-\!{\mathcal{I}}]\!+\!(s+\Gamma)%
\;\tilde{\mathcal{E}}(s+\Gamma)-{\mathcal{I}}$$
(12)
where for $s$ complex the LT is defined as
$$\tilde{F}(s)\!=\!\mathcal{L}\,[F(t)](s)\!=\!\int_{0}^{\infty}\!\!dt\;e^{-st}F(%
t)\,$$
(13)
and we have used $\mathcal{E}(0)\!=\!\mathcal{I}$ (see main text).
By rearranging terms in Eq. (12)
$$\displaystyle[{\mathcal{I}}-\Gamma\;\tilde{\mathcal{E}}(s+\Gamma)][s\tilde{{%
\Lambda}}(s)-{\mathcal{I}}]=(s+\Gamma)\;\tilde{\mathcal{E}}(s+\Gamma)-{%
\mathcal{I}}\;,$$
(14)
and hence
$$\displaystyle s\;[{\mathcal{I}}-\Gamma\;\tilde{\mathcal{E}}(s+\Gamma)]\ \tilde%
{{\Lambda}}(s)=s\;\tilde{\mathcal{E}}(s+\Gamma)\;.$$
(15)
By simplifying $s$ on both terms and introducing the inverse of ${\mathcal{I}}-\Gamma\;\tilde{\mathcal{E}}(s+\Gamma)$ we end up with Eq. (8).
IV Expansion of $\Lambda(t)$
The inverse LT of $\tilde{\mathcal{E}}(s+\Gamma)$ is $\mathcal{L}^{-1}[\tilde{\mathcal{E}}(s+\Gamma)]\!=\!e^{-\Gamma t}\mathcal{E}(t)$. Thereby, from a basic property of LT, the inverse transform of $\tilde{\mathcal{E}}^{2}(s\!+\!\Gamma)$ is the auto-convolution of $e^{-\Gamma t}\mathcal{E}(t)$, which reads
$$\displaystyle\mathcal{L}^{-1}[\tilde{\mathcal{E}}^{2}(s+\Gamma)]$$
$$\displaystyle\!=$$
$$\displaystyle\!\int_{0}^{t}\!dt^{\prime}\left[e^{-\Gamma t^{\prime}}\mathcal{E%
}(t^{\prime})\right]\left[e^{-\Gamma(t\!-\!t^{\prime})}\mathcal{E}(t\!-\!t^{%
\prime})\right]$$
(16)
$$\displaystyle\!=$$
$$\displaystyle e^{-\Gamma t}\!\!\int_{0}^{t}\!dt^{\prime}\mathcal{E}(t^{\prime}%
)\mathcal{E}(t\!-\!t^{\prime})\,.$$
The inverse LT of $\tilde{\mathcal{E}}^{3}(s\!+\!\Gamma)$ can be calculated as the convolution between Eq. (16) and $\mathcal{L}^{-1}[\tilde{\mathcal{E}}(s+\Gamma)]\!=\!\mathcal{E}(t)$, which yields
$$\mathcal{L}^{-1}[\tilde{\mathcal{E}}^{3}(s\!+\!\Gamma)]\!=\!e^{-\Gamma t}\!%
\int_{0}^{t}\!dt_{1}\!\!\int_{0}^{t_{1}}\!\!dt_{2}\,\mathcal{E}(t_{2}\!)%
\mathcal{E}(t_{1}\!\!-\!\!t_{2})\mathcal{E}(t\!-\!t_{1})\,\,.$$
(17)
Eq. (10) in MT, i.e., the case corresponding to $\tilde{\mathcal{E}}^{k}(s\!+\!\Gamma)$ for arbitrary $k$, then follows by mere induction.
V Trace preservation of $\Lambda(t)$
As discussed in the main text, the integrand in MT’s Eq. (10) is a CPT map and thus, once applied to $\rho_{0}$, it yields a state having unitary trace (clearly, ${\rm Tr}\rho_{0}\!=\!1$). Hence,
$$\displaystyle{\rm Tr}\left\{\Lambda(t)[\rho_{0}]\right\}$$
$$\displaystyle\!=$$
$$\displaystyle\sum_{k=1}^{\infty}\Gamma^{k\!-\!1}\,{\rm Tr}\left\{e^{-\Gamma t}%
\int_{0}^{t}\!\!\!dt_{1}\!\!\!\int_{0}^{t_{1}}\!\!\!\!dt_{2}\!\cdot\!\cdot\!%
\cdot\!\!\int_{0}^{t_{k\!-\!2}}\!\!dt_{k\!-\!1}\right\}$$
(18)
$$\displaystyle\!=$$
$$\displaystyle e^{-\Gamma t}\sum_{k=1}^{\infty}\Gamma^{k\!-\!1}\frac{t^{k\!-\!1%
}}{(k\!-\!1)!}\!=\!e^{-\Gamma t}e^{\Gamma t}\!=\!1\,\,.$$
References
(1)
H. P. Breuer and F. Petruccione, The Theory of Open
Quantum Systems (Oxford, Oxford University Press, 2002).
(2)
U. Weiss, Quantum Dissipative Systems, 3rd ed. (World Scientific, Singapore, 2008).
(3)
A. Rivas and S.F. Huelga, Open Quantum Systems. An Introduction (Springer, Heidelberg, 2011).
(4)
H.-P. Breuer, J. Phys. B: At. Mol. Opt. Phys. 45, 154001 (2012).
(5)
See e.g.: J. Schliemann, A. Khaetskii, and D. Loss, J. Phys.: Condens.
Matter 15, R1809 (2003); M. Michel, G. Mahler, and J. Gemmer, Phys. Rev. Lett. 95,
180602 (2005).
(6)
Due to space constraints, our introduction is not intended to provide a comprehensive overview of the many different approaches proposed in the literature so far. We therefore focus on those more closely related to the present work.
(7)
B. Vacchini and H. P. Breuer, Phys. Rev. A 81, 042103 (2010).
(8)
S. M. Barnett and S. Stenholm, Phys. Rev. A 64, 033808 (2001).
(9)
A. Shabani and D. A. Lidar, Phys. Rev. A 71, 020101(2005).
(10)
A. A. Budini, Phys. Rev. A 69, 042107 (2004).
(11)
S. Daffer et al., Phys. Rev. A 70, 010304(R) (2004).
(12)
S. Maniscalco and F. Petruccione, Phys. Rev. A 73, 12111 (2006);
S. Maniscalco, Phys. Rev. A 75, 062103 (2007).
(13)
H. P. Breuer and B. Vacchini, Phys. Rev. E 79, 041147 (2009).
and
(14)
J. Wilkie and Y. M. Wong, J. Phys. A 42, 015006 (2009).
(15)
S. Campbell et al., Phys. Rev. A 85, 032120 (2012).
(16)
D. Chruściński and A. Kossakowski, Phys. Rev. Lett. 97, 20005 (2012); D. Chruściński and A. Kossakowski, Europhys. Lett. 97, 20005 (2012).
(17)
L. Mazzola, E.-M. Laine, H.-P. Breuer, S. Maniscalco, and J. Piilo, Phys. Rev. A 81, 062120 (2010).
(18)
H.-P. Breuer, E.-M. Laine, and J. Piilo, Phys. Rev. Lett. 103, 210401 (2009).
(19)
J. Rau, Phys. Rev. 129, 1880 (1963).
(20)
R. Alicki and K. Lendi, Quantum Dynamical
Semigroups and Applications, Lecture Notes in Physics
(Springer-Verlag, Berlin, 1987).
(21)
M. Ziman et al., Phys. Rev. A65 , 042105 (2002); V. Scarani et al., Phys. Rev. Lett. 88 , 97905 (2002)
(22)
M. Ziman and V. Buzek, Phys. Rev. A 72, 022110 (2005); M. Ziman, P. Stelmachovic, and V. Buzek, Open systems and information dynamics 12, 81 (2005).
(23)
S. Attal and Y. Pautrat, Ann. Inst. Henri Poincaré 7, 59 (2006).
(24)
C. Pellegrini and F. Petruccione, J. Phys. A: Math. Theor. 42, 425304 (2009).
(25)
V. Giovannetti and M. Palma, Phys. Rev. Lett. 108, 040401 (2012); J. Phys. B: At. Mol. Opt. Phys. 45, 154006 (2012).
(26)
T. Rybar, S. N. Filippov, M. Ziman, and V. Buzek, J. Phys. B: At. Mol. Opt. Phys. 45, 154006 (2012).
(27)
M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, U. K., 2000).
(28)
Our notation is such that states of $S$ are denoted by $\rho$, while $\sigma$ is used for states of the overall system ($S$ and the bath).
(29)
We are thus assuming that each ancilla is initially in the pure state $\left|0\right\rangle$. This is not a restrictive constraint: arbitrary initial mixed states can be account for through the well-known purification picture NC .
(30)
For technical details see supplementary material at […].
(31)
B. M. Garraway, Phys. Rev. A 55, 2290 (1997).
(32)
F. Ciccarello, G. M. Palma and V. Giovannetti, in preparation. |
Possibility of $\rm tan\beta$ measurement in CMS at LHC
M. HASHEMI
Possibility of $\rm tan\beta$ measurement in CMS at LHC
Institute for Studies in Theoretical Physics and Mathematics(IPM),
Shahid Lavasani st., P.O.Box 19395-5531, Tehran, Iran
The achievable $\rm tan\beta$ determination accuracy for CMS at LHC is presented.
Using MSSM $\rm H/A\rightarrow\tau\tau$ decay in the associated production process
$\rm gg\rightarrow b\bar{b}H/A$, the event rates are measured at large $\rm tan\beta$ and the systematic and statistical errors are estimated.
Due to sensitivity of the above event rates to $\rm tan\beta$, it is shown that it is possible to determine constraints on $\rm tan\beta$ for a given set of SUSY parameters and uncetainties.
1 Introduction
Supersymmetry(SUSY)- a symmetry under interchange of bosonic and fermionic degrees of freedom- provides an elegant solution to the hierarchy problem, and thus has been considered as one of the most promising new physics scenarios among various possibilities. The Minimal Supersymmetric extention to Standard Model(MSSM) requires the introduction of two Higgs doublets in order to preserve supersymmetry. There are five physical Higgs particles, two CP-even (h,H), one CP-odd (A) and two charged ones ($\rm H^{\pm}$). All couplings and masses of the MSSM Higgs sector are determined at lowest order by two independent parameters, which are generally chosen as $\rm tan\beta=\upsilon_{2}/\upsilon_{1}$, the ratio of the vacuum expectation values of the two Higgs doublets, and the pseudoscalar Higgs boson mass $\rm m_{A}$.
One of the most important parameters to be determined in MSSM is $\rm tan\beta$ since it enters in all sectors of the theory.
In this report possibility of determining $\rm tan\beta$ value is presented taking into account different systematic and statistic uncertainties of measurements $\!{}^{{\bf?}}$. The results are presented for $\rm m_{h}^{max}$ benchmark scenario $\!{}^{{\bf?}}$ with the following set of MSSM parameters:
SU(2) gaugino mass $\rm M_{2}=200GeV/c^{2}$, $\rm\mu=300GeV/c^{2}$, gluino mass $\rm M_{\tilde{g}}=800GeV/c^{2}$, SUSY breaking mass parameter $\rm M_{SUSY}=1TeV/c^{2}$ and stop mixing parameter $\rm X_{t}=\sqrt{6}(X_{t}=A_{t}-\mu cot\beta)$. The top quark mass is set to $\rm 175GeV/c^{2}$. The Higgs boson decay to SUSY particles are allowed.
2 Summery of $\rm H/A\rightarrow\tau\tau$ analysis results and expected $5\sigma$ discovery reach
2.1 NLO cross section
The NLO cross section calculation results show that at large $\rm tan\beta$ the cross section for $\rm gg\rightarrow b\bar{b}H/A/h$ exceeds that of $\rm gg\rightarrow H/A/h$ and Higgs bosons are produced predominantly in association with two b quarks $\!{}^{{\bf?}}$.
Thus only $\rm gg\rightarrow b\bar{b}H/A/h$ has been considered as the production process.
Figure 1 shows NLO cross section times branching ratio versus $\rm tan\beta$.
Dominant parts of the cross section are proportional to $\rm tan^{2}\beta$ at leading order. At NLO there are linear terms which can be combined together with leading order terms in a term as $\rm{tan^{2}\beta}_{eff}$ which is denoted as $\rm tan\beta$ hereafter.
As a conclusion the uncertainity on the $\rm tan\beta$ measurement is half of uncertainity of rate measurement.
2.2 Signal and background identification and simulation tools
Events are generated by PYTHIA6 $\!{}^{{\bf?}}$, TopReX $\!{}^{{\bf?}}$, TAUOLA $\!{}^{{\bf?}}$, signal cross section is calculated using PPHTT $\!{}^{{\bf?}}$ and Higgs branching ratios with HDECAY $\!{}^{{\bf?}}$.
Four final states have been studied with their different braching fractions as listed in Table 1.
Main common background events for all channels are :
$\rm Z,\gamma^{*}\rightarrow\tau\tau$ Drell-Yan process, $\rm t\bar{t}$ production with real and fake $\rm\tau$’s and single top production Wt. Channels with leptonic final states suffer from $\rm b\bar{b}$ events. W+jet process is the background for final states with hadronic $\rm\tau$ decays. When both $\rm\tau$’s decay hadronically there is an aditional QCD background with possibility of having fake $\rm\tau$’s.
2.3 $\rm 5\sigma$ discovery contours
The $\rm 5\sigma$ discovery contours for $\rm H/A/h\rightarrow\tau\tau$ with different final states of $\rm e\mu,\ell\ell$ and $\rm\ell j$ are shown in Figure 2 for integrated luminosity of $\rm 30fb^{-1}$. For two jet final state the $\rm 5\sigma$ contour for $\rm 60fb^{-1}$ is shown. Also shown is the $\rm 5\sigma$ discovery contour for $\rm H/A/h\rightarrow\mu\mu$ for $\rm 60fb^{-1}$ .
As is seen in the figure this channel is one of the most promising channels for heavy neutral MSSM Higgs boson discovery.
3 Statistical and systematic uncertainties of the production cross section measurement
Systematic uncertainties of the measured production cross section come from the luminosity uncertainty, experimental selection uncertainty and background uncertainty.
The total uncertainty is the quadratic sum of statistical and systematic errors:
$$\rm\frac{\Delta\sigma_{prod.}}{\sigma_{prod.}}=\frac{\sqrt{N_{S}+N_{B}}}{N_{S}%
}\oplus\frac{\Delta L}{L}\oplus\frac{\Delta\varepsilon_{sel.}}{\varepsilon_{%
sel.}}\oplus\frac{\Delta N_{B}}{N_{S}}$$
(1)
where the first term is the statistical uncertainty and other terms are systematic errors mentioned above.
3.1 Statistical and luminosity uncertainty
Statistical uncertainty of the signal events is calculated by weighted summing over all final states.
Depending on $\rm tan\beta$ and final state the statistical errors are 5-25$\%$.
The uncertainty of the luminosity measurement is assumed to be $\rm 5\%$
3.2 Signal selection uncertainty
The uncertainty of the signal selection efficiency comes from the calorimeter energy scale(since jets and missing $\rm E_{t}$ thresholds are required), b-tagging efficiency and $\rm\tau$-tagging efficiency.
The total selection efficiency uncertainty has been obtained to be $\rm\sim 4.3\%$.
3.3 Background uncertainty
The background contribution to the signal selection efficiency is estimated by fitting the invariant mass of two $\rm\tau$’s, $\rm m_{\tau\tau}$, and varying the number of signal and background events.
The error of the fit gives the background uncertainty.
The background uncertainty is estimated to be $\rm\Delta N_{B}/N_{S}=10\%$. So the total systematic uncertainty of production cross section is $\rm\sim 12\%$ which is comparable with statistical uncertainty.
4 Estimating $\rm tan\beta$ uncertainty
As was mentioned earlier the signal production cross section is proportional to the square of $\rm tan\beta$ :
$$\rm\sigma_{prod.}=tan^{2}\beta\times X$$
(2)
so the error on $\rm tan\beta$ is :
$$\rm\frac{\Delta tan\beta}{tan\beta}=\frac{1}{2}(\frac{\Delta\sigma_{prod.}}{%
\sigma_{prod.}})\oplus\frac{1}{2}(\frac{\Delta X}{X})$$
(3)
where $\rm\Delta X$ consists of theoretical uncertainty of the production cross section and branching ratios and cross section uncertainty due to the uncertainty of the Higgs boson mass measurement.
4.1 Theoretical uncertainty of the signal selection
According to NLO cross section calculations $\!{}^{{\bf?}}$, the NLO cross section uncertainty for the signal is assumed to be $\rm 20-30\%$ for the total rate.
The branching ratio uncertainty is $\rm\sim 3\%$ which is due to SM parameters uncertainties.
4.2 Uncertainty of the signal Higgs mass measurement
The signal production cross section depends on the Higgs mass which is measured with some accuracy.
This induces some error on the cross section.
At $\rm 5\sigma$ limit where the signal statistics is lowest, the mass measurement uncertainty brings $\rm 5-6\%$ uncertainty on $\rm tan\beta$ measurement.
4.3 SUSY parameters uncertainy effects
SUSY parameters uncertainties are still unknown but to give an estimation of the rate sensitivity to SUSY parameters, those were varied by $\rm 20\%$ around the nominal values.
The variation of the rate within the discovery region is about $\rm 11\%$ which leads to at most $\rm 6\%$ uncertainty on $\rm tan\beta$ measurement.
5 $\rm tan\beta$ measurement uncertainty in different final states of $\rm H/A\rightarrow\tau\tau$
Figures 4,4 show the total uncertainty as well as statistical uncertainties of $\rm tan\beta$ measurement for $\rm 30fb^{-1}$ and $\rm 60fb^{-1}$ respectively.
6 Conclusion
The possibility of $\rm tan\beta$ measurement in CMS at LHC was presented by estimating the precision of the cross section times branching ratio measurement in $\rm H/A\rightarrow\tau\tau$ decay channel with different final states for $\rm 30fb^{-1}$.
The statistical uncertainty of production cross section is estimated to be $\rm 4-25\%$ while the systematic error is $\rm\leq 12\%$ which both depend on the signal significance.
Due to existence of large radiative corrections to the bottom Yukawa coupling, results presented correspond to $\rm tan\beta_{eff}$ which absorbs the leading part of these corrections.
Close to the $\rm 5\sigma$-discovery limit the statistical uncertainty is in the same order as the theoretical uncertainties but for $\rm tan\beta$ regions where the signal significance is more than $\rm 5\sigma$ significantly the theoretical errors dominate in the estimated total uncertainty of $\rm tan\beta$ measurement.
References
References
[1]
R. Kinnunen, S. Lehti, F. Moortgat, A. Nikitenko, M. Spira, CMS NOTE 2004/027
[2]
ALEPH, DELPHI, L3 and OPAL collaborations, LHWG Note 2004/01
[3]
S. Dittmaier, M. Spira and M. Krämer, Higgs
Radiation off Bottom Quarks at the Tevatron and the LHC, hep-ph/0309204.
[4]
T. Sjostrand et al., Comput. Phys. Commun. 135 (2001) 238.
[5]
S. R. Slabospitsky, L. Sonnenschein, Comput. Phys. Commun. 148 (2002) 87 and hep-ph/0201292
[6]
S. Jadach et. al., Comp. Phys. Commun. 76 (1993) 361 and CERN TH-6793
[7]
http://people.web.psi.ch/spira/hqq/pphtt.f
[8]
A. Djouadi, J. Kalinowski and M. Spira, HDECAY: a
Program for Higgs Boson Decays in the Standard Model and its
Supersymmetric Extension, Comput. Phys. Commun. 108 (1998) 56-74,
hep-ph/9704448. |
Representation varieties of algebras with nodes
Ryan Kinser
University of Iowa, Department of Mathematics, Iowa City, USA
[
and
András C. Lőrincz
Purdue University, Department of Mathematics, West Lafayette, IN, USA
[
Abstract.
We study the behavior of representation varieties of quivers with relations under the operation of node splitting.
Working in the “relative setting” (splitting one node at a time) allows us to
combinatorially enumerate irreducible components of representation varieties, and show they have rational singularities, for a wide class of algebras. This class contains all radical square zero algebras but also many others, as illustrated by examples throughout the paper.
We also give applications to generic decomposition within irreducible components and decomposition of moduli spaces of semistable representations of certain algebras.
Key words and phrases:Representations of algebras, quivers, moduli spaces
2010 Mathematics Subject Classification: 16G20, 14D20
\newsymbol\pp
1275
Ryan Kinser]ryan-kinser@uiowa.edu
András C. Lőrincz]alorincz@purdue.edu
Contents
1 Introduction
2 Background
3 Node splitting and bundles
4 Applications
5 Moduli spaces
1. Introduction
1.1. Context and motivation
Throughout, $\Bbbk$ is an algebraically closed field. We specialize to characteristic 0 only when necessary, for results on singularities and moduli spaces.
The algebras we study are those of the form $A=\Bbbk Q/I$ where $Q$ is a quiver and $I$ a two-sided ideal. We do not assume the ideal is admissible except when necessary, thus most of our results apply to infinite dimensional quotients of path algebras as well.
Each dimension vector $\mathbf{d}$ for $A$ determines a representation variety $\operatorname{rep}_{A}(\mathbf{d})$ with action of a product of general linear groups $GL(\mathbf{d})$ (see Section 2.3).
Orbit closures in $\operatorname{rep}_{A}(\mathbf{d})$ have remarkable connections with the representation theory of $A$ and related objects;
see surveys such as [Bon98, Zwa11, HZ14] for detailed treatments.
Interest in these varieties is not confined to representation theory of algebras, however: they also naturally arise in Lie theory, commutative algebra, and algebraic geometry. The interested reader may consult the introduction to [Kin18] for more detail and references.
Even restricting our attention to representation theory of algebras, geometric methods centered around the varieties $\operatorname{rep}_{A}(\mathbf{d})$, such as the construction of moduli spaces (see Section 5), provide a toolkit for classification of representations which is complementary to homological and functorial approaches [Kin94, Rei08].
The purpose of this paper is to systematically study connections between representation varieties for algebras related by splitting nodes: while the defining equations of these varieties can be quite different, our main results establish close connections between their irreducible components, singularities of these components, and generic decompositions (the generalization of Kac’s canonical decomposition to arbitrary quivers with relations). We also give an application of our result on singularities to the structure of certain moduli spaces of semistable representations.
We now informally summarize the idea of node splitting; see Section 2.2 for details.
A node of an algebra $A=\Bbbk Q/I$ is a vertex $x$ of $Q$ such that all the paths of length $2$ passing strictly through $x$ belong to $I$.
A node $x$ of $A$ can be split by the following local operation around $x$:
(1.1)
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resulting in a new algebra $A^{x}$ with one fewer node (disregarding sources and sinks).
The representation theory of $A$ and $A^{x}$ is essentially the same, but the geometry of their representation varieties can be drastically different. Consider for example $A$ defined by $Q=\bullet\xrightarrow{\alpha}x\xrightarrow{\beta}\bullet$ and $I=\langle\beta\alpha\rangle$ so that $x$ is a node. Representation varieties for $A$ are generally singular and can have arbitrarily large numbers of irreducible components as $\mathbf{d}$ varies,
but representation varieties for $A^{x}$ are all just affine spaces since it is hereditary (defined by a quiver with no relations). Nonetheless, it turns out that there is a close relationship between the geometry of representation varieties for $A$ and $A^{x}$, which we develop in the present work.
A special case of particular interest is when every vertex of $A$ is a node, which is precisely when $\operatorname{rad}^{2}A=0$. Such algebras are historically significant because they are the first step beyond semisimple algebras. Interesting remarks on the importance of radical square zero algebras and their associated graphs in the development of the modern representation theory of algebras can be found in the volume of Gabriel and Roiter [GR97, §§7.8, 8.7].
Turning to the history of geometry of representations of algebras, one can consider Buchsbaum-Eisenbud varieties of complexes as representation varieties of the radical square zero algebra $A$ given by quiver $\bullet\to\bullet\to\cdots\to\bullet$. These varieties were studied extensively in the 1970s and results on them were eventually generalized beyond the radical square zero case (for this particular quiver) [ADFK81]. See the introduction of [KR15] for more remarks and references on this. So from both representation theoretic and geometric perspectives, we can see the radical square zero case as an important starting point for much deeper developments.
While radical square zero algebras are an important special case, working in the relative setting allows us to apply our results to many more algebras, as demonstrated in Examples 4.1, 4.6, 4.9, 4.11, 4.12, and 5.4.
1.2. Statement of main results
We start by giving a precise geometric description of how representation varieties of $A$ and $A^{x}$ are related in Section 3, using the language of homogeneous bundles. This has applications to determining irreducible components, generic decomposition, and singularities of representation varieties.
We summarize our main results in the following theorem, noting that we actually prove more general statements later.
Theorem 1.2.
Let $A=\Bbbk Q/I$ be an algebra with node $x$.
(a) There is an injective map of sets
(1.3)
$$\coprod_{\mathbf{d}}\left\{\begin{tabular}[]{c}irreducible components\\
of $\operatorname{rep}_{A}(\mathbf{d})$\end{tabular}\right\}\hookrightarrow%
\coprod_{\mathbf{e}}\left\{\begin{tabular}[]{c}irreducible components\\
of $\operatorname{rep}_{A^{x}}(\mathbf{e})$\end{tabular}\right\}$$
from the set of all irreducible components of all $\operatorname{rep}_{A}(\mathbf{d})$ to the set of all irreducible components of all $\operatorname{rep}_{A^{x}}(\mathbf{e})$.
Given an irreducible component $C\subseteq\operatorname{rep}_{A}(\mathbf{d})$, we denote the associated irreducible component of some $\operatorname{rep}_{A^{x}}(\mathbf{e})$ by $C^{x}$.
(b) The generic decomposition (recalled in Section 4.2) of $C^{x}$ determines that of $C$. More precisely, let $C^{x}=\overline{C^{x}_{1}\oplus\cdots\oplus C^{x}_{k}}$ be the generic decomposition of the associated irreducible component for a representation variety of $A^{x}$. Then each $C^{x}_{i}$ is in the image of the map (1.3), thus uniquely determines an irreducible component $C_{i}\subseteq\operatorname{rep}_{A}(\mathbf{d}_{i})$ for some dimension vector $\mathbf{d}_{i}$, and $C=\overline{C_{1}\oplus\cdots\oplus C_{k}}$ is the generic decomposition of $C$.
(c) Assume that $\operatorname{char}\Bbbk=0$. If $C^{x}$ is a normal variety, then $C$ is a normal variety, and if $C^{x}$ has rational singularities, then $C$ has rational singularities.
The theorem is applied in practice by repeatedly splitting nodes until one arrives at an algebra whose representation varieties are already understood.
We note that the injective map of (1.3) is not canonical: there is a choice which is essentially whether the simple representation of $A$ supported at $x$ is identified with the simple representation of $A^{x}$ supported at $x_{t}$ or $x_{h}$. The choice of this paper is $x_{t}$.
We single out the case of radical square zero algebras for special attention. In this case every vertex is a node, and splitting nodes results in a quiver without relations (i.e. a hereditary algebra). Since each representation variety of a quiver without relations is isomorphic to an affine space, we can give a purely combinatorial classification of the irreducible components of radical square zero algebras this way (Theorem 4.3). We also get the following immediate corollary on singularities, which, to the best of our knowledge, is the first result limiting the singularities of all irreducible components of representation varieties for such a large class of non-hereditary algebras.
Corollary 1.4.
Let $A$ be a finite-dimensional $\Bbbk$-algebra with $\operatorname{rad}^{2}A=0$ and $\operatorname{char}\Bbbk=0$. Then for any dimension vector $\mathbf{d}$, any irreducible component $C\subseteq\operatorname{rep}_{A}(\mathbf{d})$ has rational singularities (and is thus also normal, and Cohen-Macaulay).
Such results on the singularities of irreducible components, specifically their normality, can be combined with the main theorem of [CK18] to study the geometry of decompositions of moduli spaces of semistable representations (in sense of Geometric Invariant Theory). We discuss such applications and related semistability results in Section 5.
1.3. Relation to existing literature
When considering all algebras $A=\Bbbk Q/I$, or even restricting to finite dimensional ones, there can be arbitrarily many irreducible components of the representation varieties $\operatorname{rep}_{A}(\mathbf{d})$, and their singularities can be smoothly equivalent to any singularity that appears in a finite type scheme over $\mathbb{Z}$. Thus, there is no reasonable expectation to uniformly describe these for arbitrary $A$ and $\mathbf{d}$. One instead restricts to specific classes of algebras, and even then there are very few where irreducible components of every $\operatorname{rep}_{A}(\mathbf{d})$ can be parametrized by combinatorial data, or singularities of irreducible components or orbit closures can be limited. Below we survey some literature on these problems and how our works relates.
Radical square zero: We first note that our results generalize some of those appearing in [BCHZ15], where representation varieties for radical square zero algebras are studied by rather different methods. Specific comparisons are given at relevant points in this paper.
Irreducible components: There are some techniques to find irreducible components of representation varieties for certain classes of algebras, assuming one has complete knowledge of the category of representations.
For example, Zwara’s result that Hom order and degeneration order coincide for representation finite algebras [Zwa99] can in principle be used to compute irreducible components of a given representation finite algebra, assuming one explicitly knows all its indecomposable representations and dimensions of all Hom spaces between them.
Similarly, Richmond’s results [Ric01] can be used when the number of isomorphism classes of subrepresentations of an arbitrary (finite-dimensional) projective representation is finite.
Huisgen-Zimmermann and her collaborators have classified irreducible components of representation varieties for truncated path algebras in terms of representation theoretic data in a series of papers [HZ16, HZS17, GHZ18].
Irreducible components for other classes of algebras have been classified in [BS01, GS03, Sch04, RRS11, KW14]. See also [HZG12] for a survey.
The the best of our knowledge, our results give the first purely combinatorial classification of irreducible components for such a broad class of algebras (i.e. with arbitrarily many simple representations and including wild algebras).
Generic decomposition: The generic decomposition is best understood for quivers without relations, where it is also called Kac’s canonical decomposition [Kac80, Kac82, Sch92, CBS02, DW02]. For quivers with relations, Babson, Huisgen-Zimmerman, and Thomas have studied generic behavior of modules in irreducible components in [BHZT09], obtaining the sharpest results for truncated path algebras.
Carroll has given a combinatorial method of producing the generic decomposition for acyclic gentle algebras in [Car15].
Singularities: The authoritative source on singularities of orbit closures in module varieties (equivalently, the representation varieties of this paper) is the survey of Zwara [Zwa11].
Some additional contributions to this topic can be found in more recent papers such as [Bob12, RZ13, LZ14, Sut15, Lőr15, LW18]. To the best of our knowledge, our results give the first classes of algebras (other than the trivial hereditary case) where every irreducible component of every representation variety is known to have rational singularities.
Moduli spaces:
The moduli space decomposition application which we give in Section 5 was first done for algebras of the form $A=\Bbbk Q$ with $Q$ acyclic in [DW11a] (see also [CB02]). It was extended to certain classes of non-hereditary algebras in works such as [Chi13, CC15, CCKW17].
Acknowledgements
The authors thank Birge Huisgen-Zimmermann for discussions on radical square zero algebras and Claus Ringel for encouragement to consider the relative situation of node splitting in general. We also thank Paul Muhly for discussion on the history of graphs, quivers, and radical square zero algebras.
2. Background
2.1. Quivers
We denote a quiver by $Q=(Q_{0},Q_{1},t,h)$, where $Q_{0}$ is the vertex set, $Q_{1}$ the arrow set, and $t,h\colon Q_{1}\to Q_{0}$ give the tail and head of an arrow $t\alpha\xrightarrow{\alpha}h\alpha$.
A representation $M$ of $Q$ is a collection of (finite-dimensional) $\Bbbk$-vector spaces $(M_{z})_{z\in Q_{0}}$ assigned to the vertices of $Q$, along with a collection of $\Bbbk$-linear maps $(M_{\alpha}\colon M_{t\alpha}\to M_{h\alpha})_{\alpha\in Q_{1}}$ assigned to the arrows.
We recall the some key facts here, but for a more detailed recollection we refer the interested reader to standard references such as [ASS06, Sch14, DW17].
A quiver $Q$ determines a path algebra $\Bbbk Q$.
The category of (left) modules over the algebra $\Bbbk Q/I$ is equivalent to the category of representations of the quiver with relations $(Q,R)$, where $R$ is usually taken to be a minimal set of generators of $I$.
These equivalences can be used freely without significantly affecting the geometry, as made precise in [Bon91]. The Jacobson radical $\operatorname{rad}(\Bbbk Q/I)$ is spanned by all the paths of $Q$ of length $\geq 1$ modulo $R$.
An ideal is admissible if $\operatorname{rad}^{N}(\Bbbk Q)\subseteq I\subseteq\operatorname{rad}^{2}(%
\Bbbk Q)$ for some $N\geq 2$.
Given a finite dimensional $\Bbbk$-algebra $A$, it is Morita equivalent to a quotient of a path algebra $\Bbbk Q/I$. If $I$ is taken to be admissible (which is always possible), then $Q$ is uniquely determined. We always assume that $I\subset\operatorname{rad}^{2}(\Bbbk Q)$, and that $I$ is admissible in Section 5 for our results on moduli spaces.
2.2. Node splitting
The operation of node splitting for Artin algebras was introduced by Martínez-Villa [MV80]. Here we recall this notation in language translated to quotients of quiver path algebras.
We say that $x\in Q_{0}$ is a node of an algebra $A=\Bbbk Q/I$ if $\alpha\beta\in I$ for all pairs $\alpha,\beta\in Q_{1}$ such that $h\alpha=x$ and $t\beta=x$.
In other words, any path having $x$ as an intermediate vertex is 0 in $A$.
Given a quiver $Q$ and $x\in Q_{0}$, we can consider the quiver $Q^{x}$ with vertex set
$$(Q^{x})_{0}=(Q_{0}\backslash\{x\})\cup\{x_{t},x_{h}\}$$
and arrow set $(Q^{x})_{1}=Q_{1}$. Tail and head functions $t,h\colon(Q^{x})_{1}\to(Q^{x})_{0}$ are the same as in $Q$ except that $x$ is replaced with $x_{t}$ in the codomain of $t$, and $x$ is replaced with $x_{h}$ in the codomain of $h$.
The operation locally around $x$ is illustrated below.
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Notice that the set of paths of positive length in $Q^{x}$ can be naturally identified with a subset of the set of paths of $Q$, inducing an inclusion of vector spaces $\operatorname{rad}(\Bbbk Q^{x})\subset\operatorname{rad}(\Bbbk Q)$.
Let $x$ be a node of an algebra $\Bbbk Q/I$.
The algebra $A^{x}=\Bbbk Q^{x}/I^{x}$ is defined with $Q^{x}$ as above and we set $I^{x}=I\cap\operatorname{rad}(\Bbbk Q^{x})$.
It is easily observed that if $x$ is a node of $A=\Bbbk Q/I$, then $A^{x}$ has exactly one fewer node than $A$ (not counting sources and sinks).
The representation theory of $A$ and $A^{x}$ are known to be closely related (for example, see Corollary 3.4).
An algebra $A=\Bbbk Q/I$ such that every vertex of $Q$ is a node is a radical square zero algebra, satisfying $\operatorname{rad}^{2}(A)=0$.
2.3. Representation varieties
Given a quiver $Q$ and dimension vector $\mathbf{d}\colon Q_{0}\to\mathbb{Z}_{\geq 0}$, we study the representation variety
$$\operatorname{rep}_{Q}(\mathbf{d})=\prod_{\alpha\in Q_{1}}\operatorname{Mat}(%
\mathbf{d}(h\alpha),\mathbf{d}(t\alpha)),$$
where $\operatorname{Mat}(m,n)$ denotes the variety of matrices with $m$ rows, $n$ columns, and entries in $\Bbbk$.
We consider the left action of the base change group
$$GL(\mathbf{d})=\prod_{z\in Q_{0}}GL(\mathbf{d}(z))$$
on $\operatorname{rep}_{Q}(\mathbf{d})$ given by
$$g\cdot M=(g_{h\alpha}M_{\alpha}g_{t\alpha}^{-1})_{\alpha\in Q_{1}},$$
where $g=(g_{z})_{z\in Q_{0}}\in GL(\mathbf{d})$ and $M=(M_{\alpha})_{\alpha\in Q_{1}}\in\operatorname{rep}_{Q}(\mathbf{d})$.
Now consider an algebra $A=\Bbbk Q/I$ with corresponding quiver with relations $(Q,R)$. Then the representation variety $\operatorname{rep}_{A}(\mathbf{d})$ is the closed $GL(\mathbf{d})$-stable subvariety of $\operatorname{rep}_{Q}(\mathbf{d})$ defined by
$$\operatorname{rep}_{A}(\mathbf{d})=\{M\in\operatorname{rep}_{Q}(\mathbf{d})\,%
\mid\,M(r)=0,\,\mbox{ for all }r\in R\}.$$
Thus, the points of $\operatorname{rep}_{A}(\mathbf{d})$ are in bijection with representations of $(Q,R)$ along with a fixed basis at each vertex.
Simply from the definitions, orbits in $\operatorname{rep}_{A}(\mathbf{d})$ under $GL(\mathbf{d})$ are in bijection with isomorphism classes of representations of $A$ of dimension vector $\mathbf{d}$; for a representation $M$ of $A$, we denote by $O_{M}$ the orbit of $M$ in $\operatorname{rep}_{A}(\mathbf{d})$, and by $\overline{O}_{M}$ the closure of this orbit.
2.4. Rational singularities
We say that a map between algebraic varieties $f:Z\to X$ is a resolution of singularities, if $Z$ is smooth, and $f$ is proper and birational.
An algebraic variety $X$ has rational singularities, if for some (hence, any) resolution of singularites $Z\to X$, we have
(a)
$X$ is normal, that is, the natural map $\mathcal{O}_{X}\to f_{*}\mathcal{O}_{Z}$ is an isomorphism, and
(b)
$\mathbb{R}^{i}f_{*}\mathcal{O}_{Z}=0$, for $i>0$.
It is known that if $X$ has rational singularities, then $X$ is a Cohen-Macaulay variety. For more details, we refer the reader to [Wey03, Section 1.2].
2.5. Homogeneous fiber bundles
Let $G$ be an algebraic group and $H\leq G$ a closed algebraic subgroup, and suppose we have an action of $H$ on a quasi-projective algebraic variety $S$.
We write $G\times_{H}S$ for the quotient of $G\times S$ by the free left action of $H$ given by $h\cdot(g,s)=(gh^{-1},h\cdot s)$, called an induced space or homogeneous fiber bundle.
We consider this quotient as a $G$-variety by the action $g\cdot(g^{\prime},s)=(gg^{\prime},s)$.
Furthermore, we embed $S\hookrightarrow G\times_{H}S$ via the map $s\mapsto(1,s)$.
The following lemma can be proven directly from definitions; see for example [Tim11, §2.1] for further discussion.
Lemma 2.1.
The maps below are mutually inverse, inclusion preserving bijections.
$$\begin{split}\displaystyle\left\{\begin{tabular}[]{c}$G$-stable subvarieties\\
of $G\times_{H}S$\end{tabular}\right\}&\displaystyle\leftrightarrow\left\{%
\begin{tabular}[]{c}$H$-stable subvarieties\\
of $S$\end{tabular}\right\}\\
\displaystyle Y&\displaystyle\mapsto\qquad\qquad Y\cap S\\
\displaystyle G\times_{H}Z&\displaystyle\mapsfrom\qquad\qquad Z\end{split}$$
In particular, they give a bijection on orbits and isomorphism of orbit closure posets.
3. Node splitting and bundles
Consider an algebra $A=\Bbbk Q/I$, and $\mathbf{d}$ a dimension vector of $Q$.
3.1. Node splitting on strata
Throughout we use the following notation.
Definition 3.1.
For a vertex $x\in Q_{0}$ and a representation $M\in\operatorname{rep}_{A}(\mathbf{d})$, we denote by $h_{x}(M)$ and $t_{x}(M)$ the linear maps
$$h_{x}(M)=\bigoplus_{h\alpha=x}M_{\alpha}:\,\bigoplus_{h\alpha=x}M_{t\alpha}\to
M%
_{x},\,\mbox{ and }\,t_{x}(M)=\bigoplus_{t\alpha=x}M_{\alpha}\,:M_{x}\to%
\bigoplus_{t\alpha=x}M_{h\alpha}.$$
Given subset $S\subset\operatorname{rep}_{A}(\mathbf{d})$, we define the $x$-rank of $S$ to be the number
$$r_{x}(S):=\max_{M\in S}\left\{\operatorname{rank}{h_{x}(M)}\right\}.$$
Moreover, we denote by $S^{\circ}=\{M\in S\,\mid\,r_{x}(M)=r_{x}(S)\}$.
Now assume that $x\in Q_{0}$ is a node of $A$. Let $A^{x}=\Bbbk Q^{x}/I^{x}$ be the algebra obtained by splitting the node $x$, as explained in Section 2.2.
Fix an integer $r$ with $0\leq r\leq\mathbf{d}(x)$. We denote by $\mathbf{d}^{x}_{r}$ the dimension vector of $Q^{x}$ obtained by putting $\mathbf{d}^{x}(x_{h})=r$, $\mathbf{d}^{x}(x_{t})=\mathbf{d}(x)-r$, and at the rest of the vertices $\mathbf{d}^{x}$ coincides with $\mathbf{d}$.
We realize the variety $\operatorname{rep}_{A^{x}}(\mathbf{d}^{x}_{r})$ as a $GL(\mathbf{d}^{x}_{r})$-stable closed subvariety of $\operatorname{rep}_{A}(\mathbf{d})$ by an embedding $i\colon\operatorname{rep}_{A^{x}}(\mathbf{d}^{x}_{r})\hookrightarrow%
\operatorname{rep}_{A}(\mathbf{d})$. Namely, given $M=(M_{\alpha})_{\alpha\in Q_{1}}$ in the domain, we take $i(M)=(N_{\alpha})_{\alpha\in Q_{1}}$ where
(3.2)
$$N_{\alpha}=\begin{cases}M_{\alpha}&t\alpha\neq x\neq h\alpha\\
\left[\begin{smallmatrix}M_{\alpha}\\
0\end{smallmatrix}\right]&h\alpha=x\mbox{ and }t\alpha\neq x,\\
\left[\begin{smallmatrix}0&M_{\alpha}\end{smallmatrix}\right]&t\alpha=x\mbox{ %
and }h\alpha\neq x,\\
\left[\begin{smallmatrix}0&M_{\alpha}\\
0&0\end{smallmatrix}\right]&t\alpha=x\mbox{ and }h\alpha=x.\\
\end{cases}$$
In the remainder of the paper we implicitly use this specific embedding $\operatorname{rep}_{A^{x}}(\mathbf{d}^{x}_{r})\subseteq\operatorname{rep}_{A}(%
\mathbf{d})$ without mentioning the map $i$.
If $C$ is a $GL(\mathbf{d}^{x}_{r})$-stable irreducible closed subvariety of $\operatorname{rep}_{A^{x}}(\mathbf{d}^{x}_{r})$ (hence of $\operatorname{rep}_{A}(\mathbf{d})$), we take its $GL(\mathbf{d}(x))$-saturation to obtain the subset $GL(\mathbf{d}(x))\cdot C$ of $\operatorname{rep}_{A}(\mathbf{d})$. Note that we have $r_{x}(GL(\mathbf{d}(x))\cdot C)=r_{x_{t}}(C)\leq r$.
Retaining the notation above, furthermore let $P_{r}\leq GL(\mathbf{d}(x))$ be the parabolic subgroup of block upper triangular matrices with two blocks along the diagonal, of size $r$ in the upper left and $\mathbf{d}(x)-r$ in the lower right.
Let $P^{x}_{r}(\mathbf{d})\leq GL(\mathbf{d})$ be the subgroup where the factor $GL(\mathbf{d}(x))$ is replaced by $P_{r}$, so we have also that $P^{x}_{r}(\mathbf{d})\geq GL(\mathbf{d}^{r}_{x})$.
From (3.2) we see that $\operatorname{rep}_{A^{x}}(\mathbf{d}^{x}_{r})$ is not only $GL(\mathbf{d}^{x}_{r})$-stable, but also a $P^{x}_{r}(\mathbf{d})$-stable subvariety of $\operatorname{rep}_{A}(\mathbf{d})$, since the unipotent radical of $P_{r}$ acts trivially on $\operatorname{rep}_{A^{x}}(\mathbf{d}^{x}_{r})$.
Proposition 3.3.
Let $0\leq r\leq\mathbf{d}(x)$ and consider the locally closed subvariety of $\operatorname{rep}_{A}(\mathbf{d})$ consisting of all points of $x$-rank exactly $r$:
$$\operatorname{rep}^{r}_{A}(\mathbf{d}):=\left\{N\in\operatorname{rep}_{A}(%
\mathbf{d})\mid r_{x}(N)=r\right\}.$$
If $\operatorname{rep}^{r}_{A}(\mathbf{d})$ is non-empty
then the following map is an isomorphism of varieties:
$$\Psi\colon GL(\mathbf{d})\times_{P^{x}_{r}(\mathbf{d})}\operatorname{rep}_{A^{%
x}}(\mathbf{d}^{x}_{r})^{\circ}\to\operatorname{rep}_{A}^{r}(\mathbf{d})\,,\,%
\,(g,M)\mapsto g\cdot M.$$
Proof.
The map is well-defined since $\Psi(gp^{-1},pM)=\Psi(g,M)$. To construct the inverse morphism, take any $N\in\operatorname{rep}_{A}^{r}(\mathbf{d})$, so we know $\operatorname{image}h_{x}(N)$ is an $r$-dimensional subspace of $\Bbbk^{\mathbf{d}(x)}$. Then we can find $g\in GL(\mathbf{d})$ such that
$$\operatorname{image}h_{x}(g^{-1}\cdot N)=g^{-1}_{x}\left(\operatorname{image}h%
_{x}(N)\right)=\Bbbk^{r}\subseteq\Bbbk^{\mathbf{d}(x)},$$
the subspace spanned by the first $r$ standard basis vectors. Since $x$ is a node in $A$, this means $g^{-1}\cdot N\in\operatorname{rep}_{A^{x}}(\mathbf{d}^{x}_{r})$ via the identification of (3.2). This $g$ is not unique, but any $g_{0}\in GL(\mathbf{d})$ with the same property satisfies that $g_{0}^{-1}g$ stabilizes $\Bbbk^{r}$,
which is to say $g_{0}^{-1}g\in P^{x}_{r}(\mathbf{d})$. Thus $(g,g^{-1}\cdot N)$ and $(g(g_{0}^{-1}g)^{-1},(g_{0}^{-1}g)g^{-1}\cdot N)=(g_{0},g_{0}^{-1}\cdot N)$ represent the same point in the quotient variety
$GL(\mathbf{d})\times_{P^{x}_{r}(\mathbf{d})}\operatorname{rep}_{A^{x}}^{\circ}%
(\mathbf{d}^{x}_{r})$, so $N\mapsto(g,g^{-1}\cdot N)$ as above gives a well-defined inverse morphism to $\Psi$.
∎
Using this geometric interplay, we recover the following well-known result (see [MV80]).
Corollary 3.4.
There is a bijection between the set of isomorphism classes of indecomposable representations of $A$ and the set of isomorphism classes of indecomposable representations of $A^{x}$ with the simple representation supported at $x_{h}$ removed.
Proof.
It is immediate from Proposition 3.3 that for any $\mathbf{d}$ and $0\leq r\leq\mathbf{d}(x)$ we have a bijection
(3.5)
$$\left\{\begin{tabular}[]{c}isomorphism classes of\\
representations $M$ of $\operatorname{rep}_{A^{x}}(\mathbf{d}^{x}_{r})$\\
with $r_{x_{h}}(M)=r$\end{tabular}\right\}\leftrightarrow\left\{\begin{tabular%
}[]{c}isomorphism classes of\\
representations $N$ of $\operatorname{rep}_{A}(\mathbf{d})$\\
with $r_{x}(N)=r$\end{tabular}\right\}.$$
Let $S_{x_{h}}$ denote the simple supported at $x_{h}$. Clearly, if $M\in\operatorname{rep}_{A^{x}}(\mathbf{d}^{x}_{r})$ with $r_{x_{h}}(M)<r$, then $S_{x_{h}}$ is a summand of $M$. This shows that the only indecomposable representation of $A^{x}$ that does not appear in the sets on the left hand side of (3.5) is $S_{x_{h}}$.
We are left to show that under the correspondence in (3.5), indecomposable representations are mapped to indecomposable representations. We use the well-known fact that a representation is indecomposable if and only if its stabilizer in the projective linear group is unipotent [Bri12, Cor. 2.10].
Let $H$ be the $PGL(\mathbf{d}_{x}^{r})$-stabilizer of $M\in\operatorname{rep}_{A^{x}}(\mathbf{d}_{r}^{x})$. Since $GL(\mathbf{d})\cdot O_{M}\cong GL(\mathbf{d})\times_{P^{x}_{r}(\mathbf{d})}O_{M}$, the $PGL(\mathbf{d})$-stabilizer of $M$ in $\operatorname{rep}_{A}(\mathbf{d})$ is $U\rtimes H$, where $U$ is the unipotent radical of $P_{r}^{x}\cong U\rtimes(GL(r)\times GL(\mathbf{d}(x)-r))$. Clearly, $H$ is unipotent if and only if $U\rtimes H$ is.
∎
A representation $M$ of $A$ is called Schur if $\operatorname{End}_{A}(M)=\Bbbk$. A Schur representation is indecomposable. The following shows, in particular, that a Schur representation of $A^{x}$ does not necessarily correspond to a Schur representation of $A$ via the bijection in Corollary 3.4.
Corollary 3.6.
If $M$ is a Schur representation of $A$, then either $h_{x}(M)=0$ or $t_{x}(M)=0$. Moreover, the bijection in 3.4 descends to a bijection
$$\left\{\begin{tabular}[]{c}isomorphism classes of\\
Schur representations $M\neq S_{x_{h}}$ of $A^{x}$\\
with either $M_{x_{h}}=0$ or $M_{x_{t}}=0$\end{tabular}\right\}\leftrightarrow%
\left\{\begin{tabular}[]{c}isomorphism classes of\\
Schur representations of $A$\end{tabular}\right\}.$$
Proof.
We use the notation as in the proof of Corollary 3.4. A representation is Schur if and only its stabilizer in the projective linear group is trivial. If $H$ is the $PGL(\mathbf{d}_{x}^{r})$-stabilizer of $M\neq S_{x_{h}}$ in $\operatorname{rep}_{A^{x}}(\mathbf{d}_{r}^{x})$, then $U\rtimes H$ is the $PGL(\mathbf{d})$-stabilizer of $M\in\operatorname{rep}_{A}(\mathbf{d})$. Hence $M\in\operatorname{rep}_{A}(\mathbf{d})$ is Schur if and only if $M\in\operatorname{rep}_{A^{x}}(\mathbf{d}_{r}^{x})$ is Schur and $U$ is trivial. But $U$ is trivial if and only if $r=r_{x}(M)$ is equal to either $0$ or $\mathbf{d}(x)$, hence the claim.
∎
3.2. Passage to closed subvarieties
We now turn from strata to the study of closed subvarieties.
Proposition 3.7.
Let $0\leq r\leq\mathbf{d}(x)$ and $C$ a $GL(\mathbf{d}^{x}_{r})$-stable irreducible closed subvariety of $\operatorname{rep}_{A^{x}}(\mathbf{d}^{x}_{r})$ with $r_{x_{t}}(C)=r$. Then $GL(\mathbf{d})\cdot C$ is an irreducible closed subvariety of $\operatorname{rep}_{A}(\mathbf{d})$, and the following map is a proper birational morphism of $GL(\mathbf{d})$-varieties:
$$\Psi_{C}\colon GL(\mathbf{d})\times_{P^{x}_{r}(\mathbf{d})}C\to GL(\mathbf{d})%
\cdot C\,,\,(g,M)\mapsto g\cdot M.$$
Proof.
The variety
$$GL(\mathbf{d})/P^{x}_{r}(\mathbf{d})\cong GL(\mathbf{d}(x))/P_{r}\cong%
\operatorname{Grass}(r,\Bbbk^{\mathbf{d}(x)}),$$
is projective. We have an isomorphism of varieties $GL(\mathbf{d})\times_{P_{r}^{x}(\mathbf{d})}\operatorname{rep}_{A}(\mathbf{d})%
\cong GL(\mathbf{d})/P_{r}^{x}(\mathbf{d})\times\operatorname{rep}_{A}(\mathbf%
{d})$ given by the map $(g,x)\mapsto(g,gx)$. Hence, the multiplication map
$$GL(\mathbf{d}(x))\times_{P_{r}}\operatorname{rep}_{A}(\mathbf{d})\to%
\operatorname{rep}_{A}(\mathbf{d})\,,\,\,(g,M)\mapsto g\cdot M$$
is proper. Since $C$ is closed in $\operatorname{rep}_{A}(\mathbf{d})$, it follows by Lemma 2.1 that $(GL(\mathbf{d}(x))\cdot C$ is closed in $\operatorname{rep}_{A}(\mathbf{d})$ as well. By Proposition 3.3, the map $\Psi_{C}$ induces an isomorphism on the open subsets
$$GL(\mathbf{d})\times_{P^{x}_{r}(\mathbf{d})}C^{\circ}\xrightarrow{\cong}(GL(%
\mathbf{d})\cdot C)^{\circ}.$$
Hence, $\Psi_{C}$ is birational.
∎
Proposition 3.8.
For each $0\leq r\leq\mathbf{d}(x)$, the maps below are mutually inverse, inclusion preserving bijections.
$$\begin{split}\displaystyle\left\{\begin{tabular}[]{c}irreducible closed\\
$GL(\mathbf{d}^{x}_{r})$-stable subvarieties\\
of $\operatorname{rep}_{A^{x}}(\mathbf{d}^{x}_{r})$ of $x_{h}$-rank $r$\end{%
tabular}\right\}&\displaystyle\leftrightarrow\left\{\begin{tabular}[]{c}%
irreducible closed\\
$GL(\mathbf{d})$-stable subvarieties\\
of $\operatorname{rep}_{A}(\mathbf{d})$ of $x$-rank $r$\end{tabular}\right\}\\
\displaystyle C&\displaystyle\mapsto\qquad\qquad GL(\mathbf{d})\cdot C\\
\displaystyle D\,\cap\,\operatorname{rep}_{A^{x}}(\mathbf{d}_{r}^{x})&%
\displaystyle\mapsfrom\qquad\qquad D\end{split}$$
In particular, Theorem 1.2(a) holds.
Proof.
It follows by Proposition 3.7 that $C\mapsto GL(\mathbf{d})\cdot C$ is a well-defined function between the sets above.
Each subvariety $C$ (resp $D$) in the set on the left (resp. right) hand side above is uniquely determined by $C^{\circ}$ (resp $D^{\circ}$) via $C=\overline{C^{\circ}}$ (resp. $D=\overline{D^{\circ}}$), hence the map $C\mapsto GL(\mathbf{d})\cdot C$ is bijective by Proposition 3.3 and Lemma 2.1.
To show that the inverse map is the one claimed, we are left to show that
$$(GL(\mathbf{d})\cdot C)\bigcap\operatorname{rep}_{A^{x}}(\mathbf{d}_{r}^{x})=C$$
(in fact, this holds for any $GL(\mathbf{d}^{x}_{r})$-stable subset $C$ of $\operatorname{rep}_{A^{x}}(\mathbf{d}^{x}_{r})$).
The containment $\supseteq$ is immediate, so we must show the other direction.
Take $g\in GL(\mathbf{d})$ and $M\in C$ such that $g\cdot M\in\operatorname{rep}_{A^{x}}(\mathbf{d}^{x}_{r})$. We want to show that $g\cdot M\in C$. To do so, it is enough to find $g^{\prime}\in GL(\mathbf{d}^{x}_{r})$ such that $g^{\prime}\cdot M=g\cdot M$ since $C$ is $GL(\mathbf{d}^{x}_{r})$-stable. Such a $g^{\prime}$ exists if and only if $M$ is isomorphic to $g\cdot M$ when considered as a representation of $A^{x}$.
So this containment is essentially just saying that if two representations of $A$ are isomorphic (by $g$), then they are isomorphic when considered as representations of $A^{x}$ (by $g^{\prime}$).
Let $B_{1}$ (resp. $B_{2}$) be the matrix of the map $h_{x}(M)$ (resp. $h_{x}(g\cdot M)$).Since $M,\,g\cdot M\in\operatorname{rep}_{A^{x}}(\mathbf{d}^{x}_{r})$, the images of both $\alpha_{1},\alpha_{2}$ are contained in $\Bbbk^{r}$. Hence, only the first $r$ rows of $B_{1}$ (resp $B_{2}$) non-zero. We denote the matrix formed by the first $r$ rows of $B_{1}$ (resp. $B_{2}$) by $B_{1}^{\prime}$ (resp. $B_{2}^{\prime}$). Since $B_{1}$ and $B_{2}$ are row-equivalent (i.e. have the same reduced row echelon form), the matrices $B_{1}^{\prime}$ and $B_{2}^{\prime}$ are also row-equivalent. Using the same argument with the maps with source $x$, we see that there is a matrix $g^{\prime}\in GL(r)\times GL(\mathbf{d}(x)-r)$ such that $g^{\prime}\cdot M=g\cdot M$.
For the “in particular” part, we simply note that each irreducible component $C\subseteq\operatorname{rep}_{Q}(\mathbf{d})$ has a well defined $x$-rank r which uniquely determines $\mathbf{e}=\mathbf{d}_{r}^{x}$ in the notation of the theorem statement, and then a unique irreducible component of $\operatorname{rep}_{A^{x}}(\mathbf{e})$ by the previous paragraph.
∎
The following is immediate from Proposition 3.8 above.
Corollary 3.9.
Let $N\in\operatorname{rep}_{A}^{r}(\mathbf{d})$, and consider its $GL(\mathbf{d})$-orbit $O_{N}$ in $\operatorname{rep}_{A}(\mathbf{d})$. Take any $M\in O_{N}\,\bigcap\,\operatorname{rep}_{A^{x}}(\mathbf{d}^{x}_{r})$ (such element exists by (3.5)). Then $\overline{O}_{N}=GL(\mathbf{d})\cdot\overline{O}_{M}$, where $\overline{O}_{M}$ denotes the closure of the $GL(\mathbf{d}_{r}^{x})$-orbit $O_{M}$ in $\operatorname{rep}_{A^{x}}(\mathbf{d}^{x}_{r})$.
4. Applications
4.1. Irreducible components
Given an algebra $A=\Bbbk Q/I$ which has nodes, one can use repeated application of Theorem 1.2(a) to classify irreducible components of $\operatorname{rep}_{A}(\mathbf{d})$ if splitting the nodes eventually results in representation varieties with known irreducible components. We illustrate this with examples.
Example 4.1.
Consider the algebra $A=\Bbbk Q/I$ where $Q$ is given below and $I$ is generated by relations declaring that $x$ is a node, along with the relation $cba=0$.
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Notice that $A$ does not fall within a well-studied class such as special biserial, radical square zero, etc.
The overlapping relations make direct analysis of irreducible components of each $\operatorname{rep}_{A}(\mathbf{d})$ challenging. Also, $A$ is not representation finite, so irreducible components cannot be determined by computing dimensions of Hom spaces as in [Zwa99]. But each $\operatorname{rep}_{A^{x}}(\mathbf{d}^{x}_{r})$ decomposes as the product of an affine space with a union of orbit closures (determined by the relation $cba$) in a representation variety for the subquiver of Dynkin type $\mathbb{A}_{4}$ with arrows $a,b,c$. So these can be explicitly determined for any given $\mathbf{d}^{x}_{r}$.
For example, take $\mathbf{d}=(3,2,2,1,3,3,3)$ (with the convention that $\mathbf{d}(x)$ is the last entry). By Proposition 3.7, the irreducible components of $\operatorname{rep}_{A}(\mathbf{d})$ are among the $GL(\mathbf{d})$-saturations of the irreducible components of $\operatorname{rep}_{A^{x}}(\mathbf{d}^{x}_{r})$ for $r=0,1,2,3$, which reduce to the quiver of type $\mathbb{A}_{4}$ with the following dimension vector
$$(3-r)$$23$$r$$$$a$$$$b$$$$c$$
We first describe the irreducible components of the $\mathbb{A}_{4}$ quiver above, with the convention that indecomposables correspond to roots (their dimension vectors). In the cases $r=0$ and $r=3$, the representation varieties are irreducible affine spaces $C_{0},C_{3}$. When $r=1$, the representation variety has two components $C_{1}$ and $C_{1}^{\prime}$ that are the orbit closures of the representations $(1,1,1,0)^{\oplus 2}\oplus(0,0,1,1)$ and $(1,0,0,0)\oplus(1,1,1,0)\oplus(0,1,1,1)\oplus(0,0,1,0)$, respectively. For $r=2$, there are again two components $C_{2}$ and $C_{2}^{\prime}$, that are the closures of $(1,1,1,0)\oplus(0,1,1,1)\oplus(0,0,1,1)$ and $(1,0,0,0)\oplus(0,1,1,1)^{\oplus 2}\oplus(0,0,1,0)$, respectively.
By abuse of notation, we can view the components obtained above as components for $A^{x}$. Since all the components have maximal $x_{h}$-rank, their $GL(\mathbf{d})$-saturation yield irreducible closed subsets in $\operatorname{rep}_{A}(\mathbf{d})$ according to Proposition 3.8. However, it is clear that the $GL(\mathbf{d})$-saturation of $C_{1}^{\prime}$ (resp. $C_{2}^{\prime}$) is contained in the $GL(\mathbf{d})$-saturation of $C_{2}$ (resp. $C_{3}$). This shows that $\operatorname{rep}_{A}(\mathbf{d})$ has $4$ irreducible components given by the $GL(\mathbf{d})$-saturations of $C_{0},C_{1},C_{2},C_{3}$.
∎
As seen in the example above, in Proposition 3.8 the irreducible components of $\operatorname{rep}_{A^{x}}(\mathbf{d}^{x}_{r})$ do not necessarily yield irreducible components in $\operatorname{rep}_{A}(\mathbf{d})$ under the map $C\to GL(\mathbf{d})\cdot C$. Nevertheless, we give a condition when this indeed happens.
Lemma 4.2.
Let $C$ be an irreducible component of $\operatorname{rep}_{A^{x}}(\mathbf{d}^{x}_{r})$ with $r_{x_{h}}(C)=r$, and assume that there is a representation $M\in C$ such that the map
$t_{x_{t}}(M)$ is injective. Then $GL(\mathbf{d})\cdot C$ is an irreducible component of $\operatorname{rep}_{A}(\mathbf{d})$.
Proof.
Assume by contradiction that $GL(\mathbf{d})\cdot C$ is strictly contained in an irreducible component $C^{\prime}$ of $\operatorname{rep}_{A}(\mathbf{d})$. By Proposition 3.8, we must have $r_{x}(C^{\prime})>r$. On the open subset of $C^{\prime}$ of representations $N$ with $r_{x}(N)=r_{x}(C^{\prime})$, we must have $\dim\ker t_{x}(N)>r$. Since $C^{\prime}$ is irreducible, this shows that for all representations $N\in C^{\prime}$, we have $\dim\ker t_{x}(N)>r$. But the assumptions imply that for $M$ (viewed as a representation in $\operatorname{rep}_{A}(\mathbf{d})$), we have $\dim\ker t_{x}(M)=r$. Hence $M\notin C^{\prime}$, a contradiction.
∎
Applying recursively Proposition 3.8, we give an explicit description of the irreducible components of representation varieties for radical-square algebras.
Theorem 4.3.
Consider a radical-square algebra $A=\Bbbk Q/\operatorname{rad}^{2}(\Bbbk Q)$ and a dimension vector $\mathbf{d}$. For a dimension vector $\mathbf{r}\leq\mathbf{d}$, we denote by $C_{\mathbf{r}}$ the closure of the set of representations $M\in\operatorname{rep}_{A}(\mathbf{d})$ such that $r_{x}(M)=\mathbf{r}(x)$, for all $x\in Q_{0}$. Then $C_{\mathbf{r}}$ is irreducible (possibly empty). Furthermore, set $\mathbf{s}=\mathbf{d}-\mathbf{r}$, and for $x\in Q_{0}$ let $l_{x}$ be the number of loops at $x$ and put
$$u_{x}(\mathbf{r})=\sum_{h\alpha=x}\mathbf{s}(t\alpha)\,-\mathbf{r}(x),\quad%
\mbox{ and }\quad v_{x}(\mathbf{r})=\sum_{t\alpha=x}\mathbf{r}(h\alpha)\,-%
\mathbf{s}(x).$$
Then the irreducible components of $\operatorname{rep}_{A}(\mathbf{d})$ are given precisely by the irreducibles $C_{\mathbf{r}}$ for which $\mathbf{r}$ satisfies the following for all $x\in Q_{0}$:
(4.4)
$$u_{x}(\mathbf{r})\geq 0,\mbox{ and when }u_{x}(\mathbf{r})>l_{x}\mbox{ then }v%
_{x}(\mathbf{r})\geq 0.$$
Proof.
Let $Q^{\operatorname{sp}}$ denote the quiver obtained by splitting all the nodes of $A$. Clearly, $|Q^{\operatorname{sp}}_{0}|=2|Q_{0}|$, and the vertices of $Q^{\operatorname{sp}}$ are sinks $x_{h}$ and sources $x_{t}$ corresponding to the vertices $x\in Q_{0}$. Since the quiver $Q^{\operatorname{sp}}$ has no relations, all of its representation varieties are irreducible affine spaces.
Fix $\mathbf{r}\leq\mathbf{d}$ and let $\mathbf{s}=\mathbf{d}-\mathbf{r}$. To show that $C_{\mathbf{r}}$ is irreducible, it is enough to show that $C_{\mathbf{r}}^{\circ}$ is so. Starting with the representation variety $\operatorname{rep}_{A}(\mathbf{d})$ and splitting the nodes of $A$ repeatedly w.r.t. the ranks given by the dimension vector $\mathbf{r}$, we arrive at the representation variety $\operatorname{rep}_{Q^{\operatorname{sp}}}(\mathbf{e})$, where $\mathbf{e}$ is the dimension vector given by $\mathbf{e}(x_{h})=\mathbf{r}(x)$ and $\mathbf{e}(x_{t})=\mathbf{s}(x)$ for $x\in Q_{0}$. Via the isomorphisms in Proposition 3.3 (applied recursively), $C_{\mathbf{r}}^{\circ}$ corresponds to the open subset of representations $N\in\operatorname{rep}_{Q^{\operatorname{sp}}}(\mathbf{e})$ such that $r_{x_{h}}(N)=\mathbf{e}(x_{h})=\mathbf{r}(x)$ for all $x\in Q_{0}$. Since the latter is irreducible, this shows that $C_{\mathbf{r}}^{\circ}$ is irreducible as well. Moreover, under the bijections in Proposition 3.8 (applied recursively), $C_{\mathbf{r}}$ corresponds to $\operatorname{rep}_{Q^{\operatorname{sp}}}(\mathbf{e})$.
Given $x\in Q_{0}$, it is easy to see that there is a representation $N\in\operatorname{rep}_{Q^{\operatorname{sp}}}(\mathbf{e})$ such that $r_{x_{h}}(N)=\mathbf{r}(x)$ if and only if $u_{x}(\mathbf{r})\geq 0$. Since $\operatorname{rep}_{Q^{\operatorname{sp}}}(\mathbf{e})$ is irreducible, we obtain that $C_{\mathbf{r}}$ is non-empty if and only if $u_{x}(\mathbf{r})\geq 0$ for all $x\in Q_{0}$.
Now take $C_{\mathbf{r}}$ non-empty. We show that if $\mathbf{r}$ satisfies $u_{x}(\mathbf{r})>l_{x}$ and $v_{x}(\mathbf{r})<0$ for some $x\in Q_{0}$, then $C_{\mathbf{r}}\subset C_{\mathbf{r}^{\prime}}$, where $\mathbf{r}^{\prime}(x)=\mathbf{r}(x)+1$, and $\mathbf{r}^{\prime}(y)=\mathbf{r}(y)$ for $y\in Q_{0}\setminus\{x\}$. First, we show that $C_{\mathbf{r}^{\prime}}$ is non-empty. We have $u_{x}(\mathbf{r}^{\prime})=u_{x}(\mathbf{r})-l_{x}-1\geq 0$. Since $v_{x}(\mathbf{r})<0$, we have in particular that $\mathbf{s}(x)>\mathbf{r}(y)$ for any vertex $y\neq x$ such that there is an arrow from $x$ to $y$, and so $u_{y}(\mathbf{r}^{\prime})\geq\mathbf{s}^{\prime}(x)-\mathbf{r}^{\prime}(y)=%
\mathbf{s}(x)-1-\mathbf{r}(y)\geq 0$. We obtain that $u_{z}(\mathbf{r}^{\prime})\geq 0$ for all $z\in Q_{0}$, hence $C_{\mathbf{r}^{\prime}}$ is non-empty and it corresponds to $\operatorname{rep}_{Q^{\operatorname{sp}}}(\mathbf{e}^{\prime})$ via the bijections in Proposition 3.8. Now take any $M\in C_{\mathbf{r}}$. By abuse of notation, we can view $M$ as a representation in $\operatorname{rep}_{Q^{\operatorname{sp}}}(\mathbf{e})$. We see that since $v_{x}(\mathbf{r})<0$, we have a decomposition $M\cong N\oplus S_{x_{t}}$, where $S_{x_{t}}$ denotes the simple at $x_{t}$. Hence, as representations of $A$, we have $M\cong N\oplus S_{x}$. But then $N\oplus S_{x_{h}}\in\operatorname{rep}_{Q^{\operatorname{sp}}}(\mathbf{e}^{%
\prime})$ which shows that $M\cong N\oplus S_{x}\in C_{\mathbf{r}^{\prime}}$ as well. Hence, $C_{\mathbf{r}}\subset C_{\mathbf{r}^{\prime}}$. We have showed that
$$\operatorname{rep}_{A}(\mathbf{d})=\bigcup_{\begin{subarray}{c}\mathbf{r}\leq%
\mathbf{d}\\
\mathbf{r}\mbox{ \footnotesize satisfies \normalsize}(\ref{eq:rcond})\end{%
subarray}}C_{\mathbf{r}}.$$
We are left to show that there are no containments between the irreducibles above. Take $C_{\mathbf{r}},C_{\mathbf{r}^{\prime}}$ two irreducibles from the union above, and assume that $C_{\mathbf{r}}\subset C_{\mathbf{r}^{\prime}}$. Clearly, we must have $\mathbf{r}\leq\mathbf{r}^{\prime}$, and so $\mathbf{s}^{\prime}\leq\mathbf{s}$. Assume that there is a vertex $x\in Q_{0}$ such that $\mathbf{r}(x)<\mathbf{r}^{\prime}(x)$. Since $u_{x}(\mathbf{r})>u_{x}(\mathbf{r}^{\prime})+l_{x}\geq l_{x}$, we must have $v_{x}(\mathbf{r})\geq 0$. Then there is a representation $M\in\operatorname{rep}_{Q^{\operatorname{sp}}}(\mathbf{e})$ such that map $t_{x_{t}}(M)$ is injective. We conclude as in Lemma 4.2 that $M\notin C_{\mathbf{r}^{\prime}}$, a contradiction.
∎
Remark 4.5.
We call the vectors $\mathbf{r}$ as in (4.4) rank sequences. In the case that $A=\Bbbk Q/\operatorname{rad}^{2}(\Bbbk Q)$, a description of the irreducible components of each representation variety for $A$ is given in [BCHZ15, Prop. 3.9]. Comparing our notation with that of [BCHZ15], the modules in an irreducible component $C_{\mathbf{r}}$ have generic top $\bigoplus_{x\in Q_{0}}S_{x}^{\mathbf{s}(x)}$. Their description is in terms of representation theoretic data. Our description is complementary, given purely in terms of combinatorial data from the dimension vector. Thus, it is straightforward to write a script enumerating the rank sequences of the irreducible components of $\operatorname{rep}_{A}(\mathbf{d})$ using only the adjacency matrix of $Q$ and dimension vector $\mathbf{d}$ as input.
We illustrate this in the following example.
∎
Example 4.6.
We compare the two approaches mentioned in the previous remark by computing irreducible components for the following module quiver from [BCHZ15, Example 3.11] using our methods. Let $A=\Bbbk Q/\operatorname{rad}^{2}(\Bbbk Q)$, where $Q$ is the quiver
$$\xymatrix@C+0.5pc@R+0.5pc{1\ar@/^{/}[dr]\ar@/_{/}[rr]\ar@/^{/}[rr]&&2\ar@/^{/}%
[dl]\ar[dl]\ar@/_{/}[dl]\\
&3\ar@/^{/}[ul]\ar@(dl,dr)&}$$
Consider the dimension vector $\mathbf{d}=(4,3,3)$. By Theorem 4.3, the representation variety of $\operatorname{rep}_{A}(\mathbf{d})$ has 19 irreducible components $C_{\mathbf{r}}$, where the rank sequences $\mathbf{r}$ are the following:
$$(0,0,3),\,(0,1,2),\,(0,1,3),\,(0,2,2),\,(0,2,3),\,(0,3,2),\,(0,3,3),\,(1,0,2),%
\,(1,1,1),\,(1,1,2),$$
$$(1,2,1),\,(1,2,2),\,(1,3,1),\,(1,3,2),\,(2,0,1),\,(2,1,1),\,(2,2,1),\,(2,3,1),%
\,(3,2,0).$$
To illustrate the ease of our method with a larger dimension vector, computer calculation shows that for the dimension vector $\mathbf{d}=(123,123,123)$, the variety $\operatorname{rep}_{A}(\mathbf{d})$ has 380473 irreducible components. The rank sequences can easily be output as part of this calculation, but we omit them here.
∎
4.2. Generic decomposition
We recall here the geometric version of the Krull-Schmidt decomposition.
We say that a subset $C\subseteq\operatorname{rep}_{A}(\mathbf{d})$ is indecomposable if $C$ has a nonempty open subset of indecomposable representations.
Given a collection of subvarieties $\{C_{i}\subseteq\operatorname{rep}_{A}(\mathbf{d}_{i})\}_{i=1}^{k}$, let $\mathbf{d}=\sum_{i}\mathbf{d}_{i}$, so we have the subvariety $\prod_{i}C_{i}\subseteq\operatorname{rep}_{A}(\mathbf{d})$ embedded diagonally. We define their direct sum $\overline{C_{1}\oplus\ldots\oplus C_{k}}$ to be the closure of $GL(\mathbf{d})\cdot\prod_{i}C_{i}$.
It was shown by de la Peña in [dlP91] and Crawley-Boevey and Schröer in [CBS02, Theorem 1.1] that any irreducible component $C\subseteq\operatorname{rep}_{A}(\mathbf{d})$ satisfies a Krull-Schmidt type decomposition
(4.7)
$$C=\overline{C_{1}\oplus\ldots\oplus C_{k}}$$
for some indecomposable irreducible components $C_{i}\subseteq\operatorname{rep}_{A}(\mathbf{d}_{i})$ with $\sum\mathbf{d}_{i}=\mathbf{d}$. Moreover, $C_{1},\ldots,C_{k}$ are uniquely determined by this property, up to order. The expression (4.7) is referred to as the generic decomposition of $C$, since it means $C$ has a dense subset in which each representation $M$ decomposes as $M=M_{1}\oplus\cdots\oplus M_{k}$ with each $M_{i}\in C_{i}$. This generalizes the canonical decomposition of a dimension vector $\mathbf{d}$ of a quiver without relations introduced by Kac [Kac80, Kac82] and studied extensively by Schofield, Derksen, and Weyman [Sch92, DW02, DW11b].
Proof of Theorem 1.2(b).
Let $C$ be an irreducible component of $\operatorname{rep}_{A}(\mathbf{d})$ with $r_{x}(C)=r$, and consider the corresponding irreducible component $C^{x}$ of $\operatorname{rep}_{A^{x}}(\mathbf{d}_{r}^{x})$ given by Proposition 3.8. Let $C^{x}=\overline{C^{x}_{1}\oplus\cdots\oplus C^{x}_{k}}$ be the generic decomposition of $C^{x}$, where $C^{x}_{i}$ is an irreducible component of $\operatorname{rep}_{A^{x}}(\mathbf{e}_{i})$, for some dimension vector $\mathbf{e}_{i}$. Since $r_{x_{h}}(C^{x})=r$, the map $h_{x_{h}}(M)$ is surjective for an open set of representations $M$ in $C^{x}$. Hence, the dimension vector $\underline{\dim}S_{x_{h}}$ of the simple representation $S_{x_{h}}$ cannot appear among the $\mathbf{e}_{i}$. In particular, since each $C_{i}^{x}$ is indecomposable we must have $r_{x_{h}}(C_{i}^{x})=\mathbf{e}_{i}(x_{h})$, for all $i=1,\dots,k$.
Put $C_{i}=GL(\mathbf{d})\cdot C_{i}^{x}$, which is irreducible and closed in $\operatorname{rep}_{A}(\mathbf{d}_{i})$ by Proposition 3.8, where $\mathbf{d}_{i}$ is given by $\mathbf{d}_{i}(y)=\mathbf{e}_{i}(y)$ for $y\neq x$, and $\mathbf{d}_{i}(x)=\mathbf{e}_{i}(x_{h})+\mathbf{e}_{i}(x_{t})$. By Proposition 3.3 and Corollary 3.4, each $C_{i}$ is indecomposable, and decompositions of generic representations in $C^{x}$ carry over to $C$.
We are left to show that $C_{i}$ is an irreducible component of $\operatorname{rep}_{A}(\mathbf{d}_{i})$ for all $i=1,\dots,k$. If $\mathbf{e}_{i}=\underline{\dim}S_{x_{t}}$, then $C_{i}=\operatorname{rep}_{A}(\mathbf{d}_{i})$ is a point. Otherwise, any indecomposable representation $M\in C_{i}^{x}$ satisfies the assumption in Lemma 4.2, thus finishing the proof.
∎
In [BCHZ15, Theorem 5.6], they determine the generic decomposition of each irreducible component for a radical square zero algebra. Their result can be described in our notation of the previous subsection as follows.
Given an irreducible component $C_{\mathbf{r}}$, we obtain a dimension vector $\mathbf{e}$ for the quiver $Q^{\operatorname{sp}}$ as in the proof of Theorem 4.3. Take the canonical decomposition $\mathbf{e}=\mathbf{e}_{1}\oplus\cdots\oplus\mathbf{e}_{k}$ over $Q^{\operatorname{sp}}$.
Each $\mathbf{e}_{i}$ determines the dimension vector $\mathbf{d}_{i}$ and rank sequence $\mathbf{r}_{i}$ for the original $Q$ such that $C_{\mathbf{r}_{i}}$ is an indecomposable irreducible component of $\operatorname{rep}_{A}(\mathbf{d}_{i})$. Namely, the vector $\mathbf{d}_{i}$ is given by $\mathbf{d}_{i}(x)=\mathbf{e}_{i}(x_{h})+\mathbf{e}_{i}(x_{t})$ for all $x\in Q_{0}$. The rank sequence $\mathbf{r}_{i}$ is given by forgetting the integers assigned to the sources: that is, $\mathbf{r}_{i}(x)=\mathbf{e}_{i}(x_{h})$ for all $x\in Q_{0}$. Then $C_{\mathbf{r}}=\overline{C_{\mathbf{r}_{1}}\oplus\cdots\oplus C_{\mathbf{r}_{k%
}}}$ is the generic decomposition.
We recall that an algebra has the dense orbit property in the sense of [CKW15], if each irreducible component of each of its representation varieties has a dense orbit. Since it is enough to check this property on indecomposable irreducible components, the following is an easy consequence of the considerations above.
Corollary 4.8.
The algebra $A^{x}$ has the dense orbit property if and only if $A$ has the dense orbit property.
It is immediate from this corollary that a radical square zero algebra has the dense orbit property if and only if it is already representation finite, which happens precisely when $Q^{\rm sp}$ is of Dynkin type (not necessarily connected).
This is [BCHZ15, Theorem 7.2].
Our result is more flexible than the radical square zero situation in that we can get the canonical decomposition for any irreducible component of an algebra where splitting nodes ends in a component where the generic decomposition is known.
We illustrate this in the following example where splitting nodes yields a gentle algebra.
Example 4.9.
Consider the algebra given by quiver
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}%
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with $I\subset\Bbbk Q$ generated by relations such that vertices 1 and 3 are nodes, and additionally all 2-cycles are zero.
Splitting both of the nodes 1 and 3 yields the gentle algebra of [Car15, Example 1]. Irreducible components for representation varieties of this gentle algebra can be parametrized by maximal rank sequences. Carroll gives a combinatorial method for determining the generic decomposition into string and band modules for each such irreducible component, and thus our Theorem 1.2(b) gives the generic decomposition for the corresponding irreducible component of the algebra in this example (which is not gentle).
∎
4.3. Singularities
We now prove the singularities statement of the main theorem. Assume $\operatorname{char}\Bbbk=0$ throughout this subsection.
We continue with a fixed $x\in Q_{0}$ which is a node of $A$, and $A^{x}$ is the algebra obtained by splitting the node $x$ as in Section 2.2. Theorem 1.2(c) is a special case of the following more general theorem.
Theorem 4.10.
Let $C$ be $GL(\mathbf{d}^{x}_{r})$-stable irreducible closed subvariety of $\operatorname{rep}_{A^{x}}(\mathbf{d}^{x}_{r})$, for some $0\leq r\leq\mathbf{d}(x)$. If $C$ is normal (resp. has rational singularities), then the same is true for the variety $GL(\mathbf{d})\cdot C\subseteq\operatorname{rep}_{A}(\mathbf{d})$.
Proof.
We apply Kempf’s results [Kem76] on collapsing of vector bundles to our setup, following his notation as closely as possible. Recall that $GL(\mathbf{d})$ acts on the affine space $\operatorname{rep}_{Q}(\mathbf{d})$, in which $\operatorname{rep}_{A^{x}}(\mathbf{d}^{x}_{r})$ is a closed $P^{x}_{r}(\mathbf{d})$-stable subvariety, with the unipotent radical of $P^{x}_{r}(\mathbf{d})$ acting trivially. In the language of [Kem76, Section 2], $P^{x}_{r}(\mathbf{d})$ acts on $\operatorname{rep}_{A^{x}}(\mathbf{d}^{x}_{r})$ completely reducibly.
Hence, $P^{x}_{r}(\mathbf{d})$ acts on $C$ completely reducibly as well. By Proposition 3.7, we have the birational “collapsing map”
$$\Psi_{C}\colon GL(\mathbf{d})\times_{P^{x}_{r}(\mathbf{d})}C\to GL(\mathbf{d})%
\cdot C.$$
The statement on normality now follows from Proposition 1 of [Kem76], and the statement on rational singularities follows from Theorem 3 of loc. cit.
∎
Example 4.11.
Consider again the algebra $A$ of Example 4.1.
We noted there that any representation variety for $A^{x}$ is the product of an affine space with a union of orbit closures in a type $A$ quiver representation variety.
Each of these orbit closures, thus each irreducible component of any $\operatorname{rep}_{A^{x}}(\mathbf{d}^{x}_{r})$, is known to have rational singularities [ADFK81].
Therefore every irreducible component of any $\operatorname{rep}_{A}(\mathbf{d})$ for this algebra has rational singularities by Theorem 1.2(a).
∎
When splitting nodes of an algebra $A$ results in an algebra whose orbit closures are known to be normal or have rational singularities (e.g. if $A$ is of Dynkin type $\mathbb{A}$ [LM98, BZ01] or $\mathbb{D}$ [BZ02]), then we can conclude the same for orbit closures of $A$, as illustrated in the following example.
Example 4.12.
Consider the following algebra $A=\Bbbk Q/I$ obtained by deleting vertex 6 from the algebra in Example 4.1, so again $I$ is generated by relations declaring that $x$ is a node, along with the relation $cba=0$.
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Orbit closures of $A^{x}$ are orbit closures for a type $\mathbb{D}$ quiver, and thus have rational singularities by [BZ02].
Therefore, combining Corollary 3.9 with Theorem 4.10 shows that all orbit closures for $A$ have rational singularities. Note that by Corollary 3.4 the algebra $A$ is of finite representation type.
∎
5. Moduli spaces
In this section we apply the results above to moduli spaces of semistable representations. We give only a minimal recollection of the background here, referring the reader to A. D. King’s original paper [Kin94] or [Rei08, DW17] for more detailed treatment.
The idea of King was to apply the general machinery of Geometric Invariant Theory (GIT) [MFK94, New09] to study representations of finitely generated algebras. The tools of GIT are very useful for understanding closed orbits of the action of a reductive group on a variety. However, in the situation of $GL(\mathbf{d})$ acting on $\operatorname{rep}_{A}(\mathbf{d})$, the closed orbits correspond to just the semisimple representations, so there is only one such representation per $\mathbf{d}$ when $A$ is finite dimensional.
It turns out that there are many subcategories of the category of representations of $A$ with richer collections of semisimple objects. From the representation theory perspective, each choice of weight $\theta\in\mathbb{Z}Q_{0}$ determines an abelian subcategory of $\theta$-semistable representations.
The simple objects of this category are called $\theta$-stable representations. The choice of $\theta$ can be arbitrary in our results below.
More precisely, for each $\mathbf{d}$ satisfying $\theta\cdot\mathbf{d}=0$, the collection of $\theta$-semistable points of $\operatorname{rep}_{A}(\mathbf{d})$ is defined by
$$\operatorname{rep}_{A}(\mathbf{d})^{ss}_{\theta}:=\left\{M\in\operatorname{rep%
}_{A}(\mathbf{d})\mid\forall N\leq M,\ \theta\cdot\underline{\dim}N\leq 0%
\right\}.$$
This is an open subvariety $\operatorname{rep}_{A}(\mathbf{d})$ (possibly empty!).
There is a corresponding projective variety $\mathcal{M}(\mathbf{d})^{ss}_{\theta}$ known as the moduli space of $\theta$-semistable representations of $A$ of dimension vector $\mathbf{d}$, and morphism of varieties
$$\pi\colon\operatorname{rep}_{A}(\mathbf{d})^{ss}_{\theta}\twoheadrightarrow%
\mathcal{M}(\mathbf{d})^{ss}_{\theta}$$
which is a quotient map in a sense made precise by GIT.
The $\theta$-stable points (those $\theta$-semistable $M$ such that $\theta\cdot N<0$ for all proper, nonzero $N<M$) form an open subvariety $\operatorname{rep}_{A}(\mathbf{d})^{s}_{\theta}\subseteq\operatorname{rep}_{A}%
(\mathbf{d})^{ss}_{\theta}$ on which $GL(\mathbf{d})$ acts freely; $\pi$ is an honest quotient map when restricted to this subvariety (again possibly empty).
We extend the notations above to subsets $C\subseteq\operatorname{rep}_{A}(\mathbf{d})$, writing $C_{\theta}^{ss}$ for the set of $\theta$-semistable points in $C$, and $\mathcal{M}(C)^{ss}_{\theta}$ for the image of $\pi(C_{\theta}^{ss})$.
Since the $\theta$-stable representations are the simple objects in the abelian category of $\theta$-semistable representations, every $\theta$-semistable representation $M$ has a well-defined set of $\theta$-stable composition factors from the Jordan-Hölder theorem, and associated graded representation $\operatorname{gr}_{\theta}(M)$.
The following result shows that the semistability of representations for an algebra $A$ with a node $x$ is particularly simply-behaved around $x$ and carries over to the semi-stability of the algebra $A^{x}$.
Proposition 5.1.
Assume that $A=\Bbbk Q/I$ with $x\in Q_{0}$ a node, and let $\theta\in\mathbb{Z}Q_{0}$ be a weight. Let $M$ is a $\theta$-semistable representation of $A$. Then one of the following occurs:
(a)
If $\theta(x)<0$, then $h_{x}(M)$ is surjective and $t_{x}(M)=0$;
(b)
If $\theta(x)>0$, then $h_{x}(M)=0$ and $t_{x}(M)$ is injective;
(c)
If $\theta(x)=0$, then the $\theta$-semistability of $M$ is equivalent to the $\theta$-semistability of $M^{\prime}$, where $M^{\prime}$ is obtained from $M$ by putting $M^{\prime}_{x}=0$ with $h_{x}(M^{\prime})=t_{x}(M^{\prime})=0$ and leaving the rest of the maps of $M$ unchanged.
Proof.
Clearly, we can assume $M_{x}\neq 0$. Consider first the case when $M$ is $\theta$-stable. In particular, it must be a Schur representation. By Corollary 3.6, either $h_{x}(M)$ or $t_{x}(M)$ is zero. Assume that the latter holds (the former is analogous). Then $M_{x}\neq 0$ implies that $S_{x}$ is a subrepresentation of $M_{x}$. Since $M$ is $\theta$-stable, either $S_{x}=M$ in which case $\theta\cdot\underline{\dim}S_{x}=\theta(x)=0$, or $\theta(x)<0$ in which case $h_{x}(M)$ is onto.
Now let $M$ be $\theta$-semistable. Let us prove part (a) (part (b) is analogous). Then $\theta(x)<0$ implies that for all $\theta$-stable composition factors $N$ of $M$ the map $h_{x}(N)$ is onto. Hence, $h_{x}(\operatorname{gr}_{\theta}(M))$ is onto, and then $h_{x}(M)$ is onto as well.
We are left with part (c), in which case the simple $S_{x}$ is $\theta$-stable. Then (c) follows from the fact that the set of $\theta$-semistable representations forms an abelian category which is closed under extensions, and that $x$ is a node forces every copy of $S_{x}$ to lie in the top or socle of any representation.
∎
The process of passing from $M$ to $\operatorname{gr}_{\theta}(M)$ can be carried out in a geometric setting, known as a $\theta$-stable decomposition. We follow the exposition of [CK18, §2.4] which is a slight generalization of [Chi13, Section 3C], based on the original idea of [DW11a] in the case that $A=\Bbbk Q$ for an acyclic quiver $Q$. From here on, we assume that $\operatorname{char}\Bbbk=0$.
Definition 5.2.
Let $C$ be a $GL(\mathbf{d})$-invariant, irreducible, closed subvariety of $\operatorname{rep}_{A}(\mathbf{d})$, and assume $C$ has a nonempty subset of $\theta$-semistable points. Consider a collection $\{C_{i}\subseteq\operatorname{rep}_{A}(\mathbf{d}_{i})\}_{i=1}^{k}$ of irreducible components such that each has a nonempty subset of $\theta$-stable points, $C_{i}\neq C_{j}$ for $i\neq j$, and also consider some multiplicities $m_{i}\in\mathbb{Z}_{>0}$, for $i=1,\dots,k$.
We say that $\{(C_{i},m_{i})\}_{i=1}^{k}$ is a $\theta$-stable decomposition of $C$ if, for a general representation $M\in C^{ss}_{\theta}$, its corresponding $\operatorname{gr}_{\theta}(M)$ is in $C_{1}^{\oplus m_{1}}\oplus\cdots\oplus C_{k}^{\oplus m_{k}}$, and write
$$C=m_{1}C_{1}\pp\ldots\pp m_{k}C_{k}.\qed$$
It is shown in [CK18, Prop. 3] that any $GL(\mathbf{d})$-stable, irreducible, closed subvariety of $\operatorname{rep}_{A}(\mathbf{d})$ which has as least one $\theta$-semistable point admits a $\theta$-stable decomposition.
The following result makes precise how the geometry of a moduli space of $\theta$-semistable representations is constrained (and in some cases completely determined) by the geometry of moduli spaces arising from its $\theta$-stable decomposition.
Here, the $m^{th}$ symmetric power $S^{m}(X)$ of a variety $X$ is the quotient of $\prod_{i=1}^{m}X$ by the action of the symmetric group on $m$ elements which permutes the coordinates.
Theorem 5.3.
[CK18] Let $A$ be a finite-dimensional algebra and let $C\subseteq\operatorname{rep}_{A}(\mathbf{d})^{ss}_{\theta}$ be an irreducible component such that $C^{ss}_{\theta}\neq\varnothing$.
Let $C=m_{1}C_{1}\pp\ldots\pp m_{k}C_{k}$ be a $\theta$-stable decomposition of $C$ where $C_{i}\subseteq\operatorname{rep}_{A}(\mathbf{d}_{i})$, $1\leq i\leq k$, are pairwise distinct $\theta$-stable irreducible components.
If $\mathcal{M}(C)^{ss}_{\theta}$ is an irreducible component of $\mathcal{M}(\mathbf{d})^{ss}_{\theta}$, then there is a natural morphism
$$\Psi\colon S^{m_{1}}(\mathcal{M}(C_{1})^{ss}_{\theta})\times\ldots\times S^{m_%
{r}}(\mathcal{M}(C_{k})^{ss}_{\theta})\to\mathcal{M}(C)^{ss}_{\theta}$$
which is finite, and birational. In particular, if $\mathcal{M}(C)^{ss}_{\theta}$ is normal then $\Psi$ is an isomorphism.
Every irreducible component of $\mathcal{M}(\mathbf{d})^{ss}_{\theta}$ is of the form $\mathcal{M}(C)^{ss}_{\theta}$ where $C$ is an irreducible component of $\operatorname{rep}_{A}(\mathbf{d})$, so this covers all of them.
Here we have combined the three parts of the main theorem of [CK18] for simplicity; this is enough for our application.
We also note that the map of this theorem is quite simplistic on the set-theoretical level, sending a list of representations to their direct sum. The entire content is that this induces a morphism of varieties with nice properties.
Using Corollary 1.4, the map $\Psi$ in Theorem 5.3 is always an isomorphism for radical square zero algebras. However, in this case the map is perhaps less interesting due to Proposition 5.1. Nevertheless, if we add to the quiver of a radical square zero algebra some additional arrows and vertices (without adding additional relations) then the representation varieties of the obtained algebra are still normal, hence the map $\Psi$ in Theorem 5.3 is again an isomorphism. On the other hand, such algebras have richer moduli spaces. We illustrate these considerations with the following example.
Example 5.4.
Consider a quiver of the form
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and let $Q^{\prime}\subseteq Q$ be a subquiver containing $x$. Let $I=\operatorname{rad}^{2}(\Bbbk Q^{\prime})$, and set $A=\Bbbk Q/I$ and $A^{\prime}=\Bbbk Q^{\prime}/I$.
Now let $\mathbf{d}$ be a dimension vector for $Q$, let $\mathbf{d}^{\prime}$ be its restriction to $Q^{\prime}$, and let $C\subseteq\operatorname{rep}_{A}(\mathbf{d})$ an irreducible component. Then we have that $\operatorname{rep}_{A}(\mathbf{d})$ is the product of $\operatorname{rep}_{A^{\prime}}(\mathbf{d}^{\prime})$ with an affine space, so $C$ is the product of some irreducible component $C^{\prime}\subseteq\operatorname{rep}_{A^{\prime}}(\mathbf{d}^{\prime})$ with that same affine space.
By Corollary 1.4, we know $C^{\prime}$ is a normal variety, so $C$ is as well.
Thus $\mathcal{M}(C)^{ss}_{\theta}$ is normal by applying [DK02, Prop. 2.3.11] to the definition of $\mathcal{M}(C)^{ss}_{\theta}$, and we can apply Theorem 5.3 to decompose the moduli space $\mathcal{M}(C)^{ss}_{\theta}$. We note that there is not necessarily any relation between $\mathcal{M}(C)^{ss}_{\theta}$ and $\mathcal{M}(C^{\prime})^{ss}_{\theta}$; although $C$ and $C^{\prime}$ have essentially the same singularities, their $GL(\mathbf{d})$-orbit structures can be drastically different.
∎
References
[ADFK81]
S. Abeasis, A. Del Fra, and H. Kraft.
The geometry of representations of $A_{m}$.
Math. Ann., 256(3):401–418, 1981.
[ASS06]
Ibrahim Assem, Daniel Simson, and Andrzej Skowroński.
Elements of the representation theory of associative algebras.
Vol. 1, volume 65 of London Mathematical Society Student Texts.
Cambridge University Press, Cambridge, 2006.
Techniques of representation theory.
[BCHZ15]
Frauke M. Bleher, Ted Chinburg, and Birge Huisgen-Zimmermann.
The geometry of finite dimensional algebras with vanishing radical
square.
J. Algebra, 425:146–178, 2015.
[BHZT09]
E. Babson, B. Huisgen-Zimmermann, and R. Thomas.
Generic representation theory of quivers with relations.
J. Algebra, 322(6):1877–1918, 2009.
[Bob12]
Grzegorz Bobiński.
Normality of maximal orbit closures for Euclidean quivers.
Canad. J. Math., 64(6):1222–1247, 2012.
[Bon91]
Klaus Bongartz.
A geometric version of the Morita equivalence.
J. Algebra, 139(1):159–171, 1991.
[Bon98]
Klaus Bongartz.
Some geometric aspects of representation theory.
In Algebras and modules, I (Trondheim, 1996), volume 23 of
CMS Conf. Proc., pages 1–27. Amer. Math. Soc., Providence, RI, 1998.
[Bri12]
Michel Brion.
Representations of quivers.
In Geometric methods in representation theory. I, volume 24
of Sémin. Congr., pages 103–144. Soc. Math. France, Paris, 2012.
[BS01]
M. Barot and Jan Schröer.
Module varieties over canonical algebras.
J. Algebra, 246(1):175–192, 2001.
[BZ01]
Grzegorz Bobiński and Grzegorz Zwara.
Normality of orbit closures for Dynkin quivers of type $\mathbb{A}_{n}$.
Manuscripta Math., 105(1):103–109, 2001.
[BZ02]
Grzegorz Bobiński and Grzegorz Zwara.
Schubert varieties and representations of Dynkin quivers.
Colloq. Math., 94(2):285–309, 2002.
[Car15]
Andrew T. Carroll.
Generic modules for gentle algebras.
J. Algebra, 437:177–201, 2015.
[CB02]
William Crawley-Boevey.
Decomposition of Marsden-Weinstein reductions for representations
of quivers.
Compositio Math., 130(2):225–239, 2002.
[CBS02]
W. Crawley-Boevey and J. Schröer.
Irreducible components of varieties of modules.
J. Reine Angew. Math., 553:201–220, 2002.
[CC15]
A. T. Carroll and C. Chindris.
Moduli spaces of modules of Schur-tame algebras.
Algebr. Represent. Theory, 18(4):961–976, 2015.
[CCKW17]
Andrew T. Carroll, Calin Chindris, Ryan Kinser, and Jerzy Weyman.
Moduli spaces of representations of special biserial algebras.
To appear in International Mathematics Research Notices,
rny028, https://doi.org/10.1093/imrn/rny028 (2018), 2017.
[Chi13]
C. Chindris.
On the invariant theory for tame tilted algebras.
Algebra Number Theory, 7(1):193–214, 2013.
[CK18]
Calin Chindris and Ryan Kinser.
Decomposing moduli of representations of finite-dimensional algebras.
Mathematische Annalen, 372(1):555–580, Oct 2018.
[CKW15]
Calin Chindris, Ryan Kinser, and Jerzy Weyman.
Module varieties and representation type of finite-dimensional
algebras.
Int. Math. Res. Not. IMRN, (3):631–650, 2015.
[DK02]
Harm Derksen and Gregor Kemper.
Computational invariant theory.
Invariant Theory and Algebraic Transformation Groups, I.
Springer-Verlag, Berlin, 2002.
Encyclopaedia of Mathematical Sciences, 130.
[dlP91]
J. A. de la Peña.
On the dimension of the module-varieties of tame and wild algebras.
Comm. Algebra, 19(6):1795–1807, 1991.
[DW02]
Harm Derksen and Jerzy Weyman.
On the canonical decomposition of quiver representations.
Compositio Math., 133(3):245–265, 2002.
[DW11a]
H. Derksen and J. Weyman.
The combinatorics of quiver representations.
Ann. Inst. Fourier (Grenoble), 61(3):1061–1131, 2011.
[DW11b]
Harm Derksen and Jerzy Weyman.
The combinatorics of quiver representations.
Ann. Inst. Fourier (Grenoble), 61(3):1061–1131, 2011.
[DW17]
Harm Derksen and Jerzy Weyman.
An introduction to quiver representations, volume 184 of Graduate Studies in Mathematics.
American Mathematical Society, Providence, RI, 2017.
[GHZ18]
Kenneth R. Goodearl and Birge Huisgen-Zimmermann.
Closures in varieties of representations and irreducible components.
Algebra Number Theory, 12(2):379–410, 2018.
[GR97]
P. Gabriel and A. V. Roiter.
Representations of finite-dimensional algebras.
Springer-Verlag, Berlin, 1997.
Translated from the Russian, With a chapter by B. Keller, Reprint of
the 1992 English translation.
[GS03]
Christof Geiss and Jan Schröer.
Varieties of modules over tubular algebras.
Colloq. Math., 95(2):163–183, 2003.
[HZ14]
B. Huisgen-Zimmermann.
Fine and coarse moduli spaces in the representation theory of finite
dimensional algebras.
In Expository lectures on representation theory, volume 607 of
Contemp. Math., pages 1–34. Amer. Math. Soc., Providence, RI, 2014.
[HZ16]
B. Huisgen-Zimmermann.
Irreducible components of varieties of representations: the local
case.
J. Algebra, 464:198–225, 2016.
[HZG12]
B. Huisgen-Zimmermann and K. R. Goodearl.
Irreducible components of module varieties: projective equations and
rationality.
In New trends in noncommutative algebra, volume 562 of Contemp. Math., pages 141–167. Amer. Math. Soc., Providence, RI, 2012.
[HZS17]
B. Huisgen-Zimmermann and I. Shipman.
Irreducible components of varieties of representations: the acyclic
case.
Math. Z., 287(3-4):1083–1107, 2017.
[Kac80]
Victor G. Kac.
Infinite root systems, representations of graphs and invariant
theory.
Invent. Math., 56(1):57–92, 1980.
[Kac82]
Victor G. Kac.
Infinite root systems, representations of graphs and invariant
theory. II.
J. Algebra, 78(1):141–162, 1982.
[Kem76]
George R. Kempf.
On the collapsing of homogeneous bundles.
Invent. Math., 37(3):229–239, 1976.
[Kin94]
A. D. King.
Moduli of representations of finite-dimensional algebras.
Quart. J. Math. Oxford Ser. (2), 45(180):515–530, 1994.
[Kin18]
Ryan Kinser.
K-polynomials of type a quiver orbit closures and lacing diagrams.
In Representations of Algebras, volume 705 of Contemp.
Math., pages 99–114. Amer. Math. Soc., Providence, RI, 2018.
[KR15]
Ryan Kinser and Jenna Rajchgot.
Type $A$ quiver loci and Schubert varieties.
Journal of Commutative Algebra, 7(2):265–301, 2015.
[KW14]
Ryan Kinser and Jerzy Weyman.
Irreducible components of some module varieties via the springer
resolution.
Oberwolfach Reports, 11(1):475–477, 2014.
[LM98]
V. Lakshmibai and Peter Magyar.
Degeneracy schemes, quiver schemes, and Schubert varieties.
Internat. Math. Res. Notices, (12):627–640, 1998.
[Lőr15]
András C. Lőrincz.
Singularities of zero sets of semi-invariants for quivers.
arXiv:1509.04170, 2015.
To appear in J. Commut. Algebra.
[LW18]
András C. Lőrincz and Jerzy Weyman.
Free resolutions of orbit closures of Dynkin quivers.
arXiv:1801.00193, 2018.
To appear in Trans. Amer. Math. Soc.
[LZ14]
Nguyen Quang Loc and Grzegorz Zwara.
Modules and quiver representations whose orbit closures are
hypersurfaces.
Colloq. Math., 134(1):57–74, 2014.
[MFK94]
D. Mumford, J. Fogarty, and F. Kirwan.
Geometric invariant theory, volume 34 of Ergebnisse der
Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related
Areas (2)].
Springer-Verlag, Berlin, third edition, 1994.
[MV80]
Roberto Martínez-Villa.
Algebras stably equivalent to $l$-hereditary.
In Representation theory, II (Proc. Second Internat.
Conf., Carleton Univ., Ottawa, Ont., 1979), volume 832 of Lecture Notes in Math., pages 396–431. Springer, Berlin, 1980.
[New09]
P. E. Newstead.
Geometric invariant theory.
In Moduli spaces and vector bundles, volume 359 of London
Math. Soc. Lecture Note Ser., pages 99–127. Cambridge Univ. Press,
Cambridge, 2009.
[Rei08]
Markus Reineke.
Moduli of representations of quivers.
In Trends in representation theory of algebras and related
topics, EMS Ser. Congr. Rep., pages 589–637. Eur. Math. Soc., Zürich,
2008.
[Ric01]
Nicola J. Richmond.
A stratification for varieties of modules.
Bull. London Math. Soc., 33(5):565–577, 2001.
[RRS11]
Ch. Riedtmann, M. Rutscho, and S. O. Smalø.
Irreducible components of module varieties: an example.
J. Algebra, 331:130–144, 2011.
[RZ13]
Christine Riedtmann and Grzegorz Zwara.
Orbit closures and rank schemes.
Comment. Math. Helv., 88(1):55–84, 2013.
[Sch92]
A. Schofield.
General representations of quivers.
Proc. London Math. Soc. (3), 65(1):46–64, 1992.
[Sch04]
Jan Schröer.
Varieties of pairs of nilpotent matrices annihilating each other.
Comment. Math. Helv., 79(2):396–426, 2004.
[Sch14]
Ralf Schiffler.
Quiver representations.
CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC.
Springer, Cham, 2014.
[Sut15]
Kavita Sutar.
Orbit closures of source-sink Dynkin quivers.
Int. Math. Res. Not. IMRN, (11):3423–3444, 2015.
[Tim11]
Dmitry A. Timashev.
Homogeneous spaces and equivariant embeddings, volume 138 of
Encyclopaedia of Mathematical Sciences.
Springer, Heidelberg, 2011.
Invariant Theory and Algebraic Transformation Groups, 8.
[Wey03]
Jerzy Weyman.
Cohomology of vector bundles and syzygies, volume 149 of Cambridge Tracts in Mathematics.
Cambridge University Press, Cambridge, 2003.
[Zwa99]
Grzegorz Zwara.
Degenerations for modules over representation-finite algebras.
Proc. Amer. Math. Soc., 127(5):1313–1322, 1999.
[Zwa11]
Grzegorz Zwara.
Singularities of orbit closures in module varieties.
In Representations of algebras and related topics, EMS Ser.
Congr. Rep., pages 661–725. Eur. Math. Soc., Zürich, 2011. |
GRB 081029: Understanding Multiple Afterglow Components
S. T. Holland
M. De Pasquale
J. Mao
T. Sakamoto
P. Schady
S. Covino
P. D’Avanzo
A. Antonelli
V. D’Elia
G. Chincarini
F. Fiore
S. B. Pandey
Abstract
We present an analysis of the unusual optical light curve of the
gamma-ray burst GRB 081029, which occurred at a redshift of $z=3.8479$. We combine $X$-ray and optical observations from the Swift $X$-Ray Telescope and the Swift UltraViolet
Optical Telescope with optical and infrared data obtained using the
REM and ROTSE telescopes to construct a detailed data set extending
from 86 s to $\sim$100 000 s after the BAT trigger. Our data also
cover a wide energy range, from 10 keV to 0.77 eV (1.24 Å to
16 000 Å). The $X$-ray afterglow shows a shallow initial decay
followed by a rapid decay starting at about 18 000 s. The optical
and infrared afterglow, however, shows an uncharacteristic rise at
about 5000 s that does not correspond to any feature in the $X$-ray
light curve. Our data are not consistent with synchrotron radiation
from a single-component jet interacting with an external medium. We
do, however, find that the observed light curve can be explained
using multi-component model for the jet.
Keywords:gamma-ray burst: individual: GRB 081029
: 98.70s.Rz
1 Introduction
There is growing evidence that the classical picture of a single
uniform jet cannot explain the spectral energy distributions and light
curves of some gamma-ray burst (GRB) afterglows. For example, the
unusually bright optical afterglow of the “naked-eye” burst
GRB 080319B was best explained using a two-component jet
(Racusin et al., 2008) while GRB 030329 (Berger et al., 2003) appears to require a
narrow, ultra-relativistic inner jet and a wide, mildly relativistic
outer jet to explain its light curves. This is in agreement with
results from magneto-hydrodynamic modelling that show complex structure
in GRB jets (e.g., Tchekhovskoy et al., 2010). GRB afterglows appear to be
more complex than originally thought.
An example of a GRB afterglow that appears to require a
multi-component jet is GRB 081029. This burst was detected by Swift/BAT at 01:43:56 UT on 2008 Oct 29 (Sakamoto et al., 2008).
ROTSE-IIIc identified the optical afterglow at 86 s (Rykoff, 2008), and
the REM telescope started observing the optical afterglow at 154 s
(Covino et al., 2008), so there is a well-sampled $R$-band light curve
starting less than 90 s after the BAT trigger. Due to an observing
constraing Swift was unable to slew to this burst as soon as
it was detected. XRT and UVOT observations began about 45 minute
after the BAT trigger and continued for approximately 10 days.
The VLT/UVES and Gemini-South measured a redshift of $z=3.8479$
(D’Elia et al., 2008; Cucchiara et al., 2008), which corresponds to a look back time of
11.9 Gyr. The Gemini-South GMOS spectrum shows evidence for a damped
Lyman-alpha system as well as several metal absorption features in the
host galaxy of GRB 081029.
2 Observations
2.1 BAT: Prompt Emission
The Swift/BAT discovered and observed GRB 081029. The burst
duration was $T_{90}=280\pm 50$ s, the peak flux was $(2.8\pm 1.3)\times 10^{-8}$ erg cm${}^{-2}$ s${}^{-1}$, and the spectrum was best fit
by a simple power law with a photon index of $\Gamma=1.5\pm 0.2$.
The BAT light curve was somewhat smoother and weaker than a typical
BAT-detected GRB. Figure 1 shows the BAT light cuvrve
for the prompe emission from GRB 081029.
2.2 XRT: X-Ray Light Curve and Spectrum
The Swift/XRT observed GRB 081029 from 41.4 minutes to
approximately 10 days after the BAT trigger. The $X$-ray light curve
(see Figure 2) is well fit by a broken power
law with indices ($f_{\nu}\propto t^{-\alpha}$) of $\alpha_{1}=0.56\pm 0.03$ until $18\,230\pm 346$ s, and $\alpha_{2}=2.56\pm 0.09$
after that. There is some evidence for flaring between approximately
2500 s and 5000 s. The time scales of these flares are consistent
with $\Delta t/t<1$. The $X$-ray data are not unusual, and are
consistent with the canonical $X$-ray light curve for GRB afterglows
described by (Nousek et al., 2006) and (Zhang et al., 2006).
The Swift/XRT spectrum can be fit by a single power law
($f_{\nu}\propto\nu^{-\beta}$) with an index of $\beta_{X}=0.98\pm 0.08$. There is no evidence for any evolution in the power law index
at $X$-ray energies. The estimated Galactic column density in the
direction of the burst is $N_{H}=2.8\times 10^{-20}$ cm${}^{-2}$, and
the absorption in the host is $N_{H}=4.9\times 10^{-21}$ cm${}^{-2}$.
2.3 Optical and Infrared Observations
The SwiftUVOT began observing the afterglow of GRB 081029 at
2689 s after the BAT trigger. The afterglow was detected in the UVOT
$v$, $b$, and white bands, consistent with the reported redshift of $z=3.8479$. Ground-based data was obtained using REM and ROTSE.
ROTSE began observations 86 s after the burst in the $R$ band. REM
began observing GRB 081029 156 s after the BAT trigger in the $R$,
$J$, and $H$ bands. The resulting light curves are complex, in stark
contrast to the simple $X$-ray light curve. The combined optical and
infrared observations are shown in Figure 2
along with the Swift/XRT light curve. The optical and
infrared data show a jump in the flux density of approximately a
factor of ten at approximately 5000 s. There is no corresponding
increase in the $X$-ray flux density at that time.
3 Discussion
The $X$-ray light curve is consistent with energy injection from
ongoing central engine activity until about 15 000 s followed by a
jet break at 18 230 s. However, this scenario cannot explain the
jump in the flux seen at optical and infrared wavelengths at about
5000 s. Therefore, we do not think that a change in the energy
injection is capable of explaining the light curves for this
afterglow. Similarly, the jump cannot be modelled by invoking the
passage of the synchrotron peak frequency through the optical regime,
or as the rise of the forward shock due to interaction with the
circumburst medium. We also examined the possibility that the jump is
due to density structure in the surrounding environment, but this is
unable to reproduce the speed or the magnitude of the increase in
luminosity.
In general we find that a one-component jet cannot explain the
observed light curves and spectral energy distribution of the $X$-ray,
optical, and infrared afterglows of GRB 081029. However, a
two-component jet model, similar to what is seen in some other GRB
afterglows, does provide a reasonable fit to the data Our
two-component jet model is shown in Figure 3, and the
parameters of each jet are listed in Table 1. The
half-opening angle of the jet is denoted by $\theta_{j}$, $\Gamma_{0}$ is
the Lorentz factor, $E_{K,\mathrm{iso}}$ is the isotropic equivalent
kinetic energy in the jet, $p$ is the electron index, $\epsilon_{e}$ and
$\epsilon_{B}$ are the fractions of the energy in electrons and magnetic
fields respectively, $n$ is the density of the circumburst medium, and
$z$ is the redshift.
The narrow, inner jet has a half-opening angle of $\theta_{j,n}=0.01$ rad and a Lorentz factor of 500. This component gives rise to
the $X$-ray flux and the pre-jump optical flux. The wider, outer jet
has $\theta_{j,w}=0.02$ rad and a Lorentz factor of 60. This
component dominates the afterglow after about 10 000 s. The total
electromagnetic energy in the afterglow is approximately equally
divided between the two jets.
4 Conclusions
GRB 081029 was a long–soft GRB with a redshift of $z=3.8479$. It
had a smooth gamma-ray light curve and did not appear to have any
unusual gamma-ray properties. Neither the gamma-ray nor the $X$-ray
properties of this burst showed any sign of strange behaviour. The
optical and infrared light curves, on the other hand, were not typical
of GRB afterglows. There is a brightening in the optical and infrared
light curves at about 5000 s that cannot be explained using a
single-component jet model. However, we find that a two-component jet
model fits the data reasonably well.
We conclude that the afterglow of GRB 081029 was probably powered by a
two-component jet with the energy split approximately equally between
a narrow ($\theta_{j,n}=0.01$ rad) inner jet and a wider
($\theta_{j,w}=0.02$ rad) outer jet. The inner jet has a Lorentz
factor of $\Gamma_{n}=500$ while the outer jet has $\Gamma_{w}=60$.
This result provides evidence that some (and perhaps all) GRB jets
have complex internal structure.
We acknowledge the use of public data from the Swift Data
Archive. This work is based in part on observations taken with the
ROTSE-IIIc telescope in Namibia, the REM telescope at la Silla
Observatory, and with ESO Telescopes at the Paranal Observatories.
References
Berger et al. (2003)
Berger, E., et al., 2003, Nature, 426, 154
Covino et al. (2008)
Covino, S., et al., 2008, GCNC 8441
Cucchiara et al. (2008)
Cucchiara, A., et al., 2008, GCNC 8448
D’Elia et al. (2008)
D’Elia, V., et al., 2008, GCNC 8438
Nousek et al. (2006)
Nousek, J. A., et al., 2006, ApJ, 642, 389
Racusin et al. (2008)
Racusin, J. L., et al., 2008, Nature, 455, 183
Rykoff (2008)
Rykoff, E. S., 2008, GCNC 8436
Sakamoto et al. (2008)
Sakamoto, T., et al., 2008, GCNC 8435
Tchekhovskoy et al. (2010)
Tchekhovskoy, A., et al., 2010, New Astronomy, 15, 749
Zhang et al. (2006)
Zhang, B., et al., 2006, ApJ, 642, 354 |
Sittin’On the Dock of the (WiFi) Bay:
On the Frame Aggregation under IEEE 802.11 DCF
Ricardo J. Rodríguez${}^{*}$, José Luis Salazar, and Julián Fernández-Navajas
${}^{*}$Corresponding author. This work was supported in part by the Aragonese Government under Programa de Proyectos Estratégicos de Grupos de Investigación (T21-20R and T31-20R) and by the Spanish Ministry of Economy, Industry and Competitiveness (TIN2015-64770-R).Ricardo J. Rodríguez is with Dpto. de Informática e Ingeniería de Sistemas, Universidad de Zaragoza, Calle María de Luna 1, 50018
Zaragoza, Spain. E-mail: rjrodriguez@unizar.esJosé Luis Salazar and Julián Fernández-Navajas are with Dpto. de Ingeniería Electrónica y Comunicaciones, Universidad de Zaragoza, Calle María de Luna 1, 50018
Zaragoza, Spain. E-mail: navajas@unizar.es, jsalazar@unizar.es
Abstract
It is well known that frame aggregation in Internet communications improves transmission efficiency. However, it also causes a delay that for some real-time communications is inappropriate, thus creating a trade-off between efficiency and delay. In this paper, we establish the conditions for frame aggregation under the IEEE 802.11 DCF protocol to be beneficial on average delay. To do so, we first describe the transmission time in IEEE 802.11 in a stochastic framework and then we calculate the optimal value of the frames that, when aggregated, saves transmission time in the long term. Our findings, discussed with numerical experimentation, show that frame aggregation reduces transmission congestion and transmission delays.
Index Terms:
Wireless communication, Multiplexing
I Introduction
WiFi protocols have evolved to achieve higher and higher transmission speeds, giving rise to different standards, such as 802.11a/b/g/n/ac. Older standards are superseded by newer standards and must coexist in the same implementation to ensure backward compatibility. Those with a lower rate provide greater coverage, while those with a higher rate force the terminals to be closer. When the terminals are far away, there is a lower transmission rate, so the aggregation of frames in WiFi has been seen as an improvement in efficiency (less bandwidth overhead), but it increases the risk of causing delays. For instance, this situation can be problematic in real-time services such as VoIP, video conferencing, or online gaming, where the transmission delay must not exceed certain limit values to satisfy the user experience. In this paper, we study how frame aggregation causes delays and how this deterioration can be quantified, especially for those standards (or lower rate situations) where it is more relevant to give a correct solution (e.g., 802.11b/g).
IEEE 802.11 DCF (Distributed Coordination Function) is the medium access control used by WiFi [1], which is based on Carrier Sense Multiple Access/Collision Avoidance and operates at layer 2 of the OSI model. CSMA/CA is a network arbitration protocol which regulates communication between multiple nodes communicating through a single channel.
CSMA/CA works as follows. When a node wants to transmit, it first checks if the channel is free. When the node detects that the medium is continuously free for a time defined by the DCF Interframe Space (DIFS), then it is allowed to transmit its data frame. The receiving node will send an acknowledgment (ACK) frame after a time, specified by SIFS (Short Interframe Space), when the sent frame is successfully received. Conversely, when the node detects that the medium is busy, it postpones its transmission until the end of the current transmission. After an additional DIFS interval, the node generates a random backoff period for an additional deferral time before transmitting. That is, the node generates an integer $i$ uniformly distributed in the interval $\{0,1,...,CW\}$, where $CW$ is an integer within the range of $aCWmin$ and $aCWmax$, i.e., $aCWmin\leq CW\leq aCWmax$. The $aCWmin$ and $aCWmax$ values are the minimum and maximum time for the content window and depend on the characteristics of the physical layer. Then, the node waits for the $i$ $SLOT$ intervals before transmitting. The $SLOT$ value also depends on the characteristics of the physical layer.
When another node transmits before the backoff period ends, the countdown stops, and the remaining time is used on the next transmission attempt. When two nodes have the same backoff period (or the remaining backoff) values, they will transmit at the same time and therefore their transmissions collide. The station detects the collision as it will not receive the ACK frame from the receiving node. When this happens, an exponential backoff algorithm is applied, i.e., $CW$ is doubled up to $aCWmax$ for the next transmission attempt.
In [2] a complete study of IEEE 802.11 DCF performance is presented considering throughput, fairness, and delay. As the authors indicate, CSMA/CA supports long-term fairness by achieving the best performance for a small number of nodes, although it suffers short-term unfairness when the number of nodes exceeds the optimal value, as nodes whose transmission collides will increase their content window and then they will be less likely to access the medium.
Note that CSMA/CA exhibits some randomness as a random integer is computed to calculate the backoff period and the data payload of each frame (although it is limited to a maximum of 1500 bytes, it is likely to be different in each frame). In this work, we want to show that under certain conditions, frame aggregation can reduce the average delay. In this regard, we first consider this randomness to better characterize the transmission time of a frame under IEEE 802.11, which allows us to obtain an analytical solution to our problem. Then, we calculate the optimal value of the frames that, when aggregated, saves transmission time in the long term. Finally, we present and discuss some numerical experiments.
This paper is organized as follows. Section II reviews related work. Section III details the characterization of frame transmission time by means of random variables. Then, Section IV establishes the formulae to calculate the optimal number of aggregated frames which, on average, saves transmission times. Finally, Section V concludes the paper.
II Related Work
Most of the published works study the performance of IEEE 802.11 under different conditions, such as [3, 4, 5], to name a few. Other works closer to ours are [6], where a packet aggregation scheme is proposed to maximize network performance, and [7], where the frame delay is studied under stochastic terms.
Other works have studied the savings achievable when using 802.11 aggregation for packets from one application (or different applications) to the same destination [8]. In [9], a central-controlled aggregation mechanism is proposed that limits the maximum aggregation size in applications with real-time constraints that share the medium with background traffic. This mechanism enables the provision of a low latency service for applications with real-time constraints and maximum throughput for the other applications. As an alternative, the authors also consider permanently setting a low value for the maximum A-MPDU size. An adaptive machine learning-based approach to adjusting the maximum length of A-MSDU per user is presented in [10]. In this context, the authors of [11] propose an algorithm for dynamically adjusting the maximum size of aggregated frames in 802.11 WLANs, which allows a network administrator to find an optimal balance between performance and latency in these networks.
As far as we know, we are the first to mathematically study when delaying frame transmissions saves transmission time, and how much time it saves, across all IEEE 802.11 protocols.
III Stochastic Transmission Time
In this section, we present a stochastic interpretation of the transmission time that we then use in our calculations.
Figure 1 shows the timing diagram involved in a successful single frame transmission under IEEE 802.11 (adapted from [2]). Some of these times are overhead, such as DIFS and SIFS intervals, ACK transmission, and the Physical Layer Convergence Protocol (PLCP) preamble and header that precede each frame. These values depend on the characteristics of the physical layer. The MAC header and CRC transmission times are also dependent on the physical layer bit rate.
The time boxes depicted with a dashed line in Figure 1 indicate that there is some randomness and therefore, we cannot define them as a concrete value: $t_{\text{\em backoff}}$ refers to the backoff time, which is equal to a discrete random value uniformly distributed in the interval $\{0,\text{CW}\}$, multiplied by SLOT units of time; and $t_{data}$ refers to the data payload transmission, which is equal to a discrete random value that expresses the size of the data (in bytes) multiplied by the time required to transmit a byte, considering the underlying physical characteristics.
Let us first denote $t_{\text{\em backoff}}$ in terms of random variables. Let $X\sim\mathcal{U}_{\{0,1,...,\text{CW}\}}$ be a discrete random variable following a discrete uniform distribution $\mathcal{U}$ in the interval $\{0,1,...,CW\}$. Therefore, $Y=\text{SLOT}\cdot X\sim\mathcal{U}_{\{0,\text{SLOT},\dots,\text{SLOT}\cdot\text{CW}\}}$. Let the time $t^{i}_{\text{\em backoff}}$ for a frame $i$ be denoted as $Y_{i}$. Although this model does not closely match reality (since when the medium is congested and collisions are generated, the backoff times are redefined to $\{0,1,\ldots,2\text{CW}\}$), our final goal is to compare the times of the standard model with those of our proposal (see Figure 2(b)), where there is no probability of collision since we have grouped all the sources in a single buffer, which cannot collide. Therefore, to assume that there are no collisions in a system without frame aggregation is to give it an additional advantage when comparing its performance with our proposal of frame aggregation.
Now, let us denote $t_{data}$ similarly to $t_{\text{\em backoff}}$. Let $P$ be a discrete random variable that describes the size distribution (in bytes) of the data payload of the frames. Let $Z=\frac{8}{br}P$ be the dependent variable that represents the time required to transmit a data payload, where $br$ is the physical layer bit rate, in units of bits per second. Let the transmission time $t^{i}_{data}$ for a frame $i$ be denoted as $Z_{i}$ seconds.
Putting everything together, the total time $tx_{i}$ for a successful transmission of a frame $i$ under IEEE 802.11 is defined as
$tx_{i}=\gamma+Y_{i}+Z_{i},$
where $\gamma=t_{DIFS}+2t_{pr}+t_{MAC}+t_{crc}+t_{SIFS}+t_{ack}$ is a real constant whose value depends on the underlying physical characteristics defined in IEEE 802.11.
IV On the Optimal Number of Frames Aggregated
This section first describes our proposal to aggregate frames intuitively and then details the assumptions and derivation of the mathematical formulae to calculate the number of aggregated frames that, on average, will allow us to improve the transmission time. Finally, we present a series of numerical experiments and provide our final observations.
IV-A Description of the Intuitive Idea
Figure 2(a) depicts the standard model of a node that receives $N$ transmission sources, each with a distribution rate of $\lambda_{i},1\leq i\leq N$. Each destination frame $f_{i}$ is queued, assigned a backoff time $t^{f_{i}}_{\text{backoff}}$, and sent accordingly when the backoff time expires and the medium is free, in accordance with IEEE 802.11.
Our proposed frame-aggregation model is outlined in Figure 2(b). Unlike the standard model (without frame aggregation), the destination frames are queued together. Upon the arrival of $k$ frames, they are aggregated and sent as a single multicast frame $g$ following the IEEE 802.11 specification (i.e., a backoff time $t^{g}_{\text{backoff}}$ is calculated accordingly and the frame $g$ is sent when the backoff time expires). Thus, through multicast transmission, downlink efficiency is increased by aggregating packets from different applications and various origins and destinations. Security mechanisms can be put in place to prevent group members from reading data that is not intended for them.
Our aggregation proposal, via multicast frames, is implementable in all IEEE 802.11 standards, thus improving the efficiency achieved regardless of the chosen standard. The use of multicast frames also has some drawbacks. For instance, it is more difficult to provide QoS support in the case of multicast traffic. In addition, multicast transmissions on 802.11 networks face serious reliability and scalability problems, due to the lack of acknowledgement messages and retransmissions. For this reason, the 802.11 standard recommends the use of basic data rates, which makes the channel busy for longer periods of time, affecting other network services. To solve these problems, the IEEE 802.11aa amendment introduced a set of retransmission policies to provide robust audio and video services in multicast mode, although the concrete way to use these features is left to the implementer [12].
IV-B Calculating the Optimal $k$ Value
Initially, we start from a series of sources that send frames to a transmission service. Let us focus on one of these sources, $A$, with a distribution rate $\lambda_{i}$. We will assume that all the distribution rates follow a Poisson distribution. Therefore, the distribution rate resulting from adding all of them is also a Poisson distribution whose rate is the sum of the rates, denoted as $\lambda$. We assume a payload size for each source, which follows a distribution (usually exponential), denoted as $P$. In any event, the payload size distribution is left open, because we are primarily interested in the mean.
Our scenario is the following:
1.
The frames enter and wait for a buffer of size $k$. Since they enter with a Poisson distribution, the waiting time for the arrival of $k$ packets follows an Erlang distribution of mean $\dfrac{k}{\lambda}$. Then, the mean waiting time of the $j$th frame in the buffer will be $\dfrac{k-j}{\lambda}$, on average $\text{Er}(k)=\dfrac{k-1}{2\lambda}$.
2.
With a time that we will consider negligible, an aggregated frame $f$ of size $k\text{E}[P]$ is built that will wait for a mean backoff time $t^{f}_{\text{backoff}}$, where $P$ is the random variable that determines the payload size of the frames that will be transmitted.
3.
The aggregated frame will then move to a queuing system with a Poisson input rate $\lambda_{A}(k)=\lambda/k$. The service time for each frame will be $\dfrac{1}{\mu(k)}=\dfrac{8\cdot k\text{E}[P]}{br}+\gamma+t^{f}_{\text{backoff}}$, where $\mu(k)$ is the system throughput, measured as bits per second (bps). We consider that the average time of stay of the frame in the system is $W(k)$. As it is a $M/G/1$ queue, hence $W(k)=\dfrac{(\lambda_{A}(k))^{2}(\sigma(K))^{2}+(\rho(k))^{2}}{2\lambda_{A}(k)(1-\rho(k))}$, where $\rho(k)=\dfrac{\lambda_{A}(k)}{\mu(k)}$. Another formulation (Pollaczek-Khintchine [13, 14]) can be $W(k)=\dfrac{\lambda_{A}(k)(\mu(k))^{-2}}{2\left(1-\rho(k)\right)}$.
4.
The mean time of the frame in the system will be $F(k)=\text{Er}(k)+\dfrac{1}{\mu(k)}+W(k)$.
Therefore, we can evaluate the gains of our proposed model (shown in Figure 2(b)) by evaluating $G(k)=F(k)-F(1)$. When $k\in\mathbb{Z}_{>1}$ and $G(k)>0$, then the aggregation of $k$ frames causes, on average, that the delay in receiving all frames in the destination is less than sent individually.
IV-C Numerical Experiments
To appreciate under what circumstances our proposal on frame aggregation will be most beneficial, we need to calculate $G(k)$ with network traffics that have different $\lambda$ distribution rates (measured in pps) and varying the value of $k$.
Figure 3 shows the value of $G(k)$ and $\lambda$ calculated for a WiFi transmission under IEEE 802.11b with a rate of 11Mbps, for different values of $k(k\in{2,3,...,10})$. The other IEEE 802.11b parameters remain the same for all experiments ($CW=16,t_{\text{backoff}}=20\mu s,t_{DIFS}=50\mu s,t_{pr}=96µs$ and $E[P]=800$ bits). To better understand what the values of $\lambda$ mean, remember that they are proportional to the input rate values following the expression $\lambda E[P]$, measured in bps. We can observe that, regardless of the number of frames aggregated, $G(k)$ takes positive values up to a certain value of $\lambda$. A positive value indicates that a delay in the transmission will appear if we aggregate frames. From that value of $\lambda$, $G(k)$ takes negative values, which indicates that the aggregation no longer causes delay. Furthermore, the graph also indicates that if the value of $\lambda$ continues to increase, the values of $G(k)$ tend to $-\infty$. This means that if no aggregation is done, the transmission becomes impossible because there is a situation of unacceptable congestion.
Figures 4 and 5 show the value of $\lambda$ that makes $G(k),k\in{2,3,...,20},$ become negative (and thus the aggregation no longer causes delay, but attenuates it) for the transmission rates specified by the 802.11b and 802.11g WiFi standards, respectively. The other parameters used for these numerical experiments are shown in the title of each figure. We can see that when the value of $k$ increases, the value of $\lambda$ also increases. This finding is logical, because for values greater than $k$, the delay increases unless the input rate is higher, in which case the aggregation decreases the delays.
Final Remarks. In summary, we have found that the delays caused by the aggregation of frames are compensated by the decrease in transmission congestion, which is a recommended practice to allow a better use of the radio medium in WiFi. To conclude, frame aggregation not only improves congestion situations, but also improves transmission delays. The mathematical expression proposed in this work is useful to discriminate when to use frame aggregation and save transmission time, on average. This expression can be used as a step prior to transmission, but the final decision of the value $k$ chosen must be adapted to each case.
V Conclusions
We have described the transmission time of a frame under the IEEE 802.11 protocol in a stochastic way and then we have obtained a formula to calculate the number of frames that can be aggregated to save transmission time, on average. Our numerical experiments demonstrate that WiFi frame aggregation improves the efficiency under certain IEEE 802.11 traffic conditions saving transmission time and reducing transmission congestion, and can be even proposed in those standards that do not have an aggregation mechanism defined. This formula can be used as a configuration criterion for a traffic regulator node (for instance, a router) with frame aggregation capability, to decide when frame aggregation should occur.
References
[1]
“IEEE Standard for Information Technology - (…) - Part 11: Wireless LAN
Medium Access Control (MAC) and Physical Layer (PHY) Specifications,”
IEEE Std 802.11-2007 (Revision of IEEE Std 802.11-1999), pp. 1–1076,
June 2007.
[2]
A. Duda, “Understanding the Performance of 802.11 Networks,” in
Procs. 2008 IEEE 19th Int. Symp. on Personal, Indoor and Mobile Radio
Commun., Sep. 2008, pp. 1–6.
[3]
G. Bianchi, “Performance analysis of the IEEE 802.11 distributed coordination
function,” IEEE Journal on Selected Areas in Communications,
vol. 18, no. 3, pp. 535–547, 2000.
[4]
Y. Xiao and J. Rosdahl, “Throughput and delay limits of IEEE 802.11,”
IEEE Commun. Lett., vol. 6, no. 8, pp. 355–357, Aug. 2002.
[5]
J. W. Tantra and C. H. Foh, “Achieving near maximum throughput in IEEE 802.11
WLANs with contention tone,” IEEE Commun. Lett., vol. 10, no. 9, pp.
658–660, Sep. 2006.
[6]
P. Teymoori, A. Dadlani, K. Sohraby, and K. Kim, “An Optimal Packet
Aggregation Scheme in Delay-Constrained IEEE 802.11n WLANs,” in
Procs. 2012 8th Int. Conf. Wireless Communications, Networking and
Mobile Computing, Sep. 2012, pp. 1–4.
[7]
Y. W. Kuo, W.-F. Lu, and T. L. Tsai, “A framework to approximate the delay
distribution for IEEE 802.11 DCF protocol,” in Procs. 2009 IEEE 9th
Malaysia Int. Conf. Commun. (MICC), Dec. 2009, pp. 874–879.
[8]
T. Selvam and S. Srikanth, “A frame aggregation scheduler for IEEE
802.11n,” in 2010 National Conference On Communications (NCC), 2010,
pp. 1–5.
[9]
J. Saldana, J. Ruiz-Mas, and J. Almodóvar, “Frame Aggregation in Central
Controlled 802.11 WLANs: The Latency Versus Throughput Tradeoff,”
IEEE Commun. Lett., vol. 21, no. 11, pp. 2500–2503, 2017.
[10]
E. Coronado, A. Thomas, and R. Riggio, “Adaptive ML-Based Frame Length
Optimisation in Enterprise SD-WLANs,” Journal of Network and Systems
Management, vol. 28, no. 4, pp. 850–881, Oct 2020.
[11]
J. Saldana, O. Topal, J. Ruiz-Mas, and J. Fernández-Navajas, “Finding the
Sweet Spot for Frame Aggregation in 802.11 WLANs,” IEEE Commun.
Lett., vol. 25, no. 4, pp. 1368–1372, 2021.
[12]
“IEEE Standard for Information technology– (…) – Specific requirements
Part 11: Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY)
Specifications Amendment 2: MAC Enhancements for Robust Audio Video
Streaming,” IEEE Std 802.11aa-2012, pp. 1–163, 2012.
[13]
F. Pollaczek, “Über eine Aufgabe der Wahrscheinlichkeitstheorie. I,”
Mathematische Zeitschrift, vol. 32, no. 1, pp. 64–100, Dec 1930.
[14]
A. Khintchine, “Mathematical theory of a stationary queue,”
Matematicheskii Sbornik, vol. 39, no. 4, pp. 73–84, 1932. |
Exact solution of the EM radiation-reaction problem
for classical finite-size and Lorentzian charged particles
Claudio Cremaschini
International School for Advanced Studies (SISSA), Trieste, Italy
Consortium for Magnetofluid Dynamics, University of Trieste, Trieste, Italy
Massimo Tessarotto
Department of Mathematics and Informatics, University of Trieste, Trieste,
Italy
Consortium for Magnetofluid Dynamics, University of Trieste, Trieste, Italy
(December 3, 2020)
Abstract
An exact solution is given to the classical electromagnetic (EM)
radiation-reaction (RR) problem, originally posed by Lorentz. This refers to
the dynamics of classical non-rotating and quasi-rigid finite size particles
subject to an external prescribed EM field. A variational formulation of the
problem is presented. It is shown that a covariant representation for the EM
potential of the self-field generated by the extended charge can be uniquely
determined, consistent with the principles of classical electrodynamics and
relativity. By construction, the retarded self 4-potential does not possess
any divergence, contrary to the case of point charges. As a fundamental
consequence, based on Hamilton variational principle, an exact
representation is obtained for the relativistic equation describing the
dynamics of a finite-size charged particle (RR equation), which is shown to
be realized by a second-order delay-type ODE. Such equation is proved to
apply also to the treatment of Lorentzian particles, i.e., point-masses with
finite-size charge distributions, and to recover the usual LAD equation in a
suitable asymptotic approximation. Remarkably, the RR equation admits both
standard Lagrangian and conservative forms, expressed respectively in terms
of a non-local effective Lagrangian and a stress-energy tensor. Finally,
consistent with the Newton principle of determinacy, it is proved that the
corresponding initial-value problem admits a local existence and uniqueness
theorem, namely it defines a classical dynamical system.
Classical Electrodynamics, Special Relativity, Radiation-reaction,
Variational principles
pacs: 03.50.De, 45.50.Dd, 45.50.Jj
I Introduction
An unsolved theoretical problem is related to the description of the
dynamics of classical charges with the inclusion of their electromagnetic
(EM) self-fields, the so-called radiation-reaction (RR) problem
(Dirac Dirac1938 , Pauli Pauli1958 , Feynman Feynman1988 ). Despite efforts spent by the scientific community in more than one
century of intensive theoretical research, an exact solution is still
missing (see related discussion in Ref.Dorigo2008a ; for a review see
Refs.Rohrlich1965 ; Teitel1970a ; Teitel1970b ; parrot1987 ; parrot1993 ). In
this regard, of fundamental importance is the construction of the exact (i.e., non-asymptotic) relativistic equation of
motion for a classical charged particle in the presence of its EM
self-field, also known as RR equation. This concerns, in particular,
its treatment in the context of special relativity (SR) and classical electrodynamics (CE), namely imposing the following basic
physical requirements, hereafter referred to as SR-CE Axioms:
1
Axiom #1: the Maxwell equations are fulfilled everywhere in the
flat space-time $M^{4}\mathcal{\subseteq}{\mathbb{R}}^{4}$, with metric
tensor $g_{\mu\nu}$. The Minkowski metric tensor is denoted as $\eta_{\mu\nu}\equiv diag(1,-1,-1,-1)$. In particular the EM 4-potential $A^{\mu}$
is assumed of class $C^{k}(M^{4})$, with $k\geq 2$;
2
Axiom #2: the Hamilton variational principle holds for a suitable
functional class of variations $\left\{f\right\}$. In particular, the
Hamilton principle must uniquely prescribe the particle world-line as a real
function $r^{\mu}(s)\in C^{k}(\mathbb{R}),$ with $k\geq 2$ for all $s\in\mathbb{R}$. The RR equation is then determined by the corresponding
Euler-Lagrange (E-L) equations. Hence, $\left\{f\right\}\equiv\left\{f_{i}(s),i=1,n\right\}$ is identified with the set of real functions of
class $C^{k}(\mathbb{R}),$ with $k\geq 2$:
$$\left\{f\right\}\equiv\left\{\begin{array}[]{c}f_{i}(s):f_{i}(s)\in C^{k}(%
\mathbb{R});\\
i=1,n;\emph{\ and }k\geq 2\end{array}\right\},$$
(1)
with functions $f_{i}(s)$ (for $i=1,n$) to be properly defined. In
particular, we shall require that the action functional is allowed to be of
the general form
$$S_{1}(f,\left[f\right])\equiv\int_{-\infty}^{+\infty}dsL_{1}\left(f(s),\frac{%
df(s)}{ds},\left[f(s)\right],\left[\frac{df(s)}{ds}\right]\right).$$
(2)
Here $L_{1}$ denotes a non-local variational particle Lagrangian, by
assumption defined on a finite-dimensional phase-space, which depends
at most on first-order derivatives $\frac{df(s)}{ds},$ with $f(s)$ belonging
to the functional class $\left\{f\right\},$ while $\left\{f(s),\frac{df(s)}{ds}\right\}$ and $\left\{\left[f(s)\right],\left[\frac{df(s)}{ds}\right]\right\}$ indicate respectively local and non-local dependencies in
terms of $f(s)$ and $\frac{df(s)}{ds};$
3
Axiom #3: the Newton determinacy principle (NDP) holds. This
implies the validity of an existence and uniqueness theorem for the
corresponding E-L equations. As a consequence, there exists necessarily a
classical dynamical system, namely a diffeomorphism
$$\mathbf{x}_{0}\equiv\mathbf{x}\left(s_{0}\right)\rightarrow\mathbf{x}\left(s%
\right),$$
(3)
with $\mathbf{x}\in I$ and $s$ representing respectively the state of a
classical particle and a suitable proper time, where $I\subseteq\mathbb{R}$
is an appropriate finite interval of the real axis;
4
Axiom #4: the Einstein causality principle (ECP) and the Galilei
inertia principle (GIP) both apply;
5
Axiom #5: the general covariance property of the theory, and in
particular the so-called manifest Lorentz covariance (MLC), i.e., the
covariance with respect to the group of special Lorentz transformations, are
satisfied.
Manifestly, these axioms are understood as identically fulfilled,
i.e., they must apply for arbitrary choices of both the initial conditions
for the dynamics of the charged particles and the applied external EM field.
The RR problem was first posed by Lorentz in his historical work (Lorentz,
1892 Lorentz ; see also Abraham, 1905 Abraham1905 ). Traditional
approaches are based either on the RR equation due to Lorentz, Abraham and
Dirac (first presented by Dirac in 1938 Dirac1938 ), nowadays
popularly known as the LAD equation, or the equation derived from it
by Landau and Lifschitz LL via a suitable “reduction
process” , the so-called LL equation. As recalled
elsewhere Dorigo2008a several aspects of the RR problem - and of the
LAD and LL equations - are yet to find a satisfactory formulation/solution. Common feature of all previous approaches is the adoption of an asymptotic
expansion for the EM self-field (or for the corresponding EM 4-potential),
rather than of its exact representation. This, in turn, implies that such
methods allow one to determine - at most - only an asymptotic approximation
for the correct RR equation.
For contemporary science the solution of the RR problem represents a
fundamental prerequisite for the proper formulation of all relativistic
theories, both classical and quantum ones, which are based on the
description of relativistic dynamics for classical charged particles.
Since Lorentz famous paper Lorentz several textbooks and research
articles have appeared on the subject of RR. Many of them have criticized
aspects of the RR theory, and in particular the LAD and LL equations (for a
review see Rohrlich1965 ; Teitel1970a ; Teitel1970b ; parrot1987 ; parrot1993 , where one can find the discussion of the related problems). However,
despite contrary claims Spohn2000 ; Rohrlich2001 ; Rohrlich2000 ; Medina2006 ; Rohrlich2008 ; caldirola ,
rigorous results are scarce Dorigo2008a . In particular, most of
previous investigations concern the treatment of point charges. These are
usually based either on suitable asymptotic approximations or regularization
schemes to deal with intrinsic divergences of the point-charge model. On the
other hand, there is no obvious classical physical mechanism, consistent
with the SR-CE axioms, which can explain the appearance of a finite
EM self-force acting on a point charge. This should arise as a consequence
of a finite delay time occurring between the particle position at the
time of the generation of its EM self-field and the instantaneous particle
position. It is well-known, as discovered by Lorentz himself (Lorentz, 1892
Lorentz ; see also for example Landau and Lifschitz, 1951 LL )
that such a force can act on a charged particle only if the particle itself
is actually finite-size. Therefore, although “ad
hoc” models based on the adoption of a finite delay time
have been known for a long time (see for example the heuristic approach to
the RR problem by Caldirola, 1956 caldirola leading to a delay-type
differential equation), the treatment of extended charge distributions
emerges as the only possible alternative, in analogy with the case of the
Debye screening problem in electrostatics Tessarotto2006 . In this
regard, a first approach in this direction is provided by the paper by
Nodvik (Nodvik,1964 Nodvik1964 ), where a variational treatment for
point mass particles having finite-size charge distributions was developed.
However, charge and mass are expected to have the same support, as required,
for example, by the energy-momentum conservation law in both special and
general relativity. Therefore a fully consistent relativistic theory should
actually be formulated for finite-size particles. From the analysis
of previous literature two important related problems arise:
•
Issue #1 - Existence of an exact variational RR equation: this
refers to the lack of an exact RR equation, based on Hamilton variational
principle, even for classical point-particles (or point-masses). In fact,
previous approaches have all been based on approximate (i.e., asymptotic)
estimates. Example of this type leading to the well-known LAD equation
(Lorentz, Abraham and Dirac Lorentz ; Yagh ; Abraham1905 ; Dirac1938 ) are
those due to Nodvik Nodvik1964 and Medina Medina2006 . A
critical aspect of the LAD equation, as well as of the related LL (Landau
and Lifschitz, 1951 LL ) equation, is that it does not satisfy a
variational principle in the customary sense, i.e., according to Axiom $\#2$
Dorigo2008a . In particular, the resulting LAD equation is only
asymptotic and non-variational in the sense of Axiom $\#2$. Instead,
the LL is non-variational, i.e., it does not admit a variational
action at all. However, the problem arises whether, in the context of
special relativity, an exact RR equation actually exists which holds
for suitable classical finite-size charged particles, and for Lorentzian
particles as a limiting case, namely finite-size charges having point-mass
distributions. Important related issues follow, such as the possibility for
the resulting equation to admit a standard Lagrangian form in terms
of a non-local effective Lagrangian function, and to be cast in an
equivalent conservative form, as the divergence of an effective
stress-energy tensor. Finally, the recovery of the customary LAD equation
in a suitable approximation must be verified.
•
Issue #2 - Existence and uniqueness problem: the second issue
is related to the consistency of the variational RR equation with the SR-CE
axioms and in particular with NDP. Therefore, the question arises whether an
existence and uniqueness theorem for the corresponding initial value problem
can be reached or not. Clearly the problem is relevant only for the exact RR
equation (yet to be established).
Clearly, the possible solution of these problems has potential wide-ranging
implications which are related to the description of relativistic dynamics
of systems of classical finite-size particles both in special and general
relativity.
II Goals of the paper and scheme of the presentation
The aim of the research program, of which the first part is reported here,
is to provide a consistent and exact theoretical formulation of the RR
problem for classical charged particles with finite-size charge and
mass distributions, addressing precisely issues #1 and #2. In this paper
the case is considered of extended particles having mass and charge
distributions localized on the same support, identified with a surface shell
(see Section 3 for a complete rigorous definition). The result is obtained
without introducing any perturbative or asymptotic expansion for the
evaluation of EM self 4-potential and/or “ad
hoc” regularization schemes for its point-particle limit.
In particular, finite-size charge distributions are introduced in order to
avoid intrinsic divergences (characteristic of the point-charge treatment)
and achieve an analytical description of the RR phenomena which is
consistent with the SR-CE axioms. A covariant representation for the EM self
4-potential is obtained, uniquely determined by the prescribed charge
current density. This allows us to point out the characteristic non-local
feature of the EM self-field, which is due to a causal retarded effect,
produced by the finite spatial extension of the charge. Here we shall
restrict the analysis to the treatment of charge and mass translational
motion, leaving the inclusion of rotational dynamics to a subsequent study.
Therefore, a suitable mathematical formulation of the problem is given, in
which rotational degrees of freedom are effectively excluded from the
present investigation. As a further result, it is proved that the exact RR
equation here obtained also holds for classical non-rotating
Lorentzian particles (Lorentz, 1892Lorentz ), i.e., in the case in
which the mass is regarded as point-wise localized and only the charge has a
finite spatial extension. The approach here adopted is based on the
variational formulation for finite-size charged particles earlier pointed
out by Tessarotto et al. Tessarotto2008c , in turn relying on
the hybrid form of the synchronous variational principle Pozzo1998 ; Beklemishev1999 . A key feature of this variational principle is
the adoption of superabundant dynamic variables Cremaschini2006 (see
also related discussion in Sections 5 and 7). Due to the arbitrariness of
their definition, they can always be identified with the components of the
particle position and velocity 4-vectors $r^{\mu}$ and $u^{\mu}$. This
also implies that, by construction, the variational functional necessarily
satisfies the property of covariance and MLC. Then, the corresponding E-L
equations yield both the RR equation and also the required physical
realizability constraints for $r^{\mu}$ and $u^{\mu}$, which allow one to
identify them with physical observables.
The paper is organized as follows. In Section 3 we present the derivation of
the charge and mass current densities for the particle model adopted, while
in Section 4 the exact solution for the EM self 4-potential generated by the
non-rotating charge distribution is constructed (Lemma 1). On the basis of
this result an explicit integral representation is obtained for the EM
self-potential (Lemma 2). Subsequently, in Section 5 we proceed in detail to
the construction of the variational functional. In particular, by making use
of Lemma 3, the contributions from the EM-coupling with both the EM
self-field (Subsection 5.1) and the external EM field (Subsection 5.2), as
well as the inertial mass contribution (Subsection 5.4) are determined.
Then, in Section 6 the resulting variational Lagrangian is derived. In
Section 7 the variational formulation for finite-size particles is
presented, based on a synchronous variational principle (THM.1). As a
fundamental consequence, it is found that the RR equation is a covariant
second-order delay differential equation which fulfills all SR-CE
axioms and in particular ECP, GIP and MLC. The equation is proved to apply
also to the particular case of Lorentzian particles. Then, in Section 8 the
RR equation is shown to admit both standard Lagrangian and conservative
forms (THM.2). Section 9 deals instead with the asymptotic behavior of the
RR equation, showing that in the short delay-time approximation it recovers
the customary LAD equation, while not admitting the point-charge limit
(THM.3). Finally, in Section 10 it is proved that, under suitable physical
assumptions, the RR equation here obtained fulfills also NDP and,
consequently, admits a well-posed initial value problem (i.e., there is an
existence and uniqueness theorem; see THM.4).
III Charge and mass current densities
In this section we define the particle model, prescribing its mass and
charge distributions, and determine the corresponding covariant expressions
for the charge and mass current densities, both needed for the subsequent
developments. Here we consider the treatment in the special relativity
setting.
By definition, the particle is characterized by a positive constant rest
mass $m_{o}$ and a non-vanishing constant charge $q$, with surface mass and
charge densities $\rho_{m}$ and $\rho_{c}$ respectively. We shall assume
that the mass and charge distributions have supports $\partial\Omega_{m}$
and $\partial\Omega_{\sigma}$. To define the particle mass and charge
distributions on $\partial\Omega_{m}$ and $\partial\Omega_{\sigma},$
let us assume initially that in a time interval $[-\infty,t_{o}]$ the
particle is at rest with respect to an inertial frame (i.e., that external
forces acting on the particle vanish identically). As a consequence, by
assumption in the subset of the space-time $\mathcal{M}^{4}\subseteq\mathbb{R}^{4}$ in which $t\in[-\infty,t_{o}],$ there is an inertial frame
in which both the particle mass and charge distributions are at rest
(particle rest-frame $\mathcal{R}_{o}$). In this frame, we shall assume that
there exists a point, hereafter referred to as center of symmetry
(COS), whose position 4-vector $r_{COS}^{\mu}\equiv(ct,\mathbf{r}_{o})$
spans the Minkowski space-time $\mathcal{M}^{4}\subseteq\mathbb{R}^{4}$ and
with respect to which:
1) $\partial\Omega_{\sigma}$ and $\partial\Omega_{m}$ are stationary
spherical surfaces of radii $\sigma>0$ and $\sigma_{m}>0$ of equations $\left(\mathbf{r-r}_{o}\right)^{2}=\sigma^{2}$ and $\left(\mathbf{r-r}_{o}\right)^{2}=\sigma_{m}^{2};$
2) the particle is quasi-rigid, i.e., the mass and charge
distributions are stationary and spherically-symmetric respectively on $\partial\Omega_{m}$ and $\partial\Omega_{\sigma}$111In order to warrant the condition of rigidity in a manner consistent with
the SR-CE Axioms, following the literature a possibility is to assume that
the extended particle is acted upon by a local non-EM force
“whose precise nature is left
unspecified” (see Nodvik Nodvik1964 and further
references indicated there).$;$
3) in addition, consistent with the principle of energy-momentum
conservation (see further discussion below), we shall assume the
distributions of mass and charge densities to have the same support $\partial\Omega_{\sigma}\equiv\partial\Omega_{m}$, hence letting
$$\sigma_{m}=\sigma.$$
(4)
Finally, the case in which the mass is considered localized point-wise (Lorentzian particle) is recovered letting $\sigma_{m}\neq\sigma,$
with $\sigma>0$ and $\sigma_{m}=0$. In both cases the particle mass and
charge distributions remain uniquely defined in any reference frame for
arbitrary particle motion.
In this paper, we are concerned only with the investigation of the EM RR
phenomenon on the translational dynamical motion of the charged particle.
Hence, we require that the mass density (and, as a consequence, also the
charge density) does not possess pure spatial rotation, nevertheless still
allowing for space-time rotations (i.e., Thomas precession, see below). For
definiteness, let us introduce here the Euler angles $\alpha(s)\equiv\left\{\varphi(s),\vartheta(s),\psi(s)\right\}$ which define the
orientation of the body-axis system $K^{\prime}$ with respect to the rest
system $K$ (according to the notations used by Nodvik Nodvik1964 ).
Introducing the generalized velocities $\frac{d\alpha\left(s\right)}{ds}\equiv\left\{\frac{d\varphi}{ds},\frac{d%
\vartheta}{ds},\frac{d\psi}{ds}\right\},$ the condition of vanishing mass and charge spatial rotation in a time interval $I\subseteq\mathbb{R}$ is thus prescribed imposing
that the particular solution
$$\displaystyle\alpha(s)$$
$$\displaystyle=$$
$$\displaystyle\alpha_{o},$$
$$\displaystyle\frac{d\alpha\left(s\right)}{ds}$$
$$\displaystyle\equiv$$
$$\displaystyle 0,$$
(5)
holds for all $s\in I.$ For a physical motivation for this assumption we
refer to the discussion reported by Yaghjian Yagh .
Having specified the physical properties of the particle by means of the
mass and charge distributions, we can now move on to obtaining the covariant
expression for the corresponding charge and mass current densities. Since
the charge and the mass have the same support, the mathematical derivation
is formally the same for both of them. For convenience we start with the
charge current $j^{\mu}(r)$, introducing for it the representation used by
Nodvik. For definiteness, let us denote Nodvik1964
$$\displaystyle s$$
$$\displaystyle\equiv$$
proper time of the COS,
$$\displaystyle r^{\mu}(s)$$
$$\displaystyle\equiv$$
COS 4-position,
$$\displaystyle\zeta^{\mu}$$
$$\displaystyle\equiv$$
charge element 4-position.
Then, we define the displacement vector $\xi^{\mu}$ as follows:
$$\xi^{\mu}\equiv\varsigma^{\mu}-r^{\mu}(s),$$
(6)
from which we also have that $\varsigma^{\mu}=r^{\mu}(s)+\xi^{\mu}.$
The physical meaning of the 4-vector $\xi^{\mu}$ is that of a displacement
between the particle COS and its boundary, where the charge is
located. According to this representation, $\xi^{\mu}$ is subject to the
following two constraints Nodvik1964 :
$$\displaystyle\xi^{\mu}\xi_{\mu}$$
$$\displaystyle=$$
$$\displaystyle-\sigma^{2},$$
(7)
$$\displaystyle\xi_{\mu}u^{\mu}(s)$$
$$\displaystyle=$$
$$\displaystyle 0,$$
(8)
where
$$u^{\mu}(s)\equiv\frac{d}{ds}r^{\mu}(s)$$
(9)
is the 4-velocity of the COS. The first equality (7) defines the
boundary $\partial\Omega_{\sigma}=\partial\Omega_{m}$. The second
constraint (8) represents instead the constraint of rigidity for
the particle. This implies that in the particle rest frame the 4-vector $\xi^{\mu}$ has only spatial components. We can use the information from Eq.(7) to define the internal and the external domains with respect to
the mass and charge distributions. In particular, if we define a generic
displacement 4-vector $X^{\mu}\in M^{4}$ as
$$X^{\mu}=r^{\mu}-r^{\mu}\left(s\right),$$
(10)
which is subject to the constraint
$$X^{\mu}u_{\mu}(s)=0,$$
(11)
then the following relations hold:
$$\displaystyle X^{\mu}X_{\mu}$$
$$\displaystyle\leq$$
$$\displaystyle-\sigma^{2}\emph{\ : external domain,}$$
(12)
$$\displaystyle X^{\mu}X_{\mu}$$
$$\displaystyle>$$
$$\displaystyle-\sigma^{2}\emph{\ : internal domain,}$$
$$\displaystyle X^{\mu}X_{\mu}$$
$$\displaystyle=$$
$$\displaystyle\xi^{\mu}\xi_{\mu}=-\sigma^{2}\emph{\ : boundary
location.}$$
To derive the current density 4-vector corresponding to the spherical
charged shell we follow the presentation by Nodvik Nodvik1964 .
Consider first the charge-current density $\Delta j^{\mu}(r)$ corresponding
to a charge element $\Delta q$ on the shell. This is expressed as follows:
$$\Delta j^{\mu}(r)=c\Delta q\int_{1}^{2}d\zeta^{\mu}\delta^{4}\left(r^{\mu}-%
\zeta^{\mu}\right)=c\Delta q\int_{-\infty}^{+\infty}ds\left[u^{\mu}+\frac{d\xi%
^{\mu}}{ds}\right]\delta^{4}\left(x^{\mu}-\xi^{\mu}\right),$$
(13)
where
$$x^{\mu}=r^{\mu}-r^{\mu}\left(s\right).$$
(14)
Note that, for the simplicity of the notation, here and in the rest of the
paper the symbol $r$ stands for the generic 4-vector $r^{\alpha}$ when used
as an argument of a function. Since the charge does not possess any pure
spatial rotation, the relation
$$\frac{d\xi^{\mu}}{ds}=\Gamma u^{\mu}$$
(15)
holds, where $\Gamma\equiv-\left(\frac{du_{\alpha}}{ds}\xi^{\alpha}\right)$ carries the effect associated with the Thomas precessionNodvik1964 . The expression for $\Delta j^{\mu}(r)$ then becomes
$$\Delta j^{\mu}(r)=c\Delta q\int_{-\infty}^{+\infty}dsu^{\mu}\left[1+\Gamma%
\right]\delta^{4}\left(x^{\mu}-\xi^{\mu}\right).$$
(16)
To compute the total current of the charged shell we express the charge
element $\Delta q$ according to the constraint (8) as follows: $\Delta q=qf(\left|\xi\right|)\delta(\xi^{\alpha}u_{\alpha}(s))d^{4}\xi,$ where $d^{4}\xi$ is the 4-volume element in the $\xi$-space. Moreover, $f(\left|\xi\right|)$ is referred to as the form factor, which
describes the charge distribution of the moving body. In particular, for a
spherically symmetric distribution this has the following representation:
$$f(\left|\xi\right|)=\frac{1}{4\pi\sigma^{2}}\delta(\left|\xi\right|-\sigma),$$
(17)
where $\left|\xi\right|\equiv\left|\sqrt{\xi^{\mu}\xi_{\mu}}\right|.$ The total current density $j^{\mu}(r)$ can therefore
be obtained by integrating $\Delta j^{\mu}(r)$ over $d^{4}\xi$. We get
$$\displaystyle j^{\mu}(r)$$
$$\displaystyle\equiv$$
$$\displaystyle qc\int_{-\infty}^{+\infty}dsu^{\mu}\int_{1}^{2}d^{4}\xi f(\left|%
\xi\right|)\delta(\xi^{\alpha}u_{\alpha})\left[1+\Gamma\right]\delta^{4}\left(%
x^{\mu}-\xi^{\mu}\right)=$$
(18)
$$\displaystyle=$$
$$\displaystyle qc\int_{-\infty}^{+\infty}dsu^{\mu}f(\left|x\right|)\delta(x^{%
\alpha}u_{\alpha})\left[1+\Gamma\right],$$
where
$$f(\left|x\right|)=\frac{1}{4\pi\sigma^{2}}\delta(\left|x\right|-\sigma)$$
(19)
with $\left|x\right|\equiv\left|\sqrt{x^{\mu}x_{\mu}}\right|.$ Then we notice that
$$\delta(x^{\alpha}u_{\alpha}(s))=\frac{1}{\left|\frac{d\left[x^{\alpha}u_{%
\alpha}\right]}{ds}\right|}\delta(s-s_{1})=\frac{1}{\left|1+\Gamma\right|}%
\delta(s-s_{1}),$$
(20)
where by definition $s_{1}$ is the root of the algebraic equation
$$u_{\mu}(s_{1})\left[r^{\mu}-r^{\mu}\left(s_{1}\right)\right]=0.$$
(21)
Combining these relations, it follows that the integral covariant expression
for the charge current density is given by
$$j^{\mu}(r)=\frac{qc}{4\pi\sigma^{2}}\int_{-\infty}^{+\infty}dsu^{\mu}(s)\delta%
(\left|x\right|-\sigma)\delta(s-s_{1}).$$
(22)
Finally, an analogous expression for the mass current density $j_{mass}^{\mu}(r)$ can be easily obtained from $j^{\mu}(r)$ by replacing the total
charge $q$ with the total mass $m_{o}$, thus giving
$$j_{mass}^{\mu}(r)=\frac{m_{o}c}{4\pi\sigma^{2}}\int_{-\infty}^{+\infty}dsu^{%
\mu}(s)\delta(\left|x\right|-\sigma)\delta(s-s_{1}).$$
(23)
We remark that in both equations (22) and (23):
1) the dependence in terms of the 4-position $r$ enters explicitly through $|x|=|r^{\mu}-r^{\mu}\left(s\right)|$ in the form factor and implicitly
through the root $s_{1}$;
2) consistent with assumption (5), possible charge and mass
spatial rotations have been set to be identically zero.
IV EM self 4-potential - Case of non-rotating charge distribution
A prerequisite for the subsequent developments is the determination of the
EM self-potential ($A_{\mu}^{(self)}$) produced by the spherical charged
particle shell here introduced. In principle the problem could be formally
treated by solving the Maxwell equations with the 4-potential written in
terms of a suitable Green function according to standard methods.
Remarkably, the solution can also be achieved in a more straightforward way
based on the relativity principle and the covariance of Maxwell’s equations.
This implies the possibility of obtaining a covariant representation of the
EM 4-vector in a generic reference system once its definition is known in a
particular reference frame. The approach is analogous to the derivation
presented by Landau and Lifschitz LL for the treatment of a point
charge. The solution is provided by the following Lemmas.
Lemma 1 - Covariant representation for $A_{\mu}^{(self)}(r)$
Given validity of the assumptions on the particle structure introduced
in the previous section and the results obtained for the current density,
the following statements hold:
L1${}_{1}:$ Particle at rest in an inertial frame.
Let us assume that the particle is at rest in an inertial frame $S_{0}$ and, according to (5), is non-rotating in this
frame. By definition, in $S_{0}$ the 4-vector potential of the
self-field is written as $A_{\mu}^{(self)}(r)=A_{S_{0}\mu}^{(self)}(r)\equiv\left\{\Phi^{(self)},\mathbf%
{0}\right\},$ where
$$\Phi^{(self)}(\mathbf{r},t)=\left\{\begin{array}[]{ccc}\frac{q}{R}&&(R\geq%
\sigma),\\
\frac{q}{\sigma}&&(R<\sigma),\end{array}\right.$$
(24)
(rest-frame representation) denote respectively the external and
internal solutions with respect to the boundary of the shell. Here
$$\displaystyle R$$
$$\displaystyle\equiv$$
$$\displaystyle\left|\mathbf{R}\right|,$$
(25)
$$\displaystyle\mathbf{R}$$
$$\displaystyle=$$
$$\displaystyle\mathbf{r}-\mathbf{r}\left(t^{\prime}\right),$$
(26)
with $r^{\mu}=(ct,\mathbf{r}),$ $r^{\prime\mu}=(ct^{\prime},\mathbf{r}^{\prime}\equiv\mathbf{r}\left(t^{\prime}%
\right)),$ and $\mathbf{r},\mathbf{r}^{\prime}\equiv\mathbf{r}\left(t^{\prime}\right)$ being respectively a generic position $3$-vector of $\mathbb{R}{}^{3}$ and the (stationary) position 3-vector of the particle COS. It
follows that $\Phi^{(self)}(\mathbf{r},t)$ can be equivalently
represented as
$$\Phi^{(self)}(\mathbf{r},t)=\left\{\begin{array}[]{ccc}\frac{q}{c(t-t^{\prime}%
)}\equiv\frac{q}{R}&&(R\geq\sigma),\\
\frac{q}{c(t-t^{\prime})}\equiv\frac{q}{\sigma}&&(R<\sigma),\end{array}\right.$$
(27)
where $t_{ret}\equiv t-t^{\prime}$ is the following
positive root
$$t_{ret}\equiv t-t^{\prime}=\left\{\begin{array}[]{ccc}t_{ret}^{(ext)}\equiv\pm%
\frac{R}{c}&&(R\geq\sigma),\\
t_{ret}^{(int)}\equiv\pm\frac{\sigma}{c}&&(R<\sigma).\end{array}\right.$$
(28)
L1${}_{2}:$ Particle with inertial motion in an arbitrary inertial frame.
Let us assume that when the particle is referred to an arbitrary
inertial frame $S_{I}$ it has a constant $4-$velocity $u^{\alpha}\equiv$ $\frac{dr^{\mu}(s^{\prime})}{ds^{\prime}}.$
Then, let us require that $t_{ret}\equiv t-t^{\prime}$ is
the positive root of the delay-time equation
$$\emph{\ }\widehat{R}^{\alpha}\emph{\ }\widehat{R}_{\alpha}=\rho^{2},$$
(29)
with $\widehat{R}^{\alpha}$ being the bi-vector
$$\emph{\ }\widehat{R}^{\alpha}=r^{\alpha}-r^{\alpha}(t^{\prime})$$
(30)
and
$$\rho^{2}=\left\{\begin{array}[]{ccc}0&&(X^{\alpha}X_{\alpha}\leq-\sigma^{2}),%
\\
\rho^{2}\equiv\sigma^{2}\left[1+\frac{X^{\alpha}X_{\alpha}}{\sigma^{2}}\right]%
&&(X^{\alpha}X_{\alpha}>-\sigma^{2}),\end{array}\right.$$
(31)
where the displacement vector $X^{\alpha}$ is defined by Eqs.(10) and (11). For consistency, Eq.(31) provides the solution Eq.(28) when evaluated in the COS
comoving frame.
It follows that in the reference frame $S_{I}$ the EM self
4-potential have the internal and external solutions
$$A_{\mu}^{(self)}(r)=\left\{\begin{array}[]{ccc}\left.q\frac{u_{\mu}}{\widehat{%
R}^{\alpha}u_{\alpha}}\right|_{t_{ret}=t_{ret}^{(ext)}}&&(X^{\alpha}X_{\alpha}%
\leq-\sigma^{2}),\\
\left.q\frac{u_{\mu}}{\widehat{R}^{\alpha}u_{\alpha}}\right|_{t_{ret}=t_{ret}^%
{(int)}}&&(X^{\alpha}X_{\alpha}>-\sigma^{2}),\end{array}\right.$$
(32)
where $\widehat{R}^{\alpha}$ is given by Eq.(30)$.$
L1${}_{3}:$ Particle with a non-inertial motion in an arbitrary frame.
Let us assume that the same particle is now referred to an arbitrary
frame in which it has a time-dependent velocity $u_{\mu}(t^{\prime})$. In this frame the EM self 4-potential $A_{\mu}^{(self)}(r)$ takes the form:
$$A_{\mu}^{(self)}(r)=\left\{\begin{array}[]{ccc}\left.q\frac{u_{\mu}(t^{\prime}%
)}{\widehat{R}^{\alpha}u_{\mu}(t^{\prime})}\right|_{t_{ret}=t_{ret}^{(ext)}}&&%
(X^{\alpha}X_{\alpha}\leq-\sigma^{2}),\\
\left.q\frac{u_{\mu}(t^{\prime})}{\widehat{R}^{\alpha}u_{\mu}(t^{\prime})}%
\right|_{t_{ret}=t_{ret}^{(int)}}&&(X^{\alpha}X_{\alpha}>-\sigma^{2}),\end{%
array}\right.$$
(33)
where $u_{\mu}(t^{\prime})$ is the 4-velociy of the COS with
4-position $r^{\alpha}(t^{\prime}),$ i.e.,
$$u_{\mu}(t^{\prime})\equiv\frac{dr^{\beta}(t^{\prime})}{ds^{\prime}}=\gamma(t^{%
\prime})\frac{dr^{\beta}(t^{\prime})}{cdt^{\prime}},$$
(34)
and $t_{ret}^{(ext)},$ $t_{ret}^{(int)}$ are the positive roots
of the delay-time equation (29).
Proof - L1${}_{1})$ If the particle is at rest in an
inertial frame $S_{0}$, from the form of the charge density (22) and
the condition of non-rotation (5), the EM self 4-potential is
stationary in $S_{0}$. Hence it takes necessarily the form $A_{\mu}^{(self)}(r)=A_{S_{0}\mu}^{(self)}(r)\equiv\left\{\Phi^{(self)},\mathbf%
{0}\right\}.$ Thus, denoting
$$\displaystyle R$$
$$\displaystyle\equiv$$
$$\displaystyle\left|\mathbf{R}\right|,$$
(35)
$$\displaystyle\mathbf{R}$$
$$\displaystyle=$$
$$\displaystyle\mathbf{r}-\mathbf{r}\left(t-\frac{\left|\mathbf{r}-\mathbf{r}(t-%
\frac{R}{c})\right|}{c}\right),$$
(36)
with $\mathbf{r}$ a generic position $3$-vector of $\mathbb{R}{}^{3}$ and $\mathbf{r}(t^{\prime})\equiv\mathbf{r}(t-\frac{R}{c})$ the
retarded-time position 3-vector, $\Phi^{(self)}$ is written as
$$\Phi^{(self)}(\mathbf{r},t)=\left\{\begin{array}[]{ccc}\frac{q}{R}&&(R\geq%
\sigma),\\
\frac{q}{\sigma}&&(R<\sigma).\end{array}\right.$$
(37)
In other words, in the external/internal sub-domains (respectively defined
by the inequalities $R\geq\sigma$ and $R<\sigma$) the ES potential $\Phi^{(self)}$ coincides with the ES potential of a point charge and a constant
potential. In terms of the delay time $t_{ret}=t=t^{\prime}$ determined by
Eq.(28) it is immediate to prove Eq.(27).
L1${}_{2})$ Next, let us consider the same particle referred to an arbitrary
inertial frame $S_{I}$ in which the COS position vector $r^{\alpha}(s^{\prime})$ has a constant velocity
$$u_{\alpha}\equiv u^{\alpha}(s^{\prime})=\frac{d}{ds^{\prime}}r^{\alpha}(s^{%
\prime})=const.$$
(38)
Since by definition $A_{\mu}^{(self)}(r)$ is a covariant 4-vector, its form
in $S_{I}$ is simply obtained by applying a Lorentz transformation LL
according to Eq.(38). This requires
$$A_{\mu}^{(self)}(r)=q\frac{u_{\mu}}{\widehat{R}^{\alpha}u_{\alpha}},$$
(39)
where $\widehat{R}^{\alpha}=r^{\alpha}-r^{\alpha}(s^{\prime})$. Denoting
$s^{\prime}\equiv s^{\prime}(t^{\prime})$ and $r^{\alpha}(s^{\prime})\equiv(ct^{\prime},\mathbf{r}(t^{\prime})),$ let us now impose that $t-t^{\prime}$ is the positive root of the delay-time equation (29). The external and internal solutions in this case are
given respectively by Eq.(32), as can be seen by noting
that when $u_{\mu}=(1,\mathbf{0})$ the correct external and internal
solutions (24) are recovered.
L1${}_{3})$ The proof of the third statement is a basic consequence of
the principle of relativity and of the covariance of the Maxwell equations.
In fact we notice that both the solution (32) for the
4-vector potential and Eq.(29) for the delay
time, which have been obtained for the specific case of an inertial frame,
are already written in covariant form by means of the 4-vector notation.
Hence, according to the principle of relativity, this solution is valid in
any reference system related by a Lorentz transformation, and for a generic
form of the 4-velocity $u_{\mu}$ (cf Landau and Lifshitz LL ).
Q.E.D.
We remark that Eq.(33) provides an exact
representation (defined up to a gauge transformation) for the EM self
4-potential generated by the non-rotating finite-size charge considered here.
On the base of the conclusions of Lemma 1 it follows that $A_{\mu}^{(self)}(r)$ can also be represented by means of an equivalent integral
representation as proved by the following Lemma.
Lemma 2 - Integral representation for $A_{\mu}^{(self)}(r)$
Given validity of Lemma 1, the EM self 4-potential Eq.(33) admits the equivalent integral
representation
$$A_{\mu}^{(self)}(r)=2q\int_{1}^{2}dr_{\mu}^{\prime}\delta(\widehat{R}^{\alpha}%
\widehat{R}_{\alpha}-\rho^{2}),$$
(40)
with $\rho^{2}$ defined by Eq.(31)
and $r_{\mu}^{\prime}\equiv r_{\mu}\left(s^{\prime}\right)$.
Proof - In fact in the external and internal domains
$$\delta(\widehat{R}^{\alpha}\widehat{R}_{\alpha}-\rho^{2})=\left\{\begin{array}%
[]{ccc}\frac{\delta(s-s^{\prime})}{2\left|\widehat{R}_{\alpha}\frac{dr^{\prime%
\alpha}}{ds^{\prime}}\right|}&&(X^{\alpha}X_{\alpha}\leq-\sigma^{2}),\\
\frac{\delta(s-s^{\prime})}{\left|2\widehat{R}_{\alpha}\frac{dr^{\prime\alpha}%
}{ds^{\prime}}+\frac{d\rho^{2}}{ds}\right|}&&(X^{\alpha}X_{\alpha}>-\sigma^{2}%
),\end{array}\right.$$
(41)
where $\frac{d\rho^{2}}{ds}=\frac{dX^{\alpha}X_{\alpha}}{ds^{\prime}}=2X_{\alpha}u_{%
\alpha}(s^{\prime})\equiv 0$ because of Eq.(11),
while $s^{\prime}$ is determined by the delay-time equation (29)$.$ Hence, Eq.(40)
manifestly implies Eq.(33).
Q.E.D.
V The action integral
In this section we derive the Hamilton action functional suitable for the
variational treatment of finite-size charged particles introduced here and
the investigation of their dynamics. As indicated in Section 3, the
contributions due to pure spatial charge and mass rotations will be ignored.
In this case, the action integral is conveniently expressed in hybrid
superabundant variables (see Tessarotto et al. Cremaschini2006 ) as follows:
$$S_{1}(r,u,\chi,\left[r\right])=S_{M}(r,u)+S_{C}^{\left(self\right)}(r,\left[r%
\right])+S_{C}^{\left(ext\right)}(r)+S_{\chi}(u,\chi),$$
(42)
where $S_{M}$, $S_{C}^{\left(self\right)}$, $S_{C}^{\left(ext\right)}$
and $S_{\chi}$ are respectively the inertial mass, the EM-coupling with the
self and external fields, and the kinematic constraint contributions. For
what concerns the notation, here $r$ and $u$ represent local
depepndencies with respect to the 4-vector position $r^{\mu}$ and the
4-velocity $u^{\mu}$, $\left[r\right]$ stands for non-local
dependencies on the 4-vector position $r^{\mu}$, while $\chi\equiv\chi(s)$ is a Lagrange multiplier (see also below and the related discussion in
THM.1 of Section 7).
Before addressing the explicit evaluation of $S_{1}(r,u,\chi,\left[r\right])$ we prove the following preliminary Lemma concerning the transformation
properties of 4-volume elements under Lorentz transformations.
Lemma 3 - Lorentz transformations and 4-volume elements
Let us consider a Lorentz transformation (Lorentz boost) from an
inertial reference frame $S_{I}$ to a reference frame $S_{NI}$ whose origin has 4-velocity $u_{\mu}(s_{2})$ with respect to $S_{I}$, with $s_{2}$ being considered here an arbitrary
proper time independent of $r^{\mu}\in S_{I}$. By assumption $u_{\mu}(s_{2})$ is constant both with respect to the 4-positions $r^{\mu}\in S_{I}$ and $r^{\prime\mu}\in S_{NI}$ in the two
reference frames. The relationship between the two 4-vectors $r^{\mu}\in S_{I}$ and $r^{\prime\mu}\in S_{NI}$ is expressed by the
transformation lawJackson
$$r^{\prime\mu}=\Lambda_{\nu}^{\mu}\left(u_{\mu}(s_{2})\right)r^{\nu},$$
(43)
where $\Lambda_{\nu}^{\mu}\left(u_{\mu}(s_{2})\right)$ is the matrix of the Lorentz boost, which by definition depends only on the
relative 4-velocity $u_{\mu}(s_{2})$ between $S_{I}$ and $S_{NI}$. Then it follows that the 4-volume element $d\Omega\in S_{I}$ is invariant with respect to the Lorentz boost (43), in
the sense:
$$d\Omega=d\Omega^{\prime},$$
(44)
with $d\Omega^{\prime}\in S_{NI}$ denoting the corresponding
volume element in the transformed frame $S_{NI}$.
Proof - The proof of this statement follows by considering the
general transformation property of volume elements under arbitrary change of
coordinates. Consider the invariant 4-volume element $d\Omega\in S_{I}$ and
assume a Minkowski metric tensor. By definition LL , for a generic
change of reference frame the volume element transforms according to the law
$$d\Omega=\frac{1}{J}d\Omega^{\prime},$$
(45)
where
$$J\equiv\left|\frac{\partial r^{\mu}}{\partial r^{\prime\mu}}\right|$$
(46)
is the Jacobian of the corresponding coordinate transformation. In the case
considered here, the Lorentz boost (43) is described by the matrix
$\Lambda_{\nu}^{\mu}\left(u_{\mu}(s_{2})\right)$ which depends
only on the 4-velocity $u_{\mu}(s_{2}),$ by assumption independent of the
coordinates $r^{\nu}$ and $r^{\prime\nu}$. It follows that $J\equiv 1$,
implying in turn Eq.(44).
Q.E.D.
We can now proceed to evaluate the various contributions to the action
integral $S_{1}(r,u,\chi,\left[r\right])$ defined in Eq.(42).
V.1 $S_{C}^{\left(self\right)}(r,\left[r\right])$: EM coupling
with the self-field
The action integral $S_{C}^{\left(self\right)}(r,\left[r\right])$
containing the coupling between the EM self-field and the electric 4-current
is of critical importance. For this reason and for the sake of clarity in
this subsection the steps of its evaluation are reported in detail.
According to the standard approach LL , $S_{C}^{\left(self\right)}$
is defined as the 4-scalar
$$S_{C}^{(self)}(r,\left[r\right])=\int_{1}^{2}d\Omega\frac{1}{c^{2}}A^{(self)%
\mu}(r)j_{\mu}\left(r\right),$$
(47)
where $A^{(self)\mu}(r)$ is given by Eq.(40), $j_{\mu}\left(r\right)$ by Eq.(22) and $d\Omega$ is the
invariant 4-volume element. In particular, in an inertial frame $S_{I}$ with
Minkowski metric tensor $\eta_{\mu\nu}$, this can be represented as
$$d\Omega=cdtdxdydz,$$
(48)
where $\left(x,y,z\right)$ are orthogonal Cartesian coordinates. The
functional can be equivalently represented as
$$\displaystyle S_{C}^{(self)}(r,\left[r\right])$$
$$\displaystyle=$$
$$\displaystyle\frac{q}{4\pi\sigma^{2}c}\int_{1}^{2}d\Omega A^{(self)\mu}(r)\int%
_{-\infty}^{+\infty}ds_{2}\delta(s_{2}-s_{1})\times$$
(49)
$$\displaystyle\times\int_{-\infty}^{+\infty}dsu^{\mu}(s)\delta(\left|x\left(s%
\right)\right|-\sigma)\delta(s-s_{2}),$$
where $s_{1}$ is the root of the equation
$$u_{\mu}(s_{1})\left[r^{\mu}-r^{\mu}\left(s_{1}\right)\right]=0.$$
(50)
Because of the principle of relativity, the integral (47) can be
evaluated in an arbitrary reference frame. The explicit calculation of the
integral (47) is then achieved, thanks to Lemma 3, by invoking the
Lorentz boost (43) to the reference frame $S_{NI}$ moving with
4-velocity $u_{\mu}(s_{2})$. In this frame, by construction $d\Omega^{\prime}=cdt^{\prime}dx^{\prime}dy^{\prime}dz^{\prime}\equiv d\Omega$. In particular, introducing the spherical spatial coordinates $\left(ct^{\prime},\rho^{\prime},\vartheta^{\prime},\varphi^{\prime}\right)$
it follows that the transformed spatial volume element can also be written
as $cdt^{\prime}dx^{\prime}dy^{\prime}dz^{\prime}\equiv cdt^{\prime}d\rho^{\prime}%
d\vartheta^{\prime}d\varphi^{\prime}\rho^{\prime 2}\sin\vartheta^{\prime}.$ In this frame the previous scalar equation
becomes
$$u_{\mu}^{\prime}(s_{1})\left[r^{\prime\mu}-r^{\prime\mu}\left(s_{1}\right)%
\right]=0.$$
(51)
On the other hand, performing the integration with respect to $s_{2}$ in Eq.(49), it follows that necessarily $s_{2}=s_{1}$, so that from Eq.(51) $s_{1}$ is actually given by
$$s_{1}=ct^{\prime}=s_{2}.$$
(52)
As a result, the integral $S_{C}^{(self)}$ reduces to
$$S_{C}^{(self)}(r^{\prime},\left[r^{\prime}\right])=\frac{q}{4\pi\sigma^{2}c}%
\int_{1}^{2}dx^{\prime}dy^{\prime}dz^{\prime}A^{{}^{\prime}(self)\mu}(r^{%
\prime})\int_{-\infty}^{+\infty}dsu^{\prime\mu}(s)\delta(\left|x^{\prime}\left%
(s\right)\right|-\sigma),$$
(53)
with $x^{\prime\mu}\left(s\right)=r^{\prime\mu}-r^{\prime\mu}\left(s\right)$. Moreover
$$A_{\mu}^{\prime(self)}(r^{\prime})=2q\int_{-\infty}^{+\infty}ds^{\prime\prime}%
u_{\mu}^{\prime}\left(s^{\prime\prime}\right)\delta(\widehat{R}^{\prime\alpha}%
\widehat{R}_{\alpha}^{\prime}-\rho^{\prime 2}),$$
(54)
with $\widehat{R}^{\prime\alpha}=r^{\prime\alpha}-r^{\prime\alpha}(s^{\prime\prime})$ and, thanks to Lemma 1,
$$\rho^{\prime 2}=\left\{\begin{array}[]{ccc}0&&(X^{\prime\alpha}X_{\alpha}^{%
\prime}\leq-\sigma^{2}),\\
\rho^{\prime 2}\equiv\sigma^{2}\left[1+\frac{X^{\prime\alpha}X_{\alpha}^{%
\prime}}{\sigma^{2}}\right]&&(X^{\prime\alpha}X_{\alpha}^{\prime}>-\sigma^{2})%
.\end{array}\right.$$
(55)
Notice here that in $S_{C}^{(self)}(r^{\prime},\left[r^{\prime}\right])$
the contributions of the external and internal domains for the self-field
can be explicitly taken into account letting
$$\displaystyle\delta(\widehat{R}^{\prime\alpha}\widehat{R}_{\alpha}^{\prime}-%
\rho^{\prime 2})$$
$$\displaystyle=$$
$$\displaystyle\Theta(\sigma^{2}+\xi^{\alpha}\xi_{\alpha})\delta(\widehat{R}^{%
\prime\alpha}\widehat{R}_{\alpha}^{\prime}-\sigma^{2}-X^{\prime\alpha}X_{%
\alpha}^{\prime})+$$
(56)
$$\displaystyle+\widehat{\Theta}(-\xi^{\alpha}\xi_{\alpha}-\sigma^{2})\delta(%
\widehat{R}^{\prime\alpha}\widehat{R}_{\alpha}^{\prime}),$$
with $\Theta(x)$ and $\widehat{\Theta}(x)$ denoting respectively the
strong and weak Heaviside step functions
$$\displaystyle\widehat{\Theta}(x)$$
$$\displaystyle=$$
$$\displaystyle\left\{\begin{array}[]{lll}1&&x\geq 0\\
0&&x<0\end{array}\right.$$
(57)
$$\displaystyle\Theta(x)$$
$$\displaystyle=$$
$$\displaystyle\left\{\begin{array}[]{lll}1&&x>0\\
0&&x\leq 0.\end{array}\right.$$
(58)
On the other hand, the only contribution to the integral (53) arises (because of the Dirac-delta in the current density)
from the subdomain for which $-\xi^{\alpha}\xi_{\alpha}-\sigma^{2}=0$.
Hence, $S_{C}^{(self)}$ simply reduces to the functional form:
$$\displaystyle S_{C}^{(self)}(r^{\prime},\left[r^{\prime}\right])$$
$$\displaystyle=$$
$$\displaystyle\frac{2q^{2}}{4\pi\sigma^{2}c}\int_{0}^{\pi}d\vartheta^{\prime}%
\sin\vartheta^{\prime}\int_{0}^{2\pi}d\varphi^{\prime}\int_{0}^{+\infty}d\rho^%
{\prime}\rho^{{}^{\prime}2}\times$$
(59)
$$\displaystyle\times\int_{-\infty}^{+\infty}ds^{\prime\prime}u_{\mu}^{\prime}%
\left(s^{\prime\prime}\right)\delta(\widehat{R}^{\prime\alpha}\widehat{R}_{%
\alpha}^{\prime})\int_{-\infty}^{+\infty}dsu^{\prime\mu}(s)\delta(\left|x^{%
\prime}\left(s\right)\right|-\sigma).$$
The remaining spatial integration can now be performed letting
$$\rho^{\prime}\equiv\left|x^{\prime}\left(s\right)\right|$$
(60)
and making use of the spherical symmetry of the charge distribution. The
constraints placed by the two Dirac-delta functions $\delta(\widehat{R}^{\prime\alpha}\widehat{R}_{\alpha}^{\prime})$ and $\delta(\left|x^{\prime}\left(s\right)\right|-\sigma)$ in the previous equation
imply that both $\widehat{R}^{\prime\alpha}\widehat{R}_{\alpha}^{\prime}$
and $\left|x^{\prime}\left(s\right)\right|$ are 4-scalars.
Then, introducing the representation
$$\widehat{R}^{\prime\alpha}\equiv r^{\prime\alpha}-r^{\prime\alpha}(s^{\prime%
\prime})=\widetilde{R}^{\prime\alpha}+x^{\prime\alpha}\left(s\right),$$
(61)
with
$$\displaystyle\widetilde{R}^{\prime\alpha}$$
$$\displaystyle\equiv$$
$$\displaystyle r^{\prime\alpha}\left(s\right)-r^{\prime\alpha}(s^{\prime\prime}),$$
(62)
$$\displaystyle x^{\prime\alpha}\left(s\right)$$
$$\displaystyle\equiv$$
$$\displaystyle r^{\prime\alpha}-r^{\prime\alpha}\left(s\right),$$
(63)
it follows that
$$\widehat{R}^{\prime\alpha}\widehat{R}_{\alpha}^{\prime}=\widetilde{R}^{\prime%
\alpha}\widetilde{R}_{\alpha}^{\prime}+x^{\prime\alpha}\left(s\right)x_{\alpha%
}^{\prime}\left(s\right)+2\widetilde{R}^{\prime\alpha}x_{\alpha}^{\prime}\left%
(s\right)$$
(64)
is necessarily a 4-scalar independent of the integration angles $\left(\varphi^{\prime},\vartheta^{\prime}\right)$ when evaluated on the
hypersurface $\Sigma:\widehat{R}^{\prime\alpha}\widehat{R}_{\alpha}^{\prime}=0$. Similarly, the Dirac-delta $\delta(\left|x^{\prime}\left(s\right)\right|-\sigma)$ warrants that $x^{\prime\alpha}\left(s\right)x_{\alpha}^{\prime}\left(s\right)=-\sigma^{2},$ which is manifestly a 4-scalar too. Let us now prove that necessarily
$$\widetilde{R}^{\prime\alpha}x_{\alpha}^{\prime}\left(s\right)\equiv 0.$$
(65)
In fact, on $\Sigma$ it must be
$$\displaystyle\frac{d}{ds}\left[\widehat{R}^{\prime\alpha}\widehat{R}_{\alpha}^%
{\prime}\right]$$
$$\displaystyle=$$
$$\displaystyle\frac{d}{ds^{\prime\prime}}\left[\widehat{R}^{\prime\alpha}%
\widehat{R}_{\alpha}^{\prime}\right]=0,$$
(66)
$$\displaystyle\frac{d}{ds}\left[\widetilde{R}^{\prime\alpha}x_{\alpha}^{\prime}%
\left(s\right)\right]$$
$$\displaystyle=$$
$$\displaystyle u^{\prime\alpha}\left(s\right)x_{\alpha}^{\prime}\left(s\right)-%
\widetilde{R}^{\prime\alpha}u_{\alpha}^{\prime}\left(s\right)=-\widetilde{R}^{%
\prime\alpha}u_{\alpha}^{\prime}\left(s\right)=-\frac{1}{2}\frac{d}{ds}\left[%
\widetilde{R}^{\prime\alpha}\widetilde{R}_{\alpha}^{\prime}\right],$$
(67)
$$\displaystyle\frac{d}{ds^{\prime\prime}}\left[\widetilde{R}^{\prime\alpha}x_{%
\alpha}^{\prime}\left(s\right)\right]$$
$$\displaystyle=$$
$$\displaystyle-u^{\prime\alpha}\left(s^{\prime\prime}\right)x_{\alpha}^{\prime}%
\left(s\right),$$
(68)
$$\displaystyle\frac{d}{ds^{\prime\prime}}\left[\widetilde{R}^{\prime\alpha}%
\widetilde{R}_{\alpha}^{\prime}\right]$$
$$\displaystyle=$$
$$\displaystyle-2\widetilde{R}^{\prime\alpha}u_{\alpha}^{\prime}\left(s^{\prime%
\prime}\right).$$
(69)
Therefore,
$$\frac{d}{ds}\left[\widehat{R}^{\prime\alpha}\widehat{R}_{\alpha}^{\prime}%
\right]=\frac{d}{ds}\left[\widetilde{R}^{\prime\alpha}\widetilde{R}_{\alpha}^{%
\prime}+2\widetilde{R}^{\prime\alpha}x_{\alpha}^{\prime}\left(s\right)\right]=0,$$
(70)
$$\displaystyle\frac{d}{ds^{\prime\prime}}\left[\widehat{R}^{\prime\alpha}%
\widehat{R}_{\alpha}^{\prime}\right]$$
$$\displaystyle=$$
$$\displaystyle\frac{d}{ds^{\prime\prime}}\left[\widetilde{R}^{\prime\alpha}%
\widetilde{R}_{\alpha}^{\prime}+2\widetilde{R}^{\prime\alpha}x_{\alpha}^{%
\prime}\left(s\right)\right]=$$
(71)
$$\displaystyle=$$
$$\displaystyle-2\widetilde{R}^{\prime\alpha}u_{\alpha}^{\prime}\left(s^{\prime%
\prime}\right)-2u^{\prime\alpha}\left(s^{\prime\prime}\right)x_{\alpha}^{%
\prime}\left(s\right)=0,$$
from which it follows that, on $\Sigma$, $\widetilde{R}^{\prime\alpha}$
is a 4-vector, since by definition both $u_{\alpha}^{\prime}\left(s^{\prime\prime}\right)$ and $x_{\alpha}^{\prime}\left(s\right)$ are
4-vectors too. Now we notice that
$$\widetilde{R}^{\prime\alpha}\widetilde{R}_{\alpha}^{\prime}=f\left(s,s^{\prime%
\prime}\right)=f\left(s^{\prime\prime},s\right),$$
(72)
with $f$ being a 4-scalar which is symmetric with respect to $s$ and $s^{\prime\prime}$, while by construction
$$\widetilde{R}^{\prime\alpha}x_{\alpha}^{\prime}\left(s\right)=g\left(s,s^{%
\prime\prime},\sigma\right)\neq g\left(s^{\prime\prime},s,\sigma\right),$$
(73)
where $g$ is a non-symmetric 4-scalar with respect to the same parameters.
On the other hand, Eq.(66) requires that $\widehat{R}^{\prime\alpha}\widehat{R}_{\alpha}^{\prime}$ must be symmetric in both $s$ and $s^{\prime\prime}$, so that, thanks to Eqs.(72) and (73),
we can conclude that $g=g\left(\sigma\right)$ is a constant 4-scalar
which can depend at most on $\sigma$. To determine the precise value of $g=\widetilde{R}^{\prime\alpha}x_{\alpha}^{\prime}\left(s\right)$ we
evaluate it in the COS comoving reference frame, where by definition $r_{COS}^{\mu}\left(s_{0}\right)=\left(s_{0},\mathbf{0}\right)$ for all
the COS proper times $s_{0}\in[-\infty,+\infty]$. In this frame $\widetilde{R}^{\prime\alpha}=\left(s-s^{\prime\prime},\mathbf{0}\right)$ has only time component and when $s_{0}=s$ we get $g=\widetilde{R}^{\prime\alpha}x_{\alpha}^{\prime}\left(s\right)=0$ identically. On the other
hand, since $g$ is a 4-scalar, it is independent of both $s$ and $s^{\prime\prime}$ and it is null when $s_{0}=s$, we conclude that it must be null
for all $s_{0}$ and in any reference frame, which proves Eq.(65).
Hence, as a result of the integration, the action integral $S_{C}^{(self)}$
takes necessarily the expression
$$S_{C}^{(self)}(r^{\prime},\left[r^{\prime}\right])=\frac{2q^{2}}{c}\int_{1}^{2%
}dr_{\mu}^{\prime}\left(s^{\prime\prime}\right)\int_{1}^{2}dr^{\prime\mu}(s^{%
\prime})\delta(\widetilde{R}^{\prime\alpha}\widetilde{R}_{\alpha}^{\prime}-%
\sigma^{2}).$$
(74)
Finally, since by construction $S_{C}^{(self)}$ is a 4-scalar, it follows
that the primes can be dropped thus yelding the following representation
holding in a general reference frame:
$$S_{C}^{(self)}(r,\left[r\right])=\frac{2q^{2}}{c}\int_{1}^{2}dr_{\mu}\left(s%
\right)\int_{1}^{2}dr^{\mu}(s^{\prime})\delta(\widetilde{R}^{\alpha}\widetilde%
{R}_{\alpha}-\sigma^{2}),$$
(75)
where for simplicity of notation $s^{\prime\prime}$ has been replaced with
$s^{\prime}$ and $\widetilde{R}^{\alpha}$ now denotes
$$\widetilde{R}^{\alpha}\equiv r^{\alpha}\left(s\right)-r^{\alpha}(s^{\prime}).$$
(76)
It is worth pointing out the following basic properties of the functional $S_{C}^{(self)}$:
1) it is a non-local functional in the sense that it contains a coupling
between the past and the future of the dynamical system (see Eq.(3)). In fact it can be equivalently represented as
$$S_{C}^{(self)}(r,\left[r\right])=\frac{2q^{2}}{c}\int_{-\infty}^{+\infty}ds%
\frac{dr_{\mu}\left(s\right)}{ds}\int_{-\infty}^{+\infty}ds^{\prime}\frac{dr^{%
\mu}(s^{\prime})}{ds^{\prime}}\delta(\widetilde{R}^{\alpha}\widetilde{R}_{%
\alpha}-\sigma^{2});$$
(77)
2) furthermore, it is symmetric, namely it fulfills the property
$$S_{C}^{(self)}(r_{A},\left[r_{B}\right])=S_{C}^{(self)}(r_{B},\left[r_{A}%
\right]),$$
(78)
where $r_{A}$ and $r_{B}$ are two arbitrary curves of the functional class $\left\{f\right\}$ (see Eq.(1)).
V.2 $S_{C}^{\left(ext\right)}(r)$: EM coupling with the external
field
The action integral $S_{C}^{\left(ext\right)}(r)$ of the EM coupling with
the external field is a 4-scalar defined as
$$S_{C}^{\left(ext\right)}(r)=\int_{1}^{2}d\Omega\frac{1}{c^{2}}A^{(ext)\mu}(r)j%
_{\mu}\left(r\right),$$
(79)
where $A^{(ext)\mu}(r)$ is the 4-vector potential of the external field,
assumed to be assigned, and $j_{\mu}\left(r\right)$ is the current
density given by Eq.(22). The evaluation of the action integral $S_{C}^{\left(ext\right)}$ proceeds exactly in the same way as outlined for
$S_{C}^{\left(self\right)},$ with the introduction of the Lorentz boost (43), the spherical spatial coordinates and the use of the result
from Lemma 3. The only difference now is that the vector potential $A^{(ext)\mu}(r)$ does not possess spherical symmetry when evaluated in $S_{NI}$. As a result, spatial integration over the angle variables $\vartheta^{\prime}$ and $\varphi^{\prime}$ cannot be computed
explicitly. This leads to the introduction of the surface average EM
external 4-potential $\overline{A}^{(ext)\mu},$ which is defined in $S_{NI}$ as
$$\overline{A}^{\prime(ext)\mu}\left(r^{\prime}\left(s\right),|x^{\prime}|\right%
)\equiv\frac{1}{4\pi}\int_{0}^{2\pi}d\varphi^{\prime}\int_{0}^{\pi}d\vartheta^%
{\prime}\sin\vartheta^{\prime}\left[A^{{}^{\prime}(ext)\mu}(r^{\prime\mu}(s)+x%
^{\prime\mu})\right],$$
(80)
where we have used the relation (14). With this definition, the time
and radial integrals can then be calculated using the Dirac-delta functions
as outlined for the self-coupling action integral. After performing a final
transformation to an arbitrary reference frame, this gives the following
expression for $S_{C}^{\left(ext\right)}$:
$$S_{C}^{(ext)}(r)=\frac{q}{c}\int_{1}^{2}\overline{A}^{(ext)\mu}\left(r^{\mu}%
\left(s\right),\sigma\right)dr_{\mu}(s).$$
(81)
V.3 $S_{\chi}(u,\chi)$: kinematic constraint
The kinematic constraint concerns the normalization of the extremal
4-velocity of the COS. This is defined as
$$S_{\chi}(u,\chi)\equiv\int_{-\infty}^{+\infty}ds\chi(s)\left[u_{\mu}(s)u^{\mu}%
(s)-1\right],$$
(82)
where $\chi(s)$ is a Lagrange multiplier.
V.4 $S_{M}(r,u)$: inertial mass functional
The action integral $S_{M}$ of the inertial mass for the extended particle
is here defined as the following 4-scalar:
$$S_{M}(r,u)\equiv\int_{1}^{2}d\Omega\frac{1}{c}g_{\mu\nu}T_{M}^{\mu\nu}\left(r%
\right),$$
(83)
where $d\Omega$ denotes the invariant 4-volume element and $T_{M}^{\mu\nu}$ the stress-energy tensor corresponding to the mass distribution of the
finite-size charged particle. Notice that the choice of $S_{M}$ is
consistent with the customary definition of the stress-energy tensor $T^{\mu\nu}$ (for a fluid or a field) in terms of $T^{\mu\nu}\equiv\frac{\delta L}{\delta g_{\mu\nu}}$, with $L$ being a suitable Lagrangian function and $\delta$ representing the variational derivative LL . Therefore, it is
natural to identify $S_{M}$ with the trace of the mass stress-energy tensor
for the extended particle. In particular, the explicit representation of $T_{M}^{\mu\nu}$ follows by projecting the mass current density $j_{mass}^{\mu}(r)$ given in Eq.(23) along the velocity of a
generic shell mass-element parameterized in terms of the proper time $s$ of
the COS. The procedure is completely analogous to that outlined in Section
3. Equivalently, $T_{M}^{\mu\nu}$ can also be derived by considering the
stress-energy tensor of a perfect fluid without pressure (since the mass is
located on a shell by assumption) and imposing the rigidity constraints (7) and (8). Accordingly, one obtains the following expression:
$$T_{M}^{\mu\nu}\left(r\right)\equiv\frac{m_{o}c^{2}}{4\pi\sigma^{2}}\int_{-%
\infty}^{+\infty}dsu^{\mu}(s)u^{\nu}\left(s\right)\left[1+\Gamma\right]\delta(%
\left|x\right|-\sigma)\delta(s-s_{1}),$$
(84)
with $s_{1}$ being the root of Eq.(21) and $\Gamma$ the contribution
of the Thomas precession. We notice that the stress-energy tensor thus
defined is symmetric. With this definition, the action integral $S_{M}$
becomes
$$S_{M}(r,u)\equiv\frac{m_{o}c}{4\pi\sigma^{2}}\int_{1}^{2}d\Omega\int_{-\infty}%
^{+\infty}dsu^{\mu}(s)u_{\mu}\left(s\right)\left[1+\Gamma\right]\delta(\left|x%
\right|-\sigma)\delta(s-s_{1}).$$
(85)
The integration over the 4-volume element can be performed explicitly in the
same way as explained before (for the EM coupling action integral), by using
the prescription of Lemma 3 and the transformation to local spatial
spherical coordinates. In particular, here we notice that both $u^{\mu}(s)$
and $\frac{du_{\alpha}}{ds}$ appearing in $\Gamma$ are independent of the
integration variables, while in the reference system $S_{NI}$ introduced in
Lemma 3 we have that $\int_{0}^{\pi}d\vartheta^{\prime}\sin\vartheta^{\prime}\int_{0}^{2\pi}d\varphi%
^{\prime}\xi^{\alpha}=0,$ as it follows from the property of $\xi^{\alpha}$ to be a pure spatial
vector in $S_{NI}$. Thanks to this feature, the whole integral is
straightforward, so that one obtains for $S_{M}(r,u)$ the final expression:
$$S_{M}(r,u)\equiv\int_{1}^{2}m_{o}cu_{\mu}dr^{\mu}$$
(86)
holding in an arbitrary reference frame. Concerning the solution (86), a remark is in order. The choice of $S_{M}$ given by Eq.(83)
proves to be the correct one. In fact, as expected Eq.(86) is
formally the same action integral of a point particle, with the difference
that here $u_{\mu}$ represents the 4-velocity of the COS rather than the
one of a point mass.
VI The variational Lagrangian
In this section we collect together all of the contributions to $S_{1}$
previously obtained. From the results of the previous section we can write
the action integral $S_{1}$ as a line integral in terms of a variational
Lagrangian $L_{1}(r,\left[r\right],u,\chi)$ as follows [see Eq.(2)]:
$$S_{1}=\int_{-\infty}^{+\infty}dsL_{1}(r,\left[r\right],u,\chi).$$
(87)
More precisely, $L_{1}(r,\left[r\right],u,\chi)$ is defined as:
$$L_{1}(r,\left[r\right],u,\chi)=L_{M}(r,u)+L_{\chi}(u,\chi)+L_{C}^{(ext)}(r)+L_%
{C}^{(self)}(r,\left[r\right]),$$
(88)
where
$$\displaystyle L_{M}(r,u)$$
$$\displaystyle=$$
$$\displaystyle m_{o}cu_{\mu}\frac{dr^{\mu}}{ds},$$
(89)
$$\displaystyle L_{\chi}(u,\chi)$$
$$\displaystyle=$$
$$\displaystyle\chi(s)\left[u_{\mu}(s)u^{\mu}(s)-1\right],$$
(90)
$$\displaystyle L_{C}^{(ext)}(r)$$
$$\displaystyle=$$
$$\displaystyle\frac{dr}{ds}^{\mu}\frac{q}{c}\overline{A}_{\mu}^{(ext)}(r(s),%
\sigma),$$
(91)
denote the local contributions respectively from the inertial, the
constraint and the external EM field coupling terms, while
$$L_{C}^{(self)}(r,\left[r\right])=\frac{2q^{2}}{c}\frac{dr}{ds}^{\mu}\int_{1}^{%
2}dr_{\mu}^{\prime}\delta(\widetilde{R}^{\mu}\widetilde{R}_{\mu}-\sigma^{2})$$
(92)
represents the non-local contribution arising from the EM self-field
coupling.
The conclusion is remarkable. Indeed, although the extended particle can be
regarded as a continuous system carrying mass and charge current densities,
the variational functional here determined is similar to that of a point
particle subject to appropriate interactions. In fact, because of the
rigidity constraint and the spherical symmetry imposed on the charge and
mass distributions, the variational action $S_{1}$ is actually reduced from
a volume integral to a line integral over the proper time of the COS. This
is realized by means of the volume integration performed in the reference
frame $S_{NI}$ and thanks to Lemma 3.
The procedure introduces the surface-average operator acting both on the
external and the self EM coupling terms. As a result, the Lagrangian (88) must be interpreted as prescribing the dynamics for
the COS of the charged particle in terms of averaged EM fields, integrating
all the force contributions to the translational motion on the shell.
Furthermore, we recall once again the formal analogy between the Lagrangian $L_{M}(r,u)$ and the one of a point particle, when $u_{\mu}$ is interpreted
as the 4-velocity of the point mass rather than that of the COS of the
shell. This means that the dynamics of the finite-size particle is
effectively described in terms of a point particle with a finite-size charge
distribution. Hence, the mathematical problem is formally the same of
that for a Lorentzian particle. Therefore, this proves that the particular
case of a Lorentzian particle is formally included in the present
description, in the limit in which the radius of the mass distribution $\sigma_{m}$ is sent to zero while keeping the charge spatial extension
fixed ($\sigma>0$). The conclusion manifestly follows within the framework
of special relativity, in which any possible curvature effects due to the EM
field and the mass of the particle itself are neglected.
VII The variational principle and the RR equation
In this section we shall determine the explicit form of the relativistic RR
equation for the non-rotating charged particle. As pointed out earlier Tessarotto2008c , this goal can be uniquely attained by means of a synchronous variational principle, in analogy with the approach originally
developed for point particles by Nodvik in terms of an asynchronous
principle (Nodvik, 1964 Nodvik1964 ). In particular, we intend to
prove that, in the present case, the exact RR equation can be uniquely
and explicitly obtained by using the hybrid synchronous Hamilton
variational principle defined in the previous section and given by Eq.(42). In this case the action functional is expressed by means of
superabundant hybrid (i.e., non-Lagrangian) variables and the variations are
considered as synchronous, i.e., they are performed by keeping constant the
particle COS proper time. Taking into account the results presented in the
previous sections, the appropriate form of the Hamilton variational
principle is given by the following theorem:
THM.1 - Hybrid synchronous Hamilton variational principle
In validity of the SR-CE axioms, let us assume that:
1.
the Hamilton action $S_{1}(r,u,\chi,\left[r\right])$ is defined by Eq.(42), with $A_{\mu}^{(self)}$ given by Eq.(40) and $\chi(s)$ being a suitable Lagrange multiplier;
2.
the real functions $f(s)$ in the functional class $\left\{f\right\}$ [see Eq.(1)] are
identified with
$$\emph{\ }f(s)\equiv\left[r^{\mu}(s),u_{\mu}(s),\chi(s)\right],$$
(93)
with synchronous variations $\delta f(s)\equiv$ $f(s)-$ $f_{1}(s)$ belonging to
$$\displaystyle\left\{\delta f\right\}$$
$$\displaystyle\equiv$$
$$\displaystyle\delta f_{i}(s):\delta f_{i}(s)=f_{i}(s)-f_{1i}(s);$$
(94)
$$\displaystyle\text{ }i=1,n\emph{\ and }\forall f(s),f_{1}(s)\in\left\{f\right\},$$
here referred to as the functional class of synchronous variations;
3.
the extremal curve $f\in\left\{f\right\}$ of $S_{1},$
which is the solution of the equation
$$\delta S_{1}(r,u,\chi,\left[r\right])=0,$$
(95)
exists for arbitrary variations $\delta f(s)$ (hybrid
synchronous Hamilton variational principle);
4.
if $r^{\mu}(s)$ is extremal, the line element $ds$ satisfies the constraint $ds^{2}=\eta_{\mu\nu}dr^{\mu}(s)dr^{\nu}(s)$;
5.
the 4-vector field $A_{\mu}^{(ext)}(r)$ is suitably
smooth in the whole Minkowski space-time $M^{4}$;
6.
the E-L equation for the extremal curve $r^{\mu}(s)$ is
determined subject to the constraint that the delay-time $s_{ret}$ (namely
the root of the delay-time equation (103) below) must
be chosen consistently with ECP.
It then follows that:
T1${}_{1})$ If all the synchronous variations $\delta f_{i}(s)$
(i=1,n) are considered as being independent, the E-L
equations for $\chi(s)$ and $u_{\mu}$ following from the synchronous
hybrid Hamilton variational principle (95) give respectively
$$\displaystyle\left.\frac{\delta S_{1}}{\delta\chi(s)}=u_{\mu}u^{\mu}-1=0,\right.$$
(96)
$$\displaystyle\left.\frac{\delta S_{1}}{\delta u_{\mu}}=m_{o}cdr^{\mu}+2\chi u^%
{\mu}ds=0.\right.$$
(97)
Instead, the E-L equation for $r_{\mu}$
$$\frac{\delta S_{1}}{\delta r^{\mu}(s)}=0$$
(98)
yields the following covariant (and hence also MLC) 4-vector,
second-order delay-type ODE:
$$m_{o}c\frac{du_{\mu}(s)}{ds}=\frac{q}{c}\overline{F}_{\mu\nu}^{(ext)}(r(s))%
\frac{dr^{\nu}(s)}{ds}+\frac{q}{c}\overline{F}_{\mu k}^{\left(self\right)}%
\left(r\left(s\right),r\left(s^{\prime}\right)\right)\frac{dr^{k}(s)}{ds},$$
(99)
which is identified with the RR equation of motion for the COS of a
spherical shell non-rotating charge particle. Here
$$\overline{F}_{\mu\nu}^{(ext)}\equiv\partial_{\mu}\overline{A}_{\nu}^{(ext)}-%
\partial_{\nu}\overline{A}_{\mu}^{(ext)}$$
(100)
denotes the surface-average [defined according to Eq.(80)] of the Faraday tensor carried by the externally-generated EM 4-vector and
evaluated at the particle 4-position $r^{\mu}(s).$ In addition, $\overline{F}_{\mu k}^{\left(self\right)}$ - in MLC 4-vector
representation - is the surface-averaged Faraday tensor of the corresponding
EM self-field, given by
$$\overline{F}_{\mu k}^{\left(self\right)}=-\frac{2q}{\left|\widetilde{R}^{%
\alpha}u_{\alpha}(s^{\prime})\right|}\frac{d}{ds^{\prime}}\left\{\frac{u_{\mu}%
(s^{\prime})\widetilde{R}_{k}-u_{k}(s^{\prime})\widetilde{R}_{\mu}}{\widetilde%
{R}^{\alpha}u_{\alpha}(s^{\prime})}\right\}_{s^{\prime}=s-s_{ret}}.$$
(101)
Imposing the constraint $ds^{\prime}=\gamma\left(t^{\prime}\right)cdt^{\prime}$, this implies also
$$\displaystyle\overline{F}_{\mu k}^{\left(self\right)}$$
$$\displaystyle=$$
$$\displaystyle-\frac{2q}{c\left|(t-t^{\prime})-\frac{1}{c^{2}}\frac{d\mathbf{r}%
(t^{\prime})}{dt^{\prime}}\cdot(\mathbf{r-r}\left(t^{\prime}\right))\right|}$$
(102)
$$\displaystyle\frac{d}{dt^{\prime}}\left\{\frac{v_{\mu}(t^{\prime})\widetilde{R%
}_{k}-v_{k}(t^{\prime})\widetilde{R}_{\mu}}{c^{2}\left[(t-t^{\prime})-\frac{1}%
{c^{2}}\frac{d\mathbf{r}(t^{\prime})}{dt^{\prime}}\cdot(\mathbf{r-r}\left(t^{%
\prime}\right))\right]}\right\}_{t^{\prime}=t-t_{ret}}.$$
Here $u^{\mu}=\frac{dr^{\mu}}{ds}$ denotes the COS
4-velocity and $v^{\mu}(t)=\frac{dr^{\mu}}{dt}$, while $s_{ret}=s-s^{\prime}$ is the positive root of the delay-time equation
$$\widetilde{R}^{\alpha}\widetilde{R}_{\alpha}-\sigma^{2}=0.$$
(103)
T1${}_{2})$ The E-L equations (96),(97) and (98) imply that the extremal
functional takes the form
$$S(r,\left[r\right],u,)=S_{1}(r,\left[r\right],u,\chi(s)=-\frac{m_{o}c}{2}).$$
(104)
T1${}_{3})$ If $F_{\mu}^{(ext)\nu}(r)\equiv 0$ for all $s\leq s_{1}\in\mathbb{R},$ a particular solution of Eq.(99), holding for all $s\leq s_{1}$ is provided
by the inertial motion, i.e.,
$$\displaystyle\frac{dr^{\mu}(s)}{ds}$$
$$\displaystyle=$$
$$\displaystyle u_{o}^{\mu}=const.,$$
(105)
$$\displaystyle\frac{du^{\mu}}{ds}$$
$$\displaystyle=$$
$$\displaystyle 0,$$
(106)
in agreement with the Galilei principle of inertia.
T1${}_{4})$ The RR equation Eq.(99) also
holds for a Lorentzian particle having the same charge distribution of the
finite-size particle $(\sigma>0)$ and carrying a point-mass with
position and velocity 4-vectors $r^{\mu}(s),$ $u^{\mu}(s)$.
Proof - T1${}_{1})$ and T1${}_{2})$ The proof proceeds as
follows. Since $\frac{\partial}{\partial u^{\mu}}\delta(\widetilde{R}^{\alpha}\widetilde{R}_{%
\alpha}-\sigma^{2})=\frac{\partial}{\partial u^{\prime\mu}}\delta(\widetilde{R%
}^{\alpha}\widetilde{R}_{\alpha}-\sigma^{2})\equiv 0$, the variations with respect to $\chi(s)$ and $u_{\mu}$ deliver respectively the two E-L equations (96) and (97). Hence,
the Lagrange multiplier $\chi$ must be for consistency
$$2\chi=-m_{o}c,$$
(107)
so that, ignoring gauge contributions with respect to $\chi$, the extremal
functional $S_{1}(r,u,\chi,\left[r\right])$ takes the form (104) [statement T1${}_{2}$]. To prove also Eq.(98), we notice that the synchronous
variation of $S_{C}^{(self)}$ has the form
$$\delta S_{C}^{(self)}=\delta A+\delta B,$$
(108)
where
$$\begin{array}[]{c}\delta A\equiv-\frac{4q^{2}}{c}\eta_{\mu\nu}\int_{1}^{2}%
\delta r^{\mu}d\left[\int_{1}^{2}dr^{\prime\nu}\delta(\widetilde{R}^{\alpha}%
\widetilde{R}_{\alpha}-\sigma^{2})\right],\\
\delta B\equiv\frac{4q^{2}}{c}\eta_{\alpha\beta}\int_{1}^{2}dr^{\prime\beta}%
\int_{1}^{2}dr^{\alpha}\delta r^{\mu}\frac{\partial}{\partial r^{\mu}}\delta(%
\widetilde{R}^{k}\widetilde{R}_{k}-\sigma^{2}),\end{array}$$
(109)
and $r^{\prime\nu}\equiv r^{\nu}\left(s^{\prime}\right)$ and $r^{\nu}\equiv r^{\nu}\left(s\right)$. Then we can write
$$\displaystyle\delta A$$
$$\displaystyle\equiv$$
$$\displaystyle\frac{2q}{c}\eta_{\mu\nu}\int_{1}^{2}\delta r^{\mu}dr^{k}\left[B_%
{k}^{\nu}\right]_{t^{\prime}=t-t_{ret}},$$
(110)
$$\displaystyle\delta B$$
$$\displaystyle\equiv$$
$$\displaystyle-\frac{2q}{c}\eta_{\alpha\beta}\int_{1}^{2}\delta r^{\mu}dr^{%
\alpha}\left[B_{\mu}^{\beta}\right]_{t^{\prime}=t-t_{ret}},$$
where $B_{k}^{\nu}$ is
$$B_{k}^{\nu}\equiv-\frac{q}{c\left|(t^{\prime}-t)-\frac{1}{c^{2}}\frac{d\mathbf%
{r}(t^{\prime})}{dt^{\prime}}\cdot(\mathbf{r}^{\prime}\mathbf{-r})\right|}%
\frac{d}{dt^{\prime}}\left\{\frac{v^{\nu}(t^{\prime})\widetilde{R}_{k}}{c^{2}%
\left[(t^{\prime}-t)-\frac{1}{c^{2}}\frac{d\mathbf{r}(t^{\prime})}{dt^{\prime}%
}\cdot(\mathbf{r-r}^{\prime})\right]}\right\}$$
(111)
(the details of the derivation of these identities are provided in Appendix
A). Finally, from the results given in Appendix A, the variation with
respect to $r^{\mu}$ yields
$$\frac{\delta S_{1}}{\delta r^{\mu}}=-m_{o}cdu_{\mu}(s)+\frac{q}{c}dr^{k}%
\overline{F}_{\mu k}^{\left(self\right)}+\frac{q}{c}\left[\partial_{\mu}%
\overline{A}_{\nu}^{(ext)}(r(s))-\partial_{\nu}\overline{A}_{\mu}^{(ext)}(r(s)%
)\right]dr^{\nu},$$
(112)
where
$$\overline{F}_{\mu k}^{\left(self\right)}=2(B_{k\mu}-B_{\mu k}),$$
(113)
from which Eqs.(99)-(103) follow. This
yields the RR equation being sought, i.e., the exact
relativistic equation of motion for the translational dynamics of the COS of
a finite-size spherical shell charge particle subject to the simultaneous
action of a prescribed external EM field and of its EM self-field.
T1${}_{3})$ The proof of Eqs.(105)-(106) is
straightforward. In fact, let us assume that in the interval $\left[-\infty,s_{1}\right]$ the motion is inertial, namely that $\frac{d}{ds}u_{\mu}\equiv 0$,$\forall s$ $\in\left[-\infty,s_{1}\right].$ This
implies that in $\left[-\infty,s_{1}\right]$ it must be $u_{\mu}\equiv u_{0\mu},$ with $u_{0\mu}$ denoting a constant 4-vector velocity. It
follows that $\forall s,s^{\prime}\in$ $\left[-\infty,s_{1}\right],$ $r_{\mu}(s)=r_{\mu}(s^{\prime})+u_{0\mu}(s^{\prime})(s-s^{\prime})$ and
$R_{\mu}=u_{0\mu}(s)(s-s^{\prime}).$ Hence, by direct substitution in Eq.(102) we get that $v_{\mu}(t^{\prime})\widetilde{R}{}_{k}-v_{k}(t^{\prime})\widetilde{R}_{\mu}=0$, which by consequence implies
also that $dr^{k}H_{\mu k}\equiv 0$ identically in this case.
T1${}_{4})$ The proof follows immediately from the definition of
Lorentzian particle given above by noting that in the context of SR the
variational particle Lagrangian $L_{1}(r,\left[r\right],u,\chi)$ [see Eq.(88)] formally coincides with that of a Lorentzian
particle characterized by a finite charge distribution [i.e., with $\sigma>0],$ subject to the simultaneous action of the averaged external and EM
self-fields $\overline{F}_{\mu\nu}^{(ext)}$ and $\overline{F}_{\mu k}^{\left(self\right)}.$
Q.E.D.
We notice that, by assumption, the varied functions $f(s)\equiv\left[r^{\mu}(s),u_{\mu}(s),\chi(s)\right]$ are unconstrained, namely they are solely subject to the requirement that end
points and boundary values are kept fixed. This implies that all of the 9
components of the variations $\delta f(s),$ namely $\delta r^{\mu}(s),\delta u_{\mu}(s),\delta\chi(s),$ must be considered independent. On
the other hand, the extremal curves $f(s)$ of $S_{1}(r,u,\chi,\left[r\right]),$ the solution of the hybrid Hamilton variational principle,
satisfy all of the required physical constraints, so that only 6 of them are
actually independent. In fact, the resulting E-L equations determine,
besides the RR equation (99), also the relationship
between $r^{\mu}(s)$ and $u_{\mu}(s)$, namely
$$\left.u^{\mu}(s)=\frac{dr^{\mu}(s)}{ds},\right.$$
(114)
as well as the physical constraint
$$\left.u^{\mu}(s)u_{\mu}(s)=1.\right.$$
(115)
As a consequence, $r^{\mu}(s)$ and $u^{\mu}(s)$ coincide respectively with
the physical 4-position and 4-velocity of the COS mass particle. Therefore
only 3 components of the 4-velocity are actually independent, while the
first component of the 4-position $ct$ can always be represented in terms of
the proper length $s$ (so that only the spatial part of the position
4-vector actually defines a set of independent Lagrangian coordinates)$.$
A further basic feature of the RR equation concerns the validity of GIP and
its meaning in this context. In fact, let us assume that the external EM
field is non-vanishing in the time interval $I_{12}\equiv\left[s_{1},s_{2}\right],$ while it vanishes identically in $I_{2}\equiv\left[s_{2},+\infty\right].$ Then, the inertial solution (105) and (106) does not hold, by definition, in $I_{12}$ and is only
achieved in an asymptotic sense in $I_{2}$, i.e., in the limit $s\rightarrow+\infty.$ In fact, the non-local feature of the RR effect prevents the
particle from reaching the inertial state in a finite time interval. It is
concluded, therefore, that GIP must be intended as holding in the past, namely in the time interval $s\leq s_{1}\in\mathbb{R},$ where by
assumption no external EM field is acting on the particle.
VIII Standard Lagrangian and conservative forms of the RR equation
In this section we discuss some developments about the physical properties
of the non-local RR equation obtained, which exactly describes the
translational dynamics of the COS of a spherical-shell non-rotating charged
particle. Remarkably, the variational principle (THM.1) implies that the E-L
equations (96)-(99) can
be cast in an equivalent way either:
1) in a standard Lagrangian form, namely expressed in the form of Lagrange
equations defined in terms of a suitable non-local effective Lagrangian $L_{eff};$
2) in a conservative form, as the divergence of a suitable effective
stress-energy tensor.
The result is provided by the following theorem.
THM.2 - RR equation in standard Lagrangian and conservative forms
Given validity of THM.1, it follows that:
T2${}_{1})$ Introducing the non-local real function
$$L_{eff}\equiv L_{M}(r,u)+L_{\chi}(u,\chi)+L_{C}^{(ext)}(r)+2L_{C}^{(self)}(r,%
\left[r\right]),$$
(116)
here referred to as non-local effective Lagrangian, the E-L equations (96),(97) and (112) take respectively the form
$$\displaystyle\left.\frac{\partial L_{eff}}{\partial\chi(s)}=0,\right.$$
(117)
$$\displaystyle\left.\frac{\partial L_{eff}}{\partial u_{\mu}(s)}=0,\right.$$
(118)
$$\displaystyle\left.\frac{d}{ds}\frac{\partial L_{eff}}{\partial\frac{dr^{\mu}(%
s)}{ds}}-\frac{\partial L_{eff}}{\partial r^{\mu}(s)}=0.\right.$$
(119)
These will be referred to as E-L equations in standard
Lagrangian form.
T2${}_{2})$ The stress-energy tensor of the system $T_{\mu\nu}$
is uniquely determined in terms of $L_{eff}.$ As a consequence, the
RR equation (99) can also be written in
conservative form as
$$\overline{T}_{\mu\nu,\nu}=0,$$
(120)
where $\overline{T}_{\mu\nu}\equiv\overline{T}_{\mu\nu}^{\left(M\right)}+\overline{T}%
_{\mu\nu}^{\left(EM\right)}$ is the
surface-averaged total stress energy tensor, obtained as the sum of the
corresponding tensors for the mass distribution and the EM field which
characterize the system.
Proof - T2${}_{1})$ The proof follows immediately by noting that
the Hamiltonian action (42) defines a symmetric functional with
respect to local and non-local dependencies, i.e., such that
$$S_{1}(r_{A},\left[r_{B}\right],u,\chi)=S_{1}(r_{B},\left[r_{A}\right],u,\chi).$$
(121)
Because the E-L equations (117)-(119) are written in terms of
local partial derivative differential operators, the effective Lagrangian $L_{eff}$ must be therefore distinguished from the corresponding variational
Lagrangian function $L_{1}$ which enters the Hamilton action and which
contains non-local contributions. These features imply the definition (116), which manifestly satisfies the E-L equations in
standard form (117)-(119).
T2${}_{2})$ The proof of this statement is straightforward, by first
recalling that the Lagrangian of the distributed mass is analogous to that
of a point mass particle. Moreover, the stress-energy tensor of the total EM
field $T_{\mu\nu}^{\left(EM\right)}$, to be defined in terms of $L_{eff}$
according to the standard definition (see for example Landau and Lifshitz
LL ) becomes
$$T_{\mu\nu}^{\left(EM\right)}=T_{\mu\nu}^{\left(EM-ext\right)}+T_{\mu\nu}^{%
\left(EM-self\right)}.$$
(122)
Then, given validity to the Maxwell equations, it follows that
$$T_{\mu\nu,\nu}^{\left(EM\right)}=F_{\mu\nu}j^{\nu}=\left[F_{\mu\nu}^{\left(ext%
\right)}+F_{\mu\nu}^{\left(self\right)}\right]j^{\nu}.$$
(123)
Gathering the mass and the field contributions, substituting the expressions
for $F_{\mu\nu}^{\left(ext\right)}$ and $F_{\mu\nu}^{\left(self\right)}$ obtained in THM.1, and performing the integration over the
4-volume element finally proves that the equation (120) actually
coincides with the extremal RR equation (99).
Q.E.D.
The expression (120) represents the conservative form of Eq.(99), and hence - consistent with the surface
integration procedure here adopted - it holds for the surface-averaged
EM external and self-fields $\overline{F}_{\mu\nu}^{(ext)}$ and $\overline{F}_{\mu\nu}^{\left(self\right)},$ defined respectively by Eqs.(100) and (101). It is important to remark
that the result holds both for finite-size and Lorentzian particles. On the
other hand, a local form of the conservative equation - analogous to Eq.(120) - and holding for the local EM fields is in principle achievable
too. However, this last conclusion generally applies only to finite-size
particles with the same support for the mass and charge distributions, i.e.,
for which Eq.(4) holds.
IX Short delay-time asymptotic approximation
In this section we consider the asymptotic properties of the RR equation,
considering the customary approximation in the treatment of the problem,
which leads to the LAD equation (Dirac, 1938 Dirac1938 ). This is the
power-series expansion of the retarded EM self-potential in terms of the
dimensionless parameter $\epsilon$ $\equiv\frac{(s-s^{\prime})}{s},$ to
be assumed as infinitesimal (short delay-time ordering), $s-s^{\prime}$ denoting the proper-time difference between observation ($s$) and
emission ($s^{\prime}$). The same approach was also adopted by Nodvik Nodvik1964 in the case of flat space-time and by DeWitt and Brehme DeWitt1960 and Crowley and Nodvik Crowley-Nodvik1978 in their
covariant generalizations of the LAD equation valid in curved space-time. It
is immediate to show that the following result holds:
THM.3 - First-order, short delay-time asymptotic approximation
Let us introduce the 4-vector $G_{\mu}$ defined as
$$dsG_{\mu}=\frac{q}{c}\overline{F}_{\mu k}^{\left(self\right)}dr^{k},$$
(124)
and invoke the asymptotic ordering
$$0<\epsilon\ll 1.$$
(125)
Then:
T3${}_{1})$ Neglecting corrections of order $\epsilon^{N},$
with $N\geq 1$ (first-order approximation)$,$ the
following asymptotic approximation holds for $G_{\mu}$
$$\left.G_{\mu}\cong\left\{-m_{oEM}c\frac{d}{ds}u_{\mu}+g_{\mu}\right\}\left[1+O%
(\epsilon)\right]\right.,$$
(126)
where $g_{\mu}$ denotes the 4-vector
$$g_{\mu}=\frac{2}{3}\frac{q^{2}}{c}\left[\frac{d^{2}}{ds^{2}}u_{\mu}-u_{\mu}(s)%
u^{k}(s)\frac{d^{2}}{ds^{2}}u_{k}\right],$$
(127)
with
$$m_{oEM}\equiv\frac{q^{2}}{c^{2}\sigma}\frac{1}{\left[1+\frac{(t-t^{\prime})}{2%
}\frac{d}{ds}\frac{1}{\gamma}\right]^{2}}$$
(128)
being the EM mass and $\gamma\left(t(s)\right)\equiv 1/\sqrt{\left(1-v^{2}(t(s)\right)/c^{2}}.$
T3${}_{2})$ The point-charge limit of the RR equation (99)$\ $does not exist.
Proof - T3${}_{1})$ The proof is straightforward and follows by
performing explicitly the perturbative expansion with respect to $\epsilon$. By dropping the terms which vanish in the limit $\epsilon\rightarrow 0,$
this yields Eq.(126). The proof of T3${}_{2}),$ instead,
follows by noting that the limit obtained by letting
$$\sigma\rightarrow 0^{+}$$
(129)
(point-charge limit) is not defined, since
$$\lim_{\sigma\rightarrow 0^{+}}m_{oEM}=\infty.$$
(130)
Q.E.D.
As basic consequences, in the first-order approximation the RR equation (99) recovers the LAD equation. Moreover, in a similar
way, by introducing a suitable approximate reduction scheme, also the LL
equation (Landau and Lifschitz, 1951 LL ) can be immediately obtained.
X The fundamental existence and uniqueness theorem
THMs.1 and 2 show that in the presence of RR the non-local Lagrangian
system $\left\{\mathbf{x},L\right\}$ admits E-L equations [Eq.(99)] which are of delay differential type. This
feature is not completely unexpected, since model equations of this type
have been proposed before for the RR problem (see for example caldirola ). In general, for a delay-type differential equation there is
nothing similar to the existence and uniqueness theorem holding for an
initial condition of the type
$$\mathbf{x}(s_{o})=\mathbf{x}_{o}.$$
(131)
In fact, no finite set of initial data is generally enough to determine a
unique solution. The possibility of having, under suitable physical
assumptions, an existence and uniqueness theorem therefore plays a crucial
role in the proper formulation of the RR problem. In fact, for consistency
with the SR-CE axioms, and in particular with NPD, the existence of a
classical dynamical system (3) must be warranted. The result can
be obtained by requiring that there exists an initial time $s_{o}$ before
which for all $s<s_{o}$ the particle motion is inertial (see also the
related discussion in Ref.caldirola ). The assumption has also been
invoked to define the particle mass and charge distributions (see Section
3). In view of THM.1 this happens if the external EM force vanishes
identically for all $s<s_{o}$ and is (smoothly) “turned
on” at $s=s_{o}$. In this regard, we here point out the
following theorem:
THM.4 - The fundamental theorem for the RR equation
Given validity of THM.1, let us assume that:
1.
REQUIREMENT #1: at time $t_{o}$ the initial condition (131) holds;
2.
REQUIREMENT #2: the external force $\overline{F}_{\mu\nu}^{(ext)}(r,s)$ is of the form $\overline{F}_{\mu\nu}^{(ext)}(r,s)=\Theta(s-s_{o})\overline{F}_{1\mu\nu}^{(ext%
)}(r)),$ i.e., $\overline{F}_{\mu\nu}^{(ext)}$ is “turned
on” at the proper time $s=s_{o}$. In particular we
shall take $\overline{F}_{\mu\nu}^{(ext)}(r,s)$ to be a smooth function of
$s$, of class $C^{k}\left(M^{4}\times I\right)$, with $k\geq 1$;
3.
REQUIREMENT #3: more generally, let us require that for an
arbitrary initial state $\mathbf{x}(s_{1})=\mathbf{x}_{1}$ $\in\Gamma$ there always exists $\left\{\mathbf{x}(s_{o})=\mathbf{x}_{o},s_{o}\right\}\in\Gamma\times I,$ with $s_{o}=s_{1}-s_{ret},$ such that at time $s_{o},$ $\mathbf{x}(s_{o})$ is
inertial, i.e., before s${}_{o}$ the external force $\overline{F}_{\mu\nu}^{(ext)}$ vanishes identically, so that the dynamics is of
the form provided by Eqs.(105)-(106).
It then follows that the solution of the initial-value problem (99)-(131), subject to REQUIREMENTS
#1-#3, exists at least locally in a subset $I\equiv\left[-\infty,s_{0}\right]\cup\left[s_{0},s_{n}\right]\subseteq\mathbb{R}$ with $\left[s_{0},s_{n}\right]$ a bounded interval, and is unique (fundamental theorem).
Proof - Eq.(99) can be cast in the form of a
delay-differential equation, i.e.,
$$\frac{d\mathbf{x}(s)}{ds}=\mathbf{X(x}(s),\mathbf{x}(s-s_{ret}),s),$$
(132)
subject to the initial condition
$$\mathbf{x}(s_{o})=\mathbf{x}_{o}.$$
(133)
Here $\mathbf{x}(s)$ and $\mathbf{x}(s-s_{ret})$ denote respectively the
“instantaneous” and “retarded” states $\mathbf{x}(s)$ and $\mathbf{x}(s-s_{ret}),$ while $\mathbf{X(x}(s),\mathbf{x}(s-s_{ret}),s)$ is a suitable
$C^{2}$ real vector field depending smoothly on both of them. The proof of
local existence and uniqueness for Eq.(132), with
the initial conditions (133) and the
Requirements #1-#3, requires a generalization of the fundamental
theorem holding for ordinary differential equations (in which the vector
field $\mathbf{X}$ depends only on the local state $\mathbf{x}(s)$).
Let us first consider the case in which the solution $\mathbf{x}(s)$ of the
initial-value problem (132) and (133) is defined in the half-axis $\left[-\infty,s_{o}\right]:$ by assumption this solution exists, is unique and is that
of inertial motion [see Eqs.(105)-(106)].
Next, let us consider the proper time interval $I_{o,1}\equiv\left[s_{o},s_{1}\equiv s_{o}+s_{ret}\right].$ Thanks to the Requirement #3, by
assumption in $I_{o,1}$ the particle is subject only to the action of the
external force (produced by $A_{\mu}^{(ext)}$), since $\overline{F}_{\mu\nu}^{\left(self\right)}$ vanishes by definition if $s<s_{o}+s_{ret}$.
Hence, in the same time interval the solution exists and is unique because
the differential equation (132) is of the form
$$\frac{d\mathbf{x}(s)}{ds}=\mathbf{X}^{ext}\mathbf{(x}(s),s),$$
(134)
with $\mathbf{X}^{ext}\mathbf{(x}(s),s)$ being, by assumption, a smooth
vector field (see THM.1). Eq.(134) is manifestly a local ODE for
which the fundamental theorem (for local ODEs) holds. Hence, existence and
uniqueness is warranted also in $I_{o,1}$.
Finally, let us consider the sequence of proper time intervals $I_{k,k+1}\equiv\left[s_{k},s_{k+1}=s_{k}+s_{ret}\right],$ for the integer
$k=1,2,3...n$, where $n$ $\geq 2.$ In this case, for any proper time $s\in I_{k,k+1},$ the advanced-time solution $\mathbf{x}(s-s_{ret})$ appearing in
the vector field $\mathbf{X\equiv X(x}(s),\mathbf{x}(s-s_{ret}),s)$ can be
considered as a prescribed function of $s,$ determined in the
previous time interval $I_{k,k-1}.$ Therefore, $\mathbf{X}$ is necessarily
of the form $\mathbf{X\equiv}\widehat{\mathbf{X}}\mathbf{(x}(s),s)$, so
that for $s>s_{1},$ Eq.(132) can be viewed again
as a local ODE. We conclude that, thanks to the fundamental theorem holding
for local ODEs, the local existence (in a suitable bounded proper time
interval $I\equiv[s_{1},s_{n}]$) and uniqueness of solutions of the
problem (132)-(133)
is assured under the Requirements #1-#3. This proves the statement.
Q.E.D.
XI Conclusions
In this paper we have shown that the RR problem originally posed by Lorentz
for classical non-rotating finite-size and Lorentzian particles can exactly
be solved analytically within the SR setting.
For these particles, the resulting relativistic dynamics in the presence of
the RR force, i.e., the classical RR equation, has been found
analytically by taking into account the exact covariant form of the EM self
4-potential. In particular, this has been uniquely determined consistently
with the basic principles of classical electrodynamics and special
relativity. In addition, the RR equation has been proved to be variational in the functional class of synchronous variations (1) with respect to the Hamilton variational principle,
defined in terms of a non-local variational Lagrangian function. The same
equation has been shown: 1) to admit the standard Lagrangian form in terms
of the non-local effective Lagrangian $L_{eff}$; 2) to admit a conservative
form; 3) to recover the usual asymptotic LAD and LL equations in the
first-order short delay-time approximation; 4) not to admit the point-charge
limit. From the mathematical point of view, the RR equation is a delay-type
second order ODE, which fulfills GIP in the sense of THM.1,
relativistic covariance and MLC. As a consequence, provided suitable
physical requirements are imposed, the initial-value problem for the
RR equation is well-posed, defining the classical dynamical system required
by NDP.
Acknowledgements.This work was developed in the framework of the research
projects of the Consortium for Magnetofluid Dynamics (University of Trieste,
Italy): Fundamentals and applications of relativistic Hydrodynamics
and Magnetohydrodynamics (International School for Advanced Studies
(SISSA), Trieste, Italy) and Magnetohydrodynamics in curved space:
theory and applications (Department of Mathematics and Informatics,
University of Trieste, Italy).
XII Appendix A: variational calculations
Here we report the proof of identities (108)-(111) in
THM.1. Let us first notice that
$$d\left[\int_{1}^{2}dr^{\prime\nu}\delta(\widetilde{R}^{\alpha}\widetilde{R}_{%
\alpha}-\sigma^{2})\right]=dr^{k}\int_{-\infty}^{\infty}ds^{\prime}u^{\nu}(s^{%
\prime})\frac{\partial}{\partial r^{k}}\left[\delta(\widetilde{R}^{\alpha}%
\widetilde{R}_{\alpha}-\sigma^{2})\right].$$
(135)
Hence the variations $\delta A$ and $\delta B$ given in Eqs.(110) are respectively
$$\delta A=-\frac{4q^{2}}{c}\eta_{\mu\nu}\int_{1}^{2}\delta r^{\mu}dr^{k}\int_{-%
\infty}^{\infty}ds^{\prime}u^{\nu}(s^{\prime})\frac{\partial}{\partial r^{k}}%
\left[\delta(\widetilde{R}^{\alpha}\widetilde{R}_{\alpha}-\sigma^{2})\right],$$
(136)
while
$$\delta B=\frac{4q^{2}}{c}\eta_{\alpha\beta}\int_{1}^{2}dr^{\alpha}\delta r^{%
\mu}\int_{-\infty}^{\infty}ds^{\prime}u^{\beta}(s^{\prime})\frac{\partial}{%
\partial r^{\mu}}\delta(\widetilde{R}^{k}\widetilde{R}_{k}-\sigma^{2}).$$
(137)
Let us now evaluate the partial derivative $\frac{\partial}{\partial r^{k}}\delta(\widetilde{R}^{\alpha}\widetilde{R}_{%
\alpha}-\sigma^{2}).$
Invoking the chain rule, this becomes
$$\frac{\partial}{\partial r^{k}}\delta(\widetilde{R}^{\alpha}\widetilde{R}_{%
\alpha}-\sigma^{2})=\frac{\partial(\widetilde{R}^{\alpha}\widetilde{R}_{\alpha%
})}{\partial r^{k}}\frac{d\delta(\widetilde{R}^{\alpha}\widetilde{R}_{\alpha}-%
\sigma^{2})}{d(\widetilde{R}^{\alpha}\widetilde{R}_{\alpha})}=\frac{d\delta(%
\widetilde{R}^{\alpha}\widetilde{R}_{\alpha}-\sigma^{2})}{ds^{\prime}}\frac{2%
\widetilde{R}_{k}}{\frac{d(\widetilde{R}^{\alpha}\widetilde{R}_{\alpha})}{ds^{%
\prime}}},$$
(138)
and so
$$\frac{\partial}{\partial r^{k}}\delta(\widetilde{R}^{\alpha}\widetilde{R}_{%
\alpha}-\sigma^{2})=-\frac{\widetilde{R}_{k}}{\widetilde{R}^{\alpha}u_{\alpha}%
(s^{\prime})}\frac{d}{ds^{\prime}}\left\{\frac{\delta(s-s^{\prime}-s_{ret})}{2%
\left|\widetilde{R}^{\alpha}u_{\alpha}(s^{\prime})\right|}\right\}.$$
(139)
It follows that
$$\displaystyle\frac{\partial}{\partial r^{k}}\delta(\widetilde{R}^{\alpha}%
\widetilde{R}_{\alpha}-\sigma^{2})$$
$$\displaystyle=$$
$$\displaystyle-\frac{\widetilde{R}_{k}}{c^{2}\left[(t-t^{\prime})-\frac{1}{c^{2%
}}\frac{d\mathbf{r}^{\prime}}{dt^{\prime}}\cdot(\mathbf{r-r}^{\prime})\right]}\times$$
(140)
$$\displaystyle\times\frac{d}{dt^{\prime}}\left\{\frac{\delta(t-t^{\prime}-t_{%
ret})}{2c^{2}\gamma\left(t^{\prime}\right)\left|(t-t^{\prime})-\frac{1}{c^{2}}%
\frac{d\mathbf{r}(t^{\prime})}{dt^{\prime}}\cdot(\mathbf{r-r}^{\prime})\right|%
}\right\},$$
where $\mathbf{r}^{\prime}\equiv\mathbf{r}(t^{\prime}),$ $t\equiv t(s)$
and $t^{\prime}\equiv t(s^{\prime}).$ Substituting Eq.(140) into Eqs.(136) and (137), and then directly integrating, it follows
immediately that $\delta A$ and $\delta B$ have the form (110).
References
(1)
P.A.M. Dirac, Classical Theory of Radiating
Electrons, Proc. Roy. Soc. London A167, 148 (1938).
(2)
W. Pauli, Theory of Relativity, p.99 (Pergamon,
N.Y., 1958).
(3)
R. Feynman, Lectures on Physics, Vol.2
(Addison-Wesley Publishing Company, Reading, Mass., USA, 1970; special
reprint 1988).
(4)
M. Dorigo, M. Tessarotto, P. Nicolini and A.
Beklemishev, AIP Conf. Proc. 1084, 152-157 (2008).
(5)
F. Rohrlich, Classical Charged particles
(Addison-Wesley, Reading MA, 1965).
(6)
C. Teitelboim, Phys. Rev. D1, 1572 (1970).
(7)
C. Teitelboim, Phys. Rev. D2, 1763 (1970).
(8)
S. Parrott, Relativistic Electrodynamics and
Differential Geometry, Springer-Verlag, NY (1987).
(9)
S. Parrott, Found. Phys. 23, 1093 (1993).
(10)
H.A. Lorentz, Le theorie electromagnetique de
Maxwell et son application aux corps mouvants, Archives Neederlandaises des
Sciences Exactes et Naturelles, 25, 363 (1892).
(11)
M. Abraham, Theorie der Elektrizität,
Vol.II. Elektromagnetische Strahlung (Teubner, Leiptzig, 1905).
(12)
L.D. Landau and E.M. Lifschitz, Field theory,
Theoretical Physics Vol.2 (Addison-Wesley, N.Y., 1951).
(13)
H. Spohn, Europhys. Lett. 50, 287 (2000).
(14)
F. Rohrlich, Phys. Lett. A 283, 276 (2001).
(15)
F. Rohrlich, Ann. J. Phys. 68, 1109 (2000).
(16)
R. Medina, J. Phys. A: Math. Gen. 39,
3801-3816 (2006).
(17)
F. Rohrlich, Phys. Rev. E77, 046609 (2008).
(18)
P. Caldirola, Nuovo Cim. 3, Suppl. 2, 297
(1956).
(19)
D. Sarmah, M. Tessarotto and M. Salimullah, Phys.
Plasmas, 13, 032102 (2006).
(20)
J.S. Nodvik, Ann. Phys. 28, 225 (1964).
(21)
A. Yaghjian, Relativistic dynamics of a charged sphere, Lecture notes in physics Vol. 686 (Berlin, Springer Verlag, 2006).
(22)
M. Tessarotto, C. Cremaschini, M. Dorigo, P.
Nicolini and A. Beklemishev, AIP Conf. Proc. 1084, 158-163 (2008).
(23)
M. Pozzo and M. Tessarotto, Phys. Plasmas, 5,
2232 (1998).
(24)
A. Beklemishev and M. Tessarotto, Phys. Plasmas,
6, 4487 (1999).
(25)
M. Tessarotto, C. Cremaschini, P. Nicolini and A.
Beklemishev, Proceedings of the 25th RGD International Symposium on
Rarefied Gas Dynamics, St. Petersburg, Russia, 2006, edited by M.S. Ivanov
and A.K. Rebrov (Novosibirsk Publ. House of the Siberian Branch of the
Russian Academy of Sciences, 2007).
(26)
J.D. Jackson, Classical Electrodynamics (Wiley,
2nd Edition, 1975).
(27)
B.S. DeWitt and R.W. Brehme, Ann. Phys. 9, 220
(1960).
(28)
R.J. Crowley and S.J. Nodvik , Ann. Phys.
113, 98 (1978). |
Convergence theorems for several decomposition type non-linear integrals
Ryoji Fukuda
rfukuda@oita-u.ac.jp
Oita University, 700 Dan-noharu, Oita city,
Oita, 870-1192, JAPAN
Abstract
We define several types of decomposition type non-linear integrals.
These are classified by the direction of approximation(from above or below),
the set family types (partition or covering) of simple functions,
the coefficient signature (non-negative or signed),
and cardinal number of terms of simple functions(finite or countable infinite).
We will compare these integrals
considering the monotone increasing/decreasing convergence theorems.
keywords:
convergence theorem, monotone measure,
MSC: [2010] 28E10, 28B15
28E10 Fuzzy measure theory
28B15 Set functions, measures and integrals with values in ordered spaces
††journal: Journal of LATEX Templates
1 Introduction
This is an English translation of
“Comparison of Decomposition Type Nonlinear Integrals Based on the Convergence Theorem”
FHO2020J .
We define several decomposition type non-linear integrals.
The view points are the direction of approximation,
the set family types of simple functions,
the coefficient signature,
and cardinal number of terms of simple functions.
We will give some sufficient conditions for convergence theorems:
monotone increasing convergence theorems,
monotone increasing convergence theorems,
and uniform convergence theorems.
2 Classes of Simple functions and Definitions of Integrals
We will give some concepts and notations.
Throughout the paper, $(X,\mathcal{B})$ denotes a measurable space.
$X$ is non-discrete set and $\mathcal{B}$ is a $\sigma$-algebra.
We call a set function $\mu$ ($\mathcal{B}\to{\mathbb{R}}^{+}$) “a monotone measure”
if $\mu(\emptyset)=0$ and $\mu(A)\leq\mu(B)$ if $A\subset B$.
We assume that all monotone measures $\mu$ satisfies continuity from above and below, that is:
$$A_{n}\nearrow A,\mbox{ or }A_{n}\searrow A\mbox{ as }n\to\infty\quad\Rightarrow\quad\mu(A_{n})\to\mu(A)\mbox{ as }n\to\infty.$$
Let $\varphi$ be a simple function expressed by
$\varphi(x)=\sum_{k}a_{k}{\mathrm{1}}_{A_{k}}$,
where $a_{k}\in{\mathbb{R}}$ and $A_{k}\in\mathcal{B}$ for each $k$.
The summation may be finite or infinite.
For this simple function, we define the basic sum $\mu(\varphi)$
of $\varphi$ with respect to $\mu$ by
$$\mu(\varphi)=\sum_{k}a_{k}\mu(A_{k}).$$
We assume that the series converges absolutely
when the summation is infinite.
We do not assume the additivity for a monotone measure $\mu$,
then the basic sums are not the same
among a family of simple functions
which are same as functions.
Hence, we have to distinguish simple functions when
sequences of pairs of a real number and a measurable set are not the same
when we consider the basic sums.
Definition 1
We define 8 simple function families as follows.
Let $\mu$ be a monotone measure.
$$\displaystyle\mathcal{S}^{P+}$$
$$\displaystyle=$$
$$\displaystyle\{\{(a_{k},A_{k})\}_{k=1}^{n}:n\in{\mathbb{N}},a_{k}\geq 0,\ \{A_{k}\}_{k}\mbox{ is a partition of }X\}$$
$$\displaystyle\mathcal{S}^{P\pm}$$
$$\displaystyle=$$
$$\displaystyle\{\{(a_{k},A_{k})\}_{k=1}^{n}:n\in{\mathbb{N}},a_{k}\in{\mathbb{R}},\ \{A_{k}\}_{k}\mbox{ is a partition of }X\}$$
$$\displaystyle\mathcal{S}^{P+}_{\mu}$$
$$\displaystyle=$$
$$\displaystyle\{\{(a_{k},A_{k})\}_{k=1}^{\infty}:a_{k}\geq 0,\sum_{k}a_{k}\mu(A_{k})<\infty,$$
$$\displaystyle\hskip 120.00018pt\{A_{k}\}_{k}\mbox{ is a partition of }X\}$$
$$\displaystyle\mathcal{S}^{P\pm}_{\mu}$$
$$\displaystyle=$$
$$\displaystyle\{\{(a_{k},A_{k})\}_{k=1}^{\infty}:a_{k}\in{\mathbb{R}},\sum_{k}|a_{k}|\mu(A_{k})<\infty,$$
$$\displaystyle\hskip 120.00018pt\{A_{k}\}_{k}\mbox{ is a partition of }X\}$$
$$\displaystyle\mathcal{S}^{C+}$$
$$\displaystyle=$$
$$\displaystyle\{\{(a_{k},A_{k})\}_{k=1}^{n}:n\in{\mathbb{N}},a_{k}\geq 0,\ \{A_{k}\}_{k}\mbox{ is a covering of }X\}$$
$$\displaystyle\mathcal{S}^{C\pm}$$
$$\displaystyle=$$
$$\displaystyle\{\{(a_{k},A_{k})\}_{k=1}^{n}:n\in{\mathbb{N}},a_{k}\in{\mathbb{R}},\ \{A_{k}\}_{k}\mbox{ is a covering of }X\}$$
$$\displaystyle\mathcal{S}^{C+}_{\mu}$$
$$\displaystyle=$$
$$\displaystyle\{\{(a_{k},A_{k})\}_{k=1}^{\infty}:a_{k}\geq 0,\sum_{k}a_{k}\mu(A_{k})<\infty,$$
$$\displaystyle\hskip 120.00018pt\{A_{k}\}_{k}\mbox{ is a covering of }X\}$$
$$\displaystyle\mathcal{S}^{C\pm}_{\mu}$$
$$\displaystyle=$$
$$\displaystyle\{\{(a_{k},A_{k})\}_{k=1}^{\infty}:a_{k}\in{\mathbb{R}},\sum_{k}|a_{k}|\mu(A_{k})<\infty,$$
$$\displaystyle\hskip 120.00018pt\{A_{k}\}_{k}\mbox{ is a covering of }X\}$$
A simple function $\varphi=\{(a_{k},A_{k})\}$ is a sequence of pairs of a real number and a measurable set.
We always identify $\varphi$ with the function
$$\varphi(x)=\sum a_{k}\mathrm{1}_{A_{k}}(x).$$
For a family of simple functions $\mathcal{S}$ and a measurable function $f$ on $(X,\mathcal{B})$,
we define the following families of simple functions.
$$\displaystyle L(\mathcal{S},f)$$
$$\displaystyle=$$
$$\displaystyle\{\phi\in\mathcal{S},\phi(x)\leq f(x),\ \forall x\in X\}.$$
$$\displaystyle U(\mathcal{S},f)$$
$$\displaystyle=$$
$$\displaystyle\{\phi\in\mathcal{S},\phi(x)\geq f(x),\ \forall x\in X\}.$$
Then, we define decomposition type integrals as follows.
$${\int^{\uparrow}_{\mathcal{S}}}fd\mu=\sup\{\mu(\varphi):\varphi\in L(\mathcal{S},f)\},\quad{\int^{\downarrow}_{\mathcal{S}}}fd\mu=\inf\{\mu(\varphi):\varphi\in U(\mathcal{S},f)\}.$$
${\int^{\uparrow}_{\mathcal{S}^{P+}}}$ is a Pan integral(Yang1985 ),
${\int^{\uparrow}_{\mathcal{S}^{C+}}}$ is a SD integral(MesiarPap2015 ),
${\int^{\downarrow}_{\mathcal{S}^{P+}}}$ is a concave integral(Lehrer2009 ) and
${\int^{\downarrow}_{\mathcal{S}^{C+}}}$ is a convex integral(MesiarPap2015 ).
We formally defined several integrals, however,
the simple function classes $\mathcal{S}^{c,\pm}$ and $\mathcal{S}^{c,\pm}_{\mu}$
are too wide to define the above integrals for standard measurable functions.
Next we will explain some basic properties of these integrals.
Lemma 2
Let $\mu$ be a monotone measure,
$f,g$ be measurable functions on $(X,\mathcal{B})$,
and $c$ be a positive constant.
Then, for each $\int={\int^{\uparrow}_{\mathcal{S}}},{\int^{\downarrow}_{\mathcal{S}}}$,
and $\mathcal{S}$:one of the simple function classes defined in Definition 1,
(a)
$f\leq g$ implies $\displaystyle\int fd\mu\leq\int gd\mu,$
(b)
$\displaystyle\int cfd\mu=c\int fd\mu.$
Proof. (a) $f\leq g$ implies
$$\{\varphi:\varphi\leq f\}\subset\{\varphi:\varphi\leq g\},\quad\{\varphi:\varphi\geq f\}\supset\{\varphi:\varphi\geq g\}.$$
Then
$$\sup\{\mu(\varphi):\varphi\leq f\}\leq\sup\{\mu(\varphi):\varphi\leq g\},$$
$$\inf\{\mu(\varphi):\varphi\leq f\}\geq\inf\{\mu(\varphi):\varphi\geq g\}.$$
These conclude the proof of (a).
(b) This relation can be easily obtained, from the fact that
(c)
$c\varphi\in\mathcal{S}$, $\mu(c\varphi)=c\mu(\varphi)$,
(d)
$f\leq\ (\geq)\varphi$ if and only if $cf\leq\ (\geq)c\varphi$.
$\Box$
Lemma 3
Let $\mu$ be a monotone measure, $\delta>0$ be a positive number, and $f$ be a nonnegative function on $(X,\mathcal{B})$.
When $\mathcal{S}=\mathcal{S}^{P+},\mathcal{S}^{P+}_{\mu}$,
$${\int^{\uparrow}_{\mathcal{S}}}f+\delta\mathrm{1}_{X}d\mu\leq{\int^{\uparrow}_{\mathcal{S}}}fd\mu+\delta{\int^{\uparrow}_{\mathcal{S}}}\mathrm{1}_{X}d\mu.$$
Proof. Let $\varphi=\sum_{k}a_{k}\mathrm{1}_{A_{k}}$ be an element of $L(\mathcal{S},f+\delta\mathrm{1}_{X})$.
We may assume that $a_{k}$ not less than $\delta$,
since $f+\delta\mathrm{1}_{X}$ is not less than $\delta$.
Then,
$$\varphi_{1}=\sum_{k}(a_{k}-\delta)\mathrm{1}_{A_{k}}\in L(\mathcal{S},f),\quad\varphi_{2}=\sum_{k}\mathrm{1}_{A_{k}}\in L(\mathcal{S},\mathrm{1}_{X}).$$
By the definition of $\varphi_{1}$ and $\varphi_{2}$,
$$\mu(\varphi)=\mu(\phi_{1}+\delta\varphi_{2})=\mu(\phi_{1})+\delta\mu(\varphi_{2}).$$
Therefore,
$$\displaystyle{\int^{\uparrow}_{\mathcal{S}}}(f+\delta\mathrm{1}_{X})d\mu$$
$$\displaystyle=$$
$$\displaystyle\sup\{\mu(\varphi);\varphi\in L(\mathcal{S},f+\delta\mathrm{1}_{X})\}$$
$$\displaystyle=$$
$$\displaystyle\sup\{\mu(\varphi_{1})+\delta\mu(\varphi_{2});\varphi_{1}\in L(\mathcal{S},f),\ \varphi_{2}\in L(\mathcal{S},\mathrm{1}_{X})\}$$
$$\displaystyle\leq$$
$$\displaystyle\sup\{\mu(\varphi_{1}):\varphi_{1}\in L(\mathcal{S},f)\}+\delta\sup\{\mu(\varphi_{2});\varphi_{2}\in L(\mathcal{S},\mathrm{1}_{X})\}$$
$$\displaystyle=$$
$$\displaystyle{\int^{\uparrow}_{\mathcal{S}}}fd\mu+\delta{\int^{\uparrow}_{\mathcal{S}}}\mathrm{1}_{X}d\mu.$$
$\Box$
Lemma 4
Let $\mu$ be a monotone measure, $f,g$ be nonnegative functions on $(X,\mathcal{B})$,
and $A$ is a $\mathcal{B}$-measurable set.
When $\mathcal{S}=\mathcal{S}^{P+},\mathcal{S}^{P+}_{\mu}$,
(a)
$${\int^{\uparrow}_{\mathcal{S}}}fd\mu\geq{\int^{\uparrow}_{\mathcal{S}}}f\mathrm{1}_{A}d\mu+{\int^{\uparrow}_{\mathcal{S}}}f\mathrm{1}_{A^{c}}d\mu,$$
(b)
$${\int^{\downarrow}_{\mathcal{S}}}fd\mu\leq{\int^{\downarrow}_{\mathcal{S}}}f\mathrm{1}_{A}d\mu+{\int^{\downarrow}_{\mathcal{S}}}f\mathrm{1}_{A^{c}}d\mu.$$
When $\mathcal{S}=\mathcal{S}^{C+},\mathcal{S}^{C+}_{\mu}$,
(c)
$${\int^{\uparrow}_{\mathcal{S}}}f+gd\mu\geq{\int^{\downarrow}_{\mathcal{S}}}fd\mu+{\int^{\downarrow}_{\mathcal{S}}}gd\mu,$$
(d)
$${\int^{\downarrow}_{\mathcal{S}}}f+gd\mu\leq{\int^{\downarrow}_{\mathcal{S}}}fd\mu+{\int^{\downarrow}_{\mathcal{S}}}gd\mu.$$
Proof. (a) We consider simple functions $\varphi_{1}\in L(\mathcal{S},f\ \mathrm{1}_{A})$and $\varphi_{2}\in L(\mathcal{S},f\ \mathrm{1}_{A^{c}})$,
and assume that these are expressed by
$$\varphi_{1}=\sum_{k}b_{k}\mathrm{1}_{B_{k}},\quad\varphi_{2}=\sum_{k}b^{\prime}_{k}\mathrm{1}_{B^{\prime}_{k}}.$$
Then $B_{k}\cap A^{c}\not=\emptyset\Rightarrow b_{k}=0$, and
$B^{\prime}_{k}\cap A\not=\emptyset\Rightarrow b^{\prime}_{k}=0$,
since $\mathrm{1}_{A}=0$ on $A^{c}$ and $\mathrm{1}_{A^{c}}=0$ on $A$.
After removing sets with $b_{k}=0$ or $b^{\prime}_{k}=0$, the family $\{B_{k}\}\cup\{B^{\prime}_{k}\}$ is a disjoint family.
Hence, $\varphi=\sum_{k}b_{k}\mathrm{1}_{B_{k}}+\sum_{k}b^{\prime}_{k}\mathrm{1}_{B^{\prime}_{k}}\in\mathcal{S}^{C+},\mathcal{S}^{C+}_{\mu}$ and
$\varphi\in L(\mathcal{S},f)$. This implies
$$\displaystyle{\int^{\uparrow}_{\mathcal{S}}}fd\mu$$
$$\displaystyle=$$
$$\displaystyle\sup\{\mu(\varphi):\varphi\in L(\mathcal{S},f)\}$$
$$\displaystyle\geq$$
$$\displaystyle\sup\{\mu(\varphi_{1}):\varphi_{1}\in L(\mathcal{S},f\mathrm{1}_{A})\}+\sup\{\mu(\varphi_{2}):\varphi_{2}\in L(\mathcal{S},f\mathrm{1}_{A^{c}})\}$$
$$\displaystyle=$$
$$\displaystyle{\int^{\uparrow}_{\mathcal{S}}}f\mathrm{1}_{A}d\mu+{\int^{\uparrow}_{\mathcal{S}}}f\mathrm{1}_{A^{c}}d\mu.$$
(b)
Let $\varphi_{1}$and $\varphi_{2}$ be simple functions with $\phi_{1}\in U(\mathcal{S},f\mathrm{1}_{A})$,
$\phi_{2}\in U(\mathcal{S},f\mathrm{1}_{A^{c}})$.
When, $B_{k}$ is replaced by $B_{k}\cap A^{c}$ and
$B^{\prime}_{k}$ is replaced by $B^{\prime}_{k}\cap A$, the following properties still hold.
$$\varphi_{1}\in U(\mathcal{S},f\mathrm{1}_{A}),\quad\varphi_{2}\in U(\mathcal{S},f\mathrm{1}_{A^{c}}).$$
Then, $\varphi=\varphi_{1}+\varphi_{2}\in U(\mathcal{S},f)$, and this implies
$$\displaystyle{\int^{\downarrow}_{\mathcal{S}}}fd\mu$$
$$\displaystyle=$$
$$\displaystyle\inf\{\mu(\varphi):\varphi\in U(\mathcal{S},f)\}$$
$$\displaystyle\leq$$
$$\displaystyle\inf\{\mu(\varphi_{1}):\varphi_{1}\in U(\mathcal{S},f\mathrm{1}_{A})\}+\inf\{\mu(\varphi_{2}):\varphi_{2}\in U(\mathcal{S},f\mathrm{1}_{A^{c}})\}$$
$$\displaystyle=$$
$$\displaystyle{\int^{\downarrow}_{\mathcal{S}}}f\mathrm{1}_{A}d\mu+{\int^{\downarrow}_{\mathcal{S}}}f\mathrm{1}_{A^{c}}d\mu.$$
(c) (d) Let $\varphi_{1},\varphi_{2}$ be simple functions with $\varphi_{1}\in L(\mathcal{S},f)\ (U(\mathcal{S},f))$ and $\varphi_{2}\in L(\mathcal{S},g)\ (U(\mathcal{S},g))$.
By the definition of $\mathcal{S}^{c+},\ \mathcal{S}^{c+}_{\mu}$, $f+g\in L(\mathcal{S},f+g)\ (U(\mathcal{S},f+g))$.
This implies that
$$\displaystyle{\int^{\uparrow}_{\mathcal{S}}}f+gd\mu,\quad\left({\int^{\downarrow}_{\mathcal{S}}}f+gd\mu\right)$$
$$\displaystyle=$$
$$\displaystyle\sup\ (\inf\ )\{\mu(\varphi):\varphi\in L(\mathcal{S},f+g)\ (\ U(\mathcal{S},f+g)\ )\}$$
$$\displaystyle\geq(\leq)$$
$$\displaystyle\sup\ (\inf\ )\{\mu(\varphi_{1}):\varphi_{1}\in L(\mathcal{S},f)\ (\ U(\mathcal{S},f)\ )\}$$
$$\displaystyle+\sup\ (\inf\ )\{\mu(\varphi_{2}):\varphi_{2}\in L(\mathcal{S},g)\ (\ U(\mathcal{S},g)\ )\}$$
$$\displaystyle=$$
$$\displaystyle{\int^{\uparrow}_{\mathcal{S}}}\ \left({\int^{\downarrow}_{\mathcal{S}}}\right)\ fd\mu+{\int^{\uparrow}_{\mathcal{S}}}\ \left({\int^{\downarrow}_{\mathcal{S}}}\right)\ gd\mu$$
This concludes the proof. $\Box$
3 Uniform convergence theorem.
First, we consider the uniform convergence theorem for Pan integral.
3.1 Uniform convergence theorem for Pan integral
Lemma 5
Let $\mu$ be a monotone measure
and $f$ be a measurable function.
Assume that $\displaystyle\int\mathrm{1}_{X}d\mu=M<\infty$.
(a)
When $\mathcal{S}=\mathcal{S}^{P+},\ \mathcal{S}^{P+}_{\mu}$ and $f$ is nonnegative,
$${\int^{\uparrow}_{\mathcal{S}}}fd\mu-\delta M\leq{\int^{\uparrow}_{\mathcal{S}}}(f-\delta)\vee 0d\mu\leq{\int^{\uparrow}_{\mathcal{S}}}(f+\delta)d\mu\leq{\int^{\uparrow}_{\mathcal{S}}}fd\mu+\delta M$$
for any $\delta>0$.
(b)
When $\mathcal{S}=\mathcal{S}^{P\pm},\ \mathcal{S}^{P\pm}_{\mu}$,
$${\int^{\uparrow}_{\mathcal{S}}}fd\mu-\delta M\leq{\int^{\uparrow}_{\mathcal{S}}}(f-\delta)d\mu\leq{\int^{\uparrow}_{\mathcal{S}}}(f+\delta)d\mu\leq{\int^{\uparrow}_{\mathcal{S}}}fd\mu+\delta M$$
for any $\delta>0$.
Proof. We will prove the third inequality for (a) and (b), that is,
$\mathcal{S}=\mathcal{S}^{P+},\ \mathcal{S}^{P+}_{\mu},\ \mathcal{S}^{P\pm},\ \mathcal{S}^{P\pm}_{\mu}$.
We assume that $f\geq 0$ if $\mathcal{S}=\mathcal{S}^{P+},\ \mathcal{S}^{P+}_{\mu}$.
Fix an arbitrary $\varepsilon>0$. Then there exists$\varphi\in\mathcal{S}$ such that
$$\varphi\leq f+\delta,\quad\mu(\varphi)\geq{\int^{\uparrow}_{\mathcal{S}}}(f+\delta)d\mu-\varepsilon.$$
Using the representation $\varphi=\sum_{k}a_{k}\mathrm{1}_{A_{k}}$,
we may assume that $a_{k}\geq\delta$ if $\mathcal{S}=\mathcal{S}^{P+},\ \mathcal{S}^{P+}_{\mu}$.
We define new simple function
$$\psi=\sum_{k}(a_{k}-\delta)\mathrm{1}_{A_{k}}\leq f,$$
Remark that the coefficients $(a_{k}-\delta)$ are non-negative when $\mathcal{S}=\mathcal{S}^{P+},\ \mathcal{S}^{P+}_{\mu}$.
In any cases, we have
$$\displaystyle{\int^{\uparrow}_{\mathcal{S}}}(f+\delta)d\mu-\varepsilon$$
$$\displaystyle\leq$$
$$\displaystyle\mu(\varphi)$$
$$\displaystyle\leq$$
$$\displaystyle\mu(\psi)+\mu\left(\sum_{k}\delta\mathrm{1}_{A_{k}}\right)$$
$$\displaystyle\leq$$
$$\displaystyle\int fd\mu+\delta\int\mathrm{1}_{X}d\mu.$$
$$\displaystyle=$$
$$\displaystyle\int fd\mu+\delta M.$$
This implies that
$$\int(f+\delta)d\mu\leq\int fd\mu+\delta M,$$
since $\varepsilon$ is any positive number.
The second inequality in (a) and (b) are obvious, and we will prove the first one.
We consider the case $\mathcal{S}=\mathcal{S}^{P+},\ \mathcal{S}^{P+}_{\mu}$,
For any $\varepsilon>0$, there exist $\varphi\in\mathcal{S}$ with
$$\varphi\leq f,\quad\mu(\varphi)\geq{\int^{\uparrow}_{\mathcal{S}}}fd\mu-\varepsilon.$$
Using the representation $\varphi=\sum_{k}a_{k}\mathrm{1}_{A_{k}}$,
define a new simple function
$$\psi=\sum_{k}\{(a_{k}-\delta)\vee 0\}\mathrm{1}_{A_{k}}\leq(f-\delta)\vee 0,$$
Hence, $\psi\in L(\mathcal{S},(f-\delta)\vee 0)$.
$$\displaystyle{\int^{\uparrow}_{\mathcal{S}}}fd\mu-\varepsilon$$
$$\displaystyle\leq$$
$$\displaystyle\mu(\varphi)$$
$$\displaystyle\leq$$
$$\displaystyle\mu(\psi)+\delta\sum\mu(A_{k})$$
$$\displaystyle\leq$$
$$\displaystyle{\int^{\uparrow}_{\mathcal{S}}}(f-\delta)\vee 0d\mu+\delta M.$$
Then we have
$${\int^{\uparrow}_{\mathcal{S}}}fd\mu-\delta M\leq{\int^{\uparrow}_{\mathcal{S}}}(f-\delta)\vee 0d\mu.$$
Proof of the first inequality, for the case $\mathcal{S}=\mathcal{S}^{P\pm},\ \mathcal{S}^{P\pm}_{\mu}$,
is parallel with the above proof.
$\Box$
Theorem 6
Let $\mu$ be a monotone measure, and
$\mathcal{S}=\mathcal{S}^{P+},\mathcal{S}^{P\pm},\mathcal{S}^{P+}_{\mu},\mathcal{S}^{P\pm}_{\mu}$.
Assume that $\displaystyle M={\int^{\uparrow}_{\mathcal{S}}}\mathrm{1}_{X}d\mu<\infty$ .
Then, if a sequence of nonnegative measurable functions $\{f_{n}\}$
converges to $f$ uniformly and ${\int^{\uparrow}_{\mathcal{S}}}fd\mu<\infty$,
moreover, we also assume that $f$ and $f_{n}$ ($n\in{\mathbb{N}}$) are non-negative if $\mathcal{S}=\mathcal{S}^{P+},\mathcal{S}^{P+}_{\mu}$,
then
$$\lim_{n\to\infty}{\int^{\uparrow}_{\mathcal{S}}}f_{n}d\mu={\int^{\uparrow}_{\mathcal{S}}}fd\mu.$$
Proof. For any $\delta>0$, there exists $n_{\delta}\in{\mathbb{N}}$ such that,
$$f(x)-\delta\leq f_{n}(x)\leq f(x)+\delta$$
for any $n\geq n_{\delta}$.
When $\mathcal{S}=\mathcal{S}^{P+},\mathcal{S}^{P+}_{\mu}$,
$$(f(x)-\delta)\vee 0\leq f_{n}(x)\leq f(x)+\delta$$
for any $n\geq n_{\delta}$. Using Lemma 5, we have
$${\int^{\uparrow}_{\mathcal{S}}}f_{n}d\mu\in\left[{\int^{\uparrow}_{\mathcal{S}}}fd\mu-\delta M,{\int^{\uparrow}_{\mathcal{S}}}fd\mu+\delta M\right]$$
for any $n\geq n_{\delta}$.
Hence ${\int^{\uparrow}_{\mathcal{S}}}f_{n}d\mu$ converges to ${\int^{\uparrow}_{\mathcal{S}}}fd\mu$ as $n\to\infty$. $\Box$
3.2 Uniform convergence theorem for concave integral
We consider the case $\mathcal{S}=\mathcal{S}^{c+},\mathcal{S}^{c\pm},\mathcal{S}^{c+}_{\mu},\mathcal{S}^{c\pm}_{\mu}$.
The next example illustrates that conditions for the uniformly convergence theorem
are different for concave integral.
Example 7
Set $X={\mathbb{N}}_{0}=\{0,1,2,\ldots\}$, and a monotone measure $\mu$ is defined by
$$\mu(A)\quad=\quad\left\{\quad\begin{matrix}0\ ,&\quad\mbox{$A$ is one point set or $0\not\in A$,}\\
1\ ,&\quad|A|>1\mbox{ and }0\in A.\end{matrix}\quad\right.$$
For each $n\in{\mathbb{N}}$ we define a function $f_{n}$ ($n\in{\mathbb{N}})$ as follows.
$$f_{n}(k)=\left\{\quad\begin{matrix}1\ ,&&\quad\ k=0,\\
\displaystyle\frac{\ 1\ }{\ n\ }\ ,&&\quad\ \mbox{otherwise.}\end{matrix}\right.$$
Then the following properties (a) $\sim$ (d) hold.
(a)
$\mu$ is continuous from below, and is not continuous from above.
(b)
When $\mathcal{S}=\mathcal{S}^{C+},\mathcal{S}^{C+}_{\mu}$,
$${\int^{\uparrow}_{\mathcal{S}}}\mathrm{1}_{{\mathbb{N}}_{0}}d\mu<\infty.$$
(c)
$f_{n}\searrow\mathrm{1}_{\{0\}}$ uniformly.
(d)
For all $n\in{\mathbb{N}}$,
$${\int^{\uparrow}_{\mathcal{S}}}f_{n}d\mu\mathrm{1}_{{\mathbb{N}}_{0}}=1\not={\int^{\uparrow}_{\mathcal{S}}}\mathrm{1}_{\{0\}}d\mu=0.$$
Proofs and Comments. (a) Let $\{A_{n}\}$ be a sequence of measurable sets satisfying $A_{n}\nearrow A$.
If $\mu(A)=1$, $A$ contains $0$ and other one point $a_{0}$.
Then $a_{0},0\in A_{n}$ for large enough $n\in{\mathbb{N}}$.
This implies that $\mu(A_{n})=1$ and $\mu$ is continuous from below.
Set $A_{n}=\{0,n,n+1,\ldots\}$, then
$$\bigcap_{n=1}^{\infty}A_{n}=\{0\},\ \mu(A_{n})=1,\ \mu(\{0\})=0,$$
This prove the discontinuity of $\mu$ from above.
(b) Let $\varphi$ be an element of $L(\mathcal{S},\mathrm{1}_{{\mathbb{N}}_{0}})$,
$\varphi=\sum_{k}b_{k}\mathrm{1}_{B_{k}}\leq 1$. Then,
$$\sum_{0\in B_{k}}b_{k}\leq 1,$$
since $0\not\in B_{k}$ implies $\mu(B_{k})=0$.
$$\mu(\varphi)=\sum_{k}b_{k}\mu(B_{k})\leq\sum_{0\in B_{k}}b_{k}\leq 1.$$
(1)
Hence, ${\int^{\uparrow}_{\mathcal{S}}}\mathrm{1}_{{\mathbb{N}}_{0}}d\mu<\infty$.
(c) obvious.
(d) Set $B_{k}=\{0,k\}$,
and $\varphi_{n}=\sum_{k=1}^{n}\frac{\ 1\ }{\ n\ }\mathrm{1}_{B_{k}}$, then,
$$\varphi_{n}\leq f_{n},\ \mu(\varphi_{n})=1.$$
This implies,
$${\int^{\uparrow}_{\mathcal{S}}}f_{n}d\mu\geq 1.$$
By the inequality (1) and $f_{n}\leq\mathrm{1}_{{\mathbb{N}}_{0}}$,
$${\int^{\uparrow}_{\mathcal{S}}}f_{n}d\mu\leq{\int^{\uparrow}_{\mathcal{S}}}\mathrm{1}_{{\mathbb{N}}_{0}}d\mu\leq 1.$$
Hence, ${\int^{\uparrow}_{\mathcal{S}}}f_{n}d\mu=1$.
Let $\varphi\in L(\mathcal{S},\mathrm{1}_{\{0\}})$, and $\varphi=\sum a_{k}\mathrm{1}_{A_{k}}$.
$A=\emptyset$ or $A=\{0\}$, then,
the summation is single $\varphi=a_{1}\mathrm{1}_{\{0\}}$
and $a_{1}\leq 1$.
Therefore, we have ${\int^{\uparrow}_{\mathcal{S}}}\mathrm{1}_{\{0\}}d\mu=0$, since $\mu(\{0\})=0$.
$${\int^{\uparrow}_{\mathcal{S}}}\mathrm{1}_{\{0\}}d\mu=0\not=\lim_{n\to\infty}{\int^{\uparrow}_{\mathcal{S}}}f_{n}d\mu.$$
Thus, the uniform convergence theorem is not valid. $\Box$
Theorem 8
Let $\mu$ be a monotone measure,
$\{f_{n}\}$, $f$ be non-negative measurable functions,
and $\mathcal{S}=\mathcal{S}^{c+},\mathcal{S}^{c+}_{\mu}$.
Assume that $\mu$ is continuous from below,
$$\inf_{n\in{\mathbb{N}},x\in X}f_{n}(x)=a>0,\quad\lim_{n\to\infty}\sup_{x\in X}|f_{n}(x)-f(x)|=0,$$
and
$$\int f_{n}d\mu,\int fd\mu<\infty$$
Then,
$$\int f_{n}d\mu\to\int fd\mu.$$
Proof. By the assumption that $f(x)\geq a$ ($x\in X$).
For any $\delta\in(0,1)$
there exists $\varepsilon>0$ such that
$$(1-\delta)f(x)<f(x)-\varepsilon<f(x)+\varepsilon<(1+\delta)f(x)$$
for any $x\in X$.
Using the uniform convergence to $f$,
there exists $N\in{\mathbb{N}}$ such that
$|f_{n}(x)-f(x)|<\varepsilon$ for any $x\in X$ and $n\geq N$.
This implies
$$(1-\delta)f<f_{n}<(1+\delta)f$$
Therefore,
$$(1-\delta){\int^{\uparrow}_{\mathcal{S}}}fd\mu\leq{\int^{\uparrow}_{\mathcal{S}}}f_{n}d\mu\leq(1+\delta){\int^{\uparrow}_{\mathcal{S}}}fd\mu.$$
Thus, we conclude the proof by $\delta\to 0$. $\Box$
4 Monotone Convergence Theorem
In this section, we discuss about monotone increasing and decreasing
convergence theorems, these properties are deeply
connected with the approximation direction used in
the definition of integrals.
4.1 Monotone increasing convergence theorem for ${\int^{\uparrow}_{\mathcal{S}}}$
For simple function families $\mathcal{S}=\mathcal{S}^{p+},\mathcal{S}^{p\pm},\mathcal{S}^{p+}_{\mu},\mathcal{S}^{p\pm}_{\mu},\mathcal{S}^{c+},\mathcal{S}^{c+}_{\mu}$,
we will prove the monotone increasing convergence theorem for ${\int^{\uparrow}_{\mathcal{S}}}$
using an essentially same method.
For the classes $\mathcal{S}^{c\pm},\mathcal{S}^{c\pm}_{\mu}$,
$L(\mathcal{S},f)$ or $U(\mathcal{S},f)$ are too wide and
the corresponding integrals do not make sense.
Then, we do not treat these integrals.
When $\mathcal{S}$ is a family of infinite sum,
we need the following properties,
which can be easily proved using the dominated convergence
theorem (see for example TelenceTaoBook ).
Lemma 9
Let $\{a_{k}\}_{k},\ \{x_{k}\}_{k}$ be real sequences,
$\{\ \{x^{(n)}_{k}\}_{k}\ \}_{n\in{\mathbb{N}}}$ be a sequence of real sequences.
We assume that
(a)
$a_{k}\geq 0$ for any $k\in{\mathbb{N}}$ and
$\displaystyle\sum_{k}a_{k}<\infty$.
(b)
$|x^{(n)}_{k}|\leq a_{k}$ for any $n,k\in{\mathbb{N}}$.
(c)
$\displaystyle\lim_{n\to\infty}x^{(n)}_{k}=x_{k}$ for any $k\in{\mathbb{N}}$.
Then,
$$\lim_{n\to\infty}\sum_{k}x^{(n)}_{k}=\sum_{k}x_{k}.$$
$\Box$
In the case $\mathcal{S}$ consist of non-negative functions,
we have the following theorem.
Theorem 10
When $\mathcal{S}=\mathcal{S}^{p+},\mathcal{S}^{p+}_{\mu},\mathcal{S}^{c+},\mathcal{S}^{c+}_{\mu}$, ${\int^{\uparrow}_{\mathcal{S}}}$ satisfies monotone increasing convergence theorem.
That is, increasing sequence $\{f_{n}\}$ of non-negative functions
converges to $f$.
Then we have
$$\lim_{n\to\infty}{\int^{\uparrow}_{\mathcal{S}}}f_{n}d\mu={\int^{\uparrow}_{\mathcal{S}}}fd\mu.$$
Proof. Set $M=\int fd\mu$.
Let $\varepsilon>0$ be an arbitrary positive number,
and set $M^{\prime}=M-\varepsilon$ if $M<\infty$.
If $M=\infty$, let $M^{\prime}$ be any positive number.
Then, we can select $\varphi\in L(\mathcal{S},f)$ with
$$\mu(\varphi)>M^{\prime}.$$
For $\delta>0$, we define
$$A^{(\delta)}_{n}=\{x|f_{n}(x)\geq f(x)(1-\delta)\}.$$
Then, $A^{(\delta)}_{n}\nearrow X$ as $n\to\infty$.
We define
$$\varphi_{n}=(1-\delta)\varphi\mathrm{1}_{A^{(\delta)}_{n}}.$$
Then, $\varphi_{n}\in\mathcal{S}$ and $\varphi_{n}\leq f_{n}$.
By Lemma 9, we have
$$\lim_{n\to\infty}\mu(\varphi_{n})=(1-\delta)\mu(\varphi)\geq(1-\delta)M^{\prime}.$$
This implies
$${\int^{\uparrow}_{\mathcal{S}}}f_{n}d\mu\geq\lim_{n\to\infty}\mu(\varphi_{n})\geq(1-\delta)M^{\prime}.$$
By the assumption,
${\int^{\uparrow}_{\mathcal{S}}}f_{n}d\mu\leq{\int^{\uparrow}_{\mathcal{S}}}fd\mu=M$.
Thus,
$${\int^{\uparrow}_{\mathcal{S}}}f_{n}d\mu=\lim_{n\to\infty}{\int^{\uparrow}_{\mathcal{S}}}f_{n}d\mu.$$
$\Box$
Next we consider the case with signed coefficient.
Theorem 11
Let $\mu$ be a monotone measure,
with continuity at $\emptyset$ and from below.
$\{f_{n}\}_{n}$ be an increasing sequence of measurable functions
converges to $f$.
Assume that $\mathcal{S}=\mathcal{S}^{p\pm},\mathcal{S}^{p\pm}_{\mu}$,
and ${\int^{\uparrow}_{\mathcal{S}}}f_{1}d\mu>-\infty$, ${\int^{\uparrow}_{\mathcal{S}}}\mathrm{1}_{X}d\mu<-\infty$.
Then
$$\lim_{n\to\infty}{\int^{\uparrow}_{\mathcal{S}}}f_{n}d\mu={\int^{\uparrow}_{\mathcal{S}}}fd\mu.$$
Proof. Set $M=\int fd\mu$.
Let $\varepsilon>0$ be an arbitrary positive number,
and set $M^{\prime}=M-\varepsilon$ if $M<\infty$.
If $M=\infty$, let $M^{\prime}$ be any positive number.
Then, we can select $\varphi\in L(\mathcal{S},f)$ with
$$\mu(\varphi)>M^{\prime}.$$
By the condition $\int f_{1}d\mu>-\infty$,
there exists $\varphi_{0}\in L(\mathcal{S},f_{1})$ with $\mu(\varphi_{0})>-\infty$.
We give representations for these simple functions as follows.
$$\varphi=\sum_{k}b_{k}\mathrm{1}_{B_{k}},\quad\varphi_{0}=\sum_{k}c_{k}\mathrm{1}_{C_{k}}.$$
Define $A^{(\delta)}_{n}$ for any $\delta>0$ as follows.
$$A^{(\delta)}_{n}=\{x|f_{n}(x)\geq f(x)-\delta\}.$$
Then, we define simple functions $\{\varphi_{n}\}_{n}$ as follows.
$$\displaystyle\varphi_{n}$$
$$\displaystyle=$$
$$\displaystyle(\varphi-\delta)\mathrm{1}_{A^{(\delta)}_{n}}+\varphi_{0}\mathrm{1}_{A^{(\delta)\ c}_{n}}$$
$$\displaystyle=$$
$$\displaystyle\sum_{k}(b_{k}-\delta)\mathrm{1}_{B_{k}\cap A^{(\delta)}_{n}}+\sum_{k}c_{k}\mathrm{1}_{C_{k}\cap A^{(\delta)c}_{n}}$$
Then $\varphi_{n}\in\mathcal{S}$.
Using the definition of $A^{(\delta)}_{n}$, and the fact $\varphi_{0}\leq f_{1}\leq f_{n}$,
$$\varphi_{n}(x)=\varphi\mathrm{1}_{A^{(\delta)}_{n}}(x)+\varphi_{0}\mathrm{1}_{A^{(\delta)c}_{n}}(x)\leq f_{n}(x).$$
Then,
$$\displaystyle\mu(\varphi_{n})$$
$$\displaystyle=$$
$$\displaystyle\sum_{k}(b_{k}-\delta)\mu(B_{k}\cap A^{(\delta)}_{n})+\sum_{k}c_{k}\mu(C_{k}\cap A^{(\delta)c}_{n})$$
(2)
$$\displaystyle=$$
$$\displaystyle\sum_{k}b_{k}\mu(B_{k}\cap A^{(\delta)}_{n})-\delta\sum_{k}\mu(B_{k}\cap A^{(\delta)}_{n})$$
$$\displaystyle+\sum_{k}c_{k}\mu(B_{k}\cap A^{(\delta)c}_{n}).$$
By Lemma 9 we have
$$\sum_{k}b_{k}\mu(B_{k}\cap A^{(\delta)}_{n})\to\sum_{k}b_{k}\mu(B_{k}),$$
$$\sum_{k}c_{k}\mu(B_{k}\cap A^{(\delta)c}_{n})\to 0,$$
and
$$\sum_{k}\mu(B_{k}\cap A^{(\delta)}_{n})\leq\int\mathrm{1}_{X}d\mu.$$
Terefore, for large $n$
$$\int f_{n}d\mu>\mu(\varphi_{n})\geq\int fd\mu-\delta\int\mathrm{1}_{X}d\mu-2\varepsilon$$
Then we have $\displaystyle\lim_{n\to\infty}f_{n}d\mu=\int fd\mu$.
This concludes the proof. $\Box$
REMARK. Reversing the signatures, the above theorem corresponds a monotone decreasing
convergence theorem for ${\int^{\downarrow}_{\mathcal{S}}}$.
4.2 Monotone decreasing convergence theorems for ${\int^{\downarrow}_{\mathcal{S}}}$
As we remarked in the previous section,
when $\mathcal{S}=\mathcal{S}^{p\pm},\mathcal{S}^{p\pm}_{\mu}$,
monotone decreasing convergence theorems for ${\int^{\downarrow}_{\mathcal{S}}}$
are essentially same with monotone increasing convergence theorems
${\int^{\uparrow}_{\mathcal{S}}}$. However,
when $\mathcal{S}=\mathcal{S}^{p+},\mathcal{S}^{p+}_{\mu}$, situations are quite different,
in this section we treat this case.
Lemma 12
Let $\mu$ be a monotone measure, with continuity at $\emptyset$,
$\mathcal{S}=\mathcal{S}^{p+},\mathcal{S}^{p+}_{\mu}$,
and $\{A_{n}\}$ be a decreasing set sequence, with
$${\int^{\downarrow}_{\mathcal{S}}}fd\mu<\infty,\quad\bigcap_{n}A_{n}=\emptyset$$
Then, we have
$${\int^{\downarrow}_{\mathcal{S}}}f\mathrm{1}_{A_{n}}d\mu\searrow 0.$$
Proof. ${\int^{\downarrow}_{\mathcal{S}}}fd\mu<\infty$ if and only if there exists $\varphi\in U(\mathcal{S},f)$
with $\mu(\varphi)<\infty$.
Thus,
$$f\mathrm{1}_{A_{n}}\leq\varphi\mathrm{1}_{A_{n}}=\sum_{k}b_{k}\mathrm{1}_{B_{k}\cap A_{n}}$$
implies
$$\int f\mathrm{1}_{A_{n}}d\mu\leq\mu(\varphi\mathrm{1}_{A_{n}})=\sum_{k}b_{k}\mu(B_{k}\cap A_{n}).$$
Using the continuity of $\mu$ at $\emptyset$,
$$\mu(B_{k}\cap A_{n})\searrow 0,\quad(n\to\infty)$$
for each $k\in{\mathbb{N}}$.
By Lemma 9, we have,
$$\int f\mathrm{1}_{A_{n}}d\mu\leq\sum_{k}b_{k}\mu(B_{k}\cap A_{n})\searrow 0.$$
$\Box$
Theorem 13
Let $\mu$ be a monotone measure, with continuity at $\emptyset$,
$\mathcal{S}=\mathcal{S}^{p+},\mathcal{S}^{p+}_{\mu}$.
Let $\{f_{n}\}$ be a decreasing sequence of measurable functions
converges to $f$.
Assume that ${\int^{\downarrow}_{\mathcal{S}}}f_{1}d\mu<\infty$,
we have
$${\int^{\downarrow}_{\mathcal{S}}}f_{n}d\mu\searrow\int fd\mu.$$
Proof. By Lemma 4.
$${\int^{\downarrow}_{\mathcal{S}}}f_{n}d\mu\leq{\int^{\downarrow}_{\mathcal{S}}}f_{n}\mathrm{1}_{\{f=0\}}d\mu+{\int^{\downarrow}_{\mathcal{S}}}f_{n}\mathrm{1}_{\{f>0\}}d\mu$$
(3)
${\int^{\downarrow}_{\mathcal{S}}}f_{1}d\mu<\infty$ implies that there exists $\varphi_{0}\in\mathcal{S}$ with
$\varphi_{0}\geq f_{1}$ and $\mu(\varphi_{0})<\infty$.
Then ${\int^{\downarrow}_{\mathcal{S}}}f_{n}\mathrm{1}_{\{f=0\}}d\mu\to 0$ ($n\to\infty$) as follows.
$$\displaystyle{\int^{\downarrow}_{\mathcal{S}}}f_{n}\mathrm{1}_{\{f=0\}}d\mu$$
(4)
$$\displaystyle\leq$$
$$\displaystyle{\int^{\downarrow}_{\mathcal{S}}}f_{n}\mathrm{1}_{\{f=0\}}\mathrm{1}_{\{f_{n}\leq\varepsilon\varphi_{0}\}}d\mu+{\int^{\downarrow}_{\mathcal{S}}}f_{n}\mathrm{1}_{\{f=0\}}\mathrm{1}_{\{f_{n}>\varepsilon\varphi_{0}\}}d\mu$$
$$\displaystyle\leq$$
$$\displaystyle{\int^{\downarrow}_{\mathcal{S}}}\varepsilon\varphi_{0}\mathrm{1}_{\{f=0\}}\mathrm{1}_{\{f_{n}\leq\varepsilon\varphi_{0}\}}d\mu+{\int^{\downarrow}_{\mathcal{S}}}f_{1}\mathrm{1}_{\{f=0\}}\mathrm{1}_{\{f_{n}>\varepsilon\varphi_{0}\}}d\mu.$$
We remark that $f_{1}(x)>0$ if $f_{n}(x)>0$.
Then, for every $x$ with $f(x)=0$, $f_{n}(x)\leq\varepsilon\varphi_{0}(x)$ for large enough $n$.
$$\{x:f_{n}(x)>\varepsilon\varphi_{0}(x)\}\searrow\emptyset.$$
By Lemma 9, the second term in (4) converges to $\to 0$.
$$\mbox{(1 st. term of \ref{rhs2}) }\leq\varepsilon\mu(\varphi_{0})\to 0\quad(\varepsilon\to 0).$$
Thus ${\int^{\downarrow}_{\mathcal{S}}}f_{n}\mathrm{1}_{\{f=0\}}d\mu\to 0$ ($n\to\infty$).
Next we show that ${\int^{\downarrow}_{\mathcal{S}}}f_{n}\mathrm{1}_{\{f>0\}}d\mu\to{\int^{\downarrow}_{\mathcal{S}}}fd\mu$ ($n\to\infty$).
Fix any $\delta>0$. Set
$$A^{(\delta)}_{n}=\{x:f_{n}(x)\leq(1+\delta)f(x)\}.$$
Then, using the fact that
$$A^{(\delta)}_{n}\cap\{x:f(x)>0\}\nearrow\{x:f(x)>0\}\ (n\to\infty),$$
we have:
$$\displaystyle{\int^{\downarrow}_{\mathcal{S}}}f_{n}\mathrm{1}_{\{f>0\}}d\mu$$
$$\displaystyle\leq$$
$$\displaystyle{\int^{\downarrow}_{\mathcal{S}}}f_{n}\mathrm{1}_{\{f>0\}}\mathrm{1}_{A_{n}^{(\delta)}}d\mu+{\int^{\downarrow}_{\mathcal{S}}}f_{n}\mathrm{1}_{\{f>0\}}\mathrm{1}_{A_{n}^{(\delta)c}}d\mu$$
$$\displaystyle\leq$$
$$\displaystyle(1+\delta){\int^{\downarrow}_{\mathcal{S}}}\ f\ \mathrm{1}_{\{f>0\}}\mathrm{1}_{A_{n}^{(\delta)}}d\mu+{\int^{\downarrow}_{\mathcal{S}}}\ f_{1}\ \mathrm{1}_{\{f>0\}\cap A_{n}^{(\delta)c}}d\mu$$
$$\displaystyle\leq$$
$$\displaystyle(1+\delta){\int^{\downarrow}_{\mathcal{S}}}fd\mu+{\int^{\downarrow}_{\mathcal{S}}}f_{1}\mathrm{1}_{\{f>0\}\cap A_{n}^{(\delta)c}}d\mu.$$
By Lemma 12, the second term tends to 0. Therefore,
$$\inf_{n}{\int^{\downarrow}_{\mathcal{S}}}f_{n}\mathrm{1}_{\{f>0\}}d\mu\leq(1+\delta){\int^{\downarrow}_{\mathcal{S}}}fd\mu\to{\int^{\downarrow}_{\mathcal{S}}}fd\mu,\quad(\delta\searrow 0).$$
The reverse inequality is evident.
$$\inf_{n}{\int^{\downarrow}_{\mathcal{S}}}f_{n}\mathrm{1}_{\{f>0\}}d\mu=\lim_{n\to\infty}{\int^{\downarrow}_{\mathcal{S}}}f_{n}\mathrm{1}_{\{f>0\}}d\mu={\int^{\downarrow}_{\mathcal{S}}}fd\mu$$
Thus this concludes the proof. $\Box$
4.3 Monotone decreasing convergence theorems for ${\int^{\uparrow}_{\mathcal{S}}}$
Under some special conditions, ${\int^{\uparrow}_{\mathcal{S}}}$ satisfy the monotone decreasing convergence theorem.
In this section we will give some of them.
First, we consider sub-additive case.
Lemma 14
Let $\mu$ be a monotone measure with
$$A\cap B=\emptyset\Rightarrow\mu(A\cup B)\leq\mu(A)+\mu(B).$$
(Such a monotone measure is said to be sub-additive.)
Set $(\mathcal{S}_{1},\mathcal{S}_{2})$$=(\mathcal{S}^{p+},\mathcal{S}^{c+}),$ $(\mathcal{S}^{p+}_{\mu},\mathcal{S}^{c+}_{\mu})$.
Then, for any non-negative measurable function $f$,
$${\int^{\uparrow}_{\mathcal{S}_{1}}}fd\mu={\int^{\uparrow}_{\mathcal{S}_{2}}}fd\mu.$$
Proof. For any non-negative simple function $\varphi\in\mathcal{S}_{1}$,
$\mu(\varphi)$ does not decrease when the corresponding partition is replaced by its refinement.
Moreover, for $\psi\in\mathcal{S}_{2}$, we can construct $\varphi^{\prime}\in\mathcal{S}_{1}$ with
$\psi(x)=\varphi^{\prime}(x)$ as two functions.
The sub-additivity implies also $\mu(\psi)\leq\mu(\varphi^{\prime})$.
Obviously $\mathcal{S}_{1}\subset\mathcal{S}_{2}$, and this concludes the proof. $\Box$
For a sub-additive monotone measure, a Pan integral has the following linearity
(OuyMes2017 ).
This is proved for ${\int^{\uparrow}_{\mathcal{S}^{p+}}}$, however, a similar proof valid for ${\int^{\uparrow}_{\mathcal{S}^{p+}}}_{\mu}$.
Theorem 15
(Yao Ouyang, Jun Li, Radko Mesiar OuyMes2017 ) Let $\mu$ be a sub-additive monotone measure.
$f,g$ be non-negative measurable functions,
and $a,b$ be non-negative constants.
Assume that $\mathcal{S}=\mathcal{S}^{p+},\mathcal{S}^{p+}_{\mu}$.
Then,
$$\int(af+bg)d\mu=a\int fd\mu+b\int gd\mu.$$
$\Box$
Lemma 16
Let $\mu$ be a monotone measure,
$\{A_{n}\}$ be a decreasing sequence of measurable sets with $A_{n}\searrow\emptyset$.
Assume that $\mathcal{S}=\mathcal{S}^{p+},\mathcal{S}^{p+}_{\mu}$ and
$\int fd\mu<\infty$. Then,
$$\int f\ \mathrm{1}_{A_{n}}d\mu\searrow 0$$
Proof. By Lemma 4 (a),
$$\int f\ d\mu\geq\int f\ \mathrm{1}_{A_{n}^{c}}d\mu+\int f\ \mathrm{1}_{A_{n}}d\mu.$$
By Theorem 10, we have $\int f\mathrm{1}_{A_{n}^{c}}d\mu\to\int fd\mu$.
Then, consider the limit of $n\to\infty$
$$\lim_{n\to\infty}\int f\ \mathrm{1}_{A_{n}}d\mu=0.$$
$\Box$
Theorem 17
Let $\mu$ be a sub-additive monotone measure,
$\{f_{n}\}$ be a decreasing sequence of measurable functions.
Assume $\mathcal{S}=\mathcal{S}^{p+},\mathcal{S}^{p+}_{\mu},\mathcal{S}^{c+},\mathcal{S}^{c+}_{\mu}$,
and ${\int^{\uparrow}_{\mathcal{S}}}f_{1}d\mu<\infty$. Then,
$${\int^{\uparrow}_{\mathcal{S}}}f_{n}d\mu\searrow\int fd\mu.$$
Proof. By Lemma 14, we prove the theorem for Pan integral.
Fix any $\delta>0$, and set
$$A_{n}^{(\delta)}=\{x:f_{n}(x)\leq f(x)+\delta f_{1}(x)\}$$
$f(x)=f_{n}(x)=0$ when $f_{1}(x)=0$ since the sequence is non-increasing.
Then, $A_{n}^{(\delta)}\nearrow X$ ($n\to\infty$) .
Thus,
$$\displaystyle\int f_{n}d\mu$$
$$\displaystyle=$$
$$\displaystyle\int f_{n}\ \mathrm{1}_{A_{n}^{(\delta)}}+f_{n}\ \mathrm{1}_{A_{n}^{(\delta)\ c}}d\mu$$
$$\displaystyle\leq$$
$$\displaystyle\int(f\ +\delta f_{1})\ \mathrm{1}_{A_{n}^{(\delta)}}d\mu+\int f_{1}\ \mathrm{1}_{A_{n}^{(\delta)\ c}}d\mu$$
$$\displaystyle\leq$$
$$\displaystyle\int f\ \mathrm{1}_{A_{n}^{(\delta)}}d\mu+\delta\int f_{1}d\mu+\int f_{1}\ \mathrm{1}_{A_{n}^{(\delta)\ c}}d\mu.$$
Then, using Theorem10, the first term of the above formula
converges to $\int fd\mu$.
Then, using Lemma 16 ($\delta>0$ is arbitrary small),
$$\lim_{n\to\infty}\int f_{n}d\mu=\inf_{n}\int f_{n}d\mu\leq\int fd\mu.$$
Using the reverse inequality, which is obvious
$$\lim_{n\to\infty}\int f_{n}d\mu=\int fd\mu.$$
$\Box$
Monotone decreasing convergence theorem for Pan integral is valid,
when the limit is $0$(constant function).
Theorem 18
$\mathcal{S}=\mathcal{S}^{p+},\mathcal{S}^{p+}_{\mu}$,
Let $\mu$ be a monotone measure with the continuity at $\emptyset$.
$\{f_{n}\}_{n}$ be a decreasing sequence of measurable functions.
We assume that $\displaystyle\int\mathrm{1}_{X}d\mu<\infty$,
$\int f_{1}d\mu<\infty$, and $f_{n}\searrow 0,\quad(n\to\infty)$.
Then,
$$\lim_{n\to\infty}\int f_{n}d\mu=0.$$
Proof. Fix an arbitrary $\delta>0$. Set,
$$A_{n}^{(\delta)}=\{x:f_{n}(x)>\delta\}$$
Then, $A_{n}^{(\delta)}\searrow\emptyset$.
Let $\varphi=\sum_{k}b_{k}\mathrm{1}_{B_{k}}$
be an any simple function in $L(\mathcal{S},f_{n})$.
Then,
$$\mu(\varphi)=\sum_{k}b_{k}\mu(B_{k})=\sum_{b_{k}\leq\delta}b_{k}\mu(B_{k})+\sum_{b_{k}>\delta}b_{k}\mu(B_{k}).$$
When $b_{k}>\delta$,
using $\displaystyle\inf_{x\in B_{k}}f(x)\geq b_{k}>\delta$ and
$B_{k}\subset A^{(\delta)}_{n}$,
The right hand side.
$$\displaystyle\leq$$
$$\displaystyle\delta\sum_{k}\mu(B_{k})+\int f_{n}\mathrm{1}_{A^{(\delta)}_{n}}d\mu,$$
$$\displaystyle\leq$$
$$\displaystyle\delta\int\mathrm{1}_{X}d\mu+\int f_{1}\mathrm{1}_{A^{(\delta)}_{n}}d\mu.$$
$\mu(A^{(\delta)}_{n})\to 0$ ($n\to\infty$) since $\mu$ is continuous at $\emptyset$.
Thus, the second term converges to $0$ by Lemma 16,
and
$$\inf_{n}\int f_{n}d\mu\leq\delta\int\mathrm{1}_{X}d\mu\to 0,\quad(\delta\searrow 0)$$
$\Box$
References
(1)
R. Fukuda, A. Honda, Y. Okazaki, Comparison of decomposition type nonlinear
integrals based on the convergence theorem ( in japanese), Journal of Japan
Society for Fuzzy Theory and Intelligent Informatics 32(4) (2020) 782–791.
(2)
Q. Yang, The pan-integral on the fuzzy measure space, Fuzzy Mathematics (in
Chinese) 3 (1985) 107–114.
(3)
R. Mesiar, E. Pap, Superdecomposition integrals, Fuzzy Sets and Systems, 259
(2015) 3–11.
(4)
E. Lehrer, A new integral for capacities, Economic Theory 39 (2009) 157–176.
(5)
T. Tao, An Introduction to Measure Theory, Graduate Studies in Mathematics,
American Mathematical Society, 2011.
(6)
Y. Ouyang, R. Mesiar, On linearity of pan-integral and pan-integrable functions
space, International Journal of Approximate Reasoning archive 90 Issue C
(2017) 307–318. |
hep-ph/9412311
MPI/PhT/94–91
MAD/TH/94–1
TUM–HEP–200/94
December 1994
Two-loop ${\rm O}\left(G_{F}^{2}M_{H}^{4}\right)$ corrections to the
fermionic decay rates of the Higgs boson
Loyal Durand
${}^{1}$
Bernd A. Kniehl
Electronic address:
ldurand@wishep.physics.wisc.edu
${}^{2}$
and
Kurt Riesselmann
${}^{3}$
Electronic address: kniehl@mpiw16.mppmu.mpg.deElectronic address: kurtr@physik.tu-muenchen.de
${}^{1}$ Department of Physics, University of Wisconsin,
1150 University Avenue, Madison, WI 53706, USA
${}^{2}$ Max-Planck-Institut für Physik, Werner-Heisenberg-Institut,
Föhringer Ring 6, 80805 München, Germany
${}^{3}$ Physik-Department T30, Technische Universität München,
James-Franck-Straße, 85747 Garching b. München, Germany
(November 20, 2020)
Abstract
We calculate the dominant ${\rm O}\left(G_{F}^{2}M_{H}^{4}\right)$ two-loop
electroweak corrections to the fermionic decay widths of a
heavy Higgs boson in the Standard Model.
Use of the Goldstone-boson equivalence theorem reduces the problem
to one involving only the physical Higgs boson $H$ and the
Goldstone bosons $w^{\pm}$ and $z$ of the unbroken theory.
The two-loop corrections are opposite in sign to the one-loop
electroweak corrections, exceed
the one-loop corrections in magnitude for
$M_{H}>1114\ {\rm GeV}$, and increase in relative magnitude
as $M_{H}^{2}$ for larger values of $M_{H}$.
We conclude that the perturbation expansion in
powers of $G_{F}M_{H}^{2}$
breaks down for $M_{H}\approx 1100\ {\rm GeV}$.
We discuss briefly the QCD and
the complete one-loop electroweak corrections to $H\rightarrow b\bar{b},\,t\bar{t}$, and comment on the validity of the equivalence theorem.
Finally we note how a
very heavy Higgs boson could be described in a phenomenological
manner.
pacs: PACS number(s): 12.15.Lk, 11.15.Bt, 14.80.Bn
I INTRODUCTION
One of the great puzzles of contemporary elementary particle research
is whether nature makes use of the Higgs mechanism to generate
the observed particle masses.
In the minimal standard model (SM) of electroweak interactions,
the symmetry breaking
is implemented using this mechanism with one weak-isospin doublet of complex
scalar fields with weak hypercharge $Y=1$.
With the spontaneous breaking of the SU(2)${}_{L}\otimes$U(1)${}_{Y}$ gauge
symmetry, three of the four scalar degrees of freedom are absorbed to create
the longitudinal polarization states of the intermediate bosons, $W^{\pm}$
and $Z$.
At the same time, the quarks and charged leptons acquire masses through
their Yukawa interaction with the scalar doublet.
There remains at the end
one neutral scalar boson with positive parity and charge
conjugation, the physical Higgs boson $H$.
Most of the properties of the scalar or Higgs sector of the SM are fixed
experimentally, e.g., the vacuum expectation value,
$v=2^{-1/4}G_{F}^{-1/2}\approx 246\ {\rm GeV}$,
the coupling of the Higgs boson to the gauge bosons,
$g_{VVH}=2^{5/4}G_{F}^{1/2}M_{V}^{2}$ where $V=W^{\pm},Z$,
and the coupling of the Higgs to the fermions,
$g_{f\bar{f}H}=2^{1/4}G_{F}^{1/2}m_{f}$.
However, the mass $M_{H}$ of the Higgs boson and its quartic self-coupling,
$\lambda=G_{F}M_{H}^{2}/\sqrt{2}$, are unspecified.
It is therefore of considerable interest
to analyze processes which can give theoretical limits on $M_{H}$, or test
the effects of the quartic coupling phenomenologically.
The range of possible Higgs masses is constrained
from below both experimentally and theoretically.
The non-detection of the $Z$-boson decay $Z\rightarrow f\bar{f}H$ at LEP 1
and SLC has ruled out a Higgs mass of less than 63.9 GeV at the 95%
confidence level [1]. Depending on the mass of the top quark, the
requirement that the vacuum be the true ground state could provide an
even more stringent theoretical lower bound [2].
Other theoretical arguments bound the Higgs mass from above.
Nonperturbative lattice computations [3, 4] give an
upper limit for $M_{H}$ of about 710 GeV [4].
Unitarity arguments in intermediate-boson scattering
at high energies [5, 6] and considerations concerning the
range of validity of
perturbation theory [7, 8] establish an upper bound $M_{H}<(8\pi\sqrt{2}/3G_{F})^{1/2}\approx 1\ {\rm TeV}$ in a weakly interacting theory.
The unitarity bound on $M_{H}$
is lowered significantly when the approach of [5, 6] is
extended to higher orders; see [9, 10] and
references therein.
However, the improved bound depends on the energy scale up to
which the SM is assumed to remain valid.
A violation of the unitarity bound on $M_{H}$ is presumably a
signal for the onset of strong
interactions in the Higgs sector of the SM, a possibility which is
of considerable interest in its own right [11].
It would therefore be desirable to
sharpen the bound by removing the uncertainty associated with the mass
scale at which it is applied, or to find a separate
scale-independent bound. In fact,
the work presented here on two-loop electroweak corrections to the fermionic
decay modes of the Higgs boson, $H\rightarrow f\bar{f}$,
gives a scale-independent limit on $M_{H}$ in a weakly
interacting theory.
We find that the two-loop corrections to the
fermionic decay rates exceed the one-loop corrections in magnitude for
$M_{H}\approx 1114\ {\rm GeV}$, and increase in relative magnitude
proportionally to $M_{H}^{2}$ for larger Higgs-boson masses. We conclude, as
reported previously [12, 13, 14],
that the perturbative expansion fails to converge satisfactorily, and that the
theory becomes effectively strongly interacting in the Higgs sector for
$M_{H}\gtrsim 1100\ {\rm GeV}$.
The result noted above is a consequence of
our calculation of the dominant two-loop electroweak corrections to the
fermionic decay rates of the Higgs boson, the subject of this paper.
The fermionic decay modes are of considerable
phenomenological interest. For example, the Higgs boson decays predominantly
to $b\bar{b}$ pairs if $M_{H}\lesssim 135\ {\rm GeV}$.
The search for a low- or intermediate-mass Higgs boson at future high-energy
$e^{+}e^{-}$ linear colliders [15], the Fermilab Tevatron [16],
or a possible 4-TeV upgrade thereof [17] will rely largely on this mode
by tagging the $B$ mesons.
Moreover, in this mass range, the branching fractions of all other decay
channels depend sensitively on the $H\rightarrow b\bar{b}$ decay width.
It has been argued that the low-mass Higgs boson might also be detectable
at future hadron supercolliders through the
$H\rightarrow\tau^{+}\tau^{-}$ signal [18], while the decay
$H\rightarrow t\bar{t}$ will have an appreciable branching
fraction for $M_{H}>2m_{t}$.
Future high-energy $e^{+}e^{-}$ colliders will also be able to measure
the $Ht\bar{t}$ Yukawa coupling [19].
The measurement of the Higgs-boson mass and couplings in future experiments
will require an understanding
of the radiative corrections to the fermionic decay rates of the Higgs
boson. Much work has been done in this area, and a recent review is
given in [20]. There are important differences between the
radiative corrections involved in Higgs physics, and those familiar in the
gauge sector of the SM, e.g., in $Z$-boson decays.
It is well known that the quantum effects induced by
virtual Higgs bosons are screened in $Z$-boson physics
[7, 21]: they depend only logarithmically on $M_{H}$ at one
loop, and are quadratic in $M_{H}$, but with minute coefficients,
at two loops [22].
In contrast, the one-loop electroweak corrections to the partial decay widths
[7, 23, 24, 25] and production cross sections [26]
of the Higgs boson
are already dominated for $M_{H}\gg M_{W}$ by terms quadratic in $M_{H}$.
These terms give rise to moderate enhancements
of the rates for Higgs masses of up to $1\ {\rm TeV}$.
However, it is premature to conclude that the
two-loop electroweak corrections will
also be perturbatively small in the high-$M_{H}$ range, since these
corrections have terms quartic in $M_{H}$.
It is of both theoretical and phenomenological interest to check
the importance of these potentially large corrections by explicit calculation.
In this paper, we calculate the complete ${\rm O}\left(G_{F}^{2}M_{H}^{4}\right)$
corrections to the fermionic decay
rates of a Higgs boson with $M_{H}\gg M_{W}$. These corrections,
which are the leading two-loop electroweak corrections for $M_{H}\gg M_{W}$,
are independent of the fermion flavor, and, as noted above, are larger
than the one-loop corrections of ${\rm O}\left(G_{F}M_{H}^{2}\right)$ for
$M_{H}>1114\ {\rm GeV}$.
We compare our results with other known one-loop corrections.
To obtain the full electroweak two-loop
corrections for specific fermion channels in the limit $M_{H}\gg M_{W}$,
one would have to calculate further,
flavor-dependent corrections of mixed orders in the Higgs and Yukawa couplings,
namely ${\rm O}\left(G_{F}^{2}M_{H}^{2}m_{f}^{2}\right)$
and ${\rm O}\left(G_{F}^{2}m_{f}^{4}\right)$.
These corrections, however, are not universal. For example,
different fermionic channels,
such as $H\rightarrow\tau^{+}\tau^{-},\ b\bar{b},\ t\bar{t}$, all have different dependence on $m_{t}$.
II CALCULATION OF THE ${\rm O}\left(G_{F}^{2}M_{H}^{4}\right)$ CORRECTIONS
In this section, we sketch the calculation of the dominant, flavor-independent
electroweak corrections to the decay rate $H\rightarrow f\bar{f}$.
The starting point of our analysis is the bare Lagrangian for the
Higgs-fermion interaction,
$${\cal L}_{f,0}^{\rm Yuk}=-{m_{f,0}\over v_{0}}\,\bar{\psi}_{f,0}H_{0}\psi_{f,0%
}\,,$$
(1)
where the subscript “0” denotes bare quantities.
Our aim is to obtain the leading, flavor-independent corrections to
the $Hf\bar{f}$ vertex in powers of $G_{F}M_{H}^{2}$. The mass and wave-function
renormalization constants for the fermions as well as the loop corrections
to the vertex depend on the Yukawa couplings for the fermions,
and are omitted consistently because of their subleading nature.
(We will return to these corrections at the
one-loop level in Sec. IIIB.)
We may therefore replace the fermionic quantities
$m_{f,0}$ and $\psi_{f,0}$ in Eq. (1)
by $m_{f}$ and $\psi_{f}$. The contributions to the
renormalization constants for the bare Higgs
field, $H_{0}$, and the bare vacuum expectation value, $v_{0}$,
in powers of $G_{F}M_{H}^{2}$
are determined entirely by the symmetry-breaking sector of the
SM. The contributions of fermion loops to these two quantities
depend on the Yukawa couplings
and are again omitted as they do not contribute to the ${\rm O}(G_{F}^{2}M_{H}^{4})$
corrections (see Sec. IIIB). We may therefore
calculate the desired corrections to the decay $H\rightarrow f\bar{f}$ vertex
with all Yukawa couplings set to zero.
The calculation can be simplified greatly in the limit of interest,
$M_{H}\gg M_{W}$, through the use
of the Goldstone-boson equivalence theorem [27]. This theorem
states that the leading electroweak contribution to a graph in powers
of $G_{F}M_{H}^{2}$ can be calculated by replacing the gauge bosons $W^{\pm}$,
$Z$ by the would-be Goldstone bosons $w^{\pm}$, $z$ of the symmetry-breaking
sector of the theory.
Because $M_{W}/M_{H}\propto gv/M_{H}$, we can simplify our calculation
consistently in the limit of a heavy Higgs boson
by neglecting the gauge couplings $g,g^{\prime}$ and
taking the Goldstone bosons to be massless.
Adopting the conventions of [28], we can
write the relevant Lagrangian for the symmetry-breaking sector of the SM
in terms of bare quantities as follows:
$$\displaystyle{\cal L}_{0}^{\rm SBS}$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{2}\partial_{\mu}{\bf w}_{0}\cdot\partial^{\mu}{\bf w}_{0%
}+\frac{1}{2}\partial_{\mu}H_{0}\,\partial^{\mu}H_{0}-\frac{1}{2}M_{w,0}^{2}{%
\bf w}_{0}^{2}-\frac{1}{2}M_{H,0}^{2}H_{0}^{2}$$
$$\displaystyle-{\lambda_{0}\over 4}\left({\bf w}_{0}^{2}+H_{0}^{2}\right)^{2}-%
\lambda_{0}v_{0}\left({\bf w}_{0}^{2}+H_{0}^{2}\right)H_{0}\,,$$
where the real scalar triplet,
${\bf w}=(w_{1},w_{2},w_{3})$, is related to the
Goldstone bosons, $w^{\pm}$ and $z$, by
$w^{\pm}=({w}_{1}\mp i{w}_{2})/\sqrt{2}$ and $z={w}_{3}$, respectively.
The tadpole counterterm, which cancels all tadpole contributions of
${\cal L}_{0}^{\rm SBS}$ order by order,
has been omitted in writing Eq. (II); therefore all
graphs which include tadpole contributions need to be dropped in calculations
[28]. Note that a full equivalence-theorem calculation would also require
the complete Yukawa Lagrangian for the interactions of fermions with the
massless Goldstone bosons and the Higgs boson [29]. Because we are not
interested in corrections due to the Yukawa couplings we do not give the
complete Yukawa Lagrangian here, except for the piece given in Eq. (2.1).
The on-mass-shell renormalization is carried out in such a way
that the physical
mass of the Higgs boson, defined in terms of the position of
the pole in the Higgs propagator, is
$M_{H}$. The three Goldstone bosons remain massless to all orders in the
perturbation expansion in $\lambda_{0}$, and satisfy a residual SO(3)
symmetry [30]. Requiring that
the residues of the physical on-shell propagators be unity fixes the
wave-function renormalization constants defined by the relations
$w_{0}^{\pm}=Z_{w}^{1/2}w^{\pm}$, $z_{0}=Z_{z}^{1/2}z$, and $H_{0}=Z_{H}^{1/2}H$,
where $w^{\pm}$, $z$, and $H$ are the physical fields.
The result is [28]
$$\displaystyle{1\over Z_{w}}$$
$$\displaystyle=$$
$$\displaystyle 1-\left.{d\over dp^{2}}\Pi_{w}^{0}(p^{2})\right|_{p^{2}=0}\,,$$
(3)
$$\displaystyle{1\over Z_{z}}$$
$$\displaystyle=$$
$$\displaystyle 1-\left.{d\over dp^{2}}\Pi_{z}^{0}(p^{2})\right|_{p^{2}=0}\,,$$
$$\displaystyle{1\over Z_{H}}$$
$$\displaystyle=$$
$$\displaystyle 1-\left.{d\over dp^{2}}{\rm Re}\Pi_{H}^{0}(p^{2})\right|_{p^{2}=%
M_{H}^{2}}\,,$$
where $\Pi_{w}^{0}(p^{2})$, $\Pi_{z}^{0}(p^{2})$, and $\Pi_{H}^{0}(p^{2})$ are
the self-energy functions for
the bare fields calculated from the Lagrangian in Eq. (II).
Because of the SO(3) symmetry, $\Pi_{w}^{0}=\Pi_{z}^{0}$ and $Z_{w}=Z_{z}$.
Explicit expressions for the $\Pi^{0}$’s correct to two loops
may be found in Eqs. (11) and (12) of [28].
Furthermore, we have [28]
$$\displaystyle M_{H,0}^{2}$$
$$\displaystyle=$$
$$\displaystyle M_{H}^{2}-{\rm Re}\Pi_{H}^{0}\left(M_{H}^{2}\right)\,,$$
(4)
$$\displaystyle M_{w,0}^{2}$$
$$\displaystyle=$$
$$\displaystyle-{\rm Re}\Pi_{w}^{0}(0)=-\Pi_{w}^{0}(0)\,,$$
$$\displaystyle v_{0}$$
$$\displaystyle=$$
$$\displaystyle Z_{w}^{1/2}v\,,$$
$$\displaystyle\lambda_{0}$$
$$\displaystyle=$$
$$\displaystyle{\lambda\over Z_{w}}\left(1-{{\rm Re}\Pi_{H}^{0}\left(M_{H}^{2}%
\right)-\Pi_{w}^{0}(0)\over M_{H}^{2}}\right)\,.$$
Using these results, we can write the Lagrangians above
entirely in terms of
renormalized, physical quantities.
The physical vacuum expectation value is fixed in terms of the
Fermi constant $G_{F}$ by the familiar relation
$v=2^{-1/4}G_{F}^{-1/2}$, while the physical quartic coupling is given by
$\lambda=G_{F}M_{H}^{2}/\sqrt{2}$.
The renormalized symmetry-breaking Lagrangian is given in
Eq. (7) of [28], while the renormalized form of the Higgs-fermion
Lagrangian is given for our purposes by
$${\cal L}^{\rm Yuk}_{f}=-{{Z_{H}^{1/2}}\over Z_{w}^{1/2}}\,{m_{f}\over v}\,\bar%
{\psi}_{f}H\psi_{f}\,.$$
(5)
The radiatively corrected fermionic decay rate
of the Higgs boson is consequently given by
$$\Gamma\left(H\rightarrow f\bar{f}\,\right)={{Z_{H}}\over{Z_{w}}}\,\Gamma_{B}%
\left(H\rightarrow f\bar{f}\,\right)\,,$$
(6)
where [31]
$$\Gamma_{B}\left(H\rightarrow f\bar{f}\,\right)={N_{c}m_{f}^{2}M_{H}\over 8\pi v%
^{2}}\left(1-{{4{m_{f}^{2}}}\over{M_{H}^{2}}}\right)^{3/2}$$
(7)
is the Born result.
Here $N_{c}=1$ (3) for lepton (quark) flavors.
The wave-function renormalization constants
$Z_{H}$ and $Z_{w}$ were calculated to two loops,
${\rm O}\left(G_{F}^{2}M_{H}^{4}\right)$, in [28] using dimensional
regularization.
The two-loop diagrams that contribute to $Z_{H}$ and $Z_{w}$ at that order through
the derivatives of the self-energy functions in Eq. (3)
are shown in Fig. 1;
no single diagram gives an exceptionally large
contribution. The results of the calculation
can be written in the form
$${1\over Z_{\sigma}}=1+\hat{\lambda}\xi^{\epsilon}\left(\mbox{\rule{0.0pt}{12.8%
0374pt}}a_{\sigma}+{\rm O}(\epsilon)\right)+\hat{\lambda}^{2}\xi^{2\epsilon}%
\left({3\over\epsilon}+b_{\sigma}+{\rm O}(\epsilon)\right)+{\rm O}\!\left(\hat%
{\lambda}^{3}\right)\qquad(\sigma=w,H)\,,\\
$$
(8)
where $\hat{\lambda}=(\lambda/16\pi^{2})$, $\xi=4\pi\mu^{2}/M_{H}^{2}$,
$\epsilon=(4-D)/2$, $D$ is the dimensionality of space-time, and
$\mu$ is the arbitrary scale parameter introduced in the interaction to
keep $\lambda$ dimensionless for $\epsilon\not=0$. The coefficients
in the expansions above are:
$$\displaystyle a_{w}$$
$$\displaystyle=$$
$$\displaystyle 1\,,$$
(9)
$$\displaystyle b_{w}$$
$$\displaystyle=$$
$$\displaystyle\frac{3}{2}+2\zeta(2)-6\gamma-3\pi\sqrt{3}+12{\rm Cl_{2}\left({%
\pi\over 3}\right)}\sqrt{3}$$
$$\displaystyle\approx$$
$$\displaystyle 6.098\,,$$
$$\displaystyle a_{H}$$
$$\displaystyle=$$
$$\displaystyle-12+2\pi\sqrt{3}$$
$$\displaystyle\approx$$
$$\displaystyle-1.12\,,$$
$$\displaystyle b_{H}$$
$$\displaystyle=$$
$$\displaystyle\frac{291}{2}-96\zeta(2)+90\zeta(3)-6\gamma-48\pi{\rm Cl_{2}\left%
({\pi\over 3}\right)}+116\pi\sqrt{3}-216{\rm Cl_{2}\left({\pi\over 3}\right)}%
\sqrt{3}-162K_{5}$$
$$\displaystyle\approx$$
$$\displaystyle 41.12\,.$$
The constant $K_{5}=0.92363\ldots$
was evaluated numerically from Eq. (A86) of [28].
The Riemann $\zeta$ function takes the values
$\zeta(2)=\pi^{2}/6$ and $\zeta(3)=1.20205\ldots$,
${\rm Cl}_{2}$ is Clausen’s function, ${\rm Cl}_{2}({\pi\over 3})=1.01494\ldots$,
and $\gamma=0.57721\ldots$ is Euler’s constant.
The one-loop coefficients $a_{H}$ and $a_{w}$ are similar in magnitude, but the
two-loop coefficients $b_{H}$ and $b_{w}$ differ in magnitude by roughly a
factor of 7, despite the fact that almost the same number of diagrams,
with similar structures and magnitudes, contribute.
It is also interesting that the coefficients in $Z_{H}^{-1}$ alternate
in sign; those in $Z_{w}^{-1}$ do not. We note that the results above,
revised relative to our previous analysis [12],
are now in complete agreement with the those of Ghinculov
[13], which have also been revised [14].
Because the decay width $\Gamma\left(H\rightarrow f\bar{f}\right)$
is a physical quantity, and
all radiative corrections that depend only on $G_{F}M_{H}^{2}$ are
contained in the factor $Z_{H}/Z_{w}$ in Eq. (2.6), this factor must be finite
for $\epsilon\rightarrow 0$. The $Z$’s are finite at one loop,
but not at two loops. Hence, the parts of the two-loop contributions to
$Z_{H}$ and $Z_{w}$ that are proportional to $1/\epsilon$
must cancel in the ratio.
The cancellation is clear if the $Z$’s are written in factored form,
$${1\over Z_{\sigma}}=\left(1+a_{\sigma}\hat{\lambda}\xi^{\epsilon}+b_{\sigma}%
\hat{\lambda}^{2}\xi^{2\epsilon}\right)\left(1+{3\over\epsilon}\hat{\lambda}^{%
2}\xi^{2\epsilon}\right)+{\rm O}\left(\hat{\lambda}^{3}\right)\qquad(\sigma=w,%
H)\,,$$
(10)
and is exact to all orders in $\lambda$.
The complete cancellation of the divergent
terms allows us to take the limit $\epsilon\rightarrow 0$. In this limit,
$\xi^{\epsilon}\rightarrow 1$ with no pieces left over,
and the final ratio is independent of the scale $\mu$ introduced in the process
of dimensional regularization.
The ${\rm O}\left(G_{F}^{2}M_{H}^{4}\right)$ electroweak corrections to the
fermionic decay rates emerge naturally in this formalism as the finite ratio
$${{Z_{H}}\over{Z_{w}}}={{1+a_{w}\hat{\lambda}+b_{w}\hat{\lambda}^{2}}\over{1+a_%
{H}\hat{\lambda}+b_{H}\hat{\lambda}^{2}}}\,.$$
(11)
This expression for $Z_{H}/Z_{w}$
automatically resums one-particle-reducible Higgs-boson self-energy
diagrams in a way that conforms
with the standard procedure
in $Z$-boson physics; see, e.g., [32]. However, it
is clear that the resummation contains
only limited information on higher-order terms.
Since we actually have no control of terms beyond
${\rm O}\left(\hat{\lambda}^{2}\right)$, and are not aware of a physical
principle which would select this as an optimum resummation scheme, we
expand Eq. (11) and discard terms beyond
${\rm O}\left(\hat{\lambda}^{2}\right)={\rm O}\left(G_{F}^{2}M_{H}^{4}\right)$.
This gives the alternative representation
$$\displaystyle{{Z_{H}}\over{Z_{w}}}$$
$$\displaystyle=$$
$$\displaystyle 1+(a_{w}-a_{H})\hat{\lambda}+\left(b_{w}-b_{H}-a_{w}a_{H}+a_{H}^%
{2}\right)\hat{\lambda}^{2}$$
$$\displaystyle\approx$$
$$\displaystyle 1+2.12\hat{\lambda}-32.66\hat{\lambda}^{2}$$
$$\displaystyle\approx$$
$$\displaystyle 1+0.013{\lambda}-0.0013{\lambda}^{2}$$
$$\displaystyle\approx$$
$$\displaystyle 1+11.1\%\left({M_{H}\over 1\,{\rm TeV}}\right)^{2}-8.9\%\left({M%
_{H}\over 1\,{\rm TeV}}\right)^{4}\,.$$
The result agrees at ${\rm O}\left(\hat{\lambda}\right)$
with the known one-loop result [7, 23],
$${Z_{H}\over Z_{w}}=1+{G_{F}M_{H}^{2}\over 8\pi^{2}\sqrt{2}}\left({13\over 2}-%
\pi\sqrt{3}\,\right)\,.$$
(13)
III RESULTS
III.1 Limits on perturbation theory
We are now in a position to explore the phenomenological implications
of our results.
In Fig. 2, we show the leading electroweak
corrections to $\Gamma\left(H\rightarrow f\bar{f}\,\right)$ in the one- and
two-loop approximations with and without resummation of
one-particle-reducible higher-order terms plotted as functions of $M_{H}$.
We will concentrate first on the expanded results given in Eq. (II).
While the ${\rm O}\left(G_{F}^{2}M_{H}^{4}\right)$ term
(upper solid line in Fig. 2) gives
a modest increase of the rates, e.g., by 11% at $M_{H}=1\ {\rm TeV}$,
the situation changes when the two-loop term is included.
The importance of this term, which grows as $M_{H}^{4}$, increases
with $M_{H}$ in such a way that it cancels the one-loop term completely for
$M_{H}=1114\ {\rm GeV}$, and is twice the size of the one-loop term, with the
opposite sign, for $M_{H}=1575\ {\rm GeV}$. The total correction, shown by the
lower solid line in Fig. 2, is then negative and has the same
magnitude as the one-loop correction alone. The perturbation series for the
corrections to
$\Gamma\left(H\rightarrow f\bar{f}\,\right)$ clearly ceases to converge
usefully, if at all, for $M_{H}\approx 1100\ {\rm GeV}$, or
equivalently, for $\lambda\approx 10$.
A Higgs boson with a mass larger than about 1100 GeV
effectively becomes a strongly
interacting particle.
Conversely, $M_{H}$ must not exceed approximately 1100 GeV if the standard
electroweak perturbation theory is to be predictive for the decays
$H\rightarrow f\bar{f}$.
Note that one cannot use the usual unitarization
schemes invoked in studies of $W_{L}^{\pm},Z_{L},H$ scattering [6, 33]
to restore the predictiveness for the heavy-Higgs width, as no
unitarity violation is involved.
One might expect to improve the perturbative result
in the upper range of $M_{H}$ somewhat
by resumming the one-particle-reducible contributions to the Higgs-boson
wave-function renormalization by using Eq. (11) rather than
Eq. (II).
This leads to an increase of the one-loop correction
(upper dotted line in Fig. 2),
while the negative effect of the two-loop correction
is lessened (lower dotted line) for large values of $M_{H}$.
However, in the mass range below $M_{H}=1400\ {\rm GeV}$,
this effect is too small
to change our conclusions concerning the breakdown of perturbation theory.
Moreover, the resummed expression for the one-loop terms in the perturbation
expansion, when reexpanded to ${\rm O}\left(G_{F}^{2}M_{H}^{4}\right)$,
does not yield a proper estimate
for the size of the two-loop terms. There is consequently no reason to
favor this approach to the present problem.
It might be argued that the apparent breakdown in the perturbation
expansion as judged by a comparison of the one- and two-loop terms
is an artifact of a small one-loop contribution rather than
a consequence of large two-loop terms. However,
we see no evidence in the calculation that there are unusual cancellations
in the one-loop corrections. In fact, the one-loop contributions $a_{H}$
and $a_{w}$ add
in magnitude in the
ratio $Z_{H}/Z_{w}$, whereas the two-loop contribution $b_{w}$ is in magnitude
subtracted from $b_{H}$ [see Eqs. (9) and (II)].
In a previous publication, we looked at other processes to which $Z_{w}$
and $Z_{H}$ contribute, e.g., the scattering of Higgs bosons and
longitudinally polarized $W^{\pm}$ and $Z$ bosons [10].
In the latter case, even
larger coefficients appear in the perturbation expansion when it is
expressed, as above, in a series in the (running)
parameter $\hat{\lambda}_{\rm s}=(\lambda_{\rm s}/16\pi^{2})$
[34]. The contributions due to the finite parts of $Z_{H}$ and $Z_{w}$ are
rather insignificant in comparison with the finite parts of the
unrenormalized two-loop scattering graphs.
While
the factors of $(1/16\pi^{2})$ occur naturally at each order in
perturbation theory, their incorporation into the natural parameter
$\hat{\lambda}$ is misleading: the coefficients of the zero-,
one-, and two-loop
terms in the diagonal partial-wave scattering amplitudes, and the one-
and two-loop terms in the Higgs-boson decay rate calculated above, all
have similar magnitudes when the series are
rewritten as expansions in the physical parameters
$\lambda_{\rm s}$ and $\lambda=G_{F}M_{H}^{2}/\sqrt{2}$
as seen in Eq. (II) and the comment [34].
It appears, therefore, that $\lambda$, and not
$\hat{\lambda}=\lambda/(16\pi^{2})$, is
the natural expansion
parameter.
The high-energy scattering processes give strong evidence for a breakdown of
the perturbation series for a running coupling
$\lambda_{\rm s}(\sqrt{s})\approx 2.3$
at either the one-loop [9] or two-loop [10] level. This
translates to $M_{H}\approx 380\ {\rm GeV}$
if the SM is assumed to remain valid for energies $\sqrt{s}$
up to $\sim 5$ TeV [10]. The $M_{H}$ upper bound obtained
in the present analysis is considerably less stringent than the one found in
[10].
However, we emphasize that the result obtained here does not
depend on the extra assumption about an energy scale;
the breakdown of perturbation theory is fixed solely
by the physical value of $M_{H}$.
III.2 Comparison with complete one-loop corrections
The leading corrections discussed so far are independent of the flavor of
the final-state fermion. However,
from an experimental point of view, they are relevant only for the
$t\bar{t}$ and, perhaps, the $b\bar{b}$ and $\tau^{+}\tau^{-}$
decays of the Higgs boson.
It is therefore interesting to compare
the corrections calculated here with the full one-loop
electroweak corrections [25], and the QCD corrections that are
available for these decay channels [35, 36].
In particular, the subleading two-loop electroweak corrections, those of
${\rm O}\left(G_{F}^{2}M_{H}^{2}m_{t}^{2}\right)$
and ${\rm O}\left(G_{F}^{2}m_{t}^{4}\right)$, are still unknown, but one may
estimate their likely importance by comparing the
top-quark Yukawa-coupling correction
to the Higgs-coupling correction at one loop.
The QCD corrections to the $H\rightarrow q\bar{q}$ modes,
where $q$ denotes a quark flavor, are known to ${\rm O}(\alpha_{\rm s})$ for
arbitrary values of $m_{q}$ [35] and
to ${\rm O}\left(\alpha_{\rm s}^{2}\right)$
in the limit $m_{q}\ll M_{H}$ [36].
Their main effect is to replace the pole mass of
the quark $q$ by its $\overline{\rm MS}$ mass evaluated at the scale $M_{H}$.
For completeness, we mention that the
${\rm O}\left(\alpha_{\rm s}G_{F}m_{t}^{2}\right)$
corrections to the fermionic decay rates are also now
available [37].
The results in the preceding section were derived using the
Goldstone-boson equivalence theorem and neglecting further electroweak
corrections of orders $g^{2}$, $\lambda g^{2}$, etc., as well as contributions
that involve the Yukawa and QCD couplings of the fermions. Since $\lambda$,
$g^{2}$, $\alpha_{\rm s}$, and the Yukawa couplings are independent
parameters of the theory, our conclusion that the perturbation series in
$\lambda=G_{F}M_{H}^{2}/\sqrt{2}$ fails to converge satisfactorily is
independent of further corrections involving powers of $g$ and the other
independent couplings, though those further corrections may be numerically
important in applications of the results.
The use of the equivalence theorem
provides a correct framework for calculating the leading electroweak
corrections—those enhanced by the maximum powers of $M_{H}/M_{W}$—at
each order. By neglecting subleading corrections that involve $g$ or
the Yukawa couplings $g_{f}=\sqrt{2}m_{f}/v$, we expect to obtain
a good approximation to the full
result provided that $M_{W}/M_{H}\propto gv/M_{H}\ll 1$ and $m_{f}/M_{H}\ll 1$.
Because of the
high mass of the $t$ quark [38],
it is interesting
to test the accuracy of the approximation.
In Fig. 3, we compare the
${\rm O}\left(G_{F}M_{H}^{2}\right)$ correction
to $\Gamma\left(H\rightarrow t\bar{t}\,\right)$,
already shown in Fig. 2, to the full
one-loop electroweak correction including the effects of fermions
[25]. The full correction was evaluated in the on-shell renormalization
scheme using $m_{t}=174\ {\rm GeV}$ [38].
We see that the ${\rm O}\left(G_{F}M_{H}^{2}\right)$ term underestimates the full
one-loop electroweak correction term by 32% (24%) at
$M_{H}=500\ {\rm GeV}$ (1 TeV). To check that the difference arises primarily
from the inclusion of the top quark in the full calculation,
and not from the supposedly small
contributions from the gauge sector of the SM, that is, from a failure of the
equivalence theorem, we have carried out a complete one-loop calculation
using the equivalence theorem with the gauge couplings set to zero [29],
but the top-quark Yukawa coupling retained.
The extra contributions are ${\rm O}\left(G_{F}m_{t}^{2}\right)$, and are
independent of those considered above. As shown in Fig. 3,
the result of the calculation reproduces the full one-loop electroweak result
very well. The result obtained using the equivalence theorem with $g_{t}\not=0$
is only 3.9% (1.8%) larger than the full
electroweak one-loop term at $M_{H}=500\ {\rm GeV}$ (1 TeV) for $m_{t}=174\ {\rm GeV}$. The use of the equivalence theorem therefore gives a quite
accurate approximation to the full theory, even for the rather low
values of $M_{H}$ with which we are concerned. The small residual differences
away from the decay threshold at $M_{H}=2m_{t}$
can be accounted for by the transverse gauge couplings,
the nonzero masses of the
$W$ and $Z$ bosons, and the finite masses and Yukawa couplings for the
remaining
fermions. The extra structure close to the
threshold is the result of
virtual-photon exchange in QED.
This generates a Coulomb singularity
and a correction that behaves near threshold as
$1+\alpha_{\rm em}Q_{t}^{2}[(\pi/2\beta)+{\rm O}(1)]$,
where $Q_{t}$ and $\beta$ are the top-quark electric charge and velocity;
see left end of the dashed line in Fig. 3.
In Fig. 4 we compare the electroweak and QCD corrections.
The latter were calculated with the asymptotic scale parameter of QCD
adjusted to give $\alpha_{\rm s}(M_{Z})=0.118$ [39]. The
one-loop QCD correction to $\Gamma\left(H\rightarrow t\bar{t}\right)$
in Fig. 4(a)
shows the expected color-Coulomb threshold singularity, with
$\alpha_{\rm em}Q_{t}^{2}$
replaced in the expression above by $(4/3)\alpha_{\rm s}(M_{H})$.
This singularity is associated with the nonrelativistic motion of the
quarks. The set of correction terms in powers of $(2\pi\alpha_{\rm s}/3\beta)$
corresponds to the expansion of a Coulomb wave function at zero quark
separation. For $M_{H}\gg 2m_{t}$, the one-loop QCD correction is negative, with
a magnitude which increases logarithmically. The two-loop QCD corrections to
$\Gamma\left(H\rightarrow t\bar{t}\,\right)$ are unknown.
They are expected to be large close to the $t\bar{t}$ production threshold,
at $M_{H}\gtrsim 2m_{t}$.
At $M_{H}\gg 2m_{t}$,
the potentially large logarithmic contributions in all higher orders
can be resummed by using the top-quark $\overline{\rm MS}$ mass
evaluated at the scale $M_{H}$,
and the residual corrections should be small (see the discussion given
below for the $b\bar{b}$ decay).
In Fig. 4(b), we repeat the comparison of
Fig. 4(a) for the case $H\rightarrow b\bar{b}$ assuming $m_{b}=4.72\ {\rm GeV}$ [40].
The difference between the full one-loop electroweak correction and
the ${\rm O}\left(G_{F}M_{H}^{2}\right)$ result is again accounted for at
large values of $M_{H}$ by the omission of top-quark effects in the latter.
The spikes in the full correction at $M_{H}=2M_{W}$ and $2M_{Z}$ originate in
threshold singularities of the Higgs-boson wave-function renormalization.
The dent at $M_{H}=2m_{t}$ is not accompanied by such a divergence.
These features may be understood as artifacts of the underlying approximation
of treating the unstable Higgs boson as an asymptotic state.
The QCD corrections are calculated to
${\rm O}\left(\alpha_{\rm s}^{2}\right)$
in the $\overline{\rm MS}$ scheme [35, 36]; see Eq. (37) of [41].
When the quark pole mass is used as a basic parameter,
the largest part of the one-loop QCD correction comes from a large
logarithmic term $-(4\alpha_{\rm s}/\pi)\ln\left(M_{H}/m_{b}\right)$ [35].
In general, the large logarithms are of the form
$(\alpha_{s}/\pi)^{n}\ln^{m}(M_{H}/m_{b})$,
with $n\geq m$. By
exploiting renormalization-group techniques, these logarithms may be
absorbed completely into the $\overline{\rm MS}$ quark mass, $m_{b}(\mu)$,
evaluated at the scale $\mu=M_{H}$ [20].
The logarithms are resummed to all orders, and the remaining perturbative
expansion converges more rapidly.
The offset seen in the Fig. 4(b), with the QCD-corrected
decay considerably below the Born decay rate, results mainly from the use
of $m_{b}(M_{H})$ instead of $m_{b}$ in the prefactor in Eq. (7).
While the effect is large, it is controlled by the resummation.
The remaining part of the QCD correction at two loops is rather small.
III.3 Handling a nonperturbative Higgs
One would like to be able to describe the Higgs-boson decay to fermions
phenomenologically even in the case of a strongly interacting Higgs sector.
In the case of the QCD corrections discussed above, the renormalization
group provided a physical principle that could be used to motivate
and organize a resummation of higher-order effects to obtain a
controlled final expression, even though the corrections could be
large when viewed order-by-order in $\alpha_{\rm s}$.
We do not know of a similar physical organizing principle
to use in the resummation of higher-order corrections in $\lambda$,
especially as the leading corrections only depend on one energy scale, namely
the mass of the Higgs boson. Any
summation of the perturbation series will therefore be speculative, and
will necessarily be based
on the mathematical structure of the series rather a physical argument.
We note in this connection that the correction terms in Eq. (II)
alternate in sign.
This suggests that Padé summation of the series might be reasonable.
In particular, we can use the two-loop information given in
Eq. (II) to rewrite the perturbative series
using a [1,1] Padé approximant
[42], that is, as a ratio of two first-degree polynomials
in $\lambda$, with the coefficients adjusted to fit the expansion
in Eq. (II) through order $\lambda^{2}$. We obtain
$${{Z_{H}}\over{Z_{w}}}=1+{{(a_{w}-a_{H})\hat{\lambda}}\over{1-[{{(b_{w}-b_{H}-a%
_{w}a_{h}+a_{H}^{2})}/{(a_{w}-a_{H})}}]\hat{\lambda}}}\,.$$
(14)
In Fig. 5, we compare the Padé-summed correction factor,
Eq. (14), with the earlier results from Fig. 2
or Eqs. (11) and (II). The result suggests
that the leading electroweak corrections to $\Gamma\left(H\rightarrow f\bar{f}\right)$ will be quite small
even for values of $M_{H}\gtrsim 1.5$ TeV.
How far the result can be trusted is a matter of speculation.
A similar Padé summation of the partial-wave scattering amplitudes for
$W_{L}^{\pm},\ Z_{L},\ H$ scattering turns out to give a fairly good prediction
for the two-loop contribution in terms of the zero- and one-loop terms
[43],
so the method may be more reliable in this rather similar case than our
limited input information would suggest. If so, the leading electroweak
corrections to $H\rightarrow f\bar{f}$
in powers of $\lambda$ or $G_{F}M_{H}^{2}$ will be negligible, when resummed,
relative to the corrections introduced by the Yukawa
couplings and QCD. Only experimental results or reliable nonperturbative
calculations can resolve this speculation.
We note in this connection that recent lattice simulations of certain
Yukawa models for the interaction of the Higgs boson
with mirror or reduced staggered fermions suggest that
the Yukawa couplings cannot be strong, unless the regularization scale is
unacceptably low [44].
IV CONCLUSIONS
In summary, we have calculated the leading two-loop electroweak
corrections to the fermionic decay rates of a high-mass Higgs boson
in the SM, which are of ${\rm O}\left(G_{F}^{2}M_{H}^{4}\right)$. The corrections
are negative and exceed the positive ${\rm O}\left(G_{F}M_{H}^{2}\right)$ one-loop
corrections in magnitude for $M_{H}>1114\ {\rm GeV}$.
For larger values of $M_{H}$, the perturbation series is clearly unreliable,
and the theory becomes effectively strongly interacting.
We conclude, given the
lack of a physical principle that would allow a convincing resummation
of the perturbation expansion, that a value $M_{H}\sim 1100\ {\rm GeV}$
has to be considered as a theoretical
upper bound on $M_{H}$ beyond which the fermionic decay width of the Higgs cannot
be calculated perturbatively.
This result is independent of speculations regarding
the energy scale up to which the SM is valid, as the center-of-mass energy in
the Higgs decay is fixed,
$\sqrt{s}=M_{H}$.
However, there is indication that high-energy interactions in the
Higgs sector of the SM can become effectively strong, and are not usefully
calculable in perturbation theory, for even smaller Higgs mass.
For
scattering processes with $\sqrt{s}\sim 5$ TeV, the critical value for $M_{H}$
is about
$380$ GeV, and for scattering at GUT energies the critical value is less than
$160$ GeV [10].
Clearly, the present-day precision
tests of the gauge sector of the SM are not affected by such nonperturbative
effects.
Acknowledgements.One of us (BAK) would like to express his gratitude to the Physics
Department of UW-Madison for supporting his visit, during which part of
this work was carried out, and for the great hospitality extended to him.
This work was supported in part by the U.S. Department of Energy under
Contract No. AC02–76ER00881.
References
[1]
P. Janot, LAL Report No. 94-59 (August 1994).
[2]
A.D. Linde, JETP Lett. 23, (1976) 64; S. Weinberg,
Phys. Rev. Lett. 36, 294 (1976);
M. Lindner, M. Sher, and H.W.Zaglauer,
Phys. Lett. B 228, 139 (1989);
M.J. Duncan, R. Phillipe, and M. Sher, Phys. Lett. B 153, 165 (1985);
G. Altarelli and G. Isidori, Phys. Lett. B 337, 141 (1994);
J.A. Casas, J.R. Espinosa, and M. Quiros, Madrid University Report No. IEM-FT-93-94, hep-ph/9409458 (September 1994).
[3]
A. Hasenfratz and T. Neuhaus, Nucl. Phys. B297, 205
(1988);
W. Langguth and I. Montvay, Z. Phys. C 36, 725 (1987);
A. Hasenfratz, T. Neuhaus, K. Jansen, H. Yoneyama, and C.B. Lang,
Phys. Lett. B 199, 531 (1987);
P. Hasenfratz and J. Nager, Z. Phys. C 37, 477 (1988);
M. Lüscher and P. Weisz, Phys. Lett. B 212, 472 (1988);
U.M. Heller, H. Neuberger, and P. Vranas, Nucl. Phys. B399, 271 (1993).
[4]
U.M. Heller, M. Klomfass, H. Neuberger, and P. Vranas,
Nucl. Phys. B405, 555 (1993).
[5]
D.A. Dicus and V.S. Mathur, Phys. Rev. D 7, 3111 (1973).
[6]
B.W. Lee, C. Quigg, and H.B. Thacker, Phys. Rev. Lett. 38, 883 (1977); Phys. Rev. D 16, 1519 (1977).
[7]
M. Veltman, Acta Phys. Pol. B8, 475 (1977).
[8]
M. Veltman, Phys. Lett. 70B, 253 (1977).
[9]
L. Durand, J.M. Johnson, and J.L. Lopez, Phys. Rev. Lett. 64, 1215 (1990); Phys. Rev. D 45, 3112 (1992).
[10]
L. Durand, P.N. Maher, and K. Riesselmann,
Phys. Rev. D 48, 1084 (1993); erratum (to appear).
[11]
For early discussions of the possibility of
strong interactions in the Higgs sector of the SM, see
[6, 7, 8] and M.S. Chanowitz and M.K. Gaillard, Nucl. Phys. B261, 379 (1985).
[12]
L. Durand, B.A. Kniehl, and K. Riesselmann, Phys. Rev. Lett. 72, 2534 (1994); erratum (to appear).
[13]
A. Ghinculov, Phys. Lett. B 412, 523 (1994); erratum (to appear).
[14]
The results in [12] and [13] have both been revised, and now
agree completely. We would like to thank Dr. A. Ghinculov for his
cooperation comparing the two independent calculations.
[15]
Z. Kunszt and W.J. Stirling, Phys. Lett. B 242,
507 (1990);
V. Barger, K. Cheung, A. Djouadi, B.A. Kniehl, and P.M. Zerwas,
Phys. Rev. D 49, 79 (1994).
[16]
A. Stange, W. Marciano, and S. Willenbrock,
Phys. Rev. D 49, 1354 (1994).
[17]
J.F. Gunion and T. Han, UCD Report No. UCD–94–10,
hep-ph/9404244 (April 1994), Phys. Rev. D (to appear);
A. Stange, W. Marciano, and S. Willenbrock,
Phys. Rev. D 50, 4491 (1994).
[18]
R.K. Ellis, I. Hinchliffe, M. Soldate, and J.J. van der Bij,
Nucl. Phys. B297, 221 (1988).
[19]
K. Hagiwara, H. Murayama, and I. Watanabe,
Nucl. Phys. B367, 257 (1991).
[20]
B.A. Kniehl,
Phys. Rep. 240, 211 (1994).
[21]
M. Lemoine and M. Veltman, Nucl. Phys. B164, 445 (1980).
[22]
J. van der Bij and M. Veltman, Nucl. Phys. B231, 205
(1984);
J.J. van der Bij, ibid. B248, 141 (1984);
F. Halzen and B.A. Kniehl, ibid. B353, 567 (1991);
F. Halzen, B.A. Kniehl, and M.L. Stong, Z. Phys. C 58, 119 (1993);
R. Barbieri, P. Ciafaloni, and A. Strumia, Phys. Lett. B 317,
381 (1993).
[23]
W.J. Marciano and S.S.D. Willenbrock,
Phys. Rev. D 37, 2509 (1988).
[24]
J. Fleischer and F. Jegerlehner,
Phys. Rev. D 23, 2001 (1981);
B.A. Kniehl, Nucl. Phys. B352, 1 (1991); ibid. B357,
439 (1991).
[25]
D.Yu. Bardin, B.M. Vilenskiĭ, P.Kh. Khristov,
Yad. Fiz. 53, 240 (1991)
[Sov. J. Nucl. Phys. 53, 152 (1991)];
B.A. Kniehl, Nucl. Phys. B376, 3 (1992);
A. Dabelstein and W. Hollik, Z. Phys. C 53, 507 (1992).
[26]
J. Fleischer and F. Jegerlehner, Nucl. Phys. B216,
469 (1983);
B.A. Kniehl, Z. Phys. C 55, 605 (1992);
A. Denner, J. Küblbeck, R. Mertig, and M. Böhm,
ibid. 56, 261 (1992).
[27]
J.M. Cornwall, D.N. Levin, and G. Tiktopoulos,
Phys. Rev. D 10, 1145 (1974); (E) 11, 972 (1975);
C.E. Vayonakis, Lett. Nuovo Cim. 17, 383 (1976);
M.S. Chanowitz and M.K. Gaillard, Nucl. Phys. B261, 379 (1985);
G.J. Gounaris, R. Kögerler, and H. Neufeld, Phys. Rev. D 34,
3257 (1986);
Y.-P. Yao and C.-P. Yuan, ibid. 38, 2237 (1988);
J. Bagger and C. Schmidt, ibid. 41, 264 (1990);
H. Veltman, ibid. 41, 2294 (1990); H.-J. He, Y.-P. Kuang,
and X. Li, Phys. Rev. Lett. 69, 2619 (1992).
[28]
P.N. Maher, L. Durand, and K. Riesselmann,
Phys. Rev. D 48, 1061 (1993); erratum (to appear).
[29]
L. Durand and K. Riesselmann, University of Wisconsin
report MAD/TH/94–9.
[30]
The SO(3) symmetry of the $(w^{\pm},z)$ Goldstone bosons
is broken by the
interactions of the $w^{\pm}$ and $z$ with fermions, but this does not affect
the dominant two-loop corrections. The complete renormalization scheme
for use with the Goldstone-boson equivalence theorem is discussed in
[29].
[31]
L. Resnick, M.K. Sundaresan, and P.J.S. Watson,
Phys. Rev. D 8, 172 (1973).
[32]
W. Beenakker and W. Hollik, Z. Phys. C 40, 141 (1988).
[33]
W.W. Repko and C.J. Suchyta, Phys. Rev. Lett. 62,
859 (1989);
D.A. Dicus and W.W. Repko, Phys. Lett. B 228, 503 (1989);
Phys. Rev. D 42, 3660 (1990);
G. Valencia and S. Willenbrock, Phys. Rev. D 42, 853 (1990);
H. Veltman and M. Veltman, Acta Phys. Pol. B22, 669 (1991);
K. Hikasa and K. Igi, Phys. Lett. B 261, 285 (1991).
[34]
For example, the largest of the diagonalized $J=0$ amplitudes for
$W_{L}^{\pm},\ Z_{L},\ H$ scattering is given, up to a negligible anomalous-dimension
correction, by
$$\displaystyle a_{1}(s)$$
$$\displaystyle=$$
$$\displaystyle\hat{\lambda}_{\rm s}\Biggl{[}-6+\hat{\lambda}_{\rm s}(185.6+113.%
1i)-\hat{\lambda}_{\rm s}^{2}(9523.5+6892.3i)+{\rm O}\left(\hat{\lambda}_{\rm s%
}^{3}\right)\Biggr{]}$$
$$\displaystyle=$$
$$\displaystyle\frac{-6\lambda_{\rm s}}{16\pi}\Bigl{[}1-\lambda_{\rm s}(0.1959+0%
.1194i)+\lambda_{\rm s}^{2}(0.0637+0.0461i)+{\rm O}\left(\lambda_{\rm s}^{3}%
\right)\Bigr{]}.$$
[35]
E. Braaten and J.P. Leveille, Phys. Rev. D 22, 715 (1980);
N. Sakai, ibid. 22, 2220 (1980);
T. Inami and T. Kubota, Nucl. Phys. B179, 171 (1981);
M. Drees and K. Hikasa, Phys. Rev. D 41, 1547 (1990);
Phys. Lett. B 240, 455 (1990);
(E) 262, 497 (1990).
[36]
S.G. Gorishny, A.L. Kataev, S.A. Larin, and L.R. Surguladze,
Mod. Phys. Lett. A 5, 2703 (1990); Phys. Rev. D 43,
1633 (1991);
A.L. Kataev and V.T. Kim, Mod. Phys. Lett. A9, 1309 (1994);
L.R. Surguladze, University of Oregon Report No. OITS 543, hep-ph/9405325
(May 1994);
K.G. Chetyrkin, J.H. Kühn, and A. Kwiatkowski,
Karlsruhe University Report No. TTP94-10, hep-ph/9407271 (July 1994),
to appear in Proceedings of the 1994 Zeuthen Workshop on Elementary
Particle Theory: Physics at LEP200 and Beyond, Teupitz, April 10–15, 1994,
ed. by J. Blümlein and T. Riemann, Nucl. Phys. B (Proc. Suppl.);
B.A. Kniehl, MPI Report No. MPI/PhT/94-69, hep-ph/9410356 (October 1994).
[37]
B.A. Kniehl and A. Sirlin, Phys. Lett. B 318, 367 (1993);
B.A. Kniehl, Phys. Rev. D 50, 3314 (1994);
A. Kwiatkowski and M. Steinhauser, Phys. Lett. B 338, 66 (1994);
A. Djouadi and P. Gambino, Report No. UdeM-GPP-TH-94-02 and NYU-TH-94/06/01,
hep-ph/9406431 (June 1994);
B.A. Kniehl and M. Spira,
DESY Report No. 94-102, hep-ph/9410319 (October 1994),
Nucl. Phys. B (to appear).
[38]
CDF Collaboration, F. Abe et al.,
Phys. Rev. D 50, 2966 (1994);
Phys. Rev. Lett. 73, 225 (1994).
[39]
S. Bethke, to appear in
Proceedings of the Tennessee International Symposium on Radiative
Corrections: Status and Outlook, Gatlinburg, Tennessee, June 27–July 1,
1994, ed. by B.F.L. Ward (World Scientific, Singapore, 1994).
[40]
C.A. Dominguez and N. Paver,
Phys. Lett. B 293, 197 (1992).
[41]
B.A. Kniehl, KEK Report No. KEK-TH-412, hep-ph/9410330
(September 1994), Int. J. Mod. Phys. A (to appear).
[42]
C.M. Bender and S.A. Orzag,
Advanced Mathematical Methods for
Scientists and Engineers (McGraw-Hill, New York, 1978), Sec. 8.4;
J.L. Basdevant, Fort. Phys. 20, 283 (1972).
[43]
K. Riesselmann, Ph.D. thesis, University of
Wisconsin–Madison, 1994 (unpublished).
[44]
C. Frick, L. Lin, I. Montvay, G. Münster, M. Plagge,
T. Trappenberg, and H. Wittig, Nucl. Phys. B397, 431 (1993);
W. Bock, C. Frick, J. Smit, and J.C. Vink,
ibid. B400, 309 (1993). |
Early stage massive star formation near the Galactic Center: Sgr C
S. Kendrew11affiliation: University of Oxford, Department of Astrophysics, Keble Road, Oxford OX1 3RH, United Kingdom
Max-Planck-Institut für Astronomie, Königstuhl 17, 69117 Heidelberg, Germany
kendrew@mpia.de
A. Ginsburg
CASA, University of Colorado at Boulder, UCB 389, Boulder, CO 80309, USA
K. Johnston
Max-Planck-Institut für Astronomie, Königstuhl 17, 69117 Heidelberg, Germany
H. Beuther
Max-Planck-Institut für Astronomie, Königstuhl 17, 69117 Heidelberg, Germany
J. Bally
CASA, University of Colorado at Boulder, UCB 389, Boulder, CO 80309, USA
C.J. Cyganowski22affiliation: NSF Astronomy and Astrophysics Postdoctoral Fellow
Harvard-Smithsonian Center for Astrophysics, Cambridge, MA 02138, USA
C. Battersby
CASA, University of Colorado at Boulder, UCB 389, Boulder, CO 80309, USA
Abstract
We present near-infrared spectroscopy and 1 mm line and continuum observations of a recently identified site of high mass star formation likely to be located in the Central Molecular Zone near Sgr C. Located on the outskirts of the massive evolved H ii region associated with Sgr C, the area is characterized by an Extended Green Object measuring $\sim$10″ in size (0.4 pc), whose observational characteristics suggest the presence of an embedded massive protostar driving an outflow. Our data confirm that early-stage star formation is taking place on the periphery of the Sgr C H ii region, with detections of two protostellar cores and several knots of H${}_{2}$ and Brackett $\gamma$ emission alongside a previously detected compact radio source. We calculate the cores’ joint mass to be $\sim$10${}^{3}$ M${}_{\odot}$, with column densities of 1-2$\times$10${}^{24}$ cm${}^{-2}$. We show the host molecular cloud to hold $\sim$10${}^{5}$ M${}_{\odot}$ of gas and dust with temperatures and column densities favourable for massive star formation to occur, however, there is no evidence of star formation outside of the EGO, indicating that the cloud is predominantly quiescent. Given its mass, density, and temperature, the cloud is comparable to other remarkable non-star-forming clouds such as G0.253 in the Eastern CMZ.
Subject headings:Infrared: ISM — ISM: jets and outflows — Stars: formation
††slugcomment: Submitted to ApJ Letters, 21 June 2013
1. Introduction
The Central Molecular Zone (CMZ) of the Milky Way Galaxy is a chemically and dynamically complex region containing up to 10% of the Galaxy’s molecular gas, concentrated in dense and turbulent molecular clouds ($-2.5^{\circ}\leq l\leq 3.5^{\circ}$, $|b|\leq 0.5^{\circ}$). Densities and turbulent velocities are known to be approximately an order of magnitude higher than clouds in the MW disk (Morris & Serabyn 1996), and shocks are widely observed (Riquelme et al. 2010). The CMZ presents an ideal laboratory for the study of star formation in extreme environments.
The distribution of molecular gas in the CMZ is asymmetric around the Galactic Center (GC), with roughly two-thirds of the gas found at positive (Eastern) longitudes; this is mirrored in the observed distribution of CMZ star formation (Yusef-Zadeh et al. 2009, YZ09 hereafter). The Eastern CMZ is home to several star-forming regions, e.g. Sgr B2.
The only known star forming region in the Western CMZ is Sgr C (Figure 1). Lang et al. (2010) identify the region’s main components as a 10-pc H ii region at $(l,b)=$(359.43, -0.09) (assuming a distance of 8.5 kpc; Reid et al. 2009), accompanied by a distinctive non-thermal filament (Liszt & Spiker 1995). The H ii region contains more than 250 M${}_{\odot}$ of ionized gas, powered by at least one O4-O6 star (Liszt & Spiker 1995; Odenwald & Fazio 1984), however its stellar population remains poorly characterized. To the East of the H ii region lies a pillar-shaped cloud, whose mass has been estimated at 10${}^{5}$ M${}_{\odot}$ (Lis et al. 1991; Lis & Carlstrom 1994).
Molinari et al. (2011) recently proposed that Sgr C lies at the Western vertex of a twisted elliptical ring of molecular gas and dust, 100 $\times$ 60 pc in size. The Eastern longitude extremum is formed by Sgr B2. They propose that the collision of orbit systems at the extrema give rise to strong shocks, driving enhanced star formation. Given the prolific star formation rate of Sgr B2, Sgr C must be examined more closely to understand the observed asymmetry in the CMZ.
The first evidence of high mass star formation in the Sgr C cloud was reported by Forster & Caswell (2000), who detected a faint 8-9 GHz radio source measuring 0.06 $\times$ 0.01 pc near its tip. Using data from the Spitzer GALCEN survey (Stolovy et al. 2006), YZ09 identify a region of extended 4.5 µm emission - a so-called Extended Green Object (EGO; Cyganowski et al. 2008) - within $\sim$5″ from the radio source. Numerous studies have found EGOs to be strongly associated with early-stage high mass star formation and outflows (De Buizer & Vacca 2010; Cyganowski et al. 2009). Three CH${}_{3}$OH masers are seen within the EGO at velocities consistent with the Sgr C systemic velocity (-55 to -65 km s${}^{-1}$), supporting this scenario (Caswell 2009; Chambers et al. 2011). In this Letter, we present new infrared and millimeter data towards the EGO, which we refer to as G359.44-0.102, showing the presence of two massive protostellar cores and emission knots indicative of an outflow.
2. Data and observations
TripleSpec near-infrared spectroscopy. To identify the origin of G359.44-0.102’s infrared emission, we carried out medium-resolution (R$\sim$100 km s${}^{-1}$) spectroscopy in the near-infrared K-band, using TripleSpec on the 3.5-m ARC Telescope at Apache Point Observatory, in August 2012 (PI: Ginsburg). The K-band contains several H${}_{2}$ emission lines, most notably the 1-0 S(1) line at 2.12 µm, frequently observed in outflows from YSOs (Varricatt et al. 2010; Davis et al. 2010). Spectra were obtained at 4 positions with a 1.1″ slit. The data were taken in ABBA slit-nodding mode with a series of 30s exposures, for a total integration time of 2-6 minutes per pointing.
Submillimeter Array (SMA). High-resolution data to search for dense star-forming gas toward the G359.44-0.102 EGO were obtained in June 2009 with the compact-north configuration of SMA at 280 GHz (PI: Kauffmann). These data consist of a mosaic of five $\sim$45″ fields covering the EGO and its surrounding clump with spectral resolution of 0.44 km s${}^{-1}$. The observations covered the N${}_{2}$H${}^{+}$ (3-2) line at 279.512 GHz, commonly used to trace dense gas, with RMS noise of 0.23 Jy beam${}^{-1}$ when imaged at 2 km s${}^{-1}$ resolution. A combined self-calibrated continuum image was produced from the 2 GHz sidebands centred at 278.8 and 288.8 GHz, with combined RMS noise of 0.012 mJy beam${}^{-1}$. The synthesized beam of the continuum and line images is $\sim$2″. Data reduction and analysis were performed using CASA.
Survey data. The CMZ was covered by a number of Galactic Plane surveys. The Herschel Hi-GAL survey (Molinari et al. 2010) was used to derive temperature estimates toward the EGO and its host cloud, as described in Battersby et al. (2011). A Galactic cirrus emission model was estimated using an improved iterative technique which varies the source threshold value until it converges, classifying everything below that threshold as part of the diffuse cirrus. The emission model at each wavelength was subtracted from each of the five Hi-GAL bands, which were fit with a modified blackbody assuming $\beta$ = 1.75 to derive the dust temperature at each point (Figure 1; $\beta$ derived from Ossenkopf & Henning (1994) opacity law). The 870 µm ATLASGAL survey (Schuller et al. 2009) traces cold dense gas at a spatial resolution of $\sim$19″(Figure 1). Fluxes at this wavelength were used to determine the physical properties of the EGO’s host cloud (Contreras et al. 2013). Spectral line data at 3-mm from the Mopra CMZ survey (Jones et al. 2012) were used to determine the velocity of the EGO’s region. Data on masers in the region were obtained from YZ09, Chambers et al. (2011) and Caswell (2009).
3. Results
3.1. TripleSpec near-infrared spectroscopy
The near-infrared spectra reveal a number of emission locations towards the EGO and in its vicinity (Figure 2). Within the extent of the 4.5 µm emission we find 4 knots of H${}_{2}$ emission at 2.12 µm in 2 clusters approximately 5″ (0.2 pc) apart. The Eastern cluster contains three knots, with Gaussian-fitted v${}_{LSR}$ of -30, -50 and -70 km s${}^{-1}$; the Western knot has a velocity of -40 km s${}^{-1}$. The lines are unresolved at the TripleSpec spectral resolution.
Near the Western H${}_{2}$ knot we find a site of Br $\gamma$ emission at -25 km s${}^{-1}$. Its proximity to the radio source (1″) appears consistent with the presence of a hyper- or ultra-compact H ii region; the velocity discrepancy may be the result of dynamical interaction between UCH ii region and the host cloud.
Several emission features are detected along the slits away from the EGO. Their association with the region is unclear, and we do not include these further in our analysis. Properties of the detected emission features are summarised in Table 1. As the slit positions do not cover the extent of the IR emission, the detected knots may extend beyond the slit boundaries. Further knots may be found given better coverage.
3.2. SMA 280 GHz data
The 280 GHz SMA data show two sources in the continuum bands associated with the EGO. Figure 3 shows the location of these sources with respect to the infrared emission region.
The brightest, MM1, is located $\sim$6″ (0.25 pc) South from the peak of the 4.5 µm emission, coincident with a 6.7 GHz maser (Caswell et al. 2010). It measures 5.5 $\times$ 5″ in size (0.2 $\times$ 0.2 pc), measured to the 5-$\sigma$ contour, and is elongated in the North-South direction (different from the beam elongation direction). At the location of the IR emission peak lies a second fainter source, MM2, also associated with a 6.7 GHz CH${}_{3}$OH maser. Measuring 3 $\times$ 3″ in size (0.12 $\times$ 0.12 pc), MM2 shows a similar elongation along the N-S axis. Neither core appears associated with the radio source, suggesting the presence of at least 3 distinct star-forming sites. The integrated continuum fluxes measured within the 1-$\sigma$ contours are 1073 mJy for MM1 and 623 mJy for MM2. We assumed uncertainties on these values of 15%, dominated by the flux calibration accuracy.
In addition, the SMA data show N${}_{2}$H${}^{+}$ line emission near the brighter core, with v${}_{LSR}$ ranging from -58 to -54 km s${}^{-1}$. Channel maps and continuum data are shown in Figure 3.
4. Discussion
4.1. Distance to G359.44-0.102
Dynamics of Sgr C are complex, with evidence of strong velocity gradients (Liszt & Spiker 1995) and numerous absorption components in H i spectra (Lang et al. 2010). The velocity of the near 3 kpc arm at a distance of 5.5 kpc is very similar to the Sgr C systemic value at $l$ = -0.5 (Oka et al. 1998), causing potential confusion; we note that Green et al. (2009) place the CH${}_{3}$OH masers in the 3 kpc arm.
3-mm spectra from the Mopra CMZ survey (Jones et al. 2012) were extracted in a 40″ aperture (the Mopra beam size) centred on the EGO for HCO${}^{+}$ (1-0) at 89.19 GHz, H${}^{13}$CO${}^{+}$ (1-0) at 86.75 GHz and SiO (2-1) at 86.85 GHz (Figure 4; the poor baseline calibration in weaker species is discussed by Jones et al. 2012). We show integrated velocity maps (-45 to -75 km s${}^{-1}$) alongside.
The spectra show strong emission peaks near -54 km s${}^{-1}$ with fitted peak velocities consistent to within 1 km s${}^{-1}$. Linewidth measurements indicate full widths at half maximum (FWHM) of $>$10 km s${}^{-1}$. Such broad linewidths are highly characteristic of CMZ clouds (Oka et al. 1998; Shetty et al. 2012); observed velocities for disk clouds are $\leq$ 5 km s${}^{-1}$ (Shetty et al. 2012).
The SiO (2-1) integrated velocity map shows widespread emission throughout the cloud. Extended SiO emission is a well-documented feature of CMZ clouds (Martin-Pintado et al. 1997; Riquelme et al. 2010); in the disk SiO emission is typically observed in outflow shocks locally.
These observed characteristics suggest that the EGO and its host cloud are located at the GC distance. Some uncertainty does however remain, the masses calculated in the following sections would in this case be reduced by 40%.
4.2. Physical conditions in G359.44-0.102
Integrated flux measurements from the SMA continuum data were used to estimate masses of the protostellar cores in the EGO using the following relation (Hildebrand 1983):
$$M=\frac{gS_{\lambda}D^{2}}{\kappa_{\lambda}B_{\lambda}(T_{d})}$$
(1)
where $g$ is the gas-to-dust conversion ratio, $S_{\lambda}$ is the integrated flux at the observing wavelength $\lambda$, $D$ is the distance to the source, $\kappa_{\lambda}$ the dust mass opacity coefficient at $\lambda$, and $B_{\lambda}(T_{d})$ is the blackbody flux for a dust temperature $T_{d}$ evaluated at $\lambda$. From the Hi-GAL-derived temperature map (Figure 1) we see that the EGO lies near the coldest part of the cloud at a temperature of $\sim$20K. We assume $g$ of 76 (Draine 2011, chap. 23), calculated using metallicity $Z_{GC}=2Z_{\odot}$ (Launhardt et al. 2002), and D of 8.5 kpc.
For the dust opacity coefficient $\kappa$ we log-interpolate the data in Table 1 of Ossenkopf & Henning (1994). For a gas density of 10${}^{6}$ cm${}^{-3}$ and grains with thin ice mantles, we find $\kappa_{1.06}$ of 1.25 cm${}^{2}$ g${}^{-1}$ for the SMA frequency of 280 GHz.
For the brighter continuum source MM1 the integrated flux yields a mass of 668 M${}_{\odot}$. Source MM2 has a mass of 380 M${}_{\odot}$. These values are averages of the masses calculated from the individual sideband fluxes. We estimate the combined uncertainty of these mass estimates to be a factor of 2-4, based on estimates on uncertainty in flux calibration, T${}_{d}$, $\kappa$ and g.
Similarly, peak column densities were computed using
$$N(H_{2})^{peak}=\frac{S_{\lambda}\;g}{B_{\lambda}(T_{d})\;\Omega\;\kappa_{%
\lambda}\;\mu m_{H}}$$
(2)
where $\Omega$ is the beam solid angle, $\mu$ is the mean molecular weight of the interstellar medium (assumed to be 2.8), $m_{H}$ the mass of a hydrogen atom, and $g$, $S_{\lambda}$ and $B_{\lambda}(T_{d})$ as defined above. Using the peak measured fluxes and taking a 2″ SMA beam, we find peak column density estimates of 2 $\times$ 10${}^{24}$ cm${}^{-2}$ and 1.1 $\times$ 10${}^{24}$ cm${}^{-2}$ for sources MM1 and MM2, respectively.
4.3. Evidence for star formation in G359.44-0.102
The observational characteristics and derived properties of G359.44-0.102 are similar to those of other EGOs reported in the literature. In a sample of 28 objects, Cyganowski et al. (2009) found 64% of EGOs to be associated with 6.7 GHz Class II CH${}_{3}$OH maser emission, and of those 89% additionally have 44 GHz Class I masers. Using mm and radio observations, Cyganowski et al. (2011a, b, 2013) demonstrate that EGOs are reliable markers of massive star formation, often harbouring multiple star forming cores with strong active outflows.
The SMA continuum data show the presence of two star forming cores, each carrying several 100 M${}_{\odot}$ in mass with high column densities, alongside the UCH ii region proposed by Forster & Caswell (2000) . There is no evidence of significant dust heating towards these sources, nor of free-free or Br $\gamma$ emission suggesting a UCH ii region has begun to form. The two main sites of H${}_{2}$ emission are clustered near the mm cores (Figure 2), with velocities both blue- and redshifted with respect to the systemic value.
In the absence of additional line detections, we cannot establish the excitation mechanism of these lines; however, the properties and locations of the knots show compelling evidence for the presence of at least one massive outflow. The detected mm cores are in close proximity ($\leq$ 0.5 pc) to the compact radio source, indicating that the EGO hosts at least three distinct sites of massive star formation.
4.4. G359.44-0.102 in a quiescent host cloud
The EGO is located at the tip of a dense molecular cloud seen in absorption against the bright IR background. Survey data from the ATLASGAL and Hi-GAL surveys give an insight into the nature of this cloud.
The cloud measures roughly 7 $\times$ 4′ in ATLASGAL maps, corresponding to $\sim$16 $\times$ 9 pc at the GC distance. The Hi-GAL temperature map suggests temperatures across the cloud lie in the range of 20-25 $\pm$ 2 K, with a minimum of 19 K coinciding with the peak of the sub-mm emission.
Source extraction from ATLASGAL data by Contreras et al. (2013) identify three sources within the cloud (AGAL359.437-00.102, AGAL359.474-00.152, AGAL359.514-00.154; Figure 1), measuring 3$\times$1 pc, 4$\times$2 pc and 1$\times$0.6 pc. Contreras et al. (2013) report integrated fluxes of 119 Jy, 168 Jy and 25 Jy. Assuming parameters for Equation 1 as listed above and finding $\kappa_{0.87}$ of 1.75 cm${}^{2}$ g${}^{-1}$, these values yield gas masses of $\sim$4, $\sim$5 and $\sim$0.8$\times$10${}^{4}$ M${}_{\odot}$. The total mass enclosed in the dense cloud is thus of the order of 10${}^{5}$ M${}_{\odot}$, consistent with previous estimates (Lis et al. 1991; Lis & Carlstrom 1994). The mass contained in cores MM1 and MM2 ($<$10${}^{3}$ M${}_{\odot}$) represents just a small fraction of this total mass.
The clump-averaged column densities of the ATLASGAL sources were calculated using the effective radii from the Contreras catalog, and found to be in the range of 1.5-3.5 $\times$ 10${}^{22}$ cm${}^{-2}$. These values indicate that the entire cloud shows favourable physical conditions for massive stars to form (Lada et al. 2012). Apart from the data presented here for the EGO in source AGAL359.437-00.102, however, we found no 8, 24 or 70 µm sources in the cloud indicative of further star formation activity. No radio emission is observed and no CH${}_{3}$OH masers are reported in the literature. The EGO’s location at the cloud tip is likely to be pertinent, with star formation perhaps accelerated at the interface of a cloud-cloud collision or by feedback from the H ii region. A detailed study of the dynamics of the region is required to understand which mechanism is at work.
The cloud shares similarities with the massive compact cloud G0.253+0.016 (Lis et al. 1994; Longmore et al. 2012; Kauffmann et al. 2013). The latter has a similar mass (1-2 $\times$ 10${}^{5}$ M${}_{\odot}$) as the Sgr C cloud, albeit in an apparently smaller volume, yet shows no evidence of significant star formation activity. Immer et al. (2012) report the discovery of a further 4 compact 10${}^{5}$ M${}_{\odot}$ clouds in the CMZ that appear devoid of star formation.
At the other extreme, the Sgr B2 star forming region is one of the most prolific and well-studied in the Galaxy. Using 1.1 mm maps from the BGPS, Bally et al. (2010) find a mass of 5 $\times$ 10${}^{5}$ M${}_{\odot}$ and average H${}_{2}$ column densities of $>$10${}^{24}$ cm${}^{-2}$ towards its main star-forming sites. Its chemistry is however far richer, many of its star formation sites being more evolved hot cores and UCH ii regions.
Based on our current data, Sgr C lies between the CMZ star-forming extrema represented by the G0.253 and Sgr B2. Despite the new evidence of star formation presented here, the Sgr C cloud as a whole still appears to be quiescent compared with non-CMZ clouds showing similar physical conditions (Lada et al. 2010). This adds to the evidence presented by Longmore et al. (2013) and Kauffmann et al. (2013) for an anomalously low star formation rate in the CMZ. A more detailed study of Sgr C is clearly required to establish the region’s global star formation properties.
5. Conclusions
We have presented new, archival and survey data of the molecular cloud thought to be associated with Sgr C. Our high-resolution infrared and millimeter data show two massive protostellar cores associated with an EGO alongside a previously detected hyper- or ultra-compact H ii region, which appear to be driving at least one outflow. The absence of evidence for dust heating indicates that these cores are at an early evolutionary stage.
The cores are deeply embedded in the densest region of a $\sim$ 16 $\times$ 9 pc molecular cloud abutting the Sgr C H ii region. We note that the measured velocities also allow this site to be associated with the 3 kpc arm at 5.5 kpc, however linewidths and widespread SiO emission indicate a likely distance of 8.5 kpc. The cloud harbours $\sim$10${}^{5}$ M${}_{\odot}$ of gas and dust with typical column densities of $>$10${}^{22}$ cm${}^{-2}$ and temperatures of 19-25 K. It shows no evidence of further star formation activity despite presenting suitable physical conditions.
The presence of three star-forming cores in the EGO contrasts with the absence of further star formation in the host molecular cloud, placing Sgr C at an interesting mid-point between the quiescent G0.253 and starburst-like Sgr B2. The driving and inhibiting forces at work in the CMZ clouds and the relevance of their location along the proposed 100-pc elliptical ring are fascinating open questions.
6. Acknowledgements
SK thanks Jens Kauffmann, Sergio Molinari, Cornelia Lang and Yanett Contreras for sharing and discussing data. The Apache Point Observatory 3.5-m telescope is owned and operated by the Astrophysical Research Consortium. The Submillimeter Array is a joint project between the Smithsonian Astrophysical Observatory and the Academia Sinica Institute of Astronomy and Astrophysics. The Mopra radio telescope is part of the ATNF, funded by the Commonwealth of Australia.
Facilities: Spitzer (IRAC, MIPS), ARC, Mopra, APEX, UKIRT
References
Bally et al. (2010)
Bally, J., et al. 2010, ApJ, 721, 137
Battersby et al. (2011)
Battersby, C., et al. 2011, A&A, 535, A128
Caswell (2009)
Caswell, J. L. 2009, Pub. Astron. Soc. Austr., 26, 454
Caswell et al. (2010)
Caswell, J. L., et al. 2010, MNRAS, 404, 1029
Chambers et al. (2011)
Chambers, E. T., Yusef-Zadeh, F., & Roberts, D. 2011, ApJ, 733, 42
Contreras et al. (2013)
Contreras, Y., et al. 2013, A&A, 549, A45
Cyganowski et al. (2009)
Cyganowski, C. J., Brogan, C. L., Hunter, T. R., & Churchwell, E. 2009, ApJ,
702, 1615
Cyganowski et al. (2011a)
—. 2011a, ApJ, 743, 56
Cyganowski et al. (2011b)
Cyganowski, C. J., Brogan, C. L., Hunter, T. R., Churchwell, E., & Zhang, Q.
2011b, ApJ, 729, 124
Cyganowski et al. (2013)
Cyganowski, C. J., Koda, J., Rosolowsky, E., Towers, S., Meyer, J. D., Egusa,
F., Momose, R., & Robitaille, T. P. 2013, ApJ, 764, 61
Cyganowski et al. (2008)
Cyganowski, C. J., et al. 2008, AJ, 136, 2391
Davis et al. (2010)
Davis, C. J., Gell, R., Khanzadyan, T., Smith, M. D., & Jenness, T. 2010,
A&A, 511, A24
De Buizer & Vacca (2010)
De Buizer, J., & Vacca, W. 2010, AJ, 140, 196
Draine (2011)
Draine, B. T. 2011, Physics of the Interstellar and Intergalactic Medium
Forster & Caswell (2000)
Forster, J. R., & Caswell, J. L. 2000, ApJ, 530, 371
Green et al. (2009)
Green, J. A., McClure-Griffiths, N. M., Caswell, J. L., Ellingsen,
S. P., Fuller, G. A., Quinn, L., & Voronkov, M. A. 2009, ApJ, 696,
L156
Hildebrand (1983)
Hildebrand, R. 1983, Quarterly Journal of the Royal Astronomical …, 24,
267
Immer et al. (2012)
Immer, K., Menten, K. M., Schuller, F., & Lis, D. C. 2012, A&A, 548, A120
Jones et al. (2012)
Jones, P. A., et al. 2012, MNRAS, 419, 2961
Kauffmann et al. (2013)
Kauffmann, J., Pillai, T., & Zhang, Q. 2013, arXiv preprint arXiv:1301.1338
Lada et al. (2012)
Lada, C. J., Forbrich, J., Lombardi, M., & Alves, J. a. F. 2012, ApJ, 745, 190
Lada et al. (2010)
Lada, C. J., Lombardi, M., & Alves, J. a. F. 2010, ApJ, 724, 687
Lang et al. (2010)
Lang, C. C., Goss, W. M., Cyganowski, C., & Clubb, K. I. 2010, ApJSS, 191, 275
Launhardt et al. (2002)
Launhardt, R., Zylka, R., & Mezger, P. G. 2002, A&A, 384, 112
Lis et al. (1991)
Lis, D., Carlstrom, J., & Keene, J. 1991, ApJ, 380, 429
Lis & Carlstrom (1994)
Lis, D. C., & Carlstrom, J. E. 1994, ApJ, 424, 189
Lis et al. (1994)
Lis, D. C., Menten, K. M., Serabyn, E., & Zylka, R. 1994, ApJ, 423,
L39
Liszt & Spiker (1995)
Liszt, H., & Spiker, R. 1995, ApJSS, 98, 259
Longmore et al. (2013)
Longmore, S. N., Bally, J., & Testi, L. 2013, MNRAS, 429, 987
Longmore et al. (2012)
Longmore, S. N., et al. 2012, ApJ, 746, 117
Martin-Hernandez et al. (2003)
Martin-Hernandez, N. L., Bik, A., Kaper, L., Thielens, A. G. G. M., & Hanson, M. M. 2003, A&A, 405, 175
Martin-Pintado et al. (1997)
Martin-Pintado, J., de Vicente, P., Fuente, A., & Planesas, P. 1997,
ApJ, 482, L45
Molinari et al. (2010)
Molinari, S., Swinyard, B., & Bally, J. e. a. 2010, PASP, 122, 314
Molinari et al. (2011)
Molinari, S., et al. 2011, ApJ
Morris & Serabyn (1996)
Morris, M., & Serabyn, E. 1996, ARA&A, 34, 645
Odenwald & Fazio (1984)
Odenwald, S., & Fazio, G. 1984, ApJ, 283, 601
Oka et al. (1998)
Oka, T., Hasegawa, T., Hayashi, M., Handa, T., & Sakamoto, S. 1998,
ApJ, 493, 730
Ossenkopf & Henning (1994)
Ossenkopf, V., & Henning, T. 1994, A&A, 291, 943
Reid et al. (2009)
Reid, M. J., et al. 2009, ApJ, 700, 137
Riquelme et al. (2010)
Riquelme et al. 2010, A&A, 523, A45
Schuller et al. (2009)
Schuller, F., et al. 2009, A&A, 504, 415
Shetty et al. (2012)
Shetty, R., Beaumont, C. N., Burton, M. G., Kelly, B. C., & Klessen,
R. S. 2012, MNRAS, 425, 720
Stolovy et al. (2006)
Stolovy, S., et al. 2006, J. Phys. Conf. Ser., 54, 176
Varricatt et al. (2010)
Varricatt, W. P., Davis, C. J., Ramsay, S., & Todd, S. P. 2010, MNRAS, 64
Yusef-Zadeh et al. (2009)
Yusef-Zadeh, F., et al. 2009, ApJ, 702, 178 |
Infinity Algebras, Cohomology and Cyclic Cohomology,
and Infinitesimal Deformations
Michael Penkava
University of California
Davis, CA 95616
michae@@math.ucdavis.edu
Abstract.
An $A_{\infty}$ algebra is given by a codifferential on the tensor
coalgebra of a (graded) vector space. An associative algebra is a
special case of an $A_{\infty}$ algebra, determined by a quadratic
codifferential. The notions of Hochschild and cyclic cohomology
generalize from associative to $A_{\infty}$ algebras, and classify
the infinitesimal deformations of the algebra, and those
deformations preserving an invariant inner product, respectively.
Similarly, an $L_{\infty}$ algebra is given by a codifferential on
the exterior coalgebra of a vector space, with Lie algebras being
special cases given by quadratic codifferentials. There are
natural definitions of cohomology and cyclic cohomology,
generalizing the usual Lie algebra cohomology and cyclic
cohomology, which classify deformations of the algebra and those
which preserve an invariant inner product. This article explores
the definitions of these infinity algebras, their cohomology and
cyclic cohomology, and the relation to their infinitesimal
deformations.
Key words and phrases:
$A_{\infty}$ algebras, $L_{\infty}$ algebras, coalgebras, coderivations
1991 Mathematics Subject Classification: 17B56
Partially supported by NSF grant DMS-94-0411.
The author would also like to thank the University of Washington,
Seattle for hosting him.
1. Introduction
In a joint paper with Albert Schwarz [16], we gave
definitions of Hochschild cohomology and cyclic cohomology of an
$A_{\infty}$ algebra, and showed that these cohomology theories
classified the infinitesimal deformations of the $A_{\infty}$ structure
and those deformations preserving an invariant inner product,
respectively. Then we showed that the Hochschild cohomology of an
associative algebra classifies the deformations of the algebra
into an $A_{\infty}$ algebra, and the cyclic cohomology of an algebra
with an invariant inner product classifies the deformations of the
algebra into an $A_{\infty}$ algebra preserving the inner product.
Following some ideas of Maxim Kontsevich from [9], we
applied these results to show that cyclic cocycles of an
associative algebra determine homology cycles in the complex of
metric ribbon graphs. In [16], for sake of simplicity we
avoided the more standard description of $A_{\infty}$ algebras in terms
of codifferentials.
In this article, the definitions of infinity algebras will be
given in terms of codifferentials on coalgebras, so that the
theory of $L_{\infty}$ algebras is seen to be closely analogous to that
of $A_{\infty}$ algebras. The ideas in this text lead immediately to a
simple formulation of the cycle in the complex of metric ribbon
graphs associated to an $A_{\infty}$ algebra with an invariant inner
product. This same method can be applied to show that $L_{\infty}$ algebras with an invariant inner product give rise to a cycle in
the homology of the complex of metric ordinary graphs (see
[15]). This paper is a revision of [14].
The notion of an $A_{\infty}$ algebra, also called a strongly homotopy
associative algebra, was introduced by J. Stasheff in
[19, 20], and is a generalization of an associative
algebra. From a certain point of view, an associative algebra is
simply a special case of a codifferential on the tensor coalgebra
of a vector space. An $A_{\infty}$ algebra is given by taking an
arbitrary coderivation; in particular associative algebras and
differential graded associative algebras are examples of $A_{\infty}$ algebras.
$L_{\infty}$ algebras, also called strongly homotopy Lie algebras, first
appeared in [17], and are generalizations of Lie algebras. A
Lie algebra can be viewed as simply a special case of a
codifferential on the exterior coalgebra of a vector space, and
$L_{\infty}$ algebras are simply arbitrary codifferentials on this
coalgebra.
A bracket structure was introduced on the space of cochains of an
associative algebra by Murray Gerstenhaber in [4]. From
the coalgebra point of view, the Gerstenhaber bracket turns
out to be essentially the commutator of coderivations, as was
first pointed out by James Stasheff in [21]. In the
presence of an invariant inner product, the cyclic cochains can be
identified as a subalgebra of the space of ordinary cochains, so
that the cyclic cohomology also is equipped with a natural Lie
bracket. These notions generalize to the $A_{\infty}$ case.
The space of cochains of a Lie algebra with coefficients in the
adjoint representation has a natural bracket, which is again the
commutator of coderivations. In the case of a Lie algebra, cyclic
cohomology corresponds to the cohomology of the Lie algebra with
trivial coefficients, and in the presence of an invariant inner
product, the space of cyclic cochains is also equipped with a
natural bracket. These notions generalize to the $L_{\infty}$ algebra
case.
In our considerations, we shall be interested in $\mbox{$\mathbb{Z}$}_{2}$-graded
spaces, but we should point out that the results presented here
hold in the $\mathbb{Z}$-graded case as well, because the signs in the
$\mathbb{Z}$-graded case coincide with the signs from the associated
$\mbox{$\mathbb{Z}$}_{2}$-grading. $A_{\infty}$ algebras were first defined as $\mathbb{Z}$-graded
objects, but for the applications we have in mind, the $\mbox{$\mathbb{Z}$}_{2}$-grading
is more appropriate, and the generalization of the results here to
the $\mathbb{Z}$-graded case is straightforward. We shall find it necessary
to consider the parity reversion of a $\mbox{$\mathbb{Z}$}_{2}$-graded space. This is
the same space with the parity of elements reversed. (In the
$\mathbb{Z}$-graded case, the corresponding notion is that of suspension.)
There is a natural vector space isomorphism between the tensor
coalgebra of a $\mbox{$\mathbb{Z}$}_{2}$-graded space and the tensor coalgebra of its
parity reversion. But in the case of the exterior coalgebra, the
isomorphism is to the symmetric coalgebra of the parity
reversion.
A notion that will play a crucial role in what follows is that of
a grading group. An abelian group $G$ is said to be a grading
group if it possesses a symmetric $\mbox{$\mathbb{Z}$}_{2}$-valued bilinear form $\left<\cdot,\cdot\right>$.
Any abelian group with a subgroup of index 2 possesses a natural
grading form. An element $g$ of $G$ is called odd if $\left<g,g\right>=1$. A
grading on a group will be called good if $\left<g,h\right>=1$,
whenever $g$ and $h$ are both odd. Groups equipped with the
natural inner product induced by a subgroup of index 2 are good,
and these include both $\mbox{$\mathbb{Z}$}_{2}$ and $\mathbb{Z}$. If $G$ and $H$ are grading
groups, then $G\times H$ has an induced inner product, given by
$\left<(g,g^{\prime}),(h,h^{\prime})\right>=\left<g,g^{\prime}\right>+\left<h,h%
^{\prime}\right>$. But the induced grading
on $G\times H$ is never good when the gradings on $G$ and $H$ are
good.
If $V$ is a $G$-graded vector space, then one can define the
symmetric and exterior algebras of the tensor algebra $T(V)$. The
symmetric algebra is $G$-graded commutative, but the exterior
algebra is not. On the other hand, the tensor, symmetric and
exterior algebras also are graded by $G\times\mbox{$\mathbb{Z}$}$, and with respect
to the induced inner product on $G\times\mbox{$\mathbb{Z}$}$, the exterior algebra
is graded commutative. Thus the natural grading associated to the
exterior algebra is not good. The consequences of this fact
complicates the definition of $A_{\infty}$ and $L_{\infty}$ algebras.
In the following, we shall always consider the case $G=\hbox{$\mbox{$\mathbb{Z}$}_{2}$}$,
although the case $G=\mbox{$\mathbb{Z}$}$ is similar.
2. The Exterior and Symmetric Algebras
Suppose that $V$ is a $\mbox{$\mathbb{Z}$}_{2}$-graded k-module. The (reduced) tensor
algebra $T(V)$ is given by $T(V)=\bigoplus_{n=1}^{\infty}V^{n}$,
where $V^{n}$ is the $n$-th tensor power of $V$. The full tensor
algebra is defined to include the term $V^{0}=\mbox{\bf k}$ as well, and in
[5] a notion of an $A_{\infty}$ algebra on the full tensor
algebra is given, but we do not treat this idea here.
For an element $v=v_{1}\otimes\cdots\otimes v_{n}$ in $T(V)$, define its parity
$\mbox{$|v|$}=\mbox{$|v_{1}|$}+\cdots+\mbox{$|v_{n}|$}$, and its degree by $\operatorname{deg}(v)$=n. We
define the bidegree of $v$ by $\operatorname{bid}(v)=(\mbox{$|v|$},\operatorname{deg}(v))$. If $\alpha,\beta\in T(V)$, then $\operatorname{bid}(\alpha\otimes\beta)=\operatorname{bid}(\alpha)+%
\operatorname{bid}(\beta)$,
so that $T(V)$ is naturally $\mbox{$\mathbb{Z}$}_{2}\times\mbox{$\mathbb{Z}$}$-graded by the bidegree, and
$\mbox{$\mathbb{Z}$}_{2}$-graded if we consider only the parity. For simplicity, for a
$\mbox{$\mathbb{Z}$}_{2}$-graded space, we will denote
$\mbox{$(-1)^{\alpha\beta}$}=\mbox{$(-1)^{\mbox{$|\alpha|$}\mbox{$|\beta|$}}$}$, and for a $\mbox{$\mathbb{Z}$}_{2}\times\mbox{$\mathbb{Z}$}$-graded space
(equipped with the product grading), we shall denote
$\mbox{$(-1)^{\left<\alpha,\beta\right>}$}=\mbox{$(-1)^{\left<\operatorname{bid%
}\alpha,\operatorname{bid}\alpha\right>}$}$. (In formulas
like the above, the elements $\alpha$ and $\beta$ are assumed to
be homogeneous.)
The (graded) symmetric algebra $S(V)$ is defined as the quotient of
the tensor algebra of $V$ by the bigraded ideal generated by all
elements of the form $u\otimes v-\mbox{$(-1)^{uv}$}v\otimes u$. The resulting
algebra has a decomposition $S(V)=\bigoplus_{n=1}^{\infty}S^{n}(V)$,
where $S^{n}(V)$ is the image of $V^{n}$ in $S(V)$. We shall denote the
induced product in $S(V)$ by juxtaposition. The symmetric algebra
is both $\mbox{$\mathbb{Z}$}_{2}$ and $\mbox{$\mathbb{Z}$}_{2}\times\mbox{$\mathbb{Z}$}$-graded, but is graded commutative only
with respect to the $\mbox{$\mathbb{Z}$}_{2}$-grading.
If $\sigma\in\Sigma_{n}$, then
(1)
$$v_{1}\cdots v_{n}=\epsilon(\sigma;v_{1},\cdots,v_{n})v_{\sigma(1)}\cdots v_{%
\sigma(n)}$$
where $\epsilon(\sigma;v_{1},\cdots,v_{n})$ is a sign which can be
determined by the following. If $\sigma$ interchanges $k$ and
$k+1$, then $\epsilon(\sigma;v_{1},\cdots,v_{n})=\mbox{$(-1)^{v_{k}v_{k+1}}$}$. In
addition, if $\tau$ is another permutation, then
(2)
$$\epsilon(\tau\sigma;v_{1},\cdots,v_{n})=\epsilon(\sigma;v_{\tau(1)},\cdots,v_{%
\tau(n)})\epsilon(\tau;v_{1},\cdots,v_{n}).$$
It is conventional to abbreviate $\epsilon(\sigma;v_{1},\cdots,v_{n})$
as $\epsilon(\sigma)$.
The (graded) exterior algebra $\bigwedge V$ is defined as the
quotient of $T(V)$ by the bigraded ideal generated by all elements
of the form $u\otimes v+\mbox{$(-1)^{uv}$}v\otimes u$. The resulting algebra has a
decomposition $\bigwedge V=\bigoplus_{n=0}^{\infty}\bigwedge^{n}V$,
and the induced product is denoted by $\wedge$. The exterior
algebra is both $\mbox{$\mathbb{Z}$}_{2}$ and $\mbox{$\mathbb{Z}$}_{2}\times\mbox{$\mathbb{Z}$}$-graded, but is graded commutative
only with respect to the $\mbox{$\mathbb{Z}$}_{2}\times\mbox{$\mathbb{Z}$}$-grading. Let $\alpha,\beta\in\bigwedge V$. Furthermore it is easy to see that
(3)
$$v_{1}\wedge\cdots\wedge v_{n}=\mbox{$(-1)^{\sigma}$}\epsilon(\sigma)v_{\sigma(%
1)}\wedge\cdots\wedge v_{\sigma(n)},$$
where $(-1)^{\sigma}$ is the sign of the permutation $\sigma$.
3. The Tensor, Exterior and Symmetric Coalgebras
A more formal treatment of the symmetric and exterior
coalgebras would introduce the coalgebra structure on the tensor
algebra, and then describe these coalgebras in terms of coideals.
Instead, we will describe these coalgebra structures directly.
Recall that a coalgebra structure on a k-module $C$ is given by a
diagonal mapping $\Delta:C\rightarrow C\bigotimes C$. We consider only
coassociative coalgebras (we do not consider counits). The axiom
of coassociativity is
$$(1\otimes\Delta)\circ\Delta=(\Delta\otimes 1)\circ\Delta.$$
A grading on $C$ is compatible with the coalgebra structure if for
homogeneous $c\in C$, $\Delta(c)=\sum_{i}u_{i}\bigotimes v_{i}$, where
$\mbox{$|u_{i}|$}+\mbox{$|v_{i}|$}=\mbox{$|c|$}$ for all $i$; in other words, $\Delta$ has
degree 0. We also mention that a coalgebra is graded cocommutative
if $S\circ\Delta=\Delta$, where $S:C\bigotimes C\rightarrow C\bigotimes C$ is the
symmetric mapping given by
$$S(\alpha\otimes\beta)=\mbox{$(-1)^{\alpha\beta}$}\beta\otimes\alpha.$$
The tensor coalgebra structure is given by defining the (reduced)
diagonal $\Delta:T(V)\rightarrow T(V)$ by
(4)
$$\Delta(v_{1}\otimes\cdots\otimes v_{n})=\sum_{k=1}^{n-1}(v_{1}\otimes\cdots%
\otimes v_{k})\otimes(v_{k+1}\otimes\cdots\otimes v_{n}).$$
(We use here the reduced diagonal, because we are not including
the zero degree term in the tensor coalgebra. As a consequence,
$\Delta$ is not injective; in fact, its kernel is $V$.) The tensor
coalgebra is not graded cocommutative under either the $\mbox{$\mathbb{Z}$}_{2}$ or the
$\hbox{$\mbox{$\mathbb{Z}$}_{2}$}\times\mbox{$\mathbb{Z}$}$-grading, but both gradings are compatible with the
coalgebra structure.
The symmetric coalgebra structure on $S(V)$ is given by defining
(5)
$$\Delta(v_{1}\cdots v_{n})=\sum_{k=1}^{n-1}\sum_{\sigma\in\operatorname{Sh}(k,n%
-k)}\epsilon(\sigma)v_{\sigma(1)}\cdots v_{\sigma(k)}\otimes v_{\sigma(k+1)}%
\cdots v_{\sigma(n)}.$$
where $\operatorname{Sh}(p,q)$ is the set of all unshuffles of type
$(p,q)$, that is, the permutations of $p+q$ such that
$\sigma(k)<\sigma(k+1)$ if $k\neq p$.
With this coalgebra structure, and the $\mbox{$\mathbb{Z}$}_{2}$-grading, $S(V)$ is a
cocommutative, coassociative coalgebra.
Similarly, we define the exterior coalgebra structure on
$\bigwedge V$ by
(6)
$$\Delta(v_{1}\wedge\cdots\wedge v_{n})=\sum_{\begin{subarray}{c}k=1\dots n-1\\
\sigma\in\operatorname{Sh}(k,n-k)\end{subarray}}\!\!\!\!\!\!\mbox{$(-1)^{%
\sigma}$}\epsilon(\sigma)v_{\sigma(1)}\wedge\cdots\wedge v_{\sigma(k)}\otimes v%
_{\sigma(k+1)}\wedge\cdots\wedge v_{\sigma(n)}.$$
The coalgebra structure on $\bigwedge V$ is coassociative, and is
cocommutative with respect to the $\mbox{$\mathbb{Z}$}_{2}\times\mbox{$\mathbb{Z}$}$-grading.
4. Coderivations
A coderivation on a graded coalgebra $C$ is a map $d:C\rightarrow C$ such
that
(7)
$$\Delta\circ d=(d\otimes 1+1\otimes d)\circ\Delta.$$
Note that this definition depends implicitly on the grading group,
because
$$(1\otimes d)(\alpha\otimes\beta)=\mbox{$(-1)^{\left<\alpha,d\right>}$}\alpha%
\otimes d(\beta).$$
The k-module $\operatorname{Coder}(C)$ of graded coderivations has a natural
structure of a graded Lie algebra, with the bracket given by
(8)
$$[m,n]=m\circ n-\mbox{$(-1)^{\left<m,n\right>}$}n\circ m.$$
A codifferential on a coalgebra $C$ is a coderivation $d$ such
that $d\circ d=0$. We examine the coderivation structure of the
tensor, symmetric and exterior coalgebras.
4.1. Coderivations of the Tensor Coalgebra
Suppose that we wish to extend $d_{k}:V^{k}\rightarrow V$ to a coderivation
$\hbox{{$\hat{d}$}}_{k}$ of $T(V)$. We are interested in extensions satisfying the
property that $\hbox{{$\hat{d}$}}_{k}(v_{1},\cdots,v_{n})=0$ for $n<k$. How this
extension is made depends on whether we consider the $\mbox{$\mathbb{Z}$}_{2}$ or the
$\hbox{$\mbox{$\mathbb{Z}$}_{2}$}\times\mbox{$\mathbb{Z}$}$-grading. First we consider the $\mbox{$\mathbb{Z}$}_{2}$-grading, so that
only the parity of $d$ is relevant. Then if we define
$$\hbox{{$\hat{d}$}}_{k}(v_{1},\cdots,v_{n})=\sum_{i=0}^{n-k}\mbox{$(-1)^{(v_{1}%
+\cdots+v_{i})d_{k}}$}v_{1},\cdots,v_{i},d_{k}(v_{i+1},\cdots,v_{i+k}),v_{i+k+%
1},\cdots,v_{n},$$
one can show that $\hbox{{$\hat{d}$}}_{k}$ is a coderivation on $T(V)$ with respect
to the $\mbox{$\mathbb{Z}$}_{2}$-grading. More generally, one can show that any
coderivation $\hat{d}$ on $T(V)$ is completely determined by the
induced mappings $d_{k}:V^{k}\rightarrow V$, and in fact, one obtains
(9)
$$\hbox{{$\hat{d}$}}(v_{1},\cdots,v_{n})=\hskip-7.227pt\sum_{\begin{subarray}{c}%
1\leq k\leq n\\
\\
0\leq i\leq n-k\end{subarray}}\hskip-7.227pt\mbox{$(-1)^{(v_{1}+\cdots+v_{i})d%
_{k}}$}v_{1},\cdots,v_{i},d_{k}(v_{i+1},\cdots,v_{i+k}),v_{i+k+1},\cdots,v_{n}.$$
Also, one can show that $\hat{d}$ is a codifferential with respect to
the $\mbox{$\mathbb{Z}$}_{2}$-grading precisely when for $n\geq 1$
$$\sum_{\begin{subarray}{c}k+l=n+1\\
\\
0\leq i\leq n-k\end{subarray}}\mbox{$(-1)^{(v_{1}+\cdots+v_{i})d_{k}}$}d_{l}(v%
_{1},\cdots,v_{i},d_{k}(v_{i+1},\cdots,v_{i+k}),v_{i+k+1},\cdots,v_{n})=0.$$
The module $\operatorname{Coder}(T(V))$ of coderivations of $T(V)$ with respect
to the $\mbox{$\mathbb{Z}$}_{2}$-grading is naturally isomorphic to $\mbox{\rm Hom}(T(V),V)$, so
$\mbox{\rm Hom}(T(V),V)$ inherits a natural structure of a $\mbox{$\mathbb{Z}$}_{2}$-graded Lie
algebra.
Let us examine the bracket structure on $\mbox{\rm Hom}(T(V),V)$ more
closely. Suppose that for an arbitrary element $d\in\mbox{\rm Hom}(T(V),V)$,
we denote the restriction of $d$ to $V^{k}$ by $d_{k}$, and the
extensions of $d$ and $d_{k}$ as coderivations of $T(V)$ by $\hat{d}$
and $\hbox{{$\hat{d}$}}_{k}$, resp. Also denote by $d_{kl}$ the restriction of of
$\hbox{{$\hat{d}$}}_{k}$ to $V^{k+l-1}$, so that $d_{kl}\in\mbox{\rm Hom}(V^{k+l-1},V^{l})$.
The precise expression for $d_{kl}$ is given by equation
(9) with $n=k+l-1$. It is easy to see that the bracket
of $d_{k}$ and $\delta_{l}$ is given by
(10)
$$[d_{k},\delta_{l}]=d_{k}\circ\delta_{lk}-\mbox{$(-1)^{d_{k}\delta_{l}}$}\delta%
_{l}\circ d_{kl}.$$
Furthermore, we have $[d,\delta]_{n}=\sum_{k+l=n+1}[d_{k},\delta_{l}].$
The point here is that $d_{kl}$ and $\delta_{kl}$ are determined
in a simple manner by $d_{k}$ and $\delta_{k}$, so we have given a
description of the bracket on $\mbox{\rm Hom}(T(V),V)$ in a direct fashion.
The fact that the bracket so defined has the appropriate
properties follows from the fact that if $\rho=[d,\delta]$, then
$\hat{\rho}=[\hbox{{$\hat{d}$}},\hbox{$\hat{\delta}$}]$.
Now we consider how to extend a mapping $m_{k}:V^{k}\rightarrow V$ to a
coderivation $\hbox{$\hat{m}$}_{k}$ with respect to the $\mbox{$\mathbb{Z}$}_{2}\times\mbox{$\mathbb{Z}$}$-grading. In this
case, note that the bidegree of $m_{k}$ is given by
$\operatorname{bid}(m_{k})=(\mbox{$|m_{k}|$},k-1)$. The formula for the extension is the
same as before, but with the bidegree in place of the parity. In
other words,
(11)
$$\displaystyle\hbox{$\hat{m}$}_{k}(v_{1},\cdots,v_{n})=\\
\displaystyle\sum_{i=0}^{n-k}\mbox{$(-1)^{(v_{1}+\cdots+v_{i})m_{k}+i(k-1)}$}v%
_{1},\cdots,v_{i},m_{k}(v_{i+1},\cdots,v_{i+k}),v_{i+k+1},\cdots,v_{n}.$$
Similarly, if we consider an arbitrary coderivation $\hat{m}$ on
$T(V)$, then it is again determined by the induced mappings
$m_{k}:V^{k}\rightarrow V$, and we see that
(12)
$$\displaystyle\hbox{$\hat{m}$}(v_{1},\cdots,v_{n})=\\
\displaystyle\sum_{\begin{subarray}{c}1\leq k\leq n\\
\\
0\leq i\leq n-k\end{subarray}}\hskip-7.227pt\mbox{$(-1)^{(v_{1}+\cdots+v_{i})m%
_{k}+i(k-1)}$}v_{1},\cdots,v_{i},m_{k}(v_{i+1},\cdots,v_{i+k}),v_{i+k+1},%
\cdots,v_{n}.$$
Also, one obtains that $\hat{m}$ is a codifferential with respect to
the $\mbox{$\mathbb{Z}$}_{2}\times\mbox{$\mathbb{Z}$}$-grading is equivalent to the condition that for $n\geq 1$,
(13)
$$\hskip-7.227pt\sum_{\begin{subarray}{c}k+l=n+1\\
\\
0\leq i\leq n-k\end{subarray}}\hskip-7.227pt\mbox{$(-1)^{(v_{1}+\cdots+v_{i})m%
_{k}+i(k-1)}$}m_{l}(v_{1},\cdots,v_{i},m_{k}(v_{i+1},\cdots,v_{i+k}),v_{i+k+1}%
,\cdots,v_{n})=0.$$
The module $\operatorname{Coder}(T(V))$ of coderivations of $T(V)$ with respect
to the $\mbox{$\mathbb{Z}$}_{2}\times\mbox{$\mathbb{Z}$}$-grading is naturally isomorphic to
$\bigoplus_{k=1}^{\infty}\mbox{\rm Hom}(V^{k},V)$, rather than $\mbox{\rm Hom}(T(V),V)$,
because the latter module is the direct product of the modules
$C^{k}(V)=\mbox{\rm Hom}(V^{k},V)$. However, we would like to consider elements
of the form $\hbox{$\hat{m}$}=\sum_{k=1}^{\infty}\hbox{$\hat{m}$}_{k}$, where $\hbox{$\hat{m}$}_{k}$ has
bidegree $(\mbox{$|m_{k}|$},k-1)$. Such an infinite sum is a well defined
element of $\mbox{\rm Hom}(T(V),T(V))$, so by abuse of notation, we will
define $\operatorname{Coder}(T(V))$ to be the module of such infinite sums of
coderivations.
With this convention, we now have a natural
isomorphism between $\operatorname{Coder}(T(V))$ and $\mbox{\rm Hom}(T(V),V)$.
Furthermore, the bracket of coderivations is still well defined,
and we consider $\operatorname{Coder}(T(V)$ to be a $\mbox{$\mathbb{Z}$}_{2}\times\mbox{$\mathbb{Z}$}$-graded Lie algebra. The
reason that the bracket is well defined is that any homogeneous
coderivation has bidegree $(m,n)$ for some $n\geq 0$, so the
grading is given by $\hbox{$\mbox{$\mathbb{Z}$}_{2}$}\times\mbox{$\mathbb{N}$}$ rather than the full group $\mbox{$\mathbb{Z}$}_{2}\times\mbox{$\mathbb{Z}$}$.
In structures where a $\mathbb{Z}$-grading reduces to an $\mathbb{N}$-grading, it is
often advantageous to replace direct sums with direct products.
Using the same notation convention as in the $\mbox{$\mathbb{Z}$}_{2}$-graded case, we
note that if $m,\mu\in\mbox{\rm Hom}(T(V),V)$, then we have
(14)
$$[m_{k},\mu_{l}]=m_{k}\circ\mu_{lk}-\mbox{$(-1)^{\left<m_{k},\mu_{l}\right>}$}%
\mu_{l}\circ m_{kl},$$
and $[m,\mu]_{n}=\sum_{k+l=n+1}[m_{k},\mu_{l}]$.
4.2. Coderivations of the Symmetric Coalgebra
Suppose that we want to extend $m_{k}:S^{k}(V)\rightarrow V$ to a coderivation
$\hbox{$\hat{m}$}_{k}$ of $S(V)$ such that $\hbox{$\hat{m}$}_{k}(v_{1}\cdots v_{n})=0$ for $k<n$.
Define
(15)
$$\hbox{$\hat{m}$}_{k}(v_{1}\cdots v_{n})=\sum_{\sigma\in\operatorname{Sh}(k,n-k%
)}\epsilon(\sigma)m_{k}(v_{\sigma(1)},\cdots,v_{\sigma(k)})v_{\sigma(k+1)}%
\cdots v_{\sigma(n)}.$$
Then $\hbox{$\hat{m}$}_{k}$ is a coderivation with respect to the $\mbox{$\mathbb{Z}$}_{2}$-grading. In
general, suppose that $\hat{m}$ is a coderivation on the symmetric
coalgebra. It is not difficult to see that if $m_{k}:S^{k}(V)\rightarrow V$ is
the induced map, then $\hat{m}$ can be recovered from these maps by
the relations
(16)
$$\hbox{$\hat{m}$}(v_{1}\cdots v_{n})=\sum_{\begin{subarray}{c}1\leq k\leq n\\
\\
\sigma\in\operatorname{Sh}(k,n-k)\end{subarray}}\epsilon(\sigma)m_{k}(v_{%
\sigma(1)},\cdots,v_{\sigma(k)})v_{\sigma(k+1)}\cdots v_{\sigma(n)}.$$
From this, we determine that there is a natural isomorphism
between $\operatorname{Coder}(S(V))$, the module of coderivations of $V$, and
$\mbox{\rm Hom}(S(V),V)$. Thus $\mbox{\rm Hom}(S(V),V)$ inherits the structure of a
graded Lie algebra. Also, $\hat{m}$ is a codifferential when for all
$n$,
(17)
$$\sum_{\begin{subarray}{c}k+l=n+1\\
\sigma\in\operatorname{Sh}(k,n-k)\end{subarray}}\epsilon(\sigma)m_{l}(m_{k}(v_%
{\sigma(1)},\cdots,v_{\sigma(k)}),v_{\sigma(k+1)},\cdots,v_{\sigma(n)})=0.$$
In general, it is not possible to extend a map $m_{k}:S^{k}(V)\rightarrow V$
as a coderivation with respect to the $\mbox{$\mathbb{Z}$}_{2}\times\mbox{$\mathbb{Z}$}$-grading; the signs
won’t work out correctly. Similarly, the $\mbox{$\mathbb{Z}$}_{2}$-grading is necessary
to have a good theory of coderivations of the symmetric coalgebra.
4.3. Coderivations of the Exterior Coalgebra
Suppose that we want to extend $l_{k}:\bigwedge^{k}(V)\rightarrow V$ to a
coderivation $\hbox{$\hat{l}$}_{k}$ of $\bigwedge(V)$ such that $\hbox{$\hat{l}$}_{k}(v_{1},\cdots,v_{n})=0$
for $k<n$. Define
(18)
$$\hbox{$\hat{l}$}_{k}(v_{1},\cdots,v_{n})=\sum_{\sigma\in\operatorname{Sh}(k,n-%
k)}\mbox{$(-1)^{\sigma}$}\epsilon(\sigma)l_{k}(v_{\sigma(1)},\cdots,v_{\sigma(%
k)})\wedge v_{\sigma(k+1)}\wedge\cdots\wedge v_{\sigma(n)}.$$
Then $\hbox{$\hat{l}$}_{k}$ is a coderivation with respect to the $\mbox{$\mathbb{Z}$}_{2}\times\mbox{$\mathbb{Z}$}$-grading.
In general, suppose that $\hat{l}$ is a coderivation on the exterior
coalgebra. If $l_{k}:\bigwedge^{k}(V)\rightarrow V$ is the induced map, then
$\hat{l}$ can be recovered from these maps by the relations
(19)
$$\hbox{$\hat{l}$}(v_{1}\cdots v_{n})=\sum_{\begin{subarray}{c}1\leq k\leq n\\
\\
\sigma\in\operatorname{Sh}(k,n-k)\end{subarray}}\mbox{$(-1)^{\sigma}$}\epsilon%
(\sigma)l_{k}(v_{\sigma(1)},\cdots,v_{\sigma(k)})\wedge v_{\sigma(k+1)}\wedge%
\cdots\wedge v_{\sigma(n)}.$$
From this, we determine that there is a natural isomorphism
between $\operatorname{Coder}(\bigwedge(V))$, the module of coderivations of $V$, and
$\mbox{\rm Hom}(\bigwedge(V),V)$. Thus $\mbox{\rm Hom}(\bigwedge(V),V)$ inherits the structure of a
graded Lie algebra. Also, $\hat{l}$ is a codifferential when for all
$n$,
(20)
$$\sum_{\begin{subarray}{c}k+l=n+1\\
\sigma\in\operatorname{Sh}(k,n-k)\end{subarray}}\mbox{$(-1)^{\sigma}$}\epsilon%
(\sigma)l_{l}(l_{k}(v_{\sigma(1)},\cdots,v_{\sigma(k)}),v_{\sigma(k+1)},\cdots%
,v_{\sigma(n)})=0.$$
As the $\mbox{$\mathbb{Z}$}_{2}$-grading is necessary for coderivations of the symmetric
algebra, so the $\mbox{$\mathbb{Z}$}_{2}\times\mbox{$\mathbb{Z}$}$-grading is essential for a good theory of
coderivations of the exterior coalgebra.
5. Cohomology of $A_{\infty}$ algebras
In [16], a generalization of an associative algebra, called
a strongly homotopy associative algebra, or $A_{\infty}$ algebra was
discussed, and cohomology and cyclic cohomology of this structure
was defined. $A_{\infty}$ algebras were introduced by J. Stasheff in
[19, 20]. An $A_{\infty}$ algebra structure is essentially an
odd codifferential on the tensor coalgebra; an associative algebra
is simply an odd codifferential determined by a single map
$m_{2}:V^{2}\rightarrow V$.
If $V$ is a $\mbox{$\mathbb{Z}$}_{2}$-graded k-module, then the parity reversion $\Pi V$ is the same module, but with the parity of elements reversed.
In other words, $(\Pi V)_{0}=V_{1}$ and $(\Pi V)_{1}=V_{0}$, where $V_{0}$
and $V_{1}$ are the submodules of even and odd elements of $V$,
resp. The map $\pi:V\rightarrow\Pi V$, which is the identity as a map of
sets, is odd. There is a natural isomorphism $\eta:T(V)\rightarrow T(\Pi V)$ given by
(21)
$$\eta(v_{1}\otimes\cdots\otimes v_{n})=\mbox{$(-1)^{(n-1)v_{1}+(n-2)v_{2}+%
\cdots+v_{n-1}}$}\pi v_{1}\otimes\cdots\otimes\pi v_{n}$$
Denote the restriction of $\eta$ to $V^{k}$ by $\eta_{k}$. Note that
$\eta_{k}$ is odd when $k$ is odd and even when $k$ is even, so that
$\eta$ is neither an odd nor an even mapping.
Let $W=\Pi V$. Define a bijection between $C(W)=\mbox{\rm Hom}(T(W),W)$ and
$C(V)=\mbox{\rm Hom}(T(V),V)$ by setting
$\mu=\eta^{-1}\circ\delta\circ\eta$, for $\delta\in C(W)$. Then
$\mu_{k}=\eta_{1}^{-1}\circ\delta_{k}\circ\eta_{k}$ and
$\mbox{$|\mu_{k}|$}=\mbox{$|\delta_{k}|$}+(k-1)$. In particular, note that if
$\delta_{k}$ is odd in the $\mbox{$\mathbb{Z}$}_{2}$-grading, then $\operatorname{bid}(\mu_{k})=(k,k-1)$,
so that $\mu_{k}$ is odd in the $\mbox{$\mathbb{Z}$}_{2}\times\mbox{$\mathbb{Z}$}$-grading. Now extend
$\delta_{k}:W^{k}\rightarrow W$ to a coderivation $\hbox{$\hat{\delta}$}_{k}$ on $T(W)$ with
respect to the $\mbox{$\mathbb{Z}$}_{2}$ grading, so that
(22)
$$\displaystyle\hbox{$\hat{\delta}$}_{k}(w_{1},\cdots,w_{n})=\\
\displaystyle\sum_{i=0}^{n-k}\mbox{$(-1)^{(w_{1}+\cdots+w_{i})\delta_{k}}$}w_{%
1}\otimes\cdots\otimes w_{i}\otimes\delta_{k}(w_{i+1},\cdots,w_{i+k})\otimes w%
_{i+k+1}\otimes\cdots\otimes w_{n}.$$
Let $\hbox{$\bar{\mu}$}_{k}:T(V)\rightarrow T(V)$ be given by $\hbox{$\bar{\mu}$}_{k}=\eta^{-1}\circ\hbox{$\hat{\delta}$}_{k}\circ\eta$. Let $\hat{\mu}$ be the extension of $\mu$ as a
$\mbox{$\mathbb{Z}$}_{2}\times\mbox{$\mathbb{Z}$}$-graded coderivation of $T(V)$. We wish to investigate the
relationship between $\hbox{$\bar{\mu}$}_{k}$ and $\hbox{$\hat{\mu}$}_{k}$. For simplicity, write
$w_{i}=\pi v_{i}$. Note that $\eta_{1}=\pi$ is the parity reversion
operator. So we have $\delta_{k}=\pi\circ\mu_{k}\circ\eta_{k}^{-1}$.
Thus
(23)
$$\displaystyle\hbox{$\bar{\mu}$}_{k}(v_{1},\cdots,v_{n})=\mbox{$(-1)^{r}$}\eta^%
{-1}\delta(w_{1},\cdots,w_{n})=\\
\displaystyle\eta^{-1}(\sum_{i=0}^{n-k}\mbox{$(-1)^{r+s_{i}}$}w_{1}\otimes%
\cdots\otimes w_{i}\otimes\delta_{k}(w_{i+1},\cdots,w_{i+k})\otimes w_{i+k+1}%
\otimes\cdots\otimes w_{n})=\\
\displaystyle\eta^{-1}(\sum_{i=0}^{n-k}\mbox{$(-1)^{r+s_{i}+t_{i}}$}w_{1}%
\otimes\cdots\otimes w_{i}\otimes\pi\mu_{k}(v_{i+1},\cdots,v_{i+k})\otimes w_{%
i+k+1}\otimes\cdots\otimes w_{n})=\\
\displaystyle\sum_{i=0}^{n-k}\mbox{$(-1)^{r+s_{i}+t_{i}+u_{i}}$}v_{1}\otimes%
\cdots\otimes v_{i}\otimes\mu_{k}(v_{i+1},\cdots,v_{i+k})\otimes v_{i+k+1}%
\otimes\cdots\otimes v_{n},$$
where
$$\displaystyle r$$
$$\displaystyle=$$
$$\displaystyle(n-1)v_{1}+\cdots+v_{n-1}$$
$$\displaystyle s_{i}$$
$$\displaystyle=$$
$$\displaystyle(w_{1}+\cdots+w_{i})\delta_{k}$$
$$\displaystyle=$$
$$\displaystyle(v_{1}+\cdots+v_{i})\mu_{k}+i(\mu_{k}+1-k)+(1-k)(v_{1}+\cdots+v_{%
i})$$
$$\displaystyle t_{i}$$
$$\displaystyle=$$
$$\displaystyle(k-1)v_{i+1}+\cdots+v_{i+k-1}$$
$$\displaystyle u_{i}$$
$$\displaystyle=$$
$$\displaystyle(n-k)v_{1}+\cdots+(n-k-i+1)v_{i}$$
$$\displaystyle+(n-k-i)(\mu_{k}+v_{i+1}+\cdots+v_{i+k})+(n-k-i-1)v_{i+k+1}+%
\cdots+v_{n-1}$$
Combining these coefficients we find that
(24)
$$r+s_{i}+t_{i}+u_{i}=(v_{1}+\cdots+v_{i})\mu_{k}+i(k-1)+(n-k)\mu_{k},$$
so that
(25)
$$\displaystyle\hbox{$\bar{\mu}$}_{k}(v_{1},\cdots,v_{n})=\sum_{i=0}^{n-k}\mbox{%
$(-1)^{(v_{1}+\cdots+v_{i})\mu_{k}+i(k-1)+(n-k)\mu_{k}}$}\\
\displaystyle\times v_{1}\otimes\cdots\otimes v_{i}\otimes\mu_{k}(v_{i+1},%
\cdots,v_{i+k})\otimes v_{i+k+1}\otimes\cdots\otimes v_{n}.$$
Thus we see that
(26)
$$\hbox{$\bar{\mu}$}_{k}(v_{1},\cdots,v_{n})=\mbox{$(-1)^{(n-k)\mu_{k}}$}\hbox{$%
\hat{\mu}$}_{k}(v_{1},\cdots,v_{n}).$$
Denote the restriction of $\hbox{$\hat{\mu}$}_{k}$ to $V^{k+l-1}$ by $\hbox{$\hat{\mu}$}_{kl}$.
Set $n=k+l-1$. Denote
$\hbox{$\bar{\mu}$}_{kl}=\eta_{l}^{-1}\circ\delta_{kl}\circ\eta_{n}$, so that
$\hbox{$\bar{\mu}$}_{kl}$ is the restriction of $\hbox{$\bar{\mu}$}_{k}$ to $V^{n}$. Then we
can express equation (26) in the form
$\hbox{$\bar{\mu}$}_{kl}=\mbox{$(-1)^{(l-1)\mu_{k}}$}\mu_{kl}$.
More generally, if $\hat{\delta}$ is an arbitrary derivation on $T(W)$,
induced by the maps $\delta_{k}:V^{k}\rightarrow V$, then it determines maps
$\mu_{k}:V^{k}\rightarrow V$ , and $\hbox{$\bar{\mu}$}:T(V)\rightarrow T(V)$, in a similar manner,
and we have
(27)
$$\displaystyle\hbox{$\bar{\mu}$}(v_{1},\cdots,v_{n})=\sum_{\begin{subarray}{c}1%
\leq k\leq n\\
\\
0\leq i\leq n-k\end{subarray}}\mbox{$(-1)^{(v_{1}+\cdots+v_{i})\mu_{k}+i(k-1)+%
(n-k)\mu_{k}}$}\\
\displaystyle\times v_{1}\otimes\cdots\otimes v_{i}\otimes\mu_{k}(v_{i+1},%
\cdots,v_{i+k})\otimes v_{i+k+1}\otimes\cdots\otimes v_{n}.$$
The condition that $\hat{\delta}$ is a codifferential on $T(W)$ can be
expressed in the form
(28)
$$\sum_{k+l=n+1}\delta_{l}\circ\delta_{kl}=0$$
for all $n\geq 1$. This condition is equivalent to the condition
(29)
$$\sum_{k+l=n+1}\mbox{$(-1)^{(l-1)\mu_{k}}$}\mu_{l}\circ\mu_{kl}=\sum_{k+l=n+1}%
\mu_{l}\circ\hbox{$\bar{\mu}$}_{kl}=0.$$
We can express this condition in the form
(30)
$$\displaystyle\sum_{\begin{subarray}{c}k+l=n+1\\
0\leq i\leq n-k\end{subarray}}\mbox{$(-1)^{(v_{1}+\cdots+v_{i})\mu_{k}+i(k-1)+%
(n-k)\mu_{k}}$}\\
\displaystyle\times\mu_{l}(v_{1},\cdots,v_{i},\mu_{k}(v_{i+1},\cdots,v_{i+k}),%
v_{i+k+1},\cdots,v_{n})=0.$$
When $\hat{\delta}$ is an odd codifferential, $\mbox{$|\mu_{k}|$}=k$,so the sign in
the expression above is simply $(v_{1}+\cdots+v_{i})k+i(k-1)+nk-k$.
This is the form in which the signs appear in the definition of an
$A_{\infty}$ algebra in [11, 10].
An $A_{\infty}$ algebra structure on $V$ is nothing more than an odd
codifferential on $T(W).$ This can also be expressed in terms of
the bracket on $C(W)$. An odd element $\delta\in C(W)$ satisfying
$[\delta,\delta]=0$ determines a codifferential $\hat{\delta}$ on $T(W)$.
More precisely, the condition $[\delta,\delta]=0$ is equivalent to
the condition $\hbox{$\hat{\delta}$}^{2}=0$. Thus $\mu\in C(V)$ determines an $A_{\infty}$ algebra structure on $V$ when $\delta=\eta\circ\mu\circ\eta^{-1}$
satisfies $[\delta,\delta]=0$. This is not the same condition as
$[\mu,\mu]=0$, nor even the condition $\hbox{$\hat{\mu}$}^{2}=0$, although we
shall have more to say about this later.
If one considers the situation $V=\Pi W$ instead of $W=\Pi V$,
then an odd codifferential $d$ on $T(W)$ would give rise to an a
mapping $m:T(V)\rightarrow V$ satisfying the relations
(31)
$$\displaystyle\sum_{\begin{subarray}{c}0\leq 1\leq n-k\\
\\
k+l=n+1\end{subarray}}\mbox{$(-1)^{(x_{1}+\cdots+x_{i})k+i(k-1)+n-k}$}\\
\displaystyle\times m_{l}(x_{1},\cdots,x_{i},m_{k}(x_{i+1},\cdots,x_{i+k}),x_{%
i+k+1},\cdots,x_{n})=0.$$
The signs in the expression above agree with the signs in the
definition of a $A_{\infty}$ algebra as given in [16, 9].
From these observations, we see that the two sign conventions
for a homotopy associative algebra originate because there are two
natural choices for the relationship between the space $W$, which
carries the structure of an odd differential with respect to the
usual $\mbox{$\mathbb{Z}$}_{2}$-grading, and $V$, which is its dual. One may choose
either $W=\Pi V$, to get the signs in [11, 10], or $V=\Pi W$,
to get the signs in [16, 9]. (Actually, one can vary the
definition of $\eta$ to obtain both sets of signs from either one
of these models.)
Let us examine the bracket structure on the space $\operatorname{Coder}(T(V))$.
In [4], M. Gerstenhaber defined a bracket on the space of
cochains of an associative algebra, which we shall call the
Gerstenhaber bracket. When $V$ is concentrated in degree zero, in
other words, in the non $\mbox{$\mathbb{Z}$}_{2}$-graded case, the Gerstenhaber bracket
is just the bracket of coderivations, with the $\mathbb{Z}$-grading. Thus
the Gerstenhaber bracket is given by
(32)
$$[\varphi_{k},\psi_{l}]=\varphi_{k}\psi_{l}-\mbox{$(-1)^{(k-1)(l-1)}$}\psi_{l}%
\varphi_{k},$$
for $\varphi\in C^{k}(V)$ and $\psi\in C^{l}(V)$.
One of the main results of [4] is that the differential
$D$ of cochains in the cohomology of an associative algebra can be
expressed in terms of the bracket. It was shown that
$D(\varphi)=[\varphi,m]$ where $m$ is the cochain representing the
associative multiplication. This formulation leads to a simple
proof that $D^{2}=0$, following from the properties of $\mathbb{Z}$-graded Lie
algebras. The associativity of $m$ is simply the condition
$[m,m]=0$. Let us recall the proof that an odd homogeneous element
$m$ of a graded Lie algebra satisfying $[m,m]=0$ gives rise to a
differential $D$ on the algebra by defining $D(\varphi)=[\varphi,m]$. In
other words, we need to show that $[[\varphi,m],m]=0$. Recall that $m$
is odd when $\left<\mbox{$|m|$},\mbox{$|m|$}\right>=1$. The graded Jacobi bracket gives
(33)
$$[[\varphi,m]m]=[\varphi[m,m]]+\mbox{$(-1)^{\left<m,m\right>}$}[[\varphi,m]m]=-%
[[\varphi,m],m],$$
which shows the desired result, in characteristic zero, when the
grading is good. Moreover, we point out that the Jacobi identity
also shows that $D([\varphi,\psi])=[\varphi,D(\psi)]+\mbox{$(-1)^{\psi D}$}[D(\varphi),\psi]$, so the differential in the cohomology of an
associative algebra acts as a graded derivation of the Lie
algebra, equipping $C(V)$ with the structure of a differential
graded Lie algebra.
We wish to generalize the Gerstenhaber bracket to the $A_{\infty}$ algebra case, where we are considering a more general
codifferential $m$ on $T(V)$, in such a manner that the bracket
with $m$ yields a differential graded Lie algebra structure on
$C(V)$. If we consider the bracket of coderivations, then a problem
arises when the codifferential is not homogeneous. First of all,
$\hbox{$\hat{m}$}^{2}=0$ is not equivalent to the condition $[m,m]=0$. Secondly,
if we define $D(\varphi)=[\varphi,m]$, then we do not obtain in general
that $D^{2}=0$. To see this, note that $[m,m]=0$ is equivalent to
$\sum_{k+l=n+1}[m_{k},m_{l}]=0$ for all $n\geq 1$. Let
$\varphi_{p}\in\mbox{\rm Hom}(V^{p},V)$. Then
(34)
$$\displaystyle[[\varphi_{p},m],m]_{n+p-1}=\sum_{k+l=n+1}[[\varphi_{p},m_{k}],m_%
{l}]=\\
\displaystyle\sum_{k+l=n+1}[\varphi_{p},[m_{k},m_{l}]]+\mbox{$(-1)^{\left<m_{k%
},m_{l}\right>}$}[[\varphi_{p},m_{l}],m_{k}]=\\
\displaystyle\sum_{k+l=n+1}\mbox{$(-1)^{k+l+1}$}[[\varphi_{p},m_{l}],m_{k}]=%
\mbox{$(-1)^{n}$}[[\varphi_{p},m],m]_{n+p-1}$$
Thus we only obtain cancellation of terms when $n$ is odd.
However, this is sufficient to show that in the particular case
where $m_{k}=0$ for all even or all odd $k$, then $D^{2}=0$. In this
case we also can show that $\hbox{$\hat{m}$}^{2}=0$ is equivalent to $[m,m]=0$ as
well. These problems occur because the product grading on $\mbox{$\mathbb{Z}$}_{2}\times\mbox{$\mathbb{Z}$}$ is
not good.
Since these problems do not arise if we are considering $\mbox{$\mathbb{Z}$}_{2}$-graded
codifferentials (or $\mathbb{Z}$-graded codifferentials, in the $\mathbb{Z}$-graded
case), it is natural to consider a codifferential on the parity
reversion(or suspension) of $V$.
For the remainder of this section, let us assume for definiteness
that $W=\Pi V$ and that $d\in C(W)$ satisfies $\hbox{{$\hat{d}$}}^{2}=0$. Because $W$
is $\mbox{$\mathbb{Z}$}_{2}$-graded, $d^{2}=0$ is equivalent to $[d,d]=0$ for $d$ odd, and
moreover, $C(W)$ is a differential graded Lie algebra with
differential $D(\varphi)=[\varphi,d]$. Let $m_{k}=\eta_{1}^{-1}\circ d_{k}\circ\eta_{k}$ and $\mu_{l}=\eta_{1}^{-1}\circ\delta_{l}\circ\eta_{l}$.
Define a new bracket $\left\{\cdot,\cdot\right\}$ on $C(V)$ by
$\left\{m_{k},\mu_{l}\right\}=\eta_{1}^{-1}\circ[d_{k},\delta_{l}]\circ\eta_{k+%
l-1}$.
It follows easily that
(35)
$$\left\{m_{k},\mu_{l}\right\}=\mbox{$(-1)^{(k-1)\mu_{l}}$}[m_{k},\mu_{l}].$$
Of course this new bracket no longer satisfies the graded
antisymmetry or graded Jacobi identity with respect to the inner
product we have been using on $\mbox{$\mathbb{Z}$}_{2}\times\mbox{$\mathbb{Z}$}$. However, the bracket does
satisfy these properties with respect to the a different inner
product on $\mbox{$\mathbb{Z}$}_{2}\times\mbox{$\mathbb{Z}$}$. We state this result in the form of a lemma,
whose proof is straightforward.
Lemma 1.
Let $V$ be equipped with a $\mbox{$\mathbb{Z}$}_{2}\times\mbox{$\mathbb{Z}$}$-graded Lie bracket $[\cdot,\cdot]$
with respect to the inner product $\left<(\bar{m},n),(\bar{m}^{\prime},n^{\prime})\right>=\bar{m}\bar{m}^{\prime}%
+nn^{\prime}$ on $\mbox{$\mathbb{Z}$}_{2}\times\mbox{$\mathbb{Z}$}$. Then the bracket $\left\{\cdot,\cdot\right\}$ on $V$
given by $\left\{u,v\right\}=\mbox{$(-1)^{\operatorname{deg}(u)|v|}$}[u,v]$ defines the structure of a
$\mbox{$\mathbb{Z}$}_{2}\times\mbox{$\mathbb{Z}$}$-graded Lie algebra on $V$ with respect to the inner product
$\left<(\bar{m},n),(\bar{m}^{\prime},n^{\prime})\right>=(\bar{m}+n)(\bar{m}^{%
\prime}+n^{\prime})$ on $\mbox{$\mathbb{Z}$}_{2}\times\mbox{$\mathbb{Z}$}$.
The bracket $\left\{\cdot,\cdot\right\}$ given by $\left\{u,v\right\}=\mbox{$(-1)^{\operatorname{deg}(u)(|v|+\operatorname{deg}(v%
))}$}[u,v]$ also defines the structure of a
$\mbox{$\mathbb{Z}$}_{2}\times\mbox{$\mathbb{Z}$}$-graded Lie algebra on $V$ with respect to the second inner
product.
The bracket $\left\{\cdot,\cdot\right\}$ on $C(V)$ essentially coincides with the bracket
of coderivations on $C(W)$, so $[d,d]=0$implies that $\left\{m,m\right\}=0$.
Define a differential on $C(V)$ by $D(\varphi)=\left\{\varphi,m\right\}$. The homology
of this differential is called the cohomology of the $A_{\infty}$ algebra.
We have been considering
the picture $W=\Pi V$. This means we have been considering $A_{\infty}$ algebras as defined in [11, 10]. If we consider instead the
picture $V=\Pi W$, then the bracket induced on $C(V)$ by that on
$C(W)$ will be given by $\left\{m_{k},\mu_{l},=\right\}\mbox{$(-1)^{(k-1)(\mu_{l}+l-1)}$}[m_{k},\mu_{l}]$, which is the second
modified bracket described in the lemma. Thus we have similar
results. We shall call either one of these two brackets the
modified Gerstenhaber bracket.
The Hochschild cohomology of $A_{\infty}$ algebras was defined in
[16] as the cohomology given by $D(\varphi)=\left\{\varphi,m\right\}$, and it
was shown that this cohomology classifies the infinitesimal
deformations of an $A_{\infty}$ algebra. We shall not go into the
details here. It is important to note however, that unlike the
cohomology theory for an associative algebra, where the $\mathbb{N}$-grading
on the tensor product gives rise to an $\mathbb{N}$-grading on the
cohomology, for an $A_{\infty}$ algebra there is only one cohomology
group $H(V)$. (However, $H(V)$ does inherit a natural
$\mbox{$\mathbb{Z}$}_{2}$-grading.) The reason is that the image of $\varphi\in C^{n}(V)$
under the coboundary operator has a part in all $C^{k}(V)$ with
$k\geq n$. Only in the case of an associative or differential
graded associative algebra do we get a grading of the cohomology.
5.1. Cyclic Cohomology of $A_{\infty}$ Algebras
Now let us suppose that $V$ is equipped with a nondegenerate even
graded symmetric inner product $\left<\cdot,\cdot\right>$. Graded symmetry means that
$\left<v,w\right>=\mbox{$(-1)^{vw}$}\left<w,v\right>$. The inner product induces an isomorphism
between $C^{k}(V)=\mbox{\rm Hom}(V^{k},V)$ and $C^{k}(V,\mbox{\bf k})=\mbox{\rm Hom}(V^{k+1},\mbox{\bf k})$,
given by $\varphi\mapsto\tilde{\varphi}$, where
(36)
$$\tilde{\varphi}(v_{1},\cdots,v_{k+1})=\left<\varphi(v_{1},\cdots,v_{k}),v_{k+1%
}\right>.$$
Non-degeneracy means that the map $\lambda:\mbox{V}\rightarrow\mbox{V}^{*}=\mbox{\rm Hom}(\mbox{V},\mbox{\bf k})$, given by $\lambda(v)(w)=\left<v,w\right>$ is an
isomorphism. (When k is a field, this is equivalent to the usual
definition of a non degenerate bilinear form.)
An element $\varphi\in\mbox{\rm Hom}(V^{k},V)$ is said to be cyclic with respect
to the inner product if
(37)
$$\left<\varphi(v_{1},\cdots,v_{k}),v_{k+1}\right>=\mbox{$(-1)^{k+v_{1}\varphi}$%
}\left<v_{1},\varphi(v_{2},\cdots,v_{k+1})\right>.$$
Then $\varphi$ is cyclic if and only if $\tilde{\varphi}$ is cyclic in the
sense that
(38)
$$\tilde{\varphi}(v_{1},\cdots,v_{k+1})=\mbox{$(-1)^{n+v_{k+1}(v_{1}+\cdots+v_{k%
})}$}\tilde{\varphi}(v_{k+1},v_{1},\cdots,v_{k}).$$
If $m\in C(V)$, then we say that $m$ is cyclic if $m_{k}$ is cyclic
for all $k$. If $m$ determines an $A_{\infty}$ algebra structure on $V$,
then we say that the inner product is invariant if $m$ is cyclic
with respect to the inner product. Denote the submodule of cyclic
elements in $C(V)$ by $CC(V)$, and the submodule of cyclic elements
in $C(V,\mbox{\bf k})$ by $CC(V,\mbox{\bf k})$. The following lemma shows how to
construct cyclic elements from arbitrary elements of $C^{n}(V,\mbox{\bf k})$.
Lemma 2.
Suppose that $\hbox{$\tilde{f}$}\in C^{n}(V,\mbox{\bf k})$. Define $C(\hbox{$\tilde{f}$})\in C^{n}(V,\mbox{\bf k})$ by
(39)
$$C(f)(v_{1},\cdots,v_{n+1})=\sum_{0\leq i\leq n}\mbox{$(-1)^{(v_{1}+\cdots+v_{i%
})(v_{i+1}+\cdots+v_{n+1})+ni}$}f(v_{i+1},\cdots,v_{i})$$
Then $C(\hbox{$\tilde{f}$})$ is cyclic. Furthermore, $C(\hbox{$\tilde{f}$})=(n+1)\hbox{$\tilde{f}$}$ if $\tilde{f}$
is cyclic.
The lemma above follows from the technical lemma below, which
simplifies computations with cyclic elements.
Lemma 3.
If $\tilde{f}\in C^{n}(V,\mbox{\bf k})$, then $\tilde{f}$ is cyclic iff whenever
$\alpha=v_{1}\otimes\cdots\otimes v_{i}$ and $\beta=v_{i}\otimes\cdots\otimes v_{n+1}$,
(40)
$$\tilde{f}(\alpha\otimes\beta)=\mbox{$(-1)^{\alpha\beta+in}$}\tilde{f}(\beta%
\otimes\alpha).$$
The following lemma records the fact that $CC(V)$ is a graded Lie
subalgebra of $C(V)$, with respect to the bracket of coderivations.
Lemma 4.
Let $\varphi_{k}\in C^{k}(V)$ and $\psi_{l}\in C^{l}(V)$. If $\varphi$ and $\psi$
are cyclic, then so is $[\varphi,\psi]$. Moreover, if $n=k+l-1$, then
(41)
$$\displaystyle\widetilde{[\varphi,\psi]}(v_{1},\cdots,v_{n+1})=\\
\displaystyle\sum_{\begin{subarray}{c}0\leq i\leq n\end{subarray}}\mbox{$(-1)^%
{(v_{1}+\cdots+v_{i})(v_{i+1}+\cdots+v_{n+1})+in}$}\tilde{\varphi}_{k}(\psi_{l%
}(v_{i+1},\cdots,v_{i+l}),v_{i+l+1},\cdots,v_{i}),$$
where in expressions of this type, indices should be interpreted
$\mod{n+1}$.
Since $\left\{\varphi,\psi\right\}=\mbox{$(-1)^{(k-1)\psi_{l}}$}[\varphi,\psi]$, it follows that
the modified Gerstenhaber bracket of cyclic elements is also
cyclic. Thus finally, we can state the main theorem, which allows
us to define cyclic cohomology of an $A_{\infty}$ algebra.
Theorem 1.
i)Suppose that $V$ is a $\mbox{$\mathbb{Z}$}_{2}$-graded k-module with an inner product
$\left<\cdot,\cdot\right>$. Suppose that $\varphi,\psi\in C(V)$ are cyclic. Then
$\left\{\varphi,\psi\right\}$ is cyclic. Furthermore, the formula below holds.
(42)
$$\displaystyle\widetilde{\left\{\varphi,\psi\right\}}(v_{1},\cdots,v_{n+1})=\\
\displaystyle\sum_{\begin{subarray}{c}k+l=n+1\\
\\
0\leq i\leq n\end{subarray}}\hskip-7.227pt\mbox{$(-1)^{(v_{1}+\cdots+v_{i})(v_%
{i+1}+\cdots+v_{n+1})+in+(k-1)\psi_{l}}$}\tilde{\varphi}_{k}(\psi_{l}(v_{i+1},%
\cdots,v_{i+l}),v_{i+l+1},\cdots,v_{i}).$$
Thus the inner product induces a structure of a $\mbox{$\mathbb{Z}$}_{2}\times\mbox{$\mathbb{Z}$}$-graded Lie
algebra in $CC(V,\mbox{\bf k})$ given by
$\left\{\tilde{\varphi},\tilde{\psi}\right\}=\widetilde{\left\{\varphi,\psi%
\right\}}$.
ii) If $m$ is an $A_{\infty}$ structure on $V$, then there is a
differential in $CC(V,\mbox{\bf k})$, given by
(43)
$$\displaystyle D(\tilde{\varphi})(v_{1},\cdots,v_{n+1})=\\
\displaystyle\sum_{\begin{subarray}{c}k+l=n+1\\
\\
0\leq i\leq n\end{subarray}}\hskip-7.227pt\mbox{$(-1)^{(v_{1}+\cdots+v_{i})(v_%
{i+1}+\cdots+v_{n+1})+in+(k-1)l}$}\tilde{\varphi}_{k}(m_{l}(v_{i+1},\cdots,v_{%
i+l}),v_{i+l+1},\cdots,v_{i}).$$
iii) If the inner product is invariant, then
$D(\tilde{\varphi})=\left\{\tilde{\varphi},\tilde{m}\right\}$. Thus $CC(V,\mbox{\bf k})$ inherits
the structure of a differential graded Lie algebra.
Proof.
The first statement follows from lemma 4, and the third
follows from the first two. The second assertion follows
immediately from the first when the inner product is invariant.
The general case is a routine verification, which we omit.
∎
Note that the second statement in the theorem holds even in the
absence of an inner product, because the definition of cyclicity
in $CC(V,\mbox{\bf k})$ does not depend on the inner product. Thus cyclic
cohomology of an $A_{\infty}$ algebra can be defined independently of
the inner product.
We define $HC(V)$ to be the cohomology associated to the
coboundary operator on $CC(V)$. As in the case of Hochschild
cohomology, the cyclic cohomology is $\mbox{$\mathbb{Z}$}_{2}$-graded, but does not
inherit an $\mathbb{N}$-grading, except in the associative algebra case.
6. Cohomology of $L_{\infty}$ algbras
There is a natural isomorphism $\eta$ between $\bigwedge V$ and
$S(\Pi V)$ which is given by
(44)
$$\eta(v_{1}\wedge\cdots\wedge v_{n})=\mbox{$(-1)^{(n-1)v_{1}+(n-2)v_{2}+\cdots+%
v_{n-1}}$}\pi v_{1}\cdots\pi v_{n},$$
Note that $\eta$ is neither even nor odd. The restriction $\eta_{k}$
of $\eta$ to $\bigwedge^{k}V$ has parity $k$. Of course, $\eta$
does preserve the exterior degree. For simplicity in the
following, let $W=\Pi V$ and let $w_{i}=\pi v_{i}$, and denote
$C(W)=\mbox{\rm Hom}(S(W),W)$, and $C(V)=\mbox{\rm Hom}(\bigwedge(V),V)$. We will use
notational conventions as in section 5, so that for
$d\in C(W)$, $d_{k}$ will denote the restriction of this map to
$\bigwedge^{k}W$, $d_{lk}$ will denote the restriction of the
associated coderivation $\hbox{{$\hat{d}$}}_{l}$ to $V^{k+l-1}$ etc.
The following lemma will be useful later on.
Lemma 5.
Suppose that $\sigma$ is a permutation of $n$ elements. Then
(45)
$$\displaystyle\mbox{$(-1)^{(n-1)v_{1}+\cdots+v_{n-1}}$}\mbox{$(-1)^{\sigma}$}%
\epsilon(\sigma;v_{1},\cdots,v_{n})=\\
\displaystyle\mbox{$(-1)^{(n-1)v_{\sigma(1)}+\cdots+v_{\sigma(n-1)}}$}\epsilon%
(\sigma;w_{1},\cdots,w_{n}).$$
Proof.
From the properties of the graded exterior algebra, we have
$$\displaystyle\eta(v_{\sigma(1)}\wedge\cdots\wedge v_{\sigma(n)})$$
$$\displaystyle=\mbox{$(-1)^{\sigma}$}\epsilon(\sigma;v_{1},\cdots,v_{n})\eta(v_%
{1},\cdots,v_{n})$$
$$\displaystyle=\mbox{$(-1)^{\sigma}$}\epsilon(\sigma;v_{1},\cdots,v_{n})\mbox{$%
(-1)^{(n-1)v_{1}+\cdots+v_{n-1}}$}w_{1}\cdots w_{n}.$$
On the other hand, by direct substitution, we have
$$\displaystyle\eta(v_{\sigma(1)}\wedge\cdots\wedge v_{\sigma(n)})$$
$$\displaystyle=\mbox{$(-1)^{(n-1)v_{\sigma(1)}+\cdots+v_{\sigma(n-1)}}$}w_{%
\sigma(1)}\cdots w_{\sigma(n)}$$
$$\displaystyle=\mbox{$(-1)^{(n-1)v_{\sigma(1)}+\cdots+v_{\sigma(n-1)}}$}%
\epsilon(\sigma;w_{1},\cdots,w_{n})w_{1}\cdots w_{n}$$
Comparing the coefficients of the two expressions yields the
desired result.
∎
Let $d\in C(W)$ and define $l_{k}=\eta_{1}^{-1}\circ d_{k}\circ\eta_{k}$.
so that
(46)
$$d_{k}(w_{\sigma(1)},\cdots,w_{\sigma(k)}=\mbox{$(-1)^{(k-1)v_{\sigma(1)}+%
\cdots+v_{\sigma(k-1)}}$}\pi m_{k}(v_{\sigma(1)},\cdots,v_{\sigma(k-1)}).$$
Let us abbreviate $\epsilon(\sigma;v_{1},\cdots,v_{n})$ by
$\epsilon(\sigma;v)$ and $\epsilon(\sigma;w_{1},\cdots,w_{n})$ by
$\epsilon(\sigma;w)$. Let $n=k+l-1$. Define $\bar{l}_{lk}=\eta_{k}^{-1}\circ d_{lk}\circ\eta_{n}$, where $d_{lk}$ is
given by
(47)
$$d_{lk}(w_{1},\cdots,w_{n})=\sum_{\sigma\in\operatorname{Sh}(k,n-k)}\epsilon(%
\sigma,w)d_{l}(w_{\sigma(1)},\cdots,w_{\sigma(l)})w_{\sigma(l+1)}\cdots w_{%
\sigma(n)}.$$
We wish to compute $\bar{l}_{lk}$ in terms of $l_{l}$. Now
(48)
$$\displaystyle l_{lk}(v_{1},\cdots,v_{n})=\\
\displaystyle\sum_{\sigma\in\operatorname{Sh}(l,n-l)}\!\!\!\epsilon(\sigma,w)%
\mbox{$(-1)^{(n-1)v_{1}+\cdots+v_{n-1}}$}\eta_{k}^{-1}(d_{l}(w_{\sigma(1)},%
\cdots,w_{\sigma(l)})w_{\sigma(l+1)}\odot\cdots\odot w_{\sigma(n)})\\
\displaystyle=\sum_{\sigma\in\operatorname{Sh}(l,n-l)}\mbox{$(-1)^{\sigma}$}%
\epsilon(\sigma,v)\mbox{$(-1)^{r}$}\eta_{k}^{-1}(d_{l}(w_{\sigma(1)},\cdots,w_%
{\sigma(l)})w_{\sigma(l+1)}\cdots w_{\sigma(n)})\\
\displaystyle=\sum_{\sigma\in\operatorname{Sh}(l,n-l)}\mbox{$(-1)^{\sigma}$}%
\epsilon(\sigma,v)\mbox{$(-1)^{r+s}$}\eta_{k}^{-1}(l_{l}(v_{\sigma(1)},\cdots,%
v_{\sigma(l)})w_{\sigma(l+1)}\cdots w_{\sigma(n)})\\
\displaystyle=\sum_{\sigma\in\operatorname{Sh}(l,n-l)}\mbox{$(-1)^{\sigma}$}%
\epsilon(\sigma,v)\mbox{$(-1)^{r+s+t}$}l_{l}(v_{\sigma(1)},\cdots,v_{\sigma(l)%
})v_{\sigma(l+1)}\cdots v_{\sigma(n)}\\
\displaystyle=\sum_{\sigma\in\operatorname{Sh}(l,n-l)}\mbox{$(-1)^{\sigma}$}%
\epsilon(\sigma,v)\mbox{$(-1)^{(k-1)l_{l}}$}l_{l}(v_{\sigma(1)},\cdots,v_{%
\sigma(l)})v_{\sigma(l+1)}\cdots v_{\sigma(n)}.$$
where
(49)
$$\displaystyle r$$
$$\displaystyle=$$
$$\displaystyle(n-1)v_{\sigma(1)}+\cdots+v_{\sigma(n-1)}$$
(50)
$$\displaystyle s$$
$$\displaystyle=$$
$$\displaystyle(l-1)v_{\sigma(1)}+\cdots+v_{\sigma(l-1)}$$
(51)
$$\displaystyle t$$
$$\displaystyle=$$
$$\displaystyle(k-1)(l_{l}+v_{\sigma(1)}+\cdots+v_{\sigma(l-1)}(k-2)v_{\sigma(l+%
1)}+\cdots+v_{\sigma(n-1)}$$
Thus we deduce immediately that $\bar{l}_{lk}=\mbox{$(-1)^{(k-1)l_{l}}$}l_{lk}$,
where $l_{lk}$ is the restriction of the coderivation $\hat{l}_{k}$
to $V^{k+l-1}$. This formula is identical to the formula we
deduced in section 5 connecting $\hbox{$\bar{m}$}_{lk}$ to
$\hbox{$\hat{m}$}_{lk}$.
Suppose that $d$ is an odd codifferential, so that $d^{2}=0$. This
is equivalent to $\sum_{k+l=n+1}d_{k}\circ d_{lk}=0$, which is
equivalent to the relations $\sum_{k+l=n+1}\mbox{$(-1)^{(k-1)l}$}l_{k}\circ l_{lk}=0$, since $\mbox{$|l_{l}|$}=l$. This last relation can be put in the
form
(52)
$$\displaystyle\sum_{\begin{subarray}{c}k+l=n+1\\
\sigma\in\operatorname{Sh}(l,n-l)\end{subarray}}\mbox{$(-1)^{\sigma}$}\epsilon%
(\sigma,v)\mbox{$(-1)^{(k-1)l}$}l_{k}(l_{l}(v_{\sigma(1)},\cdots,v_{\sigma(l)}%
),v_{\sigma(l+1)},\cdots,v_{\sigma(n)}))=0.$$
We say that the maps $l_{k}$ induce the structure of an $L_{\infty}$ algebra, or strongly homotopy Lie algebra on $V$. In [11],
the sign $k(l-1)$ instead of $(k-1)l$ appears in the definition,
but since $k(l-1)-(k-1)l=n+1$, this makes no difference in the
relations.
If we define $[\varphi,\psi]$ to be the bracket of coderivations, then
we can define the modified bracket
$\left\{\varphi,\psi\right\}=\mbox{$(-1)^{\operatorname{deg}\varphi\mbox{$|\psi%
|$}}$}[\varphi,\psi]$. The definition of an
$L_{\infty}$ algebra can be recast in terms of the bracket. In this
language, $l\in C(V)$ determines an $L_{\infty}$ structure on $V$ when
$\left\{l,l\right\}=0$. The cohomology of an $L_{\infty}$ algebra is defined to be
the cohomology on $C(W)$ induced by $l$, in other words,
$D(\varphi)=[\varphi,l]$. This definition makes $C(V)$ a differential
graded Lie algebra, with respect to the second inner product on
$\mbox{$\mathbb{Z}$}_{2}\times\mbox{$\mathbb{Z}$}$. These results are completely parallel to the $A_{\infty}$ case.
In [16], the relationship between infinitesimal deformations
of an $A_{\infty}$ algebra and the cohomology of the $A_{\infty}$ algebra was
explored. The basic result is that the cohomology classifies the
infinitesimal deformations. Since we did not explore this matter
here for $A_{\infty}$ algebras, we shall discuss the parallel result for
$L_{\infty}$ algebras.
An infinitesimal deformation $l_{u}$ of an $L_{\infty}$ algebra is given
by taking $l_{u}=l+u\lambda$, where $u$ is an infinitesimal
parameter whose parity chosen so that $(l_{u})_{k}$ has parity $k$.
It follows that $\mbox{$|\lambda_{k}|$}=\mbox{$|u|$}+k$. Since $u$ must have fixed
parity, this determines the parity of $\lambda_{k}$.
The situation is more transparent if we switch to the $W$ picture,
so suppose that $l=\eta^{-1}\circ d\circ\eta$, and
$\lambda=\eta^{-1}\circ\delta\circ\eta$. Let $d_{u}=d+u\delta$, and
let us suppose that $d^{2}=0$, which is equivalent to $l$ giving an
$L_{\infty}$ structure on $V$. Then $d_{u}$ is an infinitesimal
deformation of $d$ if $d_{u}^{2}=0$. Since $u$ is an infinitesimal
parameter, $u^{2}=0$, but we also want the parity of $d_{u}$ to be
odd, so that $\mbox{$|u|$}=1-\mbox{$|\delta|$}$. (We assume here that $\delta$ is
homogeneous.) Now $d_{u}^{2}=0$ is equivalent to $d^{2}+u\delta d+du\delta=0$. Also, $du=\mbox{$(-1)^{ud}$}ud=-\mbox{$(-1)^{\delta d}$}ud$, so this
condition reduces to $[\delta,d]=0$; in other words,
infinitesimal deformations are given by cocycles in the cohomology
of the $L_{\infty}$ algebra.
Trivial deformations are more complicated in the $L_{\infty}$ case than
for Lie algebras. Consider $d$ as a codifferential of $T(W)$. Two
codifferentials are said to be equivalent if there is an
automorphism of $T(W)$ which takes one of them to the other. In
the Lie algebra case, one only considers automorphisms of $T(W)$
induced by linear isomorphism of $W$ into itself. One can show
that $d_{u}$ is a trivial infinitesimal deformation precisely when
$\delta$ is a coboundary. Thus the cohomology of $C(W)$ classifies
the infinitesimal deformations of the $L_{\infty}$ algebra. When we
transfer this back to the $V$ picture, note that $m+\mbox{$(-1)^{u}$}u\lambda$
is the deformed product associated to $d_{u}$. The condition for
$l_{u}$ to be an $L_{\infty}$ algebra still is that $\left\{\lambda,m\right\}=0$. We
sumarize these results in the theorem below.
Theorem 2.
Let $l$ be an $L_{\infty}$ algebra structure on $V$. Then the cohomology
$H(V)$ of $C(V)$ classifies the infinitesimal deformations of the
$L_{\infty}$ algebra.
Suppose that $l$ is a Lie algebra structure on $V$. Then the Lie
algebra coboundary operator on $V$ coincides up to a sign with the
$L_{\infty}$ algebra coboundary operator on $V$. This gives a nice
interpretation of the cohomology of a Lie algebra.
Theorem 3.
Let $l$ be an Lie algebra structure on $V$. Then the Lie algebra
cohomology $H(V)$ of $V$ classifies the infinitesimal deformations
of the Lie algebra into an $L_{\infty}$ algebra.
6.1. Cyclic Cohomology of $L_{\infty}$ Algebras
Suppose $V$ is equipped with a nondegenerate graded even symmetric
inner product. An element $\varphi\in C^{n}(V)$ if the tensor
$\tilde{\varphi}:V^{n}\rightarrow\mbox{\bf k}$, given by
$$\tilde{\varphi}(v_{1},\cdots,v_{n+1})=\left<\varphi(v_{1},\cdots,v_{n}),v_{n+1%
}\right>$$
is (graded) antisymmetric; i.e., $\tilde{\varphi}\in CC^{n}(V,\mbox{\bf k})=\mbox{\rm Hom}(\bigwedge^{n+1},\mbox%
{\bf k})$. Note that the antisymmetry
condition is equivalent to the cyclicity condition given in the
definition of cyclicity for $A_{\infty}$ algebras, given that $\varphi$ is
antisymmetric. Since the inner product is non-degenerate, the map
$\varphi\mapsto\tilde{\varphi}$ is an even isomorphism between the submodule
$CC^{n}(\mbox{V})$ of $C^{n}(\mbox{V})$ consisting of cyclic elements, and
$CC^{n}(\mbox{V},\mbox{\bf k})$.
Theorem 4.
i)Suppose that $V$ is a $\mbox{$\mathbb{Z}$}_{2}$-graded k-module with an inner product
$\left<\cdot,\cdot\right>$. Suppose that $\varphi,\psi\in C(V)$ are cyclic. Then
$\left\{\varphi,\psi\right\}$ is cyclic. Furthermore, the formula below holds.
(53)
$$\displaystyle\widetilde{\left\{\varphi,\psi\right\}}(v_{1},\cdots,v_{n+1})=\\
\displaystyle\sum_{\begin{subarray}{c}k+l=n+1\\
\\
\sigma\in\operatorname{Sh}((,l),k)\end{subarray}}\mbox{$(-1)^{\sigma}$}%
\epsilon(\sigma)\mbox{$(-1)^{(k-1)\psi_{l}}$}\tilde{\varphi}_{k}(\psi_{l}(v_{%
\sigma(1)},\cdots,v_{\sigma(l)}),v_{\sigma(l+1)},\cdots,v_{\sigma(n+1)}),$$
Thus the inner product induces a structure of a $\mbox{$\mathbb{Z}$}_{2}\times\mbox{$\mathbb{Z}$}$-graded Lie
algebra in the module $CC(V)$ consisting of all cyclic elements in
$C(V))$, by defining
$\left\{\tilde{\varphi},\tilde{\psi}\right\}=\widetilde{\left\{\varphi,\psi%
\right\}}$.
ii) If $l$ is an $L_{\infty}$ structure on $V$, then there is a
differential in $CC(V)$, given by
(54)
$$\displaystyle D(\tilde{\varphi})(v_{1},\cdots,v_{n+1})=\\
\displaystyle\sum_{\begin{subarray}{c}k+l=n+1\\
\\
\sigma\in\operatorname{Sh}((,l),k)\end{subarray}}\mbox{$(-1)^{\sigma}$}%
\epsilon(\sigma)\mbox{$(-1)^{(k-1)l}$}\tilde{\varphi}_{k}(l_{l}(v_{\sigma(1)},%
\cdots,v_{\sigma(l)}),v_{\sigma(l+1)},\cdots,v_{\sigma(n+1)}),$$
iii) If the inner product is invariant, then
$D(\tilde{\varphi})=\left\{\tilde{\varphi},\tilde{l}\right\}$. Thus $CC(V)$ inherits
the structure of a differential graded Lie algebra.
We denote the cohomology given by the cyclic coboundary operator
on $CC(V)$ as $HC(V)$. It is interesting to note that
$HC^{n}(V)\equiv H^{n+1}(V,\mbox{\bf k})$, where $H(V,\mbox{\bf k})$ is interpreted as
the cohomology of $V$ with coefficients in the trivial module
k. Suppose that $V$ is an $L_{\infty}$ algebra with an invariant
inner product. Then an infinitesimal deformation $l_{t}=l+t\varphi$
preserves the inner product, that is the inner product remains
invariant under $l_{t}$, precisely when $\varphi$ is cyclic. Thus we see
that cyclic cocycles correspond to infinitesimal deformations of
the $L_{\infty}$ structure which preserve the inner product. In a
similar manner as before, cyclic coboundaries correspond to
trivial deformations preserving the inner product. Thus we have
the following classification theorem.
Theorem 5.
Let $l$ be an $L_{\infty}$ algebra structure on $V$, with an invariant
inner product. Then the cyclic cohomology $HC(V)$ classifies the
infinitesimal deformations of the $L_{\infty}$ algebra preserving the
inner product.
When $l$ determines a Lie algebra structure on $V$, the cohomology
$HC(V)$ has a natural $\mathbb{Z}$-grading, and the group $HC^{2}(V)$ controls
deformations of the Lie algebra structure.
Theorem 6.
Let $l$ be a Lie algebra structure on $V$, with an invariant inner
product. Then the second cyclic cohomology group $HC^{2}(V)$
classifies the infinitesimal deformations of the Lie algebra
preserving the inner product.
For a Lie algebra, we have $HC^{n}(V)\equiv H^{n+1}(V,\mbox{\bf k})$, the Lie
algebra cohomology of $V$ with trivial coefficients. Thus the
deformations preserving an invariant inner product are classified
by the cohomology of the Lie algebra with trivial coefficients.
Finally, we give a nice interpretation of the cyclic cohomology of
a Lie algebra.
Theorem 7.
Let $l$ be an Lie algebra structure on $V$, with an invariant
inner product. Then the Lie algebra cyclic cohomology $H(V)$ of
$V$ classifies the infinitesimal deformations of the Lie algebra
into an $L_{\infty}$ algebra preserving the inner product.
Note that as in the case of $A_{\infty}$ algebras, $C(V)=\mbox{\rm Hom}(\bigwedge V,V)$ is really the direct product of its graded subspaces
$C^{k}(V)$, so it is not a graded Lie algebra in the strict sense of
the definition, because it is not the direct sum of the graded
subspaces. Moreover, the cohomology $H(V)$ does not inherit a
natural $\mathbb{Z}$-grading in general. Similarly, the space $CC(V)$ of
cyclic cochains is a direct product, and $HC(V)$ does not have a
natural $\mathbb{Z}$-grading. Nevertheless, the good $\mbox{$\mathbb{Z}$}_{2}\times\mbox{$\mathbb{Z}$}$-grading does equip
these space with a natural $\mbox{$\mathbb{Z}$}_{2}$-grading, so that the cohomology is
$\mbox{$\mathbb{Z}$}_{2}$-graded. These issues are addressed in more detail in
[3].
It is interesting to note that the cyclic cohomology of $L_{\infty}$ algebras has more symmetry than that of $A_{\infty}$ algebras; the full
symmetric group acts on the space of cyclic cochains for $L_{\infty}$ algebras. This difference has some consequences in terms of the
actions of these algebras on graph complexes. In [15], it
is shown that $L_{\infty}$ algebras with an invariant inner product act
on the ordinary graph complex, while $A_{\infty}$ algebras with an
invariant inner product act on the space of ribbon graphs, which
are given by graphs equipped with a cyclic order at each vertex.
7. Acknowledgements
The author would like to thank Albert Schwarz, Dmitry Fuchs and
James Stasheff for reading the original version [14] of this article
and providing useful suggestions.
References
[1]
M. Alexandrov, M. Kontsevich, A. Schwarz, and O. Zaboronsky,
The geometry
of the master equation and topological quantum field theory, Internat.
Journ. Modern Phys. A12 (1997), 1405–1423.
[2]
A. Connes, Non-commutative differential geometry, Institute
Des Hautes
Etudes Scientifiques 62 (1985), 41–93.
[3]
A. Fialowski and M. Penkava, Deformation theory of infinity
algebras,
preprint math.RT/0101097, 2000.
[4]
M. Gerstenhaber, The cohomology structure of an associative
ring, Annals
of Mathematics 78 (1963), 267–288.
[5]
E. Getzler and J.D.S. Jones, $A_{\infty}$-algebras and
the cyclic
bar complex, Illinois Journal of Mathematics 34 (1990), no. 2,
256–283.
[6]
by same author, Operads, homotopy algebra, and iterated integrals
for double
loop spaces, Preprint, 1995.
[7]
G. Hochschild, On the cohomology groups of an associative
algebra,
Annals of Mathematics 46 (1945), 58–67.
[8]
D. Kastler, Cyclic cohomology within the differential
envelope, Travaux
En Cours, 1988.
[9]
M. Kontsevich, Feynman diagrams and low dimensional
topology, First
European Congress of Mathematics, Paris, 1992, Birkhauser, Basel, 1994,
pp. 97–121.
[10]
T. Lada and M. Markl, Strongly homotopy Lie algebras,
Comm. in Algebra
23 (1995), 2147–2161.
[11]
T. Lada and J. Stasheff, Introduction to sh Lie algebras
for
physicists, Intern. J. Theor. Phys 32 (1993), 1087–1103, Preprint
hep-th 9209099.
[12]
J. Loday, Cyclic homology, Springer-Verlag, 1992.
[13]
M. Markl, A cohomology theory for A($m$)-algebras and
applications,
Journal of Pure and Applied Algebra 83 (1992), no. 6, 141–175.
[14]
M. Penkava, $L_{\infty}$ algebras and their cohomology,
Preprint q-alg/9512014.
[15]
M. Penkava, Infinity algebras and the homology of graph
complexes,
Preprint q-alg/9601018.
[16]
M. Penkava and A. Schwarz, $A_{\infty}$ algebras and
the cohomology
of moduli spaces, Dynkin Seminar, vol. 169, American Mathematical Society,
1995, pp. 91–107.
[17]
M. Schlessinger and J. Stasheff, The Lie algebra structure
of tangent
cohomology and deformation theory, Journal of Pure and Applied Algebra
38 (1985), 313–322.
[18]
P. Seibt, Cyclic homology of algebras, World Scientific,
1987.
[19]
J.D. Stasheff, On the homotopy associativity of H-spaces
I,
Transactions of the AMS 108 (1963), 275–292.
[20]
by same author, On the homotopy associativity of H-spaces II,
Transactions
of the AMS 108 (1963), 293–312.
[21]
by same author, The intrinsic bracket on the deformation complex of
an
associative algebra, Journal of Pure and Applied Algebra 89 (1993),
231–235.
[22]
by same author, Closed string field theory, strong homotopy Lie
algebras and
the operad actions of moduli spaces, Conf. Proc. Lecture Notes Math. Phys.,
III, Internat. Press, Cambridge, MA, 1994, Perspectives in mathematical
physics, pp. 265–288.
[23]
R. Umble, The deformation complex for differential graded
hopf algebras,
Preprint, 1994. |
Are there arbitrarily long arithmetic progressions in the sequence of twin primes? II
János Pintz
Supported by OTKA Grants K72731, K67676 and ERC-AdG.228005.
()
1 Introduction
Until very recently the twin prime conjecture seemed to be completely inaccessible with available methods of number theory.
Four years ago, in a joint work with D. Goldston and C. Yıldırım [GPY] we proved that assuming a very regular distribution of primes in arithmetic progressions we obtain a somewhat weaker result
(1.1)
$$\liminf_{n\to\infty}(p_{n+1}-p_{n})\leq 16,$$
where $p_{n}$ denotes the $n$th prime.
The condition was that the level $\vartheta$ of distribution of primes, that is, an exponent, such that for $\varepsilon>0$, $A>0$
(1.2)
$$\sum_{q\leq N^{\vartheta-\varepsilon}}\max_{\begin{subarray}{c}a\\
(a,q)=1\end{subarray}}\biggl{|}\sum_{\begin{subarray}{c}p\equiv a\pmod{q}\\
p\leq N\end{subarray}}\log p-\frac{N}{\varphi(q)}\biggr{|}\ll_{\varepsilon,A}%
\frac{N}{(\log N)^{A}}$$
satisfies $\vartheta\geq 0.971$; an assumption, just slightly weaker than the strongest possible hypothesis $\vartheta=1$, the well-known Elliott–Halberstam [EH] conjecture.
The best known admissible value for $\vartheta$, the relation $\vartheta=1/2$, is the celebrated Bombieri–Vinogradov theorem.
In the same work we proved that any $\vartheta>1/2$ would yield infinitely many bounded gaps between primes, that is
(1.3)
$$\liminf_{n\to\infty}(p_{n+1}-p_{n})\leq C(\vartheta).$$
Since we can not suppose concerning the level of distribution anything beyond the Elliott–Halberstam conjecture (which would yield also (1.1)) the question arises, whether we can deduce the twin prime conjecture itself – or perhaps even a positive answer for the question in the title of our paper – from a hypothetical very regular behaviour of some related sequences (possibly including the primes themselves) similarly to the Elliott–Halberstam conjecture for primes.
We will give under some plausible hypotheses an affirmative answer for this question, including the existence of arbitrarily long arithmetic progressions in the sequence of twin primes.
Surprisingly the required distribution level is just $\vartheta>\frac{3}{4}$ (or with more precise arguments even slightly less, $\vartheta\geq 0.7284$), but the sequences for which a suitable analogue of (1.3) is needed, are not just the primes but all the following ones:
(1.4)
$$\log p,\ \lambda(n),\ \lambda(n)\lambda(n+2),\ \lambda(p+2)\log p,\ \lambda(p-%
2)\log p,$$
where $p$ denotes always primes, $\mathcal{P}$ the set of all primes.
We have to note that while for $\lambda(n)$ we know the analogue of the Bombieri–Vinogradov theorem
(1.5)
$$\sum_{q\leq N^{\vartheta-\varepsilon}}\max_{a}\biggl{|}\sum_{\begin{subarray}{%
c}n\equiv a\pmod{q}\\
n\leq N\end{subarray}}\lambda(n)\biggr{|}\ll_{\varepsilon,A}\frac{N}{\log^{A}N}$$
with $\vartheta=1/2$ (the proof of Vaughan [Vau], Theorem 4, with $\mu(n)$ in place of $\lambda(n)$ can be easily modified to yield (1.5)), our knowledge about the other sequences is much more limited, since the following problems are still open (see [Cho], [Hil], [Iwa]).
Problem 1.
Is $\sum\limits_{n\leq x}\lambda(n)\lambda(n+2)=o(x)$ (see [Cho, (341), p. 96; the quantity $O(X)$ there is a misprint, it has to be replaced by $o(x)$]), or even whether we have an absolute constant $c$ such that
(1.6)
$$\biggl{|}\sum_{n\leq x}\lambda(n)\lambda(n+2)\biggr{|}<(1-c)x\ \text{ for }\ x%
>x_{0}.$$
Problem 2.
Are there infinitely many primes with $\lambda(p+2)=-1$ (or $\lambda(p-2)=-1$)?
Problem 2’.
Are there infinitely many primes with $\lambda(p+2)=1$ (or $\lambda(p-2)=1$)?
It may be worth to mention that the author succeeded to show very recently [Pin2] the existence of a positive even $d\leq 18$ such that $\lambda(p+d)=-1$ for infinitely many primes $p$.
We can more generally work with any fixed positive even integer $h$ in place of $2$, so the same argument works for the generalized twin prime conjecture too.
Theorem 1.
Suppose that with a $\vartheta=\vartheta_{1}>3/4$, the relations (1.2), (1.5), further the analogues of (1.5) with $\lambda(n)$ replaced by $\lambda(n)\lambda(n+h)$, $\lambda(p-h)\log p$ and $\lambda(p+h)\log p$ hold, where $h$ is any positive even integer.
Then $p+h$ is prime for infinitely many primes $p$.
The result can be proved with somewhat more effort under a slightly weaker condition for the distribution of the above sequences $(\vartheta\geq 0.7284)$.
Further we can give a lower estimate for the number of generalized twin primes up to $N$ which is just a constant factor weaker than the expected number
(1.7)
$$\mathfrak{S}_{0}(h)\frac{N}{\log^{2}N},\quad\mathfrak{S}_{0}(h):=\prod\limits_%
{p\mid h}\left(1-\frac{1}{p}\right)^{-1}\prod_{p\nmid h}\left(1-\frac{1}{(p-1)%
^{2}}\right).$$
Theorem 2.
Suppose that the conditions of Theorem 1 are valid for a $\vartheta\geq 0.7284$.
Then with an absolute constant $c$ we have for any even $h>0$
(1.8)
$$\#\{p\leq N;\ p,p+h\in\mathcal{P}\}\geq\frac{c\mathfrak{S}_{0}(h)N}{\log^{2}N}%
\ \text{ for }N>N_{1}.$$
The estimate (1.8) implies that as shown in [Zho], or in greater generality in [Pin1], the method of proof of Green–Tao [GT] can be adapted to this situation, yielding
Theorem 3.
Suppose the conditions of Theorem 1 for a $\vartheta_{1}\geq 0.7231$, that is, that all $5$ functions in (1.4) have distribution level $\vartheta_{1}$, with $2$ replaced by $h$.
Then for any even $h>0$ there are arbitrarily long arithmetic progressions such that $p+h$ is also prime for all elements of the progression.
2 Proof of Theorem 1
In the work [GPY] we introduced for $k$-element sets $\mathcal{H}=\{h_{i}\}^{k}_{i=1}$ the function
(2.1)
$$\Lambda_{R}(n;\mathcal{H},l):=\frac{1}{(k+l)!}\sum_{\begin{subarray}{c}d\leq R%
\\
d\mid P_{\mathcal{H}}(n)\end{subarray}}\mu(d)\left(\log\frac{R}{d}\right)^{k+l%
},\quad P_{\mathcal{H}}(n)=\prod^{k}_{i=1}(n+h_{i}),$$
which we will use now in the special case $k=2$, $\mathcal{H}=\{0,h\}$.
However, instead of $a_{n}=\Lambda^{2}_{R}(n)$ as in [GPY] we will weight now the integers with
(2.2)
$$b_{n}=a_{n}(1-\lambda(n))(1-\lambda(n+h))\geq 0,\quad a_{n}=\Lambda_{R}(n;%
\mathcal{H},l)^{2}.$$
The singular series
(2.3)
$$\mathfrak{S}(\mathcal{H})=\prod_{p}\left(1-\frac{\nu_{p}(\mathcal{H})}{p}%
\right)\left(1-\frac{1}{p}\right)^{-k},$$
where $\nu_{p}(\mathcal{H})$ denotes the number of residue classes occupied by $\mathcal{H}\,\text{\rm mod}\;p$, reduces now to $\mathfrak{S}_{0}(h)$, given in (1.7).
The $k$-tuple $\mathcal{H}$ is called in general admissible if $\nu_{p}=\nu_{p}(\mathcal{H})<p$ for all primes $p$, equivalently, if $\mathfrak{S}(\mathcal{H})\neq 0$.
We remark that for any admissible $k$-tuple $\mathcal{H}=\mathcal{H}_{k}$, hence also in our case $\mathcal{H}=\{0,h\}$ we have
(2.4)
$$\mathfrak{S}(\mathcal{H}_{k})\geq\prod_{p\leq 2k}\frac{1}{p}\prod_{p>k}\left(1%
-\frac{k}{p}\right)\left(1-\frac{1}{p}\right)^{-k}>c_{0}(k).$$
We quote from [GPY] as our first two lemmas Propositions 1 and 2 (see (2.14)–(2.15)), which will form the base of our argument.
We will restrict ourselves for the case $\mathcal{H}=\mathcal{H}_{1}=\mathcal{H}_{2}$, but keep the parameter $l$, which will be used in Section 3 to show the stronger Theorem 2.
We will use the notation $\theta(n)=\log p$ if $n=p\in\mathcal{P}$, $\theta(n)=0$ otherwise, $n\sim N$ for $n\in[N,2N)$, $C$ an absolute constant whose value may be different at different occurrences.
Lemma 1.
If $R\ll N^{1/2}(\log N)^{-C}$ then
(2.5)
$$\frac{1}{N}\sum_{n\sim N}\Lambda_{R}(n;\mathcal{H},l_{1})\Lambda_{R}(n;%
\mathcal{H},l_{2})=(\mathfrak{S}(\mathcal{H})+o(1)){l_{1}+l_{2}\choose l_{1}}%
\frac{(\log R)^{k+l_{1}+l_{2}}}{(k+l_{1}+l_{2})!}.$$
Lemma 2.
If
$R\ll N^{(\vartheta-\varepsilon)/2}$ then for any $h\in\mathcal{H}$ we have
(2.6)
$$\displaystyle\frac{1}{N}\sum_{n\sim N}\Lambda_{R}(n;\mathcal{H},l_{1})\Lambda_%
{R}(n;\mathcal{H},l_{2})\theta(n+h)=$$
$$\displaystyle=(\mathfrak{S}(\mathcal{H})+o(1)){l_{1}+l_{2}+2\choose l_{1}+1}%
\frac{(\log R)^{k+l_{1}+l_{2}+1}}{(k+l_{1}+l_{2}+1)!}.$$
We will need an analogous lemma for the sequences
(2.7)
$$f(n)=\lambda(n),\ \ \lambda(n)\lambda(n+h),\ \ \theta(n)\lambda(n+h),\ \ %
\lambda(n)\theta(n+h),$$
where we use the hypothesis that $f(n)$ satisfies the analogue of (1.5), that is,
(2.8)
$$\sum_{q\leq N^{\vartheta-\varepsilon}}\max_{a}\biggl{|}\sum_{\begin{subarray}{%
c}n\equiv a\pmod{q}\\
n\leq N\end{subarray}}f(n)\biggr{|}\ll_{\varepsilon,A}\frac{N}{\log^{A}N}.$$
Lemma 3.
Suppose (2.8) and $f(n)\ll(\log N)^{C}$.
If $A>0$ arbitrary, $R\ll N^{(\vartheta-\varepsilon)/2}$, then we have for any $\mathcal{H}=\{h_{i}\}^{k}_{i=1}$
(2.9)
$$S_{f}(N)=\frac{1}{N}\sum_{n\sim N}\Lambda_{R}(n;\mathcal{H},l_{1})\Lambda_{R}(%
n;\mathcal{H},l_{2})f(n)\ll\frac{N}{\log^{A}N},$$
where the constant implied by the $\ll$ symbol depends on $k,l_{i},C,A,\varepsilon$.
Proof.
For any squarefree $m$ and $\mathcal{H}=\{h_{i}\}^{k}_{i=1}$ the number $\nu_{m}=\nu_{m}(\mathcal{H})$ of the solution of the congruence
(2.10)
$$\prod^{k}_{i=1}(n+h_{i})\equiv 0\pmod{m}$$
satisfies by the Chinese remainder theorem
(2.11)
$$\nu_{m}=\prod_{p\mid m}\nu_{p}\leq k^{\omega(m)}=d_{k}(m),$$
where $\omega(m)$ denotes the number of prime factors of $m$, $d_{k}(m)$ the number of ways to write $m$ as a product of $k$ integers.
Interchanging the order of summation we can write $S_{f}(N)$ with the notation $K=2k+l_{1}+l_{2}$ as
(2.12)
$$\displaystyle\frac{1}{N}\sum_{d\leq R}\sum_{e\leq R}\frac{\mu(d)\mu(e)\left(%
\log\frac{R}{d}\right)^{k+l_{1}}\left(\log\frac{R}{d}\right)^{k+l_{2}}}{(k+l_{%
1})!(k+l_{2})!}\sum_{\begin{subarray}{c}n\sim N\\
\mid P_{\mathcal{H}}(n)\end{subarray}}f(n)$$
$$\displaystyle\ll\frac{\log^{K}R}{N}\sum_{q\leq R^{2}}\biggl{(}\sum_{q=[d,e]}1%
\biggr{)}\nu_{q}E_{N}(q),$$
where (for $q\leq N$)
(2.13)
$$E_{N}(q):=\max_{a}\biggl{|}\sum_{\begin{subarray}{c}n\sim N\\
n\equiv a\pmod{q}\end{subarray}}f(n)\biggr{|}\ll\frac{N(\log N)^{C}}{q}.$$
Using our hypotheses we obtain as in (9.13) of [GPY]
(2.14)
$$\displaystyle S_{f}(N)\ll\frac{\log^{K}R}{N}\biggl{(}\sum_{q\leq R^{2}}\frac{d%
_{3k}(q)^{2}}{q}\sum_{q\leq R^{2}}qE^{2}_{N}(q)\biggr{)}^{1/2}\ll$$
$$\displaystyle\ll\frac{\log^{K}R}{N}\left((\log N)^{9k^{2}}N(\log N)^{C}\frac{N%
}{\log^{A}N}\right)^{1/2}\ll(\log N)^{K+(9k^{2}+C-A)/2}.$$
∎
Using the notation
(2.15)
$$B_{0}:=B_{0}(R,\mathcal{H},k,l)={2l\choose l}\frac{(\log R)^{k+2l}}{(k+2l)!}%
\mathfrak{S}(\mathcal{H}),$$
we have by Lemmas 1 and 3 in the special case $\mathcal{H}_{1}=\mathcal{H}_{2}=\{0,h\}$, $k=2$, $l_{1}=l_{2}=l=0$
(2.16)
$$B:=\sum_{n\sim N}b_{n}\sim\sum_{n\sim N}a_{n}\sim B_{0}:=\frac{\mathfrak{S}_{0%
}(h)N\log^{2}R}{2}.$$
On the other hand we obtain from Lemmas 2 and 3 with the same choice $\mathcal{H}=\{0,h\}$, $l_{1}=l_{2}=0$
(2.17)
$$\displaystyle P^{*}:$$
$$\displaystyle=\sum_{n\sim N}b_{n}(\theta(n)+\theta(n+h))=$$
$$\displaystyle=2\sum_{n\sim N}a_{n}\bigl{\{}(1-\lambda(n+h))\theta(n)+(1-%
\lambda(n))\theta(n+h)\bigr{\}}\sim 4\cdot 2B_{0}\cdot\frac{\log R}{3}.$$
In order to have at least one prime pair $p,p+h$ with $p\in[N,2N)$ we need to show with $R=N^{(\vartheta-\varepsilon)/2}$
(2.18)
$$P^{*}-B\log(3N)>0,$$
which is really true if
(2.19)
$$\frac{8}{3}\frac{\vartheta-\varepsilon}{2}>1+\varepsilon.$$
This is trivially true for any fixed $\vartheta>3/4$ if $\varepsilon$ is sufficiently small and $N$ sufficiently large.
This proves Theorem 1.
3 Proof of Theorem 2
The proof of Theorem 2 needs a relatively simple modification, which allows to weaken slightly the constraint $\vartheta>3/4$.
This can be achieved – similarly to Section 3 of [GPY] – by applying a linear combination of the weights $\Lambda_{R}(n;\mathcal{H},l)$ with $l=0$ and $l=1$.
More precisely we define
(3.1)
$$a^{\prime}_{n}:=a^{\prime}_{n}(\mathcal{H};u)=\left(\Lambda_{R}(n;\mathcal{H},%
0)+\frac{u(k+1)}{\log R}\Lambda_{R}(n;\mathcal{H},1)\right)^{2},$$
where $u$ is a real parameter to be chosen optimally later.
In our case $k=2$ we obtain with the notation $B_{0}$ in (2.16) for $\mathcal{H}=\{0,h\}$ from Lemmas 1 and 2 in this case with the analogue $b^{\prime}_{n}=a^{\prime}_{n}(1-\lambda(n))(1-\lambda(n+h))$:
(3.2)
$$B^{\prime}(N,\mathcal{H},u):=\sum_{n\sim N}b^{\prime}_{n}\sim\sum_{n\sim N}a^{%
\prime}_{n}\sim B_{0}\left(1+2u+2u^{2}\cdot\frac{3}{4}\right).$$
The analogue of the evaluation of (2.17) is now
(3.3)
$$\displaystyle P^{\prime}:$$
$$\displaystyle=\sum_{n\sim N}b^{\prime}_{n}(\theta(n)+\theta(n+h))\sim$$
$$\displaystyle\sim 2\sum_{n\sim N}a^{\prime}_{n}(\theta(n)+\theta(n+h))\sim$$
$$\displaystyle\sim 4B_{0}\log R\left(\frac{2}{3}+\frac{6u}{4}+\frac{18u^{2}}{20%
}\right).$$
This means that we have to assure
(3.4)
$$P^{\prime}-B^{\prime}\log(3N)>0$$
which will hold if we can find a $u$ with
(3.5)
$$g_{u}(\vartheta)=\vartheta\left(\frac{4}{3}+3u+\frac{9u^{2}}{5}\right)-\biggl{%
(}1+2u+\frac{3u^{2}}{2}\biggr{)}>0$$
if we choose $\varepsilon$ sufficiently small.
The optimal choice for $u$ is $u=u_{0}=\bigl{(}\sqrt{34}-2\bigr{)}/9$, which yields a fixed positive lower bound $c_{0}$ for $g(\vartheta)=g_{u_{0}}(\vartheta)$ if
(3.6)
$$\vartheta\geq\vartheta_{1}=0.7231.$$
This is enough to obtain a weighted estimate for the number of generalized twin primes in $[N,2N)$
(3.7)
$$\frac{1}{N}\sum_{\begin{subarray}{c}n\sim N\\
n,n+h\in\mathcal{P}\end{subarray}}a^{\prime}_{n}\log(3N)\geq c_{1}\mathfrak{S}%
_{0}(h)\log^{3}R.$$
However, if $n$ and $n+h$ are both primes then for $\mathcal{H}=\{0,h\}$ clearly
(3.8)
$$\Lambda_{R}(n;\mathcal{H},l)=\frac{1}{(2+l)!}(\log R)^{k+l}=\frac{1}{(2+l)!}(%
\log R)^{2+l},$$
consequently
(3.9)
$$a_{n}(\mathcal{H},u_{0})=\left(\frac{1+u_{0}}{2}\right)^{2}\log^{4}R,$$
which by (3.6) and (3.7) leads to the estimate
(3.10)
$$\#\bigl{\{}p\in[N,2N),\ p,p+h\in\mathcal{P}\bigr{\}}\geq\frac{c_{2}\mathfrak{S%
}_{0}(h)N}{\log R\log N}\geq\frac{c_{3}\mathfrak{S}_{0}(h)N}{\log^{2}N}.$$
Remark.
If we are allowed to choose a bigger $\vartheta$, then the lower estimate (3.10) will improve but we do not reach the expected number corresponding to $c_{3}=1$ even supposing $\vartheta=1$, the Elliott–Halberstam conjecture.
References
[Cho]
S. Chowla,
The Riemann Hypothesis and Hilbert’s tenth problem,
Gordon and Breach, New York, 1965.
[EH]
P. D. T. A. Elliott, H. Halberstam,
A conjecture in prime number theory,
Symposia Mathematica 4 INDAM, Rome,
59–72, Academic Press, London, 1968/69.
[GPY1]
D. A. Goldston, J. Pintz, C. Y. Yıldırım,
Primes in tuples I,
Ann. of Math. (2) 170
(2009), no. 2, 819–862.
[Hil]
A. J. Hildebrand,
Erdős’ problems on consecutive integers,
Paul Erdős and his Mathematics I, Bolyai Society Mathematical
Studies 11, 305–317, Budapest, 2002.
[Iwa]
H. Iwaniec,
Prime numbers and $L$-functions,
International Congress of Mathematicians, Vol. I,
279–306, Eur. Math. Soc., Zürich, 2007.
[Pin1]
J. Pintz, Are there arbitrarily long arithmetic progressions
in the sequence of twin primes?, preprint, arXiv:1002.2899
[Pin2]
J. Pintz,
An approximation to the twin prime conjecture and the parity
phenomenon, preprint.
[Vau]
R. C. Vaughan, An Elementary Method in Prime Number Theory,
Recent progress in analytic number theory, Vol. 1 (Durham, 1979),
341–348, Academic Press, London–New York, 1981.
János Pintz
Rényi Mathematical Institute of the Hungarian Academy
of Sciences
Budapest
Reáltanoda u. 13–15
H-1053 Hungary
E-mail: pintz@renyi.hu |
No detection of large-scale magnetic fields at the surfaces of Am and HgMn stars††thanks: Based on data obtained using the Télescope Bernard Lyot at Observatoire du Pic du Midi, CNRS/INSU and Université de Toulouse, France.
M. Aurière
1Laboratoire d’Astrophysique de Toulouse-Tarbes, CNRS, Université de Toulouse, 57 Avenue d’Azereix, 65008 Tarbes, France
11email: ligniere@ast.obs-mip.fr11email: alain.hui@ast.obs-mip.fr11email: donati@ast.obs-mip.fr11email: petit@ast.obs-mip.fr11email: roudier@ast.obs-mip.fr11email: stheado@ast.obs-mip.fr
1michel.auriere@ast.obs-mip.fr
G.A. Wade
2Department of Physics, Royal Military College of Canada,
PO Box 17000, Station ’Forces’, Kingston, Ontario, Canada K7K 4B4
2gregg.wade@rmc.ca
F. Lignières
1Laboratoire d’Astrophysique de Toulouse-Tarbes, CNRS, Université de Toulouse, 57 Avenue d’Azereix, 65008 Tarbes, France
11email: ligniere@ast.obs-mip.fr11email: alain.hui@ast.obs-mip.fr11email: donati@ast.obs-mip.fr11email: petit@ast.obs-mip.fr11email: roudier@ast.obs-mip.fr11email: stheado@ast.obs-mip.fr
1michel.auriere@ast.obs-mip.fr
A. Hui-Bon-Hoa
1Laboratoire d’Astrophysique de Toulouse-Tarbes, CNRS, Université de Toulouse, 57 Avenue d’Azereix, 65008 Tarbes, France
11email: ligniere@ast.obs-mip.fr11email: alain.hui@ast.obs-mip.fr11email: donati@ast.obs-mip.fr11email: petit@ast.obs-mip.fr11email: roudier@ast.obs-mip.fr11email: stheado@ast.obs-mip.fr
1michel.auriere@ast.obs-mip.fr
J.D. Landstreet
3Department of Physics & Astronomy, The University of Western Ontario, London, Ontario, Canada, N6A 3K7
3jlandstr@uwo.ca
4Armagh Observatory, College Hill, Armagh, Northern Ireland BT61 9DG
4jls@arm.ac.uk
I. Iliev
5Institute of Astronomy, Bulgarian Academy of Sciences, 72 Tsarigradsko shose, 1784 Sofia, Bulgaria
5iliani@astro.bas.bg
J.-F. Donati
1Laboratoire d’Astrophysique de Toulouse-Tarbes, CNRS, Université de Toulouse, 57 Avenue d’Azereix, 65008 Tarbes, France
11email: ligniere@ast.obs-mip.fr11email: alain.hui@ast.obs-mip.fr11email: donati@ast.obs-mip.fr11email: petit@ast.obs-mip.fr11email: roudier@ast.obs-mip.fr11email: stheado@ast.obs-mip.fr
1michel.auriere@ast.obs-mip.fr
P. Petit
1Laboratoire d’Astrophysique de Toulouse-Tarbes, CNRS, Université de Toulouse, 57 Avenue d’Azereix, 65008 Tarbes, France
11email: ligniere@ast.obs-mip.fr11email: alain.hui@ast.obs-mip.fr11email: donati@ast.obs-mip.fr11email: petit@ast.obs-mip.fr11email: roudier@ast.obs-mip.fr11email: stheado@ast.obs-mip.fr
1michel.auriere@ast.obs-mip.fr
T. Roudier
1Laboratoire d’Astrophysique de Toulouse-Tarbes, CNRS, Université de Toulouse, 57 Avenue d’Azereix, 65008 Tarbes, France
11email: ligniere@ast.obs-mip.fr11email: alain.hui@ast.obs-mip.fr11email: donati@ast.obs-mip.fr11email: petit@ast.obs-mip.fr11email: roudier@ast.obs-mip.fr11email: stheado@ast.obs-mip.fr
1michel.auriere@ast.obs-mip.fr
S. Théado
1Laboratoire d’Astrophysique de Toulouse-Tarbes, CNRS, Université de Toulouse, 57 Avenue d’Azereix, 65008 Tarbes, France
11email: ligniere@ast.obs-mip.fr11email: alain.hui@ast.obs-mip.fr11email: donati@ast.obs-mip.fr11email: petit@ast.obs-mip.fr11email: roudier@ast.obs-mip.fr11email: stheado@ast.obs-mip.fr
1michel.auriere@ast.obs-mip.fr
(Received ??; accepted ??)
Key Words.:
stars: chemically peculiar – stars: magnetic field
††offprints: M. Aurière, michel.auriere@ast.obs-mip.fr
Abstract
Context:
Aims:We investigate the magnetic dichotomy between Ap/Bp and other A-type stars by carrying out a deep spectropolarimetric study of Am and HgMn stars.
Methods:Using the NARVAL spectropolarimeter at the Télescope Bernard
Lyot (Observatoire du Pic du Midi, France), we obtained high-resolution circular polarisation spectroscopy of 12 Am stars and 3 HgMn stars.
Results:Using Least Squares Deconvolution (LSD), no magnetic field is detected in any of the 15 observed stars. Uncertaintiies as low as 0.3 G (respectively 1 G) have been reached for surface-averaged longitudinal magnetic field measurements for Am (respectively HgMn) stars.
Conclusions:Associated with the results obtained previously for Ap/Bp stars, our study confirms the existence of a magnetic dichotomy among A-type stars. Our data demonstrate that there is at least one order of magnitude difference in field strength between Zeeman detected stars (Ap/Bp stars) and non Zeeman detected stars (Am and HgMn stars). This result confirms that the spectroscopically-defined Ap/Bp stars are the only A-type stars harbouring detectable large-scale surface magnetic fields.
1 Introduction
The spectroscopically-selected “magnetic Ap/Bp stars” (hereafter Ap/Bp stars), corresponding to about 5% of main sequence (MS) A and B stars (Wolff 1968), are known to host relatively strong, ordered magnetic fields. On the other hand, the remaining 95% of MS stars at these spectral types appear to have no detectable magnetic field (with the exception of the very small magnetic field recently detected on Vega by Lignières et al. 2009, Petit et al. 2010). This is the so-called magnetic dichotomy. Using the MuSiCoS and NARVAL spectropolarimeters, Aurière et al. (2007) studied the weak part of the magnetic field distribution of Ap/Bp stars. They found, as had previously been assumed, that all confidently spectroscopically-classified Ap/Bp stars, when observed with sufficient precision and tenacity, show evidence for organised magnetic fields with model dipole polar strength stronger than about 300 G. However, demonstrating the existence of a magnetic field dichotomy relies not only on establishing the universal presence of large scale fields in Ap/Bp stars, but also showing confidently that no such fields are detectable in the non-Ap/Bp stars. The most recent
sensitive surveys of apparently non magnetic A and B stars have led to non-detection of magnetic fields at the level of a few tens of G (Shorlin et al. 2002, Bagnulo et al. 2006). Shorlin et al. (2002) used the high-resolution MuSiCoS spectropolarimeter to search for Stokes $V$ Zeeman signatures in spectra of 63 non-Ap/Bp intermediate-mass stars, finding no evidence of magnetic fields, with a median longitudinal field ($B_{\ell}$) formal error of just 22 G. Bagnulo et al. (2006) used the low-resolution FORS1 spectropolarimeter to measure magnetic fields of a large sample of intermediate-mass stars in open clusters. In their sample of 138 non-Ap/Bp stars, no magnetic field was detected, with a median longitudinal field error bar of 136 G. To refine our understanding of the dichotomy, using the possibilities of new instruments (Donati & Landstreet 2009), in this study we employ NARVAL to observe bright slow rotators among the Am and HgMn stars previously studied by Shorlin et al (2002).
In the following, we will describe our observations (Sect. 2) and our results for each category of stars (Sect. 3), then give our discussion of the magnetic dichotomy and our conclusions.
2 Observations and reduction
2.1 Observations with NARVAL
The observations took place in March 2007, at the 2-m
Télescope Bernard Lyot (TBL) of Pic du
Midi Observatory with the NARVAL high-resolution spectropolarimeter (Aurière 2003). In operation since December 2006, NARVAL is a copy of ESPaDOnS
installed at CFHT at the end of 2004
(Donati et al. 2006). NARVAL is a fiber–fed echelle spectropolarimeter
with which the whole spectrum from 370 nm to 1000 nm is recorded in each
exposure. The 40 grating orders are aligned on the CCD frame using two cross-disperser prisms.
NARVAL was used in polarimetric mode with a spectral resolution of
about 65000. Stokes $I$ (unpolarised) and Stokes $V$ (circular polarisation)
parameters were obtained by means of 4 sub-exposures between which the
retarders (Fresnel rhombs) were rotated in order to exchange the beams
in the whole instrument and to reduce spurious polarization signatures.
We aimed to get long exposures, up to 6400s, on our bright targets in order to be able to detect ultra-weak or complex magnetic fields. In order to avoid saturation of the CCD we made short sub exposures (e.g. 4 or 8 second subexposures for each Stokes $V$ series in the case of Sirius).
2.2 Reduction and magnetic field detection
During the technical tests and science demonstration time, magnetic and non magnetic stars were observed
which showed that NARVAL works properly and is 30 times more efficient than the previous instrument, MuSiCoS
(Baudrand & Böhm 1992, Donati et al. 1999), which was used by Shorlin et al. (2002). Since then, a great number of new results have been obtained that confirm the high scientific efficiency of ESPaDOnS and NARVAL (e.g. in Donati & Landstreet 2009).
The extraction of the spectra was done using Libre-ESpRIT (Donati et al. 1997),
a fully automatic reduction package installed at the TBL. In
order to carry out the Zeeman analysis, Least-Squares Deconvolution analysis
(LSD, Donati et al. 1997) was applied to all observations. We used line masks with solar abundances, $\log g$ = 4, temperatures close to the values given by Shorlin et al. (2002; see our Table 1), and included lines with a central depth greater than 10% of the continuum. For our sample, this method enabled us to average from about 500 (highest temperature HgMn star) to about 5000 (coolest Am star) lines and to obtain Stokes $V$ profiles with signal-to-noise ratio (S/N) increased by a factor of about 10 to 40. We performed a statistical test for the detection of Stokes $V$ Zeeman signatures: the reduced $\chi^{2}$ statistic is computed for zero signal in the Stokes $V$ profile, both inside and outside the spectral line (Donati et al. 1997). The statistics are then converted into detection probabilities (false alarm probability). Also included in the output are “diagnostic null” spectra $N$ (combinations of sub-exposures in which real $V$ signatures should cancel out), which are in principle featureless, and therefore serve to diagnose the presence of spurious contributions to the Stokes $V$ spectra.
We then computed the longitudinal magnetic field $B_{l}$ in G, using the
first-order moment method adapted to LSD profiles (Rees and Semel 1979, Donati et al. 1997, Wade et al. 2000). The integration range used to compute $B_{l}$ corresponds to the first and last point in the Stokes $I$ profile for which the flux was lower than 15$\%$ of the maximum depth, except for the SB2 stars for which it was optimized manually.
For a few selected stars we constructed line masks that matched the stellar spectrum in detail, by modifying individual line depths in the mask, using data provided by VALD (Kupka et al. 1999). While these custom masks naturally provided a better representation of the Stokes $I$ and $V$ spectra, they did not result in any change in the detection diagnosis, or any significant improvement in the longitudinal field upper limit. As a consequence, all results presented here correspond to solar abundance line masks.
Finally, we measured for each star (generally the primary) the radial velocity $RV$ from the averaged LSD Stokes $I$ profile, using a gaussian fit. The long term stability of NARVAL is about 30 m/s (e.g. Aurière et al. 2009a) but the absolute uncertainty of individual measurements relative to the local standard of rest is about 1 km s${}^{-1}$.
Table 1 gives for each star its $V$ magnitude, spectral class, mask temperature used, $v\sin i$, and, for each observation, the date, HJD (corresponding to the $RV$ measurement), number of exposures and total exposure time, $RV$, and the inferred longitudinal magnetic field with its standard error in G.
3 Results of the present survey
3.1 The sample
Our aim is to search for magnetic fields on non Ap/Bp A-type stars in order to establish definitively the gap of the magnetic dichotomy. Shorlin et al. (2002) showed the great influence of the value of $v\sin i$ on the sensitivity of a magnetic survey using high-resolution spectropolarimetry. In order to reduce the errors in our survey, we choose here to observe the most promising objects already observed by Shorlin et al. (2002).
Am stars are frequently found in close binaries (Abt & Levy 1985), likely because tidal interactions in such systems slow stellar rotation and thereby reduce rotational mixing. This is also the case for HgMn stars (Ryabchikova 1998). This property does not hamper our study, but the interesting cases of the SB2 stars 32 Vir and $\lambda$ Vir are discussed in detail in Sect.3.2. No Zeeman detection was obtained for any of our sample stars, since false alarm probability was always greater than 10${}^{-3}$, apart from the case of 32 Vir which is discussed in Sect. 3.2. The results are discussed further for Am and HgMn stars in Sect. 3.2 and 3.3 respectively.
3.2 Am stars
Am stars are cool A-type stars that can be considered as ”ordinary” slowly rotating A-stars (Takeda et al. 2008). A large number of Am stars deserve a sensitive magnetic survey with NARVAL; we observed 12 of them, among them the bright star Sirius. Our main selection criterion for the stars observed was low $v\sin i$, which we required to be smaller than 50 km s${}^{-1}$, and is often much smaller.
Our sample stars are generally on the main sequence, but two of them, 32 Vir and 22 Boo, have already left it. The main result of our study (no Zeeman detection and very low upper limits for a possible surface-averaged longitudinal magnetic field) is visible on Table 1, but we make comments about some stars below.
Sirius:
Besides being the brightest star in the sky after the Sun, Sirius is a hot Am star. Observing it enabled us to reach the highest precision obtained in our survey, namely 0.32 G for our $B_{\ell}$ determination. With 32 Stokes $V$ series, we got a total exposure of 1024 s. Fig.1 shows the composite LSD profiles. The huge enlargement of Stokes $V$ and Stokes $N$ show that the amplitude of the noise is currently smaller than $10^{-5}I_{c}$ . A kind of flat feature appears on Stokes $V$ profile at the position of the absorption line in the intensity profile. It is not significant with respect to the LSD detection statistical test. Splitting our spectra into two equal subsets show that this feature is more visible on our second subset, and is probably due to noise. No magnetic field is therefore detected on Sirius and the corresponding $B_{l}$ value is $0.10\pm 0.32$ G ($1\sigma$). Equally small or even smaller errors in Stokes $V$ profiles and $B_{l}$ measurements were obtained with NARVAL in the case of the normal A-star Vega (Lignières et al. 2009, Petit et al. 2010) and the red giant Pollux (Aurière et al. 2009a), and sub-G magnetic fields could be detected at a significant level in these stars.
$\alpha$ Gem B (Castor B):
Castor is a multiple system composed of three visual stars, each of which is by itself a spectroscopic binary. Castor A and Castor C were out of the slit during the NARVAL observations. Castor A and B have been now resolved in X-rays and this observation shows that the late-type secondaries within each spectroscopic binary are the sites of the X-ray production (Stelzer & Burwitz 2003). Our non-detection of a magnetic field confirms the absence of magnetic activity at the surface of the A-type star Castor B.
$\lambda$ UMa:
Our observations confirm that the 3 $\sigma$ detection ($66\pm 22$ G) of a magnetic field by Shorlin et al. (2002) is spurious, as suspected by those authors. We have improved the precision of field measurement by a factor greater than 7 with respect to the MuSiCoS result, though the corresponding error on $B_{\ell}$ is one of the least precise in this paper, 2.91 G, due to the relatively large $v\sin i$ of 50 km s${}^{-1}$.
$\theta$ Leo:
This star is considered to be a hot Am star (Smith 1974, Adelman 2004) and is a standard of radial velocity (Morse et al. 1991). Because of its moderate $v\sin i$ = 23.5 km s${}^{-1}$, the field measurement is one of the most accurate in the survey ($\sigma=$ 1.39 G) for early A-type stars.
32 Vir:
32 Vir is a well known SB2 whose primary appears to be a $\rho$ Puppis-type $\delta$ Scuti star, i.e. an evolved pulsating Am star (Mitton & Stickland 1979). Fig.2 shows the LSD Stokes profiles derived from a total integration of 3600s obtained on 12 March 2007. The Stokes $I$ LSD profile easily resolves the two components, the Am star corresponding to the sharp line. In both the Stokes $V$ and null polarization $N$ profiles, a small signal is visible at the RV of the Am star, and the LSD statistics gives a false alarm probability smaller than 10${}^{-3}$. Since this detection is obtained also, even more strongly, on the null polarization $N$ profile, it is suspected to be spurious. However weak magnetic fields in subgiant stars have been detected in the course of another magnetic survey with NARVAL (Aurière et al. 2009b). We therefore observed 32 Vir again in the same conditions on 13 March 2007 and on 02 April 2008, and got the same result: a weak signal was again visible on Stokes $V$ and $N$ profiles. Now 32 Vir is both a binary star and a pulsating star (Lampens & Boffin 2000). Bertiau (1957) derived an orbit with a period of 38.3 days and a semi amplitude of 48 km s${}^{-1}$. As a $\delta$ Scuti star, 32 Vir has a period of about 0.07 day (Bartolini et al. 1983, Kurtz et al. 1976). The RV amplitude variation due to pulsations is unknown, but could be similar to that observed for $\rho$ Puppis itself, i.e. 8.6 km s${}^{-1}$ (Mathias et al. 1997). These rapid $RV$ variations due to the binary and pulsating status of the star are expected to induce shifts in $RV$ between the LSD profiles of the four (900 s) sub-exposures of up to about 1 km/s. Such large $RV$ shifts were actually measured on our data, which can lead to detection of spurious polarization signals (Donati et al. 1997). Because of the different time-lags between combinations of sub-exposures used for getting $N$ and Stokes $V$ profiles, the spurious signal is expected to be stronger on the former than on the latter profiles. This process is probably the reason for the signal observed on the 3 dates.
$\lambda$ Vir:
This star is a well-known double lined spectroscopic Am binary: both stars are very similar in chemical abundances but the primary component is broad-lined and the secondary is sharp-lined (Zhao et al. 2007). Our NARVAL observations enabled us to resolve the two components on our LSD Stokes $V$ profiles, as already presented by Shorlin et al. (2002). In Table 1 we show that neither of the two components indicates a Zeeman detection and we have included individual $B_{\ell}$ measurements for each of the two components.
22 Boo:
22 Boo is considered to be an Am star which has already left the main sequence (Burkart et al. 1980, Bertet 1990). This is a particularly interesting object for a magnetic survey since a dynamo driven magnetic field may appear during the subgiant phase (Aurière et al. 2009b). However no Zeeman detection occur at a level of $\sigma=2.18$ G for $B_{\ell}$.
3.3 HgMn stars
The HgMn stars are generally considered as having the most stable atmospheres among intermediate mass stars (Vauclair & Vauclair, 1982). However, some binary HgMn stars have been shown to display spectroscopic variations (Adelman et al. 2002, Kochukhov et al. 2005, Hubrig et al. 2006a, Briquet et al. 2010). The non-uniform surface abundances invoked to explain these variations appear to evolve with time (Kochukhov et al. 2007). It has been proposed that they could host strong magnetic fields of peculiar topology (Hubrig et al. 2006b, 2008), and that such fields could be responsible for the surface structures. Wade et al. (2006) performed a sensitive magnetic study of the brightest HgMn star, $\alpha$ And, and placed a 3$\sigma$ upper limit of about 100 G on the possible presence of any undetected pure dipolar, quadrupolar or octupolar surface magnetic fields. Because of the rather large $v\sin i$ (52 km s${}^{-1}$), the 1$\sigma$ error bars reached 6 G at the smallest, even with ESPaDOnS. We have observed here with NARVAL 3 of the brightest of the HgMn stars having $v\sin i$ 5 times smaller than $\alpha$ And. The resulting uncertainties of $B_{\ell}$ are finally 2 to 4 times smaller than those obtained for $\alpha$ And.
$\kappa$ Cnc:
For this classical HgMn star (Zöchling & Muthsam, 1987), our non detection with a 1 $\sigma$ error of 3 G for the longitudinal magnetic field confirms the result of Shorlin et al. 2002, that a strong surface magnetic field, as suggested by older observations, is not present.
$\iota$ CrB:
Observations of this star with the Gecko spectrograph attached to the Canada-France-Hawaii Telescope have resolved the two components of the spectroscopic binary (Dubaj et al. 2005). The $v\sin i$ of the HgMn component was measured to be only about 1 km s${}^{-1}$; Shorlin et al. (2002) were only able to find an upper limit of $v\sin i$ $<$ 10 km s${}^{-1}$. Our measurement of this star has the best precision obtained for the HgMn stars of our sample, about 1 G. Figure 3 shows the Stokes $V$ and Stokes $I$ LSD profiles for $\iota$ CrB.
$\phi$ Her:
This star is a spectroscopic binary which has recently been resolved (Zavala et al. 2007) and for which the mass of the CP star has been refined (Torres 2007). No field is detected, with an longitudinal field uncertainty of about 2 G.
4 The magnetic dichotomy
No Zeeman detection was obtained for any of the 15 stars of our sample, although we have achieved a precision improvement of more than one order of magnitude with respect to the work of Shorlin et al. (2002). Although we have obtained only one observation for the majority of the stars of our sample, the non-detection of significant Stokes $V$ signatures is a strong negative result because magnetic configurations can produce detectable $V$ signatures through the line profile even for zero longitudinal magnetic field. The observation of the crossover effect requires non-negligible rotational Doppler broadening (Mathys 1995), but it could be observed in the case of HN And ($vsini$ = 2 km s${}^{-1}$, Aurière et al. 2007), and therefore could be observed on all our stars apart from $\iota$ CrB. Error bars in the range of 0.3 to 3 G have been obtained for our measurements of the surface-average longitudinal magnetic fields and can therefore be used to set upper limits of this component of the magnetic field of about 10 G (3 $\sigma$). Table 1 of Aurière et al. (2007) shows that for the weak magnetic Ap/Bp stars, $|B_{\ell}|^{max}$ is generally above 100 G, i.e. about 10 times stronger than the present upper limit. Therefore, a very significant gap of at least one order of magnitude is now established between upper limits of fields that might be present in non-detected Am/HgMn stars and the fields consistently detected in Ap/Bp stars.
To interpret this result in term of magnetic intensity, some assumption has to be made for the magnetic topology. Taking into account the results of Aurière et al. (2007) who deduced the existence of a threshold magnetic field of about 300 G at the surface of Ap/Bp stars, and for geometrical configurations similar to those observed in weakly Ap/Bp stars, large scale magnetic fields with dipole field strength greater than about 30 G are not present at the surface of Am and HgMn stars. Moreover, the high-resolution spectropolarimetric techniques used in this study have been shown to be sensitive to both the large-scale (e.g. Aurière et al. 2009a) and smaller-scale (e.g. Petit et al. 2004) magnetic fields of active late-type stars. While there is certainly a spatial resolution limit to this sensitivity, the very high precision obtained in our survey definitely does not support previous reports of strong, complex fields in Am (Lanz & Mathys 1993) and HgMn stars (Hubrig et al. 2006b, 2008).
The report of a weak magnetic field (of about one G) in Vega (Lignières et al. 2009, Petit et al. 2010) is consistent with the existence of a magnetic dichotomy in the A-type stars. The instability scenario of Aurière et al. (2007) gives a possible explanation of this gap. The Ap/Bp stars are those for which the surface magnetic field is strong enough to resist to differential rotation and instabilities such as the Tayler instability (Spruit 1999). Conversely, stars with a large scale magnetic field of lower strength are subjected to instabilities that will strongly reduce their surface-averaged longitudinal field through cancellation effects. This dichotomy between stable and unstable large scale field configurations naturally leads to a gap in the values of the longitudinal fields.
5 Conclusion
Our limited survey of 15 A-type star of peculiarity types other than Ap/Bp shows that none of them appear to host a large scale magnetic field having a surface-averaged longitudinal magnetic field of more than 3 G. Taken together with the result of Aurière et al. (2007), who showed the existence of a magnetic threshold of about 300 G for dipole strength in Ap/Bp stars, this result confirms the existence of a magnetic dichotomy, and shows that it corresponds to a gap of more than one order of magnitude in field strength. In fact, up to now a magnetic field has been detected by spectropolarimetry for a non Ap/Bp star only in Vega, and the surface averaged-longitudinal magnetic field appears to be smaller than one G (Lignières et al. 2009, Petit et al. 2010). Our result can be simply explained by the instability scenario described in Aurière et al. (2007).
Acknowledgements.
This research has made use of databases operated by CDS, Strasbourg, France, and of VALD (Vienna, Austria). GAW and JDL acknowledge Discovery Grant support from the Natural Sciences and Engineering Research Council of Canada (NSERC). II acknowledges support from Bulgarian NSF grants D002-85 and D002-362.
References
(1)
Abt H.A., Levy S.G. 1985, ApJS, 59, 229
(2)
Adelman S.J., Gulliver A.F., Kochukhov O., Ryabchikova T.A. 2002, ApJ, 575, 449
(3)
Adelman, S.J. 2004, in ”The A-Star Puzzle”, J Zverko, J. Ziznovsky, S.J. Adelman & W.W.Weis, eds, 1
(4)
Aurière M., 2003 in ”Magnetism and Activity of the Sun and Stars”, Eds J. Arnaud and N. Meunier, EAS Publ. Series 9,105
(5)
Aurière M, Wade G.A. , Silvester J., Lignières et al. 2007, A&A, 475, 1053
(6)
Aurière M, Wade G.A. , Konstantinova-Antova R. et al. 2009a, A&A, 504, 231
(7)
Aurière M, Konstantinova-Antova R., Petit P. et al. 2009b, in Cosmic Magnetic fields: From Planets, to Stars and Galaxies, K.G. Strassmeier, A.G. Kosovichev & J.E. Beckman, eds, 431
(8)
Bagnulo S., Landstreet J.D., Mason E., Andretta V., et al. 2006, A&A, 450, 777
(9)
Bartolini C., Grilli F., Parmeggiani G., Piccioni A., Silveri P. 1983, A&ASS, 53, 139
(10)
Baudrand J., Böhm T. 1992, A&A, 259, 711
(11)
Berthet S. 1990, A&A, 227, 156
(12)
Bertiau F.C. 1957, ApJ, 125, 696
(13)
Burkhart C., Van’t Veer C., Coupry M.F. 1980, A&A, 92, 132
(14)
Briquet M., Korhonen H., González J.F., Hubrig S., Hackman T. T. 2010, A&A, 511, in press
(15)
Donati J.-F., Semel M., Carter B.D., Rees D.E., Cameron A.C. 1997, MNRAS, 291, 658
(16)
Donati J.-F., Catala C., Wade G.W., et al. 1999, A&AS, 134, 149
(17)
Donati J.-F., Landstreet J.D. 2009, ARA&A, 47, 333
(18)
Donati J.-F., Catala C., Landstreet J., Petit P. 2006, in Casini R., Lites B., eds, Solar Polarization Workshop n4 Vol.358 of ASPC series, 362
(19)
Dubaj D., Monier R., Alecian G., LeBlanc F. 2005, in Scientific Highlights 2005 of SF2A, EDP SCiences, F. Casoli, T. Contini, J.M. Hameury and L. Pagani, eds, 335
(20)
Fekel F.C. 2003, PASP, 115, 807
(21)
Hubrig S., North P., Savanov I. et al. 2006a, MNRAS, 371, 1953
(22)
Hubrig S., North P., Schöller M., Mathys G. 2006b, AN, 327, 289
(23)
Hubrig S., González J.F., Arlt R. 2008, CoSka, 38, 415
(24)
Kochukhov O., Piskunov N., Sachkov M., Kudryavtsev D. 2005, A&A, 439, 1093
(25)
Kochukhov O., Adelman S.J., Gulliver A.F., Piskunov N. 2007, NatPh, 3, 526
(26)
Kupka F.G., Piskunov N.E., Ryabchikova T.A et al. 1999, A&AS, 138, 119
(27)
Kurtz D.W., Breger M., Evans S.W. 1976, ApJ, 207, 181
(28)
Lampens P., BoffinH.M.J. 2000, in ”Delta Scuti and Related Stars”, ASPCS, 210, 309, M. Breger & M.H. Montgomery, eds.
(29)
Landstreet J.D., Kupka F., Ford H.A. et al. 2009, A&A, 503, 973
(30)
Lanz, T., Mathys G. 1993, A&A, 280, 486
(31)
Lignières F., Petit P., Böhm T., Aurière M. 2009, A&A, 5000, L41
(32)
Mathias P., Gillet D., Aerts C., Breitfellner M.G. 1997, A&A, 327, 1077
(33)
Mathys G. 1995, A&A, 293, 733
(34)
Mitton J., Stickland D.J. 1979, MNRAS, 186, 189
(35)
Morse J.A., Mathieu R.D., Levine S.E. 1991, AJ, 101, 1495
(36)
Petit P., Donati J.-F., Oliveira J., Aurière M. et al. 2004, MNRAS, 351, 826
(37)
Petit P., Lignières F., Wade G.A. et al. 2010, A& A, in press
(38)
Rees D.E., Semel M.D. 1979, A&A, 74, 1
(39)
Ryabchikova T.A. 1998, CoSka, 27, 319
(40)
Shorlin S.L.S., Wade G.A., Donati J.-F. et al. 2002, A&A 392, 637
(41)
Smith M.A. 1974, ApJ, 189, 101
(42)
Spruit H.C. 1999, A&A 349, 189
(43)
Stelzer B., Burwitz V. 2003, A&A, 402, 719
(44)
Takeda Y., Han I., Kang D-I et al. 2008, JTKAS, 41, 83
(45)
Torres G. 2007, AJ, 133, 2684
(46)
Vauclair S., Vauclair G. 1982, ARA&A 20, 37
(47)
Wade G.A., Donati J.F., Landstreet J.D& Shorlin S.L.S. 2000, MNRAS, 313, 851
(48)
Wade G.A., Aurière M., Bagnulo S. et al. 2006, A&A, 451, 293
(49)
Wolff S.C. 1968, PASP 80, 281
(50)
Zhao M., Monnier J.D., Torres G. et al. 2007, ApJ, 659, 626
(51)
Zavala R.T., Adelman S.J., Hummel C.A. et al. 2007, APJ, 655, 1046
(52)
Zöchling J., Muthsam H. 1987, A&A, 176, 75 |
Vector Meson Photoproduction with an Effective
Lagrangian in the Quark Model II: $\omega$ Photoproduction
Qiang Zhao
Zhenping Li
Department of Physics
Peking University
Beijing
100871
P.R.China
C. Bennhold
Department of Physics
Center for Nuclear Studies
The George Washington University
Washington
D.C.
20052
USA
Abstract
An investigation of $\gamma p\to\omega p$ is presented in a constituent
quark model approach. The sparse data in the large $t$ region where
the resonances dominate is well described within the model,
while the diffractive
behavior in the small $t$ region requires an additional t-channel exchange.
Taking into
account the t-channel $\pi^{0}$ exchange, we find a good
overall agreement with the available data with only 3 free parameters.
Our study shows that the differential cross section is not sensitive to
s-channel resonances, however, the polarization observables are demonstrated
to be very sensitive. Thus, measuring polarization observables
is the crucial part of the vector meson
photoproduction program in the search for “missing” resonances.
PACS numbers: 13.75.Gx, 13.40.Hq, 13.60.Le,12.40.Aa
1. Introduction
The newly established electron and photon facilities have made it possible to
investigate the mechanism of vector meson
photoproduction
on nucleons with much improved experimental accuracy. This has been
motivated in part by
the puzzle that the NRQM [1, 2] predicts a much richer resonance
spectrum than has been observed in $\pi N\to\pi N$ scattering
experiments.
Quark model studies have suggested that those resonances missing in the
$\pi N$
channel may couple strongly to, i.e., the $\omega N$ and $\rho N$
channels.
Experiments have been performed at ELSA [3] and will be done
at TJNAF in the near future [4].
Therefore, a theory on the reaction mechanism
that highlights the dynamical role of
s-channel resonances is crucial in order to interpret new
experimental data.
Building on recent successes of the quark model
approach to pseudoscalar meson
photoproduction we have proposed[5] extending this
approach to vector meson
photoproduction. The focus of this paper is to present the numerical
implementation of the
reaction $\gamma p\to\omega p$ in the quark model framework.
The reaction $\gamma p\to\omega p$ shares similar features with the
reaction
$\gamma p\to\eta p$ in the pseudoscalar sector, since
both are isospin zero states.
This eliminates contributions from all isospin 3/2 intermediate
resonances, thus the signals from
isospin 1/2 states, in particular those resonances “missing” in
the $\pi N$
channels, can be isolated .
Thus, investigating the dynamical role of the
s-channel resonances and examining which
experimental observables are best suited to demonstrate
the existence of these resonances is the
primary goal of
our study. However, an important question related to this discussion
is whether additional t-channel exchanges responsible
for the diffractive behavior
in the small $t$ region are needed.
It has been proposed in the pseudoscalar sector that the extra t-channel
exchanges could be excluded with the duality
hypothesis[6]. In a study of
kaon photoproduction by
J.C. David et al [7], the inclusion of the additional
s-, u-channel resonances, particularly the higher partial wave resonances
leads to smaller coupling constants for the t-channel $K^{*}$ exchanges,
indicating some phenomenological support for
the duality argument. Similarly, one could also
investigate the role of the t-channel
exchanges in $\omega$ photoproduction by including the t-channel
exchanges and
treating the coupling constants as parameters.
If the diffractive behavior could be
described by contributions from a complete set of s-
and u-channel resonances, the coupling constants for the
t-channel exchanges should
be very small. Thus, we include in our calculations the $\pi^{0}$ exchange
suggested by Friman and Soyeur [8],
who showed that the diffractive behavior
in the low energy region
can be well described through $\pi^{0}$ exchange.
In section 2, we give a brief description of our model, of which
a complete framework has been given in Ref. [5]. The $\pi^{0}$
exchange model
by Friman and Soyeur will be introduced in the standard helicity representation. The
results of our calculations for the s- and u-channel transition amplitudes are given in section
3, and the conclusion are given in Section 4.
2. The model
The starting point of our quark model approach to vector meson photoproduction is the
effective Lagrangian [5],
$$L_{eff}=-\overline{\psi}\gamma_{\mu}p^{\mu}\psi+\overline{\psi}\gamma_{\mu}e_{%
q}A^{\mu}\psi+\overline{\psi}(a\gamma_{\mu}+\frac{ib\sigma_{\mu\nu}q^{\nu}}{2m%
_{q}})\phi^{\mu}_{m}\psi,$$
(1)
where $\psi$ and $\phi^{\mu}_{m}$ are the quark and the vector meson fields, and $a$ and $b$ are
coupling constants which will be determined by experimental data.
At the tree level, the transition matrix element based on this
Lagrangian in Eq.(1) can be written as the sum of contributions
from s-, u- and t- channels,
$$M_{fi}=M^{s}_{fi}+M^{u}_{fi}+M^{t}_{fi}.$$
(2)
The first and second terms in Eq.(2) represent the s- and u-channel
contributions, and a complete set of helicity amplitudes for each
of the s-channel resonance below 2 GeV have been evaluated
in the $SU(6)\otimes O(3)$
symmetry limit [5]. Resonances above 2 GeV
are treated as degenerate in order to express contributions from
resonances with a certain quantum number $n$ in a
compact form. The contributions from
the u-channel resonances are divided into two parts as well. The
first part contains the contributions from the resonances with
the quantum number $n=0$, which includes the spin 1/2 baryons,
such as the $\Lambda$, $\Sigma$ and the nucleons, and the spin
3/2 resonances, such as the $\Sigma^{*}$ in $K^{*}$ photoproduction
and the $\Delta(1232)$ resonance in $\rho$ photoproduction. Because
the mass splitting between the spin 1/2 and spin 3/2 resonances
with $n=0$ is significant, they have to be treated separately.
Since in $\omega$ photoproduction isospin 3/2 resonance
cannot contribute due to
isospin conservation, construction of the resonance contribution is simpler than
in the case of $\rho$ meson photoproduction. The second part comes from the excited
resonances with quantum number $n\geq 1$. Since the contributions
from the u-channel resonances are not sensitive to the precise
mass positions, they are treated as degenerate as well. The transition matrix elements are written
in the form of helicity amplitudes, which are 12 independent amplitudes and defined
as in ref.[5, 9],
$$\displaystyle H_{1\lambda_{V}}$$
$$\displaystyle=$$
$$\displaystyle\langle\lambda_{V},\lambda_{2}=+1/2|T|\lambda=1,\lambda_{1}=-1/2\rangle$$
$$\displaystyle H_{2\lambda_{V}}$$
$$\displaystyle=$$
$$\displaystyle\langle\lambda_{V},\lambda_{2}=+1/2|T|\lambda=1,\lambda_{1}=+1/2\rangle$$
$$\displaystyle H_{3\lambda_{V}}$$
$$\displaystyle=$$
$$\displaystyle\langle\lambda_{V},\lambda_{2}=-1/2|T|\lambda=1,\lambda_{1}=-1/2\rangle$$
$$\displaystyle H_{4\lambda_{V}}$$
$$\displaystyle=$$
$$\displaystyle\langle\lambda_{V},\lambda_{2}=-1/2|T|\lambda=1,\lambda_{1}=+1/2\rangle,$$
(3)
where the helicity states are denoted by $\lambda=\pm 1$ for the incident photon,
$\lambda_{V}=0,\pm 1$ for the outgoing vector meson, and $\lambda_{1}=\pm 1/2$, $\lambda_{2}=\pm 1/2$
for the initial and final state nucleons, respectively. The various experimental observables,
such as the cross section and polarization observables,
have been discussed in
Ref.[9] in terms of these amplitudes.
In particular, the differential cross section can be written as
$$\frac{d\sigma}{d\Omega_{c.m.}}=\frac{\alpha_{e}\omega_{m}(E_{f}+M_{N})(E_{i}+M%
_{N})}{8\pi s}{|{\bf q}|}\frac{1}{2}\sum^{4}_{a=1}\sum_{\lambda_{V}=0,\pm 1}|H%
_{a\lambda_{V}}|^{2}$$
(4)
in the center of mass frame, where $\sqrt{s}$ is the total energy of the
system, $M_{N}$ represents the mass of the nucleon,
$\omega_{m}$ denotes the energy of the meson with momentum ${\bf q}$,
and $E_{i}$, $E_{f}$
denote the energies of the initial and final nucleon states.
As defined in Ref. [9], the four single-spin observables in
bilinear helicity product (BHP) form have the following expressions:
$$\check{T}=-\frac{1}{2}\langle H|\framebox{{\hbox{\Gamma^{10}}}}\framebox{{%
\hbox{\omega^1}}}|H\rangle,$$
(5)
for the target polarization,
$$\check{P}_{N^{\prime}}=\frac{1}{2}\langle H|\framebox{{\hbox{\Gamma^{12}}}}%
\framebox{{\hbox{\omega^1}}}|H\rangle,$$
(6)
for the polarization of the final nucleon,
$$\check{\Sigma}=\frac{1}{2}\langle H|\Gamma^{4}\omega^{A}|H\rangle,$$
(7)
for the polarized photon asymmetry, and
$$\check{P}_{V}=\frac{1}{2}\langle H|\framebox{{\hbox{\Gamma^1}}}\framebox{{%
\hbox{\omega^3}}}|H\rangle,$$
(8)
for the polarization of the final vector meson,
where the explicit expressions and conventions for
the $\Gamma$ and $\omega$ matrices have been given in Ref. [9].
The phase space factor for these four observables are the same as in
the differential cross section, therefore,
they are normalized by being divided by the differential cross section.
The t-channel exchange of $M^{t}_{fi}$ in Eq.(2) would correspond to $\omega$
exchange which is absent since the photon cannot couple to the $\omega$.
However, an additional t-channel exchange is commonly included even though
it is not a part of the Lagrangian in Eq.(1).
This is the $\pi^{0}$ exchange
which is known to be needed in order to describe
the diffractive behavior in the small $t$ region.
The Lagrangian for the
$\pi^{0}$ exchange model has the following form [8],
$$L_{\pi NN}=-ig_{\pi NN}\overline{\psi}\gamma_{5}(\mbox{\boldmath$\tau$ %
\unboldmath}\cdot\mbox{\boldmath$\pi$ \unboldmath})\psi$$
(9)
for the $\pi NN$ coupling vertex, and
$$L_{\omega\pi^{0}\gamma}=e_{N}\frac{g_{\omega\pi\gamma}}{M_{\omega}}\epsilon_{%
\alpha\beta\gamma\delta}\partial^{\alpha}A^{\beta}\partial^{\gamma}\omega^{%
\delta}\pi^{0}$$
(10)
for the $\omega\pi\gamma$ coupling vertex, where the $\omega^{\delta}$ and $\pi^{0}$ represent
the $\omega$ and $\pi^{0}$ fields, the $A^{\beta}$ denotes the electromagnetic field,
and $\epsilon_{\alpha\beta\gamma\delta}$ is the Levi-Civita tensor, and $M_{\omega}$
is the mass of $\omega$ meson. The $g_{\pi NN}$ and $g_{\omega\pi\gamma}$ in Eqs.
(9) and (10) denote the coupling constants at the two
vertices, respectively. Therefore, the transition amplitudes of t-channel $\pi^{0}$
exchange have the following expression,
$$M^{t}_{T}=\frac{e_{N}g_{\pi NN}g_{\omega\pi\gamma}}{2M_{\omega}(t-m^{2}_{\pi})%
}\{\omega\mbox{\boldmath$\epsilon$ \unboldmath}\cdot({\bf q}\times\mbox{%
\boldmath$\epsilon$ \unboldmath}_{v})+\omega_{m}{\bf k}\cdot(\mbox{\boldmath$%
\epsilon$ \unboldmath}\times\mbox{\boldmath$\epsilon$ \unboldmath}_{m})\}\mbox%
{\boldmath$\sigma$ \unboldmath}\cdot{\bf A}e^{-\frac{({\bf q}-{\bf k})^{2}}{6%
\alpha_{\pi}^{2}}}$$
(11)
for the transverse transition, and
$$M^{t}_{L}=-\frac{e_{N}g_{\pi NN}g_{\omega\pi\gamma}}{2M_{\omega}(t-m^{2}_{\pi}%
)}\frac{M_{\omega}}{|{\bf q}|}(\mbox{\boldmath$\epsilon$ \unboldmath}\times{%
\bf k})\cdot{\bf q}\mbox{\boldmath$\sigma$ \unboldmath}\cdot{\bf A}e^{-\frac{(%
{\bf q}-{\bf k})^{2}}{6\alpha_{\pi}^{2}}}$$
(12)
for the longitudinal transition, where
$\omega$ in the transition amplitudes
denotes the energy of the photon with momentum ${\bf k}$, and
${\bf A}=-\frac{{\bf q}}{E_{f}+M_{N}}+\frac{{\bf k}}{E_{i}+M_{N}}$,
and $t=(q-k)^{2}=M_{\omega}^{2}-2k\cdot q$.
The factor $e^{-\frac{({\bf q}-{\bf k})^{2}}{6\alpha_{\pi}^{2}}}$ in Eqs.
(11) and (12) is the form factor for both $\pi NN$ and $\omega\gamma\pi$ vertices, if we assume
that the wavefunctions for nucleon, $\omega$ and $\pi$ have a Gaussian form. The constant
$\alpha_{\pi}^{2}$ in this form factor is treated as a parameter. Following the same procedure
as in Ref.[5], the explicit expressions for the operators in terms of the helicity
amplitudes can be obtained. They are listed in Tables 1 and 2 for the transverse and
longitudinal amplitudes respectively.
3. Results and Discussion
Before discussing the details of our numerical results, we point out that the nonrelativistic
wavefunction in the quark model become more inadequate as the energy of the system
increases. A procedure to partly remedy this problem is to introduce the Lorentz
boost factor in the spatial integrals that involve the spatial wavefunctions of nucleons
and baryon resonances,
$$R(q,k)\to\gamma_{q}\gamma_{k}R(q\gamma_{q},k\gamma_{k}),$$
(13)
where $\gamma_{q}=\frac{M_{f}}{E_{f}}$ and $\gamma_{k}=\frac{M_{i}}{E_{i}}$. A similar procedure had
been used in the numerical evaluation of pseudoscalar meson photoproduction[10].
There are four free parameters in the quark model approach to the s- and u-channel
resonance contributions: the quark mass $m_{q}$, the harmonic oscillator strength $\alpha$,
and the coupling constants $a$ and $b^{\prime}=b-a$ from Eq. (1).
Because the quark mass $m_{q}$ and the parameter $\alpha$ are commonly used in
the quark model, they are fixed at
$$\displaystyle m_{q}$$
$$\displaystyle=$$
$$\displaystyle 330\ \mbox{MeV}$$
$$\displaystyle\alpha$$
$$\displaystyle=$$
$$\displaystyle 410\ \mbox{MeV}.$$
(14)
In addition to the free parameters in the s- and u- channels, there are also parameters
in the t-channel $\pi^{0}$ exchange: the coupling constants for the $\pi NN$ and $\omega\pi\gamma$
vertices, $g_{\pi NN}$ and $g_{\omega\pi\gamma}$, and the parameter $\alpha_{\pi}$ in Eqs. (11)
and (12). We find that the s- and u-channel resonance contributions
alone are unable to describe the diffractive behavior in the small $t$ region.
We therefore include the $\pi^{0}$ exchange using for the coupling constants
$g_{\pi NN}$ and $g_{\omega\pi\gamma}$[8]:
$$\displaystyle\frac{g^{2}_{\pi NN}}{4\pi}$$
$$\displaystyle=$$
$$\displaystyle 14,$$
$$\displaystyle g^{2}_{\omega\pi\gamma}$$
$$\displaystyle=$$
$$\displaystyle 3.315,$$
(15)
and find a good description of the diffractive behavior. Note that the
values of $g_{\pi NN}$ and $g_{\omega\pi\gamma}$ were fixed by separate experiments
and, therefore, are not free parameters in Ref. [8].
This suggest that the duality hypothesis may not work in vector meson photoproduction.
The parameter $\alpha_{\pi}$ will be determined from $\omega$
photoproduction data
along with the coupling constants $a$ and $b^{\prime}$. Qualitatively, we would
expect that $\alpha_{\pi}$ be smaller than the parameter $\alpha=410$ MeV, since it represents
the combined form factors for both $\pi NN$ and $\omega\pi\gamma$ vertices while the parameter
$\alpha$ only corresponds to the form factor for the $\pi NN$ or $\omega NN$ vertex alone.
In Table 3, we list the s-channel resonances that contribute to the $\omega$ photoproduction
in $SU(6)\otimes O(3)$ symmetry limit. The masses and widths of these resonances come from the
recent Particle Data Group[11]. It should be noted that the Moorhouse selection[12]
rule have eliminated the states belonging to $[70,1^{-}]_{1}$ and $[70,2^{+}]_{2}$ representation with
symmetric spin structure from contributing to the $\omega$ photoproduction with the proton
target so that the s-channel states $S_{11}(1650),D_{13}(1700),D_{15}(1650)$ are not present
in our numerical evaluations. Of course, configuration mixing will lead to additional
contributions from these resonances which, however, cannot be determined at present due
to the poor quality of data.
In Table 3 we also list the partial widths of the resonances
decaying into the $\omega N$ channel and their photon decay
hecility amplitudes.
In terms of the helicity amplitudes, the partial width $\Gamma_{\omega}$ of a resonance with spin $J$
decaying into an $\omega$ meson and a nucleon is
$$\Gamma_{\omega}=\frac{|{\bf q}|\omega_{m}}{4\pi}\frac{M_{N}}{M_{R}}\frac{8}{2J%
+1}\{|A_{\frac{1}{2}}|^{2}+|A_{\frac{3}{2}}|^{2}+\frac{M^{2}_{\omega}}{|{\bf q%
}|^{2}}|S_{\frac{1}{2}}|^{2}\},$$
(16)
where $A_{\frac{1}{2}}$, $A_{\frac{3}{2}}$ and $S_{\frac{1}{2}}$ represent the vector
meson helicity amplitudes and $M_{R}$ denotes the mass of the intermediate
resonance.
The partial
width $\Gamma_{\gamma}$ of the resonances decaying into $\gamma N$,
and contributing to $\omega$ photoproduction, is
$$\Gamma_{\gamma}=\frac{{\bf k}^{2}}{\pi}\frac{2M_{N}}{(2J+1)M_{R}}\{|A^{\gamma}%
_{\frac{1}{2}}|^{2}+|A^{\gamma}_{\frac{3}{2}}|^{2}\},$$
(17)
where $A^{\gamma}_{\frac{1}{2}}$ and $A^{\gamma}_{\frac{3}{2}}$ denote the photon helicity amplitudes.
Only the resonances $P_{13}(1900)$ and
$F_{15}(2000)$, at present classified as 2-star resonances
in the 1996 PDG listings, have masses above the $\omega$ decay threshold,
and therefore have branching ratio into the $\omega N$ channel. We find that
the $F_{15}(2000)$ has a larger decay width than the $P_{13}(1900)$.
This finding differs from Ref. [2],
where the $P_{13}(1900)$ was calculated to have the larger
width into the $\omega N$ channel. The photon decay helicity amplitudes
in Table 3 are consistent with previous theoretical results
of the NRQM approach in Ref. [1]. Qualitatively, the resonance
$F_{15}(2000)$ plays a very important role in
$\omega$ photoproduction. Of course, since our investigation
here is exploratory it
can only provide an approximate description of the
resonance contributions; a more accurate approach to
the intermediate resonance decay should be adopted in a
systematic analysis [5].
We have not performed a rigorous numerical fit to the available data
because of the poor quality of the data.
Our study suggests that the parameters $a$, $b^{\prime}$
and $\alpha_{\pi}$ with the values
$$\displaystyle a$$
$$\displaystyle=$$
$$\displaystyle-2.2,$$
$$\displaystyle b^{\prime}$$
$$\displaystyle=$$
$$\displaystyle 3.0,$$
$$\displaystyle\alpha_{\pi}$$
$$\displaystyle=$$
$$\displaystyle 300\ \mbox{MeV}$$
(18)
gives a good description of the differential cross section
data [3] in the resonance region. Clearly, these
parameters have considerable uncertainties.
Fig. 1 shows our calculations for the differential cross section
at the average photon energies of $E_{\gamma}=$1.225, 1.45, 1.675
and 1.915 GeV,
in comparison with the data [3]. The results for
the t-channel $\pi^{0}$ exchange and contributions from only the s- and u-channel processes
are also shown separately.
Our results with the $\pi^{0}$ exchange are consistent with the findings of
Ref. [8] although the form factor in our calculation is different.
Fig. 1 clearly demonstrates that
the t-channel $\pi^{0}$ exchange is dominant in the small angle region, while the s- and u-channel
resonance contributions become more important as the scattering
angle $\theta$ increases.
To test the sensitivity of s-channel resonances to the differential
cross section, the angular distribution at 1.675 GeV is presented
with and without the contribution from the resonance $F_{15}(2000)$;
its threshold energy is around 1.675 GeV in the lab. frame. The results indicate that
the differential cross section data alone are not sufficient to
determine the presence of this resonance considering the theoretical
and experimental uncertainties. Since the resonance couplings of the
$F_{15}(2000)$ are larger than those of other resonances in this
mass region, the sensitivity of the differential cross section to
other resonances around 2 GeV is even smaller.
In contrast to the differential cross section,
the polarization observables show a much more dramatic dependence
on the presence of the s-channel resonances. We present
results of four single polarizations at 1.675 GeV in Fig. 2. The absence
of the resonance $F_{15}(2000)$ leads to a sign change in the
target polarization, and the variations in the recoil as well
as the meson polarization observable are very significant as well.
The absence of the resonance
$P_{13}(1900)$, also shown in Fig. 2,
leads to very significant changes in the recoil
polarization. Although we
do not expect our numerical results to give a quantitative
prediction of
polarization observables at the present stage, since the calculations are limited
to the $SU(6)\otimes O(3)$ symmetry limit that should be broken in
more realistic quark model wavefunction, our results clearly suggest that the
polarization observables may be the best place to determine s-channel
resonance properties.
Our results for the total cross section are shown in Fig. 3,
in which the contributions from the s- and
u-channel resonances alone are compared to the full calculation.
Our results indicate
an increasing discrepancy between theory and
the data [3, 19, 20]
with increasing energy $E_{\gamma}$,
This discrepancy
comes mainly from the small angle region where the $\pi^{0}$
exchange alone is not sufficient to describe the
diffractive behavior at higher energies. One might expect
that Pomeron exchange[14, 17]
plays a more important role in the higher energy region.
However, Fig. 1 shows that our results for
the differential cross section at the large angle region are
in good agreement with the data, and it
suggests that contributions from the s- and u- channel resonances
which are the main focus of our study, give
an appropriate description of the reaction mechanism.
It is interesting to note that the small bump around 1.7 GeV in the total cross section
comes from the contributions of the resonance $F_{15}(2000)$.
As discussed above, our calculations
find that the resonance $F_{15}(2000)$
has a strong coupling to the $\omega$ N channel. Thus,
this resonance is perhaps the best candidate
whose existance as a “missing” resonance can be
established through $\omega$ photoproduction.
5. Conclusion
In conclusion, we have presented
a study for $\omega$ photoproduction on the nucleon at low and
intermediate energies
Our results indicate that s- and u-channel resonances alone are insufficient
at small $t$ and that additional t-channel contributions
are necessary to describe the large diffractive behavior.
We find that the s- and u-channel resonance contributions are
important in the large scattering angle region.
The differential cross section alone is
insufficient to determine s-channel resonance properties in $\omega$
photoproduction, however, polarization observables
are demonstrated to be very sensitive
to the presence of individual s-channel resonances.
It is therefore imperative to perform polarization measurements
in future programs on vector meson
photoproduction. Our numerical results suggest that properties of the resonance
$F_{15}(2000)$ could be well determined with precise
$\omega$ photoproduction data.
Given the few free parameters in this
approach, the agreement with the data, especially
in the large scattering angle region, is satisfactory.
Clearly, an improved determination of s-channel resonance properties requires a
systematic analysis of all photoprduction data.
Due to the small number of free parameters
the quark model approach is an appropriate
tool for this important task.
Acknowledgments
Zhenping Li acknowledges the hospitality of the Saclay
Nuclear Physics Lab, and discussions with B. Saghai.
We are grateful to F.J. Klein for helpful discussions regarding the data.
This work is supported by the grant
from Chinese Education Commission and US-DOE grant DE-FG02-95-ER40907.
References
[1]
N. Isgur and G. Karl, Phys. Letts. 72B, 109(1977); Phys. Rev. D23,
817(1981); R. Koniuk and N. Isgur, Phys. Rev. D21, 1868(1980).
[2]
S. Capstick, and W. Roberts, Phys. Rev. D49, 4570(1994).
[3]
F. Klein et al., to appear in the Proceedings of the GW/TJNAF Workshop
on $N^{*}$ Physics, (Washington, D.C.) 1997, F.J. Klein, Ph.D. thesis, University of Bonn (1996).
[4]
TJNAF experimental proposals
E94-109 (P. Cole, R. Whitney, Co-spokesmen);
E91-024 (V. Burkert, H. Funsten. M. Manley, B. Mecking, Co-spokesmen).
[5]
Q. Zhao, Z.P.Li and C. Bennhold, nucl-th/9711061.
[6]
R. Dolen, D. Horn, and C. Schmid, Phys. Rev. 166, 1768(1966).
[7]
J. C. David, et al. Phys. Rev. C53, 2613(1996).
[8]
B. Friman and M. Soyeur, Nucl. Phys. A100, 477(1996).
[9]
M. Pichowsky, Ç. Şavklı, F. Tabakin, Phys. Rev.
C53, 593 (1996).
[10]
Zhenping Li, Phys. Rev. D52, 4961 (1995).
[11]
E.J.Weinberg, et al, Phys. Rev. D54, 1(1996).
[12]
R.G.Moorhouse,Phys. Rev. Lett. 16, 772(1966).
[13]
P. Söding, Phys. Lett. 19, 702(1965).
[14]
M. Pichowsky, “Nonperturbative Quark Dynamics in
diffractive processes”, Ph.D. Dissertation, The University Of Pittsburgh,
(1996); and references therein.
[15]
Zhenping Li, Ye Hongxing and Lu Minghui, Phys. Rev. C56, 1099 (1997).
[16]
Zhenping Li, Phys. Rev. D48, 3070(1993).
[17]
G. Wolf, Proc. 1971 Int. Symp. on Electron and Photon Interactions at High
Energies, Cornell University, Ithaca(USA), August 23-27,1971,p.190.
[18]
S. Capstick and B. D. Keister, Phys. Rev. D46,
84 (1992).
[19]
Aachen-Berlin-Bonn-Hamburg-Heidelberg-München Collabration,
Phys. Rev. 175, 1669(1968).
[20]
Crouch et al, Phys. Rev. 155, 1468(1967);
Y. Eisenberg et al, Phys. Rev. D5, 15(1972);
Y. Eisenberg et al, Phys. Rev. Lett. 22, 669(1969);
W. C. Barber et al, Z. Phys. C26, 343(1984);
J. Ballam et al, Phys. Rev. D7, 3150(1973);
W.Struczinski et al, Nucl. Phys. B108, 45(1976).
[21]
Zhenping Li, “Photo- and Electroproduction of
Baryon Resonances in A Potential Quark Mode”, Ph.D. Dissertation,
The University of Tennessee, (1991).
[22]
V. D. Burkert, The N* Program at CEBAF, CEBAF-PR-92-021.
[23]
B. Saghai and F. Tabakin, Phys. Rev. C55, 917(1995).
Figure Captions:
1.
The differential cross sections (solid curves) for $\gamma p\to\omega p$
at $E_{\gamma}=$1.225, 1.45, 1.675
and 1.915 GeV. The data come from Ref. [3].
The $\pi^{0}$ exchanges are shown by the dashed curves
and the contributions from s- and u-channel
exclusively are shown by the dotted curves.
In (c), the dot-dashed curve represents the
differential cross section without
contributions from the resonance $F_{15}(2000)$.
2.
The four single-spin polarization observables in $\gamma p\to\omega p$
are given by the solid curves at $E_{\gamma}=1.675$GeV. The dotted curves
correspond to the asymmetries without the resonance $F_{15}(2000)$,
while the dashed curves to those without the resonance $P_{13}(1900)$.
3.
The total cross section of $\gamma p\to\omega p$ are fitted by the
solid curve with $\pi^{0}$ exchange taken into account. The dotted curve
describes the pure contributions from s- and u-channel. The data are
from [3](triangle), [19] and other
experiments(square). |
Surface spin-electron acoustic waves in magnetically ordered metals
Pavel A. Andreev
andreevpa@physics.msu.ru
L. S. Kuz’menkov
lsk@phys.msu.ru
Faculty of physics, Lomonosov Moscow State University, Moscow, Russian Federation.
(December 4, 2020)
Abstract
Degenerate plasmas with motionless ions show existence of three surface waves: the Langmuir wave, the electromagnetic wave, and the zeroth sound. Applying the separated spin evolution quantum hydrodynamics to half-space plasma we demonstrate the existence of the surface spin-electron acoustic wave (SSEAW). We study dispersion of the SSEAW. We show that there is hybridization between the surface Langmuir wave and the SSEAW at rather small spin polarization. In the hybridization area the dispersion branches are located close to each other. In this area there is a strong interaction between these waves leading to the energy exchange. Consequently, generating the Langmuir waves with the frequencies close to hybridization area we can generate the SSEAWs. Thus, we report a method of creation of the SEAWs.
surface waves, spin waves, acoustic waves, quantum plasmas, quantum hydrodynamics
pacs: 73.22.Lp; 52.30.Ex;
52.35.Dm; 68.35.Ja
Plasmonics is a great field of technological application of plasmas Barnes Nat 03 , Giannini CR 11 . Key role in plasmonics belongs to the surface Langmuir waves or surface plasmons Fang APL 08 , Wang APL 11 . On the over hand, the materials with the partial spin polarization of the carriers reveal the existence of spin-electron acoustic waves (SEAWs) Andreev PRE 15 , Andreev AoP 15 . Quantum of the SEAWs is called spelnon. The spelnons in the bulk materials exist simultaneously with the bulk plasmons, but the spelnons have smaller energies than the energy of plasmons as it is shown in Ref. Andreev PRE 15 . SEAWs in 2D structures, they are also called spin-plasmons, are considered in Refs. Andreev spin-up and spin-down 1408 2D -Agarwal PRB 14 . Their spectrum for plane-line two-dimensional structures is described in Refs. Andreev spin-up and spin-down 1408 2D , Ryan PRB 91 for the external magnetic field perpendicular to the plane. Contribution of the cyclotron motion in the spectrum is described in Andreev spin-up and spin-down 1408 2D . Influence of the disorder on properties of spin-plasmons is described in Ref. Agarwal PRB 14 . A possibility of the spin-electron acoustic soliton formation due to the non-linear evolution of perturbations in the partially spin polarized electron gas was demonstrated in Ref. Andreev 1504 . It was shown that the electron-spelnon interaction leads to the Cooper pair formation ensuring a mechanism for the high-temperature superconductivity Andreev HTSC 15 . Experimental analysis of surface spin waves and spin waves in thin films is a subject of current research (see for instance Michel PRB 15 ) leading to a possibility of the experimental analysis of the SEAWs.
Moving towards applications of the SEAWs, in this paper, we study a possibility of existence of the surface SEAWs (SSEAWs) and their properties. Surface waves in plasmas were studied long time ago Tend PRL 67 -Jones PRL 83 . However, the zero-sound mode, which is well-known for the bulk degenerate perturbations of fermions (see for instance Landau v9 and JETP 80th ), was just recently described for the half space regime Tyshetskiy JPP 13 , Tyshetskiy PP 14 .
Collisional and collisionless Landau damping of the surface Langmuir waves in the degenerate electron gas are considered in Tyshetskiy PP 12 .
There is still the fundamental interest to the quantum effects caused by the quantum Bohm potential in surface plasma waves Lazar PP 07 -Moradi PL A 15 .
To find the SEAW a new method called the separate spin evolution quantum hydrodynamics
(SSE-QHDs) was developed in Ref. Andreev PRE 15 . This method is a
generalization of usual spin-1/2 QHD developed in Refs. MaksimovTMP 2001 , Marklund PRL07 , Andreev VestnMSU 2007 (some applications are described in review Shukla RMP 11 ). A generalization of the SSE-QHD was developed in Ref. Trukhanova PLA 15 , where the spin current evolution equation was derived in terms of separate evoltion of spin-up and spin-down electrons.
The SSE-QHD was developed in Ref. Andreev PRE 15 , however it does not consider the thermal part of the spin current. The thermal part of the spin current was recently derived in Ref. Andreev TerSpinCurent for the degenerate electron gas, where the thermal part of the spin current is reduced to the Pauli blocking contribution, similarly to the Fermi pressure.
We do not present full set of the SSE-QHD here, we restrict our presentation with the linearized equations for the wave propagating parallel to the boundary surface. Full set of the SSE-QHD equations can be found in Refs. Andreev PRE 15 , Andreev AoP 15 .
To describe the surface SEAWs we need the continuity and Euler equations for the separate spin evolution of the spin-up and spin-down electrons.
Corresponding linearized and Fourier transformed set of equations arises as
$\omega\delta n_{s}=n_{0s}k_{z}\delta v_{sz}$, $-mn_{0s}\omega\delta\textbf{v}_{s}+(6\pi^{2})^{\frac{2}{3}}n_{0s}^{\frac{2}{3}%
}\hbar^{2}k_{z}\delta n_{s}\textbf{e}_{z}/3m=q_{e}n_{0s}\delta\textbf{E}$ we solve the hydrodynamic equations in each half space independently. We consider them
together with the Maxwell equations
$\omega\delta B_{y}=k_{z}ñ\delta E_{x}+\imath ñ\partial_{x}\delta E_{z}$,
$\omega\delta E_{x}=k_{z}ñ\delta B_{y}+4\pi\imath(n_{0u}\delta v_{ux}+n_{0d}%
\delta v_{dx})$,
and
$\omega\delta E_{z}=\imath ñ\partial_{x}\delta B_{y}+4\pi\imath(n_{0u}\delta v_%
{uz}+n_{0d}\delta v_{dz})$ solved in whole space.
We consider the following equations of state for the spin-up and spin-down electrons $P_{s}=(6\pi^{2})^{2/3}n_{0s}^{5/3}\hbar^{2}/5m$. These equations include the deformation of the Fermi step distribution under the action of the external magnetic field or the strong exchange interaction in the magnetically ordered materials.
The effect of spin polarization on the Fermi pressure has been considered in literature in the application to the wave phenomena in the single
fluid model of electrons JETP 80th , Andreev AoP 14 and the two-fluid model Andreev PRE 15 , Andreev spin-up and spin-down 1408 2D , Ryan PRB 91 .
We consider the regime of zero magnetic field and assume that the partial spin polarization of conducting electrons in magnetic materials is caused by inner equilibrium effects like the exchange interaction.
To describe the main properties of the SSEAWs we consider a surface of the ferromagnetic or ferrimagnetic materials containing the conducting electrons with the partial spin polarization as it is shown in Fig. 1.
Our model works if both subspecies of electrons are degenerate: $T\ll T_{Fu},T_{Fd}$, where $T$ is the temperature of the system and $T_{Fs}=(6\pi^{2}n_{0s})^{2/3}\hbar^{2}/2m$ are the Fermi temperatures for the spin-up and spin-down electrons. It gives a restriction on the large spin polarizations $\eta=|n_{0d}-n_{0u}|/(n_{0d}+n_{0u})$, since $T_{Fu}$ would be very small and system could be described at low temperatures only. At the high spin polarization contributions of the Coulomb exchange interaction between the conductivity electrons and ion dynamics also affect evolution of electrons. The concentration of conductivity electrons in metals is large $n_{0}\sim 10^{22}$ cm${}^{-3}$, hence at the spin polarization $\eta$ below $\eta_{0}=0.99$ we can consider temperatures in area from 10 K to 300 K.
Hydrodynamic equations give the following expressions for the velocity field in terms of the electric field perturbation
$\delta v_{sx}=-(e/m)\imath\delta E_{x}/\omega,$
and
$\delta v_{sz}=-(e/m)\imath\omega\delta E_{z}/(\omega^{2}-k_{z}^{2}U_{s}^{2}),$
where $U_{s}^{2}=\frac{\hbar^{2}}{3m_{e}^{2}}(6\pi^{2}n_{0\uparrow})^{\frac{2}{3}}$. These formulae can be substituted to the Maxwell equations to find an equation for electromagnetic field in our system.
At the zero external magnetic field and zero spin polarization the dielectric permittivity arises as a diagonal tensor with the following nonzero elements $\varepsilon_{xx}=\varepsilon_{yy}\neq\varepsilon_{zz}$. If we include the spin polarization the structure of the dielectric tensor is the same, but the explicit form of $\varepsilon_{zz}$ modifies becoming more complex:
$$\varepsilon_{ud}\equiv\varepsilon_{zz}=1-\frac{\omega_{Lu}^{2}}{\omega^{2}-k_{%
z}^{2}U_{u}^{2}}-\frac{\omega_{Ld}^{2}}{\omega^{2}-k_{z}^{2}U_{d}^{2}},$$
(1)
while
$$\varepsilon\equiv\varepsilon_{xx}=\varepsilon_{yy}=1-\frac{\omega_{Le}^{2}}{%
\omega^{2}}$$
(2)
is not changed. Here $\omega_{Ls}^{2}=4\pi e^{2}n_{0s}/m$ are the partial Langmuir frequencies, with $s=u,d$ for the spin-up and spin-down electrons, and $\omega_{Le}^{2}=4\pi e^{2}n_{0e}/m$ is the full Langmuir frequency, $n_{0}=n_{0u}+n_{0d}$ is the full equilibrium concentration of the electrons. Difference of the equilibrium concentrations arises due to the spin polarization $\eta$, hence $n_{0u}=(1-\eta)n_{0}/2$ and $n_{0d}=(1+\eta)n_{0}/2$.
The surface waves under consideration are described by
$$c^{2}\partial_{x}^{2}\delta E_{z}+\omega^{2}\frac{\varepsilon_{ud}}{%
\varepsilon}\biggl{(}\varepsilon-\frac{k^{2}c^{2}}{\omega^{2}}\biggr{)}\delta E%
_{z}=0.$$
(3)
It describes waves of $E$-type.
In vacuum, at $x<0$, we have $\varepsilon_{ud}=\varepsilon=1$
Equation (3) gives the following solution for left and right half-spaces:
$$\delta E_{z}=\Biggl{\{}\begin{array}[]{cc}C_{1}\exp\biggl{(}-\sqrt{\frac{%
\varepsilon_{ud}}{\varepsilon}k_{z}^{2}-\varepsilon_{ud}\frac{\omega^{2}}{c^{2%
}}}x\biggr{)},&x>0\\
C_{2}\exp\biggl{(}\sqrt{k_{z}^{2}-\frac{\omega^{2}}{c^{2}}}x\biggr{)},&x<0.%
\end{array}$$
(4)
Boundary conditions are the continuity of $\delta E_{z}$ and $\delta B_{y}$
$$\begin{array}[]{cc}\{\delta E_{z}\}\mid_{x=0}=0,&\{\delta B_{y}\}\mid_{x=0}=0.%
\end{array}$$
(5)
The linearized Maxwell equations presented above give the following relation between $\delta B_{y}$ and $\delta E_{z}$:
$$\delta B_{y}=\frac{\imath c}{\omega}\frac{\varepsilon}{\varepsilon-\frac{k^{2}%
c^{2}}{\omega^{2}}}\partial_{x}\delta E_{z}.$$
(6)
Condition $\{\delta E_{z}\}\mid_{x=0}=0$ leads us to $C_{1}=C_{2}$. The following application of the condition $\{\delta B_{y}\}\mid_{x=0}=0$ gives the following dispersion equation:
$$\sqrt{k_{z}^{2}c^{2}-\omega^{2}}+\varepsilon\sqrt{\frac{\varepsilon_{ud}}{%
\varepsilon}}\sqrt{k_{z}^{2}c^{2}-\varepsilon\omega^{2}}=0.$$
(7)
This equation has real solutions if $\varepsilon<0$ and $\varepsilon_{ud}<0$.
Dispersion equation (7) arises at the application of the full set of the Maxwell equations. Coupling of waves appears in the limit of the large wave vectors $k$ as we show it below at the numerical analysis. Hence, it corresponds to the quasi-electrostatic regime. We can simplify the dispersion equation (7) in this regime to $1-\varepsilon\varepsilon_{ud}=0$.
Equation $1-\varepsilon\varepsilon_{ud}=0$ is a generalization of the dispersion equation existing for unpolarized degenerate electron gas, in the quasi-electrostatic regime,
$$1-\biggl{(}1-\frac{\omega_{Le}^{2}}{\omega^{2}}\biggr{)}\biggl{(}1-\frac{%
\omega_{Le}^{2}}{\omega^{2}-\frac{1}{3}v_{Fe}^{2}k^{2}}\biggr{)}=0,$$
(8)
which has the following solution $\omega^{2}=0.5(\omega_{Le}^{2}+v_{Fe}^{2}k^{2}/3)$.
Usually we have two branches of dispersion dependence of the surface waves, since the hydrodynamics does not describe the zero sound. One of them is the Langmuir wave, which is highly nonlongitudinal wave in contrast with its bulk analog. Its frequency tends to $kc$ at $k\rightarrow 0$. In the opposite limit of large wave vectors $k\rightarrow\infty$ (its frequency tends to $\omega_{Le}/\sqrt{2}$ if we drop the pressure contribution) this wave become longitudinal and can be described in the quasistatic limit. The second branch is the analog of the three dimensional electromagnetic wave.
Separate spin evolution leading to equation (7) gives three branches of the dispersion dependence. New branch is related to the spin polarization. It is the surface SEAWs or surface spelnons, similar to the bulk SEAWs considered in Refs. Andreev PRE 15 , Andreev AoP 15 .
Fig. 2 shows that at rather large spin polarization the SEAW dispersion dependence lies below the Langmuir wave. Decrease of the spin polarization increases the frequency of the SEAW. Thus, we find crossing of the dispersion dependencies of the Langmuir wave and the SEAW, which leads to the hybridization of their spectrums as it shown in Fig. 3. In figures we use the dimensionless frequency $\Omega\equiv\omega/\omega_{Le}$ and the dimensionless wave vector $\kappa=v_{Fe}k_{z}/(\omega_{Le}\sqrt{3})$, where $v_{Fe}=(3\pi^{2}n_{0})^{\frac{1}{3}}\hbar/m$ is the Fermi velocity.
Analytical solutions for the Langmuir wave and the SEAW can be found in the quasi-electrostatic limit
$$\omega^{2}=\frac{1}{4}\Biggl{\{}\omega_{Le}^{2}+k_{z}^{2}\biggl{[}U_{u}^{2}%
\biggl{(}\frac{n_{0}+n_{0d}}{n_{0}}\biggr{)}+U_{d}^{2}\biggl{(}\frac{n_{0}+n_{%
0u}}{n_{0}}\biggr{)}\biggr{]}$$
$$\pm\sqrt{\Biggl{[}\omega_{Le}^{2}+k_{z}^{2}\biggl{[}U_{u}^{2}\biggl{(}\frac{n_%
{0}+n_{0d}}{n_{0}}\biggr{)}+U_{d}^{2}\biggl{(}\frac{n_{0}+n_{0u}}{n_{0}}\biggr%
{)}\biggr{]}\Biggr{]}-8k_{z}^{4}U_{u}^{2}U_{d}^{2}-8k_{z}^{2}(\omega_{Lu}^{2}U%
_{d}^{2}+\omega_{Ld}^{2}U_{u}^{2})}\Biggr{\}}.$$
(9)
At the small wave vectors we numerically find the linear dispersion dependence for the surface SEAWs. Therefore, we find an approximate analytical solution of equation (7) for this regime
$$\Omega^{2}=\frac{1}{2}[(1-\eta)(1+\eta)^{\frac{2}{3}}+(1+\eta)(1-\eta)^{\frac{%
2}{3}}]\xi^{2},$$
(10)
which coincides with the dispersion dependence of the bulk SEAWs in the small wave vector regime.
Numerical analysis shows that the Fermi pressure and the spin separation do not affect the high frequency electromagnetic wave. The separate spin evolution does not affect it either. On the other hand spin separation could affect other branches. For instance the surface Langmuir waves is affected by the modification of the Fermi pressure at the spin polarization.
The spin polarization increases the Fermi pressure contributing as $P_{pol}=P_{u}+P_{d}=0.5[(1+\eta)^{5/3}+(1-\eta)^{5/3}]P_{Fe}$ (see Andreev PRE 15 , JETP 80th , Andreev AoP 14 ), where $P_{Fe}=(3\pi^{2})^{2/3}n_{0}^{5/3}\hbar^{2}/5m$ is the Fermi pressure at the zero spin polarization. At the full spin polarization $P_{pol}$ increases up to $1.6\cdot P_{Fe}$. Therefore, it increases the frequency of the surface Langmuir wave.
We can explicitly see this effect in Fig. 2. It reveals in the shift of the dashed blue line from the continuous red line.
For small spin polarization this effects rather small. Consequently, we see coincidence of the continuous red line and dashed red line in Fig. 3 at the large wave vectors $k$.
The area of the spectrum hybridization is located at the wave vectors $\kappa\approx\kappa_{0}=1$. Hybridization reveals in the fact that the Langmuir wave branch at $\kappa\ll\kappa_{0}$ fuses with the SEAW wave branch at $\kappa\gg\kappa_{0}$ and the SEAW branch at $\kappa\gg\kappa_{0}$ continuously turn in the Langmuir wave branch at $\kappa\ll\kappa_{0}$. Switching of branches happens at $\kappa\approx\kappa_{0}$. Branches of the Langmuir waves and the SEAWs would cross each other, but interaction of waves changes the structure of the spectrum.
Area of hybridization of the spectrum of two waves shows an area of strong interaction of these waves. In this area two waves are similar to two bound pendulums. Excitation of one of waves with $\kappa\approx\kappa_{0}$ leads to the energy transition to the second wave. Thus, excitation of the Langmuir waves and the increase of their energy to reach point $(k_{0},\omega(k_{0}))$ allows to generate the SSEAWs. This is the first report on the possibility of the SEAW generation, while several papers have been published to describe properties of two- and three-dimensional SEAWs Andreev PRE 15 , Andreev AoP 15 , Ryan PRB 91 , Agarwal PRL 11 .
The SEAWs are longitudinal waves, where concentration of the spin-up and spin-down electrons oscillate with opposite phase. It leads to the local collective motion of the spin-up and spin-down fluids in opposite directions and may be considered as a spin wave in the electron gas. Mostly, consideration of spin waves are related to the magnetic moments of lattice containing bound electron on unfilled d and f shells, even at the analysis of conducting materials. The SEAWs are examples of spin waves in the polarized electron gas along with other transverse spin waves reported in Refs. Andreev VestnMSU 2007 , Andreev TerSpinCurent .
Surface spelnons simultaneously show similarity and difference to the plasmons. Both of them have linear spectrum at the small wave vectors. However, plasmons dynamics is related to the electrodynamic effects and their phase velocity is close to the speed of light. Surface plasmons strongly interact with the small frequency light $\omega<\omega_{Le}$, which cannot penetrate in metals. Spelnons are also related to the electron oscillations, but the spelnons are longitudinal even at $k\rightarrow 0$. Judging on this property they show similarity to phonons (see an example of resent work on phonons Schuetz PRX 15 ), but the spelnons have intermediate phase velocities $c_{sn}\sim 0.01c$. Having spin nature and having intermediate properties between phonons and surface plasmons the spelnons become potentially useful and interesting subject of research.
Hybridization of the dispersion branches occurring at the linear wave coupling is a widely spread phenomenon Zheleznyakov SPU 83 . It happens in many cases including plasma physics Bogdankevich SUP 81 , Birau PU 97 , appearance of exciton-polaritons in semiconductors Deng RMP 10 , Byrnes NP 14 , metamaterials Rupin SR 15 . As we have shown above the SSEAWs in the spin polarized electron gas of ferromagnetic materials reveals a hybridization of the spectrum either.
In conclusion we want summarize that we described spectrum of collective excitations in the half spaced partially spin polarized electron gas of magnetically ordered materials containing the spin-electron acoustic wave and found regime of hybridization of the Langmuir wave and SEAW. We stress attention that strong interaction of waves in area of hybridization of wave dispersion dependencies leads to the energy transition from the Langmuir wave to the SEAW. It gave a mechanism of generation of the SSEAWs.
Work of P.A. is supported by the Dynasty foundation.
References
(1)
W. L. Barnes, A. Dereux and T. W. Ebbesen, Nature 424, 824 (2003).
(2)
V. Giannini, A. I. Fernandez-Dominguez, S. C. Heck, and S. A. Maier, Chem. Rev. 111, 3888 (2011).
(3)
Zh. Fang, X. Zhang, D. Liu, and X. Zhu, Appl. Phys. Lett. 93, 073306 (2008).
(4)
B. Wang, J. Teng, and X. Yuan, Appl. Phys. Lett. 98, 263111 (2011).
(5)
P. A. Andreev, Phys. Rev. E 91, 033111 (2015).
(6)
P. A. Andreev, L. S. Kuz’menkov, Annals of Physics 361, 278 (2015).
(7)
P. A. Andreev, L. S. Kuz’menkov, Eur. Phys. Lett. 113, 17001 (2016).
(8)
J. C. Ryan, Phys. Rev. B 43, 4499 (1991).
(9)
F. Perez, Phys. Rev. B 79, 045306 (2009).
(10)
A. Agarwal, M. Polini, R. Fazio, and G. Vignale, Phys. Rev. Lett. 107, 077004 (2011).
(11)
A. Agarwal, M. Polini, G. Vignale, M. E. Flatte, Phys. Rev. B 90, 155409 (2014).
(12)
P. A. Andreev, arXiv:1504.08234 (accepted in Phys. Plasmas).
(13)
Pavel A. Andreev, P. A. Polyakov, and L. S. Kuz’menkov, arXiv:1507.03295.
(14)
E. Michel, H. Ibach, and C. M. Schneider, Phys. Rev. B 92, 024407 (2015).
(15)
Y. Y. Teng and E. Stern, Phys. Rev. Lett. 19, 511 (1967).
(16)
R. H. Ritchie, Phys. Rev. 106, 874 (1957).
(17)
A. W. Trivelpiece, R. W. Gould, J. Appl. Phys. 30, 1784 (1959).
(18)
K. L. Ngai, N. N. Economou, and M. H. Cohen, Phys. Rev. Lett. 24, 61 (1970).
(19)
J. G. Endriz and W. E. Spicer, Phys. Rev. Lett. 24, 64 (1970).
(20)
N. Marschall, B. Fischer, and H. J. Queisser, Phys. Rev. Lett. 27, 95 (1971).
(21)
W. E. Anderson, R. W. Alexander, and R. J. Bell, Phys. Rev. Lett. 27, 1057 (1971).
(22)
Ralph L. Guernsey, Phys. Fluids 12, 1852 (1969).
(23)
K. W. Chiu and J. J. Quinn, Phys. Lett. 35A, 469 (1971).
(24)
K. W. Chiu, J. J. Quinn, Phys. Rev. B 5, 4707 (1972).
(25)
Roger D. Jones, Phys. Rev. Lett. 51, 1269 (1983).
(26)
L. D. Landau and E. M. Lifshitz, Statistical Physics, Part 2:
Theory of the Condensed State, Course of theoretical Physics
Vol. 9 (Pergamon Press, London, 1987).
(27)
A. D. Margulis and VI. A.
Marguli, Zh. Eksp. Teor. Fiz. 93, 1800 (1987).
(28)
Yu. O. Tyshetskiy, S. V. Vladimirov and R. Kompaneets,
J. Plasma Physics 79, 387 (2013).
(29)
Yu. Tyshetskiy, S. V. Vladimirov and R. Kompaneets
Phys. Plasmas 21, 122114 (2014).
(30)
Yu. Tyshetskiy, R. Kompaneets and S. V. Vladimirov,
Phys. Plasmas 19, 112107 (2012).
(31)
M. Lazar, P. K. Shukla and A. Smolyakov,
Phys. Plasmas 14, 124501 (2007).
(32)
A. Moradi, Phys. Plasmas 22, 014501 (2015).
(33)
M. Shahmansouri, Phys. Plasmas 22, 092106 (2015).
(34)
A. Moradi, Phys. Scr. 90, 085601 (2015).
(35)
A. Moradi, Phys. Lett. A 379, 1139 (2015).
(36)
L. S. Kuz’menkov, S. G. Maksimov, V. V. Fedoseev, Theor. Math.
Phys. 126, 110 (2001).
(37)
M. Marklund and G. Brodin,
Phys. Rev. Lett. 98, 025001 (2007).
(38)
P. A. Andreev, L. S. Kuz’menkov,
Moscow University Physics Bulletin 62, 271 (2007).
(39)
P. K. Shukla, B. Eliasson, Rev. Mod. Phys. 83, 885 (2011).
(40)
M. Iv. Trukhanova, Phys. Lett. A 379, 2777 (2015).
(41)
P. A. Andreev, L. S. Kuz’menkov, arXiv:1510.03468.
(42)
P. A. Andreev, Annals of Physics 350, 198 (2014).
(43)
M. J. A. Schuetz, E. M. Kessler, G. Giedke, L. M. K. Vandersypen,
M. D. Lukin, and J. I. Cirac, Phys. Rev. X 5, 031031 (2015).
(44)
V. V. Zheleznyakov, Vit. V. Kocharovskii and Vlad. V. Kocharovskii, Sov. Phys. Usp. 26, 877 (1983).
(45)
L. S. Bogdankevich, M. V. Kuzelev, A. A. Rukhadze, Sov. Phys. Usp. 24, 1 (1981).
(46)
M. Birau, M. A. Krasil’nikov, M. V. Kuzelev, A. A. Rukhadze, Phys. Usp. 40, 975 (1997).
(47)
H. Deng, H. Haug, Y. Yamamoto, Rev. Mod. Phys. 82, 1489 (2010).
(48)
T. Byrnes, N. Y. Kim, and Y. Yamamoto, Nature Physics 10, 803
(2014).
(49)
M. Rupin, P. Roux, G. Lerosey, F. Lemoult, Sci. Rep. 5, 13714 (2015). |
Phase transition in multicomponent field theory at finite temperature
JINR
E-mail
Elizaveta P. Yukalova
JINR
E-mail
yukalova@theor.jinr.ru
Abstract:
Nuclear matter at finite temperature and barion density exhibits several
phase transitions that could happen at the early stages of the Universe
evolution and could be realized in heavy-ion or hadron-hadron collisions.
Microscopic description of phase transitions is notoriously difficult
because of the absence of small parameters. Here we present a general
approach allowing to treat situations, when there are no small parameters.
The approach is based on optimized perturbation theory and self-similar
approximation theory. It allows, starting with divergent perturbation
series in powers of an asymptotically small parameter, to construct
expressions extrapolating asymptotic series to arbitrary values of the
parameter, including its infinite limit. Examples of such approximants are:
right root approximants, left root approximants, continued root approximants,
exponential approximants, and factor approximants. The approach is
illustrated by the phase transition of gauge symmetry breaking in a
multicomponent field theory. The found critical indices are in very good
agreement with Monte Carlo simulations as well as with complicated methods
of Padé-Borel summation, while our approach is much simpler. The nice
feature of the approach is that it gives exact values for the cases where
exact solutions are known.
1 Introduction
Varying temperature and baryon density, it is possible to realize several
different phases of nuclear matter. A qualitative phase portrait of admissible
phases, on temperature-baryon density plane, is shown in Fig. 1, where
$\rho_{0}=0.167$ fm${}^{-3}$ is the normal baryon density (e.g., [1]).
The variation of temperature and baryon density can be achieved in hadron-hadron
and heavy-ion collisions. The corresponding values of these thermodynamic
variables could also exist at the early stages of the Universe evolution or in
the cores of neutron stars. There can occur phase transitions of first order,
second order, as well as crossovers [2-4].
Phase transitions are known to be difficult for description because of the
absence of small parameters in the transition region. This especially concerns
the description of phase transitions in microscopic theory, where calculations
are possible only by resorting to a kind of perturbation theory. However,
perturbation theory usually results in divergent series that, in the best case,
have meaning in the limit of asymptotically small parameters, while the physical
parameters could be rather large, or even infinite. The notorious question is:
How it would be possible to extract information from perturbative series,
derived for asymptotically small parameters, for the values of finite and large
parameters?
There exist methods for effective summation of divergent series, such as Padé
summation [5] and Borel summation [6]. However, the former exhibits a number of
deficiencies, including the appearance of spurious poles, while the second is quite
complicated and requiring the knowledge of large-order terms in power law
expansions. Both these methods not always are applicable, as is discussed in [7].
In the present report, we describe an original approach to treating divergent
perturbative series, extracting from them meaningful answers, providing good
accuracy, with being much simpler and more general than Padé-Borel summation.
We illustrate the approach by the example of a phase transition in
multicomponent field theory at finite temperature.
2 Optimized perturbation theory
The first step of the approach is optimized perturbation theory, based on the
definition of control functions reorganizing divergent series to
convergent ones. The optimized perturbation theory was advanced in 1973, in
Thesis [8], submitted for publication in 1974, and published [9,10] in 1976.
The power of this method was illustrated by treating some anharmonic models
and strongly anharmonic quantum crystals [8-18]. Later this theory has been
applied to numerous models, under different guises and under different names,
such as modified perturbation theory, variational perturbation theory,
renormalized perturbation theory, oscillator representation, delta expansion,
optimized expansion, nonperturbative expansion, and so on (e.g., [19-24]).
All these works have used the variants of the same idea [8-18] of introducing
control functions renormalizing divergent perturbative series into convergent
series. In this section, we briefly delineate the idea of optimized perturbation
theory, as advanced in [8-18] and reviewed in [25,26].
Suppose we are interested in finding a function $f(x)$ satisfying a complicated
equation that cannot be solved exactly, but can be treated only by a kind of
perturbation theory. To explain the main idea, we consider here, for simplicity,
a real function of a real variable. A generalization to complex functions and
variables is straightforward.
Let perturbation theory give a divergent perturbative sequence $\{f_{k}(x)\}$,
with $k=0,1,2,\ldots$ being an approximation-order index. The basic idea of
optimized perturbation theory is to introduce a set $\{u_{k}(x)\}$ of control
functions that would allow us to reorganize the divergent perturbative sequence
$\{f_{k}(x)\}$ into a convergent sequence $\{F_{k}(x,u_{k})\}$, where $u_{k}=u_{k}(x)$.
A sequence is convergent if and only if it satisfies the Cauchy criterion:
For each positive $\epsilon$, there exists an order $k_{\epsilon}$, such that
$$|F_{k+p}(x,u_{k+p})-F_{k}(x,u_{k})|<\varepsilon\;,$$
(1)
for $k>k_{\epsilon}$ and any positive $p=0,1,2,\ldots$.
Control functions can be introduced in several ways, three of which are the
most common: (i) through initial conditions, (ii) through variable changes,
and (iii) through functional transformations.
Introduction of control functions through initial conditions
The simplest illustration of this way is when the sought function $f(x)$ is
a solution of a functional equation
$${\cal E}[f(x)]=0\;.$$
(2)
Then, starting with an initial approximation $F_{0}(x,u)$, including a control
function $u$, it is admissible to represent the functional equation (2) as an
iterative procedure
$$F_{k}(x,u_{k})=F_{k-1}(x,u_{k-1})+{\cal E}[F_{k-1}(x,u_{k-1})]\;.$$
(3)
As a rule, physical systems are characterized by their Hamiltonians, or
Lagrangians. Say, $H$ is a Hamiltonian of a complicated system that can be
treated only by means of perturbation theory. Taking for the zero approximation
a simple Hamiltonian $H_{0}(u)$, containing control functions, one can rewrite
the system Hamiltonian as
$$H=H_{0}(u)+[H-H_{0}(u)]\;.$$
(4)
Then, by perturbation theory with respect to the difference $(H-H_{0})$, one
obtains higher approximations, depending on the considered problem, either for
wave functions or for Green functions. Knowing the latter, one can calculate
the corresponding approximations for observables $\hat{A}(x)$ as the averages
$$F_{k}(x,u_{k})=\langle\hat{A}(x)\rangle_{k}\;,$$
(5)
defined for the related $k$-order approximate wave, or Green, functions.
Introduction of control functions through variable changes
It is possible to change the variable $x$ through the relations
$$x=x_{k}(z,u_{k})\;,\qquad z=z_{k}(x,u_{k})\;,$$
(6)
involving control functions, thus, getting
$$f_{k}(x)=f_{k}(x_{k}(z,u_{k}))\;.$$
(7)
The latter expression can be expanded in powers of the new variable $z$, so
that
$$f_{k}(x_{k}(z,u_{k}))\simeq\overline{f}_{k}(z,u_{k})\qquad(z\rightarrow 0)\;.$$
(8)
With the inverse variable change (6), one has
$$\overline{f}_{k}(z,u_{k})=\overline{f}_{k}(z_{k}(x,u_{k}),u_{k})\;.$$
(9)
This allows us to define
$$F_{k}(x,u_{k})=\overline{f}_{k}(z_{k}(x,u_{k}),u_{k})\;.$$
(10)
Introduction of control functions through functional transformations
The sought function can be subject to a transformation containing control
functions,
$$\hat{T}(u)f(x)=F(x,u)\;,$$
(11)
with the inverse transformation
$$f(x)=\hat{T}^{-1}(u)F(x,u)\;.$$
(12)
Then we define
$$F_{k}(x,u_{k})=\hat{T}(u_{k})f_{k}(x)\;.$$
(13)
Formulation of equations for optimal control functions
After control functions are incorporated into $F_{k}(x,u_{k})$, it is necessary
to formulate explicit equations for their calculations. By their meaning, the
control functions are to be defined in such a way that to induce convergence
for the sequence $\{F_{k}(x,u_{k})\}$. Since convergence is characterized by the
Cauchy criterion (1), the optimal control functions, in the spirit of optimal
control theory, can be defined as the minimizers of the Cauchy cost functional
$${\cal C}_{p}[u]=\frac{1}{2}\;\sum_{k}|F_{k+p}(x,u_{k+p})-F_{k}(x,u_{k})|^{2}\;,$$
(14)
whose minimization provides the fastest convergence of the sequence
$\{F_{k}(x,u_{k})\}$. That is, we need to look for the minimal value
$$\min_{u}|F_{k+p}(x,u_{k+p})-F_{k}(x,u_{k})|$$
(15)
of the difference $F_{k+p}-F_{k}$ for any given $p$.
The minimization condition (15) involves two control functions
$u_{k+p}$ and $u_{k}$, which makes it impossible to define both of them
simultaneously. Hence, we cannot find the exact absolute minimum of the
Cauchy cost functional (14), but we can try to find its approximate minimum.
Assuming that the control functions $u_{k+p}$ and $u_{k}$, and the related
terms $F_{k+p}$ and $F_{k}$ are close to each other, we can express $F_{k+p}$ as
$$F_{k+p}(x,u_{k+p})\approx F_{k+p}(x,u_{k})+\frac{\partial F_{k}(x,u_{k})}{%
\partial u_{k}}\;(u_{k+p}-u_{k})\;.$$
(16)
Then minimization (15) becomes
$$\min_{u}\left|F_{k+p}(x,u_{k})-F_{k}(x,u_{k})+\frac{\partial F_{k}(x,u_{k})}{%
\partial u_{k}}\;(u_{k+p}-u_{k})\right|\;.$$
(17)
Depending on the relation between the difference $F_{k+p}-F_{k}$ and the
term containing the derivative, there can be two cases. When the derivative
term is smaller than the difference term, then minimization (17) is
approximately satisfied under the minimal difference condition
$$F_{k+p}(x,u_{k})-F_{k}(x,u_{k})=0\;.$$
(18)
But when the difference term is smaller than the derivative term, then
minimization (17) is approximately valid under the condition
$$(u_{k+p}-u_{k})\;\frac{\partial F_{k}(x,u_{k})}{\partial u_{k}}=0\;.$$
(19)
This has to be understood as the minimal derivative condition
$$\frac{\partial F_{k}(x,u_{k})}{\partial u_{k}}=0\;,$$
(20)
provided the latter possesses a solution. In case there are no solutions,
one has to set $u_{k+p}=u_{k}$.
As is evident, the minimal difference and minimal derivative conditions
are absolutely equivalent. Of course, in particular cases, one of them can
yield a better accuracy than the other. However, in general, it is impossible
to conclude that one is preferable to the other.
In this way, the optimized perturbation theory [8-18] consists of the
following steps. The divergent perturbative sequence $\{f_{k}(x)\}$ is
reorganized into the sequence $\{F_{k}(x,u_{k})\}$ incorporating control functions
$u_{k}=u_{k}(x)$ making the latter sequence convergent. Control functions can
be introduced in three ways, through initial conditions, through variable
changes, or through functional transformations. Explicit equations for the
control functions are derived from the minimization of the Cauchy cost
functional. Approximate minimization can be done by means of either minimal
difference or minimal derivative conditions.
3 Self-similar approximation theory
Optimized perturbation theory has been used for various physical systems.
It is not our aim here to give a review if these numerous applications.
Just let us mention a couple of review-type articles [25,26], where further
citations can be found. Despite a variety of very successful and wide
applications of optimized perturbation theory, several questions remained
unanswered:
(i) How it would be possible to improve accuracy within the given number of
perturbative terms?
(ii) What is a necessary condition that the Cauchy cost functional could
reach its absolute minimum, that is zero?
(iii) How to decide which of the ways of introducing control functions would
be the best one, when there are several such ways, say, by choosing different
initial approximations?
(iv) Could it be feasible to check the stability of the calculational
procedure, when no exact solutions are available, that would allow for the
explicit comparison of these exact solutions with the obtained approximations?
(v) Is it possible to define general approximants, enjoying a fixed prescribed
structure, extrapolating the series, derived for an asymptotically small
variable $x\rightarrow 0$, to the arbitrary values of this variable from the whole
interval $[0,\infty)$?
All these questions have been answered in self-similar approximation theory
advanced in [27-33]. The main idea of this approach is to reformulate
perturbation theory to the language of dynamical theory, considering the
approximation order $k$ as discrete time, so that the approximation sequence
$\{F_{k}(x,u_{k})\}$ be isomorphic to the trajectory of a cascade. Then the
effective sequence limit will correspond to the cascade fixed point. And
the control of the calculational procedure stability will be equivalent
to the analysis of the dynamical system stability.
Let us define the expansion function $x=x_{k}(\varphi)$ by the
reonomic constraint
$$F_{0}(x,u_{k}(x))=\varphi\;,\qquad x=x_{k}(\varphi)\;.$$
(21)
Introduce the endomorphism
$$y_{k}(\varphi)\equiv F_{k}(x_{k}(\varphi),u_{k}(x_{k}(\varphi)))$$
(22)
that, owing to constraint (21), enjoys the initial condition
$$y_{0}(\varphi)=\varphi\;.$$
(23)
The inverse to this endomorphism is
$$F_{k}(x,u_{k}(x))=y_{k}(F_{0}(x,u_{k}(x)))\;.$$
(24)
In terms of this endomorphism, the Cauchy cost functional (14) takes the form
$${\cal C}_{p}[u]=\frac{1}{2}\;\sum_{k}|y_{k+p}(\varphi)-y_{k}(\varphi)|^{2}\;.$$
(25)
This functional is exactly zero, provided that
$$y_{k+p}(\varphi)=y_{k}(\varphi)$$
(26)
for all $k\geq 0$. In particular, for $k=0$, we have
$$y_{p}(\varphi)=y_{0}(\varphi)=\varphi\;.$$
(27)
Combining (26) and (27) yields the functional self-similarity relation
$$y_{k+p}(\varphi)=y_{k}(y_{p}(\varphi))\;.$$
(28)
Since transformation (28) possesses the semi-group property
$y_{k}\circ y_{p}=y_{k+p}$, it can also be called group self-similarity.
Thus, the self-similarity relation (28) is a necessary condition for
the Cauchy cost functional to be zero. Although it is not a sufficient condition.
The family of the endomorphisms $\{y_{k}\}$, with the group relation (28), forms
a dynamical system in discrete time, termed cascade. By construction,
the cascade trajectory $\{y_{k}(\varphi):k=0,1,2,\ldots\}$ is bijective to the
approximation sequence $\{F_{k}(x,u_{k}(x)):k=0,1,2,\ldots\}$.
The bijectivity of the cascade trajectory and approximation sequence means the
following. If there exists the limit
$$y^{*}(\varphi(x))\equiv\lim_{k\rightarrow\infty}y_{k}(\varphi)\;,$$
(29)
where
$$\varphi(x)\equiv\lim_{k\rightarrow\infty}F_{0}(x,u_{k}(x))\;,$$
(30)
then there also exists the limit
$$F^{*}(x)\equiv\lim_{k\rightarrow\infty}F_{k}(x,u_{k}(x))\;,$$
(31)
such that
$$F^{*}(x)=y^{*}(\varphi(x))\;.$$
(32)
Note that the existence of a trajectory limiting point $y^{*}$ guarantees the
existence of a sequence limit $F^{*}$, however this does not guarantee that $F^{*}$
necessarily corresponds to the sought function $f(x)$. Such a correspondence
is an assumption typical of calculational procedures dealing with nonlinear
problems [34,35], for which the accuracy of approximations at each step cannot
be explicitly established.
For a dynamical system, the existence of a limiting trajectory point is
equivalent to the existence of a stable fixed point. Therefore, if the cascade
trajectory $\{y_{k}\}$, with increasing $k$, tends to a limiting point $y^{*}$,
the latter is a fixed point, such that
$$y_{k}(y^{*})=y^{*}\;.$$
(33)
It is more convenient to deal with a dynamical system in continuous time,
instead of a system in discrete time. This can be realized by embedding the
approximation cascade into an approximation flow,
$$\{y_{k}(\varphi):\;k\in\mathbb{Z}_{+}\}\subset\{y_{t}(\varphi):\;t\in\mathbb{R%
}_{+}\},$$
(34)
with the flow trajectory passing through all points of the cascade trajectory:
$$y_{t}(\varphi)=y_{k}(\varphi)\qquad(t=k)\;.$$
(35)
For the dynamical system in continuous time, it is straightforward to write
down the flow evolution equation that is the Lie equation
$$\frac{\partial}{\partial t}\;y_{t}(\varphi)=v(y_{t}(\varphi))\;,$$
(36)
where the right-hand side is the flow velocity
$$v(\varphi)\equiv\left[\frac{\partial}{\partial t}\;y_{t}(\varphi)\right]_{t=0}\;.$$
Integrating (36) between a given point of the cascade trajectory
$y_{k}=y_{k}(\varphi)$ and a point $y_{k}^{*}=y_{k}^{*}(\varphi)$, we get the
evolution integral
$$\int_{y_{k}}^{y_{k}^{*}}\;\frac{dy}{v_{k}(y)}=t_{k}\;,$$
(37)
in which $t_{k}$ is the time of motion from $y_{k}$ to $y_{k}^{*}$, while $v_{k}$
is the flow velocity on this time interval.
Remembering that the cascade is embedded into the flow, the flow velocity,
near the time $t=k$, employing the Euler discretization, can be expressed
through the cascade velocity
$$v_{k}(\varphi)=F_{k+1}(x_{k},u_{k})-F_{k}(x_{k},u_{k})+(u_{k+1}-u_{k})\;\frac{%
\partial}{\partial u_{k}}\;F_{k}(x_{k},u_{k})\;,$$
(38)
where $x_{k}=x_{k}(\varphi)$ and $u_{k}=u_{k}(x_{k})$. Invoking the bijective
relation between $y_{k}$ and $F_{k}$, the evolution integral can be represented as
$$\int_{F_{k}}^{F_{k}^{*}}\;\frac{d\varphi}{v_{k}(\varphi)}=t_{k}\;.$$
(39)
The motion time $t_{k}$ can be treated as an additional control function.
In the simplest cases, it can be set to one or $1/k$ or defined through
additional conditions [25,26].
If $v_{k}$ were zero, then $y_{k}^{*}$ would be an exact fixed point of the cascade.
Unfortunately, it is difficult to set $v_{k}$ zero, since it contains two, yet
unknown, control functions. But we can require the minimal possible velocity,
in that way defining the control functions $u_{k}=u_{k}(x)$ by the condition
$$\min_{u_{k}}\left|F_{k+1}(x,u_{k})-F_{k}(x,u_{k})+(u_{k+1}-u_{k})\;\frac{%
\partial}{\partial u_{k}}\;F_{k}(x,u_{k})\right|\;.$$
(40)
This minimization is equivalent to minimization (17), with $p=1$. Analogously
to the previous consideration, an approximate minimization can be done by one
of the conditions (18) or (19). Condition (19) seems to be more convenient,
which results in the velocity
$$v_{k}(\varphi)=F_{k+1}(x_{k},u_{k})-F_{k}(x_{k},u_{k})\;.$$
Employing this in the evolution integral (39) yields the renormalized
approximant $F_{k}^{*}$.
Defining the control functions from the minimization of the cascade velocity
implies that $y_{k}^{*}(\varphi)$ is an approximate fixed point, or quasi-fixed
point. Respectively, $F_{k}^{*}(x,u_{k})$ corresponds to an effective limit $f_{k}^{*}(x)$
that is named the self-similar approximation of $f(x)$. In this way,
we have the correspondence
$$y_{k}^{*}(\varphi(x))=F_{k}^{*}(x,u_{k}(x))\rightarrow f_{k}^{*}(x)\;.$$
(41)
The improvement of the accuracy of the self-similar approximation $f_{k}^{*}(x)$,
as compared to the optimized approximation $F_{k}(x,u_{k}(x))$, is due to the
following reason. The minimization of the Cauchy cost functional is equivalent
to the minimization of the cascade velocity, which gives the optimized
approximant $F_{k}(x,u_{k}(x))$. However, the cascade velocity is not exactly zero.
The Lie equation (36) describes the motion from the given optimized approximant
$F_{k}(x,u_{k}(x))$ to the quasi-fixed point $F_{k}^{*}(x,u_{k})$ that improves the
accuracy of the former approximant.
As is mentioned above, to represent the effective limit of the approximation
sequence, the fixed point has to be stable. The stability of the procedure
here coincides with the stability of motion of the dynamical system, which is
characterized by the map multiplier
$$\mu_{k}(\varphi)\equiv\frac{\partial}{\partial\varphi}\;y_{k}(\varphi)\;.$$
(42)
The motion at the point $y_{k}(\varphi)$ is locally stable, provided that
$$|\mu_{k}(y_{k}(\varphi))|<1\;.$$
(43)
The multiplier at the quasi-fixed point is
$$\mu_{k}^{*}(\varphi)\equiv\mu_{k}(y_{k}^{*}(\varphi))\;.$$
(44)
The quasi-fixed point is stable, when
$$|\mu_{k}(F_{k}^{*}(x,u_{k}(x))|<1\;,$$
(45)
where relation (41) is used.
Because in the treated case, the fixed point is a function of $x$, it is
possible to consider the maximal multiplier
$$\mu_{k}^{*}\equiv\sup_{\varphi}|\mu_{k}^{*}(\varphi)|=\sup_{x}|\mu_{k}(F_{k}^{%
*}(x,u_{k}(x)))|\;.$$
(46)
Then we say that a quasi-fixed point is uniformly stable, when
$$|\mu_{k}^{*}|<1\;.$$
(47)
The analysis of the procedure stability makes it possible to answer the
question on which of the procedures is preferable, when there are several
admissible procedures differing by the way of introducing control functions.
For example, it is possible to introduce control functions by different
initial approximations, as has been analyzed for anharmonic models [25].
For strongly anharmonic quantum crystals, it is possible to choose different
initial approximations, say, Hartree or Hartree-Fock [17,36,37]. Or one
can introduce control functions by different changes of variables [6].
The answer is: That procedure is preferable that is more stable, since a
more stable procedure is assumed to be faster convergent [25,26].
Finally, we give the answer to the problem whether it is feasible to construct
general expressions extrapolating the series in powers of an asymptotically
small variable $x\rightarrow 0$ to its arbitrary values in the whole range
$x\in[0,\infty)$.
Suppose, the sought function can be found only for an asymptotically small
variable,
$$f(x)\simeq f_{k}(x)\qquad(x\rightarrow 0)\;,$$
(48)
where it is given by the asymptotic expansion
$$f_{k}(x)=f_{0}(x)\left(1+\sum_{n=1}^{k}a_{n}x^{n}\right)\;.$$
(49)
Such series are usually divergent for any finite $x$.
It is convenient to consider the normalized function defined by the ratio
$$\frac{f_{k}(x)}{f_{0}(x)}=1+\sum_{n=1}^{k}a_{n}x^{n}$$
(50)
that, by construction, satisfies the limit
$$\lim_{x\rightarrow 0}\;\frac{f_{k}(x)}{f_{0}(x)}=1\;.$$
Control functions can be introduced by applying to series (50) the method
of fractal transforms [26,38-45] defined as
$$F_{k}(x,u)=\frac{f_{k}(x)}{f_{0}(x)}\;x^{u}\;.$$
Then, following the self-similar approximation theory by accomplishing
several times the renormalization procedure, we come, depending on the
available boundary conditions, to one of the following approximants.
Right root approximants
$$\frac{f_{k}^{*}(x)}{f_{0}(x)}=\left(\left(\ldots(1+A_{1}x)^{n_{1}}+A_{2}x^{2}%
\right)^{n_{2}}+\ldots+A_{k}x^{k}\right)^{n_{k}}$$
(51)
that can be used, when a number of terms in the large-variable expansion
$x\rightarrow\infty$ are known. All parameters $A_{i}$ and $n_{i}$ are uniquely defined
through this expansion [26,41,43,44].
Left root approximants
$$\frac{f_{k}^{*}(x)}{f_{0}(x)}=\left(\left(\left(\ldots(1+A_{1}x)^{2}+A_{2}x^{2%
}\right)^{3/2}+A_{3}x^{3}\right)^{4/3}+\ldots+A_{k}x^{k}\right)^{n_{k}}\;,$$
(52)
in which the sole power $n_{k}$ is defined from the large-variable behavior
$x\rightarrow\infty$, while all parameters $A_{i}$ are found from the
accuracy-through-order procedure after re-expanding (52) in powers of $x\rightarrow 0$
and comparing this with the initial expansion (49). We may note that (52) is
a particular case of (51), with
$$n_{j}=\frac{j+1}{j}\qquad(j=1,2,\ldots,k-1)\;,$$
which involves all $A_{i}$ in the definition of the large-variable amplitude [7,46].
Continued root approximants
$$\frac{f_{k}^{*}(x)}{f_{0}(x)}=\left(1+A_{1}x\left(1+A_{2}x\ldots(1+A_{k}x)^{s}%
\right)^{s}\ldots\right)^{s}\;,$$
(53)
in which the power $s$ is prescribed by the large-variable behavior, while
all parameters $A_{i}$ are given by the accuracy-through-order procedure at
$x\rightarrow 0$. In the particular case of $s=-1$, approximants (53) are reduced to
continued fractions and, hence, to Padé approximants [47].
Exponential approximants
$$\frac{f_{k}^{*}(x)}{f_{0}(x)}=\exp(b_{1}x\;\exp(b_{2}x\ldots\exp(b_{k}x))%
\ldots)\;,$$
(54)
where the control functions $b_{i}$ are defined by additional conditions [26,40-42],
like the minimal difference condition (18). More elaborated variants of defining
these control functions are also possible [26], e.g.,
$$b_{n}=\frac{a_{n}(1+a_{1}^{2})}{na_{n-1}(1+a_{n}^{2})}\qquad(n=1,2,\ldots,k)\;,$$
where $a_{n}$ are the parameters of expansion (49).
Factor approximants
$$\frac{f_{k}^{*}(x)}{f_{0}(x)}=\prod_{i=1}^{N_{k}}(1+A_{i}x)^{n_{i}}\;,$$
(55)
in which
$$\displaystyle N_{k}=\left\{\begin{array}[]{ll}k/2,&~{}k=2,4,\ldots\\
(k+1)/2,&~{}k=3,5,\ldots\end{array}\right.$$
and all parameters $A_{i}$ and $n_{i}$ are defined from the re-expansion procedure
at $x\rightarrow 0$, equating the like-order terms [48-52].
4 Multicomponent field theory
Let us illustrate the application of self-similar approximation theory to
describing a phase transition in $N$-component $\varphi^{4}$ field theory in
$d$-dimensional space. The Hamiltonian of this field theory is
$$H[\varphi]=\int\left\{\frac{1}{2}\left[\frac{\partial\varphi(x)}{\partial x}%
\right]^{2}+\frac{m^{2}}{2}\;\varphi^{2}(x)+\frac{\lambda}{4!}\;\varphi^{4}(x)%
\right\}\;dx\;,$$
(56)
where the standard notations are employed:
$$\varphi(x)=\{\varphi_{n}(x):\;n=1,2,\ldots,N\}\;,\qquad x=\{x_{\alpha}:\;%
\alpha=1,2,\ldots,d\}\;,$$
$$\varphi^{2}(x)\equiv\sum_{n=1}^{N}\varphi_{n}^{2}(x)\;,\qquad\left[\frac{%
\partial\varphi(x)}{\partial x}\right]^{2}\equiv\sum_{n=1}^{N}\;\sum_{\alpha=1%
}^{d}\left[\frac{\partial\varphi_{n}(x)}{\partial x_{\alpha}}\right]^{2}\;.$$
Hamiltonian (56) is invariant under the reflection
$$\varphi_{n}(x)\rightarrow-\varphi_{n}(x)\qquad(n=1,2,\ldots N)\;,$$
(57)
so that $H[-\varphi]=H[\varphi]$. This means that the statistical average
of the field is zero: $\langle\varphi\rangle=0$. The thermodynamic
potential
$$F[\langle\varphi\rangle]=-T\ln{\rm Tr}e^{-\beta H[\varphi]}\;,$$
(58)
where $T$ is temperature and $\beta\equiv 1/T$, is also invariant under
reflection (57).
It turns out that there exists a critical temperature $T_{c}$, such that above
this temperature, the order parameter $\langle\varphi\rangle$ is zero, while
below $T_{c}$ it is nonzero:
$$\displaystyle\begin{array}[]{ll}\langle\varphi(x)\rangle=0&~{}~{}~{}~{}~{}~{}~%
{}(T>T_{c}),\\
\langle\varphi(x)\rangle\neq 0&~{}~{}~{}~{}~{}~{}~{}(T<T_{c}),\end{array}$$
(59)
which implies that
$$F[\langle\varphi(x)\rangle\neq 0]<F[\langle\varphi(x)\rangle=0]\;,$$
(60)
for $T<T_{c}$. The zero $\langle\varphi\rangle$ means that all
$\langle\varphi_{n}\rangle$ are zero. And nonzero $\langle\varphi\rangle$
assumes that at least some of $\varphi_{n}$ are nonzero. This phase transition
is accompanied by the inversion symmetry breaking.
The properties of thermodynamic quantities in the critical region, where
the relative temperature
$$\tau\equiv\frac{|T-T_{c}|}{T_{c}}\rightarrow 0$$
(61)
is small, are characterized by the critical indices describing the behavior of
the specific heat,
$$C_{V}\propto\tau^{-\alpha}\;,$$
order parameter
$$\langle\varphi\rangle\propto\tau^{\beta}\;,$$
and isothermic compressibility
$$\kappa_{T}\propto\tau^{-\gamma}\;.$$
The dependence of an external field on the order parameter, at the critical
temperature, is of the type
$$h\propto|\langle\varphi\rangle|^{\delta}\qquad(T=T_{c})\;.$$
The pair correlation function, at large distance $r\equiv|{\bf r}|$, behaves
as
$$g({\bf r})\propto\frac{\exp(-r/\xi)}{r^{d-2+\eta}}\qquad(r\rightarrow\infty)\;,$$
with the correlation length
$$\xi\propto\tau^{-\nu}\;.$$
And the vertex at $T_{c}$ exhibits the behavior
$$\Gamma(k)\propto 1+ck^{\omega}\qquad(T=T_{c})\;.$$
Not all seven critical indices,
$\alpha,\;\beta,\;\gamma,\;\delta,\;\eta,\;\nu,\;\omega$, are
independent. There are the so called scaling relations [53], due to Griffith,
$$\alpha+\beta(1+\delta)=2$$
(62)
and Widom,
$$\gamma+\beta(1-\delta)=0\;,$$
(63)
from which the Rushbrook relation
$$\alpha+2\beta+\gamma=2$$
(64)
follows. Also, the hyperscaling relations are known:
$$\alpha=2-\nu d\;,\qquad\beta=(d-2+\eta)\;\frac{\nu}{2}\;,$$
$$\gamma=(2-\eta)\nu\;,\qquad\delta=\frac{d+2-\eta}{d-2+\eta}\;.$$
(65)
Thus, only three critical indices, say $\eta,\;\nu,\;\omega$, can be
treated as independent, and all others can be expressed through them.
The critical indices $\eta,\;\nu,\;\omega$ can be represented [53] as
expansions in powers of the variable $\varepsilon\equiv 4-d$, formally
valid for asymptotically small $\varepsilon\rightarrow 0$. We extrapolate such
$\varepsilon$-expansions by means of the factor approximants (55) and set
$\varepsilon=1$ corresponding to $d=3$-dimensional space. The results for
all critical indices and different $N$ are presented in the Table. The found
values are in perfect agreement with experimental results, when these are
available, and with numerical Monte Carlo calculations, as well as with
complicated Padé-Borel summations, as discussed in [54]. It is interesting
that for the limits $N=-2$ and $N\rightarrow\infty$ our method provides exact
known results.
Table: Critical indices for $N$-component $\varphi^{4}$ field theory.
$$N$$
$$\alpha$$
$$\beta$$
$$\gamma$$
$$\delta$$
$$\eta$$
$$\nu$$
$$\omega$$
-2
0.5
0.25
1
5
0
0.5
0.80118
-1
0.36844
0.27721
1.07713
4.88558
0.019441
0.54385
0.79246
0
0.24005
0.30204
1.15587
4.82691
0.029706
0.58665
0.78832
1
0.11465
0.32509
1.23517
4.79947
0.034578
0.62854
0.78799
2
-0.00625
0.34653
1.31320
4.78962
0.036337
0.66875
0.78924
3
-0.12063
0.36629
1.38805
4.78953
0.036353
0.70688
0.79103
4
-0.22663
0.38425
1.45813
4.79470
0.035430
0.74221
0.79296
5
-0.32290
0.40033
1.52230
4.80254
0.034030
0.77430
0.79492
6
-0.40877
0.41448
1.57982
4.81160
0.032418
0.80292
0.79694
7
-0.48420
0.42676
1.63068
4.82107
0.030739
0.82807
0.79918
8
-0.54969
0.43730
1.67508
4.83049
0.029074
0.84990
0.80184
9
-0.60606
0.44627
1.71352
4.83962
0.027463
0.86869
0.80515
10
-0.65432
0.45386
1.74661
4.84836
0.025928
0.88477
0.80927
50
-0.98766
0.50182
1.98402
4.95364
0.007786
0.99589
0.93176
100
-0.89650
0.48334
1.92981
4.99264
0.001229
0.96550
0.97201
1000
-0.99843
0.49933
1.99662
4.99859
0.000235
0.99843
0.99807
10000
-0.99986
0.49993
1.99966
4.99986
0.000024
0.99984
0.99979
$$\infty$$
-1
0.5
2
5
0
1
1
5 Discussion
We have considered the challenge of defining effective limits of divergent
series by means of renormalization techniques. The necessity of this
renormalization is dictated by the frequent occurrence of such divergent
series in complicated physical problems, e.g., arising when investigating
phase transitions. The main ideas of optimized perturbation theory and
self-similar approximation theory are presented.
The name self-similarity comes from the self-similar relation (28)
that is a necessary condition for the absolute minimum of the Cauchy cost
functional (25). The property of group self-similarity is equivalent to
renormalization group in field theory [55]. To show this, let us introduce
the variable
$$\tau_{k}\equiv e^{k}\qquad(k=0,1,2,\ldots)\;.$$
(66)
Instead of (22), we can define
$$y(\tau_{k},\varphi)\equiv F_{k}(x_{k}(\varphi),u_{k}(x_{k}(\varphi))\;.$$
(67)
Since $\tau_{0}=1$ at $k=0$, the initial condition (23) becomes
$$y(1,\varphi)=\varphi\;.$$
(68)
In the place of the self-similar relation (28), we now have
$$y(\tau_{k}\tau_{p},\varphi)=y(\tau_{k},y(\tau_{p},\varphi))\;,$$
(69)
which defines a cascade. Embedding the cascade into a flow, according to (34),
with the discrete $\tau_{k}$ changing to the continuous $\tau\in[0,\infty)$,
we obtain the scaling property
$$y(\mu\tau,\varphi)=y(\tau,y(\mu,\varphi))\;.$$
(70)
This group property is typical of the renormalization group equations in
field theory [56]. Instead of the Lie equation (36), we get
$$\frac{\partial y(\tau,\varphi)}{\partial\ln\tau}=\beta(y(\tau,\varphi))\;,$$
(71)
where the right-hand side
$$\beta(\varphi)\equiv\left[\frac{\partial}{\partial\tau}\;y(\tau,\varphi)\right%
]_{\tau=1}$$
(72)
is analogous to the Gell-Mann-Low function [57].
As a simple example of the application of self-similar approximation theory,
let us briefly mention the calculation of the ground-state energy level for
the one-dimensional anharmonic oscillator [58,59] with the Hamiltonian
$$\hat{H}=-\;\frac{1}{2}\;\frac{d^{2}}{dx^{2}}+\frac{1}{2}\;x^{2}+gx^{4}\;,$$
(73)
where $g\in[0,\infty)$ is the anharmonicity, or coupling parameter. The
introduction of a control function can be done through initial conditions,
by starting perturbation theory with the Hamiltonian
$$\hat{H}_{0}=-\;\frac{1}{2}\;\frac{d^{2}}{dx^{2}}+\frac{u^{2}}{2}\;x^{2}\;.$$
(74)
The control function $u_{k}(g)$ is defined by the quasi-fixed point condition (20).
To first order of optimized perturbation theory, we have the energy $E_{1}=E_{1}(g)$.
Using in the evolution integral (39), the cascade velocity $v_{1}$ and $\tau_{1}=1$,
we find for the self-similar approximation of the ground-state energy $E^{*}=E^{*}(g)$
the equation
$$\frac{4(E^{*})^{2}-1}{4E_{1}^{2}-1}=\exp\left\{\frac{1}{4(E^{*})^{2}-1}\;-\;%
\frac{1}{4E_{1}^{2}-1}\;-\;\frac{1}{24}\right\}\;.$$
(75)
The comparison of $E^{*}(g)$ with numerical calculations from the direct solution
of the Schrödinger equation [60] shows that the found self-similar
approximation $E^{*}(g)$ is applicable for all values of $g\in[0,\infty)$,
yielding quite accurate results, whose maximal error does not exceed $0.1\%$.
In Sec. 4, we have illustrated the approach by calculating the critical indices
for the inversion symmetry-breaking phase transition in an $N$-component
$\varphi^{4}$-field theory. The results are in prefect agreement with experimental
measurements as well as with complicated numerical techniques, such as Monte Carlo
simulations or Padé-Borel summation. The advantage of our theory, as compared
with other numerical methods, is the combination of much greater simplicity and
very good accuracy.
References
[1]
K. Furushima and C. Sasaki,
Prog. Part. Nucl. Phys. 72, 99 (2013).
[2]
V.I. Yukalov and E.P. Yukalova,
Phys. Part. Nucl. 28, 37 (1997).
[3]
V.I. Yukalov and E.P. Yukalova,
Physica A 243, 382 (1997).
[4]
V.I. Yukalov and E.P. Yukalova,
POS ISHEPP 21, 046 (2012).
[5]
G.A. Baker and P. Graves-Moris,
Padé Approximants (Cambridge University, Cambridge, 1996).
[6]
H. Kleinert,
Path Integrals (World Scientific, Singapore, 2006).
[7]
S. Gluzman and V.I. Yukalov,
Eur. J. Appl. Math. 25, 595 (2014).
[8]
V.I. Yukalov,
Ph.D. Thesis (Moscow State University, Moscow, 1973).
[9]
V.I. Yukalov,
Moscow Univ. Phys. Bull. 31, 10 (1976).
[10]
V.I. Yukalov,
Theor. Math. Phys. 28, 652 (1976).
[11]
V.I. Yukalov,
Deponent VINITI 3684-75 (1975).
[12]
V.I. Yukalov,
Physica A 89, 363 (1977).
[13]
V.I. Yukalov,
Ann. Physik 36, 31 (1979).
[14]
V.I. Yukalov,
Ann. Physik 37, 171 (1980).
[15]
V.I. Yukalov,
Ann. Physik 38, 419 (1981).
[16]
V.I. Yukalov,
Phys. Lett. A 81, 433 (1981).
[17]
V.I. Yukalov and V.I. Zubov,
Fortschr. Phys. 31, 627 (1983).
[18]
V.I. Yukalov,
Phys. Rev. B 32, 436 (1985).
[19]
W.E. Caswell,
Ann. Phys. (N.Y.) 123, 153 (1979).
[20]
I. Holliday and P. Suranyi,
Phys. Rev. D 21, 1529 (1980).
[21]
G. Grunberg,
Phys. Lett. B 95, 70 (1980).
[22]
P.M. Stevenson,
Phys. Rev. D 23, 2916 (1981).
[23]
I.D. Feranchuk and L.I. Komarov,
Phys. Lett. A 88, 211 (1982).
[24]
A. Okopinska,
Phys. Rev. D 35, 1835 (1987).
[25]
V.I. Yukalov and E.P. Yukalova,
Ann. Phys. (N.Y.) 277, 219 (1999).
[26]
V.I. Yukalov and E.P. Yukalova,
Chaos Solit. Fract. 14, 839 (2002).
[27]
V.I. Yukalov,
Int. J. Mod. Phys. B 3, 1691 (1989).
[28]
V.I. Yukalov,
Int. J. Theor. Phys. 28, 1237 (1989).
[29]
V.I. Yukalov,
Physica A 167, 833 (1990).
[30]
V.I. Yukalov,
Phys. Rev. A 42, 3324 (1990).
[31]
V.I. Yukalov,
J. Math. Phys. 32, 1235 (1991).
[32]
V.I. Yukalov,
J. Math. Phys. 33, 3994 (1992).
[33]
V.I. Yukalov,
Int. J. Mod. Phys. B 7, 1711 (1993).
[34]
J.M. Ortega and W.C. Rheinboldt,
Iterative Solution of Nonlinear Equations in Several Variables
(Academic, New York, 1970).
[35]
N.S. Bakhvalov,
Numerical Methods (Mir, Moscow, 1977).
[36]
V.I. Yukalov,
Moscow Univ. Phys. Bull. 27, 59 (1972).
[37]
V.I. Yukalov,
Proc. Moscow Univ. Phys. 59, 91 (1972).
[38]
V.I. Yukalov and S. Gluzman,
Phys. Rev. Lett. 79, 333 (1997).
[39]
S. Gluzman and V.I. Yukalov,
Phys. Rev. E 55, 3983 (1997).
[40]
V.I. Yukalov and S. Gluzman,
Phys. Rev. E 55, 6552 (1997).
[41]
V.I. Yukalov, E.P. Yukalova, and S. Gluzman,
Phys. Rev. A 58, 96 (1998).
[42]
V.I. Yukalov and S. Gluzman,
Phys. Rev. E 58, 1359 (1998).
[43]
S. Gluzman and V.I. Yukalov,
Phys. Rev. E 58, 4197 (1998).
[44]
V.I. Yukalov and S. Gluzman,
Physica A 273, 401 (1999).
[45]
V.I. Yukalov,
Mod. Phys. Lett. B 14, 791 (2000).
[46]
S. Gluzman and V.I. Yukalov,
J. Math. Chem. 48, 883 (2010).
[47]
S. Gluzman and V.I. Yukalov,
Phys. Lett. A 377, 124 (2012).
[48]
V.I. Yukalov, S. Gluzman, and D. Sornette,
Physica A 328, 409 (2003).
[49]
S. Gluzman, V.I. Yukalov, and D. Sornette,
Phys. Rev. E 67, 026109 (2003).
[50]
V.I. Yukalov and S. Gluzman,
Int. J. Mod. Phys. B 18, 3027 (2004).
[51]
V.I. Yukalov and E.P. Yukalova,
Phys. Lett. A 368, 341 (2007).
[52]
V.I. Yukalov and S. Gluzman,
Mol. Phys. 107, 2237 (2009).
[53]
H. Kleinert and V. Schulte-Frohlinde,
Critical Properties of $\varphi^{4}$ - Theories
(World Scientific, Singapore, 2001).
[54]
V.I. Yukalov and E.P. Yukalova,
Eur. Phys. J. B 55, 93 (2007).
[55]
V.I. Yukalov,
Renormalization Group under Iterative Procedure
(JINR, Dubna, 1989).
[56]
N.N. Bogolubov and D.V. Shirkov,
Quantum Fields (Benjamin, London, 1983).
[57]
H. Gell-Man and F. Low,
Phys. Rev. 95, 1300 (1954).
[58]
V.I. Yukalov and E.P. Yukalova,
Laser Phys. 5, 154 (1995).
[59]
V.I. Yukalov and E.P. Yukalova,
Physica A 225, 336 (1996).
[60]
F.T. Hioe, D. MacMillen, and E.W. Montroll,
Phys. Rep. 43, 305 (1978). |
Learning Optimal Deep Projection of ${}^{18}$F-FDG PET Imaging for Early Differential Diagnosis of Parkinsonian Syndromes
Shubham Kumar${}^{1,2,}$
Equal Contribution; ${\dagger}$ corresponding to zuochuantao@fudan.edu.cn${}^{1}$Computer Aided Medical Procedures, Technical University of Munich, Germany
${}^{2}$Indian Institute of Technology, Roorkee, India
${}^{3}$Artificial Intelligence in Medical Imaging (AI-Med), KJP, LMU Munich, Germany
${}^{4}$PET Center, Huashan Hospital, Fudan University, Shanghai, China
${}^{5}$German Center for Neurodegenerative Diseases (DZNE), Bonn, Germany
${}^{6}$Dept. Neurology, Huashan Hospital, Fudan University, Shanghai, China
${}^{7}$Dept. Nuclear Medicine, Technical University of Munich, Germany
${}^{8}$Dept. Molecular and Medical Pharmacology, University of California, LA, USA
${}^{9}$Dept. Nuclear Medicine, University of Bern, Switzerland
Abhijit Guha Roy${}^{1,3,\star}$
${}^{1}$Computer Aided Medical Procedures, Technical University of Munich, Germany
${}^{2}$Indian Institute of Technology, Roorkee, India
${}^{3}$Artificial Intelligence in Medical Imaging (AI-Med), KJP, LMU Munich, Germany
${}^{4}$PET Center, Huashan Hospital, Fudan University, Shanghai, China
${}^{5}$German Center for Neurodegenerative Diseases (DZNE), Bonn, Germany
${}^{6}$Dept. Neurology, Huashan Hospital, Fudan University, Shanghai, China
${}^{7}$Dept. Nuclear Medicine, Technical University of Munich, Germany
${}^{8}$Dept. Molecular and Medical Pharmacology, University of California, LA, USA
${}^{9}$Dept. Nuclear Medicine, University of Bern, Switzerland
Ping Wu${}^{4}$
${}^{1}$Computer Aided Medical Procedures, Technical University of Munich, Germany
${}^{2}$Indian Institute of Technology, Roorkee, India
${}^{3}$Artificial Intelligence in Medical Imaging (AI-Med), KJP, LMU Munich, Germany
${}^{4}$PET Center, Huashan Hospital, Fudan University, Shanghai, China
${}^{5}$German Center for Neurodegenerative Diseases (DZNE), Bonn, Germany
${}^{6}$Dept. Neurology, Huashan Hospital, Fudan University, Shanghai, China
${}^{7}$Dept. Nuclear Medicine, Technical University of Munich, Germany
${}^{8}$Dept. Molecular and Medical Pharmacology, University of California, LA, USA
${}^{9}$Dept. Nuclear Medicine, University of Bern, Switzerland
Sailesh Conjeti${}^{1,5}$
${}^{1}$Computer Aided Medical Procedures, Technical University of Munich, Germany
${}^{2}$Indian Institute of Technology, Roorkee, India
${}^{3}$Artificial Intelligence in Medical Imaging (AI-Med), KJP, LMU Munich, Germany
${}^{4}$PET Center, Huashan Hospital, Fudan University, Shanghai, China
${}^{5}$German Center for Neurodegenerative Diseases (DZNE), Bonn, Germany
${}^{6}$Dept. Neurology, Huashan Hospital, Fudan University, Shanghai, China
${}^{7}$Dept. Nuclear Medicine, Technical University of Munich, Germany
${}^{8}$Dept. Molecular and Medical Pharmacology, University of California, LA, USA
${}^{9}$Dept. Nuclear Medicine, University of Bern, Switzerland
R. S. Anand${}^{2}$
${}^{1}$Computer Aided Medical Procedures, Technical University of Munich, Germany
${}^{2}$Indian Institute of Technology, Roorkee, India
${}^{3}$Artificial Intelligence in Medical Imaging (AI-Med), KJP, LMU Munich, Germany
${}^{4}$PET Center, Huashan Hospital, Fudan University, Shanghai, China
${}^{5}$German Center for Neurodegenerative Diseases (DZNE), Bonn, Germany
${}^{6}$Dept. Neurology, Huashan Hospital, Fudan University, Shanghai, China
${}^{7}$Dept. Nuclear Medicine, Technical University of Munich, Germany
${}^{8}$Dept. Molecular and Medical Pharmacology, University of California, LA, USA
${}^{9}$Dept. Nuclear Medicine, University of Bern, Switzerland
Jian Wang${}^{6}$
${}^{1}$Computer Aided Medical Procedures, Technical University of Munich, Germany
${}^{2}$Indian Institute of Technology, Roorkee, India
${}^{3}$Artificial Intelligence in Medical Imaging (AI-Med), KJP, LMU Munich, Germany
${}^{4}$PET Center, Huashan Hospital, Fudan University, Shanghai, China
${}^{5}$German Center for Neurodegenerative Diseases (DZNE), Bonn, Germany
${}^{6}$Dept. Neurology, Huashan Hospital, Fudan University, Shanghai, China
${}^{7}$Dept. Nuclear Medicine, Technical University of Munich, Germany
${}^{8}$Dept. Molecular and Medical Pharmacology, University of California, LA, USA
${}^{9}$Dept. Nuclear Medicine, University of Bern, Switzerland
Igor Yakushev${}^{7}$
${}^{1}$Computer Aided Medical Procedures, Technical University of Munich, Germany
${}^{2}$Indian Institute of Technology, Roorkee, India
${}^{3}$Artificial Intelligence in Medical Imaging (AI-Med), KJP, LMU Munich, Germany
${}^{4}$PET Center, Huashan Hospital, Fudan University, Shanghai, China
${}^{5}$German Center for Neurodegenerative Diseases (DZNE), Bonn, Germany
${}^{6}$Dept. Neurology, Huashan Hospital, Fudan University, Shanghai, China
${}^{7}$Dept. Nuclear Medicine, Technical University of Munich, Germany
${}^{8}$Dept. Molecular and Medical Pharmacology, University of California, LA, USA
${}^{9}$Dept. Nuclear Medicine, University of Bern, Switzerland
Stefan Förster${}^{7}$
${}^{1}$Computer Aided Medical Procedures, Technical University of Munich, Germany
${}^{2}$Indian Institute of Technology, Roorkee, India
${}^{3}$Artificial Intelligence in Medical Imaging (AI-Med), KJP, LMU Munich, Germany
${}^{4}$PET Center, Huashan Hospital, Fudan University, Shanghai, China
${}^{5}$German Center for Neurodegenerative Diseases (DZNE), Bonn, Germany
${}^{6}$Dept. Neurology, Huashan Hospital, Fudan University, Shanghai, China
${}^{7}$Dept. Nuclear Medicine, Technical University of Munich, Germany
${}^{8}$Dept. Molecular and Medical Pharmacology, University of California, LA, USA
${}^{9}$Dept. Nuclear Medicine, University of Bern, Switzerland
Markus Schwaiger${}^{7}$
${}^{1}$Computer Aided Medical Procedures, Technical University of Munich, Germany
${}^{2}$Indian Institute of Technology, Roorkee, India
${}^{3}$Artificial Intelligence in Medical Imaging (AI-Med), KJP, LMU Munich, Germany
${}^{4}$PET Center, Huashan Hospital, Fudan University, Shanghai, China
${}^{5}$German Center for Neurodegenerative Diseases (DZNE), Bonn, Germany
${}^{6}$Dept. Neurology, Huashan Hospital, Fudan University, Shanghai, China
${}^{7}$Dept. Nuclear Medicine, Technical University of Munich, Germany
${}^{8}$Dept. Molecular and Medical Pharmacology, University of California, LA, USA
${}^{9}$Dept. Nuclear Medicine, University of Bern, Switzerland
Sung-Cheng Huang${}^{8}$
${}^{1}$Computer Aided Medical Procedures, Technical University of Munich, Germany
${}^{2}$Indian Institute of Technology, Roorkee, India
${}^{3}$Artificial Intelligence in Medical Imaging (AI-Med), KJP, LMU Munich, Germany
${}^{4}$PET Center, Huashan Hospital, Fudan University, Shanghai, China
${}^{5}$German Center for Neurodegenerative Diseases (DZNE), Bonn, Germany
${}^{6}$Dept. Neurology, Huashan Hospital, Fudan University, Shanghai, China
${}^{7}$Dept. Nuclear Medicine, Technical University of Munich, Germany
${}^{8}$Dept. Molecular and Medical Pharmacology, University of California, LA, USA
${}^{9}$Dept. Nuclear Medicine, University of Bern, Switzerland
Axel Rominger${}^{9}$
${}^{1}$Computer Aided Medical Procedures, Technical University of Munich, Germany
${}^{2}$Indian Institute of Technology, Roorkee, India
${}^{3}$Artificial Intelligence in Medical Imaging (AI-Med), KJP, LMU Munich, Germany
${}^{4}$PET Center, Huashan Hospital, Fudan University, Shanghai, China
${}^{5}$German Center for Neurodegenerative Diseases (DZNE), Bonn, Germany
${}^{6}$Dept. Neurology, Huashan Hospital, Fudan University, Shanghai, China
${}^{7}$Dept. Nuclear Medicine, Technical University of Munich, Germany
${}^{8}$Dept. Molecular and Medical Pharmacology, University of California, LA, USA
${}^{9}$Dept. Nuclear Medicine, University of Bern, Switzerland
Chuantao Zuo${}^{4,{\dagger}}$
${}^{1}$Computer Aided Medical Procedures, Technical University of Munich, Germany
${}^{2}$Indian Institute of Technology, Roorkee, India
${}^{3}$Artificial Intelligence in Medical Imaging (AI-Med), KJP, LMU Munich, Germany
${}^{4}$PET Center, Huashan Hospital, Fudan University, Shanghai, China
${}^{5}$German Center for Neurodegenerative Diseases (DZNE), Bonn, Germany
${}^{6}$Dept. Neurology, Huashan Hospital, Fudan University, Shanghai, China
${}^{7}$Dept. Nuclear Medicine, Technical University of Munich, Germany
${}^{8}$Dept. Molecular and Medical Pharmacology, University of California, LA, USA
${}^{9}$Dept. Nuclear Medicine, University of Bern, Switzerland
Kuangyu Shi${}^{1,9}$
${}^{1}$Computer Aided Medical Procedures, Technical University of Munich, Germany
${}^{2}$Indian Institute of Technology, Roorkee, India
${}^{3}$Artificial Intelligence in Medical Imaging (AI-Med), KJP, LMU Munich, Germany
${}^{4}$PET Center, Huashan Hospital, Fudan University, Shanghai, China
${}^{5}$German Center for Neurodegenerative Diseases (DZNE), Bonn, Germany
${}^{6}$Dept. Neurology, Huashan Hospital, Fudan University, Shanghai, China
${}^{7}$Dept. Nuclear Medicine, Technical University of Munich, Germany
${}^{8}$Dept. Molecular and Medical Pharmacology, University of California, LA, USA
${}^{9}$Dept. Nuclear Medicine, University of Bern, Switzerland
Abstract
Several diseases of parkinsonian syndromes present similar symptoms at early stage and no objective widely used diagnostic methods have been approved until now. Positron emission tomography (PET) with ${}^{18}$F-FDG was shown to be able to assess early neuronal dysfunction of synucleinopathies and tauopathies. Tensor factorization (TF) based approaches have been applied to identify characteristic metabolic patterns for differential diagnosis. However, these conventional dimension-reduction strategies assume linear or multi-linear relationships inside data, and are therefore insufficient to distinguish nonlinear metabolic differences between various parkinsonian syndromes. In this paper, we propose a Deep Projection Neural Network (DPNN) to identify characteristic metabolic pattern for early differential diagnosis of parkinsonian syndromes. We draw our inspiration from the existing TF methods. The network consists of a (i) compression part: which uses a deep network to learn optimal 2D projections of 3D scans, and a (ii) classification part: which maps the 2D projections to labels. The compression part can be pre-trained using surplus unlabelled datasets. Also, as the classification part operates on these 2D projections, it can be trained end-to-end effectively with limited labelled data, in contrast to 3D approaches. We show that DPNN is more effective in comparison to existing state-of-the-art and plausible baselines.
1 Introduction
Approximately 7 to 10 million people worldwide are suffering from Parkinson’s disease (PD). On the other hand, very similar clinical signs can appear in patients with atypical parkinsonian syndromes, such as multiple system atrophy (MSA) and progressive supranuclear palsy (PSP) and these conditions account for approximately 25-30% of all cases of parkinsonian syndromes [1]. Diagnosis of parkinsonian patients based on longitudinal clinical follow up remains problematic with a large number of misdiagnoses in early stage [2]. Thus, early differential diagnosis is essential for determining adequate treatment strategies and for achieving the best possible outcome for these patients [3].
Positron emission tomography (PET) captures neuronal dysfunction of PD using specific in-vivo biomarkers [4, 5, 6, 7, 8] and has been shown to be more advantageous in early diagnosis, far before structural damages to the brain tissue occurs [7, 9, 10, 11]. Automated approaches such as Principal component analysis (PCA) has been successfully applied on ${}^{18}$F-FDG PET to extract PD-related pattern (PDRP), MSA-related pattern (MSARP), and PSP-related pattern (PSPRP) [12, 13]. These patterns have been found as effective surrogates to discriminate between classical PD, atypical parkinsonian syndromes and healthy control subjects [13]. To account for heterogeneous physiology and enable individual pattern visualization, a tensor-factorization based method was developed by projecting the 3D data into 2D planes containing the discriminative information [3]. However, these conventional dimension-reduction based methods assume linear or multi-linear relationship inside data. In contrast, different subtypes of parkinsonian syndromes, caused by different protein aggregation ($\alpha$-synuclein or Tau), show a non-linear relationship to the anatomical changes. Thus difference of metabolic patterns between PD, MSA and PSP can be nonlinear due to these diverse pathological manifestations and heterogeneous propagation among complex brain connectomes. Therefore, either PCA or tensor factorization is insufficient to identify nonlinear metabolic differences of various parkinsonian syndromes, and is susceptible to providing sub-optimal solutions.
Deep learning based approaches have recently been shown to be very effective in discovering non-linear characteristic patterns within data in an end-to-end fashion [14, 15].
It has been shown to surpass human performance in different complicated tasks, like image classification.
It has also gained a lot of popularity in the bio-medical community [16] for computerized diagnosis on medical imaging, such as differential diagnosis [15, 17, 18]. Inspired by these recent successes, we use a deep learning based architecture for early diagnosis of parkinsonism.
One of the major challenge associated with this task is that our input data is 3D in nature, with limited amount of labelled training samples. Standard approaches of going for 3D based CNN models (very high number of learnable parameters) are prone to overfitting when trained on limited samples. To circumvent this issue, we draw inspiration from the existing approaches which uses Tensor Factorization (TF) to project the 3D scans to 2D, and use them for diagnosis. Towards this end, we propose a deep projection neural network (DPNN), which has two parts, (i) Compression Part and (ii) Classification Part. The Compression Part basically mimics TF projection from 3D to 2D. This part can be pre-trained on a large amount of unlabelled dataset, which is easily available. This pre-trained model is added to the 2D-CNN based Classification part (lower model complexity), which is trained end-to-end with limited annotated data. Although in this paper, we present its application for PET scans, the concept is fairly generic and can be easily extended to any 3D data.
2 Materials & Methods
2.1 Data Preparation & Preprocessing
A cohort of 257 patients (Dataset-1) with clinically suspected parkinsonian features were included in this study. The patients were referred for ${}^{18}$F-FDG PET imaging and then assessed by blinded movement disorders specialists for more than 2 years. Finally 136 of them were diagnosed with PD, 91 with MSA and 30 with PSP. All the 3D PET volumes were preprocessed using intensity-normalized by global mean and spatially normalized to Montreal Neurological Institute (MNI) space using SPM8111Statistical Parametric Mapping, http://www.fil.ion.ucl.ac.uk/spm/software/spm8/, 2009 according to a standard PET processing procedure [3]. For optimizing deep networks, the limited availability of PET images of patients at early stage of parkinsonism could be a bottleneck. Therefore, a database of 1077 subjects (Dataset-2) with 41 various non-parkinsonian neurological diseases with brain FDG PET images is further included to enhance the data pool.
2.2 DPNN Architecture
We draw our inspiration from prior work which estimated tensor factorized projection of 3D PET scans and processed them for classification task. In this regard, we formulated to solve the problem in two parts: (i) Learn a separate network to mimic the tensor factorization from 3D data, i.e. learning to compress the data (Compression Part), and (ii) Learn a 2D CNN model to map the compressed input to one of the classes (Classification Part). A detailed description of both the parts are provided below with the architectural design in Fig. 1.
Compression Part:
Given a 3D PET scan $I_{P}\in\mathbb{R}^{H\times W\times D}$, here we estimate a function $f_{p}(\cdot)$ which compresses the data to a 2D projection map $P_{t}$, so that $f_{p}:I_{P}\rightarrow P_{t}$, where $P_{t}\in\mathbb{R}^{H\times W}$. This non-linear function $f_{p}(\cdot)$ is approximated by a series of blocks consisting of a $3\times 3$ convolutional layer, batch normalization and a ReLU activation function. A set of 5 such blocks are stacked together, which compresses $I_{P}$ sequentially to $P_{t}$. The final block uses a sigmoidal non-linearity instead of ReLU to rescale the activations between $[0,1]$.
Classification Part:
This part takes $P_{t}$, the compressed projection map as input. It learns a mapping $f_{c}(\cdot)$, which maps $P_{t}$ to the one of the class labels $y$. The first 5 blocks consist of a $3\times 3$ convolutional layer, batch norm, ReLU activation and a max pooling layer, reducing the spatial dimensions by a factor of 2 at every step. The final block consists of a global average pooling instead of max pooling, squeezing the feature map along spatial dimensions.This is followed by a $1\times 1$ convolutional layer, softmax layer to project the learnt features to the label probalility space $\mathbb{R}^{3}$, from where $y$ is estimated as the class with highest probability. More details regarding the size of intermediate feature maps and stride are indicated in Fig. 1.
2.3 Training Procedure
To tackle the issue of learning such a highly complex model with limited training data, we propose to address the training procedure in two stages: (i) We leverage unlabelled PET data corpus to pre-train the Compression Part, (ii) limited labelled data is used to learn the weights of the Classification Part, with the Compression Part initialized to the pre-trained weights.
Pretraining:
In this part, we use the unlabelled Dataset-2 $\{I_{i}\}$ for pre-training. We compute the tensor factorized 2D maps of all the volumes as $\{\mathcal{G}_{i}\}$. In this stage, we train the compression part $f_{p}(\cdot)$, using this dataset, with the goal of mimicking $\{\mathcal{G}_{i}\}$ as the output of the network. We hypothesize that this provides a strong initialization to the network for the classification stage. The network is learnt by jointly optimizing a combination of Mean Square Error (MSE) and Structural Similarity Index (SSIM) between the target and prediction, defined as,
$$\mathcal{L}=\underbrace{\frac{1}{2N_{p}}\sum_{\mathbf{r}}{(\mathcal{P}(\mathbf%
{r})-\mathcal{G}(\mathbf{r}))^{2}}}_{\mathrm{MSE}}-\underbrace{\frac{1}{N_{w}}%
\sum_{\mathbf{w}}{\rm SSIM}({\mathbf{w}_{p}},{\mathbf{w}_{g}}),}_{\mathrm{SSIM%
\ Index}}$$
(1)
$${\rm SSIM}({\mathbf{w}_{p}},{\mathbf{w}_{g}})={\frac{(2\mu_{p}\mu_{g}+C_{1})(2%
\sigma_{pg}+C_{2})}{(\mu_{p}^{2}+\mu_{g}^{2}+C_{1})(\sigma_{p}^{2}+\sigma_{g}^%
{2}+C_{2})}},$$
(2)
where, $\mathcal{P}$, $\mathcal{G}$, $\mathbf{r}$ and $N_{r}$ are the predicted map, target projection map, pixel-position, and the total number of pixels respectively. ${\mathbf{w}_{p}}$ and ${\mathbf{w}_{g}}$ represent a local $6\times 6$ window in $\mathcal{P}$ and $\mathcal{G}$, and ${N_{w}}$ is the total number of such windows. SSIM is calculated on all the ${N_{w}}$ windows and their average value is used in the cost function. $\mu_{p}$, $\sigma_{p}^{2}$, and $\sigma_{pg}$ are the mean of ${\mathbf{w}_{p}}$, the variance of ${\mathbf{w}_{p}}$, and the covariance of ${\mathbf{w}_{p}}$ and ${\mathbf{w}_{g}}$, respectively. $C_{1}$ and $C_{2}$ are set to $\sim 10^{-4}$ and $\sim 9\times 10^{-4}$, respectively. We use SSIM based loss function to preserve the quality of the predicted map similar to actual Tensor Factorized map. The weights of the convolutional kernels are initialized using Xavier initialization and Adam optimizer with a learning rate of $10^{-4}$ is used for the weight updates. The $\beta_{1}$, $\beta_{2}$ and $\epsilon$ parameters of the optimizer are set to $0.9$, $0.999$, and $10^{-8}$, respectively. The training is continued until the validation-cost saturates.
Fine-Tuning:
In this stage, the pre-trained compression network is combined with the classification part, and the weights of the classification part are initialized using Xavier initialization. The whole network is trained in an end-to-end fashion, minimizing 3-class Cross-Entropy loss function using Adam optimizer with $\beta_{1}$, $\beta_{2}$ and $\epsilon$ set to $0.9$, $0.999$, and $10^{-8}$, respectively. The learning rate used for the classification part is $10^{-4}$, while for the compression part it is kept $10^{-5}$. The learning rate of the compression part is kept one order low to prevent high perturbation in those layer. The weights are regularized with a decay constant of $10^{-5}$, preventing over-fitting. A mini-batch of $10$ PET scans are used. The training is continued until the convergence of the validation loss.
3 Experiments and Results
3.0.1 Experiments:
We evaluate our proposed DPNN model by a 5-fold cross-validation experiment on Dataset-1. An equal distribution of each of samples from the three classes were ensured in each of the folds. For evaluation, we used the standard metrics, (i) True Positive Rate (TPR), (ii) True Negative Rate (TNR), (iii) Positive Predictive Value (PPV), and (iv) Negative Predictive Value (NPV), consistent with [3].
Baselines:
We compare our proposed method against state-of-the-art method which uses Tensor Factorization (TF), followed by SVM for classification [3]. Apart from this, we define two other baselines to substantiate our claims:
1.
BL-1: DPNN, with pre-training using only MSE, to observe the effect of including SSIM in the cost function.
2.
BL-2: DPNN, without pre-training, trained end-to-end, to observe the effect of pre-training.
For all the experiments we used five-folded cross-validation for evaluation. All the networks were trained on NVIDIA Titan-Xp GPU with 12GB RAM.
3.0.2 Results:
Tab. 1 reports the results of our proposed model (DPNN), with defined baselines and state-of-the-art method, in terms of the mentioned evaluation metrics. Comparing with state-of-the-art [3], DPNN outperforms it in most of the evaluation scores (8 out of 12). Comparing with BL-1, DPNN outperforms it. This substantiates our previous hypothesis that MSE+SSIM based pre-training is more effective in providing stronger initialization than MSE alone, which fails to capture the quality based features in the compression stage. It can be attributed to the fact that SSIM applies stricter constraint on similarity which forces the network to learn better representations. Also, comparing to BL-2, we prove our previous claim that pre-training is necessary when training such a complicated model with limited annotated data. It has improved the specificities of MSA by 0.79%, PSP by 17.97% and PD by 14.37%, which can play a critical role for differential diagnosis. It is worth noting that DPNN shows consistent good performance across all the metrics for the PD class which has the highest number of samples (viz. 136). While all the models show greatest inconsistency in the scores for PSP class, which has just $30$ representative samples in the dataset. This is indicative of the fact that given enough data the performance of DPNN can be increased to an ideal level.
Next, we take a closer look at the learnt Projection Maps in Fig. 2, which shows example pattern images of MSA, PSP and PD. Patterns similar to tensor-factorization have been observed in the DPNN Projection results, for example, visible cerebellum and striatum activities in PD, vanishing cerebellum and striatum activities in MSA and decreasing striatum activity and visible cerebellum activity in PSP [3]. This confirms that DPNN is capable of extracting physiologically meaningful patterns, and use it for final decision making.
4 Conclusion
We developed a deep learning method to extract characteristic metabolic pattern for differential diagnosis of parkinsonian syndrome. In contrast to linear or multi-linear data-reduction methods of the state-of-the-art, the proposed DPNN, processes 3D-data using 2D-convolutions, can explore the non-linear metabolic differences between the subtypes. Furthermore, we introduced a training procedure based on the optimization of SSIM along with MSE which leverages tensor-factorized maps of inputs, from a domain similar to the task-input domain, to overcome the difficulties posed by a small dataset. With limited amount of data, the novel method has already achieved superior accuracy compared to the state-of-the-art. The advanced pre-training strategies play a critical role in the success of this novel method, which prevent the abort of cutting-edge developments before approaching to a large data-bank. The positive performance of deep learning in this study encourages a multi-center study, which is actively in preparation. Although the DPNN patterns extracted in this proof-of-concept study look similar to the previous tensor factorization, an extensive inspection by clinicians may discover the characteristic difference matching to improve accuracy. With the increase of data access, the ability of the deep learning methods to discover new discriminative features will be enhanced, which may provide the potential for a diagnosis at even earlier stage before motor impairment appears, i.e. at prodromal parkinsonian stage such as rapid eye movement (REM) sleep behavior disorder (RBD).
References
[1]
A. J. Hughes, Y. Ben-Shlomo, S. E. Daniel, and A. J. Lees.
What features improve the accuracy of clinical diagnosis in
parkinson’s disease: a clinicopathologic study.
Neurology, 57(10 Suppl 3):S34–8, 2001.
[2]
S. Fahn, D. Oakes, I. Shoulson, K. Kieburtz, A. Rudolph, A. Lang, C. W. Olanow,
C. Tanner, and K. Marek.
Levodopa and the progression of parkinson’s disease.
New England Journal of Medicine, 351(24):2498–508, 2004.
[3]
R. Li, P. Wu, I. Yakushev, J. Wang, S. I. Ziegler, S. Förster, S. C.
Huang, M. Schwaiger, N. Navab, C. Zuo, and K. Shi.
Pattern visualization and recognition using tensor factorization for
early differential diagnosis of parkinsonism.
In MICCAI 2017 - Part III, pages 125–133, 2017.
[4]
F. Gao, H. Liu, and P. Shi.
Patient-adaptive lesion metabolism analysis by dynamic PET images.
In MICCAI 2012, Part III, pages 558–565, 2012.
[5]
Z. Xu, U. Bagci, J. Seidel, D. Thomasson, J. M. Solomon, and D. J. Mollura.
Segmentation based denoising of PET images: An iterative approach
via regional means and affinity propagation.
In MICCAI 2014, Part I, pages 698–705, 2014.
[6]
U. Bagci, J. K. Udupa, N. Mendhiratta, B. Foster, Z. Xu, J. Yao, X. Chen, and
D. J. Mollura.
Joint segmentation of anatomical and functional images: Applications
in quantification of lesions from pet, pet-ct, mri-pet, and MRI-PET-CT
images.
Medical Image Analysis, 17(8):929–945, 2013.
[7]
J. Jiao, G. E. Searle, A. C. Tziortzi, C. A. Salinas, R. N. Gunn, and J. A.
Schnabel.
Spatio-temporal pharmacokinetic model based registration of 4d PET
neuroimaging data.
NeuroImage, 84:225–235, 2014.
[8]
L. Bi, J. Kim, D. Feng, and M. J. Fulham.
Multi-stage thresholded region classification for whole-body PET-CT
lymphoma studies.
In MICCAI 2014, Part I, pages 569–576, 2014.
[9]
D. Zhang and D. Shen.
Multi-modal multi-task learning for joint prediction of multiple
regression and classification variables in alzheimer’s disease.
Neuroimage, 59(2):895–907, 2012.
[10]
L. Zhou, O. Salvado, V. Doré, P. Bourgeat, P. Raniga, V. Villemagne,
C. Rowe, and J. Fripp.
MR-less surface-based amyloid estimation by subject-specific atlas
selection and bayesian fusion.
In MICCAI 2012, Part II, pages 220–227, 2012.
[11]
S. Lu, Y. Xia, W. Cai, M. J. Fulham, and D. D. Feng.
Early identification of mild cognitive impairment using incomplete
random forest-robust support vector machine and FDG-PET imaging.
Comp. Med. Imag. and Graph., 60:35–41, 2017.
[12]
D. Eidelberg.
Metabolic brain networks in neurodegenerative disorders: a functional
imaging approach.
Trends in Neurosciences, 32(10):548–57, 2009.
[13]
C. C. Tang, K. L. Poston, T. Eckert, A. Feigin, S. Frucht, M. Gudesblatt,
V. Dhawan, M. Lesser, J. P. Vonsattel, S. Fahn, and D. Eidelberg.
Differential diagnosis of parkinsonism: a metabolic imaging study
using pattern analysis.
Lancet Neurology, 9(2):149–58, 2010.
[14]
Y. LeCun, Y. Bengio, and G. Hinton.
Deep learning.
Nature, 521(7553):436–44, 2015.
[15]
V. K. Ithapu, V. Singh, O. C. Okonkwo, R. J. Chappell, N. M. Dowling, and S. C.
Johnson.
Imaging-based enrichment criteria using deep learning algorithms for
efficient clinical trials in mild cognitive impairment.
Alzheimers Dement, 11(12):1489–99, 2015.
[16]
Geert Litjens, Thijs Kooi, Babak Ehteshami Bejnordi, Arnaud Arindra Adiyoso
Setio, Francesco Ciompi, Mohsen Ghafoorian, Jeroen A.W.M. van der Laak, Bram
van Ginneken, and Clara I. Sánchez.
A survey on deep learning in medical image analysis.
Medical Image Analysis, 42:60 – 88, 2017.
[17]
L. Zhou, Y. Wang, Y. Li, P. T. Yap, and D. Shen.
Hierarchical anatomical brain networks for MCI prediction by
partial least square analysis.
In CVPR, pages 1073–1080, 2011.
[18]
H. I. Suk, S. W. Lee, and D. Shen.
Latent feature representation with stacked auto-encoder for ad/mci
diagnosis.
Brain Struct Funct, 220(2):841–59, 2015. |
A CONSEQUENCE OF LITTLEWOOD’S CONDITIONAL ESTIMATES
FOR THE RIEMANN ZETA-FUNCTION
Sergei N. Preobrazhenskiĭ111Preobrazhenskii Sergei Nikolayevich —
Department of Mathematical Analysis, Faculty of Mechanics and Mathematics,
Lomonosov Moscow State University.
Assuming the Riemann hypothesis (RH)
and using Littlewood’s conditional estimates
for the Riemann zeta-function, we provide an estimate
related to an approach of Y. Motohashi
to the zero-free region.
Key words:
Riemann zeta-function, Riemann hypothesis.
1. Introduction. The approach of Y. Motohashi [1]
to the zero-free region of the Riemann zeta-function
extended by the author in [2] may be modified to give regions
free of large values of some products, which contain finite products $\prod_{j}\zeta(s_{j})$.
On the Riemann hypothesis, one can obtain upper bounds for
such products for $s_{j}=1+t_{j}$ using the method of Littlewood.
To prove our result on regions free of large values we also use
an $\Omega$-theorem for $\prod_{j}\frac{1}{\zeta(s_{j})}$, where
$s_{j}=\sigma_{j}+i(t_{j}+h_{j})$
with $h_{j}$ lying in short intervals around $t_{j}$
and $\sigma_{j}\geqslant 1$.
The $\Omega$-theorem depends on a version of Kronecker’s theorem with an explicit upper bound.
2. Lemmas.
Lemma 1.
On the Riemann hypothesis,
uniformly for $\frac{1}{2}<\sigma_{0}\leqslant\sigma\leqslant\frac{9}{8}$ and $t\geqslant e^{27}$ we have
$$\log\zeta(s)\ll\begin{cases}\log\frac{1}{\sigma-1}&\text{if}\quad 1+\frac{1}{%
\log\log t}\leqslant\sigma\leqslant\frac{9}{8},\\
\frac{(\log t)^{2-2\sigma}-1}{(1-\sigma)\log\log t}+\log\log\log t&\text{if}%
\quad\sigma_{0}\leqslant\sigma\leqslant 1+\frac{1}{\log\log t},\end{cases}$$
and for $\sigma>1-\frac{E}{\log\log t}$, $E>0$ fixed,
$$\zeta(s)\ll e^{Le^{(2+\varepsilon)E}}(\log\log t),$$
(1)
where $L=L(t)=\log\log\log\log t$
and the implied constant in the ${\ll}$ depends on $E$.
For the first estimate, see [3], Chapter XIV, §14.33.
The second estimate is similar to the first and is obtained along the lines of [3],
Chapter XIV, §14.9. For a more precise estimate, see [4].
Lemma 2.
For $\alpha\leqslant\sigma\leqslant\beta$ and $t>1$ we have
$$\Gamma(\sigma+it)=t^{\sigma+it-1/2}\exp\left(-\frac{\pi}{2}t-it+i\frac{\pi}{2}%
\left(\sigma-\frac{1}{2}\right)\right)\sqrt{2\pi}\left(1+O\left(\frac{1}{t}%
\right)\right),$$
with the constant in the big-$O$ depending only on $\alpha$ and $\beta$.
For the proof, see e.g. [5], Appendix, §3.
Lemma 3.
Let $\sigma_{a}(n)$, $a\in\mathbb{C}$, be the sum of $a$th powers of the divisors of $n$.
Let $\xi(d)$ be an arbitrary bounded arithmetical function
with the support in the set of square-free integers.
Then for $\sigma>1$, $T_{1},T_{2}\in\mathbb{R}$ we have the identity
$$\begin{split}\displaystyle\sum_{n=1}^{\infty}&\displaystyle\sigma_{iT_{1}}(n)%
\sigma_{-iT_{2}}(n)\left(\sum_{d\mid n}\xi(d)\right)n^{-s}\\
&\displaystyle=\frac{\zeta(s)\zeta(s-iT_{1})\zeta(s+iT_{2})\zeta(s-i(T_{1}-T_{%
2}))}{\zeta(2s-i(T_{1}-T_{2}))}\left(\xi(1)+\sum_{d=2}^{\infty}\xi(d)P_{d}(s,T%
_{1},T_{2})\right),\end{split}$$
where
$$\begin{split}&\displaystyle P_{d}(s,T_{1},T_{2})\\
&\displaystyle=\prod_{p\mid d}\left(1-\left(1-\frac{1}{p^{s}}\right)\left(1-%
\frac{1}{p^{s-iT_{1}}}\right)\left(1-\frac{1}{p^{s+iT_{2}}}\right)\left(1-%
\frac{1}{p^{s-i(T_{1}-T_{2})}}\right)\left(1-\frac{1}{p^{2s-i(T_{1}-T_{2})}}%
\right)^{-1}\right).\end{split}$$
Proof. This is a version of Lemma 3 of Y. Motohashi [1].
Let
$$Z=\frac{\zeta(s)\zeta(s-iT_{1})\zeta(s+iT_{2})\zeta(s-i(T_{1}-T_{2}))}{\zeta(2%
s-i(T_{1}-T_{2}))}.$$
Changing the order of summation, we have
$$\begin{split}&\displaystyle\sum_{n=1}^{\infty}\sigma_{iT_{1}}(n)\sigma_{-iT_{2%
}}(n)\left(\sum_{d\mid n}\xi(d)\right)n^{-s}\\
&\displaystyle=\xi(1)Z+\sum_{\begin{subarray}{c}d\geqslant 2,\text{$d$ square-%
free}\\
d=p_{d_{1}}\cdots p_{d_{r}}\end{subarray}}\xi(d)\left(\sum_{k=1}^{\infty}\frac%
{\sigma_{iT_{1}}(kp_{d_{1}}\cdots p_{d_{r}})\sigma_{-iT_{2}}(kp_{d_{1}}\cdots p%
_{d_{r}})}{k^{s}p_{d_{1}}^{s}\cdots p_{d_{r}}^{s}}\right)\\
&\displaystyle=\xi(1)Z+\sum_{d\geqslant 2,\text{$d$ square-free}}\xi(d)\prod_{%
p\mid d}\left(\frac{\left(1+p^{iT_{1}}\right)\left(1+p^{-iT_{2}}\right)}{p^{s}%
}+\frac{\left(1-p^{i3T_{1}}\right)\left(1-p^{-i3T_{2}}\right)}{\left(1-p^{iT_{%
1}}\right)\left(1-p^{-iT_{2}}\right)}\frac{1}{p^{2s}}+\ldots\right)\\
&\displaystyle\times\prod_{p\nmid d}\left(1+\frac{\left(1+p^{iT_{1}}\right)%
\left(1+p^{-iT_{2}}\right)}{p^{s}}+\frac{\left(1-p^{i3T_{1}}\right)\left(1-p^{%
-i3T_{2}}\right)}{\left(1-p^{iT_{1}}\right)\left(1-p^{-iT_{2}}\right)}\frac{1}%
{p^{2s}}+\ldots\right)\\
&\displaystyle=\xi(1)Z+\sum_{d\geqslant 2,\text{$d$ square-free}}\xi(d)Z\prod_%
{p\mid d}\frac{\frac{\left(1+p^{iT_{1}}\right)\left(1+p^{-iT_{2}}\right)}{p^{s%
}}+\frac{\left(1-p^{i3T_{1}}\right)\left(1-p^{-i3T_{2}}\right)}{\left(1-p^{iT_%
{1}}\right)\left(1-p^{-iT_{2}}\right)}\frac{1}{p^{2s}}+\ldots}{1+\frac{\left(1%
+p^{iT_{1}}\right)\left(1+p^{-iT_{2}}\right)}{p^{s}}+\frac{\left(1-p^{i3T_{1}}%
\right)\left(1-p^{-i3T_{2}}\right)}{\left(1-p^{iT_{1}}\right)\left(1-p^{-iT_{2%
}}\right)}\frac{1}{p^{2s}}+\ldots}.\end{split}$$
By an identity of Ramanujan—Wilson [3], (1.3.3),
$$\begin{split}\displaystyle\prod_{p\mid d}&\displaystyle\frac{\frac{\left(1+p^{%
iT_{1}}\right)\left(1+p^{-iT_{2}}\right)}{p^{s}}+\frac{\left(1-p^{i3T_{1}}%
\right)\left(1-p^{-i3T_{2}}\right)}{\left(1-p^{iT_{1}}\right)\left(1-p^{-iT_{2%
}}\right)}\frac{1}{p^{2s}}+\ldots}{1+\frac{\left(1+p^{iT_{1}}\right)\left(1+p^%
{-iT_{2}}\right)}{p^{s}}+\frac{\left(1-p^{i3T_{1}}\right)\left(1-p^{-i3T_{2}}%
\right)}{\left(1-p^{iT_{1}}\right)\left(1-p^{-iT_{2}}\right)}\frac{1}{p^{2s}}+%
\ldots}\\
&\displaystyle=\prod_{p\mid d}\left(\frac{1-p^{i(T_{1}-T_{2})-2s}}{\left(1-p^{%
-s}\right)\left(1-p^{iT_{1}-s}\right)\left(1-p^{-iT_{2}-s}\right)\left(1-p^{i(%
T_{1}-T_{2})-s}\right)}-1\right)\\
&\displaystyle\times\frac{\left(1-p^{-s}\right)\left(1-p^{iT_{1}-s}\right)%
\left(1-p^{-iT_{2}-s}\right)\left(1-p^{i(T_{1}-T_{2})-s}\right)}{1-p^{i(T_{1}-%
T_{2})-2s}}.\end{split}$$
This obviously ends the proof of the lemma.
Lemma 4.
Assume the truth of the Riemann hypothesis.
Fix $E>0$. Let
$$\exp(A\log\log T\log\log\log T)\leqslant N\leqslant\exp(DA\log\log T\log\log%
\log T),\quad T\geqslant e^{27},$$
with $A=\frac{18+\varepsilon}{E}$ and a sufficiently large positive constant $D$,
and let us put $T_{1}=T$, $T_{2}=T+H$, with $H=c(\log\log T)^{-1}$.
Then we have
$$\begin{split}\displaystyle\sum_{n\leqslant N}&\displaystyle|\sigma_{iT_{1}}(n)%
|^{2}|\sigma_{iT_{2}}(n)|^{2}\\
&\displaystyle\mathrel{{\ll}_{A,D}}N\\
&\displaystyle\times\left((\log\log\log T)^{3}(\log\log T)^{7}|\zeta(1+iT_{1})%
|^{4}|\zeta(1+iT_{2})|^{4}\right.\\
&\displaystyle+(\log\log T)^{7}\zeta(1+i(T_{1}+H))^{2}\zeta(1-i(T_{1}-H))^{2}%
\zeta(1+i(T_{2}+H))^{2}\zeta(1-i(T_{2}-H))^{2}\\
&\displaystyle\left.+(\log\log T)^{7}\zeta(1+i(T_{1}-H))^{2}\zeta(1-i(T_{1}+H)%
)^{2}\zeta(1+i(T_{2}-H))^{2}\zeta(1-i(T_{2}+H))^{2}\right)\\
&\displaystyle+O\left(N(\log\log T)^{-1}\right).\end{split}$$
Proof. Let
$$F_{0}(s,T_{1},T_{2})=\sum_{n=1}^{\infty}|\sigma_{iT_{1}}(n)|^{2}|\sigma_{iT_{2%
}}(n)|^{2}n^{-s}\quad(\sigma>1).$$
By the identity of U. Balakrishnan [6], we have
$$\begin{split}\displaystyle F_{0}(s,T_{1},T_{2})&\displaystyle=\zeta(s)^{4}%
\zeta(s+iT_{1})^{2}\zeta(s-iT_{1})^{2}\zeta(s+iT_{2})^{2}\zeta(s-iT_{2})^{2}\\
&\displaystyle\times\zeta(s+i(T_{1}-T_{2}))\zeta(s-i(T_{1}-T_{2}))\zeta(s+i(T_%
{1}+T_{2}))\zeta(s-i(T_{1}+T_{2}))G(s,T_{1},T_{2}),\end{split}$$
where $G(s,T_{1},T_{2})$ is regular and bounded for $\sigma\geqslant\sigma_{0}>1/2$,
uniformly in $T_{1}$, $T_{2}$.
The limiting case $T_{1}=T_{2}$ gives the identity of Y. Motohashi,
which is connected with the famous nonnegative trigonometric polynomial
$3+4\cos\varphi+\cos 2\varphi$ and the inequality of Mertens.
Littlewood’s bound (1) and Perron’s inversion formula for the height $U=N^{1+\varepsilon}$ give
$$\begin{split}\displaystyle\sum_{n\leqslant N}&\displaystyle|\sigma_{iT_{1}}(n)%
|^{2}|\sigma_{iT_{2}}(n)|^{2}=\mathop{\text{{Res}}}\left(F_{0}(s,T_{1},T_{2})N%
^{s}s^{-1}\right)_{s=1,1\pm iH}\\
&\displaystyle+O\left(\left(e^{L(T)e^{(2+\varepsilon)E}}\log\log T\right)^{10}%
(\log\log T)^{6}N^{\eta}\log U\right)\\
&\displaystyle=\mathop{\text{{Res}}}\left(F_{0}(s,T_{1},T_{2})N^{s}s^{-1}%
\right)_{s=1,1\pm iH}+O\left(N(\log\log T)^{-1-\varepsilon}\right),\end{split}$$
where we have put
$$\eta=1-\frac{E}{\log\log T}.$$
Also,
$$\mathop{\text{{Res}}}\left(F_{0}(s,T_{1},T_{2})N^{s}s^{-1}\right)_{s=1}\ll N%
\sum_{k=0}^{3}|(\partial s)^{k}_{s=1}H(s,T_{1},T_{2})|(\log N)^{3-k},$$
where
$$\begin{split}\displaystyle H(s,T_{1},T_{2})&\displaystyle=\zeta(s+iT_{1})^{2}%
\zeta(s-iT_{1})^{2}\zeta(s+iT_{2})^{2}\zeta(s-iT_{2})^{2}\\
&\displaystyle\times\zeta(s+i(T_{1}-T_{2}))\zeta(s-i(T_{1}-T_{2}))\zeta(s+i(T_%
{1}+T_{2}))\zeta(s-i(T_{1}+T_{2})).\end{split}$$
By taking the logarithmic derivative, we get
$$(\partial s)^{k}_{s=1}H(s,T_{1},T_{2})\ll H(1,T_{1},T_{2})(\log\log T\log\log%
\log T)^{k}.$$
From the theorem of Littlewood and the definition of $H$ we see that
$$\zeta(1+i(T_{1}-T_{2}))\zeta(1-i(T_{1}-T_{2}))\zeta(1+i(T_{1}+T_{2}))\zeta(1-i%
(T_{1}+T_{2}))\ll(\log\log T)^{4},$$
which implies the assertion of the lemma.
Lemma 5.
Let $\mu(d)$ be the Möbius function, and let
$$\lambda_{d}(z)=\begin{cases}\mu(d)&\text{if}\quad d<z,\\
\mu(d)\frac{\log\left(z^{2}/d\right)}{\log z}&\text{if}\quad z\leqslant d<z^{2%
},\\
0&\text{otherwise},\end{cases}$$
where $z>1$ is arbitrary. Then we have, uniformly in $N>1$ and in $z$,
$$\sum_{n\leqslant N}\left(\sum_{d\mid n}\lambda_{d}(z)\right)^{2}\ll\frac{N}{%
\log z}.$$
This lemma is due to Barban—Vehov [7]
and appears as Lemma 5 in Y. Motohashi [1].
For the proof, see [8] and [9].
Lemma 6.
For any large $y$, and fixed $a$, $q>1$, $(a,q)=1$,
$$\sum_{\begin{subarray}{c}p\leqslant y\\
p\equiv a\pmod{q}\end{subarray}}\mathop{\text{{sgn}}}\left(\cos(2h\log p)%
\right)\frac{\cos(h\log p)}{p}=\frac{1}{\varphi(q)}\log\left(\min\left(h^{-1},%
\log y\right)\right)+O(1)\quad\text{{for }}0<h<c.$$
This and related estimates can be proved by using PNT in arithmetic progressions
and Stieltjes integration. A similar lemma can be found in [10].
3. Proof of Theorem.
We put
$$X=\exp(0{.}5DA\log\log T\log\log\log T),\quad z=\exp(A\log\log T\log\log\log T)$$
(2)
with the same $A$ and $D$ as in Lemma 4,
set $\xi(d)=\lambda_{d}(z)$ in Lemma 3
and for $T_{1}=T$, $T_{2}=T+H$ with $H=c(\log\log T)^{-1}$,
write
$$\displaystyle J(s,T_{1},T_{2})=\frac{\zeta(s)\zeta(s-iT_{1})\zeta(s+iT_{2})%
\zeta(s-i(T_{1}-T_{2}))}{\zeta(2s-i(T_{1}-T_{2}))},$$
$$\displaystyle K(s,T_{1},T_{2})=\sum_{d\leqslant z^{2}}\lambda_{d}(z)P_{d}(s,T_%
{1},T_{2}).$$
Theorem 1.
Assume the Riemann hypothesis.
Then there exists an infinite sequence of pairs of real numbers $(T_{1},T_{2})$,
$T_{1}=T$, $T_{2}=T+H$, with arbitrarily large values of $T$ and $H=c(\log\log T)^{-1}$,
such that
$$|\zeta(1+iT_{1})||\zeta(1+iT_{2})|\ll(\log\log T)^{-2}$$
and
$$\begin{split}&\displaystyle(\log\log T)^{7}|\zeta(1+iT_{1})|^{4}|\zeta(1+iT_{2%
})|^{4}\\
\displaystyle{}+&\displaystyle(\log\log T)^{7}\zeta(1+i(T_{1}+H))^{2}\zeta(1-i%
(T_{1}-H))^{2}\zeta(1+i(T_{2}+H))^{2}\zeta(1-i(T_{2}-H))^{2}\\
\displaystyle{}+&\displaystyle(\log\log T)^{7}\zeta(1+i(T_{1}-H))^{2}\zeta(1-i%
(T_{1}+H))^{2}\zeta(1+i(T_{2}-H))^{2}\zeta(1-i(T_{2}+H))^{2}\\
\displaystyle{}\ll&\displaystyle(\log\log T)^{-1}.\end{split}$$
Let $s_{0}=\sigma_{0}+it_{0}$ be a point such that
$$|J(s_{0},T_{1},T_{2})K(s_{0},T_{1},T_{2})|\geqslant(\log\log T)^{\varepsilon}$$
(3)
with arbitrarily small fixed $\varepsilon>0$, and
$$\sigma_{0}=1-\frac{E_{0}}{\log\log T}\geqslant 1-\frac{E}{\log\log T},\quad C%
\log\log\log T\leqslant|t_{0}|\leqslant T/2.$$
(4)
Then $E_{0}\geqslant c_{2}(\varepsilon)>0$.
Proof. By Mellin’s inversion formula, when $c-\sigma_{0}>0$,
$$e^{-n/X}=\frac{1}{2\pi i}\int\limits_{c-i\infty}^{c+i\infty}\Gamma(s-s_{0})%
\frac{X^{s-s_{0}}}{n^{s-s_{0}}}\,ds.$$
Hence for $c>1$ and $c>\sigma_{0}$ by Lemma 3 we have that
$$\begin{split}\displaystyle e^{-1/X}&\displaystyle+\sum_{n\geqslant z}\sigma_{%
iT_{1}}(n)\sigma_{-iT_{2}}(n)n^{-s_{0}}a(n)e^{-n/X}\\
&\displaystyle=\frac{X^{-s_{0}}}{2\pi i}\int\limits_{(\sigma=c)}J(s,T_{1},T_{2%
})K(s,T_{1},T_{2})\Gamma(s-s_{0})X^{s}\,ds,\end{split}$$
where
$$a(n)=\sum_{d\mid n}\lambda_{d}(z).$$
We now move the line of integration to the line
$$\sigma=\eta=1-\frac{E}{\log\log T}.$$
There are simple poles at $s=1$, $1+iT_{1}$, $1-iT_{2}$, $1+i(T_{1}-T_{2})$,
but by (4) and Lemma 2 they leave residues
that are all bounded by $O\left((\log\log T)^{-2}\right)$. Now we consider the estimation of the integral along $\sigma=\eta$.
For the estimation of $K(s,T_{1},T_{2})$ we define the generating
Dirichlet series
$$\begin{split}\displaystyle M_{w}(s,T_{1},T_{2})&\displaystyle=1+\sum_{d=2}^{%
\infty}\mu(d)P_{d}(s,T_{1},T_{2})d^{-w}\\
&\displaystyle=\prod_{p}\left(1-\frac{1}{p^{w}}\left(1-\left(1-\frac{1}{p^{s}}%
\right)\left(1-\frac{1}{p^{s-iT_{1}}}\right)\left(1-\frac{1}{p^{s+iT_{2}}}%
\right)\left(1-\frac{1}{p^{s-i(T_{1}-T_{2})}}\right)\right.\right.\\
&\displaystyle\left.\left.\times\left(1-\frac{1}{p^{2s-i(T_{1}-T_{2})}}\right)%
^{-1}\right)\right).\end{split}$$
Using a version of Perron’s inversion formula, we get
$$\frac{1}{1!}\sum_{d\leqslant z^{2}}\mu(d)P_{d}(s,T_{1},T_{2})\log\left(z^{2}/d%
\right)=\frac{1}{2\pi i}\int\limits_{c-i\infty}^{c+i\infty}M_{w}(s,T_{1},T_{2}%
)\frac{z^{2w}}{w^{2}}\,dw,$$
with $c=1-\Re s+\frac{1}{\log z}$,
which implies that on the line $\Re s(=\sigma)=\eta$ we have
$$K(s,T_{1},T_{2})\ll z^{2(1-\eta)}(\log z)^{10}\ll\exp(2AE\log\log\log T)(\log%
\log T\log\log\log T)^{10}.$$
Thus recalling (4), (3) and (2) we get,
as in the proof of Lemma 4, that
$$\begin{split}&\displaystyle\left|\mathop{\text{{Res}}}\left(X^{-s_{0}}J(s,T_{1%
},T_{2})K(s,T_{1},T_{2})\Gamma(s-s_{0})X^{s}\right)_{s=s_{0}}\right.\\
&\displaystyle\left.+\frac{X^{-s_{0}}}{2\pi i}\int\limits_{(\sigma=\eta)}J(s,T%
_{1},T_{2})K(s,T_{1},T_{2})\Gamma(s-s_{0})X^{s}ds-e^{-1/X}\right|\\
&\displaystyle\geqslant(\log\log T)^{\varepsilon}+O\left(\exp\left(0{.}5DA\log%
\log\log T(E_{0}-E)\right)\frac{\log\log T}{E-E_{0}}\right.\\
&\displaystyle\left.\times\left(e^{L(T)e^{(2+\varepsilon)E}}\log\log T\right)^%
{4}(\log\log T)^{2AE+10+\varepsilon}\right).\end{split}$$
Hence there is an $N$ such that $z\leqslant N\leqslant X^{2}$,
and
$$\sum_{N\leqslant n\leqslant 2N}|\sigma_{iT_{1}}(n)||\sigma_{-iT_{2}}(n)||a(n)|%
n^{-\sigma_{0}}\gg(\log\log T)^{-1+\varepsilon},$$
since the range of the summation $z\leqslant n\leqslant X^{2}$ may be divided
into the intervals $N\leqslant n\leqslant 2N$ so that the number of the intervals
is $\ll\log X^{2}/z\ll\log\log T\log\log\log T$ and the sum over the entire range
must be $\gg(\log\log T)^{\varepsilon}$.
By the Cauchy inequality and by Lemma 5, we get
$$(\log\log T)^{-2+\varepsilon}\log z\ll\sum_{N\leqslant n\leqslant 2N}|\sigma_{%
iT_{1}}(n)|^{2}|\sigma_{iT_{2}}(n)|^{2}N^{1-2\sigma_{0}}.$$
Finally, by Lemma 4 with $T_{1}=T$, $T_{2}=T+H$ we establish that
$$\begin{split}\displaystyle N^{2(1-\sigma_{0})}&\displaystyle\gg\left((\log\log%
\log T)^{3}(\log\log T)^{7}|\zeta(1+iT_{1})|^{4}|\zeta(1+iT_{2})|^{4}\right.\\
&\displaystyle+(\log\log T)^{7}\zeta(1+i(T_{1}+H))^{2}\zeta(1-i(T_{1}-H))^{2}%
\zeta(1+i(T_{2}+H))^{2}\zeta(1-i(T_{2}-H))^{2}\\
&\displaystyle\left.+(\log\log T)^{7}\zeta(1+i(T_{1}-H))^{2}\zeta(1-i(T_{1}+H)%
)^{2}\zeta(1+i(T_{2}-H))^{2}\zeta(1-i(T_{2}+H))^{2}\right)\\
&\displaystyle\left.+O\left((\log\log T)^{-1}\right)\right)^{-1}(\log\log T)^{%
-1+\varepsilon}.\end{split}$$
Next we prove existence of the infinite sequence of pairs of real numbers $(T_{1},T_{2})$,
claimed in the theorem.
We may choose $T_{1}=T$ and $T_{2}=T+H$ in the following way:
As in [3], Chapter VIII, §8.6, for $\sigma>1$
$$\log\frac{1}{|\zeta(s)|}=-\sum\frac{\cos(t\log p_{n})}{p_{n}^{\sigma}}+O(1).$$
Also, we have the identity
$$\cos((t+h)\log p_{n})=\cos(t\log p_{n})\cos(h\log p_{n})-\sin(t\log p_{n})\sin%
(h\log p_{n}).$$
So, we want to choose $t$ such that for, say, every $p_{n}\equiv\pm 1\pmod{7}$ and $n\leqslant N_{2}$
$$\cos(t\log p_{n})<-1+\frac{1}{N_{2}},$$
for every $p_{n}\equiv\pm 2\pmod{7}$ and $n\leqslant N_{2}$
$$\cos(t\log p_{n})\begin{cases}{}<-1+\frac{1}{N_{2}}&\text{if}\quad\cos(H\log p%
_{n})\geqslant 0,\\
{}>1-\frac{1}{N_{2}}&\text{if}\quad\cos(H\log p_{n})<0,\end{cases}$$
and for every $p_{n}\equiv\pm 3\pmod{7}$ and $n\leqslant N_{2}$
$$\cos(t\log p_{n})\begin{cases}{}<-1+\frac{1}{N_{2}}&\text{if}\quad\cos(2H\log p%
_{n})\geqslant 0,\\
{}>1-\frac{1}{N_{2}}&\text{if}\quad\cos(2H\log p_{n})<0.\end{cases}$$
This may be done as in Lemma $\delta$ of [3], Chapter VIII, §8.8.
Now existence of the sequence $(T_{1},T_{2})$ follows from this
and estimates as in Lemma 6
by the Phragmén–Lindelöf method.
Thus,
$$\begin{split}\displaystyle N^{2(1-\sigma_{0})}&\displaystyle\gg\left((\log\log%
\log T)^{3}(\log\log T)^{-1}+O\left((\log\log T)^{-1}\right)\right)^{-1}\\
&\displaystyle\times(\log\log T)^{-1+\varepsilon}.\end{split}$$
This ends the proof of the theorem.
References
[1]
Y. Motohashi,
An observation on the zero-free region
of the Riemann zeta-function,
Periodica Mathematica Hungarica 42 (1–2) (2001), 117–122.
[2]
S. N. Preobrazhenskii,
An extension of Motohashi’s observation on the zero-free region
of the Riemann zeta-function,
Preprint (2011).
[3]
E. C. Titchmarsh,
The Theory of the Riemann Zeta-function,
2nd ed., revised by D. R. Heath-Brown, Oxford Science Publications, Oxford, 1986.
[4]
E. Carneiro and V. Chandee,
Bounding $\zeta(s)$ on the critical strip,
Preprint (2010).
[5]
A. A. Karatsuba and S. M. Voronin,
The Riemann Zeta-function,
trans. from the Russian by Neal Koblitz, de Gruyter, Berlin; New York, 1992.
[6]
U. Balakrishnan,
On the sum of divisors function,
J. Number Theory 51 (1995), 147–168.
[7]
M. B. Barban and P. P. Vehov,
An extremal problem,
Trans. Moscow Math. Soc. 18 (1968), 91–99.
[8]
S. Graham,
Applications of sieve methods,
Thesis, University of Michigan, 1977.
[9]
S. Graham,
An asymptotic estimate related to Selberg’s sieve,
J. Number Theory 10 (1978), 83–94.
[10]
K. Tsang,
The distribution of the values of the Riemann zeta-function,
Dissertation, Princeton University, 1984. |
Spin-wave logic devices based on isotropic forward volume magneto-static waves
S. Klingler
stefan.klingler@wmi.badw-muenchen.de
Current Address: Walther-Meißner-Institut, Bayerische Akademie der Wissenschaften, 85748 Garching, Germany
P. Pirro
Current address: Institut Jean Lamour, Université de Lorraine, 54011 Nancy, France
T. Brächer
B. Leven
B. Hillebrands
A. V. Chumak
Fachbereich Physik and Landesforschungszentrum OPTIMAS, Technische Universität
Kaiserslautern, 67663 Kaiserslautern, Germany
(December 6, 2020)
Abstract
We propose the utilization of isotropic forward volume magneto-static spin waves in modern wave-based logic devices and suggest a concrete design for a spin-wave majority gate operating with these waves. We demonstrate by numerical simulations that the proposed out-of-plane magnetized majority gate overcomes the limitations of anisotropic in-plane magnetized majority gates due to the high spin-wave transmission through the gate, which enables a reduced energy consumption of these devices. Moreover, the functionality of the out-of-plane majority gate is increased due to the lack of parasitic generation of short-wavelength exchange spin waves.
Steadily advancing progress in modern information technology pushes the computation capabilities of common silicon-based digital logics towards fundamental quantum limits. This raises the problem of how to overcome these inherent problems. Furthermore, issues such as waste heat production demand for new approaches.ITR (2013); Galatsis et al. (2009) A potential alternative is wave-based computing. In particular, logic elements and devices based on collective excitations of the magnetization in the solid state - spin waves and their quanta magnons - have attracted attention in the recent years.Stamps et al. (2014); Schneider et al. (2008); Khitun, Bao, and Wang (2010); Brächer et al. (2013); Klingler et al. (2014) In a spin-wave device the information can be encoded into the phase or amplitude of the wave, and the information can be processed by employing interference between different waves as well as nonlinear interactions.
This field is still in its infant state, but recent progress has demonstrated its large potential.Chumak et al. (2012); Chumak, Serga, and Hillebrands (2014)
A cornerstone of wave-based logic elements is the majority gate, since it allows for a simple implementation of complex logic circuits and the processing of several boolean operations with a single gate structure.Khitun and Wang (2008); Klingler et al. (2014)
In a previous study, we introduced the design of an in-plane magnetized spin-wave majority gate.Klingler et al. (2014) It consists of three input waveguides where spin waves are excited, a symmetric spin-wave combiner which merges the different input waveguides, and an output waveguide where a spin wave propagates with the same phase as the majority of the input waves. In the combiner region scattering processes into higher-order dipolar spin-wave modes and small-wavelength exchange spin waves occur due to the broken translational symmetry of the waveguide system and the anisotropic dispersion relation of the spin waves in this magnetization configuration. We showed, that parasitic scattering processes into higher dipolar modes can be suppressed with a suitable waveguide geometry. But still, the output signal is influenced by exchange spin waves.Klingler et al. (2014)
To overcome these limitations the use of isotropic forward volume magneto-static spin waves (FVMSW) is an interesting option, which thus implies the necessity of a new suitable majority gate design. Here, we demonstrate the functionality of an out-of-plane magnetized spin-wave majority gate which operates with isotropic forward volume magneto-static spin waves (FVMSW) and, thus, overcomes the limitations of the in-plane magnetized gates. We employ numerical simulations to prove the characteristics of the gate and we find a high spin-wave transmission through the gate of up to 64 %, which is about three times larger than for the in-plane magnetized gate.Klingler et al. (2014)
The simulations are performed for Yttrium-Iron-Garnet-(YIG)-structures with a thickness of 100 nm.Klingler et al. (2015); Pirro et al. (2014) For this purpose, the following material parameters have been used: a saturation magnetization of $M_{\mathrm{s}}=140$ kA/m, an exchange constant of $A=3.5$ pJ/m and a Gilbert damping of $\alpha=5\cdot 10^{-4}$. Due to the small Gilbert damping parameter of YIG, spin-wave propagation distances in the millimeter range are observed.Serga, Chumak, and Hillebrands (2010) This is larger than the size of conventional microstructures and, thus, makes it a very suitable material for the construction of complex magnonic networks.Chumak, Serga, and Hillebrands (2014) Furthermore, the relatively small saturation magnetization of YIG allows to switch the magnetization out-of-plane with moderate external fields ($\mu_{0}H\gtrsim 180$ mT), which can be easily applied by, for instance, a permanent magnet.
In Fig. 1 the design of an out-of-plane magnetized majority gate is shown. All waveguides have a width of 1 $\mu$m, so that the possibility of a practical realization of the gate structures is ensured.Chumak, Serga, and Hillebrands (2014); Pirro et al. (2014); Hahn et al. (2014)
The majority gate structure consists of three parallel input waveguides, where the spin waves are excited and the information is encoded into the spin-wave phase. In the combiner the input waveguides are bent under an angle $\alpha$ towards the center waveguide, to overlay the spin waves and allow for their interference with each other. The bent parts have a length of 5 $\mu$m and merge with the center waveguide at positions $x_{1}$ and $x_{2}$, respectively. With this asymmetric design the spin waves can be forced to propagate into the direction of the connected output waveguide, which guarantees a high energy transmission. In the output waveguide the output spin wave propagates with the same phase as the majority of the input waves.
The majority gate structure is investigated with numerical simulations using MuMax2.Vansteenkiste et al. (2014) The software performs the simulations on the graphics cards of the computer and, thus, allows for a highly parallelized calculation of the spin-wave dynamics in a mesoscopic magnetic system with a resolution in the order of the exchange length. Spin-wave reflections at the end of the waveguides are avoided by increasing the damping by a factor of 300 over the last 4 $\mu$m in the $x$-direction. The external field was applied parallel to the $z$-axis with a strength of $\mu_{0}H=200$ mT. The cell size in the simulated area was chosen to be $14\times 8\times 100$ nm${}^{3}$, so that the resolution is in the order of the exchange length ($\lambda_{\mathrm{ex}}=18$ nm)Abo et al. (2013) of YIG and smaller than the spin-wave wave length in the microstructures. In $z$-direction the usage of only one cell is justified since no perpendicular standing spin-wave modes are excited at the working frequency of $f=1.5$ GHz. To excite the spin waves, individual coplanar waveguides (CPW) were modeled for each input arm, to suppress dynamic magnetic fields outside of the excitation area. The center conductor of the CPW has a width of 400 nm and a height of 250 nm. The ground plates have widths of 200 nm and heights of 250 nm. The center-to-center distance between conductor and ground plate is 400 nm. The excitation field is then calculated using Biot-Savart’s law for an AC-current with an amplitude of 0.1 mA in the conductor and -0.05 mA in the ground plates and a frequency of 1.5 GHz.
After an excitation time of 100 ns the spin-wave amplitude reached its steady state. Subsequently, the magnetization distribution in the waveguide system was saved with a time resolution of 4 ps.
To study the performance of the majority gate, a single input arm was excited and the transmission of energy into the output was investigated.
In Fig. 2a) the energy transmission is shown in a logarithmic colorscale for the spin-wave excitation in input 1 when the merging angle of the waveguide is $\alpha=20\,^{\circ}$ and the merging position is different for the outer input arms ($x_{1}\neq x_{2}$). With this design the energy transmission from input 1 to the output waveguide is 64 %, which is the ratio of the maximum value in the output waveguide to the maximum value around $x_{2}$. The spin waves from input 3 have the smallest transmission of 32 % and spin waves from input 2 exhibit an energy transmission of 45 % into the output waveguide.
The transmission strongly depends on the bend angle $\alpha$ and the number of merging areas which have to be passed to reach the output. This first point can be understood when the wavevector of the spin waves in the combiner region is split into its components parallel to the $x$- and $y$-direction. With a small bend angle $\alpha$ the $x$-component of the wavevector increases and the $y$-component decreases, so that the spin-wave propagation into the direction of the output waveguide is favored. At the same time, the propagation of the spin waves into the opposed waveguide is suppressed due to the non-overlapping merging areas ($x_{1}\neq x_{2}$).
The second point becomes clear when the merging areas are considered as source of spin-wave reflections (e.g. in the vicinity of $x_{1}$ and $x_{2}$) into the other input arms. This reflection process gives rise to the standing interference patterns visible in Fig. 2a). The more merging areas are passed, the more reflections can occur.
In Fig. 2b) another interesting feature of isotropic FVMSW is shown, for the extreme situation when the waveguides are merged under an angle of $90^{\circ}$ at the same positions ($x_{1}=x_{2}$) and when the spin waves are excited in the center waveguide. In this case the wavevector of the incoming spin wave points directly into the output waveguide, and a high energy transmission of 72 % through the gate can be achieved, while only 14 % of the energy are transmitted to the adjacent input waveguides. This example shows that spin-wave networks can be realized with two-dimensional rectangular crossings in the out-of-plane magnetized geometry. This allows for via-free crossings of spin-wave waveguides, and it is a major advantage of the spin-wave technology, which can examplarily be used in magnonic holographic memories Gertz et al. (2014) or magnonic full adders, based on majority gates.Ibrahim, Beiu, and Sulieman (2008)
As mentioned above, the interference patterns in Fig. 2a) result from back reflections in the combiner regions and are not originating from scattering processes into higher dipolar spin waves or exchange spin waves as in the in-plane magnetized gates.Klingler et al. (2014) This can be understood by examining the dispersion relationsKalinikos and Slavin (1986) for the first three width modes $n$ of the waveguide which are shown in Fig. 3. Here, $n$ is the number of anti-nodes across the waveguide width. The dispersions were calculated for an effective stripe width of 1.42 $\mu$m to include dynamic demagnetization effects and effective dipolar pinning.Guslienko et al. (2002) Additionally, a demagnetization factor of 0.92 was used to correct the internal static field in the waveguides.Joseph and Schlömann (1965) It can be seen, that no higher width modes can exist in the waveguides at an excitation frequency $f=1.5$ GHz. In contrast to the situation in in-plane magnetized waveguides, the direct scattering processes under energy conservation into exchange dominated spin waves are also not possible. Furthermore, one can extract a wavevector of $k=3.8$ rad/$\mu$m, a wavelength of $\lambda=1.7\,\mu$m and a group velocity of $v_{\text{g}}=0.13\,\mu$m/ns from the dispersion relation at the excitation frequency $f$. Together with the decay time of the spin wave of $\tau=566$ ns, which can be calculated from the material parameters,Patton (1975) one obtains a decay length $\lambda_{\text{dec}}=v_{\text{g}}\tau=73\,\mu$m, which is much larger than the size of the microstructures.
For the majority operation it is important to account for the different propagation losses and phase shifts resulting from the different propagation distances. For this, we compare the output phase and amplitude of all single arm excitations of the gate in Fig. 2a). As a result we reveal as equalizing parameters for input 2 an attenuation factor of 0.7 and a phase shift of $\Delta\phi=-0.35\pi$ with respect to input 3. The spin-wave excitation in input 1 has to be attenuated by a factor of 0.5 and the excitation phase has to be shifted by $\Delta\phi=-0.64\pi$ relative to input 3. The different equalizing factors can be understood by the asymmetric gate geometry. In a real device these shifts can be realized by a small displacement of the antenna positions or by the use of nanomagnets as phase shifters.Au et al. (2012)
With this adjustment parameters the output signals of every single input coincide, and all possible input combinations can be simulated.Klingler et al. (2014)
In Fig. 4a) the $x$-component of the magnetization distribution $m_{x}$ is shown for a logic 1-0-0 excitation (logic values from top to bottom) at a fixed time step. This means, that the spin waves in the upper input are excited with an additional phase shift of $\pi$, whereas the spin waves in the center and bottom waveguide are excited without an additional phase shift. At the edge of the waveguides one can see a small area where the magnetization is inverted. This is the result of an overlapping edge-modeDamon and Eshbach (1961); Lisenkov et al. (2014) which propagates on the edge of the waveguide and decays exponentially into the waveguide width. Due to its small amplitude, no negative influences of this effect are expected.
In Fig. 4b) $m_{x}$ of every possible input combination is shown for the same time step. The sinusoidal shape of the magnetization distributions is clearly visible. No disturbing influences of exchange waves can be seen, compared to in-plane magnetized majority gates.Klingler et al. (2014) All spin-waves with a majority phase of “0” are in phase and are exactly out of phase to the spin waves with a majority phase of “1”. This is the clear signature of the majority function, which can be seen by comparing the results with the truth table in the legend in Fig. 4b). From the magnetization distributions one can easily extract a wavelength of about 1.8 $\mu$m, which is in good agreement with the prediction of the dispersion relation. Additionally, the amplitudes of the spin waves in the output waveguide fit very well to the theoretical predictions: if all three input waves are in phase, the amplitude in the output waveguide is three times larger than for every other case. This confirms the adjustment parameters in the input waveguides. Since the information is encoded in the spin-wave phase, the different output amplitudes are not essential for the information content of the spin wave. However, if the output signal should be forwarded to another logic gate, the amplitude has to be normalized before, e.g. by the use of parametric amplification Serga et al. (2003); Khitun, Nikonov, and Wang (2009); Brächer et al. (2014) or by use of a magnon transistor.Chumak, Serga, and Hillebrands (2014)
To show that the spin-wave majority gate allows for a full set of logic operations, input 3 can be chosen as a control input. At a fixed readout position, e.g. at $x=15.5\,\mu$m, one can perform AND-operations between inputs 1 and 2 when the spin waves in input 3 are excited without any additional phase shift. At the same readout position OR-operations can be performed when the spin waves in input 3 are excited with an additional phase shift of $\pi$. By shifting the readout position by one half of the wavelength, e.g. to $16.4\,\mu$m, the output phase of the spin wave is shifted by $\pi$ and thus, NAND- and NOR-operations are performed dependent on the control input signal. In total, four different logic operations can be processed with a single majority gate.
In conclusion, a fully functional, asymmetric magnonic majority gate design for the use with isotropic FVMSW has been presented. It was shown, that the output signal is not disturbed by parasitic scattering processes into exchange spin waves.
This results in a very clear and undistorted interference pattern in the output waveguide. It was verified, that the majority of the input values defines the output signal, which is a clear signature of a working majority gate. Furthermore, the spin-wave logic approach was used to perform AND-, OR-, NAND-, and NOR-operations with a single gate structure. The data processing and transmission occured in the pure spin-wave system, which makes the gate suitable for an integration into complex magnonic networks with various daisy-chained spin-wave devices.Chumak, Serga, and Hillebrands (2014); Brächer et al. (2013); Vogt et al. (2014) Finally, it was shown that the utilization of isotropic FVMSW allows for the realization of orthogonal waveguide crossings, where spin waves can pass the crossing region with a high transmission without the need of a via-implementation which would demand for several layers and patterning steps. Such spin-wave waveguide crossings are necessary for spin-wave logic device networks.Gertz et al. (2014); Ibrahim, Beiu, and Sulieman (2008)
This research has been supported by the EU-FET grant InSpin 612759.
References
ITR (2013)
“International technology roadmap for semiconductors (ITRS),” (2013).
Galatsis et al. (2009)
K. Galatsis, A. Khitun,
R. Ostroumov, K. L. Wang, W. R. Dichtel, E. Plummer, J. F. Stoddart, and J. I. Zink, IEEE Trans. Nanotechnol. 8, 66 (2009).
Stamps et al. (2014)
R. L. Stamps, S. Breitkreutz,
J. Å kerman, A. V. Chumak, Y. Otani, G. E. W. Bauer, J.-U. Thiele, M. Bowen, S. A. Majetich,
M. Kläui, I. L. Prejbeanu, B. Dieny, N. M. Dempsey, and B. Hillebrands, Journal of Physics D: Applied Physics 47, 333001 (2014).
Schneider et al. (2008)
T. Schneider, A. A. Serga, B. Leven,
B. Hillebrands, R. L. Stamps, and M. P. Kostylev, Appl. Phys.
Lett. 92, 22505
(2008).
Khitun, Bao, and Wang (2010)
A. Khitun, M. Bao, and K. L. Wang, J. Phys. D. Appl. Phys. 43, 264005 (2010).
Brächer et al. (2013)
T. Brächer, P. Pirro,
J. Westermann, T. Sebastian, B. Lägel, B. Van de Wiele, A. Vansteenkiste, and B. Hillebrands, Appl. Phys.
Lett. 102, 132411
(2013).
Klingler et al. (2014)
S. Klingler, P. Pirro,
T. Brächer, B. Leven, B. Hillebrands, and A. V. Chumak, Applied
Physics Letters 105, 152410 (2014), arXiv:1408.3235 .
Chumak et al. (2012)
A. V. Chumak, V. I. Vasyuchka, A. A. Serga, M. P. Kostylev,
V. S. Tiberkevich, and B. Hillebrands, Phys. Rev. Lett. 108, 257207 (2012).
Chumak, Serga, and Hillebrands (2014)
A. V. Chumak, A. A. Serga, and B. Hillebrands, Nat. Commun. 5, 4700 (2014).
Khitun and Wang (2008)
A. Khitun and K. L. Wang, IEEE Trans. Magn. 44, 2141 (2008).
Klingler et al. (2015)
S. Klingler, A. V. Chumak, T. Mewes,
B. Khodadadi, C. Mewes, C. Dubs, O. Surzhenko, B. Hillebrands, and A. Conca, Journal of Physics D: Applied
Physics 48, 15001
(2015), arXiv:1408.5772 .
Pirro et al. (2014)
P. Pirro, T. Brächer,
A. V. Chumak, B. Lägel, C. Dubs, O. Surzhenko, P. Görnert, B. Leven, and B. Hillebrands, Appl. Phys. Lett. 104, 12402 (2014).
Serga, Chumak, and Hillebrands (2010)
A. A. Serga, A. V. Chumak, and B. Hillebrands, J. Phys. D. Appl. Phys. 43, 264002 (2010).
Hahn et al. (2014)
C. Hahn, V. V. Naletov,
G. de Loubens, O. Klein, O. d’Allivy Kelly, A. Anane, R. Bernard, E. Jacquet, P. Bortolotti, V. Cros, J. L. Prieto, and M. Muñoz, Appl. Phys. Lett. 104, 152410 (2014).
Vansteenkiste et al. (2014)
A. Vansteenkiste, J. Leliaert, M. Dvornik,
F. Garcia-Sanchez, and B. Van Waeyenberge, AIP Adv. 4, 107133 (2014), arXiv:1406.7635 .
Abo et al. (2013)
G. Abo, Y. Hong, J. Park, J. Lee, W. Lee, and B. Choi, IEEE Trans. Magn. 49, 4937 (2013).
Gertz et al. (2014)
F. Gertz, A. Kozhevnikov,
Y. Filimonov, and A. Khitun, IEEE Transactions on Magnetics PP, 1 (2014), arXiv:1401.5133 .
Ibrahim, Beiu, and Sulieman (2008)
W. Ibrahim, V. Beiu, and M. H. Sulieman, IEEE Trans. Nanotechnol. 7, 56 (2008).
Kalinikos and Slavin (1986)
B. A. Kalinikos and A. N. Slavin, J. Phys. C Solid State Phys. 19, 7013 (1986).
Guslienko et al. (2002)
K. Guslienko, S. Demokritov, B. Hillebrands, and A. Slavin, Phys. Rev. B 66, 132402 (2002).
Joseph and Schlömann (1965)
R. I. Joseph and E. Schlömann, J. Appl. Phys. 36, 1579 (1965).
Patton (1975)
C. E. Patton, in Magnetic Oxides, edited by D. J. Craik (John Wiley, London, 1975) pp. 575–645.
Au et al. (2012)
Y. Au, M. Dvornik,
O. Dmytriiev, and V. V. Kruglyak, Appl. Phys. Lett. 100, 172408 (2012).
Damon and Eshbach (1961)
R. W. Damon and J. R. Eshbach, J. Phys. Chem. Solids 19, 308 (1961).
Lisenkov et al. (2014)
I. Lisenkov, V. Tyberkevych, A. Slavin,
P. Bondarenko, B. A. Ivanov, E. Bankowski, T. Meitzler, and S. Nikitov, Phys.
Rev. B 90, 104417
(2014).
Serga et al. (2003)
A. A. Serga, S. O. Demokritov, B. Hillebrands, S.-G. Min, and A. N. Slavin, J. Appl. Phys. 93, 8585 (2003).
Khitun, Nikonov, and Wang (2009)
A. Khitun, D. E. Nikonov,
and K. L. Wang, Journal of Applied Physics 106, 123909 (2009).
Brächer et al. (2014)
T. Brächer, P. Pirro,
T. Meyer, F. Heussner, B. Lägel, A. A. Serga, and B. Hillebrands, Appl. Phys.
Lett. 104, 202408
(2014).
Vogt et al. (2014)
K. Vogt, F. Y. Fradin,
J. E. Pearson, T. Sebastian, S. D. Bader, B. Hillebrands, A. Hoffmann, and H. Schultheiss, Nat.
Commun. 5, 3727
(2014). |
Spatio-temporal Dynamics in the Origin of Genetic Information
Pan-Jun Kim
[
Hawoong Jeong
Corresponding author. Tel:+82-42-869-2543; fax:+82-42-869-2510.
hjeong@kaist.ac.kr
Department of Physics, Korea Advanced Institute of Science and Technology,
Daejeon 305-701, South Korea
Abstract
We study evolutionary processes induced by spatio-temporal dynamics in prebiotic evolution.
Using numerical simulations we demonstrate that hypercycles emerge from
complex interaction structures in multispecies systems. In this work
we also find that ‘hypercycle hybrid’ protects the hypercycle from its environment
during the growth process.
There is little selective advantage for one hypercycle to maintain coexistence with others.
This brings the possibility of the outcompetition between hypercycles
resulting in the negative effect on information diversity. To enrich the information
in hypercycles, symbiosis with parasites is suggested. It is shown that symbiosis with
parasites can play an important role in the prebiotic immunology.
keywords:
prebiotic evolution, complex networks, self-structuring
PACS: 89.75.-k, 05.65.+b, 87.23.-n, 82.40.Ck, 82.20.Wt
url]http://stat.kaist.ac.kr/~pj/origin.html,
1 Introduction
The appearance of a molecule which is capable of replicating itself is probably
the most fundamental event in the history of life.
One of the candidates for the prebiotic genes, RNA is both the carrier
of genetic information and the molecule with
biological activities of which variety becomes an interesting topic recently [1].
Eigen and coworkers were
the first to address the existence of the information threshold
in prebiotic evolution that the length of a
molecule (polynucleotide) is limited due to the finite replication accuracy per nucleotide
[2].
The maximum length of the molecules attained by the process of Darwinian selection
seems to be too short for a genetic message to encode a functional protein.
In their hypercycle theory Eigen suggests that
if molecules catalyze the replication of each other in a cyclic way,
the information threshold can be crossed [2].
No molecule in the hypercycle can outcompete another
because they are forced to cooperate. Each molecule is still
bound to the maximum string-length, but the molecules can combine their information and
thus the information threshold can be crossed.
However, there are two major problems with this idea. The first problem is that hypercycles with five or more species show a limit cycle behavior.
This implies that large hypercycles are unstable because some
species may become extinct. The second problem is that providing catalytic support
to other species is in fact an altruistic behavior, therefore, they are extremely vulnerable
to the presence of parasites which are species that do not reciprocate the catalytic support
they receive.
Boerlijst and Hogeweg [3] have shown that spatial self-structuring can solve
these extinction and parasite problems. They added the diffusive behavior of each
species to the early model of hypercycle. In this spatially diffusive model, hypercycles with
five or more species spontaneously generate spiral waves, rotating pattern of all species
in the hypercycle. With the help of this rotating spiral wave global
extinction of species no longer takes place.
Furthermore, it turns out that spiral waves are resistent to parasite invasions.
Because the molecules in the center of a spiral generate an offspring of the entire spiral
in radial direction,
it is difficult for a parasite to grow towards the center of the spiral.
Their pioneering work presents three important points. First, natural selection is
effectively driven by spatio-temporal dynamics
[3, 4, 5, 6], not only by the
population-driven competition between species. For example, Boerlijst
and Hogeweg observed that increasing of diffusion rate makes the spiral patterns bigger,
enhancing the resistence to invasion of parasites. Second, as a consequence of
self-structuring, natural selection occurs at the level of the community but not
of the individual [7]. A community maintains some minimum level of
integrity for a long period enough for natural selection to act on. Integrity can be maintained
as a form of nonrandom pattern in the spatial arrangement of individuals.
Each spiral behaves like super-organism
whose boundary is determined by the molecules in the periphery. Therefore the evolutionary
attractors do not convey a fitness benefit to individual species but to the community
determined by the spiral
[4, 8]. Third, they showed spatio-temporal dynamics of interaction
networks can bring
out nontrivial behaviors. The hypercycle which consists
of a single cyclic interaction structure seems to be fatal to parasite invasions, but it
turns out that the hypercycle is very robust when we consider its
spatio-temporal dynamics. However, little is known about spatio-temporal dynamics
for the case of more complex interaction networks
in this respect, while many research groups have studied
several properties of complex networks
recently [9, 10].
In this paper, we addressed several issues which should be resolved
to fully understand the spatio-temporal dynamics of multispecies interaction network.
Boerlijst and Hogeweg let parasite
invade hypercycles
after full development of their spatial structures. But if parasite-like species is
present before self-structuring of hypercycles, can hypercycles
successfully drive away their initially embedded parasites? Also, if
the multispecies system has the complex interaction structure (network) rather than a single
cyclic structure as in the hypercycle model, is it possible that some cyclic sub-structures
outcompete other sub-structures by the process of spiral-formation? We expect
that the complex interaction structure would be separated
into many cyclic sub-structures by self-structuring at first.
And then competition between these cyclic sub-structures would select sub-structure
with more efficient species.
Our scenario on prebiotic evolution is different from that of Jain and Krishna
[11, 12] in which selection on species in the same giant structure containing
autocatalytic sets is suppressed during structure development. In their scenario, evolutionary
unstability is unavoidable by a collapse of the giant structure due to
the selection at the level
of individuals in the complex interaction network.
In the next section we investigate the competition between initially embedded parasites
and hypercycles and find the mechanism of
separation of cyclic sub-structures using a simple model structure.
Using numerical simulations, in Section 3 we demonstrate an
emergence of information communities in prebiotic evolution. And we find
that the ‘hypercycle hybrid’ protects the hypercycle from its environment
during the development process.
In Section 4 by investigating interaction between developed communities
we find that
self-interest of each community can discourage inheritance of diverse information.
To recover information diversity, in Section 5 we introduce
symbiosis with parasites and show that these embeded parasites
can play an important role on the
immunology of the community. The final section is devoted to conclusions.
2 Simple Trials
To study the spatio-temporal dynamics of hypercycles we use the following model
equation [13].
$$\frac{dX_{i}}{dt}=-\delta_{i}X_{i}+\big{(}1-\sum_{k=1}^{N}X_{k}\big{)}\big{(}%
\rho_{i}+\sum_{j}\kappa_{i,j}X_{j}\big{)}X_{i}+D_{i}\nabla^{2}X_{i}\,\,.$$
We assume there are $N$ different types
of species and $X_{i}$ denotes the concentration of species $i$ at a given site.
$\delta_{i}$ stands for the spontaneous decay rate of species $i$, $\rho_{i}$ for the self-replication
rate, and $D_{i}$ for the diffusion coefficient.
$\kappa_{i,j}$
is the rate of replication of species $i$ catalyzed by species $j$.
The first (second) term of rhs corresponds to the decay (growth) rate of population of
species.
The term $\big{(}1-\sum_{k=1}^{N}X_{k}\big{)}$ limits the population of species at the same site
due to finite resources. The final term shows diffusive behavior of species $i$
which is responsible for spatial pattern formation.
For numerical simulations we use a rectangular grid of $147\times 147$ sites with
Neumann boundary conditons. And
we assume that species $i$ is extinct when $X_{i}$ is small enough (e.g., $X_{i}<10^{-3}$).
We use
the integration time step $D_{max}\Delta t/(\Delta x)^{2}=0.1$ where we choose $D_{max}$ as the maximum
value of $D_{i}$ ( $i=1,2,\dots,N$ ), $\Delta x=1$. We use the parameters
$\delta_{i}=1$, $\rho_{i}=2$, $D_{i}=1$,
$\kappa_{i,j}=0$ or $500$ (depending on the existence of catalytic support
to species $i$ from species $j$) unless specified [14].
One necessary condition for cyclic sub-structures being favored in complex interaction
structure would be the stability of the cyclic structures to initially embedded
parasites. To investigate this stability we consider the structure in Fig. 1(a).
Each arrow indicates
the direction of catalysis, e.g. $2\rightarrow 3$ means that species 2 catalyzes
the replication of species 3 and, therefore concentration of species 3 increases.
Species 7 is a parasite because it does not
reciprocate the catalytic support given by species 6.
We study the following two cases. First, to test whether hypercycles can drive away
initially embedded parasites, we start with randomly assigned initial concentration containing
molecules of all species (case A). Second, as a comparison with the situation where
parasites invade
hypercycles after
fully development of spatial structure, we begin with randomly assigned initial concentration
of six species while parasite concentration remains zero.
And after spirals are fully developed, species in a half of the sites
are replaced by parasite species (case B) [15].
By numerical simulations it is identified that initially embedded parasites are also effectively
outcompeted although it is only possible in
the narrow parameter region when compared with case B
(see Fig. 1(b)).
It turns out that even if parasites are initially embedded,
spiral cores avoiding infection of parasites
in the early stage can drive away the parasites by radial growth
of the spiral waves (see Figs. 2(a–c)).
Next, we consider the structure in Fig. 3(a) where species 7 is equivalent to species
1 in their parameters.
Separation of the cyclic
sub-structures
is successful by the spiral formation from molecules with randomly assigned concentration
(see Fig. 3(b)). There are two kinds of spirals. One is composed of $1\rightarrow 2\rightarrow 3\rightarrow 4\rightarrow 5\rightarrow 6\rightarrow 1$
and the other is composed of $7\rightarrow 2\rightarrow 3\rightarrow 4\rightarrow 5\rightarrow 6\rightarrow 7$.
Species 1 and 7 cannot coexist in the same spiral because
the species (either 1 or 7) slightly out of the spiral core
fails to invade the spiral core and is washed out from its environment
(see Figs. 4(a–c)). As a result species 1 and 7 cannot
coexist in the same spiral except for the case
where two kinds of spirals come in contact
with each other at peripheries. But even in that case separation
at the level of spiral cores is clear (see Fig. 3(b)) and separation of cyclic
sub-structures is completed.
In this section we examined the stability of the spiral against parasites in early stage,
and explored the possibility that the specific cyclic sub-structures
can be favored in the complex interaction networks.
3 Emergence of Communities
Next, we consider more complex model structure as in Fig. 5(a). There are three cyclic sub-structures,
$1\rightarrow 6\rightarrow 5\rightarrow 4\rightarrow 3\rightarrow 2\rightarrow 1$,
$1\rightarrow 6\rightarrow 7\rightarrow 4\rightarrow 3\rightarrow 2\rightarrow 1$ and
$1\rightarrow 6\rightarrow 7\rightarrow 8\rightarrow 2\rightarrow 1$.
As expected in the previous section the cyclic
sub-structure repels other sub-structures and each cyclic sub-structure coexists in the
form of spirals. Each spiral can be considered as community where species in the same cycle
mainly interacts each other.
However, it is unlikely that every parameters of each species will
have the same value as system gets complex. We demonstrate that
selection process indeed acts on these cyclic sub-structures due to species dependent
parameters.
If we increase the diffusion coefficient of species 5 by $D_{5}=2$, it is observed
that $1\rightarrow 6\rightarrow 5\rightarrow 4\rightarrow 3\rightarrow 2\rightarrow 1$
outcompetes other cyclic sub-structures
(see Fig. 5(b)) [16]. If we increase the catalytic support from species 6 to species 7
and
from species 7 to species 4 by $\kappa_{7,6}=\kappa_{4,7}=1000$,
$1\rightarrow 6\rightarrow 7\rightarrow 4\rightarrow 3\rightarrow 2\rightarrow 1$
outcompetes other cyclic sub-structures (see Fig. 5(c)). As you can see,
small variation in local parameter can change whole dynamics.
We note that selection occurs at the community level but not at the individual level
because the individual-level selection is overridden by community
[7]. In the situation of Fig. 5(c), the species 6 gains little benefit
in
$1\rightarrow 6\rightarrow 7\rightarrow 4\rightarrow 3\rightarrow 2\rightarrow 1$
cycle because of its altruistic behavior in this community
(see the population of species 6 in Fig. 5(c)). But since the species 6 enhances the fitness
of the entire community,
the species 6 involved in
$1\rightarrow 6\rightarrow 7\rightarrow 4\rightarrow 3\rightarrow 2\rightarrow 1$
is selected in the final stage rather than the species 6 in
$1\rightarrow 6\rightarrow 5\rightarrow 4\rightarrow 3\rightarrow 2\rightarrow 1$.
In the above example, we observed competition between cyclic sub-structures and
resulting selection process. From these observations we find that
we don’t need to affect overall properties of the entire community to select a particular
community.
We just need to adjust local properties
of the interaction, e.g. diffusion coefficient of one node (Fig. 5(b))
or catalytic support across a few links (Fig. 5(c)). This ‘local
strategy’ has the important meaning for the control scheme on networks (control process has
been an attractive topic to many research groups
[17, 18, 19, 20],
but controlling networks is not such an explored field yet) and reflects the
self-structuring nature of the autocatalytic network.
However, the above work does not guarantee that single-cyclic sub-structure is
always selected as a final fittest community. In fact
we found that spiral pattern is unstable under the presence of the flat generated by
a short hypercycle of 2 or 3 species (flat is stationary spatial-structure
dominated by few, less than 3 species).
For example, if we assign the
catalytic support from species 3 to species 7 in Fig. 5(a) the simple
structure in Fig. 6(a)
outcompetes other sub-structures in Fig. 5(a). The parasite-like species 2 in Fig. 6(a)
survives because the flat generated by a short hypercycle cannot effectively drive away
its parasites contrary to the spiral generated by a long hypercycle. This is consistent
with the observation of Boerlijst and Hogeweg that a short cycle with dangled parasites
frequently emerges from randomly connected structures [8].
The fact that the flat generated by 2 or 3 species outcompetes the spiral generated
by more species conflicts with the information-threshold crossing
introduced in Section 1. It is unnatural that
short cycles are the fittests in prebiotic evolution.
Let us consider the structure assigning the catalytic support from species 5
to species 6 in the structure of Fig. 5(a). Then two sub-structures are emerging; one is
the short cycle with a parasite as expected (see Fig. 6(b)), and the other is the
composite structure (see Fig. 6(c)) which can be separated into
two structures as in Fig. 6(d).
We find that species 5 in the structure of Fig. 6(c) is eventually eliminated, and the flat formed by the short cycle in Fig. 6(b) is outcompeted by the spiral in Fig. 6(c) (see Figs. 7(a) and (b)).
At the initial stage, the flat is rapidly expanded and surrounds the spiral. The flat protects the spiral from the outside environment, otherwise
the environment may intimidate the spiral to be extinct. This ‘hypercycle hybrid’
between spiral and flat state is
maintained until the spiral becomes fully developed and then the flat is
outcompeted by the spiral.
Here we present how the hypercycle hybrid emerges from the complex structure. A short cycle cannot
suppress its parasites by the mechanism of spiral-formation. These dangling parasites weaken
the activity of the short cycle, thus the short cycle cannot outcompete the long hypercycle containing these parasites. For example, species 4 and 7 in Fig. 6(c) are not driven away
by the cycle
composed of species 5 and 6, so the hypercycle hybrid can exist as we described above.
As a consequence a hypercycle hybrid emerges naturally by the
short cycle with its parasites, and the hypercycle hybrid provides the selective advantage
to the long hypercycle by protecting its early growth process from the environmental species.
Selection process on hypercycle hybrid is also expected as previously demonstrated in the case of pure hypercycles in Fig. 5(b) and (c).
4 Maintenance of Communities
After communities are developed completely, interaction between communities becomes an important factor to determine the evolutionary direction of the communities.
Let us consider interaction between two emergent hypercycles;
cycle A ($1\rightarrow 2\rightarrow 3\rightarrow 4\rightarrow 5\rightarrow 6\rightarrow 1$)
and cycle B ($1\rightarrow 2\rightarrow 7\rightarrow 5\rightarrow 6\rightarrow 1$)
in Figs. 8(a) and (b).
We find that
cycle B outcompetes cycle A in the long time limit. One difference between these two cycles is
the number of species in the cycles, however we found that length of the cycle is not
responsible for this outcompetition.
For example, if we consider another cycle C ($7\rightarrow 8\rightarrow 9\rightarrow 10\rightarrow 11\rightarrow 7$)
in Fig. 8(c) instead of cycle B,
it turns out that they cannot outcompete each other even though length of cycle C is shorter
than that of cycle A.
The important difference between these two cases is that
cycle A and C are disjoint while cycle A and B are interrelated via several species
(in this case, species 1, 2, 5, 6).
It turns out that the reason why cycle B outcompetes cycle A is related
with this coupling between two hypercycles.
More specifically we observe that population of species 4 is markedly decreased
when two kinds of spirals come in contact with each other;
at the boundary between two spirals,
species 5 bred by the short catalysis path across the species 7 (cycle B) suppresses
the population of species 4, which brings the elimination of cycle A.
The above observation can be interpreted as follows.
If there is no shared species/information between two communities (like cycle A and C),
nature prefers to conserve information by keeping two communities alive together. However,
if there is shared information (like cycle A and B),
nature prefers to select specific information rather than conserve information.
From the case between cycle A and C,
we notice that conservation of information is obtained by the
equivalent competition between spirals rather than by cooperation between spirals.
The community based on a spiral is maintained by dynamic process at its core, thus cooperation
at the spiral peripheries does not provide any qualitative changes to this system.
Therefore a spiral has little selective advantage when maintaining coexistence
with other spirals. This property imposes the possibility of the outcompetition between spirals
causing a negative effect on information diversity.
There are several ways to diversify genetic information. First, we can
allow an evolutionary change of species’ character under the competitive ecological condition.
Imposed mutation at the spiral core can be effective in this respect.
In the spiral core, replication of molecules
occurs in an active manner expected to cause a number
of mutations,
and mutants in the spiral core are not easily
outcompeted than mutants in the spiral peripheries [4].
Second, we can compartmentalize the information, which was originally proposed to solve
the parasite problem in the hypercycle theory [2, 21]. Boerlijst and Hogeweg also
showed that compartmentation obtained by spatial gradient of molecular decay rate
increases the capacity for
information accumulation [13].
Each spiral in its own compartment has to overcome the surrounding barriers
in order to compete with the spirals in other compartments,
thus resulting in the suppressed competition.
Third, we can try endosymbiosis of information.
Endosymbiosis means that a symbiont dwells within the body of its symbiotic partner.
For the best known example, Margulis noted that the main internal structures of cells such as
mitochondria did not originate inside the cell, but reflect an endosymbiotic coupling
[22]. She showed that the basic path of biological evolution
is through the symbiosis of independent forms into more efficient and
more adaptive cooperatives.
We investigate the case by endosymbiosis of information in the next section.
5 Innovation of Communities
As presented in Section 4 spirals showing selfish behavior discourage inheritance of
diverse information. Here we investigate one possible solution to increase information diversity in the hypercycle
systems.
According to Section 2, when a parasite invades spirals the result is either of
the followings. The parasite outcompetes
spirals or the spirals drive away the parasite depending on choice of parameters.
What happens if the parasites which were considered to be toxic to
the spirals can catalyze i.e. give some benefit to some species of them?
Given the structure in Fig. 1(a),
we increase the diffusion coefficient and rate of catalyzed replication
of species 7 high enough to invade the spiral easily. Due to the
enhancement the species 7 will outcompete
the spirals. However, if we introduce relatively weak catalytic support
from species 7 to species 1, symbiosis between the spirals and
their parasites is acquired (see Figs. 9(a–c)).
The endosymbiosis over the communities is obtained by the horizontal
transfer of parasites. The retrovirus and transposon are examples of the movable genes
which are important in the history of evolution [23]. In this respect,
one can suspect that it is necessary for the
movable genes to show the
‘parasitic’ behavior. If we increase the catalytic support from species
7 to species 1 by $\kappa_{1,7}=500$, the species 7 fails to invade the spirals.
Therefore, the parasitic behavior is necessary for the movable genes to be in
symbiosis with spirals.
However, because of the parasitic behavior of species 7, the original species of spirals
are having difficulties such
that the populaton becomes depressed (see Fig. 9(d)). Nevertheless, there is a
benefit given to the spirals in the symbiosis with species 7. That is, let
us consider another species 8 which has the same properties as the species 7
except that it does not catalyze any species
in the spirals. Invasion of species 8 would be very
toxic to the spirals without species 7, but with the help of the symbiosis with species 7
the spirals can drive away
species 8 (see Figs. 10(a) and (b)), i.e. species 7 acts like ‘vaccine’ against
the invasion of species 8.
This effect originates from the competition
between species 7 and species 8.
We suppose that this vaccine effect might be responsible for the
prebiotic immunology that should be much simpler than immunology of the present day
[24].
6 Conclusion
We have studied spatio-temporal dynamics on the prebiotic evolution of genetic
information. In this paper we have investigated emergence, maintenance,
and innovation of the information communities. We recognize
that symbiosis is important on selection process in prebiotic evolution;
the short cycle which consists of only few symbionts fails to organize resistant structure
against parasites invasion. However, this helps the emergence of hypercycle hybrid providing
the selective advantage for a long hypercycle.
Endosymbiosis between
a hypercycle and its parasite shows interesting vaccine effect.
It is found that this symbiotic union is robust against the invasion of pure parasite.
Self-structuring of species by the complex interaction network leads to the
separation of cyclic sub-structures.
To select a particular sub-structure we only need to control local properties of the network
rather than overall properties of the entire specific sub-structure.
The integrity of each sub-structure reflects the emergent property
from the collective autocatalytic individuals.
The essential feature of autocatalysis is independent of its precise biochemical definition.
Therefore, study on autocatalysis would also be applicable to several area including
ecosystems, immune systems,
and social networks. We also want to emphasize that the role of self-structuring is not
restricted to the specific field –
prebiotic evolution. Self-structuring shows various phenomena
unexpected by our intuition based on simple ordinary differential equations.
In fact, rich nontrivial
results are reported
[4, 5, 6, 7, 25, 26, 27] in theoretical ecology.
Therefore we believe that our work can be of interest in many fields as well.
Acknowledgements
We thank Tae-Wook Ko for useful comments on the manuscript.
The research was supported by the Ministry of Science and Technology through Korean Systems Biology Research Grant (M10309020000-03B5002-00000), and by KOSEF through Grant No. R08-2003-000-10285-0.
References
[1]
G. Riddihough, The other RNA world, Science 296 (2002) 1259.
[2]
M. Eigen, W. Gardiner, P. Schuster, R. Winkler-Oswatitsch, The origin of genetic information, Sci. Am. 244 (1981) 78–94.
[3]
M.C. Boerlijst, P. Hogeweg, Spiral wave structure in pre-biotic evolution: hypercycles stable against parasites, Physica D 48 (1991) 17–28.
[4]
M.C. Boerlijst, M.E. Lamers, P. Hogeweg, Evolutionary consequences of spiral waves in a host parasitoid system, Proc. R. Soc. London B Biol. Sci. 253 (1993) 15–18.
[5]
N.J. Savill, P. Rohani, P. Hogeweg, Self-reinforcing spatial patterns enslave evolution in a host-parasitoid system, J. Theor. Biol. 188 (1997) 11–20.
[6]
E.M. Rauch, H. Sayama, Y. Bar-Yam, Relationship between measures of fitness and time scale in evolution, Phys. Rev. Lett. 88 (2002) 228101.
[7]
C.R. Johnson, M.C. Boeriijst, Selection at the level of the community: the importance of spatial structure, Trends. Ecol. Evol. 17 (2002) 83–90.
[8]
M.C. Boerlijst, P. Hogeweg, Self-structuring and selection, in: C.G. Langton, C. Talyor, J.D. Farmer, S. Rasmussen (Eds.), Artificial Life 2, Addison-Wesléy, Redwood City, 1991, pp. 255–276.
[9]
R. Albert, A.-L. Barabási, Statistical mechanics of complex networks, Rev. Mod. Phys. 74 (2002) 47–97.
[10]
S.C. Manrubia, J.F. Poyatos, Motif selection in a model of evolving replicators: the role of surfaces and limited transport in network topology, Europhys. Lett. 64 (2003) 557–563.
[11]
S. Jain, S. Krishna, Autocatalytic sets and the growth of complexity in an evolutionary model, Phys. Rev. Lett. 81 (1998) 5684–5687.
[12]
S. Jain, S. Krishna, Large extinctions in an evolutionary model: the role of innovation and keystone species, P. Natl. Acad. Sci. USA 99 (2002) 2055–2060.
[13]
M.C. Boerlijst, P. Hogeweg, Spatial gradients enhance persistence of hypercycles, Physica D 88 (1995) 29–39.
[14]
The specific parameters were selected because
they belong to the parameter region in which hypercycles can
be properly developed and
outcompete their initially embedded parasites. See Section 2 for more detail.
[15]
Instead of replacing the existing species by the invading species,
we also tried the randomly distributed invading species at that place to imitate more realistic
situation. The identical results are reproduced, probably by the partial differential equation
nature which promotes the discretized concentration to be flattened.
Somewhat different results may be possible in the cellular automata.
[16]
The increased diffusion coefficient of species 5 promotes the transmission and
catalytic coupling of species 5 to other species in the early stage,
therefore bringing the selective advantage to the spiral development with species 5.
[17]
W.L. Ditto, M.L. Spano, J.F. Lindner, Techniques for the control of chaos, Physica D 86 (1995) 198–211.
[18]
A. Babloyantz, C. Lourenco, J.A. Sepulchre, Control of chaos in delay-differential equations, in a network of oscillators and in model cortex, Physica D 86 (1995) 274–283.
[19]
G. Hu, J.H. Xiao, L.O. Chua, L. Pivka, Controlling spiral waves in a model of two-dimensional arrays of Chua’s circuits, Phys. Rev. Lett. 80 (1998) 1884–1887.
[20]
S. Sinha, A. Pande, R. Pandit, Defibrillation via the elimination of spiral turbulence in a model for ventricular fibrillation, Phys. Rev. Lett. 86 (2001) 3678–3681.
[21]
M.B. Cronhjort, C. Blomberg, Cluster compartmentalization may provide resistance to parasites for catalytic networks, Physica D 101 (1997) 289–298.
[22]
S. Goerner, Chaos, evolution, and deep ecology, in: R. Robertson, A. Combs (Eds.), Chaos Theory in Psychology and Life Sciences, Lawrence Erlbaum Publishers, Mahwah, 1995, pp. 17–38.
[23]
L. Stryer, Biochemistry, W.H. Freeman and Company, New York, 1995.
[24]
A.S. Perelson, G. Weisbuch, Immunology for physicists, Rev. Mod. Phys. 69 (1997) 1219–1268.
[25]
M.P. Hassell, H.N. Comins, R.M. May, Species coexistence and self-organizing spatial dynamics, Nature 370 (1994) 290–292.
[26]
P. Rohani, T.J. Lewis, D. Grünbaum, G.D. Ruxton, Spatial self-organization in ecology: pretty patterns or robust reality?, Trends. Ecol. Evol. 12 (1997) 70–74.
[27]
B. Kerr, M.A. Riley, M.W. Feldman, B.J.M. Bohannan, Local dispersal promotes biodiversity in a real-life game of rock-paper-scissors, Nature 418 (2002) 171–174. |
Abstract
Consider an SDE on a foliated manifold whose trajectories lay on
compact leaves. We investigate the effective behaviour of a small
transversal perturbation of order $\epsilon$. An average principle is shown to
hold such that the component transversal to the leaves converges to
the solution of a deterministic ODE, according to the average of the
perturbing vector field with respect to invariant measures on the leaves, as
$\epsilon$ goes to zero. An estimate of the rate of convergence is given. These
results generalize the geometrical scope of previous approaches, including
completely integrable stochastic Hamiltonian system.
An averaging principle for diffusions in foliated spaces
[3mm]
Iván I. Gonzáles
Gargate111e-mail: ivan.gargate@unila.edu.br.
Paulo R.
Ruffino222Corresponding author, e-mail:
ruffino@ime.unicamp.br.
Departamento de Matemática, Universidade Estadual de Campinas,
13.083-859- Campinas - SP, Brazil.
Key words: Averaging principle, foliated diffusion, rescaled
stochastic systems, stochastic flows.
MSC2010 subject classification: 60H10, 58J65, 58J37.
1 Introduction and set up
Generally speaking, the original heuristic idea of an averaging principles
refers to an intertwining
of two dynamics where one
of them is, in some sense, much slower and is affected
somehow by the other faster dynamics. An
averaging principle in this case
refers to the possibility of approximate, in some topology, the slow dynamics
considering only the
average action or perturbation which the fast motion induces on it. These ideas
have appeared long ago and, as mentioned by V. Arnold [3, p.287],
they
were implicitly contained in the works of Laplace, Lagrange and Gauss on
celestial mechanics; literature on the matter can be found e.g. among many
others in [3], Sanders,
Verhulst and
Murdoch [13] and references therein. Presently, on what
regards
stochastic systems, averaging has been quite an active
research field on which there is also a vast literature on the topic.
Interesting quick historical overviews can
be found in X.-M. Li
[9, p.806], Kabanov and Pergamenshchikov
[7, Appendix], [13, Appendix A]. Among many
other works somehow related to the topic, we refer to
Khasminski and
Krylov [8], Sowers [14], Namachchvaya and Sowers
[10]
Borodin and Freidlin [4],
[7] and references therein.
The specific problem
that we
address in this article is a perturbation of a diffusion in a foliated
manifold $M$ such that the
unperturbed random trajectories lay on the leaves. The perturbation are
taken
transversal to the leaves of the foliation. Here, the slow system is the
transversal component and the fast system
is given
by the rescaled $y_{\frac{t}{\epsilon}}^{\epsilon}$, where
$y_{t}^{\epsilon}$
is the solution of the original SDE perturbed by a vector field $\epsilon K$.
Our results generalize the recent approach by X.M. Li
[9] on an
averaging principle for a completely
integrable stochastic Hamiltonian system. In that article, as in the classical
approach, see e.g. [3], Li has
explored the benefits of a well structured geometrical coordinates in the
state
space given by the coordinates of the Liouville torus; these benefits include
vanishing Itô-Stratonovich correction terms besides also vanishing covariant
derivative of Hamiltonian vector fields in tangent directions to the leaves.
We prove here that an
averaging principle
also holds in a generalized geometrical scope, so that this averaging phenomenon
occurs independently of symplectic
structures (but with possibly slower rates of
convergence).
Comparing to Li’s
previous result [9, Lemma
3.2], where
the estimates contain a term of order $1/\sqrt{t}$, our corresponding estimates
in
Lemma 3.1 are continuous at $\epsilon=t=0$.
Some of the rates of convergence of [9, Lemma 3.1] is
recovered as particular cases in Corollaries 2.2 and
2.3.
In the main result,
we show that in the average, the approximation goes to zero faster than
$|\ln\epsilon|^{-\frac{\beta}{p}}$, for $\beta\in(0,1/2)$.
The set up. Let $M$ be a smooth Riemannian manifold with an
$n$-dimensional
smooth foliation,
i.e. $M$ is endowed with an
integrable regular distribution of dimension $n$ (for definition
and further properties of foliated spaces see e.g. the initial chapters of
Tondeur [15], Walcak
[16] among others). We denote by $L_{x}$ the leaf
of the foliation passing through a point $x\in M$.
For simplicity, we shall assume that the leaves are compact
and that each leaf $L_{x}$ has a
tubular neighbourhood $U\subset M$ where $U$ is diffeomorphic to $L_{x}\times V$, where $V\subset\mathbf{R}^{d}$ is an open bounded neighbourhood of the origin and $d$
is the codimension of the foliation.
We shall assume an SDE in $M$ whose solution flow preserves the foliation, i.e.
we consider a Stratonovich equation
$$dx_{t}=X_{0}(x_{t})dt+\sum_{i=1}^{r}X_{i}(x_{t})\circ dB^{i}_{t}$$
(1)
where the smooth vector fields $X_{i}$ are foliated in the
sense that $X_{i}(x)\in T_{x}L_{x}$,
for $i=0,1,\ldots,r$. Here $B_{t}=(B^{1}_{t},\ldots,B^{r}_{t})$ is a standard
Brownian motion in $\mathbf{R}^{r}$
with respect to a filtered probability space $(\Omega,\mathcal{F}_{t},\mathcal{F},\mathbf{P})$. For an
initial condition $x_{0}$, the trajectories of the solution $x_{t}$ in this case
lay on the leaf $L_{x_{0}}$ a.s.. Moreover, there exists a (local) stochastic
flow of diffeomorphisms $F_{t}:M\rightarrow M$ which restricted
to the initial leaf is a flow in the compact $L_{x_{0}}$.
For a smooth vector field $K$ in $M$, we shall denote the perturbed system by
$y^{\varepsilon}_{t}$ which satisfies the SDE
$$dy^{\varepsilon}_{t}=X_{0}(y^{\varepsilon}_{t})dt+\sum_{i=1}^{r}X_{i}(y^{%
\varepsilon}_{t})\circ dB^{i}_{t}+\varepsilon K(y^{\varepsilon}_{t})\,dt,$$
(2)
with the same initial condition $y^{\varepsilon}_{0}=x_{0}$ of the unperturbed
system $x_{t}$.
Our main result, Theorem 4.1, says that
locally the transversal behaviour of
$y^{\varepsilon}_{\frac{t}{\epsilon}}$ can be approximated in the average by an
ordinary differential equation in the transversal space whose
coefficients are given by the average of the transversal component of the
perturbation $K$ with respect to the invariant measure on the leaves for the
original dynamics of Equation (1). The reader will notice by
the end of the proofs that compactness of
the leaves in fact can be substituted by some other boundedness
conditions, added also to some rather technical adjustments which we will not
address here.
In the Sections 2 and 3 we present the main lemmas. The main result
appears in
Section 4, where we also present a simple illustrative example. In particular,
under some symmetry hypothesis on a foliated system embedded in an Euclidean
space, we use the main theorem to conclude that Lyapunov exponents
in the transversal direction must tend to zero as $\epsilon$ goes to zero,
cf. Proposition 4.2.
2 Preliminaries results
Our coordinate system.
Given an initial condition $x_{0}\in M$, let $U\subset M$ be a
bounded neighboorhood of
$x_{0}$ which is diffeomorphic to $L_{x_{0}}\times V$ and whose closure $\bar{U}\subset M$. By compactness of $L_{x_{0}}$, there exists a finite number of local
foliated
coordinate systems $\varphi_{i}:U_{i}\rightarrow W_{i}\times V\subset\mathbf{R}^{n}\times\mathbf{R%
}^{d}$, where $W_{i}$ and $V$ are open sets, say with $1\leq i\leq k$ and $x_{0}\in U_{1}$ such that:
1)
$U=\cup_{i=1}^{k}U_{i}$;
2)
The leaf $L_{x_{0}}=\cup_{i=1}^{k}\varphi^{-1}(W_{i}\times\{0\})$, i.e. each $U_{i}$ is
diffeomorphic to the product of an open set (with the induced topology) in the
leaf $L_{x_{0}}$ and the vertical component $V$;
3)
If a pair of points
$p\in U_{i}$ and $q\in U_{j}$ in $U$ belong to the same leaf then their
transversal
coordinates in $V$ are the same; i.e. $\pi(\varphi_{i}(p))=\pi(\varphi_{j}(p))$
where $\pi$ is the projection on the
transversal space $V$;
4)
For $i=1,\ldots,k$, $\varphi_{i}$ has bounded derivatives (obtained
reducing open set $U$ if necessary).
Note that for a fixed $y\in V$, the finite
union $\cup^{k}\varphi_{i}^{-1}(W_{i},y)$ is the leaf
$L_{\varphi_{i}^{-1}(x,y)}$ for any $x\in W_{i}$. Natural examples
of this scheme of coordinates systems appear if we consider compact foliation
given by the inverse image of
submersions: values in the image space provide
local coordinates for the vertical space $V$.
Next lemma gives information on the order of which the perturbed
trajectories
$y^{\epsilon}_{t}$ approaches the unperturbed $x_{t}$ when one varies $\epsilon$
and $t$ in
equation (2); it will be used to prove that the
dynamics of the rescaled system $y^{\epsilon}_{\frac{t}{\epsilon}}$ is such
that its time average for any function $g$ in $M$ approximates the time average
of the spacial average of $g$ on the leaves, Lemma 3.1. An
exponential factor in the estimates
is expected, as trivial linear examples show.
We shall denote the coordinates of a point $p\in U_{1}$ by $\varphi_{1}(p)=(u,v)\in\mathbf{R}^{n}\times\mathbf{R}^{d}$.
Lemma 2.1
Let $\tau^{\epsilon}$ be the first time the
process $y^{\epsilon}_{t}$ exits the foliated coordinate neighbourhood $U_{1}$ as
above. For any locally Lipschitz continuous
function $f:M\rightarrow\mathbf{R}$ and $2\leq p<\infty$ we have
$$\left[\mathbb{E}\left(\sup_{s\leq t\wedge\tau^{\epsilon}}\left|f(y^{\epsilon}_%
{s})-f(x_{s})\right|^{p}\right)\right]^{\frac{1}{p}}\leq K_{1}\,\varepsilon\,t%
\,e^{K_{2}t^{p}}.$$
where $K_{1},K_{2}\geq 0$ are constants depending on upper bounds of the norms of
the perturbing vector field $K$, on the Lipschitz coefficients of $f$ and on
the derivatives of $X_{0},X_{1}\cdots,X_{r}$ with respect to the
coordinate system.
Proof: Initially write $x_{t}$ and
$y^{\epsilon}_{t}$, the solutions of Equations
(1) and (2) respectively, according to the
foliated coordinates $\varphi_{1}$. So we write
$(u_{t},v_{t}):=\varphi_{1}(x_{t})$ and
$(u^{\epsilon}_{t},v^{\epsilon}_{t}):=\varphi_{1}(y^{\epsilon}_{t})$. Then
$$\displaystyle\left|f(y^{\epsilon}_{t})-f(x_{t})\right|$$
$$\displaystyle=$$
$$\displaystyle\left|f\circ\varphi^{-1}_{1}(u^{\epsilon}_{t},v^{\epsilon}_{t})-f%
\circ\varphi^{-1}_{1}(u_{t},v_{t})\right|$$
$$\displaystyle\leq$$
$$\displaystyle C\left|u^{\epsilon}_{t}-u_{t}\right|+C\left|v^{\epsilon}_{t}-v_{%
t}\right|,$$
for some constant $C\geq 0$, using the fact that
$U_{1}$ is relatively compact.
We shall denote $u_{t}=(u^{1}_{t},\cdots,u^{n}_{t})$, $v_{t}=(v^{1}_{t},\cdots,v^{d}_{t})$,
$u^{\epsilon}_{t}=(u^{\epsilon,1}_{t},\cdots,u^{\epsilon,n}_{t})$,
$v^{\epsilon}_{t}=(v^{\epsilon,1}_{t},\cdots,v^{\epsilon,d}_{t})$. We also split the
horizontal and vertical component of the perturbing vector field
$\tilde{K}=(K_{u},K_{v})$, into coordinates: $K_{u}=(K_{u}^{1},\ldots,K_{u}^{n},)$ and $K_{v}=(K_{v}^{1},\ldots,K_{v}^{d},)$.
In our coordinate system, the equations of the horizontal
and
vertical
components of the perturbed system $u^{\epsilon}_{t}$ and $v^{\epsilon}_{t}$ are
given by
$$\displaystyle du^{\epsilon,i}_{t}$$
$$\displaystyle=$$
$$\displaystyle\sum^{r}_{k=1}b^{i}_{k}(u^{\epsilon}_{t},v^{\epsilon}_{t})\circ dB%
^{k}_{t}+b^{i}_{0}(u^{\epsilon}_{t},v^{\epsilon}_{t})dt+\epsilon K^{i}_{u}(u^{%
\epsilon}_{t},v^{\epsilon}_{t})dt,$$
(4)
$$\displaystyle dv^{\epsilon,j}_{t}$$
$$\displaystyle=$$
$$\displaystyle\epsilon K^{j}_{v}(u^{\epsilon}_{t},v^{\epsilon}_{t})dt,$$
(5)
with $i=1,2,\cdots,n$ and $j=1,2,\cdots,d$, for the induced vector
fields $b_{0},b_{1},\ldots b_{r}$ which, together with
their derivatives, are also bounded.
From equation $(\ref{eqcoord2})$ we have
$$\displaystyle\sup_{s\leq t\wedge\tau^{\epsilon}}\left|v^{\epsilon}_{s}-v_{s}\right|$$
$$\displaystyle\leq$$
$$\displaystyle\epsilon\sup_{s\leq t\wedge\tau^{\epsilon}}\int^{s}_{0}\left|K_{v%
}(u^{\epsilon}_{s},v^{\epsilon}_{s})\right|ds$$
(6)
$$\displaystyle\leq$$
$$\displaystyle\epsilon~{}t~{}\sup_{x\in U}\left|K_{v}(x)\right|=C_{1}\epsilon\ t,$$
where $C_{1}=\sup_{x\in U}\left|K(x)\right|$.
From equation $(\ref{eqcoord1})$ we have in each $i$-th component, for
$s<\tau^{\epsilon}$:
$$\displaystyle u^{\epsilon,i}_{s}-u^{i}_{s}=\displaystyle\sum^{r}_{k=1}%
\displaystyle\int^{s}_{0}{(b^{i}_{k}(u^{\epsilon}_{r},v^{\epsilon}_{r})-b^{i}_%
{k}(u_{r},v_{r}))\circ dB^{k}_{r}}+$$
(8)
$$\displaystyle\ \ \ \ \ \ \ \displaystyle\int^{s}_{0}{(b^{i}_{0}(u^{\epsilon}_{%
r},v^{\epsilon}_{r})-b^{i}_{0}(u_{r},v_{r}))dr}+\epsilon\displaystyle\int^{s}_%
{0}{K^{i}_{u}(u^{\epsilon}_{r},v^{\epsilon}_{r})dr}.$$
In terms of Itô integral,
$$\displaystyle\int^{s}_{0}(b^{i}_{k}(u^{\epsilon}_{r},v^{\epsilon}_{r})-b^{i}_{%
k}(u_{r},v_{r}))\circ dB^{k}_{r}$$
$$\displaystyle=$$
$$\displaystyle\int^{s}_{0}(b^{i}_{k}(u^{\epsilon}_{r},v^{\epsilon}_{r})-b^{i}_{%
k}(u_{r},v_{r}))dB^{k}_{r}$$
$$\displaystyle+\frac{1}{2}\int^{s}_{0}[\nabla b^{i}_{k}\cdot b_{k}(u^{\epsilon}%
_{r},v^{\epsilon}_{r})-\nabla b^{i}_{k}\cdot b_{k}(u_{r},v_{r})]dr.$$
Hence, taking the absolute values in both sides of Equation
(8) we get, for each
$i$:
$$\displaystyle\left|u^{\epsilon,v}_{s}-u^{i}_{s}\right|$$
$$\displaystyle\leq$$
$$\displaystyle\sum^{r}_{k=1}\left|\int^{s}_{0}{(b^{i}_{k}(u^{\epsilon}_{r},v^{%
\epsilon}_{r})-b^{i}_{k}(u_{r},v_{r}))dB^{k}_{r}}\right|+$$
(9)
$$\displaystyle\frac{1}{2}\displaystyle\sum_{k=1}^{r}\int^{s}_{0}\left|\nabla b^%
{i}_{k}\cdot b_{k}(u^{\epsilon}_{r},v^{\epsilon}_{r})-\nabla b^{i}_{k}\cdot b_%
{k}(u_{r},v_{r})\right|dr$$
$$\displaystyle+\int^{s}_{0}\left|b^{i}_{0}(u^{\epsilon}_{r},v^{\epsilon}_{r})-b%
^{i}_{0}(u_{r},v_{r})\right|dr+\epsilon\int^{s}_{0}\left|K^{i}_{u}(u^{\epsilon%
}_{r},v^{\epsilon}_{r})\right|dr.$$
Functions $b^{i}_{0}$ and $(\nabla b^{i}_{k}\cdot b_{k})$ are Lipschitz, hence for a
common constant $C_{2}$,
$$\displaystyle\left|u^{\epsilon,i}_{s}-u^{i}_{s}\right|\leq\left|\sum^{r}_{k=1}%
\int^{s}_{0}{(b^{i}_{k}(u^{\epsilon}_{r},v^{\epsilon}_{r})-b^{i}_{k}(u_{r},v_{%
r}))dB^{k}_{r}}\right|+$$
(10)
$$\displaystyle\ \ \ \ \ \ C_{2}\int^{s}_{0}\left|v^{\epsilon}_{r}-v_{r}\right|%
dr+C_{2}\int^{s}_{0}\left|u^{\epsilon}_{r}-u_{r}\right|dr+\epsilon s\ \sup_{U}%
\left|K_{u}\right|.$$
The first deterministic integral, together with inequality
$(\ref{estimativa1})$ yields:
$$\displaystyle\left|u^{\epsilon,i}_{s}-u^{i}_{s}\right|$$
$$\displaystyle\leq$$
$$\displaystyle\sum^{r}_{k=1}\left|\int^{s}_{0}{(b^{i}_{k}(u^{\epsilon}_{r},v^{%
\epsilon}_{r})-b^{i}_{k}(u_{r},v_{r}))dB^{k}_{r}}\right|+C_{1}C_{2}\epsilon s^%
{2}$$
$$\displaystyle+C_{2}\int^{s}_{0}\left|u^{\epsilon}_{r}-u_{r}\right|dr+C_{1}%
\epsilon s.$$
Now, for $p\geq 1$, there exists a constant $C_{3}$ such that
$$\displaystyle\left|u^{\epsilon,i}_{s}-u^{i}_{s}\right|^{p}$$
$$\displaystyle\leq$$
$$\displaystyle C_{3}\sum^{r}_{k=1}\left|\int^{s}_{0}{(b^{i}_{k}(u^{\epsilon}_{r%
},v^{\epsilon}_{r})-b^{i}_{k}(u_{r},v_{r}))dB^{k}_{r}}\right|^{p}+C_{3}\left(C%
_{1}C_{2}\epsilon s^{2}\right)^{p}$$
$$\displaystyle+C_{3}C^{p}_{2}\left(\int^{s}_{0}\left|u^{\epsilon}_{r}-u_{r}%
\right|dr\right)^{p}+C_{3}\left(C_{1}\epsilon s\right)^{p}.$$
Cauchy-Schwartz inequality yields:
$$\displaystyle\left|u^{\epsilon,i}_{s}-u^{i}_{s}\right|^{p}$$
$$\displaystyle\leq$$
$$\displaystyle C_{3}\sum^{r}_{k=1}\left|\int^{s}_{0}{(b^{i}_{k}(u^{\epsilon}_{r%
},v^{\epsilon}_{r})-b^{i}_{k}(u_{r},v_{r}))dB^{k}_{r}}\right|^{p}+C_{3}\left(C%
_{1}C_{2}\epsilon s^{2}\right)^{p}$$
$$\displaystyle+C_{3}C^{p}_{2}~{}s^{p-1}\int^{s}_{0}\left|u^{\epsilon}_{r}-u_{r}%
\right|^{p}dr+C_{3}\left(C_{1}\epsilon s\right)^{p}.$$
Hence,
$$\displaystyle\mathbb{E}\displaystyle\sup_{s\leq t\wedge\tau^{\epsilon}}\left|u%
^{\epsilon,i}_{s}-u^{i}_{s}\right|^{p}$$
$$\displaystyle\leq$$
$$\displaystyle C_{3}~{}\mathbb{E}\displaystyle\sup_{s\leq t\wedge\tau^{\epsilon%
}}\displaystyle\sum^{r}_{k=1}\left|\int^{s}_{0}{(b^{i}_{k}(u^{\epsilon}_{r},v^%
{\epsilon}_{r})-b^{i}_{k}(u_{r},v_{r}))dB^{k}_{r}}\right|^{p}+C_{3}\left(C_{1}%
C_{2}\epsilon t^{2}\right)^{p}$$
$$\displaystyle+C_{3}C_{2}^{p}~{}t^{p-1}~{}\mathbb{E}\displaystyle\sup_{s\leq t%
\wedge\tau^{\epsilon}}\int^{s}_{0}\left|u^{\epsilon}_{r}-u_{r}\right|^{p}dr+C_%
{3}\left(C_{1}\epsilon t\right)^{p}$$
$$\displaystyle\leq$$
$$\displaystyle C_{4}\displaystyle\sum^{r}_{k=1}\mathbb{E}\displaystyle\left[%
\int^{t\wedge\tau^{\epsilon}}_{0}{(b^{i}_{k}(u^{\epsilon}_{r},v^{\epsilon}_{r}%
)-b^{i}_{k}(u_{r},v_{r}))^{2}dr}\right]^{p/2}+C_{3}\left(C_{1}C_{2}\epsilon t^%
{2}\right)^{p}$$
$$\displaystyle+C_{3}C_{2}^{p}~{}t^{p-1}\displaystyle\int^{t}_{0}\mathbb{E}\left%
(\displaystyle\sup_{s\leq r\wedge\tau^{\epsilon}}\left|u^{\epsilon}_{r}-u_{r}%
\right|^{p}\right)dr+C_{3}\left(C_{1}\epsilon t\right)^{p}$$
where we have used classical $L^{p}$-inequality for martingales (e.g. Revuz and
Yor [11]).
Using again the Lipchitz property of each
$b_{k}$ for the terms in the brackets above:
$$\displaystyle\sum^{r}_{k=1}\int^{t\wedge\tau^{\epsilon}}_{0}{(b^{i}_{k}(u^{%
\epsilon}_{r},v^{\epsilon}_{r})-b^{i}_{k}(u_{r},v_{r}))^{2}dr}$$
$$\displaystyle\leq\ 2C^{2}_{2}\left(\displaystyle\int^{t\wedge\tau^{\epsilon}}_%
{0}\left|v^{\epsilon}_{r}-v_{r}\right|^{2}dr+\displaystyle\int^{t\wedge\tau^{%
\epsilon}}_{0}\left|u^{\epsilon}_{r}-u_{r}\right|^{2}dr\right)$$
$$\displaystyle\leq\ 2C^{2}_{2}\left(\displaystyle\int^{t}_{0}C_{1}^{2}\epsilon^%
{2}r^{2}~{}dr+\displaystyle\int^{t\wedge\tau^{\epsilon}}_{0}\displaystyle\sup_%
{s\leq r\wedge\tau^{\epsilon}}\left|u^{\epsilon}_{r}-u_{r}\right|^{2}dr\right)$$
$$\displaystyle\leq\ C^{2}_{2}C_{1}^{2}\epsilon^{2}t^{3}+2C^{2}_{2}\displaystyle%
\int^{t\wedge\tau^{\epsilon}}_{0}\displaystyle\sup_{s\leq r\wedge\tau^{%
\epsilon}}\left|u^{\epsilon}_{r}-u_{r}\right|^{2}dr.$$
(11)
We end up with:
$$\displaystyle\mathbb{E}\displaystyle\sup_{s\leq t\wedge\tau^{\epsilon}}\left|u%
^{\epsilon,i}_{s}-u^{i}_{s}\right|^{p}$$
$$\displaystyle\leq$$
$$\displaystyle C_{4}\displaystyle\sum^{r}_{k=1}\mathbb{E}\displaystyle\left[C^{%
2}_{2}C_{1}^{2}\epsilon^{2}t^{3}+2C^{2}_{2}\displaystyle\int^{t\wedge\tau^{%
\epsilon}}_{0}\displaystyle\sup_{s\leq r\wedge\tau^{\epsilon}}\left|u^{%
\epsilon}_{r}-u_{r}\right|^{2}dr\right]^{p/2}$$
(12)
$$\displaystyle+C_{3}C_{2}^{p}\,t^{p-1}~{}\displaystyle\int^{t}_{0}\mathbb{E}%
\left(\displaystyle\sup_{s\leq r\wedge\tau^{\epsilon}}\left|u^{\epsilon}_{r}-u%
_{r}\right|^{p}\right)dr$$
$$\displaystyle+C_{3}\left(C_{1}C_{2}\epsilon t^{2}\right)^{p}+C_{3}\left(C_{1}%
\epsilon t\right)^{p}.$$
For $p\geq 2$ one can use Cauchy-Schwartz again to conclude that there exists a
constant $C_{5}$ such that the last expression is less than or equal
$$\displaystyle C_{5}\left(C_{2}C_{1}\epsilon t^{3/2}\right)^{p}+C_{5}C^{p}_{2}t%
^{\frac{p-2}{2}}\displaystyle\int^{t\wedge\tau^{\epsilon}}_{0}\mathbb{E}%
\displaystyle\sup_{s\leq r\wedge\tau^{\epsilon}}\left|u^{\epsilon}_{r}-u_{r}%
\right|^{p}dr+C_{3}\left(C_{1}C_{2}\epsilon t^{2}\right)^{p}$$
$$\displaystyle+C_{3}C_{2}^{p}\,t^{p-1}~{}\displaystyle\int^{t}_{0}\mathbb{E}%
\left(\displaystyle\sup_{s\leq r\wedge\tau^{\epsilon}}\left|u^{\epsilon}_{r}-u%
_{r}\right|^{p}\right)dr+C_{3}\left(C_{1}\epsilon t\right)^{p}$$
$$\displaystyle=$$
$$\displaystyle C_{5}\left(C_{2}C_{1}\epsilon t^{3/2}\right)^{p}+C_{3}\left(C_{1%
}C_{2}\epsilon t^{2}\right)^{p}+C_{3}\left(C_{1}\epsilon t\right)^{p}$$
$$\displaystyle+\left(C_{5}C^{p}_{2}t^{\frac{p-2}{2}}+C_{3}C_{2}^{p}\,t^{p-1}%
\right)\displaystyle\int^{t}_{0}\mathbb{E}\left(\displaystyle\sup_{s\leq r%
\wedge\tau^{\epsilon}}\left|u^{\epsilon}_{r}-u_{r}\right|^{p}\right)dr.$$
Now, summing up over $i$ in the inequalities above leads to
$$\displaystyle\mathbb{E}\sup_{s\leq t\wedge\tau^{\epsilon}}\left|u^{\epsilon}_{%
r}-u_{r}\right|^{p}$$
$$\displaystyle\leq$$
$$\displaystyle C_{5}\left(C_{2}C_{1}\epsilon t^{3/2}\right)^{p}+C_{3}\left(C_{1%
}C_{2}\epsilon t^{2}\right)^{p}+C_{3}\left(C_{1}\epsilon t\right)^{p}$$
$$\displaystyle+\left(C_{5}C^{p}_{2}t^{\frac{p-2}{2}}+C_{3}C_{2}^{p}\,t^{p-1}%
\right)\displaystyle\int^{t}_{0}\mathbb{E}\left(\displaystyle\sup_{s\leq r%
\wedge\tau^{\epsilon}}\left|u^{\epsilon}_{r}-u_{r}\right|^{p}\right)dr.$$
We use now the integral form of Gronwall’s inequality to find that:
$$\displaystyle\mathbb{E}\left(\sup_{s\leq t\wedge\tau^{\epsilon}}\left|u^{%
\epsilon}_{r}-u_{r}\right|^{p}\right)$$
$$\displaystyle\leq$$
$$\displaystyle C_{6}\epsilon^{p}t^{p}\left(1+t^{p}\right)\exp\{C_{7}(t^{p/2}+t^%
{p})\}$$
$$\displaystyle\leq$$
$$\displaystyle C_{8}\epsilon^{p}t^{p}\left(1+t^{p}\right)\exp\{C_{9}t^{p}\}.$$
Going back to the inequality 2, now we have
$$\left|f(y^{\epsilon}_{t})-f(x_{t})\right|^{p}\leq C_{10}\left|v^{\epsilon}_{t}%
-v_{t}\right|^{p}+C_{10}\left|u^{\epsilon}_{t}-u_{t}\right|^{p}$$
hence:
$$\displaystyle\mathbb{E}\left(\displaystyle\sup_{s\leq t\wedge\tau^{\epsilon}}%
\left|f(y^{\epsilon}_{s})-f(x_{s})\right|^{p}\right)$$
$$\displaystyle\leq$$
$$\displaystyle C_{10}\mathbb{E}\displaystyle\sup_{s\leq t\wedge\tau^{\epsilon}}%
\left|v^{\epsilon}_{s}-v_{s}\right|^{p}+C_{10}\mathbb{E}\displaystyle\sup_{s%
\leq t\wedge\tau^{\epsilon}}\left|u^{\epsilon}_{s}-u_{s}\right|^{p}$$
$$\displaystyle\leq$$
$$\displaystyle C_{11}\epsilon^{p}t^{p}+C_{12}\epsilon^{p}t^{p}(1+t^{p})\exp(C_{%
9}t^{p})$$
$$\displaystyle\leq$$
$$\displaystyle C_{13}\epsilon^{p}t^{p}(1+t^{p})\exp(C_{9}~{}t^{p}).$$
From here, finally, one concludes that there exist constants $K_{1}$ and $K_{2}$
such that
$$\mathbb{E}\left(\displaystyle\sup_{s\leq t\wedge\tau^{\epsilon}}\left|f(y^{%
\epsilon}_{s})-f(x_{s})\right|^{p}\right)^{\frac{1}{p}}\leq K_{1}\epsilon t%
\exp(K_{2}~{}t^{p}).$$
$\Box$
Next corollary includes the case of a completely integrable
stochastic Hamiltonian system when one uses the action-angle coordinates, cf.
X.-M. Li
[9, Lemma 3.1]).
Corollary 2.2
If the vector fields $X_{0},\cdots,X_{r}$
depend only on the
vertical coordinate (null derivative in the directions of the leaves,
as in the Hamiltonian case
[9]) then the estimates above
can be improved, and for $p\geq 1$ there exists a constant $K_{1}$ such that
$$\left[\mathbb{E}\left(\sup_{s\leq t\wedge\tau^{\epsilon}}\left|f(y^{\epsilon}_%
{s})-f(x_{s})\right|^{p}\right)\right]^{\frac{1}{p}}\leq K_{1}\varepsilon(t+t^%
{2}).$$
Proof: In this case the correction term of the Stratonovich stochastic integral
in terms of Itô integral in inequality (9)
vanishes, and also so does the determinist integration of $|u^{\epsilon}_{r}-u_{r}|$ in inequalities (10) and (11). Hence
inequality (12) improves to
$$\displaystyle\mathbb{E}\displaystyle\sup_{s\leq t\wedge\tau^{\epsilon}}\left|u%
^{\epsilon,i}_{s}-u^{i}_{s}\right|^{p}$$
$$\displaystyle\leq$$
$$\displaystyle C_{5}\left(C_{2}C_{1}\epsilon t^{3/2}\right)^{p}+C_{3}\left(C_{1%
}C_{2}\epsilon t^{2}\right)^{p}+C_{3}\left(C_{1}\epsilon t\right)^{p}.$$
(13)
The argument in the rest of the proof follows straightforward for $p\geq 1$
skipping Gronwall inequality.
$\Box$
Next corollary includes the case $X_{0}\equiv 0$, cf. [9, Lemma 3.1(2)]
for stochastic Hamiltonian systems with action-angle
coordinate system.
Corollary 2.3
If in addition to conditions of Corollary
2.2 above, we have that the deterministic
vector field $X_{0}$ is constant when represented with respect to a certain local
coordinate system in $U_{1}$ (i.e. $b_{0}$ has null derivative w.r.t $u$ and $v$)
then, for $p\geq 1$ the
estimates can be improved further to $K_{1}\varepsilon(t+t^{\frac{3}{2}})$.
Proof: Besides the vanishing terms already mentioned above, the
second
deterministic integral on the right hand side of inequality
(9) also vanishes. Hence inequality
(12) simplifies further to
$$\displaystyle\mathbb{E}\displaystyle\sup_{s\leq t\wedge\tau^{\epsilon}}\left|u%
^{\epsilon,i}_{s}-u^{i}_{s}\right|^{p}$$
$$\displaystyle\leq$$
$$\displaystyle C_{5}\left(C_{2}C_{1}\epsilon t^{3/2}\right)^{p}+C_{3}\left(C_{1%
}\epsilon t\right)^{p}.$$
$\Box$
Yet, from the proof of the Lemma 2.1 we have the following
Remark 2.4
For $1\leq p<2$ and $t$ sufficiently
small, there exist constants $K_{1}$ and $K_{2}$ such that
$$\left[\mathbb{E}\left(\sup_{s\leq t\wedge\tau^{\epsilon}}\left|f(y^{\epsilon}_%
{s})-f(x_{s})\right|^{p}\right)\right]^{\frac{1}{p}}\leq K_{1}\varepsilon t\,%
\exp{(K_{2}\,t^{p})}.$$
(14)
Proof: One can no longer use Cauchy-Schwartz after inequality
(12). Alternatively, from (12), use that
$$\displaystyle\mathbb{E}\displaystyle\sup_{s\leq t\wedge\tau^{\epsilon}}\left|u%
^{\epsilon}_{s}-u_{s}\right|^{p}$$
$$\displaystyle\leq$$
$$\displaystyle C_{5}\left(C_{2}C_{1}\epsilon t^{3/2}\right)^{p}+C_{5}C^{p}_{2}t%
^{p/2}\mathbb{E}\displaystyle\sup_{s\leq t\wedge\tau^{\epsilon}}\left|u^{%
\epsilon,i}_{s}-u^{i}_{s}\right|^{p}+C_{3}\left(C_{1}C_{2}\epsilon t^{2}\right%
)^{p}$$
$$\displaystyle+C_{3}C_{2}^{p}~{}\displaystyle\int^{t}_{0}\mathbb{E}\left(%
\displaystyle\sup_{s\leq r\wedge\tau^{\epsilon}}\left|u^{\epsilon}_{r}-u_{r}%
\right|^{p}\right)dr+C_{3}C_{1}^{p}\epsilon^{p}t^{p}.$$
If we fix an $0<\delta<1$ and take $t$ sufficiently small such that
$1-C_{5}C^{p}_{2}t^{p/2}>\delta$ then
$$\displaystyle\mathbb{E}\displaystyle\sup_{s\leq t\wedge\tau^{\epsilon}}\left|u%
^{\epsilon}_{s}-u_{s}\right|^{p}$$
$$\displaystyle\leq$$
$$\displaystyle\delta^{-1}C_{5}\left(C_{2}C_{1}\epsilon t^{3/2}\right)^{p}+%
\delta^{-1}C_{3}\left(C_{1}C_{2}\epsilon t^{2}\right)^{p}$$
$$\displaystyle+\delta^{-1}C_{3}C_{2}^{p}\,t^{p-1}~{}\displaystyle\int^{t}_{0}%
\mathbb{E}\left(\displaystyle\sup_{s\leq r\wedge\tau^{\epsilon}}\left|u^{%
\epsilon}_{r}-u_{r}\right|^{p}\right)dr+\delta^{-1}C_{3}C_{1}^{p}\epsilon^{p}t%
^{p}.$$
And one completes the calculation as before using the integral version of
Gronwall inequality.
$\Box$
3 Averaging functions on the leaves
Consider a differentiable function $g:M\rightarrow\mathbb{R}$.
The leaf $L_{p}$ passing through a point $p\in M$ contains the support of an
invariant measure $\mu_{p}$ for
the unperturbed system (1);
we shall assume that $\mu_{p}$ is
ergodic. We shall work
with the following
function defined for each leaf $Q^{g}:V\subset\mathbf{R}^{d}\rightarrow\mathbf{R}$
given by the average of $g$ with respect to these measures on the
leaves. Namely, if $v$ is the vertical coordinate of $p$, i.e.
$p=\varphi(u,v)$, then:
$$Q^{g}(v)=\int_{L_{p}}g(x)\,d\mu_{p}(x).$$
We assume that $Q^{g}$ has some degree of continuity with respect to
$v$. This assumption appears in two levels in Lemma 3.4: 1) Riemann
integrability of $Q^{g}(\pi(y^{\epsilon}_{r}))$ with respect to $r$ guarantees the convergence to zero; 2)
$\alpha$-Hölder continuity guarantees the rate of convergence.
Our next step is to estimate the time average of $g$ and $Q^{g}$ along the
perturbed system $y_{t}^{\epsilon}$. To use the notations and results
we have introduced before in local coordinates, we shall write simply $\pi(p)=v$ for the composition of the projection on the second coordinates with the
local chart $\varphi(p)=(u,v)$.
Here the stopping time $\tau^{\epsilon}$ denotes the first exit time of the
open neighbourhood $U\subset M$ which is diffeomorphic to $L_{x_{0}}\times V$. We
have the following estimates for the difference of the averages of
functions $g$ and $Q^{g}$.
Lemma 3.1
Given a function $g:M\rightarrow\mathbf{R}$ let $Q^{g}:V\rightarrow\mathbf{R}$ be its average on the leaves. For $s,t\geq 0$ write
$$\delta(\epsilon,t)=\int^{(s+t)\wedge\epsilon\tau^{\epsilon}}_{s\wedge\epsilon%
\tau^{\epsilon}}g(y^{\epsilon}_{\frac{r}{\epsilon}})-Q^{g}(\pi(y^{\epsilon}_{%
\frac{r}{\epsilon}}))\ dr.$$
Then $\delta(\epsilon,t)$ goes to zero when $t$ or $\epsilon$ tend
to zero.
Moreover, if $Q^{g}$ is $\alpha$-Hölder continuous with $\alpha>0$ then for
$p\geq 1$ and any $\beta\in(0,1/2)$ we have the following estimates:
$$\left(\mathbb{E}\sup_{s\leq t}\left|\delta(\epsilon,s)\right|^{p}\right)^{%
\frac{1}{p}}\leq\sqrt{t}|\ln\epsilon|^{-\frac{\beta}{p}}h(t,\epsilon),$$
where $h(t,\epsilon)$ is continuous for $t,\epsilon>0$ and converges to
zero when $(t,\epsilon)\to 0$.
Proof:
The proof consists of considering a convenient
partition of the interval $(s/\epsilon\wedge\tau^{\epsilon},(s+t)/\epsilon\wedge\tau^{\epsilon})$ where we
can get the estimates by comparing in each subinterval the average
of the flow of the original system (on the corresponding leaf) with the
average
of the perturbed flow (possibly transversal to the leaves). These estimates in
each subinterval are obtained using Lemma 2.1. So,
a key point in the proof is a careful choice of the increments of such a
convenient partition.
For sufficiently small $\epsilon$,
we take the following assignment of
increments:
$$\Delta t=\frac{(s+t)\wedge\tau^{\epsilon}-s\wedge\tau^{\epsilon}}{|\ln\epsilon%
|^{-\frac{2\beta}{p}}}.$$
Hence, the partition $t_{n}=\frac{s}{\epsilon}\wedge\tau^{\epsilon}+n\Delta t$, for $1\leq n\leq N-1,$ is such that
$$\frac{s}{\epsilon}\wedge\tau^{\epsilon}=t_{0}<t_{1}<\cdots<t_{N-1}<\frac{s+t}{%
\epsilon}\wedge\tau^{\epsilon}.$$
with $N=N(\epsilon)=[\epsilon^{-1}|\ln\epsilon|^{-\frac{2\beta}{p}}]$
where here $[x]$ denotes the integer part of $x$.
Initially we represent the left hand side as the sum:
$$\epsilon\int^{\frac{s+t}{\epsilon}\wedge\tau^{\epsilon}}_{\frac{s}{\epsilon}%
\wedge\tau^{\epsilon}}{g(y^{\epsilon}_{r})dr}=\epsilon\sum^{N-1}_{n=0}\int^{t_%
{n+1}}_{t_{n}}{g(y^{\epsilon}_{r})dr}+\epsilon\int^{\frac{s+t}{\epsilon}\wedge%
\tau^{\epsilon}}_{t_{N}}{g(y^{\epsilon}_{r})dr}.$$
Denote by $\theta_{t}$ the canonical shift operator on the probability space. Let
$F_{t}(\cdot,\omega)$ with $t\geq 0$ be the flow of the original unperturbed
system in $M$.
Triangular inequality splits our calculation into four parts
$$|\delta(\epsilon,t)|\leq|A_{1}|+|A_{2}|+|A_{3}|+|A_{4}|,$$
(15)
where
$$\displaystyle A_{1}$$
$$\displaystyle=$$
$$\displaystyle\epsilon\sum^{N-1}_{n=0}\int^{t_{n+1}}_{t_{n}}\left[g(y^{\epsilon%
}_{r})-g(F_{r-t_{n}}(y^{\epsilon}_{t_{n}},\theta_{t_{n}}(\omega)))\right]dr,$$
$$\displaystyle A_{2}$$
$$\displaystyle=$$
$$\displaystyle\epsilon\sum^{N-1}_{n=0}\left[\int^{t_{n+1}}_{t_{n}}g(F_{r-t_{n}}%
(y^{\epsilon}_{t_{n}},\theta_{t_{n}}(\omega)))dr-\Delta tQ^{g}(\pi(y^{\epsilon%
}_{t_{n}}))\right],$$
$$\displaystyle A_{3}$$
$$\displaystyle=$$
$$\displaystyle\sum^{N-1}_{n=0}\epsilon\Delta tQ^{g}(\pi(y^{\epsilon}_{t_{n}}))-%
\int^{(s+t)\wedge\epsilon\tau^{\epsilon}}_{s\wedge\epsilon\tau^{\epsilon}}Q^{g%
}(\pi(y^{\epsilon}_{\frac{r}{\epsilon}}))dr,$$
$$\displaystyle A_{4}$$
$$\displaystyle=$$
$$\displaystyle\epsilon\int^{\frac{s+t}{\epsilon}\wedge\tau^{\epsilon}}_{t_{N}}g%
(y^{\epsilon}_{r})dr.$$
We proceed showing that each of the processes $A_{1},A_{2},A_{3}$ and $A_{4}$ above
tends to zero uniformly on compact intervals. In what follows, we will
explore many times the fact that for $a>0$ and $b\in\mathbf{R}$, $\epsilon^{a}|\ln\epsilon|^{b}$ goes to zero as $\epsilon\searrow 0$.
Hence, by construction, except when $\tau^{\epsilon}\leq s$ (where, restricted to
which the lemma is trivial) both $\Delta t$ and
$N$ go to infinity when $\epsilon$ tends to zero.
Lemma 3.2
Process $A_{1}$ converges to zero
uniformly on compact intervals when
$\epsilon$ goes to zero. More precisely, we have the following estimates on
the rate of convergence: For any $\gamma\in(0,1)$, there exists a function
$h_{1}$ such that
$$\left(\mathbb{E}\sup_{s\leq t}\left|A_{1}\right|^{p}\right)^{\frac{1}{p}}\leq K%
_{1}\,t\,\epsilon^{\gamma}\ h_{1}(t,\epsilon)$$
where $h_{1}$ is continuous in $t,\epsilon>0$ and
converges to zero
when
$(t,\epsilon)\mapsto(0,0)$.
Proof: Initially note that by triangular inequality, and putting the supremum
inside the integral we get
$$\displaystyle\left(\mathbb{E}\sup_{s\leq t}\left|A_{1}\right|^{p}\right)^{%
\frac{1}{p}}$$
$$\displaystyle\leq$$
$$\displaystyle\epsilon\sum^{N-1}_{n=0}\left(\mathbb{E}\left[\int^{t_{n+1}}_{t_{%
n}}\sup_{t_{n}\leq s\leq r}\left|g(y^{\epsilon}_{s})-g(F_{s-t_{n}}(y^{\epsilon%
}_{t_{n}},\theta_{t_{n}}(\omega)))\right|dr\right]^{p}\right)^{\frac{1}{p}}$$
If $\frac{1}{p}+\frac{1}{q}=1$, by Hölder inequality we have that the
estimate above is again bounded by
$$\epsilon\sum^{N-1}_{n=0}\displaystyle\left(\mathbb{E}\left[\left(\int^{t_{n+1}%
}_{t_{n}}dr\right)^{\frac{1}{q}}\left(\displaystyle\int^{t_{n+1}}_{t_{n}}\sup_%
{t_{n}\leq s\leq r}\left|g(y^{\epsilon}_{s})-g(F_{s-t_{n}}(y^{\epsilon}_{t_{n}%
},\theta_{t_{n}}(\omega)))\right|^{p}dr\right)^{\frac{1}{p}}\right]^{p}\right)%
^{\frac{1}{p}}$$
(16)
The increment $\Delta t$ is a random variable bounded by
$t|\ln\epsilon|^{\frac{2\beta}{p}}$. In the estimates below
$\Delta t$ will denote this very deterministic upper
bounded. Analogously to $N$
which
has order $[\epsilon^{-1}|\ln\epsilon|^{-\frac{2\beta}{p}}]$.
Hence, with this notation, last inequality is again bounded by
$$\displaystyle\leq$$
$$\displaystyle\epsilon(\Delta t)^{\frac{1}{q}}\sum^{N-1}_{n=0}\displaystyle%
\left(\mathbb{E}\left[\displaystyle\Delta t\ \sup_{t_{n}\leq s\leq t_{n+1}}%
\left|g(y^{\epsilon}_{s})-g(F_{s-t_{n}}(y^{\epsilon}_{t_{n}},\theta_{t_{n}}(%
\omega)))\right|^{p}\right]\right)^{\frac{1}{p}}$$
$$\displaystyle\leq$$
$$\displaystyle\epsilon\,\Delta t\ \sum^{N-1}_{n=0}\displaystyle\left(\mathbb{E}%
\left[\displaystyle\sup_{t_{n}\leq s\leq t_{n+1}}\left|g(y^{\epsilon}_{s})-g(F%
_{s-t_{n}}(y^{\epsilon}_{t_{n}},\theta_{t_{n}}(\omega)))\right|^{p}\right]%
\right)^{\frac{1}{p}}$$
Lemma 2.1 and its corollaries says that for all $0\leq n\leq N-1$
above, the function $g$ evaluated along trajectories of the perturbed system
compared with $g$ evaluated along the unperturbed trajectories, both starting at
$y_{t_{n}}^{\epsilon}$ satisfies:
$$\left[\mathbb{E}\sup_{t_{n}\leq s\leq t_{n+1}}\left|g(y^{\epsilon}_{s})-g(F_{s%
-t_{n}}(y^{\epsilon}_{t_{n}},\theta_{t_{n}}(\omega))\right|^{p}\right]^{\frac{%
1}{p}}\leq K_{1}\epsilon\ \Delta t\,e^{K_{2}(\Delta t)^{p}}$$
Note that here, possibly Lemma 2.1 might be applied using different
(finitely many) local coordinate systems $\varphi_{i}$ for each $n$.
$$\displaystyle\left[\mathbb{E}\sup_{s\leq t}|A_{1}|^{p}\right]^{\frac{1}{p}}$$
$$\displaystyle\leq$$
$$\displaystyle K_{1}\,N\,\epsilon^{2}(\Delta t)^{2}e^{K_{2}(\Delta t)^{p}}$$
$$\displaystyle=$$
$$\displaystyle K_{1}\epsilon^{2}[\epsilon^{-1}|\ln\epsilon|^{-\frac{2\beta}{p}}%
]\ t^{2}|\ln\epsilon|^{\frac{4\beta}{p}}e^{K_{2}(t|\ln\epsilon|^{\frac{2\beta}%
{p}})^{p}}$$
$$\displaystyle=$$
$$\displaystyle K_{1}\,t\,\epsilon^{\gamma}\ h_{1}(\epsilon,t)$$
for any $\gamma\in(0,1)$ where
$$h_{1}(\epsilon,t)=t\ \epsilon^{\frac{1-\gamma}{2}}\ |\ln\epsilon|^{\frac{2%
\beta}{p}}\ \exp\left\{\left(\frac{1-\gamma}{2}\right)\ln\epsilon+K_{2}t^{p}|%
\ln\epsilon|^{2\beta}\right\}.$$
which satisfies the required properties for $\beta\in(0,1/2)$.
$\Box$
Lemma 3.3
Process $A_{2}$ in equation
(15) goes to zero with the following rate of convergence:
$$\left[\mathbb{E}\sup_{s\leq t}|A_{2}|^{p}\right]^{\frac{1}{p}}\leq K\sqrt{t}|%
\ln\epsilon|^{-\frac{\beta}{p}}$$
for a positive constant $K$.
Proof: We have
$$\displaystyle\left[\mathbb{E}\sup_{s\leq t}|A_{2}|^{p}\right]^{\frac{1}{p}}$$
$$\displaystyle\leq$$
$$\displaystyle\epsilon\ \left[\mathbb{E}\left|\sum^{N-1}_{n=0}[\int^{t_{n+1}}_{%
t_{n}}{g(F_{r-t_{n}}(y^{\epsilon}_{t_{n}},\theta_{t_{n}}(\omega)))dr}-\Delta tQ%
^{g}(\pi(y^{\epsilon}_{t_{n}}))]\right|^{p}\right]^{\frac{1}{p}}$$
$$\displaystyle\leq$$
$$\displaystyle\epsilon\ \sum^{N-1}_{n=0}\left[\mathbb{E}\left|\int^{t_{n+1}}_{t%
_{n}}{g(F_{r-t_{n}}(y^{\epsilon}_{t_{n}},\theta_{t_{n}}(\omega)))dr}-\Delta tQ%
^{g}(\pi(y^{\epsilon}_{t_{n}}))\right|^{p}\right]^{\frac{1}{p}}$$
$$\displaystyle=$$
$$\displaystyle\epsilon\ \Delta t\sum^{N-1}_{n=0}\left[\mathbb{E}\left|\frac{1}{%
\Delta t}\int^{t_{n+1}}_{t_{n}}{g(F_{r-t_{n}}(y^{\epsilon}_{t_{n}},\theta_{t_{%
n}}(\omega)))dr}-Q^{g}(\pi(y^{\epsilon}_{t_{n}}))\right|^{p}\right]^{\frac{1}{%
p}}$$
For all $n=0,\ldots,N-1$, by construction the two terms inside the
modulus converges to each other when $\Delta t$ goes to infinity.
Moreover, as in [9, Lemma 3.2] by Markovian property and central limit
theorem,
the rate of
convergence has order $\frac{1}{\sqrt{\Delta t}}$ when $\Delta t$ goes to
infinity. Hence, for small $\epsilon$
we have
$$\displaystyle\left[\mathbb{E}\sup_{s\leq t}|A_{2}|^{p}\right]^{\frac{1}{p}}$$
$$\displaystyle\leq$$
$$\displaystyle K\epsilon\ N(\Delta t)\frac{1}{\sqrt{\Delta t}}$$
$$\displaystyle\leq$$
$$\displaystyle K\epsilon\left[\epsilon^{-1}|\ln\epsilon|^{-\frac{2\beta}{p}}%
\right]\ \frac{t|\ln\epsilon|^{\frac{2\beta}{p}}}{\sqrt{t}|\ln\epsilon|^{\frac%
{\beta}{p}}}$$
$$\displaystyle=$$
$$\displaystyle K\sqrt{t}|\ln\epsilon|^{-\frac{\beta}{p}}$$
$\Box$
Lemma 3.4
$A_{3}$
converges to zero when $t$ or $\epsilon$ go to $0$. Moreover, if $Q^{g}$ is
$\alpha$-Hölder continuous with $0<\delta<\alpha$ then the rate of
convergence is given by
$$\left(\mathbb{E}\sup_{s\leq t}\left|A_{3}\right|^{p}\right)^{\frac{1}{p}}\leq K%
\,\epsilon\,t^{1+\frac{\delta}{2}}\,|\ln\epsilon|^{\frac{\beta\delta}{p}},$$
for a positive constant $K$.
Proof: By definition
$$\displaystyle|A_{3}|$$
$$\displaystyle=$$
$$\displaystyle\left|\sum^{N-1}_{n=0}\epsilon\Delta tQ^{g}(\pi(y^{\epsilon}_{t_{%
n}}))-\epsilon\int^{\frac{s+t}{\epsilon}\wedge\tau^{\epsilon}}_{\frac{s}{%
\epsilon}\wedge\tau^{\epsilon}}Q^{g}(\pi(y^{\epsilon}_{r}))\ dr\right|$$
and the convergence to zero here corresponds to the existence of the Riemann
integral.
Moreover, assuming that $Q^{g}$ is
$\alpha$-Hölder continuous for $\delta<\alpha$; since $y^{\epsilon}_{t}$ is
also $\alpha$-Hölder continuous for $\alpha<\frac{1}{2}$ (classical
result, see e.g.
[11]), then
we have that the previous expression is again bounded by
$$\displaystyle\leq$$
$$\displaystyle\epsilon\sum^{N-1}_{n=0}\Delta t\sup_{\epsilon t_{n}<s\leq%
\epsilon t_{n+1}}|Q^{g}((\pi(y^{\epsilon}_{t_{n}})))-Q^{g}((\pi(y^{\epsilon}_{%
s})))|$$
$$\displaystyle\leq$$
$$\displaystyle K\epsilon(\Delta t)^{(1+\frac{\delta}{2})}N$$
$$\displaystyle\leq$$
$$\displaystyle K\,\epsilon\,t^{1+\frac{\delta}{2}}\ |\ln\epsilon|^{\frac{\beta%
\delta}{p}}.$$
$\Box$
Lemma 3.5
Process $A_{4}$ converges to zero
with
$$\left(\mathbb{E}\sup_{s\leq t}\left|A_{4}\right|^{p}\right)^{\frac{1}{p}}\leq C%
\,t\,\epsilon\,|\ln\epsilon|^{\frac{2\beta}{p}}.$$
Proof: Denoting
$$C=\sup_{x\in U}|g(x)|.$$
The result follows straightforward since
$$\epsilon\left|\displaystyle\int^{\frac{s+t}{\epsilon}\wedge\tau^{\epsilon}}_{t%
_{N}}{g(y^{\epsilon}_{r})dr}\right|\leq C\epsilon\Delta t=Ct\epsilon|\ln%
\epsilon|^{\frac{2\beta}{p}}.$$
Now, going back to the proof of Lemma 3.1. Note that
each of the four estimates of Lemmas 3.2–3.5 allows a
factorization which has a
common factor $\sqrt{t}|\ln\epsilon|^{-\frac{\beta}{p}}$ times a continuous function
which goes to zero when $(t,\epsilon)\to 0$ (indeed, in Lemma
3.3, use a $\beta^{\prime}\in(\beta,1/2)$). Lemma 3.1
now follows by inequality (15).
$\Box$
4 An averaging principle
We state the averaging principle in the next theorem. To use Lemma
3.1 of the previous section we have to assume
regularity in the average function $Q^{g}$, which naturally depends on $g$, on the
foliated coordinate system and on the
transversal behaviour of the invariant measures on the leaves of the original
foliated system. We are going to assume the following:
Hypothesis (H): For any Lipschitz continuous function $g$ on
$M$,
its corresponding average function $Q^{g}$
on the transversal space $V$ which indexes the leaves is also Lipschitz.
This hypothesis holds naturally if the invariant measures $\mu_{p}$ for the
unperturbed foliated system has sort of weakly continuity on $p$. For
deterministic
systems it corresponds to a certain regularity in the sense that there is no
bifurcation
with respect to the vertical parameter $v\in V$.
We use the
derivative of each component of
$$\pi(\cdot)=(\pi_{1}(\cdot),\ldots,\pi_{d}(\cdot))\in V\subset\mathbf{R}^{d}.$$
to get the averages $Q^{{d\pi_{i}(K)}}(x)$ of the real functions $d\pi_{i}(K)$, $i=1,\ldots,d$ on each leaf $L_{x}$.
Theorem 4.1
Assume that the unperturbed foliated
system (1) on $M$ satisfies hypothesis (H) above.
Let $v(t)$ be the solution of the deterministic ODE in the transversal
component $V\subset\mathbf{R}^{n}$
$$\displaystyle\frac{dv}{dt}=(Q^{{d\pi_{1}(K)}},\ldots,Q^{d\pi_{d}(K)})(v(t))$$
(17)
with initial condition $v(0)=\pi(x_{0})=0$. Let $T_{0}$ be the time that
$v(t)$
reaches the boundary of $V$, then
(1)
For all $0<t<T_{0}$, $\beta\in(0,1/2)$ and $2\leq p<\infty$ ($1\leq p$ for small $t$, cf. Remark 2.4),
$$\left[\mathbb{E}\left(\sup_{s\leq t}\left|\pi\left(y^{\epsilon}_{\frac{s\wedge%
\tau^{\epsilon}}{\epsilon}}\right)-v(s)\right|^{p}\right)\right]^{\frac{1}{p}}%
\leq\sqrt{t}|\ln\epsilon|^{-\frac{\beta}{p}}h(t,\epsilon),$$
where $h(t,\epsilon)$ is continuous and converges to zero when $\epsilon$ or
$t$ goes
to $0$.
(2)
For $\gamma\leq 0$, let
$$T_{\gamma}=\inf_{0<t}\ \{\mathrm{dist}(v(t),\partial V)\leq\gamma\}$$
The exit times of the two systems satisfies the estimates
$$\mathbb{P}(\epsilon\tau^{\epsilon}<T_{\gamma})\leq\gamma^{-p}\ t^{\frac{p}{2}}%
|\ln\epsilon|^{-\beta}h(t,\epsilon)^{p}.$$
Proof: The gradient of each real function $\pi_{i}$
is orthogonal to the leaves, hence by Itô formula, for $i=1,2,\ldots,d$ we have that
$$\pi_{i}\left(y^{\epsilon}_{\frac{t\wedge\tau^{\epsilon}}{\epsilon}}\right)=%
\int^{t\wedge\tau^{\epsilon}}_{0}d\pi_{i}(K)(y^{\epsilon}_{\frac{s}{\epsilon}}%
)\ ds.$$
Lemma 3.1 for the function $d\pi_{i}(K)$ in $M$,
triangular inequality and hypothesis (H) imply that
$$\displaystyle\left|\pi_{i}\left(y^{\epsilon}_{\frac{t\wedge\tau^{\epsilon}}{%
\epsilon}}\right)-v_{i}(t)\right|$$
$$\displaystyle\leq$$
$$\displaystyle\int^{t\wedge T^{\epsilon}}_{0}{\left|Q^{d\pi_{i}(K)}\left(\pi%
\left(y^{\epsilon}_{\frac{t\wedge\tau^{\epsilon}}{\epsilon}}\right)\right)-Q^{%
d\pi_{i}(K)}(v(s))\right|\ ds}+|\delta_{i}(\epsilon,t)|$$
$$\displaystyle\leq$$
$$\displaystyle C_{i}\int^{t}_{0}\left|\pi\left(y^{\epsilon}_{\frac{s\wedge\tau^%
{\epsilon}}{\epsilon}}\right)-v(s)\right|ds+|\delta_{i}(\epsilon,t)|,$$
where each $C_{i}$ is the Lipschitz constant of $Q^{d\pi_{i}(K)}$ and
$\delta_{i}(\epsilon,t)$ is defined in Lemma 3.1.
Summing up the $i$’s and using Gronwall lemma we have, for a constant $C$:
$$\left|\pi\left(y^{\epsilon}_{\frac{t\wedge\tau^{\epsilon}}{\epsilon}}\right)-v%
(t)\right|\leq e^{Ct}\sum_{i=1}^{n}|\delta_{i}(\epsilon,t)|.$$
The first part of the theorem follows by Lemma 3.1.
For the second part we have the following estimates
$$\displaystyle\mathbb{P}(\epsilon\tau^{\epsilon}<T_{\gamma})$$
$$\displaystyle\leq$$
$$\displaystyle\mathbb{P}\left(\ \ \sup_{s\leq T_{\gamma}\wedge\epsilon\tau^{%
\epsilon}}\ \ \left|v(s)-\pi\left(y^{\epsilon}_{\frac{t\wedge\tau^{\epsilon}}{%
\epsilon}}\right)\right|>\gamma\right)$$
$$\displaystyle\leq$$
$$\displaystyle\gamma^{-p}\mathbb{E}\left(\ \ \sup_{s\leq T_{\gamma}\wedge%
\epsilon\tau^{\epsilon}}\ \ \left|v(s)-\pi\left(y^{\epsilon}_{\frac{t\wedge%
\tau^{\epsilon}}{\epsilon}}\right)\right|^{p}\right)$$
$$\displaystyle\leq$$
$$\displaystyle\gamma^{-p}\left(\sqrt{t}|\ln\epsilon|^{-\frac{\beta}{p}}h(t,%
\epsilon)\right)^{p}$$
$$\displaystyle\leq$$
$$\displaystyle\gamma^{-p}\ t^{\frac{p}{2}}|\ln\epsilon|^{-\beta}h(t,\epsilon)^{%
p}.$$
$\Box$
4.1 A detailed example:
The following simple example illustrates the
framework
where the averaging principle described in this section holds. Consider $M=\mathbf{R}^{3}-\{(0,0,z),z\in R\}$
with the 1-dimension horizontal circular foliation of $M$ where the leaf
passing through a point $p=(x,y,z)$ is given by the circle $L_{p}=\{(\sqrt{x^{2}+y^{2}}\cos\theta,\sqrt{x^{2}+y^{2}}\sin\theta,z)$, $\theta\in[0,2\pi]$. Consider the foliated linear SDE on
$M$ consisting of random rotations:
$$dx_{t}=\left(\begin{array}[]{ccc}0&-1&0\\
1&0&0\\
0&0&0\end{array}\right)x_{t}\ (\lambda_{1}\ dt+\lambda_{2}\ dB_{t}).$$
For an initial condition $p_{0}=(x_{0},y_{0},z_{0})$, say with $x_{0}\geq 0$ consider
the local foliated coordinates in the neighbourhood $U=\mathbf{R}^{3}\setminus\{(x,0,z);x\leq 0;z\in\mathbf{R}\}$ given by cylindrical coordinates.
Hence, using
the same notation as before $\varphi=(u,v)$ will be defined by $\varphi:U\subset M\rightarrow(-\pi,\pi)\times\mathbf{R}_{>0}\times\mathbf{R}$, where $u\in(-\pi,\pi)$
is angular and $v=(r,z)\in\mathbf{R}_{>0}\times\mathbf{R}$ such that $\varphi^{-1}:(u,v)\mapsto(r\cos u,r\sin u,z)\in M$. In this coordinates system, the transversal projections $\pi_{1}$ and
$\pi_{2}$
correspond to the radial $r$-component and the $z$-coordinate,
respectively.
For $\lambda_{1},\lambda_{2}\in\mathbf{R}$ with $|\lambda_{1}|+|\lambda_{2}|>0$, the
invariant measures $\mu_{p}$ in the leaves $L_{p}$ passing through points $p\in M$
are
given by normalized Lebesgue measures in $L_{p}$, which here corresponds
to the normalized angle 1-form. Note that Hypothesis (H) is satisfied. We
investigate the
effective behaviour of a small
transversal perturbation of order $\epsilon$:
$$dy^{\epsilon}_{t}=\lambda_{1}\left(\begin{array}[]{ccc}0&-1&0\\
1&0&0\\
0&0&0\end{array}\right)y^{\epsilon}_{t}\ (\lambda_{1}\ dt+\lambda_{2}\ dB_{t})%
+\epsilon K(y^{\epsilon}_{t})\ dt.$$
with initial condition $x_{0}=(1,0,0)$. In this example we shall consider two classes of perturbing vector
field
$K$.
Constant perturbation.
(A) Assume that the perturbation is given
by
a vector
field which is constant $K=(k_{1},k_{2},k_{3})$ with respect to Euclidean coordinates in $M$. Initially, to fix the
ideas, assume that $k_{3}=0$. Then, not
only the average on the $z$-component vanishes,
i.e. $Q^{d\pi_{2}K}=0$, but also, by the geometrical symmetry of $K$ with
respect to the invariant measure, the average radial $r$-component
also vanishes, i.e. $Q^{d\pi_{1}K}=0$. Hence the transversal
component
in the main Theorem 4.1 is constant
$v(t)=(r(0),z(0))$ for all $t\geq 0$.
Theorem 4.1 establishes a rate of
convergence to zero of the difference between the initial radius $r(0)=1$ and
$r(\frac{t\wedge\tau^{\epsilon}}{\epsilon})=\pi_{1}(y^{\epsilon}_{\frac{t\wedge%
\tau^{\epsilon}}{\epsilon}})$, the radial
component of the perturbed systems, precisely, that
$$\left[\mathbb{E}\left(\sup_{s\leq t}\left|r(\frac{s\wedge\tau^{\epsilon}}{%
\epsilon})-1\right|^{p}\right)\right]^{\frac{1}{p}}$$
goes to zero as
$\epsilon$ and $t$ goes to zero with the prescribed rate of convergence.
For comparison, indeed, the perturbed systems actualy has solution
$$y^{\epsilon}_{\frac{t}{\epsilon}}=\left(\begin{array}[]{c}\cos\left(\frac{%
\lambda_{1}t}{\epsilon}+\lambda_{2}B_{\frac{t}{\epsilon}}\right)\\
\\
\sin\left(\frac{\lambda_{1}t}{\epsilon}+\lambda_{2}B_{\frac{t}{\epsilon}}%
\right)\\
\\
0\end{array}\right)+\epsilon\left(\begin{array}[]{c}k_{1}\sin\left(\frac{%
\lambda_{1}t}{\epsilon}+\lambda_{2}B_{\frac{t}{\epsilon}}\right)+k_{2}\cos%
\left(\frac{\lambda_{1}t}{\epsilon}+\lambda_{2}B_{\frac{t}{\epsilon}}\right)-k%
_{2}\\
\\
-k_{1}\cos\left(\frac{\lambda_{1}t}{\epsilon}+\lambda_{2}B_{\frac{t}{\epsilon}%
}\right)+k_{2}\sin\left(\frac{\lambda_{1}t}{\epsilon}+\lambda_{2}B_{\frac{t}{%
\epsilon}}\right)-k_{1}\\
\\
0\end{array}\right)$$
One can easily get an explicit description of the $r$-component: By
normalization and using the symmetry, one can fix any $k_{1}$ and $k_{2}$; for
simplicity, we shall fix $K=(1,0,0)$, hence, in this case, for $t\leq\tau^{\epsilon}$
$$r(t)=1+\epsilon^{2}\left[2+\cos\left(\frac{\lambda_{1}t}{\epsilon}+\lambda_{2}%
B_{\frac{t}{\epsilon}}\right)\right]-\epsilon\left[\cos 2\left(\frac{\lambda_{%
1}t}{\epsilon}+\lambda_{2}B_{\frac{t}{\epsilon}}\right)+\sin\left(\frac{%
\lambda_{1}t}{\epsilon}+\lambda_{2}B_{\frac{t}{\epsilon}}\right)\right]$$
Now, using that $r(t)+1$ is bounded and that $|r(t)-1|=|r^{2}(t)-1|/|r(t)+1|$
one finally finds that
$$|r(t)-1|<K(t)\sqrt{\epsilon}$$
for any $t\leq\tau^{\epsilon}$, according to the boundaries on the the rate of convergence
stated in the theorem.
(B) Vertical perturbations. Now, for constant and vertical $K=(0,0,k_{3})$, the radial average $Q^{d\pi_{1}K}$ is null but $Q^{d\pi_{2}K}$ equals $k_{3}$ for every leaf in $M$. Hence the averaged system $v(t)=(r(0),k_{3}t)$ is constant in the radial component and increases linearly in the
$z$-coordinate. The perturbed
systems has the simple solution
$$y^{\epsilon}_{\frac{t}{\epsilon}}=\left(\begin{array}[]{c}\cos\left(\frac{%
\lambda_{1}t}{\epsilon}+\lambda_{2}B_{\frac{t}{\epsilon}}\right)\\
\\
\sin\left(\frac{\lambda_{1}t}{\epsilon}+\lambda_{2}B_{\frac{t}{\epsilon}}%
\right)\\
\\
k_{3}t\end{array}\right)$$
Hence, the comparison
$$|\pi_{2}\left(y^{\epsilon}_{\frac{t\wedge\tau^{\epsilon}}{\epsilon}}\right)-v(%
t)|\equiv 0$$
for all $t\geq 0$ and the convergence of the theorem is trivially verified.
Linear perturbation.
Consider a linear perturbation of the
form $K(x,y,z)=(x,0,0)$. In this case, again, the $z$-coordinate average
vanishes trivially. For the radial component, we have that $d\pi_{1}K=r_{0}\cos^{2}u$,
where $u$ is the angular coordinate of $p$ whose distance to the $z$-axis
($r$-coordinate) is $r_{0}$. Hence the average with respect to the invariant
measure on the leaves is given by $Q^{d\pi_{1}K}=r/2$ for leaves with radius
$r$.
The transversal system stated in the Theorem is then $v(t)=(e^{\frac{t}{2}}r(0),z(0))$. Hence the result guarantees that the radial part of
$y^{\epsilon}_{\frac{t\wedge\tau^{\epsilon}}{\epsilon}}$ must have a behaviour
close to the exponential $e^{\frac{t}{2}}$ in the sense that
$$\left[\mathbb{E}\left(\sup_{s\leq t}\left|r\left(\frac{t}{\epsilon}\right)-e^{%
\frac{t}{2}}\right|^{p}\right)\right]^{\frac{1}{p}}$$
goes to zero when $\epsilon$ goes to zero.
The fundamental
solution of the linear perturbed Stratonovich systems $y_{t}^{\epsilon}$ is given
by the
exponential
of the matrix for each fixed $t$
$$\left(\begin{array}[]{ccc}\epsilon t&-(\lambda_{1}t+\lambda_{2}B_{t})&0\\
\lambda_{1}t+\lambda_{2}B_{t}&0&0\\
0&0&0\end{array}\right)$$
where the eigenvalues corresponding to the first two coordinates (horizontal
plane)
are
$$\lambda_{1,2}=\frac{-\epsilon t\pm\sqrt{\epsilon^{2}t^{2}-4(\lambda_{1}t+%
\lambda_{2}B_{t})^{2}}}{2}.$$
whose real part is given by $\epsilon t/2$ with probability
increasing to 1 as $\epsilon$ goes to
zero. This exponential rates coincides with that one above guaranteed by the
Theorem 4.1.
Lyapunov exponents.
For foliated manifolds embedded in $\mathbf{R}^{N}$, the
symmetry of the perturbing vector
field $K$ with respect to the geometry of the leaves, hence also with
respect to Lebesgue invariant measure, as presented in the case of constant $K$,
has implied that the transversal average $Q^{d\pi K}$
vanishes. This phenomenon also appears in a couple of other examples where the
leaves are not only diffeomorphic to each other, but also has this symmetry in
the
sense that the integration of a constant perturbation $K\in\mathbf{R}^{N}$ with respect
to the Lebesgue measure is zero. To mention a
couple of
simple examples: the spherical foliation of $\mathbf{R}^{n}\setminus\{0\}$, nested
torus (increasing the smaller radius)
foliation of the solid torus minus the central circle $S^{1}\times D^{2}\setminus S^{1}\subset\mathbf{R}^{3}$ or more generally (when they exists) tubular
foliation of
$\mathbf{R}^{n}\setminus\{C\}$, with $C$ a compact set (this context also includes
the
Hamiltonian case with the Lyouville foliation of the symplectic structure of
$\mathbf{R}^{2n}$ as in [9]). In these symmetric geometrical configuration, if
the invariant measure on the leaves are the Lebesgue measures (taking gradient
Brownian motion on the leaves for instance, as in [6]) the averages
of
$Q^{d\pi}$ vanishes, hence our main theorem says that on the average, the
trajectories of the perturbed system stay somehow close to the initial leaf.
Lyapunov exponent of the system in the direction of a tangent vector $v\in T_{x_{0}}M$ contains the long time behaviour of points close to $x_{0}$ in the
direction of $v$, for details on the definition, properties,
existence
conditions, multiplicative ergodic theory, etc see e.g. among many others L.
Arnold [2] and the references therein. In particular, under
the symmetric geometrical circumstances above, if there exists the Lyapunov
exponents of the perturbed system $y_{t}^{\epsilon}$, Theorem
4.1 will imply that in transversal directions the Lyapunov
exponent can not be too far from zero. This vanishing property must happen with
multiplicity given by the codimension of the foliation, as in the examples of
the paragraphs above, where the asymptotic relevant
parameters (rotation
number ([12], [1], [5] and Lyapunov
exponents) do exist. In short:
Proposition 4.2 (Continuity of Lyapunov exponents)
Assume that the perturbed system $y^{\epsilon}_{t}$ does have
Lyapunov exponents a.s. at the assigned initial condition. If the averaged
perturbation in the leaves vanishes, i.e. $Q^{d\pi K}=0$, then a number, given
by the codimension of the foliation, of Lyapunov exponents in the spectrum
goes to zero as $\epsilon$ goes to zero.
Proof: In fact, Theorem 4.1 says that the perturbed system
increases in the transversal coordinates with order $|\pi(y^{\epsilon}_{t})|\sim\sqrt{\epsilon t}\ h(\epsilon t,\epsilon)$. Hence, any exponential
behaviour, if it exists, must be in the direction of the leaf $L_{x_{0}}$.
$\Box$
Acknowledgments: This article has been written while the
second author is in a sabbatical visit to University of Humboldt. He would
like to express his gratitude to Prof. Peter Imkeller and his research group for
the nice and friendly hospitality. Author I. G.-G. has been supported by CNPq
555241/2009-2. and P. R. has been partially supported by FAPESP 11/50151-0 and
12/03992-1.
References
[1]
L. Arnold and P. Imkeller – Rotation numbers for linear
stochastic differential equations. Ann. Probab. 27, 130–149 (1999).
[2]
L. Arnold – Random Dynamical Systems,
Springer-Verlag (1998).
[3]
V. Arnold – Mathematical Methods in Classical
Mechanics. 2nd ed. Berlin, Springer (1989).
[4]
A. Borodin and M. Friedlin – Fast oscillating random
perturbations of dynamical systems with conservation laws. Ann. Inst. H.
Poincaré. Prob. Statist., 31 (1995) 485-525.
[5]
P. Catuogno, D. Ledesma and P. Ruffino –
Relative rotation
number for stochastic systems: dynamical and topological applications.
Dynamical
Systems, 23 (2008) 425-435.
[6]
P. Catuogno, D. Ledesma and P. Ruffino – Harmonic
measures in embedded foliated manifolds. Submitted. (2012) ArXiv 1208.0629.
[7]
Y. Kabanov and S. Pergamenshchikov –Two-Scale Stochastic Systems: asymptotic analysis and control.
Springer-Verlag, 2003.
[8]
R. Khasminski and N. Krylov – On averaging
principle for diffusion processes with null-recurrent fast component. Stoch
Proc Appl. 93 (2001) 229-240.
[9]
Xue-Mei Li – An averaging principle for a completely
integrable stochastic Hamiltonian systems. Nonlinearity, 21 (2008)
803-822.
[10]
S. Namachchvaya and R. Sowers – Rigorous
stochastic averaging at a center with additive noise. Meccanica, 37
(2002) 85-114.
[11]
D. Revuz and M. Yor –Brownian Motion and
Continuous
Martingales. Springer-Verlag, 3rd ed. Berlin 1999.
[12]
P. R. Ruffino – Decomposition of stochastic flow and
rotation matrix. Stochastics and Dynamics, 2 (2002) 93-108.
[13]
J.A. Sanders, F. Verhulst and J. Murdock – Averaging
Methods in Nonlinear dynamical Systems. 2nd Ed. Springer, 2007.
[14]
R. Sowers – Stochastic averaging with a flattened
Hamiltonian: a Markov process on a stratified space (a whiskered sphere). Trans. Am. Math. Soc, 354 (2002) 853-900.
[15]
P. Tondeur. Foliations on Riemannian manifolds.
Universitext, Springer Verlag, Berlin-Heidelberg-New York, 1988.
[16]
P. Walcak – Dynamics of foliations, groups and
pseudogroups. Birkhäuser Verlag 2004. |
astro-ph/0604488
On the One Loop Corrections to Inflation
and the CMB Anisotropies
Martin S. Sloth***sloth@physics.ucdavis.edu
Department of Physics, University of California
Davis, CA 95616, USA
Abstract
We investigate the one loop effective potential of
inflation in a standard model of chaotic inflation. The leading
one loop corrections to the effective inflaton potential are
evaluated in the quasi de Sitter background, and we estimate the
one loop correction to the two-point function of the inflaton
perturbations in the Hartree approximation. In this approximation,
the one loop corrections depends on the total number of e-foldings
of inflation and the maximal effect is estimated to be a correction to the
power spectrum of a few percent. However, such a correction may be
difficult to disentangle from the background in the simplest scenario.
1 Introduction
The measurements of the Cosmic Microwave Background (CMB)
anisotropies proceed with increasing accuracy, which together with
Super Novae (SN) and large scale structure (LSS) data yields an
ever more detailed picture of the evolution of the universe.
Inflation has earned its place in this picture as a successful
paradigm, which, in addition to solving the theoretical problems
of the standard Big-Bang theory, explains a number of
observational data [1]. However, we know little about
the theory of inflation itself, essentially due to our ignorance
of physics at the very short distance scales. In fact, even the
energy scale of inflation is at present unknown within $10$ orders
of magnitude. Future experiments might change that situation
dramatically. Especially a detection of primordial gravitational
waves could pin down the energy scale at which inflation takes
place, and as we measure the spectrum of CMB perturbations to
higher and higher accuracy we will soon be able to discriminate
between different models of inflation and rule out some of the
popular ones [2]. This is especially exciting
because it is one of our few windows to fundamental physics well
above the TeV scale.
Naturally there is a large interest in the theoretical limitations
to what we can learn about the physics of inflation and thus
physics at very short distance scales from CMB measurements. In
addition to attempts to reconstruct parts of the effective
potential of inflation from data [3], it has also been
attempted to find possible signatures of the $UV$ scale where the
effective theory of inflation breaks down, using modified
dispersion relations [4], an effective minimum length
[5], non-commutativity [6], a new-physics
hyper-surface [7], or effective field theory
[8, 9, 10, 11], combined with
constraints from more theoretical considerations [12].
While these considerations are important, it is not yet clear
whether such effects are relevant in the real universe.
Here, we investigate the effect of one loop corrections to
inflation. If inflation has lasted very long, the Hubble rate at
the beginning of inflation $H_{i}$, is a new $UV$ scale in between
the Planck scale $M_{p}$ and the Hubble scale, $H$, when observable
modes exits the horizon. The physical modes corresponding to
length scales smaller than $1/H_{i}$, initially smaller than the
inflating patch, are pushed outside the horizon during inflation.
When integrated over in the loops, they might lead to a
significant enhancement of the one loop
effects[13, 14, 15, 16, 17, 18, 19, 20, 21].
The physical $IR$ cutoff, given by the initial physical radius of
the inflating patch, is approximately equivalent to the apparent
particle horizon during inflation, and much larger than the Hubble
radius. This is the standard choice in the literature
[13, 14, 15, 16, 17, 18, 19, 20, 21].
We expect that due to causality, to a local observer such $IR$
contributions can never uniquely be identified to be due
inhomogeneities, but will look very similar to an addition to the
background parameters of the effective theory and thus only
indirectly influence our measurements. On the other hand, the
effects are sensible to the $UV$ scale of the theory, which is the
Hubble rate at the beginning of inflation.
As an example, we will evaluate the one loop corrections to the
effective potential and the two-point function in a specific
$\lambda\phi^{4}$ type of chaotic inflation. We estimate that the
maximal effect in this particular model is a $1\%$ correction to
the power spectrum.
In the first part of the paper we will consider the effective one
loop potential and one loop corrections to the background
parameters of the theory. The effect appears to be maximally about
$1\%$. In the second part of the paper, we estimate the one loop
correction to the two-point function of the quantum fluctuations
in the Hartree approximation. We find a similar effect of order
$1\%$. This will lead to a small correction to the slow-roll
parameters and a possible change in the cosmological
scalar-to-tensor perturbation ratio, which is often also expressed
in the so called consistency relation.
Some of the earliest work on the one loop effects in inflation is
the work of Vilenkin [13], who studied the effective
potential of inflation in the approximation of an exact de Sitter
background. Some similar techniques to those applied in the second
half of the present paper, was applied in [10] in order
to understand the effect of a non-standard vacuum state on the
$UV$ cutoff on the one loop effects***An other approach to loop effects, the stochastic approach, has been applied by Linde [22] (see also [23]), in order to understand the global structure of space-time in eternal inflation.. Finally, the effective
potential of inflation has been calculated in a slow-roll
approximation in [24, 25]. They
found, as expected on dimensional grounds, that in an effective
field theory of inflation, one loop effects are suppressed by a
factor $H^{2}/M_{p}^{2}$ [8], where $H$ is the
Hubble-rate when the observable modes exit and $M_{p}$ is the
reduced Planck scale. Our approach here is somewhat similar in
spirit. However, the effects we find are enhanced by a factor of
$H_{i}^{4}/H^{4}$, where $H_{i}$ is the Hubble-rate at the beginning of
inflation. This is because we go beyond the approximation of a
constant $H$. Our results are consistent with those of
[25], when we take $H_{i}\to H$.
Thus, it is not always that $H^{2}/M_{p}^{2}$ is the correct expansion
parameter for quantum effects in the effective field theory. In
long inflation there could be a significant enhancement, because
the loops are effectively suppressed by only a factor of
$H^{4}_{i}/(H^{2}M^{2}_{p})$.
One might note, already in
[26, 27, 28] it
was shown that even within the effective field theory approach, one
can have significant enhancements of quantum effects compared to the
dimensional $H^{2}/M_{p}^{2}$ estimate from potential bumps or other
non-adiabatic effects during inflation.
In the next subsection we will introduce the uniform curvature
gauge, which we will chose for our analysis in the next sections.
In section 2, we calculate the one loop corrections to the
equation of motion of the inflaton and the quantum corrections to
the slow-roll parameters. Our analysis is to first order in the
slow-roll and effective field theory expansion. In section 3, we
estimate the quantum corrections to the two point function of the
inflaton fluctuations in the Hartree approximation. In section 4,
we discuss our findings.
1.1 Uniform curvature gauge
We find that it is simplest to understand the one loop correction
to the inflaton potential in the uniform curvature gauge, because
here the residual degrees of the freedom of the perturbations can
be identified with the inflaton field fluctuations. This makes it
more transparent to understand the perturbations as quantum
fluctuations of the inflaton and to renormalize the divergences
with simple counter terms in the renormalized inflaton potential.
In order to calculate the renormalized two point function
$\left<\right.\delta\phi^{2}\left.\right>_{0}$ in the effective theory, we
will thus write the perturbed metric to first order in the uniform
curvature gauge where the metric takes the form
[29, 30]
$$ds^{2}=-(1+2\varphi)dt^{2}+2aB_{,i}dtdx^{i}+a^{2}\delta_{ij}dx^{i}dx^{j}~{}.$$
(1)
In the absence of anisotropic stress $\dot{B}+2HB+2\varphi/a=0$, we
expect that there should be only one independent scalar degree of
freedom like in the longitudinal gauge if we have fixed the gauge
correctly. In fact one finds using the constraints from the
Einstein equations that the equation of motion for the inflaton
perturbations becomes
$$\ddot{\delta\phi}+3H\dot{\delta\phi}-\frac{1}{a^{2}}\nabla^{2}\delta\phi+\left%
(V_{\phi\phi}-6\epsilon H^{2}\right)\delta\phi=0~{},$$
(2)
to first order in the slow-roll parameters. This shows that in
this gauge the quantum fluctuations can be identified with the
gauge invariant Sasaki-Mukhanov variable, $Q$, which satisfies the
same equation [31]. In appendix A, we have
generalized this to third order in perturbations and computed the
effective action of the perturbations to fourth order, since we
will need it to compute the self-interactions of the quantum
fluctuations of the inflaton.
Thus, in the zero curvature gauge we can describe the generation
of perturbations from quantum fluctuations of the inflaton in
Fourier space, by expanding the quantum field $\delta\phi(t,{\bf x})$ in c-number mode functions with respect to the Bunch-Davis
vacuum
$$\delta\phi(t,{\bf x})=\int\frac{d^{3}{\bf k}}{(2\pi)^{3/2}}\left[U_{k}(t)e^{i{%
\bf k\cdot x}}a_{\bf k}+U^{*}_{k}(t)e^{-i{\bf k\cdot x}}a^{\dagger}_{\bf k}%
\right]~{},$$
(3)
where the operator $a_{\bf k}$ annihilates the Bunch-Davis vacuum
$\left|0\right>_{0}$. The mode functions satisfy the Fourier transform of
eq.(2), which in conformal coordinates yields
$$\left[\eta^{2}\frac{\partial^{2}}{\partial\eta^{2}}-2\eta\frac{\partial}{%
\partial\eta}+\eta^{2}k^{2}+\frac{V_{\phi\phi}-6\epsilon H^{2}}{H^{2}}\right]U%
_{k}(\eta)=0~{},$$
(4)
and the conformal time is defined as $a(\eta)d\eta=dt$. So, with
the usual normalization to the Minkowski vacuum in the infinite
past the solution for the modes becomes
$$U_{k}(\eta)=\frac{\sqrt{\pi}}{2}H\eta^{3/2}H_{\nu}^{(2)}(k\eta)~{},$$
(5)
where $H_{\nu}^{(2)}(k\eta)$ is the usual second Hankel function
and $\nu=3/2+3\epsilon-\eta$. The spectral index of the perturbations
is defined as
$$\mathcal{P}_{\delta\phi}(k,t)=\frac{k^{3}}{2\pi^{2}}\left<\right.\left|\delta%
\phi_{k}(t)\right|^{2}\left.\right>_{0}=\frac{k^{3}}{2\pi^{2}}\left|U_{k}(t)%
\right|^{2}~{},$$
(6)
which on super-Hubble scales approximately gives
$$\mathcal{P}_{\delta\phi}(k,t)\simeq\frac{H^{2}}{4\pi^{2}}\left(\frac{k}{aH}%
\right)^{n-1}~{},$$
(7)
with $n-1=3-2\nu$. It is important to note that this solution is
to first order in the slow-roll parameters and treating them as
constant.
2 One loop effective field theory of inflation
Inflation is most often formulated in terms of a minimally coupled
scalar inflaton quantum field, $\phi$, with a Lagrangian
$$\mathcal{L}=-\frac{1}{2}(\partial_{\mu}\phi)^{2}-V(\phi)~{},$$
(8)
in a quasi de Sitter space with metric
$$ds^{2}=-dt^{2}+a^{2}(t)\delta_{ij}dx^{i}dx^{j}~{},$$
(9)
where the scale factor is conveniently written on the form $a(t)=\exp(Ht)$ and the Hubble rate $H$ is almost constant, a statement
which we can quantify in terms of the slow-roll parameter $\epsilon=-\dot{H}/H^{2}<<1$.
For definiteness we will consider a simple version of chaotic
inflation driven by an inflaton field with potential
$$V(\phi)=\frac{1}{4}\lambda\phi^{4}~{}.$$
(10)
The resulting equation of motion is
$$\Box_{g}\phi+\lambda\phi^{3}=0~{}.$$
(11)
The dynamics is conveniently analyzed in an effective field theory
for the classical background field $\phi_{c}$, defined by
$$\phi=\phi_{c}+\delta\phi~{},$$
(12)
where $\delta\phi$ is a quantum field with a vanishing vacuum
expectation value enforced through the tadpole condition
$$\left<\delta\phi\right>_{0}\equiv\left<0\right|\delta\phi\left|0\right>=0~{},$$
(13)
such that
$$\left<\phi\right>_{0}\equiv\left<0\right|\phi\left|0\right>=\phi_{c}~{}.$$
(14)
From Wick’s theorem it follows that also
$\left<\right.\delta\phi^{3}\left.\right>_{0}=0$. The effective equation
of motion for the classical field $\phi_{c}$ can then be obtained by
taking the expectation value of the equation of motion in eq.
(11), which yields
$$\left<\right.\Box_{g+\delta g}\phi\left.\right>_{0}+3\lambda\left<\right.%
\delta\phi^{2}\left.\right>_{0}\phi_{c}+\lambda\phi_{c}^{3}=0~{}.$$
(15)
To calculate $\left<\right.\Box_{g}\phi\left.\right>_{0}$, we need to take
into account also the metric perturbations in the uniform
curvature gauge as discussed in the introduction. The constraint
equations relating the scalar metric perturbations to the scalar
field fluctuations are in general complicated. Thus, we find that
the effective equation of motion to second order, is most easily
computed from the effective action of the perturbations expanded
to third order using the ADM formalism [34]. In
the appendix we have in fact derived the effective action of the
perturbations to fourth order, which we will need later. The
expressions are given in eq.(74) and eq.(75). Using
the linear perturbation equation eq.(2), we find from
eq.(74), on super-Hubble scales where we can neglect
gradient terms, to leading order
$$\ddot{\phi}_{c}+3H\dot{\phi}_{c}+6\lambda\left<\right.\delta\phi^{2}\left.%
\right>_{0}\phi_{c}+\lambda\phi_{c}^{3}=0~{}.$$
(16)
The contributions from sub-Hubble scales, contributing to the
trace-anomaly, where computed in [25].
However, since they are not amplified by long inflation, those
contributions are always suppressed by a factor $H^{2}/M_{p}^{2}$ and we
will therefore ignore gradient terms and second order time
derivatives of the perturbations. Thus, to obtain an order of
magnitude estimate of the minimal one loop effects, it should be
sufficient to consider only the contributions in
eq.(16). In section 3.2 we will see more formally how
the effective one loop equation of motion in eq.(16)
follows from the tadpole renormalization condition of the quantum
fluctuations.
From the effective equation of motion in eq.(16), we
can see that the effective mass gets a one loop contribution
$$\delta m^{2}_{eff}=6\lambda\left<\right.\delta\phi^{2}\left.\right>_{0}~{}.$$
(17)
The effective potential, which is consistent with the effective
energy density derived from the effective energy-momentum tensor
computed in [15, 16], is
$$V_{eff}(\phi_{c})=V(\phi_{c})+2V^{\prime}\left<\right.\delta\phi\varphi\left.%
\right>_{0}+\frac{1}{2}V^{\prime\prime}\left<\right.\delta\phi^{2}\left.\right%
>_{0}~{}.$$
(18)
The inflationary expansion is then given by the slow-roll of the
classical field in the effective potential, and in the slow-roll
approximation the effective Hubble rate is determined by the
effective Friedman equations
$$3H^{2}\simeq\frac{1}{M_{p}^{2}}V_{eff}(\phi_{c})~{},\qquad 3H\dot{\phi}_{c}%
\simeq-V^{\prime}(\phi_{c})-\delta m^{2}_{eff}\phi_{c}~{},$$
(19)
as can be seen from the effective equation of motion
eq.(16), with $\ddot{\phi}_{c}\simeq 0$. The effective
slow-roll parameters receives a one loop correction
$\delta\epsilon_{eff}$, $\delta\eta_{eff}$, when compared to the
tree-level slow-roll parameters $\epsilon$, $\eta$, such that
$$\epsilon_{eff}=\epsilon+\delta\epsilon_{eff}~{},\qquad\eta_{eff}=\eta+\delta%
\eta_{eff}~{}.$$
(20)
With the following definition of the slow-roll parameter,
$$\epsilon_{eff}\equiv\frac{1}{2M_{p}^{2}}\frac{\dot{\phi}_{c}^{2}}{H^{2}}~{},$$
(21)
we can compute
$$\delta\epsilon_{eff}\simeq\frac{4}{3}\frac{\delta m_{eff}^{2}}{H^{2}}~{},$$
(22)
and we assumed that the one loop contribution to the effective
potential is small compared to the tree-level value, i.e. $\delta m_{eff}^{2}\phi_{c}^{2}<<V(\phi_{c})$.
To make the approximations self-consistent, we should use the
effective slow-roll parameters in eq.(20) to calculate
the quantum fluctuations in eq.(2). This is similar to
the cactus or Hartree approximation
[13], which corresponds to an infinite summation of
self-energy diagrams of the form shown if fig.(1). In section 3,
we shall calculate explicitly the one loop diagram in fig.(1),
which is first order in $\lambda$ and similar to computing the
loop correction to $\eta$ using Dyson’s equation. To second order
in $\lambda$ there are also other one loop diagrams, not included
in the cactus approximation, that contributes, since the vacuum
expectation value of the background field is non-vanishing.
Assuming, according to the standard lore, that the initial vacuum
is the Euclidean vacuum, the spectral index $n_{eff}$ of the
primordial scalar curvature perturbations generated during
inflation follows
$$n_{eff}-1=2\eta_{eff}-6\epsilon_{eff}~{},$$
(23)
as we will discuss in more details in the next section where we
will also calculate the magnitude of the corrections. Especially
we need to calculate $\left<\right.\delta\phi^{2}\left.\right>_{0}$ and
extract the dominant finite part.
2.1 Renormalized quantum fluctuations
If one assumes that the inflationary state is preceded by a
radiation dominated one, then the expectation value $\left<\right.\delta\phi^{2}\left.\right>_{0}$ will receive two contributions. A
thermal contribution and a vacuum contribution. As inflation kicks
in, only the vacuum contribution will grow, and one can ignore the
thermal contribution. It is the modes corresponding to physical
wavelengths which are stretched to super-Hubble scales by the
inflationary expansion, that are responsible for the growth. As
modes are continuously pushed outside the horizon, they freeze and
give a little accumulating contribution to the expectation value.
Thus, the modes that where already outside the horizon at the
beginning of inflation, gives only a constant contribution, which
one can ignore a while after inflation has started.
Above, we calculated the one loop induced mass in terms of
$\left<\right.\delta\phi^{2}\left.\right>_{0}$, which we can now compute
directly from the power spectrum
$$\left<\right.\left|\delta\phi(t,{\bf x})\right|^{2}\left.\right>_{0}=\int\frac%
{d^{3}{\bf k}}{(2\pi)^{3/2}}\left<\right.\left|\delta\phi_{k}(t)\right|^{2}%
\left.\right>_{0}=\int_{a_{i}H_{i}}^{a\Lambda}\frac{dk}{k}\mathcal{P}_{\delta%
\phi}(k,t)~{}.$$
(24)
The $IR$ cutoff on comoving momenta $a_{i}H_{i}$ is dynamically given
by the scale that exits the horizon at the beginning of inflation,
as explained above, and the $UV$ cutoff has been introduced by
hand to regulate a quadratic $UV$ divergence, which should be
subtracted by a proper renormalization. When the $UV$ divergences
has been subtracted, one finds that $\left<\right.\delta\phi^{2}\left.\right>_{0}$ is dominated by the $IR$ modes.
In order to actually evaluate the integral, we will split it in an
$IR$ part and a $UV$ part that we will evaluate separately
$$\left<\right.\left|\delta\phi(t,{\bf x})\right|^{2}\left.\right>_{0}=\left<%
\right.\left|\delta\phi(t,{\bf x})\right|^{2}\left.\right>_{IR}+\left<\right.%
\left|\delta\phi(t,{\bf x})\right|^{2}\left.\right>_{UV}~{},$$
(25)
with
$$\left<\right.\left|\delta\phi(t,{\bf x})\right|^{2}\left.\right>_{IR}=\int_{a_%
{i}H_{i}}^{a_{0}H_{0}}\frac{dk}{k}\mathcal{P}_{\delta\phi}(k,t)~{},\qquad\left%
<\right.\left|\delta\phi(t,{\bf x})\right|^{2}\left.\right>_{UV}=\int_{a_{0}H_%
{0}}^{a\Lambda}\frac{dk}{k}\mathcal{P}_{\delta\phi}(k,t)~{}.$$
(26)
Here we let $a_{0}$, $H_{0}$ denote the values of $a$, $H$ at the time
when the relevant scales for the CMB crosses outside the horizon.
Let us first evaluate the more important $IR$ part. To properly
account for the running of the spectral index with the scale, we
will solve the mode function, $U_{k}$, in pure de Sitter, but using
the value of $H=H_{k}$ and $\dot{\phi}=\dot{\phi}_{k}$ when the given mode
crosses the horizon. If we specify to the specific $\lambda\phi^{4}$
model of chaotic inflation, this implies that we obtain the right
$k$ scaling from (See appendix B)
$$U_{k}(\eta)=\left(\frac{H_{k}}{H(\eta)}\right)^{3/2}U^{ds}_{k}(\eta)=\frac{%
\sqrt{\pi}}{2}H\eta^{3/2}\left(\frac{H_{k}}{H(\eta)}\right)^{3/2}H_{3/2}^{(2)}%
(k\eta)~{},$$
(27)
where the mode solution in pure de Sitter was denoted by
$U^{ds}_{k}(\eta)$.
If we let $N=\ln(a/a_{i})$ denote the number of e-foldings of
expansions since the beginning of inflation, it is convenient to
recast the relevant integral into an integral over $N$. To do so,
we use the relations $d\ln k=dN$, $\ln(aH/k)=N$ and
$$H_{k}^{2}=\frac{\lambda}{12M_{p}^{2}}\left(\phi_{i}^{2}-64M_{p}^{2}N\right)^{2%
}~{}.$$
(28)
Now we can easily evaluate the dominant part of the $IR$
contribution to the relevant correlator, we obtain from
eq.(83) in appendix B
$$\left<\right.\left|\delta\phi(t,{\bf x})\right|^{2}\left.\right>_{IR}\simeq%
\frac{H^{2}}{16\pi^{2}}\left(\lambda\frac{1024}{3}\frac{M_{p}^{2}}{H^{2}}%
\right)^{3/2}N^{4}~{}.$$
(29)
If we had used same type of argument for a $m^{2}\phi^{2}$ theory, the
$IR$ part of the power spectrum would instead of $H_{k}^{3/2}$ scale
as $H_{k}^{2}$. If one applies the relation $\varphi=-\sqrt{\epsilon/2}\delta\phi/M_{p}$, the result is consistent with
[15, 16, 17].
The $UV$ part is much easier to evaluate. Inside the horizon, the
modes are not sensitive to the details of the expansion
$$\displaystyle\left<\right.\left|\delta\phi(t,{\bf x})\right|^{2}\left.\right>_%
{UV}$$
$$\displaystyle\simeq$$
$$\displaystyle\frac{1}{8\pi}H^{2}\int_{1}^{\Lambda_{H}}\frac{dp_{H}}{p_{H}}p_{H%
}^{3}\left|H_{3/2}^{(2)}(p_{H})\right|^{2}$$
(30)
$$\displaystyle=$$
$$\displaystyle\frac{1}{8\pi^{2}}H^{2}\left(\Lambda_{H}^{2}+\ln(\Lambda_{H}^{2})%
\right)~{}.$$
where $p_{H}=|k/(aH)|=|k\eta|$.
Finally we can add the $IR$ and the $UV$ contributions to obtain
the full correlator to leading order
$$\displaystyle\left<\right.\left|\delta\phi(t,{\bf x})\right|^{2}\left.\right>_%
{0}$$
$$\displaystyle\simeq$$
$$\displaystyle\frac{1}{8\pi^{2}}H^{2}\left(\Lambda_{H}^{2}+\ln(\Lambda_{H}^{2})%
+\Delta_{N}\right)~{},$$
(31)
where we for convenience have defined
$$\Delta_{N}=\frac{1}{2}\left(\lambda\frac{1024}{3}\frac{M_{p}^{2}}{H^{2}}\right%
)^{3/2}N^{4}~{}.$$
(32)
Another way of writing $\Delta_{N}$, which is illuminating, is in
terms of the Hubble rate at the beginning of inflation $H_{i}$,
which gives
$$\Delta_{N}=\frac{\sqrt{3}}{64}\frac{1}{\sqrt{\lambda}}\frac{H_{i}^{4}}{M_{p}H^%
{3}}~{}.$$
(33)
The $UV$ divergent parts are canceled in the renormalized
potential introducing appropriate counter-terms. We shall not
elaborate on this aspect, but note that the physical $IR$
contributions survives as the dominant one loop correction to the
effective renormalized potential after subtracting the $UV$
divergences, as discussed in more details in
[24].
2.2 Quantum corrections to slow-roll parameters
We can now evaluate the quantum corrections to the slow-roll
parameters and the spectral index of the CMB anisotropies, using
the relations derived in eq.(17) and eq.(22). The
corrections to the slow-roll parameters can then be written in
terms of $\Delta_{N}$ in the following way
$$\delta\epsilon_{eff}=\frac{\lambda}{\pi^{2}}\Delta_{N}~{}.$$
(34)
From eq.(33) we obtain
$$\delta\epsilon_{eff}=\frac{\sqrt{3}}{64\pi^{2}}\sqrt{\lambda}\frac{H_{i}^{4}}{%
M_{p}H^{3}}~{}.$$
(35)
If we use the background values for the slow-roll parameters
$$\epsilon=\sqrt{\frac{16\lambda}{3}}\frac{M_{p}}{H}~{},$$
(36)
we can easily write the effective slow-roll parameter $\epsilon_{eff}$
in terms of a fractional correction to the background value
$$\epsilon_{eff}=\epsilon\left(1+\frac{\delta\epsilon_{eff}}{\epsilon}\right)=%
\epsilon\left(1+\frac{3}{256\pi^{2}}\frac{H^{2}}{M_{p}^{2}}\frac{H_{i}^{4}}{H^%
{4}}\right)~{}.$$
(37)
From an effective field theory point of view we generically expect
that corrections are of order $H^{2}/M_{p}^{2}$
[8, 24]. In fact, if we take
$H_{i}\to H$ that is exactly the type of correction we find.
However, due to the variation in the expansion rate over time, the
effect is amplified by the potentially large term $H_{i}^{4}/H^{4}$. It
is also interesting to note that
$$\frac{H_{i}^{4}}{M_{p}^{4}}\simeq 10^{5}\lambda^{2}N^{4}~{},$$
(38)
so the correction depends very sensitively on the total number of
e-foldings. This leads us to the possibility that the spectral
index carries a tiny imprint from the beginning of inflation
through the loop effects, such that we in principle indirectly can
observe the total number of e-foldings of inflation in this model.
In the model of chaotic inflation with a potential
$V(\phi)=1/4\lambda\phi^{4}$, in the regime when $V(\phi)>M_{p}^{4}$
quantum gravity effects are large and no classical description of
space is possible within the effective field theory framework.
When $10^{-4}M_{p}^{4}~{}\mbox{\raisebox{-2.58pt}{$\stackrel{{\scriptstyle<}}{{\sim}%
}$}}~{}V(\phi)~{}\mbox{\raisebox{-2.58pt}{$\stackrel{{\scriptstyle<}}{{\sim}}$%
}}~{}M_{p}^{4}$, the amplitude
of inflaton fluctuations is large compared to the mean and the
perturbations can not be treated perturbatively. This is the
self-reproduction regime. For our effective field theory
description of inflation to be consistent, we must therefore at
least require that the energy density of inflation, $\rho$, is
safely below the Planck scale,
$$\rho^{1/4}~{}\mbox{\raisebox{-2.58pt}{$\stackrel{{\scriptstyle<}}{{\sim}}$}}~{%
}0.1M_{p}~{}.$$
(39)
This implies that the Hubble rate at the beginning of inflation
must satisfy $H_{i}^{4}~{}\mbox{\raisebox{-2.58pt}{$\stackrel{{\scriptstyle<}}{{\sim}}$}}~{}%
10^{-9}M_{p}^{4}$. If we assume that the
Hubble rate, when the observable modes left the horizon, is given
by $H\simeq 10^{-5}M_{p}$, then the maximal correction to the first
slow-roll parameter, $\epsilon$, is
$$\frac{|\delta\epsilon_{eff}|}{\epsilon}~{}\mbox{\raisebox{-2.58pt}{$\stackrel{%
{\scriptstyle<}}{{\sim}}$}}~{}\frac{30}{256\pi^{2}}\simeq 0.01~{}.$$
(40)
Thus, the maximal correction to the slow-roll parameter, $\epsilon$, is
less than about $1\%$ in this approximation.
3 One loop contributions to the power spectrum
In order to evaluate the one loop contributions self-consistently,
we need to compute the one loop contribution to the two-point
function of the quantum fluctuations. It is convenient to apply
the Schwinger-Keldysh formalism
[32, 33] for this purpose. Below, we
will first briefly review the Schwinger-Keldysh formalism and then
calculate the two-point function in the Hartree approximation.
This is similar to computing the one loop correction to the second
slow-roll $\eta_{eff}$ using Dyson’s equation.
3.1 Schwinger-Keldysh formalism
This formalism has recently been reviewed and applied to the specific
problem of a self-interacting scalar field in de Sitter space,
in order to understand the $UV$ properties of inflaton
fluctuations originating in an alpha vacuum state [10].
Here we will therefore only review it briefly in order to
introduce the proper notation, which will essentially follow
[10].
The Schwinger-Keldysh formalism is a perturbative approach for
solving the evolution of a matrix element over a finite time
interval. This is useful in the inflationary coordinates of de
Sitter space, where there is no well defined asymptotic
in and out state in which to define an
S-matrix, essentially due to the explicit time-dependence
of the metric. Instead of specifying an initial state in the
infinite past, one develops a given state forward in time from a
specified initial time $\eta_{infl}$, which we can think of as
being the beginning of inflation.
In the interaction picture, the evolution of operators is given by
the free Hamiltonian, $H_{0}$, while the part of the Hamiltonian
containing the interactions, $H_{I}$, is used to evolve the states
in the theory. If one specifies a density of state
$\rho(\eta_{infl})$ at a specific moment in time, then one can
define a unitary time evolution operator $U_{I}(\eta,\eta^{\prime})$ that
evolves the state, and which is given by Dyson’s equation
$$U_{I}(\eta,\eta_{infl})=T\left\{e^{-i\int_{\eta_{infl}}^{\eta}d\eta^{\prime}H_%
{I}(\eta^{\prime})}\right\}~{},$$
(41)
where $T$ is the time ordering of the product in the curly
brackets. Then one can write
$$\rho(\eta)=U_{I}(\eta,\eta_{infl})\rho(\eta_{infl})U_{I}^{-1}(\eta,\eta_{infl}%
)~{}.$$
(42)
Absorbing a step function $\Theta(\eta-\eta_{infl})$ into the
interaction Hamiltonian, $H_{I}$, such that interaction only turn on
after $\eta_{infl}$, one can then write the evolution of the
expectation value of some operator, $\mathcal{O}$, as
$$\left<\right.\mathcal{O}\left.\right>(\eta)=\frac{\textrm{Tr}\left[U_{I}(-%
\infty,0)U_{I}(0,\eta)\mathcal{O}U_{I}(\eta,-\infty)\rho(\eta_{infl})\right]}{%
\textrm{Tr}\left[U_{I}(-\infty,0)U_{I}(\eta,-\infty)\rho(\eta_{infl})\right]}~%
{}.$$
(43)
This matrix element describes a system in the initial state
$\rho(\eta_{infl})$, evolved from conformal time $-\infty$ to $0$
with an operator inserted at $\eta$, and back again from $0$ to
$-\infty$. To evaluate this matrix element, one can formally
double the field content of the theory, with a set of ”+” fields
on the increasing-time contour and a set of ”-” fields on the
decreasing-time contour and then group the evolution operators
into a single time-ordered exponential. One can then write the
interacting part of the action appearing in Dyson’s equation
together in a single time-contour, as
$$S_{I}=-\int_{-\infty}^{0}d\eta\left[H_{I}(\psi^{+})-H_{I}(\psi^{-})\right]~{},$$
(44)
where contractions between different pairs of the two types of
fields now yields four kinds of propagators
$$\displaystyle\left<0\right|T\left[\psi^{\pm}(x)\psi^{\pm}(x^{\prime})\right]%
\left|0\right>$$
$$\displaystyle=$$
$$\displaystyle-iG^{\pm\pm}(x,x^{\prime})$$
(45)
$$\displaystyle=$$
$$\displaystyle-i\int\frac{d^{3}k}{(2\pi)^{3}}e^{i\vec{k}\cdot(\vec{x}-\vec{x}^{%
\prime})}G_{k}^{\pm\pm}(\eta,\eta^{\prime})~{}.$$
The time-ordering of the contractions then yields
$$\displaystyle G^{++}_{k}(\eta,\eta^{\prime})$$
$$\displaystyle=$$
$$\displaystyle G^{>}_{k}(\eta,\eta^{\prime})\Theta(\eta-\eta^{\prime})+G^{<}_{k%
}(\eta,\eta^{\prime})\Theta(\eta^{\prime}-\eta)$$
$$\displaystyle G^{--}_{k}(\eta,\eta^{\prime})$$
$$\displaystyle=$$
$$\displaystyle G^{>}_{k}(\eta,\eta^{\prime})\Theta(\eta^{\prime}-\eta)+G^{<}_{k%
}(\eta,\eta^{\prime})\Theta(\eta-\eta^{\prime})$$
$$\displaystyle G^{-+}_{k}(\eta,\eta^{\prime})$$
$$\displaystyle=$$
$$\displaystyle G^{>}_{k}(\eta,\eta^{\prime})$$
$$\displaystyle G^{+-}_{k}(\eta,\eta^{\prime})$$
$$\displaystyle=$$
$$\displaystyle G^{<}_{k}(\eta,\eta^{\prime})~{},$$
(46)
where
$$\displaystyle G^{>}_{k}(\eta,\eta^{\prime})$$
$$\displaystyle=$$
$$\displaystyle iU_{k}(\eta)U^{*}_{k}(\eta^{\prime})$$
$$\displaystyle G^{<}_{k}(\eta,\eta^{\prime})$$
$$\displaystyle=$$
$$\displaystyle iU^{*}_{k}(\eta)U_{k}(\eta^{\prime})~{}.$$
(47)
One can of course also define $G^{>}(x,x^{\prime})$, $G^{<}(x,x^{\prime})$ from
which $G^{>}_{k}(\eta,\eta^{\prime})$, $G^{<}_{k}(\eta,\eta^{\prime})$ can obtained
by a Fourier transform.
From Dyson’s equation eq.(41) and eq.(44), one
finds that eq.(43) yields [10]
$$\left<0\right|\mathcal{O}\left|0\right>=\frac{\left<0\right|T\left\{\mathcal{O%
}~{}e^{-i\int_{-\infty}^{0}d\eta\left[H_{I}(\phi_{c},\psi^{+})-H_{I}(\phi_{c},%
\psi^{-})\right]}\right\}\left|0\right>}{\left<0\right|T\left\{e^{-i\int_{-%
\infty}^{0}d\eta\left[H_{I}(\phi_{c},\psi^{+})-H_{I}(\phi_{c},\psi^{-})\right]%
}\right\}\left|0\right>}~{},$$
(48)
if the initial state is the vacuum state $\left|0\right>$. Since we
absorbed the step function $\Theta(\eta-\eta_{infl})$ in $H_{I}$,
the time integral effectively have $\eta_{infl}$ as lower limit.
3.2 Hartree approximation
In appendix A, we have expanded the action for the inflaton field
fluctuations to fourth order. From the action given in
eq.(74) and eq.(75), it is simple to compute the
interaction Hamiltonian. Like in section 2, we shall constrain
ourself for simplicity, to consider only interactions that do not
contain space derivatives. These are the interactions that we
naively expect can give a large one loop contribution after a long
period of inflation. After using the results in eq.(74),
eq.(75), some partial integrations, and applying the linear
perturbation equation in eq.(2), we obtain with
$\psi\equiv\delta\phi$, that the effective interaction Hamiltonian
in the present approximation can be given approximately as
$$\displaystyle H_{I}(\phi_{c},\psi^{\pm})$$
$$\displaystyle\simeq$$
$$\displaystyle\int\frac{d^{3}y}{\eta^{4}H^{4}}\left[\psi^{\pm}\left(\phi^{%
\prime\prime}_{c}+2\mathcal{H}\phi_{c}^{\prime}+\lambda\phi_{c}^{3}\right)\right.$$
(49)
$$\displaystyle\qquad+\left.2\lambda\phi_{c}{\psi^{\pm}}^{3}(y)+\frac{15}{4}%
\lambda{\psi^{\pm}}^{4}(y)\right]~{},$$
in order to estimate the one loop correction to the two-point
function of the inflaton field fluctuations. Our equations are
consistent with section two. In fact, to first order in $\lambda$
one can verify that the tadpole renormalization condition
[10],
$$\displaystyle 0$$
$$\displaystyle=$$
$$\displaystyle\left<\psi^{\pm}(x)\right>_{0}$$
(50)
$$\displaystyle=$$
$$\displaystyle-\int_{-\infty}^{\eta_{0}}d\eta\int\frac{d^{3}\vec{y}}{\eta^{4}H^%
{4}(\eta)}\left[\left(G^{>}(x,y)-G^{<}(x,y)\right)\left(\phi^{\prime\prime}_{c%
}+2\mathcal{H}\phi_{c}^{\prime}+\lambda\phi_{c}^{3}-i6\lambda\phi_{c}G^{>}(y,y%
)\right)\right]~{}.~{},$$
yields the effective one loop equation of motion in
eq.(16).
The two-point function to one loop order can be organized in terms
of contributions to zero $T^{(0)}$, first $T^{(1)}$, second
$T^{(2)}$ and second order in $\lambda$,
$$\left<\psi^{+}(\eta_{0},\vec{x}_{1}),\psi^{+}(\eta_{0},\vec{x}_{2})\right>=T^{%
(0)}(\eta_{0},|\vec{x}_{1}-\vec{x}_{2}|)+T^{(1)}(\eta_{0},|\vec{x}_{1}-\vec{x}%
_{2}|)+T^{(2)}(\eta_{0},|\vec{x}_{1}-\vec{x}_{2}|)~{},$$
(51)
where $T^{(0)}$ is the lowest order free tree-level contribution
to the two-point function. The first order contribution in
$\lambda$ receives a contribution
$$\displaystyle T^{(1)}(x_{1},x_{2})$$
$$\displaystyle=$$
$$\displaystyle i\int_{-\infty}^{\eta_{0}}d\eta\int\frac{d^{3}\vec{y}}{\eta^{4}H%
^{4}(\eta)}\left(G^{>}(x_{1},y)G^{>}(x_{2},y)-G^{<}(x_{1},y)G^{<}(x_{2},y)\right)$$
(52)
$$\displaystyle\times 45\lambda G^{>}(y,y)~{}.$$
To second order in $\lambda$ there is two one loop contributions
$$T^{(2)}(x_{1},x_{2})=T^{(2)}_{1}(x_{1},x_{2})+T^{(2)}_{2}(x_{1},x_{2})~{},$$
(53)
and three two-loop contributions $\tilde{T}_{i}(x_{1},x_{2})$. These
diagrams are given in appendix D. Here we will focus on the
contributions to linear order in $\lambda$.
It is convenient to define the Fourier transform of the diagrams,
which is relevant when computing the corrections to the power
spectrum
$$T^{(i)}(x_{1},x_{2})=\int\frac{d^{3}k}{(2\pi)^{3}}e^{i\vec{k}\cdot(\vec{x}_{1}%
-\vec{x}_{2})}T^{(i)}(\eta_{0},k)~{}.$$
(54)
In this notation the one loop corrected power spectrum becomes
$$\mathcal{P}(\eta_{0},k)=\frac{k^{3}}{2\pi^{2}}\sum_{i}T^{(i)}(\eta_{0},k)~{}.$$
(55)
Let us evaluate specifically the diagram which is first order in
$\lambda$. That is the seagull diagram $T^{(1)}(x_{1},x_{2})$
shown in fig.(2). In Fourier space we obtain
$$T^{(1)}(\eta_{0},k)=-2\int_{\eta_{infl}}^{\eta_{0}}\frac{d\eta}{\eta^{4}H^{4}}%
\textrm{Im}\left[G^{>}_{k}(\eta_{0},\eta)G^{>}_{k}(\eta_{0},\eta)\right]\times
4%
5\lambda\int\frac{d^{3}k^{\prime}}{(2\pi)^{3}}G^{>}_{k^{\prime}}(\eta,\eta)$$
(56)
To compute the seagull diagram $T^{(1)}(\eta_{0},k)$ given by
$$T^{(1)}_{2}(\eta_{0},k)=-90\lambda\int_{\eta_{infl}}^{\eta_{0}}\frac{d\eta}{%
\eta^{4}H^{4}}\textrm{Im}\left[U_{k}^{2}(\eta)U_{k}^{*2}(\eta^{\prime})\right]%
\int\frac{d^{3}k^{\prime}}{(2\pi)^{3}}G^{>}_{k^{\prime}}(\eta,\eta)~{},$$
(57)
we have to understand the behavior of the integral. We will first
make some approximations, such that we can solve it analytically.
Since the physically relevant (observable) $k$-modes only spent
less than $60$ e-folds outside the horizon, as a first
approximation, we will take
$$U_{k}(\eta)=\frac{iH}{k\sqrt{2k}}(1+ik\eta)e^{-ik\eta}~{}.$$
(58)
In appendix C we have shown the computation, when the mode
function is taken to be given by a Hankel function with index
$\nu\neq 3/2$ and reproduced the correct scaling for the seagull
contribution. We have shown that we are making a very small error
above, when estimating the magnitude of the seagull contribution,
by approximating the mode function with the scale invariant one.
Changing variables to $x=-k\eta$ and letting $\eta_{infl}\to-\infty$, we obtain the following integral
$$\displaystyle T^{(1)}_{1}(\eta_{0},k)$$
$$\displaystyle=$$
$$\displaystyle-45\frac{\lambda}{k^{3}}\frac{H^{2}}{4\pi^{2}}\Delta_{N}\int_{x_{%
0}}^{\infty}\frac{dx}{x^{4}}\left[(-1+x^{2}+x_{0}^{2}-x_{0}^{2}x^{2}+4x_{0}x)%
\sin(2(x-x_{0}))\right.$$
(59)
$$\displaystyle\left.+2(-x_{0}+x_{0}x^{2}+x-x_{0}^{2}x)\cos(2(x-x_{0}))\right]~{}.$$
Since the integrand falls of as a power on sub-Hubble scales
$x>>1$, the dominant contribution to the integral is from
$x~{}\mbox{\raisebox{-2.58pt}{$\stackrel{{\scriptstyle<}}{{\sim}}$}}~{}1$ and since
$$\int\frac{d^{3}k^{\prime}}{(2\pi)^{3}}G^{>}_{k^{\prime}}(\eta,\eta)=\frac{H^{2%
}}{2\pi^{2}}\Delta_{N}~{},$$
(60)
is almost constant on super-Hubble scales for physically
observable $k$-modes, we have approximated it with a constant in
the integral in eq.(59). Now the integral is easy to solve
and it yields
$$\displaystyle T^{(1)}_{1}(\eta_{0},k)$$
$$\displaystyle=$$
$$\displaystyle-45\frac{\lambda}{k^{3}}\frac{H^{2}}{4\pi^{2}}\Delta_{N}\left\{%
\frac{20}{3}\right.$$
(61)
$$\displaystyle+\left[\frac{1}{3}x_{0}^{2}\pi-\frac{1}{3}\pi+\frac{2}{3}\textrm{%
Si}(2x_{0})-\frac{2}{3}x_{0}^{2}\textrm{Si}(2x_{0})-\frac{44}{3}x_{0}\textrm{%
Ci}(2x_{0})\right]\sin(2x_{0})$$
$$\displaystyle\left.+\left[-\frac{22}{3}x_{0}\pi+\frac{44}{3}x_{0}\textrm{Si}(2%
x_{0})-\frac{2}{3}x_{0}^{2}\textrm{Ci}(2x_{0})+\frac{2}{3}\textrm{Ci}(2x_{0})%
\right]\cos(2x_{0})\right\}~{}.$$
Taking the limit $x_{0}\to 0$, one finds
$$T^{(1)}(\eta_{0},k)\simeq-45\frac{\lambda}{k^{3}}\frac{H^{2}}{4\pi^{2}}\Delta_%
{N}\left[\frac{20}{3}+\frac{2}{3}\textrm{Ci}(2x_{0})\right]~{},$$
(62)
which shows that this one loop contribution to first order in
$\lambda$ is constant on super-Hubble scales up to a logarithmic
time-dependence, as we would have expected within this
approximation.
From eq.(55) we find on super-Hubble scales
$$\mathcal{P}(\eta_{0},k)\simeq\frac{H^{2}}{4\pi^{2}}\left[1-\frac{15}{\pi^{2}}%
\lambda\Delta_{N}\left(10+\textrm{Ci}(-2k\eta_{0})\right)+\mathcal{O}(\lambda^%
{2})\right]~{}.$$
(63)
To estimate the logarithmic correction to the power spectrum, we
can again use $H\simeq 10^{-5}M_{p}$, $\lambda\simeq 10^{-14}$,
$H_{i}^{4}~{}\mbox{\raisebox{-2.58pt}{$\stackrel{{\scriptstyle<}}{{\sim}}$}}~{}%
10^{-9}M_{p}^{4}$, which implies
$$\frac{45}{3\pi^{2}}\lambda\Delta_{N}=\frac{15\sqrt{3}}{16\pi^{2}}\sqrt{\lambda%
}\frac{H_{i}^{4}}{M_{p}H^{3}}~{}\mbox{\raisebox{-2.58pt}{$\stackrel{{%
\scriptstyle<}}{{\sim}}$}}~{}0.016~{}.$$
(64)
Again we find that within the present approximation, the one loop
correction to the power spectrum is maximally of order $1\%$, if
inflation is sufficiently long. We might also note again, that in the
limit $H_{i}\to H$ the contribution becomes negligible as expected.
We conclude that our results are consistent with a maximal effect
of a few percent.
4 Discussion
We have calculated the one loop corrections to a $\lambda\phi^{4}$
type of chaotic inflation. We found that in the effective field
theory the quantum corrections are not always suppressed by a
factor $H^{2}/M_{p}^{2}$ as generally expected. In scenarios with very
long inflation there can be a significant enhancement of the one
loop effects. This is because the phase-space volume of $IR$
modes, to be integrated over in the loops, grows with the total
number of e-folds of inflation. Hence, the loops are effectively
suppressed only by a factor of $H^{4}_{i}/(H^{2}M^{2}_{p})$, where $H_{i}$ is
the Hubble rate at the beginning of inflation. We estimated that
if inflation is very long, or equivalently the Hubble scale at the
beginning of inflation was very close to the Planck scale, the
effects can be as large as perhaps a few percent.
Since the effect comes from an integration of $IR$ modes in the
loop integrals, we expect that it looks similar to a redefinition
of the background parameters of the theory up to a small time
dependence. However, there is no reason a priori why a mass term
of the same magnitude as the one loop induced mass term should
present at tree-level. Such a mass term could affect the slow-roll
parameters and the cosmological scalar-to-tensor consistency
relation, and one can speculate about the theoretical possibility
of an observational indirect evidence for very long inflation.
One aspect which we find also deserves further exploration is if
there could be other new effects from an $IR$ enhancement of one
loop corrections. For instance if it could lead to an enhancement
of the effects found in
[4, 6, 7, 5, 9, 10]. It is as
well tempting to speculate about non-gaussianities, which will
also be enhanced by one loop effects in models with long
inflation. In an effective $\lambda\phi^{4}$ theory, we have to go
to second order in $\lambda$ to have a one loop integral that
contributes to the three-point function, so it will be suppressed
with a factor of $\lambda$ compared to the two-point seagull
contribution. Going to a $\lambda\phi^{6}$ type interaction, the
suppression is $H^{2}/M_{p}^{2}$ compared to the two-point seagull
contribution, from dimensional considerations.
Acknowledgments
I would like to thank Kari Enqvist, Massimo Giovannini,
Steen Hannestad, Nemanja Kaloper, Massimo Porrati and David Seery for
discussions and comments. Especially I would like to thank Nemanja
Kaloper for suggesting to explore loop effects in long inflation.
The work was supported in part by the DOE Grant DE-FG03-91ER40674.
Appendix A Inflaton perturbations in the ADM formalism
It is convenient to use the ADM formalism [34]
to derive the action for the inflaton perturbations. Let us
consider the scalar action of the inflaton field
$$S=\frac{1}{2}\int\sqrt{g}\left[R-(\partial\phi)^{2}-2V(\phi)\right]~{},$$
(65)
in the ADM metric, given by
$$ds^{2}=-\mathcal{N}^{2}dt^{2}+h_{ij}(dx^{i}+\mathcal{N}^{i}dt)(dx^{j}+\mathcal%
{N}^{j}dt)~{}.$$
(66)
In this metric, the action becomes [34]
$$S=\frac{1}{2}\int\sqrt{h}\left[\mathcal{N}R^{(3)}-2\mathcal{N}V+\mathcal{N}^{-%
1}\left(E_{ij}E^{ij}-E^{2}\right)+\mathcal{N}^{-1}\left(\dot{\phi}-\mathcal{N}%
^{i}\partial_{i}\phi\right)^{2}-\mathcal{N}h^{ij}\partial_{i}\phi\partial_{j}%
\phi\right]~{},$$
(67)
where
$$E_{ij}=\frac{1}{2}\left(\dot{h}_{ij}-\nabla_{i}\mathcal{N}_{j}-\nabla_{j}%
\mathcal{N}_{i}\right)~{}.$$
(68)
As mentioned in the introduction, we find it convenient to discuss
the effective action of the inflaton perturbations in the uniform
curvature gauge, where, when ignoring vector and tensor modes, we have
$$\phi=\phi_{c}+\delta\phi~{},\qquad h_{ij}=a^{2}\delta_{ij}~{},\qquad\mathcal{N%
}=1+\alpha~{},\qquad\mathcal{N}^{i}=\partial_{i}\chi~{}.$$
(69)
The strength of the ADM formalism, is that the constraint
equations are easily obtained by varying the action in $N$ and
$N_{i}$, which acts as Lagrange multipliers. In this way the
constraint equations in the uniform curvature gauge becomes
$$-a^{2}\delta^{ij}\partial_{i}\phi\partial_{j}\phi-2V-\mathcal{N}^{-2}\left(E_{%
ij}E^{ij}-E^{2}+\left(\dot{\phi}-\mathcal{N}^{i}\partial_{i}\phi\right)^{2}%
\right)=0~{},$$
(70)
and
$$\nabla_{j}\left[\mathcal{N}^{-1}\left(E^{i}_{j}-\delta^{i}_{j}E\right)\right]=%
\mathcal{N}^{-1}\left(\dot{\phi}-\mathcal{N}^{j}\partial_{j}\phi\right)%
\partial_{i}\phi~{}.$$
(71)
If we perturb the action by taking
$$\phi=\phi_{c}+\delta\phi~{},\qquad\alpha=\alpha_{1}+\alpha_{2}+\dots~{},\qquad%
\chi=\chi_{1}+\chi_{2}+\dots~{},$$
(72)
and solve the constraints equations order by order, one finds to
first order [35]
$$\alpha_{1}=\frac{1}{2}\frac{\dot{\phi}_{c}}{H}\delta\phi~{},\qquad\partial^{2}%
\chi_{1}=-\frac{1}{2}\frac{\dot{\phi}_{c}}{H}\dot{\delta\phi}-\frac{1}{2}\dot{%
\phi}_{c}\frac{\dot{H}}{H}\delta\phi+\frac{1}{2}\frac{\ddot{\phi}}{H}\delta%
\phi~{}.$$
(73)
Generally, in order to obtain the action to order $n$, we need
only to derive the constraint equations to order $n-1$, since the
$n$’th order terms multiplies the constraint equation to zero’th
order. In fact, it also turns out that to obtain the action to
third order in perturbations, one only needs the first order terms
in eq.(73), since $\alpha_{2}$, $\chi_{2}$ cancels out to leading
order in the slow-roll expansion. One obtains
$$\displaystyle S_{3}$$
$$\displaystyle=$$
$$\displaystyle\int a^{3}\left[-\frac{1}{4}\frac{\dot{\phi}_{c}}{H}\dot{\delta%
\phi}^{2}\delta\phi-\frac{a^{2}}{4}\frac{\dot{\phi}_{c}}{H}\delta\phi(\partial%
\delta\phi)^{2}-\dot{\delta\phi}\partial_{i}\chi_{1}\partial_{i}\delta\phi\right.$$
(74)
$$\displaystyle+\frac{3}{8}\frac{\dot{\phi}_{c}^{3}}{H}\delta\phi^{3}-\frac{1}{4%
}\frac{\dot{\phi}_{c}}{H}V^{\prime\prime}\delta\phi^{3}-\frac{1}{6}V^{\prime%
\prime\prime}\delta\phi^{3}+\frac{1}{4}\frac{\dot{\phi}_{c}^{3}}{H^{2}}\delta%
\phi^{2}\dot{\delta\phi}+\frac{1}{4}\frac{\dot{\phi}_{c}^{2}}{H}\delta\phi^{2}%
\partial^{2}\chi_{1}$$
$$\displaystyle~{}\left.+\frac{1}{4}\frac{\dot{\phi}_{c}}{H}\left(-\delta\phi%
\partial_{i}\partial_{j}\chi_{1}\partial_{i}\partial_{j}\chi_{1}+\delta\phi%
\partial^{2}\chi_{1}\partial^{2}\chi_{1}\right)\right]~{},$$
as first derived by Maldacena [35], and
subsequently generalized in
Ref. [36, 37, 38]. By going
one order further, we can in a similar fashion obtain the action
to fourth order in perturbations
$$\displaystyle S_{4}$$
$$\displaystyle=$$
$$\displaystyle\int a^{3}\left[\frac{1}{16}\frac{\dot{\phi}_{c}^{2}}{H^{2}}\dot{%
\delta\phi}^{2}\delta\phi^{2}-\frac{a^{2}}{16}\frac{\dot{\phi}_{c}^{2}}{H^{2}}%
\delta\phi^{2}(\partial\delta\phi)^{2}+\frac{1}{2}\frac{\dot{\phi}_{c}}{H}%
\delta\phi\dot{\delta\phi}\partial_{i}\chi_{1}\partial_{i}\delta\phi+\frac{1}{%
24}\frac{\dot{\phi}_{c}^{3}}{H^{2}}\delta\phi^{3}\partial^{2}\chi_{1}\right.$$
(75)
$$\displaystyle-\frac{15}{64}\frac{\dot{\phi}_{c}^{4}}{H^{2}}\delta\phi^{4}-%
\frac{1}{16}\frac{\dot{\phi}_{c}^{2}}{H^{2}}V^{\prime\prime}\delta\phi^{4}-%
\frac{1}{12}\frac{\dot{\phi}_{c}}{H}V^{\prime\prime\prime}\delta\phi^{4}-\frac%
{1}{24}V^{\prime\prime\prime\prime}\delta\phi^{4}+\frac{1}{2}(\partial_{i}\chi%
_{1})^{2}(\partial_{j}\delta\phi)^{2}$$
$$\displaystyle~{}\left.-\dot{\delta\phi}\partial_{i}\chi_{2}\partial_{i}\delta%
\phi-\frac{1}{16}\frac{\dot{\phi}_{c}^{2}}{H^{2}}\left((\partial^{2}\chi_{1})^%
{2}-(\partial_{i}\partial_{j}\chi_{1})^{2}\right)\delta\phi^{2}-\frac{1}{2}%
\left((\partial^{2}\chi_{2})^{2}-(\partial_{i}\partial_{j}\chi_{2})^{2}\right)\right.$$
$$\displaystyle\left.+\frac{\dot{\phi}_{c}}{2H}\left(\delta\phi\partial^{2}\chi_%
{1}\partial^{2}\chi_{2}-\delta\phi\partial_{i}\partial_{j}\chi_{1}\partial_{i}%
\partial_{j}\chi_{2}\right)-\partial^{2}\chi_{1}\partial^{2}\chi_{3}+\partial_%
{i}\partial_{j}\chi_{1}\partial_{i}\partial_{j}\chi_{3}\right.$$
$$\displaystyle\left.-\left(2H\partial^{2}\chi_{2}+\frac{1}{2}V^{\prime\prime}%
\delta\phi^{2}+\frac{1}{2}(\partial\delta\phi)^{2}+\frac{1}{4}\frac{\dot{\phi}%
_{c}^{2}}{H^{2}}V\delta\phi^{2}+VF(\delta\phi,\dot{\delta}\phi)\right.\right.$$
$$\displaystyle\left.\left.-\frac{1}{2}(\partial^{2}\chi_{1})^{2}+\frac{1}{2}(%
\partial_{i}\partial_{j}\chi_{1})^{2}+\frac{1}{2}\dot{\delta\phi}^{2}-\dot{%
\phi}_{c}\partial_{i}\chi_{1}\partial_{i}\delta\phi\right)F(\delta\phi,\dot{%
\delta}\phi)\right]~{},$$
valid up to total derivative terms. We also note that to leading
order in slow-roll $\alpha_{3}$ has cancelled out of the action. Above,
we also used the solution to the constraint equations to second
order,
$$\alpha_{2}=\frac{\dot{\phi}_{c}^{2}}{8H^{2}}+F(\delta\phi,\dot{\delta\phi})~{},$$
(76)
and
$$\displaystyle\partial^{2}\chi_{2}$$
$$\displaystyle=$$
$$\displaystyle\frac{3}{8}\frac{\dot{\phi}_{c}^{2}}{H}\delta\phi^{2}+\frac{3}{4}%
\frac{\ddot{\phi}_{c}}{H\dot{\phi}_{c}}\delta\phi^{2}-\frac{a^{2}}{4H}(%
\partial\delta\phi)^{2}-\frac{1}{4H}\dot{\delta\phi}^{2}+\frac{\dot{\phi}_{c}}%
{2H}\partial_{i}\chi_{1}\partial_{i}\delta\phi$$
(77)
$$\displaystyle+\frac{1}{4H}\left((\partial^{2}\chi_{1})^{2}-(\partial_{i}%
\partial_{j}\chi_{1})^{2}\right)-\frac{V}{H}F(\delta\phi,\dot{\delta\phi})~{},$$
where we have for convenience defined
$$F(\delta\phi,\dot{\delta\phi})=\frac{1}{2H}\partial^{-2}\left[\partial^{2}%
\alpha_{1}\partial^{2}\chi_{1}-\partial_{i}\partial_{j}\alpha_{1}\partial_{i}%
\partial_{j}\chi_{1}+\partial_{i}\dot{\delta\phi}\partial_{i}\delta\phi+\dot{%
\delta\phi}\partial^{2}\delta\phi\right]~{}.$$
(78)
The extra terms in the $S_{4}$ action involving $\alpha_{3}$, which cancels to leading order in slow-roll, are $(2\alpha_{1}\alpha_{2}-\alpha_{3})(4H\partial^{2}\chi_{1}+2\dot{\phi}_{c}\dot{%
\delta\phi})$, $-(V^{\prime}+\frac{\dot{\phi}_{c}}{H}V)\delta\phi\alpha_{3}~{}.$ By using $V$ to leading order in slow-roll we also neglected a term $(5/128)(\dot{\phi}_{c}^{6}/H^{4})\delta\phi^{4}$ in the action $S_{4}$, and in a similar fashion we have also neglected terms of higer order in slow-roll in the expression for $\chi_{2}$.
Appendix B IR scaling behavior
In this section of the appendix, we will evaluate the $IR$ part of
the correlator $\left(\delta\phi^{2}\right)_{0}$. We know that in the
chosen gauge $|U_{k}|=(\dot{\phi}/H)\mathcal{R}_{k}$, where
$\mathcal{R}_{k}=-\zeta_{k}$ is the gauge invariant curvature
perturbation, constant on super-Hubble scales. Thus, we can solve
the mode equations in pure de Sitter ($\epsilon=0$) and then evaluate
$\mathcal{R}_{k}$, but using the value of $H=H_{k}$ and
$\dot{\phi}=\dot{\phi}_{k}$ when the given mode crosses the horizon.
Since we know that $\mathcal{R}_{k}$ and $U_{k}$ scales in the same
way, we can first integrate over the power spectrum of
$\mathcal{R}_{k}$ and then relate that to the two-point function of
$\delta\phi$ at the end. In pure de Sitter on super-Hubble scales
$|U_{k}(\eta)|=H/2\pi$, so the curvature perturbation in this
approach can be written as
$$\mathcal{R}_{k}\simeq\frac{1}{2\pi}\left(\frac{H_{k}^{2}}{\dot{\phi}_{k}}%
\right)~{}.$$
(79)
If we specify to the specific $\lambda\phi^{4}$ model of chaotic
inflation, this implies that we obtain the right $k$ scaling from
$$U_{k}(\eta)=\left(\frac{H_{k}}{H(\eta)}\right)^{3/2}U^{ds}_{k}(\eta)=\frac{%
\sqrt{\pi}}{2}H\eta^{3/2}\left(\frac{H_{k}}{H(\eta)}\right)^{3/2}H_{3/2}^{(2)}%
(k\eta)~{},$$
(80)
where the mode solution in pure de Sitter was denoted by
$U^{ds}_{k}(\eta)$.
If we let $N=\ln(a/a_{i})$ denote the number of e-foldings of
expansions since the beginning of inflation, it is convenient to
recast the relevant integral into an integral over $N$. To do so,
we use the relations $d\ln k=dN$, $\ln(aH/k)=N$ and
$$H_{k}^{2}=\frac{\lambda}{12M_{p}^{2}}\left(\phi_{i}^{2}-64M_{p}^{2}N\right)^{2%
}~{}.$$
(81)
Now we can easily evaluate the dominant part of the $IR$
contribution to the relevant correlator
$$\displaystyle\left<\right.\left|\delta\phi(t,{\bf x})\right|^{2}\left.\right>_%
{IR}$$
$$\displaystyle\simeq$$
$$\displaystyle\frac{H^{2}}{4\pi^{2}}\int_{a_{i}H_{i}}^{aH}\frac{dk}{k}\left(%
\frac{H_{k}}{H}\right)^{3}$$
(82)
$$\displaystyle=$$
$$\displaystyle\frac{H^{2}}{4\pi^{2}}\left(\frac{\lambda}{12M_{p}^{2}H^{2}}%
\right)^{3/2}\int_{0}^{N}dN\left(\phi_{i}^{2}-64M_{p}^{2}N\right)^{3}$$
Using $N\simeq\phi_{i}^{2}/(64M_{p}^{2})$, we obtain
$$\displaystyle\left<\right.\left|\delta\phi(t,{\bf x})\right|^{2}\left.\right>_%
{IR}$$
$$\displaystyle\simeq$$
$$\displaystyle\frac{H^{2}}{4\pi^{2}}\left(\frac{\lambda}{12M_{p}^{2}H^{2}}%
\right)^{3/2}\frac{1}{256M_{p}^{2}}\phi_{i}^{8}$$
(83)
$$\displaystyle=$$
$$\displaystyle\frac{H^{2}}{16\pi^{2}}\left(\lambda\frac{1024}{3}\frac{M_{p}^{2}%
}{H^{2}}\right)^{3/2}N^{4}~{}.$$
As mentioned also in section (2.1), our results are consistent
with [15, 16, 17], when
comparison is possible.
Appendix C Seagull integral
Consider the seagull integral in eq.(57). Instead of using
the approximation in eq.(58), we want to use the exact
expression with the correct scaling behavior
$$U_{k}(\eta)=\frac{\sqrt{\pi}}{2}H(-\eta)^{3/2}H_{\nu}^{(2)}(-k\eta)~{}.$$
(84)
In the approximation where we can treat $H^{2}\Delta_{N}$ as constant,
the relevant integral becomes
$$\displaystyle I_{1}(\eta_{0},k)$$
$$\displaystyle=$$
$$\displaystyle\int\frac{d\eta}{\eta^{4}H^{4}}\textrm{Im}\left[U_{k}^{2}(\eta_{0%
})U_{k}^{*2}(\eta)\right]$$
(85)
$$\displaystyle=$$
$$\displaystyle\frac{\pi^{2}}{16}\int d\eta~{}\eta^{2}\textrm{Im}\left[H_{\nu}^{%
(2)}(-k\eta_{0})H_{\nu}^{(2)}(-k\eta_{0})H_{\nu}^{(1)}(-k\eta)H_{\nu}^{(1)}(-k%
\eta)\right]~{}.$$
It is useful to make a transformation to a dimensionless parameter
$x=-k\eta$, such that the integral becomes
$$I_{1}(x_{0},k)=-\frac{\pi^{2}}{16}\int dx~{}x^{2}\textrm{Im}\left[H_{\nu}^{(2)%
}(x_{0})H_{\nu}^{(2)}(x_{0})H_{\nu}^{(1)}(x)H_{\nu}^{(1)}(x)\right]~{}.$$
(86)
We know from our exact calculation in the case $\nu=3/2$, that in
the limit $x_{0}\to 0$, the dominant contribution to the integral
are a constant contribution from $x\simeq 1$ and a logarithmic
$x_{0}$ dependent contribution from $x<<1$. That means, that we can
obtain a good approximation to the integral by expanding the
Hankel function in the small argument limit, reproducing the $x_{0}$
dependent contribution to the integral, and add the constant
constant contribution computed in the $\nu=3/2$ case. In this way
we obtain
$$\displaystyle I_{1}(x_{0},k)$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{k^{3}}\frac{\pi^{2}}{8}\frac{2^{2\nu}}{\sin^{3}(\nu\pi)%
\Gamma^{3}(1-\nu)\Gamma(\nu+1)}\left(\frac{1}{3}-\frac{1}{3-2\nu}\right)x_{0}^%
{3-2\nu}+\frac{20}{12}~{},\qquad\textrm{for}\qquad\nu\neq 3/2~{},$$
$$\displaystyle I_{1}(x_{0},k)$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{k^{3}}\frac{-\pi^{2}}{8}\frac{2^{2\nu}}{\Gamma^{3}(1-\nu%
)\Gamma(\nu+1)}\left(\frac{1}{3}-\ln(x_{0})\right)+\frac{20}{12}~{},\qquad%
\textrm{for}\qquad\nu=3/2~{}.$$
(87)
In fig.(4), we have compared the exact solution for the integral
in the case $\nu=3/2$ with the approximate solution given in
eq.(87). We find that the approximation is reasonably
good on very super-Hubble scales $x_{0}<<10^{-3}$.
Appendix D One loop diagrams
The Bubble diagram, $T^{(2)}_{1}$ shown in fig.(4), is given by
$$\displaystyle T^{(2)}_{1}(k,\eta_{0})$$
$$\displaystyle=$$
$$\displaystyle 36\lambda^{2}\int_{-\infty}^{\eta_{0}}d\eta\int\frac{d^{3}y}{%
\eta^{4}H^{4}}\int_{-\infty}^{\eta_{0}}d\eta^{\prime}\int\frac{d^{3}y^{\prime}%
}{\eta^{\prime 4}H^{\prime 4}}\phi_{c}(\eta)\phi_{c}(\eta^{\prime})\left[\left%
(G^{>}(x_{1},y)-G^{<}(x_{1},y)\right)\right.$$
(88)
$$\displaystyle\times$$
$$\displaystyle\left.\left(G^{>}(x_{2},y^{\prime})G^{>}(y,y^{\prime})G^{>}(y,y^{%
\prime})-G^{<}(x_{2},y^{\prime})G^{<}(y,y^{\prime})G^{<}(y,y^{\prime})\right)%
\right]~{}.$$
The diagram $T^{(2)}_{2}$, shown in fig.(5), is
given by
$$\displaystyle T^{(2)}_{2}(k,\eta_{0})$$
$$\displaystyle=$$
$$\displaystyle 24(48\lambda)^{2}\int_{-\infty}^{\eta_{0}}d\eta\int\frac{d^{3}y}%
{\eta^{4}H^{4}}\int_{-\infty}^{\eta_{0}}d\eta^{\prime}\int\frac{d^{3}y^{\prime%
}}{\eta^{\prime 4}H^{\prime 4}}\phi_{c}(\eta)\phi_{c}(\eta^{\prime})$$
(89)
$$\displaystyle\times$$
$$\displaystyle\left[\left(G^{>}(x_{1},y)G^{>}(x_{2},y^{\prime})-G^{<}(x_{1},y)G%
^{<}(x_{2},y^{\prime})\right)\right.$$
$$\displaystyle\times$$
$$\displaystyle\left.\left(G^{>}(y,y^{\prime})G^{>}(y^{\prime},y^{\prime})-G^{<}%
(y,y^{\prime})G^{<}(y^{\prime},y^{\prime})\right)\right]~{}.$$
However, since the tadpole subgraph is canceled by the tadpole
renormalization condition, this diagram will also cancel out.
Finally there is three two loop diagrams contributing to second
order in $\lambda$. They are labeled $\tilde{T}^{(2)}_{1}$, $\tilde{T}^{(2)}_{2}$, $\tilde{T}^{(2)}_{3}$ in fig.(7), and are given below
$$\displaystyle\tilde{T}^{(2)}_{1}(k,\eta_{0})$$
$$\displaystyle=$$
$$\displaystyle 3(48\lambda)^{2}\int_{-\infty}^{\eta_{0}}d\eta\int\frac{d^{3}y}{%
\eta^{4}H^{4}}\int_{-\infty}^{\eta_{0}}d\eta^{\prime}\int\frac{d^{3}y^{\prime}%
}{\eta^{\prime 4}H^{\prime 4}}$$
(90)
$$\displaystyle\times$$
$$\displaystyle\left[\left(G^{>}(x_{1},y)G^{>}(x_{2},y^{\prime})-G^{<}(x_{1},y)G%
^{<}(x_{2},y^{\prime})\right)\right.$$
$$\displaystyle\times$$
$$\displaystyle\left.\left(G^{>}(y,y^{\prime})G^{>}(y,y^{\prime})G^{>}(y^{\prime%
},y^{\prime})-G^{<}(y,y^{\prime})G^{<}(y,y^{\prime})G^{<}(y^{\prime},y^{\prime%
})\right)\right]~{}.$$
$$\displaystyle\tilde{T}^{(2)}_{2}(k,\eta_{0})$$
$$\displaystyle=$$
$$\displaystyle 4(48\lambda)^{2}\int_{-\infty}^{\eta_{0}}d\eta\int\frac{d^{3}y}{%
\eta^{4}H^{4}}\int_{-\infty}^{\eta_{0}}d\eta^{\prime}\int\frac{d^{3}y^{\prime}%
}{\eta^{\prime 4}H^{\prime 4}}$$
(91)
$$\displaystyle\times$$
$$\displaystyle\left[\left(G^{>}(x_{1},y)G^{>}(y,y)-G^{<}(x_{1},y)G^{<}(y,y)%
\right)\right.$$
$$\displaystyle\times$$
$$\displaystyle\left.\left(G^{>}(x_{2},y^{\prime})G^{>}(y,y^{\prime})G^{>}(y^{%
\prime},y^{\prime})-G^{<}(x_{2},y^{\prime})G^{<}(y,y^{\prime})G^{<}(y^{\prime}%
,y^{\prime})\right)\right]~{}.$$
$$\displaystyle\tilde{T}^{(2)}_{3}(k,\eta_{0})$$
$$\displaystyle=$$
$$\displaystyle 4(48\lambda)^{2}\int_{-\infty}^{\eta_{0}}d\eta\int\frac{d^{3}y}{%
\eta^{4}H^{4}}\int_{-\infty}^{\eta_{0}}d\eta^{\prime}\int\frac{d^{3}y^{\prime}%
}{\eta^{\prime 4}H^{\prime 4}}$$
(92)
$$\displaystyle\times$$
$$\displaystyle\left[\left(G^{>}(x_{1},y)G^{>}(y,y^{\prime})-G^{<}(x_{1},y)G^{<}%
(y,y^{\prime})\right)\right.$$
$$\displaystyle\times$$
$$\displaystyle\left.\left(G^{>}(x_{2},y^{\prime})G^{>}(y,y^{\prime})G^{>}(y,y^{%
\prime})-G^{<}(x_{2},y^{\prime})G^{<}(y,y^{\prime})G^{<}(y,y^{\prime})\right)%
\right]~{}.$$
References
[1]
A. H. Guth,
Phys. Rev. D 23, 347 (1981);
A. D. Linde,
Phys. Lett. B 108, 389 (1982);
B 114, 431 (1982);
B 116, 335 (1982);
B 116, 340 (1982);
A. Albrecht and P. J. Steinhardt,
Phys. Rev. Lett. 48, 1220 (1982).
[2]
D. N. Spergel et al.,
arXiv:astro-ph/0603449.
[3]
S. Dodelson, W. H. Kinney and E. W. Kolb,
Phys. Rev. D 56, 3207 (1997)
[arXiv:astro-ph/9702166].
W. H. Kinney, E. W. Kolb, A. Melchiorri and A. Riotto,
Phys. Rev. D 69, 103516 (2004)
[arXiv:hep-ph/0305130];
D. H. Lyth and L. Alabidi,
arXiv:astro-ph/0510441.
H. J. de Vega and N. G. Sanchez,
arXiv:astro-ph/0604136.
[4]
R. H. Brandenberger and J. Martin,
Mod. Phys. Lett. A 16, 999 (2001)
[arXiv:astro-ph/0005432];
J. Martin and R. H. Brandenberger,
Phys. Rev. D 63, 123501 (2001)
[arXiv:hep-th/0005209];
J. C. Niemeyer,
Phys. Rev. D 63, 123502 (2001)
[arXiv:astro-ph/0005533];
T. Tanaka,
arXiv:astro-ph/0012431.
J. C. Niemeyer and R. Parentani,
Phys. Rev. D 64, 101301 (2001)
[arXiv:astro-ph/0101451];
A. A. Starobinsky,
Pisma Zh. Eksp. Teor. Fiz. 73, 415 (2001)
[JETP Lett. 73, 371 (2001)]
[arXiv:astro-ph/0104043];
S. Shankaranarayanan,
Class. Quant. Grav. 20, 75 (2003)
[arXiv:gr-qc/0203060];
R. H. Brandenberger and J. Martin,
Phys. Rev. D 71, 023504 (2005)
[arXiv:hep-th/0410223];
R. Easther, W. H. Kinney and H. Peiris,
JCAP 0505, 009 (2005)
[arXiv:astro-ph/0412613].
[5]
A. Kempf and J. C. Niemeyer,
Phys. Rev. D 64, 103501 (2001)
[arXiv:astro-ph/0103225];
S. F. Hassan and M. S. Sloth,
Nucl. Phys. B 674, 434 (2003)
[arXiv:hep-th/0204110].
[6]
C. S. Chu, B. R. Greene and G. Shiu,
Mod. Phys. Lett. A 16, 2231 (2001)
[arXiv:hep-th/0011241];
R. Easther, B. R. Greene, W. H. Kinney and G. Shiu,
Phys. Rev. D 64, 103502 (2001)
[arXiv:hep-th/0104102];
R. Easther, B. R. Greene, W. H. Kinney and G. Shiu,
Phys. Rev. D 67, 063508 (2003)
[arXiv:hep-th/0110226];
F. Lizzi, G. Mangano, G. Miele and M. Peloso,
JHEP 0206, 049 (2002)
[arXiv:hep-th/0203099];
R. Brandenberger and P. M. Ho,
Phys. Rev. D 66, 023517 (2002)
[AAPPS Bull. 12N1, 10 (2002)]
[arXiv:hep-th/0203119].
[7]
U. H. Danielsson,
Phys. Rev. D 66, 023511 (2002)
[arXiv:hep-th/0203198];
J. C. Niemeyer, R. Parentani and D. Campo,
Phys. Rev. D 66, 083510 (2002)
[arXiv:hep-th/0206149];
V. Bozza, M. Giovannini and G. Veneziano,
JCAP 0305, 001 (2003)
[arXiv:hep-th/0302184];
M. Giovannini,
Class. Quant. Grav. 20, 5455 (2003)
[arXiv:hep-th/0308066].
[8]
N. Kaloper, M. Kleban, A. E. Lawrence and S. Shenker,
Phys. Rev. D 66, 123510 (2002)
[arXiv:hep-th/0201158].
[9]
C. P. Burgess, J. M. Cline, F. Lemieux and R. Holman,
JHEP 0302, 048 (2003)
[arXiv:hep-th/0210233];
B. R. Greene, K. Schalm, G. Shiu and J. P. van der Schaar,
JCAP 0502, 001 (2005)
[arXiv:hep-th/0411217];
K. Schalm, G. Shiu and J. P. van der Schaar,
AIP Conf. Proc. 743, 362 (2005)
[arXiv:hep-th/0412288];
U. H. Danielsson,
Phys. Rev. D 71, 023516 (2005)
[arXiv:hep-th/0411172];
F. Nitti, M. Porrati and J. W. Rombouts,
Phys. Rev. D 72, 063503 (2005)
[arXiv:hep-th/0503247];
R. Easther, W. H. Kinney and H. Peiris,
JCAP 0508, 001 (2005)
[arXiv:astro-ph/0505426];
U. H. Danielsson,
arXiv:hep-th/0511273;
B. Greene, M. Parikh and J. P. van der Schaar,
arXiv:hep-th/0512243.
[10]
H. Collins, R. Holman and M. R. Martin,
Phys. Rev. D 68, 124012 (2003)
[arXiv:hep-th/0306028];
H. Collins and M. R. Martin,
Phys. Rev. D 70, 084021 (2004)
[arXiv:hep-ph/0309265];
H. Collins and R. Holman,
Phys. Rev. D 71, 085009 (2005)
[arXiv:hep-th/0501158];
H. Collins and R. Holman,
arXiv:hep-th/0507081.
[11]
S. Weinberg,
Phys. Rev. D 72, 043514 (2005).
[12]
A. Albrecht, N. Kaloper and Y. S. Song,
arXiv:hep-th/0211221;
U. H. Danielsson,
JCAP 0303, 002 (2003)
[arXiv:hep-th/0301182];
E. Keski-Vakkuri and M. S. Sloth,
JCAP 0308, 001 (2003)
[arXiv:hep-th/0306070].
[13]
A. Vilenkin,
Nucl. Phys. B 226, 504 (1983);
A. Vilenkin,
Nucl. Phys. B 226, 527 (1983).
[14]
L. R. W. Abramo and R. P. Woodard,
Phys. Rev. D 60, 044010 (1999)
[arXiv:astro-ph/9811430].
L. R. W. Abramo and R. P. Woodard,
Phys. Rev. D 60, 044011 (1999)
[arXiv:astro-ph/9811431].
L. R. Abramo and R. P. Woodard,
Phys. Rev. D 65, 063515 (2002)
[arXiv:astro-ph/0109272].
L. R. Abramo and R. P. Woodard,
Phys. Rev. D 65, 063516 (2002)
[arXiv:astro-ph/0109273].
[15]
V. F. Mukhanov, L. R. W. Abramo and R. H. Brandenberger,
Phys. Rev. Lett. 78, 1624 (1997)
[arXiv:gr-qc/9609026].
[16]
L. R. W. Abramo, R. H. Brandenberger and V. F. Mukhanov,
Phys. Rev. D 56, 3248 (1997)
[arXiv:gr-qc/9704037].
[17]
N. Afshordi and R. H. Brandenberger,
Phys. Rev. D 63, 123505 (2001)
[arXiv:gr-qc/0011075].
[18]
F. Finelli, G. Marozzi, G. P. Vacca and G. Venturi,
Phys. Rev. D 65, 103521 (2002)
[arXiv:gr-qc/0111035];
F. Finelli, G. Marozzi, G. P. Vacca and G. Venturi,
Phys. Rev. D 69, 123508 (2004)
[arXiv:gr-qc/0310086];
F. Finelli, G. Marozzi, G. P. Vacca and G. Venturi,
arXiv:gr-qc/0604081.
[19]
B. Losic and W. G. Unruh,
Phys. Rev. D 72, 123510 (2005)
[arXiv:gr-qc/0510078].
[20]
P. Martineau and R. H. Brandenberger,
“The effects of gravitational back-reaction on cosmological
Phys. Rev. D 72, 023507 (2005)
[arXiv:astro-ph/0505236].
[21]
C. H. Wu, K. W. Ng, W. Lee, D. S. Lee and Y. Y. Charng,
arXiv:astro-ph/0604292.
[22]
A. D. Linde,
Phys. Lett. B 175, 395 (1986).
[23]
A. S. Goncharov, A. D. Linde and V. F. Mukhanov,
Int. J. Mod. Phys. A 2, 561 (1987).
[24]
D. Boyanovsky, H. J. de Vega and N. G. Sanchez,
arXiv:astro-ph/0503669;
[25]
D. Boyanovsky, H. J. de Vega and N. G. Sanchez,
Phys. Rev. D 72, 103006 (2005)
[arXiv:astro-ph/0507596].
[26]
A. A. Starobinsky,
JETP Lett. 55 (1992) 489
[Pisma Zh. Eksp. Teor. Fiz. 55 (1992) 477].
[27]
J. A. Adams, B. Cresswell and R. Easther,
Phys. Rev. D 64, 123514 (2001)
[arXiv:astro-ph/0102236].
[28]
N. Kaloper and M. Kaplinghat,
Phys. Rev. D 68, 123522 (2003)
[arXiv:hep-th/0307016].
[29]
V. F. Mukhanov, H. A. Feldman and R. H. Brandenberger,
Phys. Rept. 215, 203 (1992).
[30]
J. C. Hwang,
Class. Quant. Grav. 11, 2305 (1994).
[31]
M. Sasaki,
Prog. Theor. Phys. 76, 1036 (1986).
V. F. Mukhanov,
Sov. Phys. JETP 67, 1297 (1988)
[Zh. Eksp. Teor. Fiz. 94N7, 1 (1988)].
[32]
J. S. Schwinger,
J. Math. Phys. 2, 407 (1961).
[33]
L. V. Keldysh,
Zh. Eksp. Teor. Fiz. 47, 1515 (1964)
[Sov. Phys. JETP 20, 1018 (1965)].
[34]
R. Arnowitt, S. Deser and C. W. Misner,
arXiv:gr-qc/0405109.
[35]
J. Maldacena,
JHEP 0305 (2003) 013
[arXiv:astro-ph/0210603].
[36]
P. Creminelli,
JCAP 0310 (2003) 003
[arXiv:astro-ph/0306122].
[37]
D. Seery and J. E. Lidsey,
JCAP 0506 (2005) 003
[arXiv:astro-ph/0503692].
[38]
D. Seery and J. E. Lidsey,
JCAP 0509 (2005) 011
[arXiv:astro-ph/0506056]. |
Accelerating and Improving AlphaZero Using Population Based Training
Ti-Rong Wu1, Ting-Han Wei1,3, I-Chen Wu1,2
1Department of Computer Science, National Chiao Tung University, Taiwan
2Pervasive Artificial Intelligence Research (PAIR) Labs, Taiwan
3Department of Computing Science, University of Alberta, Edmonton, Canada
{kds285, ting, icwu}@aigames.nctu.edu.tw
Abstract
AlphaZero has been very successful in many games. Unfortunately, it still consumes a huge amount of computing resources, the majority of which is spent in self-play. Hyperparameter tuning exacerbates the training cost since each hyperparameter configuration requires its own time to train one run, during which it will generate its own self-play records. As a result, multiple runs are usually needed for different hyperparameter configurations. This paper proposes using population based training (PBT) to help tune hyperparameters dynamically and improve strength during training time. Another significant advantage is that this method requires a single run only, while incurring a small additional time cost, since the time for generating self-play records remains unchanged though the time for optimization is increased following the AlphaZero training algorithm. In our experiments for 9x9 Go, the PBT method is able to achieve a higher win rate for 9x9 Go than the baselines, each with its own hyperparameter configuration and trained individually. For 19x19 Go, with PBT, we are able to obtain improvements in playing strength. Specifically, the PBT agent can obtain up to 74% win rate against ELF OpenGo, an open-source state-of-the-art AlphaZero program using a neural network of a comparable capacity. This is compared to a saturated non-PBT agent, which achieves a win rate of 47% against ELF OpenGo under the same circumstances.
Accelerating and Improving AlphaZero Using Population Based Training
Ti-Rong Wu1, Ting-Han Wei1,3, I-Chen Wu1,2
1Department of Computer Science, National Chiao Tung University, Taiwan
2Pervasive Artificial Intelligence Research (PAIR) Labs, Taiwan
3Department of Computing Science, University of Alberta, Edmonton, Canada
{kds285, ting, icwu}@aigames.nctu.edu.tw
Copyright © 2020,
Association for the Advancement of Artificial Intelligence (www.aaai.org).
All rights reserved.
Introduction
Up until recently, games such as chess, Go, and shogi had crucial roles as interesting and challenging measures of development in artificial intelligence research. DeepMind’s work, starting with AlphaGo (?), followed up by AlphaGo Zero (?), and culminating in AlphaZero (?), demonstrated that reinforcement learning can be a powerful tool in solving difficult problems, first with the help of human expert knowledge, then, without any human intervention.
While this family of algorithms were able to deal with the challenge posed by these benchmarks, interest in classical games research remains high. Starting from ELF OpenGo, a reimplementation of the AlphaGo Zero/AlphaZero algorithm, Facebook AI Research is also moving ahead with the more general ELF project (?), which is aimed at covering a wider range of games for reinforcement learning research. In late August, 2019, DeepMind also announced the OpenSpiel framework, with the goal of incorporating various games with different properties (?).
Given the continued interest in using games as a reinforcement learning environment, there are still issues that need to be resolved even with a powerful algorithm such as AlphaZero. First, AlphaZero training requires a significant amount of computing resources, at a scale that is prohibitively costly for smaller research teams. As an example, both DeepMind and Facebook AI research use several thousand GPUs to train their Go agents (?). Recently, Wu (?) tested and proposed several techniques that can accelerate AlphaZero training. First, the number of network outputs was increased, including multiple value outputs for different komi111In Go, komi is the number of points added to the second player to balance the game. values, and also having new outputs for ownership of board intersections (?). Additionally, Wu also proposed a new method for Monte-Carlo tree search (MCTS) exploration and variation of search parameters, among many other techniques which we will not cover in detail in this paper. While the acceleration factor was reported to be 50, the new techniques introduced a variety of hyperparameters, which were all set to specific values without further explanation.
This leads to the next issue that remains to be resolved for AlphaZero training, namely hyperparameter choice or design. Each hyperparameter configuration requires a significant amount of computing resource commitment before its effects on the trained agent can be observed. As an improvement on manually tuning hyperparameters through experience, Wang et al. (?) investigated designing hyperparameter configurations by sweeping each hyperparameter and evaluating their different combinations for 6x6 Othello. By comprehensively listing 12 hyperparameters, then testing each with three different values, they were able to arrive at some intuition on what good hyperparameter design entails in about 36 runs. However, considering its comprehensiveness and the fact that exactly one best hyperparameter configuration is needed ultimately, this method is inefficient in practice. Furthermore, hyperparameter tuning in this case is performed in an offline manner.
In this paper, we propose an online hyperparameter tuning method based on population based training (PBT) (?). We can then perform hyperparameter adjustment while the AlphaZero algorithm trains, saving precious computing resources. Another significant advantage of using PBT is that this method requires a single run only while incurring a small additional cost for the optimization and evaluation phases of AlphaZero training.
We test our PBT hyperparameter adjustment method on 9x9 and 19x19 Go, where the PBT method is tested against a baseline of 8 AlphaZero agents, each with its own hyperparameter configuration. We pick two hyperparameters to adjust dynamically, the learning rate and the value loss ratio. Judging by the win rate against the 8 baselines, the PBT method is able to achieve a higher win rate for 9x9 Go. For 19x19 Go, with PBT, we were able to obtain improvements (up to 74% win rate against ELF OpenGo) in playing strength from a saturated agent (about 47% win rate against ELF OpenGo). For the two hyperparameters, PBT is shown to be able to decay the learning rate accordingly, while also adjusting the value loss ratio dynamically during training.
Background
In this section, first, we review the family of AlphaZero-like algorithms. Second, we review the PBT method.
AlphaZero-Like Algorithms
Since AlphaGo successfully defeated the top human Go player in 2016 (?), DeepMind followed up with the algorithms AlphaGo Zero (?) and AlphaZero (?). The main breakthrough for these two successor algorithms is that they do not require any human expert knowledge or input, other than the basic rules of the relevant game. First, we review the AlphaGo Zero algorithm, then we briefly point out the differences between AlphaGo Zero and AlphaZero.
There are three phases during each iteration of AlphaGo Zero training: self-play, optimization, and evaluation. In the self-play phase, the current best network weights (as according to the evaluation phase, which we will discuss shortly) are used to generate self-play records via MCTS. The generated self-play records are then stored in a replay buffer. During the optimization phase, the algorithm samples random positional data from this replay buffer, and uses the sampled batch of data to update the network weights, such that:
•
the error between the output value $v$ and the sampled data’s ground truths $z$ is minimized, and
•
the similarity of the output policy $p$ (consisting of a probability distribution of all moves) and the sampled MCTS search probabilities $\pi$ is maximized.
More specifically, during optimization, the parameterized network $\theta$ is updated to minimize the loss
$$L=(z-v)^{2}-\pi^{\rm{T}}\log p+10^{-4}\|\theta\|^{2}.$$
(1)
where the last term is the L2 weight regularization.
For every 1,000 training steps during optimization, the network weights are saved as a checkpoint, each representing an agent that is capable of playing Go with different levels of ability. In the evaluation phase, the new checkpoint is evaluated against the current network. If the checkpoint is superior to the current network, namely, if it surpasses the current network by a win rate of 55% and above, it replaces the current network. For the next iteration, the best of all networks will be used to generate a new collection of self-play game records.
While AlphaGo Zero was designed to tackle Go, the AlphaZero algorithm (?) was proposed to generalize to other games, where DeepMind focused on the games of shogi and chess. AlphaZero shares most of the same routines as in AlphaGo Zero, with a few differences. We do not list the details completely, but instead focus on what is one of the most major differences in terms of implementation. In the AlphaZero algorithm, the evaluation phase is removed, and the self-play games are generated instead by the latest network, rather than the superior one, as described above.
Since the publication of AlphaGo Zero and AlphaZero, there have been numerous open-source projects that have tried to replicate their results. These include Facebook AI Research’s ELF OpenGo (?), the crowd-sourced LeelaZero (?), MiniGo (?), written by Google engineers, and KataGo (?), which was trained using resources from the company Jane Street. KataGo is of particular interest since it can accelerate training by a factor of 50. On the other hand, to achieve this acceleration, a significant number of techniques were used, introducing many new hyperparameters to the overall algorithm.
Population Based Training
Hyperparameter choice is highly critical to whether a neural network based approach to solving a problem succeeds or not. In many cases, the importance of picking the right hyperparameters is made even more apparent since a configuration can only be evaluated after a long period of network training. Despite this, hyperparameter choice often relies on human experience or computationally expensive search algorithms. Recently, population based training (PBT) was proposed to support online training, which adjusts hyperparameters dynamically (?). Thereafter, PBT was successfully applied to many problems, most notably on Quake III Arena Capture the Flag (?).
Using similar concepts to genetic algorithms, PBT works by training multiple neural networks with initially random hyperparameters. The entire population of networks pool information together to improve the hyperparameters, and concentrates more computational resources on the better-performing individuals. A straight-forward method that implements exploitation involves replacing a lower-performing network with a better-performing network by copying its hyperparameters directly. Similar to mutation in genetic algorithms, there are also several ways of changing current hyperparameter values to explore different configurations.
By constantly performing both exploitation and exploration as the overall network training proceeds, PBT ensures that each individual in the population can perform reasonably well while also guaranteeing that previously unseen configurations are attempted at a specific rate. As a result, PBT can commit computational resources towards more promising hyperparameter configurations, all while training proceeds normally.
We specifically mention four mechanisms here, two for exploitation and two for exploration. For exploitation, the T-test selection method involves randomly sampling a target from the population, and evaluating the means of the last 10 episodic rewards for both the network itself and the target. If the target outperforms the network, and it also satisfies Welch’s t-test, the network is replaced by the target’s parameters and hyperparameters. Next, the truncation selection method involves ranking all networks in the population, and replacing the bottom $20\%$ individuals with the top $20\%$.
For exploration, a method called perturb is used to randomly multiply hyperparameters by 0.8 or 1.2. The resample method uses a predefined prior probability distribution and resamples hyperparameters from it. Both exploration mechanisms take place after exploitation, where only the replaced individuals are eligible for exploration.
Our Method
We now present our method, which incorporates PBT into the AlphaZero algorithm.
In AlphaZero training, the self-play games are typically generated by a single agent with the latest network parameters. However, in our approach, self-play games are generated by a population of $P=16$ agents. This setting for $P$ is an attempt to follow the findings by Jaderberg et al. (?), where they tested a variety of values for $P$ for the Atari benchmark and found that $P>20$ begins to yield an improvement. In our case, $16$ is the closest power of 2 to $20$; for the remainder of this paper, we will use $P=16$.
In each iteration of self-play, we randomly choose 8 pairs from these 16 agents, where each agent uses its own latest network parameters. Namely, each agent will play with exactly one other agent in each iteration of self-play. Compared to the original AlphaZero method, where the games were played by a single agent, who acts both as Black and White, in each game we randomly choose the two agents in a pair so that they alternate as Black and White, to ensure better balancing between the two roles. In fact, diversity can be also increased in this way, since different agents may have their own playing styles. A possible advantage we expect is that the competition among different agents would make it easier to explore the weaknesses of other agents, relative to using a single agent. For simplicity, we avoid performing extra computations in each self-play iteration by generating $1/8$ of the total number of games for each pair. Namely, the total number of self-play games in our method will be equal to the original AlphaZero-like method.
Among the 16 agents, we use PBT to optimize the hyperparameters of the learning rate and the value loss ratio $x$. The value loss ratio $x$ indicates the ratio between the policy loss term and the value loss term. Namely, the parameterized network of agents are updated by minimizing the loss
$$L=x(z-v)^{2}-\pi^{\rm{T}}\log p+10^{-4}\|\theta\|^{2}.$$
(2)
Compared to the original AlphaZero algorithm, where only one agent was trained in the process, we train all 16 agents in parallel. In addition, in PBT, while different agents use their own sets of hyperparameters to generate self-play games, each agent uses all of the game records generated by all agents for optimization.
Since all 16 agents need to be trained following this method, an additional computation cost for optimization is incurred. More specifically, with $P=16$, the optimization computation cost will be 16 times the cost in the original AlphaZero algorithm. Fortunately, the optimization process tends to be much less costly compared to self-play. To illustrate this, we refer to the paper by ELF OpenGo in which it is stated that 2000 GPUs were used for self-play, while only 8 GPUs were needed for optimization; as a comparison, with half as many simulations, the original AlphaZero training involved 5000 TPUs for self-play and 64 TPUs for optimization (?). For this reason, the cost of 16 times as much optimization computations only incurs a relatively small additional time cost in the entire training scheme.
One of the key differences between AlphaZero and its predecessors such as AlphaGo and AlphaGo Zero is that it does not perform evaluation, and instead simply replaces the self-player with the newest optimized agent. In our method, to take account of the multiple number of agents, it is important to evaluate the strengths of all the agents, so that the weaker agents will be replaced by the stronger agents.
It is worth noting that agent strength is not a one dimensional measure and that cycles of strengths are possible, e.g. some agent A may win against B, B against C, and C against A. Ultimately, the goal is to train an agent that has a higher win rate against all other agents. Therefore, during evaluation, we use a round-robin tournament for these 16 agents, where each agent plays 6 games against every other agent, with alternating roles as Black and White for fair comparisons. Namely, during evaluation a total of 90 games are played for each agent, and 720 games in total for one iteration (two agents to a pair for a total of 8 pairs). In our experiments, the total number of self-play games is 5,000 for 9x9 Go and 10,000 for 19x19 Go. Thus, the additional cost for evaluation is also minor when compared with the amount of computation spent on generating self-play records. Altogether, the overhead for the cost of optimization and evaluation in PBT is relatively small.
To optimize hyperparameters using PBT, we choose the truncation selection method for the exploitation strategy and perturbation for the exploration strategy (see the Background section on PBT) as follows. First, we rank all agents by its win rate in evaluation. If the agent is in the bottom 20% of the population (namely, if the agent belongs to the bottom 3 when $P=16$), we simply replace them by copying the weights and hyperparameters from the top 20% of the population (one-to-one). Next, exploration occurs by perturbing the hyperparameters of the replaced networks by a factor of 1.2 or 0.8.
In AlphaGo Zero and AlphaZero, the ratio between the loss function for the policy and value networks were set to be equal without any further investigation. Given that PBT allows us to adjust weights online, we wanted to see whether these weights could also be adjusted accordingly.
Experiments
In this section, we present our experiment results, performed on 9x9 and 19x19 Go.
Experiments for 9x9 Go
For the 9x9 Go experiments, the network architecture consists of 3 residual blocks with 64 filters. In our experiment, for each baseline (following the AlphaZero algorithm, trained using the loss function as in Equation (2)), we run a total of 200 iterations, where each iteration contains a self-play phase with 5000 games and an optimization phase. That is, a total of 1,000,000 games are generated for each trained network. Note that the komi of 9x9 Go is 7, leading to the possibility that the outcome may be a draw.
We train a PBT with $P=16$ agents, and also 8 AlphaZero versions individually as baselines. Although the baseline versions only consist of 8 agents, which is less than PBT which has a total of 16 agents, the computation cost of training the baseline versions is almost 8 times that of PBT. Following the example set by AlphaGo Zero (?) and AlphaZero (?), we randomly initialized the network parameters and set different learning rates and value loss ratios222In fact, we trained Agents 1, 2, 5 and 6 first, both sets of which follow the AlphaZero hyperparameters, since they are highly regarded in the community. That is, Agents 1 and 2 follow the same settings, while 5 and 6 follow similar settings, but with a constant learning rate. We then halved/doubled the value loss ratio to explore different settings., as detailed in Table 1. Note that in Table 1, the learning rate changes after 100 iterations, separated by a comma. In addition, the listed values for PBT are only the initial values, where the hyperparameters will be changed dynamically during training.
To analyze the performance of the 8 baseline versions, we made a round-robin tournament. Every five iterations, the 8 baselines each play against the other 7 baselines, where each match-up consists of 100 games (for a total of 700 game per baseline). Figure 1 shows the minimum win rate for each agent; namely, after each agent plays against the other 7 agents, we simply depict the worst win rate in the figure. Figure 2 shows the average win rate for each agent since the minimum win rate does not give the full picture of the agent’s overall strength.
From these two figures, the result shows that overall agents 4 and 8 perform better than the others. Agent 4 especially stands out, as it performs around 60% minimum win rate and around 80% win rate during iterations 100 to 150. From this result, according to the hyperparameters, we observe that the larger value loss ratio of 2 might accelerate and improve training in a total of 200 iterations, while the smaller value loss ratios such as 0.5 perform the worst in this experiment, as is the case with agents 3 and 7. The learning rate schedules are also important, since agent 8 has the same value loss ratio with agent 4, but it performs worse than agent 4 after 100 iterations, which is likely the result of its static learning rate. Interestingly, no single agent can dominate all other agents; for certain iterations (say, around iteration 160 in Figure 1), we can even see how every agent’s minimum win rate is below 50%. This is a fitting example to illustrate circular strengths (i.e. how the strength of agents is not a one-dimensional scale) that we mentioned in regards to the evaluation phase in the Method section.
Next, we analyze the performance of training by the PBT method. To determine its relative strength to the baselines, for each iteration, we simply use the top agent (out of all 16 agents in the population) that was chosen in that iteration’s evaluation phase. The chosen agent then plays against all 8 baselines, where the results are shown in Figure 3. Note that in this figure the average values are depicted as the bolded black line. From the figure, we can see that with the exception of a few iterations where PBT performs slightly worse than one or two baselines (e.g. 45.5% win rate against agent 8 at iteration 175), its win rate exceeds 50% in most iterations. Again, we want to stress that a major benefit of using PBT is that it can outperform the 8 baselines without having to generate more self-play games, and consequently saving computation resources by only having to train once.
Figure 4 and Figure 5 show the trend of the hyperparameters adjusted by PBT. First, in Figure 4, the average learning rate starts from about 0.02, increase significantly (say, to a maximum of 0.03 at iteration 34), then drops rapidly to 0.015 at around iteration 50 because of the exploit mechanism. The average of the learning rate decrease gradually and reaches 0.003 at iteration 100. After 100 iterations, the learning rate stays at a value around 0.002 to 0.003, which continues up to iteration 200. Interestingly, the schedule seems to be similar to the hand-tuned schedule, as is the case for the baselines. This shows that PBT can adjust learning rates automatically.
Second, in Figure 5, the value loss ratio starts from 1, increases to an average of 7.5 around iteration 70, and then decreases to an average of 2 around iteration 120. Beyond that point, the value loss ratio fluctuates around 2 to 4. In this experiment, we can see that a sufficiently large value loss ratio is better for training, which corroborates our experience with the baseline experiments. While PBT is able to find the best value loss ratio to be around 2 to 4, without PBT we would need more than 8 experiments to figure this out. However, according to the experiments by Tian et al. (?) (referred to in the paper as the ”dominating value gradients” experiments), if the value loss ratio is too large (say, a value of 361 for 19x19 Go), the agent strength will hit a limitation. Fortunately, PBT offers a dynamic adjustment mechanism that can decrease the value loss ratio when it becomes a limiting factor on performance.
Another interesting observation is that the learning rate and the value loss ratio seem to complement each other in the early stages of training. While the learning rate increases around iterations 0 to 30, the value loss ratio maintains the same value. During iterations 30 to 60, the learning rate decreases, but the value loss ratio increases. Our conjecture is that PBT focuses on the hyperparameters that have a higher impact on performance. Consequently, the learning rate changes more significantly in the beginning stages of training, but then the algorithm ”shifts focus” and tunes the value loss ratio once the learning rate stabilizes.
In addition, we performed two ablation experiments to further analyze the benefit of the PBT method: (1) no perturbation with replacement; (2) no perturbation, no replacement. Since these two experiments are trained without perturbation, we initialized the hyperparameters of the 16 agents for diversity and fairness such that each setting entry in Table 1 will have 2 agents. Figure 6 shows the win rates for the two ablation methods against the PBT method. Between the two non-perturbed cases, the one with replacement performs slightly better than the without replacement. Although all agents used different initial hyperparameters, in the case with replacement, the agents soon converged to the same hyperparameters. More specifically, the value loss ratio converged to 2 after only 15 iterations. As a result, the training is equivalent to multi-agent training without diversity. Next, for the case with no perturbation and no replacement, the diversity of agents remains high throughout training. However, a full sweep for optimization would involve suboptimal agents that will never be replaced. Since each agent will train using the collection of self-play records generated by all agents, optimization may be negatively impacted by low-quality game records. Conclusively, these ablation experiments show that: (1) PBT performs well not only due to the diversity of agents (as is the case with no perturbation and no replacement); and that (2) PBT with perturbation and replacement can maintain diversity as well as strength, as shown in Figure 6.
Moreover, we performed additional experiments on multi-labelled value networks (MLVN) (?) using PBT. MLVN is a simple technique that was also included in KataGo, where it was referred to as the score belief network (?). The experiment results are similar to those for the single value output network and it shows that the PBT method can generalize for different network architectures.
Experiments for 19x19 Go
Next, we apply our method to our 19x19 Go program CGI (?). We made some changes between 9x9 and 19x19 training as follows. The network is expanded to consist of 20 residual blocks with 256 filters, the same size as ELF OpenGo v2. The output of the network consists of a policy and 31 value outputs, each corresponding to a different komi ranged from -7.5 to 22.5, centered at 7.5 komi. In each iteration, 10,000 games are generated via self-play.
In this experiment, we first train a network following the AlphaGo Zero algorithm, with learning rate decreasing from 0.01 to 0.0001 (following the scheduling by AlphaGo Zero (?)), and finally fine-tuning with a learning rate of 0.00005 after the network is saturated at 0.0001. Saturation in this context refers to the situation where millions of games of additional training no longer leads to improvement on win rates against ELF OpenGo v2.
Training on the chosen network stopped at saturation with a learning rate of 0.00005.
Next, with the saturated trained network, we apply the PBT method to try and improve the network’s performance beyond saturation. For the first part of the PBT training, we increase the learning rate to 0.0001 (from the previous saturated value of 0.00005) for the following reason. Since 0.00005 is already lower than all the published learning rates from current AlphaZero-related articles, we wish to leave some room so that perturbation of the learning rate will not lead to an unreasonably small learning rate. In addition, the ratio between the losses for policy and the multiple value outputs is set to be 0.2 (the same as the baseline version). In our 19x19 training, we start all 16 agents from the same network weights, but with perturbation before training starts to increase the population diversity.
Figure 7 shows the training curve. The results show that with a network (with the same size as ELF OpenGo v2), after 30 iterations the win rate is 71.2% against it, while the version without PBT showed no improvement after an additional 100 iterations of training. Moreover, after 92 iterations the PBT version reaches 74.0% win rate against ELF OpenGo v2.
Figure 8 and Figure 9 shows the hyperparameters that are adjusted by PBT in 19x19 Go. First, in Figure 8, the average values of the learning rate start from 0.0001, and gradually drops to 0.00005 after 80 iterations. Although a few agents use a larger learning rate of, say, 0.0002 in iteration 38, the result shows that the network is at saturation with the larger learning rate, and that improvement can only be achieved by a smaller value.
Second, the average movement of value loss ratio is shown in Figure 9. The value loss ratio increases gradually and reaches about 0.3 at iteration 40, where some agents use even larger ratios such as 0.45 for a while. However, after iteration 50, the average movement of the value loss weight drops to 0.2, and only slightly increases after iteration 85. It is interesting to see that the value loss weight changes during different training stages, and that the results are similar to 9x9 Go in the previous subsection. In AlphaZero training, the policy and value complement each other; a stronger policy tends to generate strong values in self-play, and stronger values will generate a stronger policy in MCTS. Thus, in our opinion, it is reasonable that the policy and the value loss weight will fluctuate during training since the agent should pay more attention to the policy or the value at different training stages. PBT therefore offers a dynamic approach to adjust these hyperparameters.
Conclusion
AlphaZero is a powerful reinforcement learning algorithm that is able to train super-human level agents for many different games without requiring human expert knowledge. However, AlphaZero algorithms often involve many hyperparameters, especially if we wish to accelerate the overall training process.
This paper shows that PBT is a promising method to help tune the hyperparameters, and in turn can be used to improve AlphaZero-like algorithms. Using PBT, by simply adjusting two hyperparameters, the learning rate and the ratio of value to policy loss, we were able to train a 19x19 Go program with a win rate of 74.0% against Facebook’s ELF OpenGo v2, a state-of-the-art open-source 19x19 Go program with 20 residual blocks. To our knowledge, our program, which incorporates PBT into AlphaZero, is state-of-the-art in playing strength among neural networks of a comparable capacity. Other open-source AlphaZero-like reimplementations including MiniGo and KataGo are all reported to be of similar playing strength as ELF OpenGo v2 with 20 residual blocks. This implies that PBT plays a crucial role in penetrating the performance ceiling of state-of-the-art 19x19 Go programs.
We also greatly reduce computing resource usage by leveraging PBT while reaching state-of-the-art performance. Since AlphaZero was first published, many open-source computer Go projects have attempted to reproduce and improve upon it. Unfortunately, training with AlphaZero consumes a tremendous amount of computing resources. As reported by (?), it requires 2000 GPUs (V100s) over 9 days for training a single run. With different hyperparameters settings (like the hyperparameter sweep as proposed in (?)), this requirement in computing resources will increase accordingly. In this paper, our method can reap the benfits of having a wide collection of hyperparameters, while only requiring a single run with a small extra overhead.
This paper is a simple demonstration that shows how PBT can adjust hyperparameters on-line. A future direction for investigation is applying PBT to a more comprehensive list of hyperparameters. This would include hyperparameters such as the loss ratio for auxiliary outputs (intersection ownership (?; ?), long-term prediction (?), etc.), the constant $c$ in PUCT (?), virtual loss during self-play, optimization batch size, L2 regularization term weighting, among others. While it is true that with more hyperparameters, the search space becomes larger, PBT traverses this search space in a more informed manner. It is worth emphasizing that the additional computation cost for using PBT is dependent on the population size, and not directly related to the number of hyperparameters that PBT aims to adjust.
Acknowledgments
This research is partially supported by the Ministry of Science and Technology (MOST) of Taiwan under Grant Number MOST 107-2634-F-009-011 and MOST 108-2634-F-009-011 through Pervasive Artificial Intelligence Research (PAIR) Labs. The computing resource is partially supported by National Center for High-performance Computing (NCHC) of Taiwan. The authors would like to thank anonymous reviewers for their valuable comments.
Supplementary Materials: Experiments for 9x9 Go Using Multi-Labelled Value Networks
We present an alternate set of experiments for 9x9 Go, where instead of the single value output, we now use the multi-labelled value network (MLVN) (?), which was also used to improve the performance in KataGo (?) (referred to there as the score belief network). These experiments were performed to see how well the PBT method generalizes for different network architectures.
The MLVN simply changes the single value output to multiple value outputs, each of which corresponds to a different komi in the game of Go. We use a total of 11 komi settings for the value output, from komi 2 to 12 (centered at komi 7). Since it becomes uncertain about the optimal value loss ratio for the multiple outputs, it is interesting to see how PBT will adjust the ratio, when compared with that with single value only.
Similar to the previous experiment, we train using PBT with 16 agents, along with 8 AlphaZero baselines. All versions use the MLVN technique, where the initial hyperparamter values are shown in Table 2. For agents 1, 5, and the PBT version, we set the value loss ratio as 1 (the default setting by Wu et al. (?)). For other baselines, the ratio is set to 0.1, 0.2, and 0.5 since with MLVN the total value output is 11 times the original single value output, so we attempt smaller value loss ratios.
We performed round-robin tournaments for the baselines, similar to the previous experiment. Figure 10 and Figure 11 show the minimum win rate and the average win rate for each agent. Surprisingly, agent 1, with the highest value loss ratio, performs much better than all other agents, with almost over 60% minimum win rate and over 70% average win rate compared to the other agents. On the other hand, agent 2 and 6, which have smaller value loss ratios, perform worse against all other agents. This result also corroborates the previous baseline experiment: larger value loss ratios with appropriate learning rate decay often leads to good training results.
The win rate of the PBT version against the 8 baselines, where all are trained with ML value networks, is shown in Figure 12. The result seems similar to the previous PBT result in the main text, Figure 3, except where PBT loses against agent 1 (the best performing baseline agent) during around iterations 100 to 200. However, near the end of training, PBT performs better than all 8 baseline versions. In fact, we also evaluate this PBT version against the 16 baselines, 8 for single value (described in the section of Experiments for 9x9 Go in the main text) and 8 for MLVN (in this supplementary). Similar to Figure 12, near the end of training, PBT still performs comparable or better than the 16 baseline versions.
Figure 13 shows the learning rate curve. The learning rate starts from 0.02 and increases to 0.35 at iteration 50. After iteration 50, the learning rate decreases gradually for the remainder of training. Figure 14 shows the value loss ratio curve. Similarly, the value loss increases in the beginning and decreases in the middle of training. The value loss ratio stabilizes around 0.3 to 0.5 after 100 iterations. Since we train 11 labels for the ML value, the real loss ratio between the policy and the value loss should correspond to about 1:3.3 to 1:5.5, which is also close to the previous result.
The results show that PBT can adjust to the suitable hyperparameters from different initial settings.
References
[2017]
Jaderberg, M.; Dalibard, V.; Osindero, S.; Czarnecki, W.; Donahue, J.; Razavi,
A.; Vinyals, O.; Green, T.; Dunning, I.; Simonyan, K.; Fernando, C.; and
Kavukcuoglu, K.
2017.
Population based training of neural networks.
ArXiv abs/1711.09846.
[2019]
Jaderberg, M.; Czarnecki, W. M.; Dunning, I.; Marris, L.; Lever, G.; Castaneda,
A. G.; Beattie, C.; Rabinowitz, N. C.; Morcos, A. S.; Ruderman, A.; et al.
2019.
Human-level performance in 3D multiplayer games with
population-based reinforcement learning.
Science 364(6443):859–865.
[2019]
Lanctot, M.; Lockhart, E.; Lespiau, J.-B.; Zambaldi, V.; Upadhyay, S.;
Pérolat, J.; Srinivasan, S.; Timbers, F.; Tuyls, K.; Omidshafiei, S.;
et al.
2019.
OpenSpiel: A framework for reinforcement learning in games.
arXiv preprint arXiv:1908.09453.
[2019]
Lee, B.; Jackson, A.; Madams, T.; Troisi, S.; and Jones, D.
2019.
Minigo: A case study in reproducing reinforcement learning research.
In RML@ICLR.
[2017]
Pascutto, G.-C.
2017.
Leela-zero.
https://github.com/gcp/leela-zero.
[2011]
Rosin, C. D.
2011.
Multi-armed bandits with episode context.
Annals of Mathematics and Artificial Intelligence
61(3):203–230.
[2016]
Silver, D.; Huang, A.; Maddison, C. J.; Guez, A.; Sifre, L.; Van Den Driessche,
G.; Schrittwieser, J.; Antonoglou, I.; Panneershelvam, V.; Lanctot, M.;
et al.
2016.
Mastering the game of Go with deep neural networks and tree search.
nature 529(7587):484–489.
[2017]
Silver, D.; Schrittwieser, J.; Simonyan, K.; Antonoglou, I.; Huang, A.; Guez,
A.; Hubert, T.; Baker, L.; Lai, M.; Bolton, A.; et al.
2017.
Mastering the game of Go without human knowledge.
Nature 550(7676):354.
[2018]
Silver, D.; Hubert, T.; Schrittwieser, J.; Antonoglou, I.; Lai, M.; Guez, A.;
Lanctot, M.; Sifre, L.; Kumaran, D.; Graepel, T.; et al.
2018.
A general reinforcement learning algorithm that masters chess, shogi,
and Go through self-play.
Science 362(6419):1140–1144.
[2015]
Tian, Y., and Zhu, Y.
2015.
Better computer go player with neural network and long-term
prediction.
arXiv preprint arXiv:1511.06410.
[2019]
Tian, Y.; Ma, J.; Gong, Q.; Sengupta, S.; Chen, Z.; Pinkerton, J.; and Zitnick,
C. L.
2019.
ELF OpenGo: an analysis and open reimplementation of alphazero.
In ICML.
[2019]
Wang, H.; Emmerich, M.; Preuss, M.; and Plaat, A.
2019.
Hyper-parameter sweep on alphazero general.
arXiv preprint arXiv:1903.08129.
[2018]
Wu, T.-R.; Wu, I.-C.; Chen, G.-W.; Wei, T.-h.; Wu, H.-C.; Lai, T.-Y.; and Lan,
L.-C.
2018.
Multilabeled value networks for computer Go.
IEEE Transactions on Games 10(4):378–389.
[2019]
Wu, D. J.
2019.
Accelerating self-play learning in Go.
ArXiv abs/1902.10565. |
FinMatcher at FinSim-2: Hypernym Detection in the Financial Services Domain using Knowledge Graphs
Jan Portisch
University of Mannheim, Data and Web Science GroupB 6, 26MannheimGermany43017-6221
jan@informatik.uni-mannheim.de
0000-0001-5420-0663
,
Michael Hladik
SAP SE Product Engineering Financial ServicesDietmar-Hopp-Allee 16WalldorfGermany
michael.hladik@sap.com
0000-0002-2204-3138
and
Heiko Paulheim
0000-0001-5420-0663
University of Mannheim, Data and Web Science Group B 6, 26MannheimGermany43017-6221
heiko@informatik.uni-mannheim.de
0000-0003-4386-8195
(2021)
Abstract.
This paper presents the FinMatcher system and its results for the FinSim 2021 shared task which is co-located with the Workshop on Financial Technology on the Web (FinWeb) in conjunction with The Web Conference. The FinSim-2 shared task consists of a set of concept labels from the financial services domain. The goal is to find the most relevant top-level concept from a given set of concepts.
The FinMatcher system exploits three publicly available knowledge graphs, namely WordNet, Wikidata, and WebIsALOD. The graphs are used to generate explicit features as well as latent features which are fed into a neural classifier to predict the closest hypernym.
financial services, knowledge graphs, wikidata, knowledge graph embeddings, RDF2vec, hypernymy detection
††journalyear: 2021††conference: Companion Proceedings of the Web Conference 2021; April 19–23, 2021; Ljubljana, Slovenia††booktitle: Companion Proceedings of the Web Conference 2021 (WWW ’21 Companion), April 19–23, 2021, Ljubljana, Slovenia††doi: 10.1145/3442442.3451382††isbn: 978-1-4503-8313-4/21/04††submissionid: FinW02imS††ccs: Theory of computation Semantics and reasoning††ccs: Information systems Web mining††ccs: Information systems Language models
1. Introduction
A hypernym or hyperonym is a concept which is superordinate to another one. In computer science, it is often represented as an IS-A relationship. For example, animal is a hypernym of cat and equity index is a hypernym of S&P 500 Index. A hyponym, on the other hand, is a concept which is subordinate to another one. For example, cat is a hyponym of animal and S&P 500 Index is a hyponym of equity index. (Murphy, 2003) Hypernymy detection can be broadly applied in real-world applications. The detection of hypernyms in the financial services domain is particularly interesting due to a domain specific vocabulary and a lack of publicly available domain-specific resources and concept representations.
The FinSim task models the hypernym detection task as a multi class classification problem: Given a concept label (i.e., the hyponym), the correct hypernym is to be found from a set of 10 mutually exclusive classes (i.e., hypernyms). A system participating in this task can return a sorted list of classes. The task is evaluated with two performance metrics: mean rank and accuracy.
The FinMatcher system uses two very broad publicly available knowledge graphs (Wikidata and WebIsALOD) as well as a small linguistic graph resource (WordNet).
A knowledge graph contains real world entities from various domains and the relationships that hold between them in a graph format (Paulheim, 2017).
The system presented in this paper calculates multiple explicit features and uses RDF2vec embeddings obtained from WebIsALOD. The features are concatenated into a feature vector which is presented to a neural classifier which was trained with the provided FinSim training data.
In the following section, related work is introduced. Afterwards, the provided dataset is quickly described.
In Section 4, the FinMatcher system is presented.
The results of FinSim task are given in Section 5 together with an ablation study. The paper is concluded in Section 6 where future research directions are also presented.
2. Related Work
2.1. Shared Tasks for Hypernym Detection
Hypernym discovery has been addressed before as challenge, for example at SemEval-2018 (Camacho-Collados
et al., 2018). Unique to the FinSim task is the focus on the financial services industry. The evaluation campaign premiered in 2020 (El Maarouf et al., 2021) and has been extended for the 2021 campaign, also referred to as FinSim-2 (Mansar
et al., 2021): Two additional tags have been introduced and the training and evaluation datasets have been extended.
2.2. Knowledge Graphs
FinMatcher uses three external knowledge graphs as background knowledge for the task of hypernym detection.
WordNet (Fellbaum, 1998) is a well known lexical resource. It is a database of English words grouped in sets which represent a particular meaning, called synsets; further semantic relations such as hypernymy also exist in the database. The resource is publicly available.111see https://wordnet.princeton.edu/download
Wikidata is a knowledge graph hosted by the Wikimedia Foundation which is publicly available222see https://www.wikidata.org/wiki/Wikidata:Main_Page and maintained by an open community. The graph contains class-like entities, such as “stock market index”, and also instance-like entities, such as “MSCI World”. An example for a Wikidata statement would be “MSCI World” instance of “stock market index”333see https://www.wikidata.org/wiki/Q1881843. The graph can be queried using SPARQL444see https://query.wikidata.org/.
A frequent problem that occurs when working with external background knowledge in the financial services domain is the fact that less common entities – so called long tail entities – are not contained within a knowledge base. The WebIsA (Seitner et al., 2016) database is an attempt to tackle this problem by providing a dataset which is not based on a single source of knowledge – like DBpedia (Lehmann
et al., 2015) – but instead on the whole Web: The dataset consists of hypernymy relations extracted from the Common Crawl555see http://commoncrawl.org/, a freely downloadable crawl of a significant portion of the Web. For the automated extraction, lexico-syntactic patterns similar to those presented by Hearst (Hearst, 1992) were used.
Like Wikidata, the graph contains class-like and instance-like concepts. A sample triple from the dataset is “zero-coupon bond” skos:broader “bond”666see http://webisa.webdatacommons.org/concept/zero-coupon_bond_.
The dataset is also available via a Linked Open Data (LOD) endpoint777see http://webisa.webdatacommons.org/ under the name WebIsALOD (Hertling and
Paulheim, 2017) – hence, it can be queried like Wikidata using SPARQL.
2.3. Knowledge Graph Embeddings
In recent years, latent representations have gained traction not only in natural language processing but also in other data science communities. RDF2vec (Ristoski et al., 2019) is a knowledge graph embedding approach which allows to obtain a latent representation for the elements of a knowledge graph, i.e. a vector, for each node and each edge in a graph. It applies the word2vec (Mikolov
et al., 2013a; Mikolov et al., 2013b) model to RDF data: Random walks are performed for each node and are interpreted as sentences. After the walk generation, the sentences are used as input for the word2vec algorithm. As a result, one obtains a vector for each word, i.e., a concept in the RDF graph. Multiple flavors of RDF2vec have been developed in the past such as biased walks (Cochez et al., 2017) or RDF2vec Light (Portisch
et al., 2020c).888For a good overview of the RDF2vec approach and its applications, refer to
http://www.rdf2vec.org/ The calculation of knowledge graph embeddings on large graphs can require a significant amount of resources. Therefore, KGvec2go999see http://www.kgvec2go.org (Portisch
et al., 2020b) provides pre-trained RDF2vec knowledge graph embeddings through a Web API as well as via download. For the system presented in this paper, a pre-trained embedding of WebIsALOD has been downloaded from KGvec2go.
Both, RDF2vec and WebIsALOD, have been used for integration tasks in the financial services domain before (Monych
et al., 2020; Portisch, 2018).
3. FinSim Dataset Description
The FinSim dataset consists of 614 hyponym-hypernym pairs. There are 10 class labels, i.e. hypernyms:
(1)
Equity Index
(2)
Credit Index
(3)
Bonds
(4)
Swap
(5)
Option
(6)
Funds
(7)
Future
(8)
MMIs
(9)
Stocks
(10)
Forward
The class labels presented above classify concepts not according to their features but instead according to their prototypical kind.
The distribution of class labels is not balanced. As shown in Figure 1, the distribution of labels follows a power-law with 286 entries for “equity index” and only 9 entries for “forward”. This is a challenging setting for multiple reasons: (i) the training dataset is comparatively small, (ii) the hypernyms are semantically very related, (iii) industry abbreviations are used, and (iv) there are textual overlaps. The FinSim-2 test dataset consists of 212 entries, the distribution of class labels is not known.
Compared to other evaluation campaigns where participants have to submit their implementations, such as the Ontology Alignment Evaluation Initiative (OAEI), participants of the FinSim task run their system on their own premises and submit the predictions made by their system.
4. System Description
The FinMatcher system combines explicit and latent features. In total, there are five groups of features which will be presented in the following. The overall architecture is shown in Figure 2.
4.1. Features
Word Overlap
The overlap between hyponym and class label is a strong signal for a match. An example would be “Supranational Bond” which is a “Bond”. As such constellations are relatively frequent in the provided dataset, the first feature vector encodes whether the label contains the class label. For this feature minimal text pre-processing is applied including lower-casing and removal of the plural suffix “s”. As this step is performed for each class label, a vector of length 10 is obtained. The overlap feature vector is displayed in green in Figure 2.
Wikidata Hypernym Lookup
Wikidata is a large general-purpose knowledge graph which is not tailored to the financial domain. Nonetheless, the data source contains many financial concepts and relations between them. For example, the concept “UCITS” can be linked to“Undertakings for Collective Investment in Transferable Securities” via the also known as label; due to the annotated relation subclass of, it is easily recognizable that “UCITS” is an “investment fund”.101010see https://www.wikidata.org/wiki/Q25323628 This notion is exploited in this set of features: A comprehensive linking mechanism from the MELT framework111111The Matching EvaLuation Toolkit (MELT) is a framework for ontology and instance matching (development, evaluation, visualization (Portisch
et al., 2020a)). However, components can also be exploited for other tasks. For a better overview, see https://github.com/dwslab/melt/ (Hertling
et al., 2019, 2020) is used to link classes (the hypernyms) as well as labels (the hyponyms) to Wikidata concepts and then relations $P31$ (instance of) and $P279$ (sublcass of) are followed up to two hops to evaluate whether the class label appears. Distant matches receive a lower signal strength which is calculated through the inverse hop-distance: A direct hypernym annotation (as in the UCITS example stated earlier) receives the value $\frac{1}{1}=1$ whereas a two-hop match would receive a value of $\frac{1}{2}=0.5$. As this step is performed for each class label, a vector of length 10 is obtained. The Wikidata lookup feature vector is displayed in blue in Figure 2.
WordNet Hypernym Lookup
The same exploitation approach chosen for Wikidata is applied on the WordNet graph: Hypernyms and hyponyms are linked into WordNet and then the inverse hop-distance is used as feature value. This is done for each class label that could be linked. The WordNet lookup feature vector is displayed in yellow in Figure 2.
WebIsALOD Hypernym Lookup
In a similar fashion to the Wikidata hypernym lookup, class labels as well as hyponym labels are linked to the WebIsALOD graph using a linker from the MELT framework. In this graph, there exists only one significant relation: skos:broader. For each hyponym, the broader concepts are obtained and it is checked whether the hypernym appears. Due to a high level of noise, the number of upwards hops is limited to 1. As this step is performed for each class label, a vector of length 10 is obtained. The WebIsALOD lookup feature vector is displayed in purple in Figure 2.
WebIsALOD RDF2vec Similarity
For the embedding feature, each class label as well as each hyponym label is linked again into the WebIsALOD knowledge graph. Each concept in WebIsALOD has an associated embedding vector $v\in I\!R^{200}$. For comparisons, the cosine similarity between the hyponym and the class label is calculated.
If the whole concept cannot be linked, multiple sub-concepts are detected and linked. Within this linking process, longer sub-concepts are favored. For example, the string “CDX Emerging Markets” cannot be directly linked – however, the longest substring that can be linked here is “Emerging Markets”; in addition, “CDX” can also be linked. Comparisons in such cases are performed as follows:
(1)
$$\frac{\sum^{I}_{i=0}\max^{J}_{j=0}(sim(v_{i},v_{j})}{|I|}$$
where $I$ represents the set of links of the hyponym, $J$ represents the set of links of the hypernym, $v_{i}$ and $v_{j}$ correspond to the vectors of the links and $sim$ refers to a similarity function. In this case, the cosine is used as similarity function.
As this step is performed for each class label, a vector of length 10 is obtained. The WebIsALOD lookup feature vector is displayed in salmon in Figure 2.
Feature Composition
Each of the features $i$ returns a signal vector $s_{i}\in I\!R^{10}$. All vectors are concatenated to form the final signal vector $S=\mathbin{\|}^{5}_{i=1}s_{i}$, which is used as input for the classifier.
4.2. Classifier
Due to the small total number of training examples, a very simple artificial neural network architecture has been chosen. It is configured with one fully connected layer of size 10 and mean squared error as loss. The network was trained with 100 epochs and a batch size of 25 on a consumer PC. The vector that is to be predicted is of size 10 and represents the one-hot-encoded class label. The neural network classifier performed best among the classifiers evaluated: Naïve Bayes, J48 decision trees, random forests, and a regression.
As the distribution of class labels is skewed (see Figure 1), we applied the synthetic minority oversampling technique (SMOTE) (Chawla
et al., 2002) to upsample underrepresented class labels. We experimentally chose 33% of the majority class total as the upsampling barrier; this means that if the majority class in the training split totals to 229 records, upsampling for class labels with less than $\frac{1}{3}*229=76$ records will be performed so that there are 76 records for the underrepresented class label.
5. Results
5.1. Results on the Training Data and Ablation Study
We evaluated our matching system by performing a stratified five-fold cross validation on the training data. We trained each ANN configuration 10 times and report the average results for accuracy and mean rank. We further performed an ablation study by training and evaluating the performance when leaving out each of the five feature groups. The results can be found in Table 1.
It is visible that the most important feature group in terms of accuracy is word overlap. This is not surprising given the high number of labels that contain the hypernym within their name (for example “green bonds” $\rightarrow$ “bonds”) and shows that it is sensible for the task at hand to combine explicit and latent features. The observation that the inclusion of the target label in the term is a significant signal has also been made in the last FinSim campaign (El Maarouf et al., 2021).
The negligible role of WordNet in terms of accuracy is also comprehensible since this particular external background knowledge dataset contains merely general-purpose class knowledge (such as “call option”) but no knowledge about instances (such as “MSCI EMU Index”). For the FinSim dataset, very large knowledge graphs that contain class as well as instance knowledge are more beneficial due to their higher concept coverage. However, the information in the knowledge graphs used also contain some redundancy, as can be observed in Table 1: leaving out a single knowledge graph does not significantly change the results.
To further analyze the contribution of the different signals, we plotted the weights of the input features. As the weight of each input neuron $s_{i}$ relates to label $i$, we can directly observe which features the trained model considers relevant to identify which label.
Table 2 shows the summed absolute weight per feature group. This allows to analyze the overall contribution of the individual feature group. Here, it is visible that the latent RDF2vec feature group receives the highest weight – higher than the word overlap group.
While the word overlap feature is important for the majority labels (equity index, credit index), it is not equally important for all labels and does not have the overall highest weight: Figure 3 shows the summed absolute weight per feature group and class label. The class labels are sorted in descending order by frequency. Here, it is visible that the word overlap has the highest contribution for the equity index as well as a high contribution for the credit index but low weights for the remaining minority classes.
5.2. Results on the Reference Data
FinMatcher participated only with one configuration and achieved an accuracy of 81.1% and a mean rank of 1.415 on the reference data below the expected scores from the training data shown in Table 1.
6. Conclusion
In this paper, we presented FinMatcher, a hypernym detection system for the financial services domain which exploits multiple knowledge graphs by combining explicit and latent features. We could show that the task can be addressed by including external knowledge in the form of knowledge graphs and that the combination of multiple graphs is overall beneficial.
In the future, we strive to improve the results through the inclusion of more advanced embedding techniques as well as the exploration of additional external datasets.
Acknowledgements.
We would like to thank the FinSim-2 organizers (Youness Mansar, Ismaïl El Maarouf, and Juyeon Kang) for compiling the data, conducting the evaluation campaign, and for promptly answering all questions.
References
(1)
Camacho-Collados
et al. (2018)
José Camacho-Collados,
Claudio Delli Bovi, Luis Espinosa Anke,
Sergio Oramas, Tommaso Pasini,
Enrico Santus, Vered Shwartz,
Roberto Navigli, and Horacio Saggion.
2018.
SemEval-2018 Task 9: Hypernym Discovery. In
Proceedings of The 12th International Workshop on
Semantic Evaluation, SemEval@NAACL-HLT 2018, New Orleans, Louisiana, USA,
June 5-6, 2018, Marianna Apidianaki,
Saif M. Mohammad, Jonathan May,
Ekaterina Shutova, Steven Bethard, and
Marine Carpuat (Eds.). Association for
Computational Linguistics, 712–724.
https://doi.org/10.18653/v1/s18-1115
Chawla
et al. (2002)
Nitesh V. Chawla, Kevin W.
Bowyer, Lawrence O. Hall, and W. Philip
Kegelmeyer. 2002.
SMOTE: Synthetic Minority Over-sampling
Technique.
J. Artif. Intell. Res.
16 (2002), 321–357.
https://doi.org/10.1613/jair.953
Cochez et al. (2017)
Michael Cochez, Petar
Ristoski, Simone Paolo Ponzetto, and
Heiko Paulheim. 2017.
Biased graph walks for RDF graph embeddings. In
Proceedings of the 7th International Conference on
Web Intelligence, Mining and Semantics, WIMS 2017, Amantea, Italy, June
19-22, 2017, Rajendra Akerkar,
Alfredo Cuzzocrea, Jannong Cao, and
Mohand-Said Hacid (Eds.). ACM,
21:1–21:12.
https://doi.org/10.1145/3102254.3102279
El Maarouf et al. (2021)
Ismail El Maarouf, Youness
Mansar, Virginie Mouilleron, and
Dialekti Valsamou-Stanislawski.
2021.
The finsim 2020 shared task: Learning semantic
representations for the financial domain. In
Proceedings of the Second Workshop on Financial
Technology and Natural Language Processing. 81–86.
Fellbaum (1998)
Christiane Fellbaum (Ed.).
1998.
WordNet: An Electronic Lexical
Database.
MIT Press, Cambridge,
Massachusetts.
Hearst (1992)
Marti A. Hearst.
1992.
Automatic Acquisition of Hyponyms from Large Text
Corpora. In 14th International Conference on
Computational Linguistics, COLING 1992, Nantes, France, August 23-28,
1992. 539–545.
https://www.aclweb.org/anthology/C92-2082/
Hertling and
Paulheim (2017)
Sven Hertling and Heiko
Paulheim. 2017.
WebIsALOD: Providing Hypernymy Relations Extracted
from the Web as Linked Open Data. In The Semantic
Web - ISWC 2017 - 16th International Semantic Web Conference, Vienna,
Austria, October 21-25, 2017, Proceedings, Part II
(Lecture Notes in Computer Science,
Vol. 10588), Claudia
d’Amato, Miriam Fernández,
Valentina A. M. Tamma, Freddy
Lécué, Philippe Cudré-Mauroux,
Juan F. Sequeda, Christoph Lange, and
Jeff Heflin (Eds.). Springer,
111–119.
https://doi.org/10.1007/978-3-319-68204-4_11
Hertling
et al. (2019)
Sven Hertling, Jan
Portisch, and Heiko Paulheim.
2019.
MELT - Matching EvaLuation Toolkit. In
Semantic Systems. The Power of AI and Knowledge
Graphs - 15th International Conference, SEMANTiCS 2019, Karlsruhe, Germany,
September 9-12, 2019, Proceedings (Lecture Notes in
Computer Science, Vol. 11702),
Maribel Acosta, Philippe
Cudré-Mauroux, Maria Maleshkova,
Tassilo Pellegrini, Harald Sack, and
York Sure-Vetter (Eds.). Springer,
231–245.
https://doi.org/10.1007/978-3-030-33220-4_17
Hertling
et al. (2020)
Sven Hertling, Jan
Portisch, and Heiko Paulheim.
2020.
Supervised ontology and instance matching with
MELT. In Proceedings of the 15th International
Workshop on Ontology Matching co-located with the 19th International Semantic
Web Conference (ISWC 2020), Virtual conference (originally planned to be in
Athens, Greece), November 2, 2020 (CEUR Workshop
Proceedings, Vol. 2788),
Pavel Shvaiko,
Jérôme Euzenat, Ernesto
Jiménez-Ruiz, Oktie Hassanzadeh, and
Cássia Trojahn (Eds.).
CEUR-WS.org, 60–71.
http://ceur-ws.org/Vol-2788/om2020_LTpaper6.pdf
Lehmann
et al. (2015)
Jens Lehmann, Robert
Isele, Max Jakob, Anja Jentzsch,
Dimitris Kontokostas, Pablo N. Mendes,
Sebastian Hellmann, Mohamed Morsey,
Patrick van Kleef, Sören Auer,
and Christian Bizer. 2015.
DBpedia - A large-scale, multilingual knowledge
base extracted from Wikipedia.
Semantic Web 6,
2 (2015), 167–195.
https://doi.org/10.3233/SW-140134
Mansar
et al. (2021)
Youness Mansar, Juyeon
Kang, and Ismaïl El Maarouf.
2021.
FinSim-2 2021 Shared Task: Learning Semantic
Similarities for the Financial Domain. In
Proceedings of The Web Conference 2021, Virtual
Edition.
Mikolov
et al. (2013a)
Tomas Mikolov, Kai Chen,
Greg Corrado, and Jeffrey Dean.
2013a.
Efficient Estimation of Word Representations in
Vector Space. In 1st International Conference on
Learning Representations, ICLR 2013, Scottsdale, Arizona, USA, May 2-4,
2013, Workshop Track Proceedings, Yoshua
Bengio and Yann LeCun (Eds.).
http://arxiv.org/abs/1301.3781
Mikolov et al. (2013b)
Tomas Mikolov, Ilya
Sutskever, Kai Chen, Gregory S. Corrado,
and Jeffrey Dean. 2013b.
Distributed Representations of Words and Phrases
and their Compositionality. In Advances in Neural
Information Processing Systems 26: 27th Annual Conference on Neural
Information Processing Systems 2013. Proceedings of a meeting held December
5-8, 2013, Lake Tahoe, Nevada, United States,
Christopher J. C. Burges,
Léon Bottou, Zoubin Ghahramani,
and Kilian Q. Weinberger (Eds.).
3111–3119.
http://papers.nips.cc/paper/5021-distributed-representations-of-words-and-phrases-and-their-compositionality
Monych
et al. (2020)
Michael Monych, Jan
Portisch, Michael Hladik, and Heiko
Paulheim. 2020.
DESKMatcher. In
Proceedings of the 15th International Workshop on
Ontology Matching co-located with the 19th International Semantic Web
Conference (ISWC 2020), Virtual conference (originally planned to be in
Athens, Greece), November 2, 2020 (CEUR Workshop
Proceedings, Vol. 2788),
Pavel Shvaiko,
Jérôme Euzenat, Ernesto
Jiménez-Ruiz, Oktie Hassanzadeh, and
Cássia Trojahn (Eds.).
CEUR-WS.org, 181–186.
http://ceur-ws.org/Vol-2788/oaei20_paper7.pdf
Murphy (2003)
Lynne Murphy.
2003.
Semantic relations and the lexicon :
antonymy, synonymy, and other paradigms.
Cambridge University Press,
Cambridge, UK New York. 2016–236 pages.
Paulheim (2017)
Heiko Paulheim.
2017.
Knowledge graph refinement: A survey of
approaches and evaluation methods.
Semantic Web 8,
3 (2017), 489–508.
https://doi.org/10.3233/SW-160218
Portisch (2018)
Jan Portisch.
2018.
Automatic schema matching utilizing hypernymy
relations extracted from the web.
Portisch
et al. (2020a)
Jan Portisch, Sven
Hertling, and Heiko Paulheim.
2020a.
Visual Analysis of Ontology Matching Results with
the MELT Dashboard. In The Semantic Web: ESWC
2020 Satellite Events - ESWC 2020 Satellite Events, Heraklion, Crete,
Greece, May 31 - June 4, 2020, Revised Selected Papers
(Lecture Notes in Computer Science,
Vol. 12124), Andreas
Harth, Valentina Presutti, Raphaël
Troncy, Maribel Acosta, Axel Polleres,
Javier D. Fernández, Josiane Xavier
Parreira, Olaf Hartig, Katja Hose, and
Michael Cochez (Eds.). Springer,
186–190.
https://doi.org/10.1007/978-3-030-62327-2_32
Portisch
et al. (2020b)
Jan Portisch, Michael
Hladik, and Heiko Paulheim.
2020b.
KGvec2go - Knowledge Graph Embeddings as a
Service. In Proceedings of The 12th Language
Resources and Evaluation Conference, LREC 2020, Marseille, France, May
11-16, 2020, Nicoletta Calzolari,
Frédéric Béchet,
Philippe Blache, Khalid Choukri,
Christopher Cieri, Thierry Declerck,
Sara Goggi, Hitoshi Isahara,
Bente Maegaard, Joseph Mariani,
Hélène Mazo, Asunción
Moreno, Jan Odijk, and Stelios
Piperidis (Eds.). European Language Resources
Association, 5641–5647.
https://www.aclweb.org/anthology/2020.lrec-1.692/
Portisch
et al. (2020c)
Jan Portisch, Michael
Hladik, and Heiko Paulheim.
2020c.
RDF2Vec Light - A Lightweight Approachfor
Knowledge Graph Embeddings. In Proceedings of the
ISWC 2020 Demos and Industry Tracks: From Novel Ideas to Industrial
Practice co-located with 19th International Semantic Web Conference (ISWC
2020), Globally online, November 1-6, 2020 (UTC)
(CEUR Workshop Proceedings,
Vol. 2721), Kerry L.
Taylor, Rafael S. Gonçalves, Freddy
Lécué, and Jun Yan (Eds.).
CEUR-WS.org, 79–84.
http://ceur-ws.org/Vol-2721/paper520.pdf
Ristoski et al. (2019)
Petar Ristoski, Jessica
Rosati, Tommaso Di Noia, Renato De
Leone, and Heiko Paulheim.
2019.
RDF2Vec: RDF graph embeddings and their
applications.
Semantic Web 10,
4 (2019), 721–752.
https://doi.org/10.3233/SW-180317
Seitner et al. (2016)
Julian Seitner, Christian
Bizer, Kai Eckert, Stefano Faralli,
Robert Meusel, Heiko Paulheim, and
Simone Paolo Ponzetto. 2016.
A Large DataBase of Hypernymy Relations Extracted
from the Web. In Proceedings of the Tenth
International Conference on Language Resources and Evaluation LREC 2016,
Portorož, Slovenia, May 23-28, 2016,
Nicoletta Calzolari,
Khalid Choukri, Thierry Declerck,
Sara Goggi, Marko Grobelnik,
Bente Maegaard, Joseph Mariani,
Hélène Mazo, Asunción
Moreno, Jan Odijk, and Stelios
Piperidis (Eds.). European Language Resources
Association (ELRA).
http://www.lrec-conf.org/proceedings/lrec2016/summaries/204.html |
EXCITON, SPINON AND SPIN WAVE MODES
IN AN EXACTLY SOLUBLE
ONE-DIMENSIONAL
QUANTUM MANY-BODY SYSTEM
Bill Sutherland and Rudolf A. Römer
Physics Department, University of Utah, Salt Lake City, UT 84112
(April 16, 1993)
Abstract
In this paper, we present the exact solution to a one-dimensional,
two-component, quantum many-body system in which like particles interact
with a pair potential $s(s+1)/{\rm sinh}^{2}(r)$, while unlike particles
interact with a pair potential $-s(s+1)/{\rm cosh}^{2}(r)$.
We first give a proof of integrability, then derive the coupled
equations determining the complete spectrum. All singularities occur in the
ground state when there are equal numbers of the two components;
we give explicit results for the ground state and low-lying states in this
case. For $s>0$, the system is an antiferromagnet/insulator, with
excitations consisting of a pair-hole–pair continuum, a two-particle
continuum with gap, and excitons with gaps.
For $-1<s<0$, the system has excitations consisting of a hole-particle
continuum, and a two-spin wave continuum, both gap-less.
pacs:
I The S-C Model
We present the exact solution to a one-dimensional, two-component,
quantum many-body system of considerable complexity. The two kinds of
particles are distinguished by a quantum number $\sigma=\pm 1$, which
may be thought of as either spin or charge.
The system is defined by the Hamiltonian
$$H=-\sum_{1\leq j\leq N}\frac{1}{2}\frac{\partial^{2}}{\partial x_{j}^{2}}+\sum%
_{1\leq j<k\leq N}v_{jk}(x_{j}-x_{k}),$$
(1)
where the pair potential is
$$v_{jk}(x)=s(s+1)\left[\frac{1+\sigma_{j}\sigma_{k}}{2{\rm sinh}^{2}(x)}-\frac{%
1-\sigma_{j}\sigma_{k}}{2{\rm cosh}^{2}(x)}\right]$$
(2)
We assume $s\geq-1$.
We call this the S-C model, for the sinh-cosh interaction.
Thus for $s>0$, like particles repel, while unlike
particles attract. When like particles are near, the repulsive potential
increases as $1/r^{2}$, while for large separations, both potentials decay
exponentially with a decay length we take as our length scale, and hence
unity. The potentials might usefully be thought of as a screened $1/r^{2}$
potential.
This system was first introduced by Calogero [1], who showed it to be
integrable. Sutherland [2] soon afterward showed that the system could
be exactly solved, and gave the solution for a single component system. He
further showed the Toda lattice to be the low-density limit, and was able
to take the classical limit to reproduce Toda’s celebrated results,
identifying the particle-hole excitations of the quantum system with the
soliton-phonon modes of the classical system. Our present solution for the
two-component system exploits in a fundamental way the integrability of the
system, so we first discuss this point.
II Integrability
For a classical system of $N$ one-dimensional particles, Lax [3],
Moser [4] and Calogero [5] have shown that for certain potentials
one can find two Hermitean $N\times N$ matrices $L$ and $A$ that obey the
Lax equation ${\rm d}L/{\rm d}t=i[A,L]$.
Thus $L$ evolves by a unitary transformation generated by $A$, and hence
$\det[L-\omega\openone]$ is a constant of motion.
Expanding the determinant in powers of $\omega$, we find $N$ integrals of
motion, i.e.,
$\det[L-\omega\openone]=\sum_{0\leq j\leq N}J_{j}(-\omega)^{N-j},j=1,\ldots,N.$
Further, these integrals have been shown to be in involution, and thus the
system is integrable.
Under a reasonable assumption for the form of the Lax $A$ and $L$ matrices,
Calogero has shown that the most general solution to the Lax equations is
given in terms of the Jacobi elliptic function ${\rm sn}(x|m)$, and by a
suitable choice of parameters, our Hamiltonian is included.
Calogero has also demonstrated that if one replaces the classical
dynamical variables with the corresponding quantum mechanical operators,
$\det[L-\omega\openone]$ is well defined with no ordering ambiguity, and
the quantum mechanical commutator ${[\!|}H,\det[L-\omega\openone]{|\!]}=0$.
Thus, the $J_{j}$ are still constants of motion.
Finally, Calogero showed that
${[\!|}\det[L-\omega\openone],\det[L-\omega^{\prime}\openone]{|\!]}=0$, and thus the
quantum system is also completely integrable.
To sum up the situation: Our system is completely integrable. For the
general classical system integrability tells us something concrete, namely
that the motion in terms of action-angle variables is on a torus. However,
for the general quantum system integrability seems to buy one almost nothing.
The exception is for those special cases which support scattering; i.e.,
systems which fly apart when the walls of the box are removed. In these
cases, in the distant past and future the Lax matrix $L$ approaches a
diagonal matrix with the momenta as diagonal elements, so that
$\det[L-\omega\openone]=\prod_{1\leq j\leq N}(p_{j}-\omega).$
Thus the individual momenta $p_{j}$ are conserved in a collision, and hence,
as emphasized by Sutherland [2], the wavefunction is given
asymptotically by Bethe’s ansatz. Sutherland has exploited this fact to
completely determine, in the thermodynamic limit, properties of systems
interacting by such potentials including our system. We stress that no
features of the proof of integrability are needed. All we need is to know
it to be integrable, by whatever method.
The proof of Calogero, however , is very difficult, and only briefly
sketched in the literature. For that reason, we now offer an alternative
proof of integrability based on a method of Shastry [6].
Let us write the Lax matrices as
$$\displaystyle A_{jk}$$
$$\displaystyle=$$
$$\displaystyle\delta_{jk}\sum_{l\neq j}\alpha^{\prime}_{jl}+(\delta_{jk}-1)%
\alpha^{\prime}_{jl},$$
(3)
$$\displaystyle L_{jk}$$
$$\displaystyle=$$
$$\displaystyle\delta_{jk}p_{j}+i(1-\delta_{jk})\alpha_{jk},$$
(4)
where
$$\alpha_{jk}=-s\left[\frac{1+\sigma_{j}\sigma_{k}}{2}{\rm coth}(x_{j}-x_{k})+%
\frac{1-\sigma_{j}\sigma_{k}}{2}{\rm tanh}(x_{j}-x_{k})\right]$$
(5)
Then if the two-body potential potential $v_{jk}$ is given by
$v_{jk}=\alpha_{jk}^{2}+\alpha^{\prime}_{jk}-s^{2}$, we find that the
quantum Lax equation ${[\!|}H,L{|\!]}=[L,A]$ is satisfied.
Here the first fancy commutator is a quantum mechanical commutator
between operators, while the second commutator is an ordered matrix
commutator, so that the equation above is really $N^{2}$ equations of the
form
$[H,L_{jk}]=\sum_{1\leq l\leq N}(L_{jl}A_{lk}-A_{jl}L_{lk})$.
This potential, however, is exactly the potential for our system.
We now observe that the Lax $A$ matrix has the following very important
property. Defining a vector $\zeta$ with $\zeta_{j}=1$, we see
$A\zeta=\zeta^{\dagger}A=0$. This allows us to construct constants
of motion by $I_{n}=\zeta^{\dagger}L^{n}\zeta$, since
$$\displaystyle{[\!|}H,I_{n}{|\!]}$$
$$\displaystyle=$$
$$\displaystyle\zeta^{\dagger}{[\!|}H,L_{n}{|\!]}\zeta$$
(6)
$$\displaystyle=$$
$$\displaystyle\zeta^{\dagger}\sum_{0<j<N-1}\left\{L^{j}{[\!|}H,L{|\!]}L^{N-1-j}%
\right\}\zeta$$
(7)
$$\displaystyle=$$
$$\displaystyle\zeta^{\dagger}\sum_{0<j<N-1}\left\{L^{j}[A,L]L^{N-1-j}\right\}\zeta$$
(8)
$$\displaystyle=$$
$$\displaystyle\zeta^{\dagger}\left\{AL^{N-1}-L^{N-1}A\right\}\zeta=0$$
(9)
By Jacobi’s relation for commutators, ${[\!|}I_{n},I_{m}{|\!]}$ is a
constant of motion, and since this is a system that supports scattering,
we see ${[\!|}I_{n},I_{m}{|\!]}\rightarrow 0$, and hence the system
is completely integrable.
III The Two-Body Problem
Having shown the system to be integrable, we then know the asymptotic
wavefunction to be of the Bethe ansatz form, and the only input needed for the
Bethe ansatz is the solution to the $2$-body problem. Such a solution is
given in Landau and Lifshitz, Quantum Mechanics [7].
We summarize the results below.
First, we discuss like particles. In terms of the relative coordinate
$r=x_{2}-x_{1}$, the potential is $s(s+1)/{\rm sinh}^{2}(r)$, and with
the relative momentum $k=(k_{2}-k_{1})/2$, the wave function in the
center of mass is given asymptotically as
$$\Psi(r)=\left\{\begin{array}[]{ll}r^{s+1}&r\rightarrow 0+,\\
e^{-ikr}+S(2k)e^{ikr}&r\rightarrow+\infty.\end{array}\right.$$
(10)
The scattering amplitude $S(k)$ is given by
$$S(k)=-\frac{\Gamma(1+ik/2)\Gamma(1+s-ik/2)}{\Gamma(1-ik/2)\Gamma(1+s+ik/2)}$$
(11)
This scattering does not rearrange the particles. For bosons (fermions) the
wavefunction must be (anti)symmetric, so the scattering amplitude for
transmision will be $\pm S(k)$.
In what follows, we will drop factors of $-1$ in the scattering amplitudes,
assuming that they are taken care of by either the choice of statistics of
the particles, the choice of quantum numbers as half-odd-integers, or the
choice of number of particles as even or odd.
Now, we discuss unlike particles, with the potential
$-s(s+1)/{\rm cosh}^{2}(r)$. The wave function in the center of mass is
given asymptotically as
$$\Psi(r)=\left\{\begin{array}[]{ll}e^{ikr}+R(2k)e^{-ikr}&r\rightarrow-\infty,\\
T(2k)e^{ikr}&r\rightarrow+\infty.\end{array}\right.$$
(12)
The reflection and transmission amplitudes are given as $R(k)=S(k)r(k)$,
$T(k)=S(k)t(k)$, where
$$\displaystyle r(k)$$
$$\displaystyle=$$
$$\displaystyle\frac{\sin\pi s}{\sin\pi(s+ik/2)},$$
(13)
$$\displaystyle t(k)$$
$$\displaystyle=$$
$$\displaystyle-\frac{\sin\pi ik/2}{\sin\pi(s+ik/2)}.$$
(14)
Also present are bound states, labeled by an index $m=1,2,\ldots,M$,
according to increasing energy. Thus the parity of the states is
$(-1)^{m-1}$, and even parity states have spin $0$, while odd parity
states have spin $1$.
Bound states appear as poles of the reflection and
transmission amplitudes, $R(k_{1}-k_{2})$ and $T(k_{1}-k_{2})$ on the
positive half of the imaginary axis, given by $k_{1,2}=k\pm i\kappa,k>0$.
The momentum and energy of such a bound state is $P=2k$, and
$E=k^{2}-\kappa^{2}$.
From the particular form of the reflection and transmission amplitudes,
we find $\kappa_{m}=s+1-m$, where $m=1,2,\ldots,M$, and $M(s)$ is the
largest integer less than $s$.
There are no bound states for $0\leq s\leq-1$. Threshhold values of $s$
are $s=0,1,2,\dots$, and at these values, the reflection amplitude vanishes.
At the bound state poles we also find
$r(2i\kappa_{m})/t(2i\kappa_{m})=(-1)^{m-1}$.
We call the bound states pairs.
IV Yang-Baxter Equations
We know the Yang-Baxter equations must hold, and we can verify this
explicitly. For a two-component system the Yang-Baxter equations are
equivalent to $r_{2}=r_{3}r_{1}+t_{3}r_{2}t_{1}$ and
$r_{3}t_{2}=r_{3}t_{1}+t_{3}r_{2}r_{1}$, where
$r_{1}=r(k_{1}-k_{2}),r_{2}=r(k_{1}-k_{3}),r_{3}=r(k_{2}-k_{3})$
etc. for $t_{j}$.
A degenerate situation occurs at a pole in $r_{3}$ and $t_{3}$,
when $k_{2}-k_{3}=2i\kappa_{m}$.
There, since $r_{3}/t_{3}=(-1)^{m-1}$, the equations become
$0=r_{1}+(-1)^{m-1}r_{2}t_{1}$ and $t_{2}=t_{1}+(-1)^{m-1}r_{2}r_{1}$,
where $r_{2,1}=r(k\pm i\kappa_{m})$, and etc. for $t_{j}$.
These relationships will be important when we calculate phase shifts.
V Phase Shifts
If a particle of type $m$ passes through a particle of type $m^{\prime}$, without
reflection, then we have a scattering amplitude
$\exp[-i\theta_{mm^{\prime}}(k_{1}-k_{2})]$ , and a corresponding phase shift
$\theta_{mm^{\prime}}(k)$.
These phase shifts are at the heart of the Bethe ansatz. Let us label the
unbound particle by $m=0$. Then we have found
$$\theta_{00}(k)=i\log\left[\frac{\Gamma(1+ik/2)\Gamma(1+s-ik/2)}{\Gamma(1-ik/2)%
\Gamma(1+s+ik/2)}\right]$$
(15)
As we explained, we will not include factors of $-1$ in the scattering
amplitudes, so $\theta_{00}(0)=0$.
In general $\theta_{mm^{\prime}}(k)=-\theta_{mm^{\prime}}(-k)=\theta_{m^{\prime}m}(k)$,
and we will find that we always have $\theta_{mm^{\prime}}(0)=0$.
Now, consider the scattering of a particle $k_{1}$ on a pair of two
particles with momenta $k_{2}\pm i\kappa_{m}$. Let $k=k_{1}-k_{2}$.
Then using the degenerate Yang-Baxter equations, we find for the
scattering amplitude
$$S(k+i\kappa_{m})S(k-i\kappa_{m})t(k+i\kappa_{m})=\exp[-i\theta_{0m}(k)].$$
(16)
Using the explicit forms, we can verify that $\theta_{0m}(k)$ is real
for $k$ real.
Finally, we view the scattering of a pair from a pair as the scattering of two
particles with momenta $k_{1}\pm i\kappa_{m}$ from a pair with
$k_{2}\pm i\kappa_{m^{\prime}}$. This gives us a net phase shift
$\theta_{mm^{\prime}}(k)=\theta_{0m^{\prime}}(k-i\kappa_{m})+\theta_{0m^{\prime%
}}(k+i\kappa_{m}).$
Again, using the explicit forms, we can verify that $\theta_{mm^{\prime}}(k)$
is real for $k$ real, and symmetric in $m,m^{\prime}$.
To summarize: We have $N_{\uparrow}$ particles with $\sigma=+1$, and
$N_{\downarrow}$ with $\sigma=-1$, for a total of
$N=N_{\uparrow}+N_{\downarrow}$.
Let us assume $N_{\uparrow}\geq N_{\downarrow}$. Further, pairs of up-down spins
bind into a variety of bound states, or pairs, labeled by $m$,
$m=1,\ldots,M(s)$. Let there be $N_{m}$ of each type. Then the number of
unbound particles is
$N_{0}=N-2\sum_{1\leq m\leq M}N_{m},$
We will call these simply particles from now on.
They would correspond to spinons/ions in the spin/charge picture.
Of these particles, we have $N_{-1}$ with spin down; let us call them
spin waves. Clearly
$N_{-1}=N_{\downarrow}-\sum_{1\leq m\leq M}N_{m},$
and $N_{-1}\leq N_{0}/2$.
We still must treat the dynamics of the spin waves, but since they are not
“real” particles, but only correlations in the quantum numbers of particles,
they have no momentum or energy directly. Thus, defining
$$\eta_{m}=\left\{\begin{array}[]{ll}0,&m=-1,\\
1,&m=0,\\
2,&m=1,2,\ldots,M(s)\end{array}\right.$$
(17)
then we can write the momentum and energy as
$$\displaystyle P$$
$$\displaystyle=$$
$$\displaystyle\sum_{-1\leq m\leq M}\eta_{m}\sum_{k_{m}}k_{m},$$
(18)
$$\displaystyle E$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{2}\sum_{-1\leq m\leq M}\eta_{m}\sum_{k_{m}}k_{m}^{2}-%
\sum_{1\leq m\leq M}N_{m}\kappa_{m}^{2},$$
(19)
VI Spin Waves
Since particle-pair and pair-pair pass through one another with only a
phase shift and no reflection, their interaction is in some sense trivial.
However, particles do scatter from particles with reflection, and their
interaction is not trivial.
We write the asymptotic wavefunction explicitly in the Bethe ansatz
form, and for now consider only the $N_{0}$ particles. We use the spin language,
so $\sigma_{z}(j)=\pm 1$ according to whether the $j$th particle in the
ordering $x_{1}<...<x_{N_{0}}$ has spin up or down.
A choice for all $\sigma_{z}(j)$ we denote simply by $\sigma$. Then
asymptotically the wave function is given by
$$\Psi(x|\sigma)\rightarrow\sum_{P}A(P|\sigma)\exp\left[i\sum_{1\leq j\leq N_{0}%
}x_{j}k_{Pj}\right].$$
(21)
The summation is over all the $N_{0}!$ permutations of the momenta.
We arrange the $A(P|\sigma)$ for fixed $P$ as a column vector $\xi(P)$.
Then the Yang-Baxter equations ensure that we can find a consistent set of
amplitudes $A(P|\sigma)$, by finding the simultaneous eigenvector of the
$N_{0}$ equations
$$e^{ik_{j}L}\prod_{1\leq n\leq N_{0}}S(k_{j}-k_{n})X_{j,j-1}\cdots X_{j,1}X_{j,%
N_{0}}\cdots X_{j,j+1}\xi(I)=\xi(I)$$
(22)
In this equation, the $X_{j,n}$ are operators given as
$$\displaystyle X_{j,n}$$
$$\displaystyle=$$
$$\displaystyle\frac{1+t(k)}{2}\openone+\frac{1-t(k)}{2}\sigma_{z}(j)\sigma_{z}(n)$$
(23)
$$\displaystyle+\frac{r(k)}{2}\left[\sigma_{x}(j)\sigma_{x}(n)+\sigma_{y}(j)%
\sigma_{y}(n)\right],$$
where $k=k_{P_{j}}-k_{P_{n}}$.
These eigenvalue equations can in turn be solved by a Bethe ansatz for the
$N-1$ overturned spins — the spin waves — on a lattice of $N_{0}$ particles.
These equations can be solved either: (i) directly, by the methods of
Yang [8]; (ii) with commuting transfer matrices, by the methods of
Baxter [9]; or (iii) by quantum inverse scattering methods of Faddeev
and Takhtajan [10]. We are not aware that these equations have appeared
before in the solution of a quantum many-body problem, although the
low-density case has often appeared, for instance first in Yang’s original
solution for $\delta$-function fermions.
The solution is sufficiently technical that we postpone discussion to a later
publication. However, the result has many interesting physical consequences,
and that is what we want to discuss in this letter. One finds for the
eigenvalues of the previous equations,
$$e^{ik_{j}L}\prod_{1\leq n\leq N_{0}}S(k_{j}-k_{n})\prod_{1\leq q\leq N_{-1}}%
\frac{\sin\pi[s-i(k_{j}-\lambda_{q})]/2}{\sin\pi[s+i(k_{j}-\lambda_{q})]/2}=1$$
(24)
In this equation, the $\lambda$’s are the momenta of the spin waves, and are
determined from the equation
$$\prod_{1\leq q\leq N_{-1}}\frac{\sin\pi[s+i(\lambda_{j}-\lambda_{q})/2]}{\sin%
\pi[s-i(\lambda_{j}-\lambda_{q})/2]}\prod_{1\leq n\leq N_{0}}\frac{\sin\pi[s-i%
(\lambda_{j}-k_{n})]/2}{\sin\pi[s+i(\lambda_{j}-k_{n})]/2}=1$$
(25)
We now have our two final phase shifts, for particle-spin wave and spin
wave-spin wave
scattering:
$$\displaystyle\theta_{0,-1}(k)$$
$$\displaystyle=$$
$$\displaystyle i\log\left[\frac{\sin\pi[s-ik]/2}{\sin\pi[s-ik]/2}\right],$$
(26)
$$\displaystyle\theta_{-1,-1}(k)$$
$$\displaystyle=$$
$$\displaystyle i\log\left[\frac{\sin\pi[s+ik/2]}{\sin\pi[s-ik/2]}\right].$$
(27)
As noted, there is no phase shift for spin wave-pair scattering.
At the threshhold values $s={\rm integer}$, the spin wave modes uncouple
completely from the particles, and thus from the system, since they
contribute no energy or momentum directly.
In this case, we have the very high degeneracies found in the $1/r^{2}$
lattice systems [11, 12], and for the same reason — an absence of
reflection.
VII The Solution
Let us now impose periodic boundary conditions and take any particle,
pair or spin wave around a ring of large circumference. Along the
way, it suffers a phase change as it scatters from every other particle,
pair or spin wave, plus a phase change of $PL$, where $P=\eta k$ is its
own momentum.
Periodicity requires that this phase change be an integer multiple of $2\pi$,
the integer being the quantum number. We write this statement as coupled
equations in a rather symbolic form:
$$L\eta_{m}k_{m}=2\pi I_{m}(k_{m})+\sum_{-1\leq m^{\prime}\leq M}\sum_{k^{\prime%
}_{m^{\prime}}}\theta_{m,m^{\prime}}(k_{m}-k_{m^{\prime}}),\quad m=-1,0,1,%
\ldots,M.$$
(28)
Here the $I_{m}(k_{m})$ are the quantum numbers, the only subtlety being that
for the spin waves, $I_{-1}$ ranges only over $1,\ldots,N_{0}$.
VIII Results for Zero Temperature and Zero Spin/Charge
In this letter, we give explicit results for the ground state and low-lying
states when $N_{\uparrow}=N_{\downarrow}$, which we call the spin/charge
zero-sector. This certainly is the most interesting case, since all
singularities in the $(N_{\uparrow},N_{\downarrow})$ ground state
phase diagram occur for $N_{\uparrow}=N_{\downarrow}$. In fact, as we shall
see, for $s>0$, the chemical potential has a discontinuity across the line
$N_{\uparrow}=N_{\downarrow}$, and thus the system is an
antiferromagnet/insulator, although not of the Neel/Mott type.
For $-1<s<0$, there is a weak singularity at $N_{\uparrow}=N_{\downarrow}$,
without a discontinuity in the chemical potential.
For $s>0$, which we call the attractive case, the ground state consists of
a spin fluid of type $m=1$, and thus spin $0$.
This is the bound state with lowest binding energy, when
$\kappa=s$, and so $P(k)=2k$ and $E(k)=k^{2}-s^{2}$.
In the ground state, the $k$’s for the pairs distribute themselves densely
with a density $\rho(k)$, between limits $\pm B$, normalized so that
$$N_{1}/L=\int_{-B}^{B}\rho(k){\rm d}k=N/2L.$$
(29)
The energy and momentum are given by
$$\displaystyle P/L$$
$$\displaystyle=$$
$$\displaystyle 2\int_{-B}^{B}\rho(k)k{\rm d}k=0,$$
(30)
$$\displaystyle E/L$$
$$\displaystyle=$$
$$\displaystyle\int_{-B}^{B}\rho(k)k^{2}{\rm d}k-s^{2}N_{1}/L.$$
(31)
The integral equation which determines $\rho(k)$ is
$$1/\pi=\rho(k)+\frac{1}{2\pi}\int_{-B}^{B}\theta^{\prime}_{11}(k-k^{\prime})%
\rho(k^{\prime}){\rm d}k^{\prime}.$$
(32)
The kernel of the equation, $\theta_{11}^{\prime}(k)$, is the derivative of the
phase shift. In figure (1) we show $E_{0}/L$ versus $N/L$ for
selected values of $s=1/2,1,3/2$.
Having determined the ground state properties of the system, we now
determine the low-energy excited states. They are given by the following:
(i) Remove a pair from the ground state distribution, and place it outside the
limits; we call this creating a pair-hole and a pair, and it gives a two
parameter continuum.
(ii) Break a pair, to give two particles, one spin up and the other spin
down; this also gives a two parameter continuum.
(iii) Excite a pair into a higher energy bound state, if allowed; these we
call excitons, and they have single parameter dispersion relations.
(Away from the zero-sector, we can have in addition spin waves. These will be
important for $s<0$.)
By the techniques of Yang and Yang [13], the dispersion relations are
given parametrically by
$$\displaystyle\Delta P$$
$$\displaystyle=$$
$$\displaystyle\sum_{m}\left[\eta_{m}k_{m}-\int_{-B}^{B}\theta_{m1}(k_{m}-k)\rho%
(k){\rm d}k\right],$$
(33)
$$\displaystyle\Delta E$$
$$\displaystyle=$$
$$\displaystyle\sum_{m}\left[\frac{\eta_{m}k_{m}^{2}}{2}-\frac{1}{2\pi}\int_{-B}%
^{B}\theta^{\prime}_{m1}(k_{m}-k)\epsilon(k){\rm d}k\right].$$
(34)
Here $\epsilon(k)$ is the solution to the integral equation
$$k^{2}-s^{2}=\mu_{1}=\epsilon(k)+\frac{1}{2\pi}\int_{-B}^{B}\theta^{\prime}_{11%
}(k-k^{\prime})\epsilon(k^{\prime}){\rm d}k^{\prime}.$$
(35)
The chemical potential $\mu_{1}$ is the chemical potential for pairs, given by
$\partial E_{0}/\partial N_{1}$.
The results are shown in figure (2) for
$s=3/2$, $B=3/2$, $d=N/L=0.943$, $E_{0}/L=-0.691$, $\mu_{1}=1.215$.
The gap for the creation of two particles is $\Delta E=1.170$,
and is equal to the discontinuity of the chemical potential across the line
$N_{\uparrow}=N_{\downarrow}$.
The exciton with $m=2$ is the only exciton allowed at this value of $s$,
and has a gap of $\Delta E=1.017$.
For $0>s>-1$, in the zero-sector, we have two coupled equations for $N$
particles and $N/2$ spin waves. However, in the zero-sector, the limits of the
spin wave distribution are $\pm\infty$. Thus we can solve by Fourier transforms
for the spin wave distribution in terms of the particle distribution, and then
substitute this into the particle equation, giving us a single integral
equation for the distribution of particles $\rho(k)$:
$$\frac{1}{2\pi}=\rho(k)+\frac{1}{2\pi}\int_{-B}^{B}\theta^{\prime}(k-k^{\prime}%
)\rho(k^{\prime}){\rm d}k^{\prime}.$$
(36)
Here the kernel $\theta^{\prime}(k)$ is given as
$$\theta^{\prime}(k)=\theta^{\prime}_{00}(k)-2\int_{-\infty}^{\infty}{\rm d}te^{%
ikt}\frac{{\rm sinh}^{3}t(1+s){\rm cosh}ts}{{\rm sinh}^{3}t}.$$
(37)
In figure (1), we show $E_{0}/L$ versus $N/L$ for $s=-1/2$.
The excited states in the zero-sector are given by:
(i) Remove a particle from the ground state distribution, and place it
outside the limits; we call this creating a hole and a particle, and it
gives a two parameter continuum.
(ii) Remove a spin wave from the ground state distribution, and place it on
the line with imaginary part equal to $i$; we call this creating two spin
waves,
one with spin up and the other with spin down. It gives a two parameter
continuum, familiar from the Heisenberg-Ising model. The results are shown
in figure (3), for $s=-1/2$, $B=1$, $d=N/L=0.600$,
$E_{0}/L=0.094$, $\mu=0.374$.
Finally, we remark that all thermodynamics can be explicitly calculated,
since there are no ambiguities with counting states, or difficulties with
strings of length greater than two.
Acknowledgements.We would like to thank Sriram Shastry for his interest, insight and
encouragement. In addition, we would like to thank the creators of
Mathematica${}^{{\sc\tiny TM}}$ for the LogGamma[] and
PolyGamma[] functions, without which this investigation would be
where it was fifteen years ago.
References
[1]
F. Calogero, O. Ragnisco and C. Marchioro,
Lett. Nuovo Cimento 13, 383 (1975).
[2]
B. Sutherland,
Rocky Mtn. J. Math. 8, 413 (1978).
[3]
P. D. Lax,
Comm. Pure Appl. Math. 21, 467 (1968).
[4]
J. Moser,
Adv. Math. 16, 197 (1975).
[5]
F. Calogero,
Lett. Nuovo Cimento 13, 411 (1975).
[6]
B. S. Shastry and B. Sutherland,
submitted to Phys. Rev. Lett.
[7]
L. D. Landau and E. M. Lifshitz,
Quantum Mechanics (Pergamon, London, 1958).
[8]
C. N. Yang,
Phys. Rev. Lett. 19, 1312 (1967).
[9]
R. J. Baxter,
Exactly Solved Models in Statistical Mechanics
(Academic, London, 1982).
[10]
B. S. Shastry, S. S. Jha and V. Singh, ed.,
Exactly Solvable Problems in Condensed Matter and
Relativistic Field Theory (Springer, Berlin, 1985).
[11]
F. D. M. Haldane,
Phys. Rev. Lett. 66, 1529 (1991).
[12]
B. Sutherland and B. S. Shastry,
submitted to Phys. Rev. Lett.
[13]
C. N. Yang and C. P. Yang,
J. Math. Phys. 10, 1115 (1969). |
Deep Bag-of-Sub-Emotions for Depression Detection in Social Media
Juan S. Lara, Mario Ezra Aragón, Fabio A. González, Manuel Montes-y-Gómez
Abstract
This paper presents the Deep Bag-of-Sub-Emotions (DeepBoSE), a novel deep learning model for depression detection in social media. The model is formulated such that it internally computes a differentiable Bag-of-Features (BoF) representation that incorporates emotional information. This is achieved by a reinterpretation of classical weighting schemes like term frequency-inverse document frequency into probabilistic deep learning operations. An important advantage of the proposed method is that it can be trained under the transfer learning paradigm, which is useful to enhance conventional BoF models that cannot be directly integrated into deep learning architectures. Experiments were performed in the eRisk17 and eRisk18 datasets for the depression detection task; results show that DeepBoSE outperforms conventional BoF representations and it is competitive with the state of the art, achieving a F1-score over the positive class of 0.64 in eRisk17 and 0.65 in eRisk18.
1 Introduction
Nowadays, millions of people around the world are affected by different mental disorders that interfere in their thinking and behavior, damaging their quality of life [Bromet et al., 2017, Mathers and Loncar, 2006]. Timely detection of mental disorders is important to help people before the illness gets worse, minimizing disabilities and returning them to their normal life. Depression is one of the most common mental disorders and a leading cause of risk for suicide [Bromet et al., 2017]. It is a serious medical condition associated with loss of interest, a significant change in weight or appetite, and insomnia. It is a highly debilitating condition and a major cause of disability. Depression affects millions of people worldwide, no matter the culture, gender, age, race, or economic status. Currently, only around 20% of the affected people receive treatment, and most of the spending on mental health is used to maintain psychiatric hospitals instead of detection, prevention, and rehabilitation [Renteria-Rodriguez, 2018]. Based on these facts, it is imperative to create effective methods to detect depression before it causes irreparable damage to people affected by this disease.
In another order of ideas, it is a fact that for many people the majority of their social life does not take place in their immediate environment, but in a virtual world created by social media platforms like Facebook, Twitter, Reddit, Instagram, among others. This presents a great opportunity to understand depression through the analysis of social media documents, increasing the chances to detect people that present signs of depression, and providing professional assistance as soon as possible [Guntuku et al., 2017, Pestian et al., 2010]. In this matter, several natural language processing (NLP) methods have been used for depression detection [Xue et al., 2013], especially, linguistic and sentiment analysis are applied to determine the posts’ polarity, and to represent the users by histograms of the ratios of their positive, negative and neutral posts.
Although conventional NLP approaches provide a good measure of the emotions in the text data, recent works in depression detection [Aragón et al., 2019] have demonstrated that a better performance is achieved using fine-grained representations. For instance, the Bag-of-Sub-Emotions (BoSE) is a representation that creates a dictionary of sub-emotions using a clustering strategy and a lexical resource of emotions. Each word in the users’ posts is replaced with a label of its closest sub-emotion and a histogram of the sub-emotions is computed as the final representation. An important advantage of BoSE is that it achieves a very good performance while preserving interpretability, which differs from most of the state-of-the-art methods that use deep learning for improved performance but cannot be easily interpreted. The main disadvantage of BoSE is that it relies on feature engineering, i.e., it separates the representation and the prediction phases, because, a Bag-of-Features (BoF) must be offline learned and a classifier must be separately trained. In this concern, an end-to-end neural network model has the potential advantage of combining the representation and the classification phases in a fully trainable model that integrates the expressive power of the BoF representation while making it possible to fine-tune it to get higher performance.
Based on the above ideas, we present a novel deep learning model that internally computes an interpretable BoSE representation while taking advantage of deep representation learning. This is achieved using a differentiable reformulation of a BoF representation that is incorporated into an end-to-end model. The main contributions of this work are:
•
The Deep Bag of Sub-Emotions (DeepBoSE) model for depression detection that extends the BoSE representation using probabilistic deep learning components.
•
A training strategy that combines unsupervised and supervised learning. On the one hand, unsupervised information is used to enhance the clustering strategy that defines the sub-emotions; on the other hand, supervised information is incorporated to enhance the representation.
2 Related Work
The assessment of social media data is an application that mainly concerns NLP and sentiment analysis; it is a promising alternative for the automatic diagnostic of mental illness [Guntuku et al., 2017]. For instance, it has been successfully used for the automatic assessment of several mental health conditions including borderline personality disorder, bipolar disorder, schizophrenia, anxiety, depression, self-harming, suicidal tendency, addiction, alcoholism, autism, among others [Gkotsis et al., 2017]. These applications have gained special attention in recent years, considering that an automated diagnostic system can be potentially applied for early detection [Guntuku et al., 2017]. Likewise, the development of new methods is also encouraged by popular competitions like the early risk prediction on the internet (eRisk) that is annually performed in the CLEF conference [Losada et al., 2018]. Several text processing strategies have been explored for depression detection; they can be divided in two main approaches:
Feature engineering-based approaches: the fundamental component of these kinds of methods is the design of a meaningful representation that captures the general topics and sentiments that users express in their posts. More precisely, a descriptor of a document is computed and classification models like support vector machines, logistic regression, or random forests are used to predict a depression label [Guntuku et al., 2017]. Examples of the most relevant used features include word and character ngrams with TF-IDF weighting [Coppersmith et al., 2016], topic probabilities estimated through latent Dirichlet allocation [Yazdavar et al., 2017], statistical features, part-of-speech labels [Sawhney et al., 2018], linguistic structure measures, and interpersonal awareness and interaction [De Choudhury et al., 2016].
Although these general NLP descriptors provide a good overall representation of the documents, there are domain-specific features that better describe the emotional and specialized content in the posts, and therefore that are more appropriate for the depression detection task. For example, there is evidence that hypo-maniac states, levels of guilt or shame, and the presence of emotions like anger or sadness in posts of depressed users are correlated with suicide attempts [Coppersmith et al., 2016]. In order to capture this, a common strategy is to design descriptors that, for example, measure the emotional valence or polarity in the posts [De Choudhury et al., 2014, Reece et al., 2017]. Similarly, detailed descriptors like Bag-of-Emotions (BoE) or specialized Bag-of-Features are used to represent the ratios of terms associated with certain emotions or categories like pronouns, cognitive processes, health, among others [Eichstaedt et al., 2018]. In this matter, a common approach is to use the Linguistic Inquiry and Word Count (LIWC) [Tausczik and Pennebaker, 2010], which determines the most psychologically meaningful words according to categories like social relationships, thinking styles or individual differences. LIWC also includes additional lexicons that are typically used to compute these detailed representations [Reece et al., 2017, De Choudhury et al., 2014, Eichstaedt et al., 2018, Sawhney et al., 2018]. More recently, it was proposed an interesting alternative that consists of using a fine-grained representation of the emotions, referred to as Bag of Sub-Emotions [Aragón et al., 2019]. This idea extends the BoE representations, specifically, the terms are not only assigned to a specific emotion, instead, they are assigned to sub-groups that provide a better approximation of the distribution of the emotions in the texts.
Deep learning-based approaches: recently, deep learning has been used as an alternative to classical feature engineering approaches. Neural networks can automatically learn a representation through several non-linear transformations, which is useful considering that it does not require the effort and the domain knowledge for the manual design of the features. An important advantage of deep representation learning is that it can incorporate supervised information to enhance the descriptors, obtaining specialized representations for depression detection. The most remarkable examples of deep learning applied to depression detection include convolutional neural networks [Orabi et al., 2018], recurrent neural networks [Orabi et al., 2018, Gkotsis et al., 2017], and approaches that use special components like attention mechanisms [Cong et al., 2018].
There is a trade-off between feature engineering and deep learning approaches. On the one hand, feature engineering provides methods that are interpretable, and, therefore, that are useful to discover patterns in the data that may help humans to distinguish between depressed and healthy users. However, these methods require an offline computation of the representation, which restricts them to be static during the training of the classification model and to achieve non-outstanding performances. On the other hand, the deep learning approaches automatically learn an appropriate descriptor and combines the representation and classification in a single model, allowing to adapt the representations to achieve higher performances. However, the main disadvantage is that the learned representations in a neural network cannot be easily interpreted. We formulate the DeepBoSE model such that it takes advantage of the interpretability of BoSE as well as from the representation learning capabilities of neural networks. As it will be detailed in the next section, the proposed method extends BoSE using probabilistic notions that are embedded in a deep neural network that allows transfer learning.
3 Deep Bag-of-Sub-Emotions
To introduce the Deep Bag of Sub-Emotions (DeepBose), we firstly summarize the BoSE approach, which consists of three main steps: first, a set of fine-grained emotions are unsupervisedly learned from a lexical resource that contains words associated to different emotions and sentiments, this is achieved using a clustering technique that discretizes the distribution of each emotion $e$ in $K_{e}$ sub-groups (named as sub-emotions). Second, the fine-grained emotions are used to represent the documents, specifically, each word is masked or substitute by its closer sub-emotion, and each document is represented by a frequency histogram of their sub-emotions. Third, the histogram representation is used to train a classification model that predicts the depression label.
DeepBoSE uses a similar procedure and combines the second and third step, i.e., the representation or construction of the histograms and the classification phase are integrated into a single deep learning model, allowing to tune the sub-emotions to the specific task of depression detection through transfer learning. The model architecture is depicted in Fig. 1 and contains four main components: (1) embedding, a word embedding strategy is used to compute a vector representation from all the terms in a document; (2) DM-encoder, it is a deep learning layer that assigns each embedded term to a specific sub-emotion; (3) dbof, it is composed of some deep learning components that permit the intermediate calculation of the BoSE representation from the assignments; (4) prediction layers, several fully-connected layers are used to obtain the depression grade from the BoSE representation. The mathematical and probabilistic details of each component and their intermediate representations will be described in the following subsections.
3.1 Model Description
As depicted in Fig. 1, the main purpose of the DeepBoSE architecture is to compute a prediction $\tilde{y}_{i}$ from a document $\mathbf{d}_{i}=\{t_{1},t_{2},\dots,t_{N_{w}}\}$ of $N_{w}$ terms. To achieve this, let us present the mathematical details of the four components in DeepBose:
Embedding: a text embedding $f(\cdot)$ is used to compute a vector representation $\mathbf{x}_{j}\in\mathbb{R}^{1\times m}$ of each term $t_{j}$ as shown in Eq. 1, more precisely, the document $\mathbf{d}_{i}$ is represented as a matrix $\mathbf{X}_{i}\in\mathbb{R}^{N_{w}\times m}$ using $f$. An important property of this embedding is that it preserves semantic relations as numerical similarities, this will allow to numerically assign each embedded term $\mathbf{x}_{j}$ to its most similar sub-emotion.
$$\mathbf{x}_{j}=f(t_{j})\qquad\mathbf{X}_{i}=f(\mathbf{d}_{i})$$
(1)
Dissimilarity mixture encoder (DM-Encoder): we exploit the properties of the Dissimilarity Mixture Autoencoder (DMAE) [Lara and González, 2020], which is an autoencoder architecture for deep clustering that can be easily incorporated into deep neural networks. DeepBoSE incorporates a DM-Encoder to compute a matrix of soft-assignments $\mathbf{S}_{i}\in\mathbb{R}^{N_{w}\times K}$ from the embedded representations $\mathbf{X}_{i}$. Each entry $s_{j,k}\in\mathbf{S}_{i}$ represents a soft assignment of the $j$-th term in document $\mathbf{d}_{i}$ to the $k$-th sub-emotion. Sub-emotions are obtained by clustering sets of words associated with different emotions (this is discussed in detail in Subsection 3.3). There are a total of $E$ emotions and each emotion $e$ is further divided in $K_{e}$ sub-emotions, so the total number of sub-emotions is $K=\sum_{i=1}^{E}K_{e}$. The DM-Encoder calculates the soft-assignments $\mathbf{S}_{i}$ through Eq. 2, where $\sigma(\cdot)$ is the softmax activation function, $\alpha$ is a parameter that controls the sparsity of the assignments, $\mathcal{V}_{p}$ is a pairwise dissimilarity measure that compares the embedding of each term with a matrix of codevectors $\Theta\in\mathbb{R}^{K\times m}$ and $\mathbf{b}\in\mathbb{R}^{1\times K}$ are the biases or mixing coefficients. In this case, a specific sub-emotion is represented by a codevector $\theta_{k}\in\Theta$ and each emotion is represented by its $K_{e}$ codevectors which are codified in sub-matrices $\Theta_{e}\in\mathbb{R}^{K_{e}\times m}$, such that $\Theta=[\Theta_{1},\Theta_{2},\dots,\Theta_{E}]$.
$$\mathbf{S}_{i}=\sigma(-\alpha\mathcal{V}_{p}(\mathbf{X}_{i},\Theta)+\mathbf{b})$$
(2)
Differentiable Bag-of-Features (dBoF): DeepBoSE uses the dBoF to transform the soft-assignments $\mathbf{S}_{i}$ into an overall descriptor $\mathbf{h}_{i}\in\mathbb{R}^{1\times K}$ using a weights vector $\mathbf{w}_{idf}\in\mathbb{R}^{1\times K}$. Further, as it will be demonstrated in the subsection 3.2, this representation is equivalent a bag-of-features when the codevectors are not constrained or a bag of sub-emotions when the codevectors contain emotional information as it will be described in the subsection 3.3.
$$\mathbf{h}_{i}=\text{dBoF}(\mathbf{S}_{i},\mathbf{w}_{idf})$$
(3)
Prediction layers: a number of $d$ fully-connected layers that describe a function $g$ are used to obtain a prediction $\widetilde{y}_{i}$ from $\mathbf{h}_{i}$, using a set of weights $W=\{\mathbf{W}_{1},\mathbf{W}_{2},\dots,\mathbf{W}_{d}\}$:
$$\tilde{y}_{i}=\text{g}(\mathbf{h}_{i},W)$$
(4)
To summarize, DeepBoSE learns the $\Theta$, $\mathbf{b}$, $\mathbf{w}_{idf}$ and $W$ parameters. It can be trained as any other deep learning model for classification, for instance, if the depression label $y_{i}$ is binary, then, the output can be a sigmoid activation and the learning process would consist on the optimization of the binary crossentropy presented in Eq. 5. The model is optimized using $N$ samples from a training set $\mathcal{D}=\{(\mathbf{d_{1}},y_{1}),(\mathbf{d_{2}},y_{2}),\dots,(\mathbf{d_{N}},y_{N})\}~{}\forall~{}y_{i}\in\{0,1\}$) and the loss function measures how similar are the predictions $\tilde{y}_{i}$ and the ground truth $y_{i}$.
$$\mathcal{L}=-\frac{1}{N}\sum_{i=1}^{N}\left(y_{i}\log{\tilde{y}_{i}}+(1-y_{i})\log{(1-\tilde{y}_{i})}\right)$$
(5)
3.2 Probabilistic Intepretation of the Differentiable Bag-of-Features
A bag-of-features (BoF) is a probabilistic representation that extends a Bag-of-Words (BoW) and is widely used in NLP and computer vision. A conventional BoF uses a word embedding with a quantization technique to determine the distribution of a vector of discrete latent variables or codebook $\mathbf{z}\in\mathbb{R}^{1\times K}$, the idea is that a feature representation $\mathbf{h}^{\prime}_{i}$ of a document $\mathbf{d}_{i}$ is computed using this distribution as shown in Eq. 6.
$$\mathbf{h}^{\prime}_{i}=P(\mathbf{z}|\mathbf{d}_{i})$$
(6)
The BoSE representation naturally appears if we constrain the codebook with emotional information, i.e., the codebook $\mathbf{z}$ is divided into a set of $E$ emotions, such that each emotion is a sub-codebook $\mathbf{z}_{e}\in\mathbb{R}^{1\times K_{e}}$ with $K_{e}$ associated codevectors as shown in Eq. 7.
$$\mathbf{h}^{\prime}_{i}=P(\mathbf{z}_{1},\mathbf{z}_{2},\dots,\mathbf{z}_{E}|\mathbf{d}_{i})$$
(7)
We exploit a property of DMAE that allows to reinterpret the soft-assignments as probabilities, to this end, let $\mathbf{z}$ be a vector of binary latent variables $\mathbf{z}=[z_{1},z_{2},\dots,z_{K}]$ where $z_{k}=1$ if a codevector $\theta_{k}$ is representative for the term $t_{j}$ and $z_{k}=0$ otherwise. Then, each value $s_{jk}\in\mathbf{S}_{i}$ corresponds to the probability $P(z_{k}=1|t_{j},\mathbf{d}_{i})$ of a codevector $\theta_{k}\in\Theta$ to be representative for a term $t_{j}$ in a given document $\mathbf{d}_{i}$.
A conceptual diagram of the differentiable Bag-of-Features (dBOF) is presented in Fig. 2, it exploits the reinterpretation of the soft-assignments as a probability distribution to calculate a BoF representation of a document. Likewise, it includes a special activation function and an attention mechanism that are equivalent to the TF-IDF weighting schema that is typically used to improve BoF representations. The main purpose of dBoF is to compute the probability distribution $P(\mathbf{z}|\mathbf{d_{i}})$ that is shown in Eq. 6 from the soft-assignments $\mathbf{S}_{i}$ through the marginalization of the conditional distribution $P(\mathbf{z},t_{j}|\mathbf{d_{i}})$:
$$\begin{split}P(\mathbf{z},t_{j},\mathbf{d}_{i})&=P(\mathbf{d}_{i})P(t_{j}|\mathbf{d}_{i})P(\mathbf{z}|\mathbf{d}_{i},t_{j})\\
P(\mathbf{z},t_{j}|\mathbf{d}_{i})&=\frac{P(\mathbf{z},t_{j},\mathbf{d}_{i})}{P(\mathbf{d}_{i})}\\
P(\mathbf{z}|\mathbf{d}_{i})&=\sum_{t_{j}}P(t_{j}|\mathbf{d}_{i})P(\mathbf{z}|\mathbf{d}_{i},t_{j})\\
\end{split}$$
(8)
The distribution $P(t_{j}|\mathbf{d}_{i})$ corresponds to the term frequencies that are typically used in BoW representations. In fact, it can be estimated as $\frac{N_{t}}{N_{w}}$, where $N_{t}$ is the number of times that a term $t_{j}$ appears in a document $d_{i}$ of $N_{w}$ terms. Moreover, since the representation $\mathbf{X}_{i}$ contains repeated words, each row has a single contribution of $\frac{1}{N_{w}}$, thus, the distribution of the codebook can be estimated as shown in Eq. 9. It is equivalent to an one-dimensional global average pooling (GlobalAveragePool1D) over the rows of $\mathbf{S}_{i}$.
$$\begin{split}P(\mathbf{z}|\mathbf{d_{i}})&=\frac{1}{N_{w}}\sum_{j=1}^{N_{w}}P(\mathbf{z}|t_{j},\mathbf{d_{i}})\\
\mathbf{h}^{\prime}_{i}&=\text{GlobalAveragePool1D}(\mathbf{S_{i}})\end{split}$$
(9)
In a like manner, an approximation of the term frequency $\text{TF}(\mathbf{z}|\mathbf{d_{i}})$ is obtained when the constant $\frac{1}{N_{w}}$ is not considered and it is equivalent to an one-dimensional global sum pooling (GlobalSumPool1D):
$$\begin{split}\text{TF}(\mathbf{z}|\mathbf{d_{i}})&=\sum_{j=1}^{N_{w}}P(\mathbf{z}|t_{j},\mathbf{d_{i}})\\
\mathbf{h}^{\prime}_{i}&=\text{GlobalSumPool1D}(\mathbf{S_{i}})\end{split}$$
(10)
These $\mathbf{h}^{\prime}_{i}$ representations are a first approximation of a BoF, nevertheless, weighting and scaling schemes are needed to mitigate the effect of common terms as it is usually done in classical histogram representations. To this end, dBoF also includes deep learning operations that reformulate the term frequency - inverse document frequency (TF-IDF) statistic. First, a sub-linear scaling of the term frequency is calculated though the Rectifier Logarithmic Unit (ReLoU) activation that is presented in Eq. 11.
$$\text{ReLoU}(x)=\left\{\begin{array}[]{cc}\log{(x)}+1&\text{if}~{}x>1\\
0&\text{otherwise}\\
\end{array}\right.$$
(11)
Second, the Inverse Document Frequency Attention mechanism (IDF-Attention) is proposed as an alternative to the inverse document frequency (IDF) weighting. Specifically, the Hadamard product between a weights vector $\mathbf{w}_{idf}\in\mathbb{R}^{1\times N_{k}}$ and the sub-linear representation $\text{ReLoU}(\text{TF}(\mathbf{z}|\mathbf{d_{i}}))$ is used as the descriptor $\mathbf{h}_{i}$ and is a final approximation of TF-IDF as shown in Eq. 12. An important advantage of this approach is that it allows to initialize these weights using the classical IDF that is computed through counting techniques, nonetheless, $\mathbf{w}_{idf}$ can be modified during the model training and adjusted for the depression detection task.
$$\mathbf{h}_{i}=\text{TFIDF}(\mathbf{z}|\mathbf{d_{i}})=\mathbf{w}_{idf}\odot\text{ReLoU}(\text{TF}(\mathbf{z}|\mathbf{d_{i}}))$$
(12)
As it is shown, the dBoF can compute a TF-IDF approximation using the soft-assignments from DMAE and common deep learning operations. This is important considering that it allows differentiation and gradient-based optimization through backpropagation.
3.3 Learning Sub-Groups of Emotions
To include additional emotional information from lexicons in the codebook, we explore a property of DMAE that allows the initialization with other clustering strategies and its enhancement using transfer learning. Specifically, the DM-Encoder is initialized using a set of sub-codebooks $\Theta$ as shown in Fig. 1.
To extract relevant information from a specific $e$ emotion in a lexicon, a set $K_{e}$ codevectors $\Theta_{e}\in\mathbb{R}^{K_{e}\times m}$ is estimated as the parameters of the cluster distributions that are learned in a DMAE model. The complete process to determine these codevectors $\Theta=[\Theta_{1},\Theta_{2},\dots,\Theta_{E}]$ is depicted in Fig. 3, it requires to represent each of the $N_{e}$ words from the $e$ vocabulary as an embedding matrix $\mathbf{X}_{e}\in\mathbb{R}^{N_{e}\times m}$. Although the codevectors can be computed using any clustering technique, we use DMAE considering that it allows similarity-based clustering, which is important since a certain dissimilarity function would be more suitable for a given embedded space. Specifically, we use the cosine dissimilarity as in Eq. 13, it is a standard similarity for text embeddings and is the same metric originally used in the BoSE representation [Aragón et al., 2019]. This dissimilarity function is a measure of the alignment or the angle between the embedding $\mathbf{x}_{j}\in\mathbf{X}_{e}$ of a term $t_{j}$ and a codevector $\theta_{k}\in\Theta$, lower values represent a complete alignment while higher values represent a maximum degree between the vectors.
$$\mathcal{V}(\mathbf{x}_{j},\theta_{k})=1-\frac{\mathbf{x}_{j}\cdot\theta_{k}}{||\mathbf{x}_{j}||~{}~{}||\theta_{k}||}$$
(13)
Finally, all the DMAE instances are optimized using the loss function presented in Eq. 14. Each model is independently trained and the resultant parameters $\Theta_{e}$ are concatenated into a single matrix $\Theta$ which is used to initialize the DeepBoSE model.
$$\mathcal{L}_{e}=\frac{1}{N_{e}}\sum_{\mathbf{x}_{j}\in\mathbf{X}_{e},~{}\tilde{\theta}_{j}\in\tilde{\Theta}_{e}}\mathcal{V}\left(\mathbf{x}_{j},\tilde{\theta}_{j}\right)$$
(14)
4 Experimental Settings
4.1 Datasets Description
In our evaluation we used the data from the eRisk 2017 and 2018 shared task, it contains Reddit posts with binary labels that indicate if the user is depressed or not. The eRisk2017 dataset contains a training partition of 486 samples (83 users with depression and 403 healthy) and a test partition of 401 samples (52 users with depression and 349 healthy); the eRisk2018 dataset contains a training partition of 887 samples (135 with depression and 752 healthy) and a test partition of 820 samples (79 with depression and 741 healthy). In these shared tasks, the F1-score over the positive class (depression) has been used as the main evaluation measure, however, to assess the overall performance of the proposed model we also report the accuracy, macro-average precision, and recall.
4.2 Learning Approaches
The experiments carried out aim to evaluate the effects of the unsupervised and supervised learning phases of DeepBoSE. To this end, the following three cases are assessed:
Ofline learning (BoSE): it uses the original BoSE representation [Aragón et al., 2019], where the codevectors $\Theta$ are estimated using $\mathcal{E}$ instances of affinity propagation (AP) with cosine similarity, then, counts of unigram and n-gram sequences of hard-assignments are used to build a BoF that is weighted using sub-linear term frequency and TF-IDF.
Unsupervised transfer learning (BoSE+UTL): this case evaluates the performance of a BoSE representation that is computed using DMAE. The same procedure of the first case is used, but the AP models are replaced with $E$ instances of DMAE (initialized with AP). This allows us to evaluate the effects of the unsupervised transfer learning that has shown promising results in deep clustering tasks.
Supervised transfer learning (DeepBose): this corresponds to the training of the proposed model, where the codevectors $\Theta$ are initialized using the results of step 2 as depicted in Fig. 1, and a classical TF-IDF representation is computed to determine an appropriate initial value for the IDF weights $\mathbf{w}_{idf}$.
4.3 Hyperparameter Selection
To determine an appropriate combination of hyperparameters we extracted a stratified validation split of 20% from the training set. The models were trained using the remaining 80% and the best hyperparameters were selected by a grid search using the F1-score as criteria. The model’s weights were estimated using the Adam optimization algorithm with different learning rates ($lr$) that were chosen in an exploratory analysis to avoid over and underfitting: UTL $lr=1e^{-5}$, STL $lr=1e^{-6}$. We used FastText embeddings that were pretrained on WikiNews to represent the words. For the unsupervised phase we used the EmoLex lexicon [Mohammad and Turney, 2012], which is composed of eight different emotions (anger, anticipation, disgust, fear, joy, sadness, surprise, and trust) and two sentiments (negative and positive). Considering that DMAE and DeepBoSE are initialized with AP, we used the number of codevectors $K_{e}$ that AP automatically identifies for BoSE [Thavikulwat, 2008]. The softmax inverse temperature $\alpha$ parameter was explored in the range $[10,10^{3}]$. For the fully connected layers, we used two intermediate dense layers with a ReLU activation and 64 units per each layer, a dropout probability of 0.2 was added to the weights for regularization. Finally, the binary cross-entropy loss was modified using class weights to deal with the class imbalance problem.
5 Results and Analysis
Table 1 presents the results of the proposed method in the depression detection task. It also shows the results from the BoSE representation as well as the best results in both shared tasks [Losada et al., 2017, Losada et al., 2018]. The second approach (BoSE+UTL) shows the advantages of using DMAE. More precisely, the unsupervised fine-tuning allowed to enrich the original BoSE representation that consisted of AP; this is important since we are using a shallow version of DMAE and the results must be similar to other shallow approaches like AP. Moreover, one of the major disadvantages of AP is that clusters are constrained to points in the original dataset, while DMAE only uses these points as initialization and it is able to determine a new set of improved and unconstrained clusters.
The best results in both datasets were achieved using the DeepBoSE model under the supervised transfer learning approach. This model is formulated in such a way that it approximates the unigram case of BoSE, moreover, the results show that an enhanced unigram model outperforms the n-gram representation of the original BoSE. In addition, these results also show that supervised information has an important role in the depression detection task, specifically, it allows to learn a set of more representative codevectors. This is the expected performance of most neural networks that are used for transfer learning, in fact, general information from an emotion lexicon is first introduced in the model, and then, it is fine-tuned to obtain a specialized model for the specific task.
To highlight the interpretability of DeepBose, we computed the saliency $\mathcal{S}_{j}$ of two sample texts that were extracted from users with depression; the saliency maps are shown in Fig. 4. We define the saliency $\mathcal{S}_{j}$ of an specific term $t_{j}\in\mathbf{d}_{i}$ as the sum of the magnitude of the gradient [Li et al., 2016] of each component $x_{l}$ in the embedded representation $\mathbf{x}_{j}$ of $t_{j}$ as shown in Eq. 15.
$$\mathcal{S}_{j}=\sum_{x_{l}\in\mathbf{x}_{j}}\left|\left|\frac{\partial\tilde{y}_{i}}{\partial x_{l}}\right|\right|$$
(15)
In the first text, the user does not directly talk about depression, but DeepBoSE was able to focus on some specific terms that are associated with this disorder. In the second text, the user explicitly talks about depression, and DeepBoSE determined the most representative terms including some specific words like ”DM-IV”, which is a publication for the classification of different mental disorders.
Fig. 5 shows the overall learned BoSE representation for all healthy and depressed users in the test sets. As expected, the histograms show that users with depression are associated with negative emotions like anger or sadness. Moreover, the fine-grained representation provides a better emotion spectrum that allows better discrimination of the users, this is reflected in the results, which not only show an improvement on the F1-score over the positive class but in all other metrics.
6 Conclusions and future work
We presented the DeepBoSE model, which incorporates information from lexical emotion resources, preserves interpretability, and leverages from the properties of a deep neural network. This model demonstrated competitive performance with respect to state-of-the-art methods and improved the results from the original BoSE representation. For future work, we plan to exploit the deep representation learning capabilities of the deep clustering methods, and also to consider the incorporation of novel embedding techniques like transformers, which have shown to outperform other text embeddings in several similar tasks.
References
[Aragón et al., 2019]
Mario Ezra Aragón, Adrian Pastor López-Monroy, Luis Carlos
González-Gurrola, and Manuel Montes y Gómez.
2019.
Detecting depression in social media using fine-grained emotions.
Proceedings of the 2019 Conference of the North American
Chapter of the Association for Computational Linguistics: Human Language
Technologies, Volume 1 (Long and Short Papers).
[Bromet et al., 2017]
Ronald Kessler Evelyn Bromet, Peter Jonge, Victoria Shahly, and Marsha Wilcox.
2017.
The burden of depressive illness.
Public Health Perspectives on Depressive Disorders.
[Cong et al., 2018]
Qing Cong, Zhiyong Feng, Fang Li, Yang Xiang, Guozheng Rao, and Cui Tao.
2018.
Xa-bilstm: A deep learning approach for depression detection in
imbalanced data.
In 2018 IEEE International Conference on Bioinformatics and
Biomedicine (BIBM), pages 1624–1627. IEEE.
[Coppersmith et al., 2016]
Glen Coppersmith, Kim Ngo, Ryan Leary, and Anthony Wood.
2016.
Exploratory analysis of social media prior to a suicide attempt.
In Proceedings of the Third Workshop on Computational
Linguistics and Clinical Psychology, pages 106–117.
[De Choudhury et al., 2014]
Munmun De Choudhury, Scott Counts, Eric J Horvitz, and Aaron Hoff.
2014.
Characterizing and predicting postpartum depression from shared
facebook data.
In Proceedings of the 17th ACM conference on Computer supported
cooperative work & social computing, pages 626–638.
[De Choudhury et al., 2016]
Munmun De Choudhury, Emre Kiciman, Mark Dredze, Glen Coppersmith, and Mrinal
Kumar.
2016.
Discovering shifts to suicidal ideation from mental health content in
social media.
In Proceedings of the 2016 CHI conference on human factors in
computing systems, pages 2098–2110.
[Eichstaedt et al., 2018]
Johannes C Eichstaedt, Robert J Smith, Raina M Merchant, Lyle H Ungar, Patrick
Crutchley, Daniel Preoţiuc-Pietro, David A Asch, and H Andrew Schwartz.
2018.
Facebook language predicts depression in medical records.
Proceedings of the National Academy of Sciences,
115(44):11203–11208.
[Gkotsis et al., 2017]
George Gkotsis, Anika Oellrich, Sumithra Velupillai, Maria Liakata, Tim JP
Hubbard, Richard JB Dobson, and Rina Dutta.
2017.
Characterisation of mental health conditions in social media using
informed deep learning.
Scientific reports, 7:45141.
[Guntuku et al., 2017]
Sharath Chandra Guntuku, David B Yaden, Margaret L Kern, Lyle H Ungar, and
Johannes C Eichstaedt.
2017.
Detecting depression and mental illness on social media: an
integrative review.
Current Opinion in Behavioral Sciences, 18:43–49.
[Lara and González, 2020]
Juan S. Lara and Fabio A. González.
2020.
Dissimilarity mixture autoencoder for deep clustering.
ArXiv, abs/2006.08177.
[Li et al., 2016]
Jiwei Li, Xinlei Chen, Eduard H. Hovy, and Dan Jurafsky.
2016.
Visualizing and understanding neural models in nlp.
In HLT-NAACL.
[Losada et al., 2017]
David E. Losada, Fabio Crestani, and Javier Parapar.
2017.
erisk 2017: Clef lab on early risk prediction on the internet:
Experimental foundations.
Proceedings of the 8th International Conference of the CLEF
Association, CLEF 2017, Dublin, Ireland.
[Losada et al., 2018]
David E. Losada, Fabio Crestani, and Javier Parapar.
2018.
Overview of erisk 2018: Early risk prediction on the internet
(extended lab overview).
Proceedings of the 9th International Conference of the CLEF
Association, CLEF 2018, Avignon, France.
[Mathers and Loncar, 2006]
Colin Mathers and Dejan Loncar.
2006.
Projections of global mortality and burden of disease from 2002 to
2030.
PLOS Medicine, Public Library of Science.
[Mohammad and Turney, 2012]
Saif M. Mohammad and Peter D. Turney.
2012.
Crowdsourcing a word-emotion association lexicon.
Computational Intelligence.
[Orabi et al., 2018]
Ahmed Husseini Orabi, Prasadith Buddhitha, Mahmoud Husseini Orabi, and Diana
Inkpen.
2018.
Deep learning for depression detection of twitter users.
In Proceedings of the Fifth Workshop on Computational
Linguistics and Clinical Psychology: From Keyboard to Clinic, pages 88–97.
[Pestian et al., 2010]
John Pestian, Henry Nasrallah, Pawel Matykiewicz, Aurora Bennett, and Antoon
Leenaars.
2010.
Suicide note classification using natural language processing: A
content analysislin heidelberg.
Biomed Inform Insights.
[Reece et al., 2017]
Andrew G Reece, Andrew J Reagan, Katharina LM Lix, Peter Sheridan Dodds,
Christopher M Danforth, and Ellen J Langer.
2017.
Forecasting the onset and course of mental illness with twitter data.
Scientific reports, 7(1):1–11.
[Renteria-Rodriguez, 2018]
Miguel Enrique Renteria-Rodriguez.
2018.
Salud mental en mexico.
NOTA-INCyTU NÚMERO 007.
[Sawhney et al., 2018]
Ramit Sawhney, Prachi Manchanda, Raj Singh, and Swati Aggarwal.
2018.
A computational approach to feature extraction for identification of
suicidal ideation in tweets.
In Proceedings of ACL 2018, Student Research Workshop, pages
91–98.
[Tausczik and Pennebaker, 2010]
Yla R. Tausczik and James W. Pennebaker.
2010.
The psychological meaning of words: Liwc and computerized text
analysis methods.
Journal of Language and Social Psychology.
[Thavikulwat, 2008]
Precha Thavikulwat.
2008.
Affinity propagation: A clustering algorithm for computer-assisted
business simulation and experimental exercises.
Developments in Business Simulation and Experiential Learning.
[Xue et al., 2013]
Yuanyuan Xue, Qi Li, Li Jin, Ling Feng, and David A. Cliftonand Gari D.
Clifford.
2013.
Detecting adolescent psychological pressures from micro-blog.
IJCNLP.
[Yazdavar et al., 2017]
Amir Hossein Yazdavar, Hussein S Al-Olimat, Monireh Ebrahimi, Goonmeet Bajaj,
Tanvi Banerjee, Krishnaprasad Thirunarayan, Jyotishman Pathak, and Amit
Sheth.
2017.
Semi-supervised approach to monitoring clinical depressive symptoms
in social media.
In Proceedings of the 2017 IEEE/ACM International Conference on
Advances in Social Networks Analysis and Mining 2017, pages 1191–1198. |
MC-LCR: Multi-modal contrastive classification by locally correlated representations for effective face forgery detection
Gaojian Wang
Qian Jiang
Xin Jin
Wei Li
Xiaohui Cui
Abstract
As the remarkable development of facial manipulation technologies is accompanied by severe security concerns, face forgery detection has become a recent research hotspot. Most existing detection methods train a binary classifier under global supervision to judge real or fake. However, advanced manipulations only perform small-scale tampering, posing challenges to comprehensively capture subtle and local forgery artifacts, especially in high compression settings and cross-dataset scenarios. To address such limitations, we propose a novel framework named Multi-modal Contrastive Classification by Locally Correlated Representations (MC-LCR), for effective face forgery detection. Instead of specific appearance features, our MC-LCR aims to amplify implicit local discrepancies between authentic and forged faces from both spatial and frequency domains. Specifically, we design the shallow style representation block that measures the pairwise correlation of shallow feature maps, which encodes local style information to extract more discriminative features in the spatial domain. Moreover, we make a key observation that subtle forgery artifacts can be further exposed in the patch-wise phase and amplitude spectrum and exhibit different clues. According to the complementarity of amplitude and phase information, we develop a patch-wise amplitude and phase dual attention module to capture locally correlated inconsistencies with each other in the frequency domain. Besides the above two modules, we further introduce the collaboration of supervised contrastive loss with cross-entropy loss. It helps the network learn more discriminative and generalized representations. Through extensive experiments and comprehensive studies, we achieve state-of-the-art performance and demonstrate the robustness and generalization of our method.
keywords:
Face forgery detection, Multimedia forensics, Deepfake detection, Local feature correlation
††journal: arXivlabel2label2footnotetext: Corresponding author.
E-mail address: xcui@whu.edu.cn
1 Introduction
Visual forgeries are now rampant on social media and the Internet, with the remarkable progress of deep generative models, such as Variational Autoencoderspu2016variational ; kingma2013auto and Generative Adversarial Networksgoodfellow2014generative . Nowadays, even non-specialists can easily manipulate facial content through booming appsfaceapp ; deepfakesweb and open-source toolsDeepFakes ; FaceSwap ; DeepFaceLab . Although these neutral synthesis techniques can generate high-quality images or videos to help create facial visual effectsavatarify , they can also be abused by malicious users. Misinformation tends to spread quickly on the Internet, causing severe trust and security concerns in our societykwok2021deepfake ; pantserev2020malicious , given that the personal face implies sensitive identity information. To against various facial manipulations in practical scenarios, developing more robust and generalized face forgery detection methods is a paramount and urgent need.
Benefiting from the breakthrough of deeper and deeper neural networks in image classificationchollet2017xception ; tan2019efficientnet , DNN-based detection methods have sprung upafchar2018mesonet ; nguyen2019capsule ; rossler2019faceforensics++ ; guera2018deepfake ; sabir2019recurrent . Most of these models follow the backbone network stream to extract the entire facial image’s global features directly and then make the decision through a binary classifier. However, some advanced manipulation methodsthies2019deferred ; li2020celeb only produce subtle forgery traces locally, resulting in the discriminability of global features suffering from small-scale tampering. Recent works observe this phenomenon and suggest exploring discrepancies in local regions, such as creating an attention map to indicate manipulated regionsdang2020detection , focusing on local patches with patch-based classifierschai2020makes , and attending to different local parts by multiple spatial attention headszhao2021multi . Although some of the methods mentioned above emphasize local features in the spatial domain, the forgery artifacts in color-space are fragile to compression, thus limiting their robustness in high compression settings.
Meanwhile, frequency domain information introduces another perspective for mining forged clues. Prior studiesdurall2020watch ; frank2020leveraging have found that the up-sampling structure of generators leads to checkerboard artifacts in the frequency domain. They usually use Fourier transformdurall2019unmasking ; zhang2019detecting or Discrete Cosine Transformfrank2020leveraging to acquire frequency spectrums and then feed them to the detector. Liu et al.liu2021spatial further prove that the phase spectrum is more sensitive to up-sampling than the amplitude spectrum. However, their performance drops greatly when encountering manipulation methods without the up-sampling step. What is more, most of the existing frequency-based methods directly transform the entire image into the frequency spectrum for analysis. Some local tampered facesthies2016face2face ; thies2019deferred show indiscernible anomalies in the global frequency domain, such that these methods also suffer from subtle local artifacts. F3-Netqian2020thinking , which is consisted of two frequency-aware branches, keeps the importance of local frequency information in mind and achieves state-of-the-art performance in highly compressed videos. However, it cannot guarantee generalization only by mining subtle frequency patterns.
Based on these observations of current face forgery and detection methods, the key motivations behind our works are mainly: a) In addition to the high-level semantic features, subtle tampering traces should also be effectively captured in the spatial domain. Moreover, forged faces may expose similar face warpingli2018exposing and swapping artifactsli2020face across different areas. Correlated encoding of these local patterns enables concentration more on commonly counterfeit evidence than a specific manipulation method. b) The limited pixels modified by small-scale tampering methods also do not introduce distinct frequency component abnormalities, thus forged clues mined in the frequency domain should also tend to be localized. On the other hand, it is incomprehensive to utilize only the amplitudedurall2019unmasking or phaseliu2021spatial information since they complement each other, and both should be jointly exploited. c) Most DNN-based models treat face forgery detection as a vanilla binary classification problem and learn end-to-end with the supervision of cross-entropy loss. However, training a generalizable detection model with so many multi-modal features deeply mined from both spatial and frequency domains is a nontrivial task. This is mainly because some subtle artifacts are manipulation-specific, and binary cross-entropy loss does not directly encourage wider separation between cross-domain forged faces, i.e., overfitting.
Inspired by the above thoughts, we propose a novel Multi-modal Contrastive Classification by Locally Correlated Representations (MC-LCR) for face forgery detection. Corresponding to the aforementioned motivations, our framework is composed of three meticulously devised modules. The first is the shallow style representation block (SSRB), which aims at extracting low-level style features from the spatial domain. To hold subtle artifacts in local regions, SSRB truncates the feature maps from shallow layers and measures the pairwise correlation of feature maps by refining the Gram matrixgatys2015texture . Secondly, we make a crucial observation that although forged discrepancies are minimal in color-space and entire frequency spectrums, they still can be revealed in spectrums of local patches. Specifically, patch-wise amplitude spectrums expose more visible checkboard artifacts, and patch-wise phase spectrums exhibit more differences. Since amplitude and phase are complementary frequency information, we further propose Patch-wise Amplitude and Phase Dual Attention module (PAPDA) to guide interactions with each other, which aims to capture locally correlated inconsistencies in amplitude and phase spectrum. Then the frequency branch network composes more abundant and informative amplitude and phase features. Thirdly, these multi-modal features are fused and then projected into an additional embedding space. In such a restricted space, the supervised contrastive losskhosla2020supervised explicitly encourages facial embeddings of the same class to be put together while increasing the distance between different categories. With the collaboration of the classification head supervised by cross-entropy loss, our MC-LCR network learns a discriminative and generalized locally correlated representation from multi-modality.
To demonstrate the effectiveness of the proposed MC-LCR, we conduct comprehensive evaluations on the FF++rossler2019faceforensics++ dataset with different compression settings and manipulation methods. Moreover, we achieve state-of-the-art performance on the FF++ while still keeping superior generalization on the cross-dataset evaluation of challenging Celeb-DFli2020celeb and DeeperForensicsjiang2020deeperforensics . In summary, the contributions of this work are presented as follows:
$\bullet$ We propose a novel framework for effective and robust face forgery detection, named Multi-modal contrastive classification by locally correlated representations, which mines local discriminative features from different modalities.
$\bullet$ SSRB and PAPDA are developed to capture subtle artifacts in the spatial and frequency domain, respectively. We also introduce the combination of supervised contrastive loss and cross-entropy loss to learn generalizable representations.
$\bullet$ We perform a frequency analysis on local image patches and reveal the implicit frequency discrepancy on the patch-wise amplitude and phase spectrum. We collaborate amplitude and phase information based on the proposed PAPDA to model their correlation interactively.
$\bullet$ Extensive experiments demonstrate the effectiveness of our method for face forgery detection. Ablation studies and visualizations are presented to give new insights into our state-of-the-art performance.
2 Related work
Face forgery detection is a classic problem in computer vision and image forensicsfarid2009image ; piva2013overview . Recently, with the remarkable progress of computer graphics and deep generative models, conventional image forgery detection methods struggle to distinguish realistic deepfakes. As face forgery has gained more and more attention, various countermeasures have been proposed to tackle the increasing challenges. Nevertheless, improved synthesis algorithms, small-scale tampering, and high compression settings still pose threats to current detection methods in terms of effectiveness, generalization, and robustness. In this section, we briefly review these previous works.
2.1 Spatial-based forgery detection methods
Earlier facial manipulation techniques are prone to produce obviously forged artifactskorshunov2018deepfakes ; li2018ictu , which inspired many detection methods to explore anomalies in the spatial domain, such as lack of eye blinkingli2018ictu , abnormal head poseyang2019exposing , face warping artifactsli2018exposing , and visual artifactsmatern2019exploiting . However, the improved generation algorithmsli2020celeb significantly reduce visible artifacts, rendering these methods powerless. Face X-rayli2020face further observed the blending step of the face swap and located the resulting boundaries, which significantly improved the generalization performance. However, its performance suffers from the weakened boundaries in highly compressed video scenarios, and it may not be extrapolated to entire face synthesis without a blending step.
Turning to the excellent automatic feature extraction capabilities of deep neural networks, many DNN-based methods take RGB pixels as inputs and treat face forgery detection as a Vanilla binary classification problem. MesoNetafchar2018mesonet introduced two CNN architectures to detect forged faces based on the mesoscopic properties of images. Nguyen et al.nguyen2019capsule combined the capsule network with the pre-trained VGG19 model for face forensics. Rossler et al.rossler2019faceforensics++ released a benchmark for facial manipulation detection and showed the effectiveness of Xceptionchollet2017xception . Considering the temporal information in the video stream, some works adopted recurrent neural network models, such as LSTMguera2018deepfake and GRUsabir2019recurrent .
Instead of blindly utilizing deep learning directly as the detector, a series of methods focus on forgery artifacts. Dang et al.dang2020detection proposed to highlight the forged regions and locate manipulations through learned attention maps. Liu et al.liu2020global leveraged the global texture information to detect the images entirely generated by GANs, but it seemed weak in the face of the locally tampered deepfakes. Recently, Zhao et al.zhao2021multi presented the MADD framework for deepfake detection, exploring different local parts guided by multiple attention maps and combining enhanced shallow texture features. However, most of these spatial-based detection methods are susceptible to compression settings, given that subtle manipulation traces are quality-sensitive. Besides, forgery artifacts take various forms in the color-space, which also challenges the generalization ability of these models. On the other hand, many works on mining frequency clues of forged faces have been proposed.
2.2 Frequency-based forgery detection methods
In the field of digital signal processing and computer vision, frequency domain analysis has always been a classical and powerful method, which has also been exploited for forgery image detection. In fact, the up-sampling step of the deep generative models will introduce anomalies in the frequency domain, such as distribution differences of high-frequency componentsdurall2020watch and checkerboard artifactsfrank2020leveraging .
Durall et al.durall2019unmasking averaged the amplitude spectrum with DFT and used distribution differences to distinguish fake images. AutoGANzhang2019detecting simulated the checkerboard artifacts produced by GAN in the frequency domain and input the DFT spectrum into the detector. Frank et al.frank2020leveraging transformed the image from spatial to frequency domain through DCT and analyzed the frequency anomalies of various GANs. These methods are indeed effective for detecting images synthesized entirely by GAN. However, some local manipulationsthies2019deferred will not cause apparent abnormalities in the global frequency domain, and the performance of these detection methods on small-scale tampering drops greatly. F3-Netqian2020thinking reported state-of-the-art detection results on high compression deepfake videos, which designed frequency-aware decomposition and local frequency statistics to mine subtle forgery patterns in the frequency domain. Nevertheless, F3-NET showed unsatisfactory generalization ability on cross-dataset evaluation.
2.3 Forgery detection methods combining spatial and frequency features
To extract features comprehensively and complementary, considering both spatial and frequency domain information gradually becomes the research mainstreamwu2020sstnet ; masi2020two ; li2021frequency ; luo2021generalizing ; liu2021spatial ; chen2021local .
SSTNetwu2020sstnet detected manipulated face images by extracting spatial, steganalysis, and temporal features with modified Xception and LSTM. Two-branch RNmasi2020two combined information from both spatial and frequency domains, and a Laplacian of Gaussian (LOG) was used to enhance multi-band frequencies. However, the use of fixed filters or hand-crafted features limits discernibility. Recently, SPSLliu2021spatial has observed that the up-sampling structure will cause more significant pixel differences in the phase spectrum than amplitude spectrum, and combined phase information with shallow feature maps to improve the generalizability of face forgery detection. Nevertheless, SPSL also struggles to capture small-scale frequency artifacts caused by local manipulation methods in the global phase spectrum, limiting its in-domain evaluation performance and robustness.
This section reviews the state-of-the-art face forgery detection methods, and a summary of the limitation of existing forgery detection methods and the novelty of the proposed method is provided in Table 1. The detailed comparison clarifies the contributions of this work and provides insights for future research and exploration.
3 Methods
In this section, we propose the Multi-modal Contrastive Classification by Locally Correlated Representations (MC-LCR) for face forgery detection, as illustrated in Figure 1. To tease the framework systematically, we first briefly introduce the main structure and algorithmic workflow of MC-LCR in Section 3.1, and then present the design motivation, the structure of proposed modules, and details of the MC-LCR: We present the proposed SSRB in Section 3.2; We analyze the amplitude and phase anomalies of the forged faces in Section 3.3 and further design the PAPDA in Section 3.4; Finally, we introduce the contrastive classification learning via multi-modal features in Section 3.5.
3.1 Overview
As shown in Figure 1, the proposed MC-LCR is a two-stream framework, which works in both spatial and frequency domains. In the spatial branch, Xceptionchollet2017xception is employed as backbone network. The proposed SSRB (shallow style representation block) truncates feature maps from blocks 1-3 of Xception, and then extracts shallow style features $\it SSF_{b1}$, $\it SSF_{b2}$, and $\it SSF_{b3}$, respectively. These local features served to complement the global semantic feature $GF$ output by the last block of Xception. In the frequency branch, the input image $I$ is first converted to grayscale and segmented into patches $\left\{I_{i}\right\}_{i=1}^{N}$. Then, each patch $I_{i}$ is separately performed Discrete Fourier Transform, i.e., patch-wise DFT, to obtain the amplitude spectrum $AS_{i}$ and phase spectrum $PS_{i}$. The patch-wise frequency spectrums $\left\{AS_{i}\right\}_{i=1}^{N}$ and $\left\{PS_{i}\right\}_{i=1}^{N}$ are flattened and then projected linearly to frequency embeddings $E_{as}$ and $E_{ps}$, respectively, which have been encoded position relationship by an additional embedding layer. After that, the proposed PAPDA (Patch-wise amplitude and phase dual attention module) interactively captures frequency anomaly patterns from $E_{as}$ and $E_{ps}$, and generates amplitude tokens $AT$ and phase tokens $PT$. Finally, the refined local amplitude features $AF$ and phase features $PF$ are extracted by MLP-Mixer network and the last global average pooling (GAP) layer.
These multi-modal representations ($\it SSF_{b1}$, $\it SSF_{b2}$, $\it SSF_{b3}$, $GF$, $AF$, $PF$) extracted from different branches are concatenated to produce $F_{mm}$. After $F_{mm}$ is encoded as $F_{E}$, contrastive learning and classification learning are performed in the embedding space and the classifier, where the training procedure is guided by supervised contrastive loss $L_{sc}$ and cross-entropy loss $L_{ce}$, respectively. The overall learning algorithm of MC-LCR is presented in Algorithm 1.
3.2 Shallow style representation block
The shallow layers of the deep convolutional neural network have smaller receptive fields and tend to distinguish low-level features (e.g., color and texture) locally, while the deeper layers tend to be high-level abstract features. Figure 2 shows feature maps at different layers of the trained deepfake detection network. It can be observed that the feature maps output by shallower layers (blocks 1-3) are generally activated on a small scale and identify local regions such as the eyes, nose, and mouth. With the deepening of layers and the increase of receptive fields, the activated regions of feature maps become larger, which means that higher-level semantic concepts are encoded. The feature maps of the last blocks 13-14 present abstract activations rather than local filtering patterns. In deep fake detection tasks, many methods directly connect binary classifier to the last layer of CNN and decide with global features. However, given those common facial organs, there are few differences between forged and real faces in terms of high-level semantic information. Moreover, some manipulation methods only tamper with the specified region of a face. Therefore, the forgery artifacts are usually more discriminative in shallow feature maps. Based on these phenomena, we extract multi-scale low-level style representations through the Gram matrix at the shallow layers of the backbone network, and learn with other modal features, as shown in the top stream of Figure 1.
Gram matrix is usually used to encode style attributes in deep neural networksgatys2015texture ; gatys2016image , such as shading and texture patterns. Specifically, it calculates the dot product of the vectorized feature maps $F_{i}^{l}$ and $F_{j}^{l}$ of the $l$-th layer:
$$\centering G_{ij}^{l}=\sum_{k}F_{ik}^{l}F_{jk}^{l}=\left[\begin{array}[]{ccc}F_{i1}^{l}{}^{T}F_{j1}^{l}&\cdots&F_{i1}^{l}{}^{T}F_{jk}^{l}\\
\vdots&\ddots&\vdots\\
F_{ik}^{l}{}^{T}F_{j1}^{l}&\cdots&F_{ik}^{l}{}^{T}F_{jk}^{l}\end{array}\right],\@add@centering$$
(1)
where Gram matrix $G_{ij}^{l}$ is the eccentric covariance matrix, i.e., the covariance matrix of the $i$-th and $j$-th feature maps in layer $l$, but without subtracting the mean value. The Gram matrix actually measures the pairwise correlation of two feature maps: the diagonal elements provide the respective responses of each filter to itself, and the remaining elements indicate the degree of correlation between the different filter responses. We argue that such pairwise correlations that emphasize response patterns contribute to capturing forged textures between different feature maps, taking into account those stereotyped visual artifactsmatern2019exploiting or face warping artifactsli2018exposing .
Given the remarkable performance of Xception in deepfake detectionrossler2019faceforensics++ , we employ it as the backbone network, and carefully design the shallow style representation block (SSRB) to extract discriminative local texture features. As shown in Figure 3, SSRB truncates the feature map $F_{l}\in R^{H\times W\times C}$ of layer $l$ as input, and refines $F_{l}$ as a vectorized feature representation. Considering that the calculated Gram matrix $G_{l}\in R^{C\times C}$ has redundant correlation and is hard to be aligned in multiple semantic layers, we aggregate the cumulative correlation of each feature map (i.e., each row of $G_{l}$) through the global average pooling layer, and finally, obtain the shallow style feature $\textit{SSF}\in R^{C\times 1}$. There are 14 blocks in Xceptionchollet2017xception , which are denoted from $b1$ to $b14$. Specifically, SSRB truncates the feature maps with smaller receptive fields output by blocks 1-3 and obtains $\textit{SSF}_{b1}$, $\textit{SSF}_{b2}$, and $\textit{SSF}_{b3}$, respectively. As shown in Figure 1, the extracted SSF features act as multi-scale shallow style representations, given that the feature maps of different blocks correspond to receptive fields of different scales. These shallow style features complement the global semantic feature GF output by block 14, encouraging the network to capture more subtle locally correlated texture information in the spatial domain.
3.3 Amplitude and phase anomalies of forged faces
The up-sampling structure will introduce periodic checkerboard artifacts in the frequency domain. Some methodsfrank2020leveraging ; durall2019unmasking ; zhang2019detecting directly extract the frequency spectrum from the entire image and train the binary classifier to detect images generated by various GANs. However, in some faces manipulated by computer graphics-based methods and improved deepfake videosthies2019deferred ; li2020celeb , forged artifacts are insignificant in the global frequency domain of the entire image, as shown in Figure 4-(a). In view of the forged cues implied by different frequency bands (especially high-frequency components), other methodsmasi2020two ; zhou2018learning extract features through hand-crafted filters as inputs to the model, but with the concomitant consequence that the complete frequency domain is not covered. Further, the phase information of images is lost when only the amplitude spectrum is used for detectiondurall2019unmasking ; zhang2019detecting , and vice versaliu2021spatial . To address these issues, we propose to extract the patch-wise discriminative frequency representation on amplitude and phase spectrum simultaneously.
As shown in Figure 1, we convert the input image $I$ with the resolution of $H\times W\times C$ to grayscale, i.e., $C$=1. Because the use of RGB images in our experiments significantly increases the dimension of frequency features, but it does not significantly improve performance. Then we split it into $N={HW}/{P^{2}}$ non-overlapping patches, and the size of each patch is $P\times P\times 1$. After that, each patch $f_{i}(x,y)$ is independently performed discrete Fourier transform, which is:
$$\begin{gathered}F_{i}(u,v)=\frac{1}{P^{2}}\sum_{x=0}^{P-1}\sum_{y=0}^{P-1}f_{i}(x,y)e^{-2\pi j\left(\frac{ux+vy}{P}\right)},\\
\text{ for }i=1,\ldots,N,\end{gathered}$$
(2)
and expansion with Euler’s formula can be further expressed as:
$$F_{i}(u,v)=R_{i}(u,v)+jI_{i}(u,v)=\left|F_{i}(u,v)\right|e^{j\varphi_{i}(u,v)}.$$
(3)
Here, we acquire the amplitude spectrum $AS_{i}$ and phase spectrum $PS_{i}$ for each patch, as follows:
$$\begin{gathered}AS_{i}=\left|F_{i}(u,v)\right|=\left[R_{i}^{2}(u,v)+I_{i}^{2}(u,v)\right]^{\frac{1}{2}},\\
PS_{i}=\varphi_{i}(u,v)=\arctan\left[\frac{I_{i}(u,v)}{R_{i}(u,v)}\right],\\
\text{ for }i=1,\ldots,N,\end{gathered}$$
(4)
and the size of $AS_{i}$ and $PS_{i}$ is also $P\times P$.
The phase spectrum generally holds the structure and position information of the image, and the amplitude spectrum actually contains most texture and shading information. Amplitude and phase information complement the loss of each other, and both play an important role in image perceptionmorgan1991relative . A visual comparison of the $AS_{i}$ and $PS_{i}$ of all patches from an original image and its corresponding forged image is provided in Figure 4-(b). It is apparent that the $AS_{i}$ and $PS_{i}$ of the subdivided patches are more diversified than the global spectrum $AS$ and $PS$. Pay attention to the residual amplitude spectrum at Figure 4-(a), the forged image does not clearly expose frequency artifacts that distinguish it from the real image. Behind this phenomenon, the fake image has only been altered a very limited part of the pristine face, resulting in only slight brightness differences in the global frequency domain. Fortunately, the residual amplitude spectrum in Figure 4-(b) exposes many apparent checkerboard artifacts on split patches. This motivated us to explore frequency information on local patches. Further comparing the residual images of the patch-wise amplitude spectrum and the phase spectrum, the pixel difference of the phase spectrum exists in more patches, which indicates that the phase spectrum presents more inconsistencies.
Based on these observations, we propose the patch-wise amplitude and phase dual attention (PAPDA) module, which explicitly considers the correlation between the amplitude and phase information to extract more discriminative frequency features.
3.4 Patch-wise amplitude and phase dual attention module
First, the patch-wise amplitude spectrums $AS_{i}$($i=1,...,N)$ are flattened and projected linearly to a $D$-dimensional embedding $E_{as}\in\boldsymbol{R}^{N\times D}$ through a shared-weight dense layer, which keep the position relationship of image patches in the frequency domain by embedding layer. Here we set $D$ to be the same as the input dimension, i.e., $D=P\times P$. In the same way, the phase spectrums $PS_{i}$ are projected to the phase embedding $E_{ps}\in\boldsymbol{R}^{N\times D}$ through another linear layer. Aims to complement the information of the amplitude and phase fully, we designed the PAPDA module to guide the interactions between $E_{as}$ and $E_{ps}$, as shown in Figure 5.
PAPDA is inspired by the attention module of transformervaswani2017attention , and treats $E_{as}$ or $E_{ps}$ as query($Q$), key($K$), and value($V$). Specifically, in the amplitude attention branch, $K_{as}=V_{as}=LN(E_{as})$ but $Q_{ps}=LN(E_{ps})$, where $LN$ represents layer normalizationba2016layer . The output matrix is calculated as:
$$A\left(Q^{ps},K^{as},V^{as}\right)=\operatorname{softmax}\left(\frac{Q^{ps}\left(K^{as}\right)^{T}}{\sqrt{D}}\right)V^{as},$$
(5)
where $Q^{ps}$ and $K^{as}$ are used to measure the correlation between the amplitude artifacts and the phase differences, such that $Q^{ps}\left(K^{as}\right)^{T}$ as an interactive weight to enhance the amplitude embedding $E_{as}$. We use beneficial multi-head attentionvaswani2017attention : the attention function is calculated in parallel on $d$-dimensional $Q$, $K$, and $V$ after linear projection by $h$ heads, and then all the outputs are connected and projected back again. Note that $d$ is set to $D/h$ to keep the consistency of dimension and calculation. These can be summarized as follows:
$$\begin{gathered}MHA\left(Q^{ps},K^{as},V^{as}\right)=\left[A_{1},A_{2},\ldots,A_{h}\right]W^{0},\\
A_{i}=A\left(Q^{ps}W_{i}^{Q},K^{as}W_{i}^{K},V^{as}W_{i}^{V}\right),\end{gathered}$$
(6)
where $W_{i}^{Q}\in\boldsymbol{R}^{D\times d_{q}}$, $W_{i}^{K}\in\boldsymbol{R}^{D\times d_{k}}$, $W_{i}^{V}\in\boldsymbol{R}^{D\times d_{v}}$ and $W^{O}\in\boldsymbol{R}^{h\times d_{v}\times D}$ are the projection matrices. We use 4 attention heads, i.e., $d_{q}=d_{k}=d_{v}=D_{s}/h=256/4$. Finally, we obtain the amplitude token $AT$ through the residual connectionhe2016deep :
$$AT=E_{as}+MHA\left(Q^{ps},K^{as},V^{as}\right).$$
(7)
As shown in Figure 5, the phase token $PT$ are obtained through similar computation:
$$PT=E_{ps}+MHA\left(Q^{as},K^{ps},V^{ps}\right).$$
(8)
The spectral representation is not compatible with vanilla CNNs, given that the frequency domain does not match the shift-invariance and local consistency owned by natural images. Unlike other methodsliu2021spatial ; qian2020thinking ; wang2021m2tr that convert the extracted frequency representation back to color-space again, we aim at utilizing the frequency information directly: a two-stream MLP-Mixertolstikhin2021mlp takes $AT$ and $PT$ as input, respectively, and capture locally correlated inconsistencies in amplitude and phase between different patches. In specific, amplitude or phase tokens $T\in\boldsymbol{R}^{N\times D}$ are input to a sequence of Mixer layers. As shown in Figure 6, each layer is identical and consists of a token-mixing MLP block and a channel-mixing MLP block. The calculation of the token-mixing MLP block is as follows:
$$U=T+W_{2}\left[\sigma\left(W_{1}\left(LN(T)^{T}\right)\right)\right]^{T},$$
(9)
where $W_{1}$ and $W_{2}$ are linear operations by the fully connected layer. $LN$ and $\sigma$ represent layer normalizationba2016layer and GELUhendrycks2016gaussian nonlinear activation, respectively. The mapping $\boldsymbol{R}^{N}\rightarrow\boldsymbol{R}^{N}$ is shared across all patches, mixing the information of different patches by acting on each channel independently. After that, the resulting $U\in\boldsymbol{R}^{N\times D}$ is input to the channel-mixing MLP block:
$$\mathrm{Y}=\mathrm{U}+\mathrm{W}_{4}\left[\sigma\left(\mathrm{W}_{3}(\mathrm{LN}(\mathrm{U}))\right)\right].$$
(10)
Similarly, $\mathrm{Y}\in\boldsymbol{R}^{N\times D}$ aggregates all channel information of each patch and is input to the next mixer layer.
The channel-mixing block applies the same linear transformation to all channels ($D$-dimensional vectors) in each patch, i.e., parameter sharing, which is similar to $1\times 1$ convolution. The channel-mixing block extracts discriminative features from local patches, and such local representation is beneficial to capture stereotyped frequency patterns (such as checkerboard artifacts) in different patches. In contrast, the token-mixing block performs the linear transformation on different patches ($N$-dimensional vectors) across each channel, which is analogous to a depthwise convolution with shared weights on the channels. The Mixer layer fits the correlation between patch-wise amplitude and phase spectrums and provides a good way for long-distance interactions. As the branch network of our model, it complements the local structure awareness of CNN. In our experiments, we only use two Mixer layers as a compromise between detection performance and network scale. $AT$ and $PT$ are input into two MLP-Mixer networks. Finally, the amplitude feature ($AF$) and phase feature ($PF$) are output through the global average pooling layer, that is:
$$\displaystyle AF=GAP(LN(\operatorname{Mixer}(AT))),$$
(11)
$$\displaystyle PF=GAP(LN(\operatorname{Mixer}(PT))).$$
3.5 Multi-modal features contrastive classification
As shown in Figure 1, the output of our hybrid network includes multi-scale shallow style features $\textit{SSF}_{b1}$, $\textit{SSF}_{b2}$, $\textit{SSF}_{b3}$, high-level semantic features $GF$, amplitude features $AF$ and phase features $PF$. These features of multiple modalities and even different scales are concatenated and used for more comprehensively face forgery detection:
$$F_{MM}=\operatorname{concatenate}(\textit{SSF}_{b1},\textit{SSF}_{b2},\textit{SSF}_{b3},\textit{GF},\textit{AF},\textit{PF}).$$
(12)
Typically, the feature fusion $F_{MM}$ is followed by a fully connected layer to predict binary classes, supervised by the cross-entropy loss. However, forged face detection is a vanilla binary classification problem, and the classification margin learned by the dichotomy model may not be suitable for forgery methods that are not seen in the training, i.e., it suffers from overfitting. Considering the diversity of manipulation methods, disentangling the representation between different data classes will improve our model’s generalization ability. Given the performance improvement brought by supervised contrastive learningkhosla2020supervised in representation learning, we combine it with the cross-entropy loss to make our model not only learns classifier, but also learns more separate representations.
The idea behind supervised contrastive losskhosla2020supervised is to pull the samples from the same class together, and push it away from samples of different classes in the normalized embedding space. Unlike the unsupervised contrastive loss, the supervised contrastive loss takes the same class images as positives, and the label information is effectively used to control the distance. We employ it as one of the loss functions to improve the generalizability of our model. As shown in Figure 1: firstly, $F_{MM}$ is encoded as a representation vector ${F}_{E}\in\boldsymbol{R}^{E}$ through a single linear layer; and then $F_{E}$ is projected to $z\in\boldsymbol{R}^{p}$ with another linear layer. The projected output $z$ is normalized to calculate the inner product for measuring the distance in the embedding space. If the mini-batch size is $N$, the supervised contrastive loss is formulated as:
$$L_{SC}=\sum_{i=1}^{N}\frac{-1}{\left|\left\{z_{i}^{0}\right\}\right|}\sum_{z_{p}\in\left\{z_{i}^{0}\right\}}\log\frac{\exp\left(z_{i}\cdot z_{p}/\tau\right)}{\sum_{z_{n}\in\left\{z_{i}^{1}\right\}}\exp\left(z_{i}\cdot z_{n}/\tau\right)},$$
(13)
where $0$ and $1$ are used to indicate the class of images, i.e., real or fake, and $\tau\in\boldsymbol{R}^{+}$ is a scalar temperature parameter. $\left|\left\{z_{i}^{0}\right\}\right|$ denotes the number of positive samples. The index $i$ is called an anchor for comparison, $p$ is a positive of the same class as the anchor, and $n$ indicates a negative of a different class.
With the supervision of supervised contrastive loss, the encoded representations $F_{E}$ from the same class are encouraged the compactness. Finally, the prediction $\widehat{y}_{i}$ is obtained through $F_{E}$ followed by the fully connected layer, and learned by cross-entropy loss:
$$L_{CE}=-\frac{1}{N}\sum_{i=1}^{N}y_{i}\log\widehat{y}_{i}+\left(1-y_{i}\right)\log\left(1-\widehat{y}_{i}\right).$$
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Note that we did not adopt the two-stage trainingkhosla2020supervised , but let the model learn the feature representation and the classifier simultaneously, because we found that it would take into account both detection and generalization ability. That is:
$$L=\alpha L_{SC}+(1-\alpha)L_{CE},$$
(15)
where $\alpha$ is the hyperparameter used to control the trade-off between $L_{CE}$ and $L_{SC}$.
4 Experiments
In this section, we first introduce the overall experimental setup and then present extensive experimental results to demonstrate the effectiveness of the proposed method.
4.1 Experimental setup
Datasets To comprehensively evaluate the effectiveness of our method for detecting forged faces, its robustness in different compression scenarios, and its generalization across datasets, we conduct extensive experiments on three large-scale face forgery benchmarks: FaceForensics++(FF++)rossler2019faceforensics++ , Celeb-DFli2020celeb , and DeeperForensicsjiang2020deeperforensics . FF++ has been widely evaluated by recent detection models, which is composed of 1000 pristine videos as well as 4000 videos forged by four common face manipulation methods: DeepFakes(DF)DeepFakes , Face2Face(F2F)thies2016face2face , FaceSwap(FS)FaceSwap , NeuralTexturesthies2019deferred . Each video in FF++ comes in three versions according to the compression level: uncompressed c0 (RAW), lightly compressed c23(HQ), and heavily compressed c40 (LQ). We adopt the HQ and more challenging LQ versions since the benchmark already achieved nearly perfect detection performance in the RAW version. Following the official preprocessing waysrossler2019faceforensics++ , we split 720 videos for training, 140 videos for validation, and the remaining 140 videos for testing. After 30 frames of each video are sampled, we use DLIBking2009dlib to extract and align the faces from frames, and adjust facial image size to $256\times 256$.
Implementation details and hyper-parameters We use Xceptionchollet2017xception pre-trained on the imagenet as the RGB branch and MLP-Mixertolstikhin2021mlp as the frequency branch. The drop rate of all dropout layers in our network is set to 0.5. The temperature parameter in the Eq. 13 is set to 0.1, and the $\alpha$ in the Eq. 15 is set to 0.5. The batch size is set to 16. Note that each batch consists of 8 real and 8 fake facial images, and is guided for contrastive learning. The Adamloshchilov2017decoupled algorithm with initial learning rate $1e-3$ and weight decay $1e-4$ is used to optimize the network. The model is trained for 50 epochs. If the validation loss does not decrease after 5 epochs, the learning rate will be reduced to half of the original. Our experiments are performed on the NVIDIA GeForce RTX 2080Ti GPU with 128GB RAM. More model and feature representation specifications are shown in Table 2.
Metrics Following the evaluation metrics commonly used in deepfake detection tasksrossler2019faceforensics++ ; li2020celeb , we report both the frame-level ACC(accuracy) and AUC(Area Under the Curve of ROC) scores to evaluate the detection performance, which facilitate intuitive comparison with other SOTA methods.
4.2 Evaluation and comparison on FF++
Different video compression We first evaluate our method on different compression versions of the FF++ and report the comparison with previous detection methods in the Table 3. The test results show that the proposed method outperforms previous methods: 1) Obviously, our method is superior to early methods such as Steg.Featuresfridrich2012rich , LD-CNNcozzolino2017recasting , C-Convbayar2016deep , CP-CNNrahmouni2017distinguishing , and MesoNetafchar2018mesonet in terms of ACC. 2) Our results are consistently better than powerful detection methods Xceptionrossler2019faceforensics++ and Face X-rayli2020face in different quality settings. The performance gains mainly benefit from PAPDA, as shown in Section 3.3 and Figure 4: although compressed videos lose many shading and frequency information, noticeable artifacts are still exposed in the patch-wise amplitude and phase spectrums. Furthermore, the AUC of our method significantly exceeds Face X-ray by 12.25% and 28.68% at the HQ and LQ versions. Face X-ray relies on the discrepancies around blending boundaries, and heavy compression results in weakened blending traces that limit its detection performance.
Even compared with the most recent state-of-the-arts, our method achieves the leadership position on the FF++(HQ) and obtains competitive results on the LQ version. Two-branchmasi2020two and SPSLliu2021spatial also fuse features from spatial and frequency domains to isolate forged faces. Our method outperforms them in almost all metrics, which suggests the effectiveness of the proposed SSRB for extracting shallow style representations. SPSL suppresses high-level semantic information and focuses on local textures by throwing away many convolutional blocks. This straightforward way improves generalizability but limits in-domain data fitting capability, given the reduced parameter space. However, our results with LQ settings are weaker than F3-Netqian2020thinking , since F3-Net is targeted at highly compressed videos but lacks generalization ability. Our SSRB extracts style representation, such as shallow texture information, is sensitive to high compression rate, but it can improve generalization performance, as discussed in ablation studies 5.
Different manipulation methods
We further evaluate the proposed framework against different manipulation methods in FF++. In such a case, the models are trained and tested exactly on the LQ version for each manipulation method. The test results listed in Table 4 prove the effectiveness of our method in detecting various manipulated faces. The SPSLliu2021spatial makes use of the global phase differences produced by the up-sampling steps. But some local manipulations, e.g., F2Fthies2016face2face and NTthies2019deferred , would not have noticeable differences in the global phase spectrum. To address this issue, our PAPDA captures more significant amplitude artifacts and phase differences in local patches, which further boosts the performance. It is worth mentioning that NeuralTextures(NT) is the most challenging, which only modifies the lip region pixels corresponding to the facial expressions, resulting in subtle forgery artifacts in the RGB space and frequency spectrum. Compared with the baseline Xceptionrossler2019faceforensics++ , our method makes a remarkable improvement in ACC of 4.6%. Our performance improvement is mainly attributed to SSRB and PAPDA capturing forgery traces in spatial and frequency domains and concentrating on locally exposed artifacts.
4.3 Generalization evaluation on unseen datasets
To evaluate the generalization ability of the proposed model, we also conduct the cross-dataset experiment, i.e., the model is trained on FF++ with four manipulations while tested on Celeb-DFli2020celeb and DeeperForensicsjiang2020deeperforensics , respectively. Celeb-DF has significantly reduced visual artifacts with improved synthesis algorithms than FF++. DeeperForensics applies image degradations to the real videos of FF++ and varies in scenarios such as poses and illumination conditions.
We follow the evaluation settingsli2020celeb of Celeb-DF to split testing set and report the frame-level AUC scores in the Table 5. The results show that our method achieves better generalization than most existing methods while still keeping almost the top AUC in FF++ DeepFakes. The SSRB and $L_{SC}$ significantly improve the transferability of our model, which is discussed in the ablation studies 5. Two-branchmasi2020two and SPSLliu2021spatial generalize to Celeb-DF better than ours, at the expense of the in-dataset AUC scores are far behind ours. How to balance bias and variance has always been a challenge, and our method also has limitations here. In the evaluation of generalizing to DeeperForensics, we follow jiang2020deeperforensics to split 20% videos in each sub-database as the testing set and report the frame-level AUC scores in Table 6. The proposed MC-LCR achieves the best performance in both FF++(HQ) and DeepForensics. MC-LCR improves roughly 5% AUC to the baseline Xception, which demonstrates the robustness of our method for generalizing to real-world perturbations.
4.4 Evaluation of efficiency
Considering MC-LCR introduced multiple components for contrastive classification learning in spatial and frequency domains simultaneously, we also compare the efficiency with other models to measure the consumption of resources and computation. Table 7 reports the number of parameters(Param), floating-point operations(Flops), and total read-write memory cost(Total MemR+W) of the proposed MC-LCR and other Deepfake detection methods. The comparison models include the baseline networks, VGG19, EfficientNet-B4, ResNet50, and Xception, which are widely adopted as backbone networks for various deepfake detection methods. We also reproduced the state-of-the-art DSP-FWA and MADD methods with officially released models. All models are evaluated under the same environment, and the input size of the image is $256\times 256\times 3$.
As shown in Table 7, baseline classification networks have fewer parameters, Flops, and memory cost, given that there are no modules introduced or framework redesigned specifically for deepfake detection, which also limits the detection performance of these naive CNN. Compared with the baseline network Xception, the proposed MC-LCR increases the network parameters by 7.6M but is still far less than DSP-FWA and MADD. The additional parameters are mainly derived from the introduced SSRB, PAPDA, and MLP-Mixer modules. However, the size of Flops and Total MemR+W of MC-LCR does not increase significantly to Xception and is much smaller than VGG19, which indicates that the proposed method is acceptable in terms of computational cost and memory overhead. It is worth mentioning that MC-LCR has advantages in both time and space complexity compared with the MADD method that adopted Xception as the backbone network, and achieved better performance and stronger generalization in previous comparisons.
5 Ablation Studies
In this section, we perform ablation studies to demonstrate the effectiveness of each proposed module. The SSRB is designed to extract the style representation from shallow layers, given that the low-level texture artifacts introduced by different manipulation methods are more discriminative and generalizable in smaller receptive fields. To dig out forgery clues in the frequency domain and against local tampering methods, PAPDA further interactively captures implicated checkerboard artifacts and differences in the patch-wise amplitude and phase spectrum, which complement the RGB branch. After these locally correlated features are extracted from both spatial and frequency domains, the supervised contrastive loss is combined with cross-entropy loss to learn more separate encoding representations, improving generalization ability. We remove each component individually from our framework and keep other settings the same, then quantitatively evaluate these variants: 1) the baseline model(Xception) w/o SSRB, PAPDA, and SC loss $L_{SC}$, 2) our method w/o SSRB, 3) our method w/o PAPDA, 4) our method w/o SC loss, i.e., $\alpha$ in the Eq. 15 is set to $0$.
5.1 Effectiveness to in-dataset detection
As listed in Table 8, we conduct experiments on HQ and challenging LQ versions of FF++ to analyze the improved performance of each module in our MC-LCR. The comparison results verify that both SSRB and PAPDA effectively improve the detection performance. Furthermore, variants 2 and 3 show that PAPDA makes more detection performance improvement than SSRB, especially in the LQ version. This is expected, as many existing worksli2021frequency ; qian2020thinking ; masi2020two have observed that forgery cues in the frequency domain are generally more robust to compression. Moreover, variant 4 consistently outperforms variants 2 and 3 in terms of ACC and AUC scores, indicating that local features extracted from different domains by SSRB and PAPDA complement each other.
5.2 Effectiveness to cross-dataset detection
To illustrate the improved generalizability of our framework, Table 9 presents the cross-datasets ablation results. We can observe that the AUC score of our model without SSRB(variant 2) generalized to Celeb-DF is significantly reduced by 6.2%. This demonstrates that the additionally extracted shallow style representations are more generalized across different forged faces. Plugging SC Loss into training indeed contributes to improving the transferability, which introduced a wider separation margin for multi-modal fusion features between real and fake faces. The AUC score also decreased slightly with the removal of PAPDA(variant 3), although not as severe as removing SSRB and SC loss.
5.3 Effect of low-level features in different blocks
Inspired by forgery artifacts of some local face manipulations that are usually more distinct in low-level information, the proposed SSRB truncates the feature maps at the shallow layers b1, b2, and b3 of the backbone network. To illustrate the discriminativeness of low-level local features and the action scope of SSRB, we test SSRB to truncate different blocks and compare detection results in the FF++ dataset. The comparison in Figure7-(a) shows that intercepting local features in shallower layers of the network can achieve better detection performance. Figure7-(b) demonstrates the rationality of SSRB acting on blocks b1, b2, and b3.
5.4 Effect of different backbones
Given that the proposed MC-LCR in this paper is mainly based on Xception for experiments and evaluations, we also adopt different networks as the backbone to verify the universality of SSRB, PAPDA, and contrastive classification learning. Three backbones commonly employed in the deepfake detection model, ResNet50, EfficientNet-B0, and EfficientNet-B4, are used for evaluation, where SSRB acts on the shallower three layers. As shown in Table 10, except that the generalization performance is not improved on ResNet50, other comparison results demonstrate that the proposed method is generally applicable to various backbones to improve performance. We speculate that ResNet50 is larger and deeper than other networks, and such large parameter space leads to the overfitting of the model.
6 Visualization
In this section, we make many visualizations to illustrate the effectiveness of our methods, which provide new insights into the success of the proposed MC-LCR.
6.1 Feature maps
To better understand the effectiveness of our methods, we visualize the shallow feature maps extracted from blocks 1-3 of baseline(Xception) and our MC-LCR, respectively. As presented in Figure 8: (1) Almost all feature maps of baseline tend to have similar responses in different facial regions, and our model varies with manipulation methods. There are visible abnormal shadows around the eye and eyebrow of the DeepFakes-based image, and our model makes much higher activation values in this region than the baseline. (2) Our model is more sharp-sighted to subtle forgery artifacts. Note that the NeuralTextures-based image just slightly tamper the mouth, resulting in the baseline, and even human eyes fail to notice. However, our model strongly activates the tampered mouth as early as the shallow layer (b1) of the network. These phenomena are derived from the proposed SSRB and PAPDA, the former measures the pairwise correlation of feature maps, and the latter guides the interaction between the local amplitude and phase spectrums. The feature map visualization illustrates that our methods significantly emphasize the forged region through local correlation, and effectively tackle the problem that the subtle artifacts of small-scale tampering faces are difficult to be captured.
6.2 Responses of patches
In the frequency branch of MC-LCR, the facial image is split into N(256) patches of size $P\times P(16\times 16)$, which are then transformed to the frequency domain. To visualize the responses of patches to the detected face, we input faces manipulated by different methods into the corresponding trained networks, then extract the frequency embedding and remap its feature values to the input face, as shown in Figure 9. The response maps of all patches are shown in Figure 9-(b). Patches can be remapped to the input face given that the spectrum embedding was linearly projected through the fully connected layer, while the diagonal pattern presented by all patches is the embodiment of position encoding. Furthermore, Figure 9-(c) presents the average response map of all patches. It can be observed that: 1) DeepFakes and FaceSwap are mainly swapped faces, where patches are saliently responded to the face boundaries; 2) Face2Face is an expression manipulation method that controls the facial features, and the responses map successfully focus on the manipulated regions such as eyes and mouth; 3) In particular, NeuralTextures only tampered with the lip, where corresponding tampered patches are strongly activated. These response maps illustrate the effectiveness of the proposed method to capture forgery artifacts in manipulated image patches.
6.3 Heatmap of feature intensities
The ablation studies in Section 5 demonstrate the effectiveness of different modules in MC-LCR. To further clarify the influence or contribution of features from different components for detection, Figure 10 presents the heatmap of feature intensities, which is derived from the average activation ratio of different features extracted from all testing faces. The global feature $GF$ extracted from the backbone network has the most prominent intensity, especially for faces manipulated by FaceSwapFaceSwap . FaceSwap, as a manipulation method based on computer graphics, generally synthesis low-quality faces with salient forged artifacts on the entire image. The activation degree of heat in shallow style features from MC-LCR is ranked as $\textit{SSF}_{b1}>\textit{SSF}_{b2}>\textit{SSF}_{b3}$, which again verifies that the network tends to low-level style features in shallower layers for face forgery detection. As for faces locally manipulated by Face2Face and NeuralTextures, local frequency features $AF$ and $PF$ have a larger proportion than the other two face-swapping methods. This indicates that MC-LCR indeed resorts to capturing more significant amplitude artifacts and phase differences in local frequency patches.
6.4 Learned encoding representations
Furthermore, we also visualize the t-SNEvan2008visualizing feature spaces of different data in FF++(LQ) to thoroughly explore the influence of our components on the distribution of learned representations $F_{E}$. We can observe from Figure 11: (a) The features extracted from Xception are compactly gathered in the t-SNE embedding space, which limits the discrimination of the four forged faces against real faces. In particular, the features of NT fake faces and real faces are compacted together, because this method only performed small-scale manipulation. (b) After adding the local features extracted from the spatial and frequency domain by SSRB and PAPDA, the distribution of the learned fusion representations has changed. Although some NT fake faces are still confused with real faces, more manipulated faces tend to be farther away from real faces and other categories. These distribution changes prove that the local artifacts captured by our method in different domains help distinguish fake faces from real ones. (c) Further combined with SC loss to training, the representations of the same class are pulled together, while the distances between different classes are significantly boosted. These wider separation boundaries provide new insights into the success of the proposed MC-LCR.
6.5 Failure Case
Although MC-LCR achieved state-of-the-art face forgery detection performance, it is necessary to look at and analyze failure cases. Derived from frequently misclassified samples in the testing sets, Figure 12 shows some representative failure cases: 1) Poor illumination and light change environments with strong contrasts; 2) Special facial biometrics such as thicker eyebrows, heavier eye bags, and dark circles, which are rare in the training set; 3) Disturbances caused by decorations such as glasses and heavy make-up; 4) Common image variations like blurring and relatively tiny modification regions. Whether global or local, spatial or frequency, these strong perturbations bring challenges to feature extraction, resulting in difficulty to be distinguished.
7 Conclusion
In this paper, we introduce a novel method for face forgery detection, named MC-LCR, which models the correlation and interaction of local discrepancies. Given the challenges of small-scale tampered faces, high compression settings, and cross-dataset scenarios, the proposed framework is expected to effectively capture subtle forgery artifacts from both the spatial and frequency domains. To this end, SSRB is carefully devised to extract more discriminative local texture features, and PAPDA is developed to capture local inconsistencies in amplitude and phase spectrum. Meanwhile, the joint supervision of SC and CE loss is introduced to help disentangle the comprehensive representation from multi-modalities. The ablation studies and visualizations illustrated the effectiveness of each module. Extensive experiments demonstrate the robustness and superiority of our MC-LCR.
CRediT authorship contribution statement
Gaojian Wang: Conceptualization, Methodology, Software, Investigation, Visualization, Formal analysis, Validation, Writing - Original Draft, Writing - Review & Editing. Qian Jiang: Funding acquisition, Formal analysis, Writing - Review & Editing, Data Curation. Xin Jin: Writing - Review & Editing, Validation, Visualization, Resources.
Wei Li: Writing - Review & Editing, Conceptualization, Investigation, Validation.
Xiaohui Cui: Writing - Review & Editing, Project administration, Supervision, Resources.
Declaration of competing interes
The authors declare that they have no known competing financial
interests or personal relationships that could have appeared
to influence the work reported in this paper.
ACKNOWLEDGMENT
This study is supported by the National Natural Science Foundation of China (Nos. 61863036, 62002313, 62101481), Key Areas Research Program of Yunnan Province in China (No. 202001BB050076), and Key Laboratory in Software Engineering of Yunnan Province (No. 2020SE408).
References
(1)
Y. Pu, Z. Gan, R. Henao, X. Yuan, C. Li, A. Stevens, L. Carin, Variational
autoencoder for deep learning of images, labels and captions, Advances in
neural information processing systems 29 (2016) 2352–2360.
(2)
D. P. Kingma, M. Welling, Auto-encoding variational bayes, arXiv preprint
arXiv:1312.6114 (2013).
(3)
I. Goodfellow, J. Pouget-Abadie, M. Mirza, B. Xu, D. Warde-Farley, S. Ozair,
A. Courville, Y. Bengio, Generative adversarial nets, Advances in neural
information processing systems 27 (2014).
(4)
Faceapp, http://faceapp.com/app. Accessed:2021-08.
(5)
deepfakesweb, https://deepfakesweb.com/. Accessed:2021-08.
(6)
DeepFakes, https://github.com/deepfakes/faceswap. Accessed:2021-08.
(7)
FaceSwap, https://github.com/MarekKowalski/FaceSwap. Accessed:2021-08.
(8)
DeepFaceLab, https://github.com/iperov/DeepFaceLab. Accessed:2021-08.
(9)
avatarify, https://geeknizer.com/tag/avatarify/. Accessed:2021-08.
(10)
A. O. Kwok, S. G. Koh, Deepfake: a social construction of technology
perspective, Current Issues in Tourism 24 (13) (2021) 1798–1802.
(11)
K. A. Pantserev, The malicious use of ai-based deepfake technology as the new
threat to psychological security and political stability, in: Cyber defence
in the age of AI, smart societies and augmented humanity, Springer, Cham,
2020, pp. 37–55.
(12)
F. Chollet, Xception: Deep learning with depthwise separable convolutions, in:
Proceedings of the IEEE conference on computer vision and pattern
recognition, 2017, pp. 1251–1258.
(13)
M. Tan, Q. Le, Efficientnet: Rethinking model scaling for convolutional neural
networks, in: International Conference on Machine Learning, PMLR, 2019, pp.
6105–6114.
(14)
D. Afchar, V. Nozick, J. Yamagishi, I. Echizen, Mesonet: a compact facial video
forgery detection network, in: 2018 IEEE International Workshop on
Information Forensics and Security (WIFS), IEEE, 2018, pp. 1–7.
(15)
H. H. Nguyen, J. Yamagishi, I. Echizen, Capsule-forensics: Using capsule
networks to detect forged images and videos, in: ICASSP 2019-2019 IEEE
International Conference on Acoustics, Speech and Signal Processing (ICASSP),
IEEE, 2019, pp. 2307–2311.
(16)
A. Rossler, D. Cozzolino, L. Verdoliva, C. Riess, J. Thies, M. Nießner,
Faceforensics++: Learning to detect manipulated facial images, in:
Proceedings of the IEEE/CVF International Conference on Computer Vision,
2019, pp. 1–11.
(17)
D. Güera, E. J. Delp, Deepfake video detection using recurrent neural
networks, in: 2018 15th IEEE international conference on advanced video and
signal based surveillance (AVSS), IEEE, 2018, pp. 1–6.
(18)
E. Sabir, J. Cheng, A. Jaiswal, W. AbdAlmageed, I. Masi, P. Natarajan,
Recurrent convolutional strategies for face manipulation detection in videos,
Interfaces (GUI) 3 (1) (2019) 80–87.
(19)
J. Thies, M. Zollhöfer, M. Nießner, Deferred neural rendering: Image
synthesis using neural textures, ACM Transactions on Graphics (TOG) 38 (4)
(2019) 1–12.
(20)
Y. Li, X. Yang, P. Sun, H. Qi, S. Lyu, Celeb-df: A large-scale challenging
dataset for deepfake forensics, in: Proceedings of the IEEE/CVF Conference on
Computer Vision and Pattern Recognition, 2020, pp. 3207–3216.
(21)
H. Dang, F. Liu, J. Stehouwer, X. Liu, A. K. Jain, On the detection of digital
face manipulation, in: Proceedings of the IEEE/CVF Conference on Computer
Vision and Pattern recognition, 2020, pp. 5781–5790.
(22)
L. Chai, D. Bau, S.-N. Lim, P. Isola, What makes fake images detectable?
understanding properties that generalize, in: European Conference on Computer
Vision, Springer, 2020, pp. 103–120.
(23)
H. Zhao, W. Zhou, D. Chen, T. Wei, W. Zhang, N. Yu, Multi-attentional deepfake
detection, in: Proceedings of the IEEE/CVF Conference on Computer Vision and
Pattern Recognition, 2021, pp. 2185–2194.
(24)
R. Durall, M. Keuper, J. Keuper, Watch your up-convolution: Cnn based
generative deep neural networks are failing to reproduce spectral
distributions, in: Proceedings of the IEEE/CVF Conference on Computer Vision
and Pattern Recognition, 2020, pp. 7890–7899.
(25)
J. Frank, T. Eisenhofer, L. Schönherr, A. Fischer, D. Kolossa, T. Holz,
Leveraging frequency analysis for deep fake image recognition, in:
International Conference on Machine Learning, PMLR, 2020, pp. 3247–3258.
(26)
R. Durall, M. Keuper, F.-J. Pfreundt, J. Keuper, Unmasking deepfakes with
simple features, arXiv preprint arXiv:1911.00686 (2019).
(27)
X. Zhang, S. Karaman, S.-F. Chang, Detecting and simulating artifacts in gan
fake images, in: 2019 IEEE International Workshop on Information Forensics
and Security (WIFS), IEEE, 2019, pp. 1–6.
(28)
H. Liu, X. Li, W. Zhou, Y. Chen, Y. He, H. Xue, W. Zhang, N. Yu, Spatial-phase
shallow learning: rethinking face forgery detection in frequency domain, in:
Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern
Recognition, 2021, pp. 772–781.
(29)
J. Thies, M. Zollhofer, M. Stamminger, C. Theobalt, M. Nießner, Face2face:
Real-time face capture and reenactment of rgb videos, in: Proceedings of the
IEEE conference on computer vision and pattern recognition, 2016, pp.
2387–2395.
(30)
Y. Qian, G. Yin, L. Sheng, Z. Chen, J. Shao, Thinking in frequency: Face
forgery detection by mining frequency-aware clues, in: European Conference on
Computer Vision, Springer, 2020, pp. 86–103.
(31)
Y. Li, S. Lyu, Exposing deepfake videos by detecting face warping artifacts,
arXiv preprint arXiv:1811.00656 (2018).
(32)
L. Li, J. Bao, T. Zhang, H. Yang, D. Chen, F. Wen, B. Guo, Face x-ray for more
general face forgery detection, in: Proceedings of the IEEE/CVF Conference on
Computer Vision and Pattern Recognition, 2020, pp. 5001–5010.
(33)
L. Gatys, A. S. Ecker, M. Bethge, Texture synthesis using convolutional neural
networks, Advances in neural information processing systems 28 (2015)
262–270.
(34)
P. Khosla, P. Teterwak, C. Wang, A. Sarna, Y. Tian, P. Isola, A. Maschinot,
C. Liu, D. Krishnan, Supervised contrastive learning, arXiv preprint
arXiv:2004.11362 (2020).
(35)
L. Jiang, R. Li, W. Wu, C. Qian, C. C. Loy, Deeperforensics-1.0: A large-scale
dataset for real-world face forgery detection, in: Proceedings of the
IEEE/CVF conference on computer vision and pattern recognition, 2020, pp.
2889–2898.
(36)
H. Farid, Image forgery detection, IEEE Signal processing magazine 26 (2)
(2009) 16–25.
(37)
A. Piva, An overview on image forensics, International Scholarly Research
Notices 2013 (2013).
(38)
P. Korshunov, S. Marcel, Deepfakes: a new threat to face recognition?
assessment and detection, arXiv preprint arXiv:1812.08685 (2018).
(39)
Y. Li, M.-C. Chang, S. Lyu, In ictu oculi: Exposing ai created fake videos by
detecting eye blinking, in: 2018 IEEE International Workshop on Information
Forensics and Security (WIFS), IEEE, 2018, pp. 1–7.
(40)
X. Yang, Y. Li, S. Lyu, Exposing deep fakes using inconsistent head poses, in:
ICASSP 2019-2019 IEEE International Conference on Acoustics, Speech and
Signal Processing (ICASSP), IEEE, 2019, pp. 8261–8265.
(41)
F. Matern, C. Riess, M. Stamminger, Exploiting visual artifacts to expose
deepfakes and face manipulations, in: 2019 IEEE Winter Applications of
Computer Vision Workshops (WACVW), IEEE, 2019, pp. 83–92.
(42)
Z. Liu, X. Qi, P. H. Torr, Global texture enhancement for fake face detection
in the wild, in: Proceedings of the IEEE/CVF Conference on Computer Vision
and Pattern Recognition, 2020, pp. 8060–8069.
(43)
X. Wu, Z. Xie, Y. Gao, Y. Xiao, Sstnet: Detecting manipulated faces through
spatial, steganalysis and temporal features, in: ICASSP 2020-2020 IEEE
International Conference on Acoustics, Speech and Signal Processing (ICASSP),
IEEE, 2020, pp. 2952–2956.
(44)
I. Masi, A. Killekar, R. M. Mascarenhas, S. P. Gurudatt, W. AbdAlmageed,
Two-branch recurrent network for isolating deepfakes in videos, in: European
Conference on Computer Vision, Springer, 2020, pp. 667–684.
(45)
J. Li, H. Xie, J. Li, Z. Wang, Y. Zhang, Frequency-aware discriminative feature
learning supervised by single-center loss for face forgery detection, in:
Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern
Recognition, 2021, pp. 6458–6467.
(46)
Y. Luo, Y. Zhang, J. Yan, W. Liu, Generalizing face forgery detection with
high-frequency features, in: Proceedings of the IEEE/CVF Conference on
Computer Vision and Pattern Recognition, 2021, pp. 16317–16326.
(47)
S. Chen, T. Yao, Y. Chen, S. Ding, J. Li, R. Ji, Local relation learning for
face forgery detection, in: Proceedings of the AAAI Conference on Artificial
Intelligence, Vol. 35, 2021, pp. 1081–1088.
(48)
L. A. Gatys, A. S. Ecker, M. Bethge, Image style transfer using convolutional
neural networks, in: Proceedings of the IEEE conference on computer vision
and pattern recognition, 2016, pp. 2414–2423.
(49)
P. Zhou, X. Han, V. I. Morariu, L. S. Davis, Learning rich features for image
manipulation detection, in: Proceedings of the IEEE Conference on Computer
Vision and Pattern Recognition, 2018, pp. 1053–1061.
(50)
M. J. Morgan, J. Ross, A. Hayes, The relative importance of local phase and
local amplitude in patchwise image reconstruction, Biological cybernetics
65 (2) (1991) 113–119.
(51)
I. Tolstikhin, N. Houlsby, A. Kolesnikov, L. Beyer, X. Zhai, T. Unterthiner,
J. Yung, D. Keysers, J. Uszkoreit, M. Lucic, et al., Mlp-mixer: An all-mlp
architecture for vision, arXiv preprint arXiv:2105.01601 (2021).
(52)
A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez,
Ł. Kaiser, I. Polosukhin, Attention is all you need, in: Advances in
neural information processing systems, 2017, pp. 5998–6008.
(53)
J. L. Ba, J. R. Kiros, G. E. Hinton, Layer normalization, arXiv preprint
arXiv:1607.06450 (2016).
(54)
K. He, X. Zhang, S. Ren, J. Sun, Deep residual learning for image recognition,
in: Proceedings of the IEEE conference on computer vision and pattern
recognition, 2016, pp. 770–778.
(55)
J. Wang, Z. Wu, J. Chen, Y.-G. Jiang, M2tr: Multi-modal multi-scale
transformers for deepfake detection, arXiv preprint arXiv:2104.09770 (2021).
(56)
D. Hendrycks, K. Gimpel, Gaussian error linear units (gelus), arXiv preprint
arXiv:1606.08415 (2016).
(57)
D. E. King, Dlib-ml: A machine learning toolkit, The Journal of Machine
Learning Research 10 (2009) 1755–1758.
(58)
J. Fridrich, J. Kodovsky, Rich models for steganalysis of digital images, IEEE
Transactions on Information Forensics and Security 7 (3) (2012) 868–882.
(59)
D. Cozzolino, G. Poggi, L. Verdoliva, Recasting residual-based local
descriptors as convolutional neural networks: an application to image forgery
detection, in: Proceedings of the 5th ACM Workshop on Information Hiding and
Multimedia Security, 2017, pp. 159–164.
(60)
B. Bayar, M. C. Stamm, A deep learning approach to universal image manipulation
detection using a new convolutional layer, in: Proceedings of the 4th ACM
workshop on information hiding and multimedia security, 2016, pp. 5–10.
(61)
N. Rahmouni, V. Nozick, J. Yamagishi, I. Echizen, Distinguishing computer
graphics from natural images using convolution neural networks, in: 2017 IEEE
Workshop on Information Forensics and Security (WIFS), IEEE, 2017, pp. 1–6.
(62)
T. S. Gunawan, S. A. M. Hanafiah, M. Kartiwi, N. Ismail, N. F. Za’bah, A. N.
Nordin, Development of photo forensics algorithm by detecting photoshop
manipulation using error level analysis, Indonesian Journal of Electrical
Engineering and Computer Science 7 (1) (2017) 131–137.
(63)
I. Loshchilov, F. Hutter, Decoupled weight decay regularization, arXiv preprint
arXiv:1711.05101 (2017).
(64)
P. Zhou, X. Han, V. I. Morariu, L. S. Davis, Two-stream neural networks for
tampered face detection, in: 2017 IEEE Conference on Computer Vision and
Pattern Recognition Workshops (CVPRW), IEEE, 2017, pp. 1831–1839.
(65)
H. H. Nguyen, F. Fang, J. Yamagishi, I. Echizen, Multi-task learning for
detecting and segmenting manipulated facial images and videos, arXiv preprint
arXiv:1906.06876 (2019).
(66)
X. Li, Y. Lang, Y. Chen, X. Mao, Y. He, S. Wang, H. Xue, Q. Lu, Sharp multiple
instance learning for deepfake video detection, in: Proceedings of the 28th
ACM international conference on multimedia, 2020, pp. 1864–1872.
(67)
N. Bonettini, E. D. Cannas, S. Mandelli, L. Bondi, P. Bestagini, S. Tubaro,
Video face manipulation detection through ensemble of cnns, in: 2020 25th
International Conference on Pattern Recognition (ICPR), IEEE, 2021, pp.
5012–5019.
(68)
L. Van der Maaten, G. Hinton, Visualizing data using t-sne., Journal of machine
learning research 9 (11) (2008). |
\AtNextBibliography\subject
xxxxx, xxxxx, xxxx
\corres
I. V. Karlin
Reactive mixtures with the lattice Boltzmann model
N. Sawant
B. Dorschner and I. V. Karlin
Department of Mechanical and Process Engineering, ETH Zurich, 8092 Zurich, Switzerland
ikarlin@ethz.ch
Abstract
A new lattice Boltzmann model for reactive ideal gas mixtures is presented.
The model is an extension to reactive flows of the recently proposed multi-component lattice Boltzmann model for compressible ideal gas mixtures with Stefan-Maxwell diffusion for species interaction.
First, the kinetic model for the Stefan–Maxwell diffusion is enhanced to accommodate a source term accounting the change of the mixture composition due to chemical reaction. Second, by including the heat of formation in the energy equation, the thermodynamic consistency of the underlying compressible lattice Boltzmann model for momentum and energy allows a realization of the energy and temperature change due to chemical reactions.
This obviates the need for ad-hoc modelling with source terms for temperature or heat.
Both parts remain consistently coupled through mixture composition, momentum, pressure, energy and enthalpy.
The proposed model uses the standard three-dimensional lattices and is validated with a set of benchmarks including laminar burning speed in the hydrogen-air mixture and circular expanding premixed flame.
keywords: xxxx, xxxx, xxxx
{fmtext}
1 Outline
In this paper, we present derivation and analysis of the kinetic equations as well as the lattice Boltzmann formulation for Stefan–Maxwell diffusion for reactive mixtures.
Subsequently, the compressible lattice Boltzmann model is extended to reactive flows. Finally, the model is validated for a set of benchmarks ranging from flame speed simulations of premixed hydrogen-air mixtures to challenging two-dimensional simulations of outward propagating circular flames with detailed chemistry.
2 Introduction
The lattice Boltzmann method (LBM) is a recast of fluid dynamics into a fully discrete kinetic system for the populations $f_{i}(\bm{x},t)$ of designer particles, which are associated with the discrete velocities $\bm{c}_{i}$ fitting into a regular space-filling lattice. As a result, the kinetic equations for the populations $f_{i}(\bm{x},t)$ follow a simple algorithm of “stream along links $\bm{c}_{i}$ and collide at the nodes $\bm{x}$ in discrete time $t$".
LBM has been successfully applied to a range of problems in fluid dynamics including but not limited to transitional flows, flows in complex moving geometries compressible flows, multiphase flows and rarefied gas, to name a few [1, 2].
Nevertheless, in spite of extensive development, the multicomponent reactive mixtures so far resisted a significant advancement in the LBM context.
Arguably, one of the main reasons was the absence of a thermodynamically consistent LBM for mixtures. Early approaches such as [3, 4] suffer many limitations such as incompressible flow restriction, constant transport properties, rudimentary diffusion modelling.
As a remedy, a number of recent works [5, 6, 7] abandoned the construction of a kinetic model or LBM for multicomponent mixtures infavour of a so-called hybrid LBM where only the flow of the mixture is represented by an (augmented) LBM equation while the species and the temperature dynamics are modelled by conventional macroscopic equations. While the hybrid LBM approach can be potentially useful, in particular for combustion applications, our goal here is to retain a fully kinetic model and LBM for multicomponent reactive mixtures.
Recently, we proposed a novel lattice Boltzmann framework for compressible multi-component mixtures with a realistic equation of state and thermodynamic consistency [8]. The strongly coupled formulation consists of kinetic equations for momentum, energy and species dynamics and
was validated for a variety of test cases involving uphill diffusion, opposed jets and Kelvin-Helmholtz instability.
This extends the LBM to realistic mixtures and opens the door for reactive flow applications with a fully kinetic approach, which is the subject of this paper.
We propose a fully kinetic, strongly coupled lattice Boltzmann model for
compressible reactive flows as an extension of [8].
To that end, a generic $M$-component ideal gas mixture is represented by two sets of kinetic equations. A set of $M$ kinetic equations is used to model species undergoing Stefan–Maxwell diffusion is extended to include the reaction source term.
Furthermore, the mixture is described by a set of two kinetic equations, where one accounts for the total mass and momentum of the mixture and another one for the total energy of the mixture.
The kinetic equation for the mixture energy is extended to also include the internal energy of formation in addition to the sensible internal energy.
Thus, the approach presented here can accurately model a reactive $M$-component compressible mixture with $M+2$ kinetic equations. The system is fully coupled through mixture composition, momentum, pressure, and enthalpy. The thermodynamic consistency of the model allows us to automatically account for the energy changes due to chemical reactions. The Stefan–Maxwell diffusion is retained and thus complicated phenomena such as reverse diffusion, osmotic diffusion or diffusion barrier can be captured, as it was already demonstrated in the non-reactive case in [8].
The outline of the paper is as follows. In sec. 3, we extend the lattice Boltzmann model of Ref. [8] to the reactive multicomponent mixtures. This is achieved by supplying a reaction source term to the kinetic equations for the species in such a way that the Stefan–Maxwell diffusion mechanism already implemented by the model 3 stays intact. In sec. 4, we extend the two-population lattice Boltzmann model for the mixture flow and energy to include the enthalpy of formation of chemically reacting species. Thanks to the thermodynamic consistency featured by the original model [8], this final step completes the construction of the lattice Boltzmann model for the reactive mixtures. The derivation follows the path presented in detail in [8], and we indicate the differences brought about by the thermodynamics of the chemical reaction. In sec. 5, we outline the coupling of the lattice Boltzmann solver with the open source chemical kinetics package Cantera. Validation of the model is presented in sec. 6 with the simulation of detailed hydrogen/air combustion mechanism and the discussion is provided in sec. 7.
3 Lattice Boltzmann model for the species
The composition of a reactive mixture of $M$ ideal gases is described by the species densities $\rho_{a}$, $a=1,\dots,M$, while the mixture density is
$\rho=\sum_{a=1}^{M}\rho_{a}$.
The rate of change of $\rho_{a}$ due to chemical reaction $\dot{\rho}_{a}^{c}$ satisfies mass conservation,
$$\sum_{a=1}^{M}\dot{\rho}_{a}^{c}=0.$$
(1)
Introducing the mass fraction
$Y_{a}={\rho_{a}}/{\rho}$, the molar mass of the mixture $m$ is given by
${m}^{-1}=\sum_{a=1}^{M}Y_{a}/m_{a},$
where $m_{a}$ is the molar mass of the component $a$.
The equation of state of the mixture provides a relation between the pressure $P$, the temperature $T$ and the composition,
$$P=\rho RT,$$
(2)
where $R={R_{U}}/{m}$ is the specific gas constant of the mixture and $R_{U}$ is the universal gas constant. The pressure of an individual component $P_{a}$ is related to the pressure of the mixture $P$ through Dalton’s law of partial pressures,
$P_{a}=X_{a}P$, where the mole fraction of a component $X_{a}$ is related to its mass fraction $Y_{a}$ as $X_{a}={m}Y_{a}/{m_{a}}$.
Combined with the equation of state (2), the partial pressure $P_{a}$ takes the form $P_{a}=\rho_{a}R_{a}T$,
where $R_{a}={R_{U}}/{m_{a}}$ is the specific gas constant of the component.
Kinetic model for the Stefan–Maxwell diffusion in the non-reactive mixture were introduced in [8]. Here, we extend the formulation [8] to include the reaction.
To that end, we write the kinetic equation for the populations $f_{ai}$, $a=1,\dots,M$, of the component $a$, corresponding to the discrete velocities $\bm{c}_{i}$, $i=0,\dots,Q-1$,
$$\partial_{t}f_{ai}+\bm{c}_{i}\cdot\nabla f_{ai}=\sum_{b\neq a}^{M}\frac{PX_{a}X_{b}}{\mathcal{D}_{ab}}\left[\left(\frac{f_{ai}^{\rm eq}-f_{ai}}{\rho_{a}}\right)-\left(\frac{f_{bi}^{\rm eq}-f^{*}_{bi}}{\rho_{b}}\right)\right]+\dot{f}_{ai}^{c}.$$
(3)
Here $\mathcal{D}_{ab}$ are the binary diffusivity coefficients. The species’ densities $\rho_{a}$ and partial momenta $\rho_{a}\bm{u}_{a}$ are, respectively,
$$\displaystyle\rho_{a}=\sum_{i=0}^{Q-1}f_{ai},\quad\rho_{a}\bm{u}_{a}=\sum_{i=0}^{Q-1}f_{ai}\bm{c}_{i}.$$
(4)
The momenta of the components sum up to the mixture momentum,
At variance with the non-reactive mixture [8], kinetic equation (3) includes a source term $\dot{f}_{ai}^{c}$ which implements the rate of change of $\rho_{a}$ due to the reaction and satisfies the following conditions,
$$\displaystyle\sum_{i=0}^{Q-1}\dot{f}_{ai}^{c}=\dot{\rho}_{a}^{c},\quad\sum_{i=0}^{Q-1}\dot{f}_{ai}^{c}\bm{c}_{i}$$
$$\displaystyle=\dot{\rho}_{a}^{c}\bm{u}.$$
(5)
The kinetic model (3) is realized on the standard three-dimensional $D3Q27$ lattice with the discrete velocities $\bm{c}_{i}=(c_{ix},c_{iy},c_{iz}),\ c_{i\alpha}\in\{-1,0,1\}$.
Same as in [8], the equilibrium $f_{ai}^{\rm eq}$ and the quasi-equilibrium $f_{ai}^{*}$ in (3) are constructed using the product-form [9]: We define a triplet of functions in two variables, $\xi$ and $\zeta>0$,
$$\displaystyle\Psi_{0}(\xi,\zeta)=1-(\xi^{2}+\zeta),\ \Psi_{1}(\xi,\zeta)=\frac{\xi+(\xi^{2}+\zeta)}{2},\ \Psi_{-1}(\xi,\zeta)=\frac{-\xi+(\xi^{2}+\zeta)}{2}.$$
(6)
The equilibrium $f_{ai}^{\rm eq}$ and the quasi-equilibrium $f_{ai}^{*}$ populations are evaluated as the products of the functions (6), with $\xi=u_{\alpha}$ and $\xi=u_{a\alpha}$, respectively, and with $\zeta=R_{a}T$ in both cases,
$$\displaystyle f_{ai}^{\rm eq}(\rho_{a},\bm{u},R_{a}T)=\rho_{a}\Psi_{c_{ix}}\left(u_{x},R_{a}T\right)\Psi_{c_{iy}}\left(u_{y},R_{a}T\right)\Psi_{c_{iz}}\left(u_{z},R_{a}T\right),$$
(7)
$$\displaystyle f_{ai}^{*}(\rho_{a},\bm{u_{a}},R_{a}T)=\rho_{a}\Psi_{c_{ix}}\left(u_{ax},R_{a}T\right)\Psi_{c_{iy}}\left(u_{ay},R_{a}T\right)\Psi_{c_{iz}}\left(u_{az},R_{a}T\right).$$
(8)
The reaction source term $\dot{f}_{ai}^{c}$ in (3) is also represented with the product-form similar to (7),
$$\displaystyle\dot{f}_{ai}^{c}(\dot{\rho}_{a}^{c},\bm{u},R_{a}T)=\dot{\rho}_{a}^{c}\Psi_{c_{ix}}\left(u_{x},R_{a}T\right)\Psi_{c_{iy}}\left(u_{y},R_{a}T\right)\Psi_{c_{iz}}\left(u_{z},R_{a}T\right).$$
(9)
The analysis of the hydrodynamic limit of the kinetic model (3) follows the lines already presented in [8]. Note that the constraint on the momentum of the source term (5) is required. The balance equations for the densities of the species in the presence of the source term are found as follows,
$$\displaystyle\partial_{t}\rho_{a}$$
$$\displaystyle=-\nabla\cdot(\rho_{a}\bm{u})-\nabla\cdot(\rho_{a}\delta\bm{u}_{a})+\dot{\rho}_{a}^{c},$$
(10)
where the diffusion velocities, $\delta\bm{u}_{a}=\bm{u}_{a}-\bm{u}$, satisfy the Stefan–Maxwell constitutive relation,
$$P\nabla X_{a}+(X_{a}-Y_{a})\nabla P=\sum_{b\neq a}^{M}\frac{PX_{a}X_{b}}{\mathcal{D}_{ab}}\left({\delta}\bm{u}_{b}-{\delta}\bm{u}_{a}\right).$$
(11)
Summarizing, kinetic model (3) recovers both the Stefan–Maxwell law of diffusion
and the contribution of the species mass change due to chemical reaction, as presented in equation (10).
Derivation of the lattice Boltzmann equation from the kinetic model (3) proceeds along the lines of the non-reactive case [8]. Upon integration of (3) along the characteristics and application of the trapezoidal rule, we arrive
at a fully discrete lattice Boltzmann equation,
$$f_{ai}(\bm{x}+\bm{c}_{i}\delta t,t+\delta t)=f_{ai}(\bm{x},t)+2\beta_{a}[f_{ai}^{\rm eq}(\bm{x},t)-f_{ai}(\bm{x},t)]+\delta t(\beta_{a}-1)F_{ai}(\bm{x},t)+\delta t\dot{f}_{ai}^{c}.$$
(12)
The shorthand notation $F_{ai}$ for the inter-species interaction term and the relaxation parameters $\beta_{a}\in[0,1]$ are,
$$F_{ai}=Y_{a}\sum_{b\neq a}^{M}\frac{1}{\tau_{ab}}\left(f_{bi}^{\rm eq}-f_{bi}^{*}\right),\quad\beta_{a}=\frac{\delta t}{2\tau_{a}+\delta t},$$
(13)
where the characteristic times $\tau_{ab}$ and the relaxation times $\tau_{a}$ are related to the binary diffusivities,
$${\tau_{ab}}=\left(\frac{m_{a}m_{b}}{mR_{U}T}\right)\mathcal{D}_{ab},\quad\frac{1}{\tau_{a}}=\sum_{b\neq a}^{M}\frac{Y_{b}}{\tau_{ab}}.$$
(14)
Furthermore, the quasi-equilibrium populations $f_{bi}^{*}=f_{bi}^{*}(\rho_{b},\bm{u}+\delta\bm{u}_{b},R_{b}T)$ in the expression $F_{ai}$ (13) depend on the diffusion velocity $\delta\bm{u}_{b}$. The latter are found by solving the $M\times M$ linear algebraic system for each spatial component,
$$\displaystyle\left(1+\frac{\delta t}{2\tau_{a}}\right)\delta\bm{u}_{a}-\frac{\delta t}{2}\sum_{b\neq a}^{M}\frac{1}{\tau_{ab}}Y_{b}\delta\bm{u}_{b}=\bm{u}_{a}-\bm{u}.$$
(15)
The linear algebraic system was already derived in [8] for the non-reactive mixtures and is not altered by the presence of the reaction source term. The equilibrium population $f_{ai}^{eq}=f_{ai}^{eq}(\rho_{a},\bm{u},R_{a}T)$ and the reaction source term $\dot{f}_{ai}^{c}=\dot{f}_{ai}^{c}(\dot{\rho}_{a},\bm{u},R_{a}T)$ in (12) and (13) are evaluated at the mixture velocity $\bm{u}$.
Summarizing, the lattice Boltzmann system (12) delivers the extension of the species dynamics subject to the Stefan–Maxwell diffusion to the reactive mixtures. We proceed with the extension of the flow and energy dynamics of the mixture.
4 Lattice Boltzmann model of mixture momentum and energy
The mass-based specific internal energy ${U}_{a}$ and enthalpy ${H}_{a}$ of a specie $a$ are,
$$\displaystyle{U}_{a}=U^{0}_{a}+\int_{T_{0}}^{T}{C}_{a,v}(T^{\prime})dT^{\prime},\quad{H}_{a}=H^{0}_{a}+\int_{T_{0}}^{T}{C}_{a,p}(T^{\prime})dT^{\prime},$$
(16)
where $U^{0}_{a}$ and $H^{0}_{a}$ are, respectively, the energy and the enthalpy of formation at the reference temperature $T_{0}$, while $C_{a,v}$ and $C_{a,p}$ are specific heats at constant volume and at constant pressure.
The internal energy $\rho U$ and the enthalpy $\rho H$ of a mixture are,
$$\displaystyle\rho U=\sum_{a=1}^{M}\rho_{a}U_{a},\quad\rho H=\sum_{a=1}^{M}\rho_{a}H_{a}.$$
(17)
While the sensible heat was considered in the non-reactive case [8], by taking into account the heat of formation we immediately extend the model to reactive mixtures. Same as in [8], we follow a two-population approach.
One set of populations ($f$-populations) is used to represent the density and the momentum of the mixture,
$$\displaystyle\sum_{i=0}^{Q-1}f_{i}=\rho,\quad\sum_{i=0}^{Q-1}f_{i}\bm{c}_{i}=\rho\bm{u}.$$
(18)
Another set ($g$-populations) represents the total energy,
$$\displaystyle\sum_{i=0}^{Q-1}g_{i}=\rho E,\quad\rho E=\rho U+{\frac{\rho u^{2}}{2}}.$$
(19)
A coupling between the mixture and the species kinetic equations is established through energy since the mixture internal energy (17) depend on the composition.
Furthermore, the temperature is evaluated by solving the integral equation, cf. (16) and (17),
$$\sum_{a=1}^{M}Y_{a}\left[U_{a}^{0}+\int_{T_{0}}^{T}{C}_{a,v}(T^{\prime})dT^{\prime}\right]=E-\frac{u^{2}}{2}.$$
(20)
The temperature is used as the input for the equation of state (2) and hence in the equilibrium, the quasi-equilibrium and the reaction source term of the species lattice Boltzmann system which leads to a two-way coupling between the species and the mixture kinetic systems.
Same as in [8], the lattice Boltzmann equations for the $f$- and $g$-populations are realized on the $D3Q27$ discrete velocity set,
$$\displaystyle f_{i}(\bm{x}+\bm{c}_{i}\delta t,t+\delta t)-f_{i}(\bm{x},t)$$
$$\displaystyle=\omega(f_{i}^{\rm eq}-f_{i})+\bm{A}_{i}\cdot\bm{X},$$
(21)
$$\displaystyle g_{i}(\bm{x}+\bm{c}_{i}\delta t,t+\delta t)-g_{i}(\bm{x},t)$$
$$\displaystyle=\omega_{1}(g_{i}^{\rm eq}-g_{i})+(\omega-\omega_{1})(g_{i}^{*}-g_{i}),$$
(22)
where relaxation parameters $\omega$ and $\omega_{1}$ are related to the viscosity and thermal conductivity.
The equilibrium $f$-populations $f_{i}^{\rm eq}$ in (21) are evaluated using the product-form, with ${\xi}_{\alpha}={u}_{\alpha}$ and $\zeta=RT$ in (6),
$$f_{ai}^{\rm eq}(\rho,\bm{u},RT)=\rho\Psi_{c_{ix}}\left(u_{x},RT\right)\Psi_{c_{iy}}\left(u_{y},RT\right)\Psi_{c_{iz}}\left(u_{z},RT\right).$$
(23)
The last term in (21) is a correction needed to compensate for the insufficient isotropy of the $D3Q27$ lattice in the compressible flow setting [10, 8]: $\bm{X}$ is the vector with the components,
$$X_{\alpha}=-\partial_{\alpha}\left[\left(\frac{1}{\omega}-\frac{1}{2}\right)\delta t\partial_{\alpha}(\rho u_{\alpha}(1-3RT)-\rho u_{\alpha}^{3})\right],$$
(24)
while the components of vectors $\bm{A}_{i}$ are defined as,
$$A_{i\alpha}=\frac{1}{2}c_{i\alpha}\ \ \text{for}\;c_{i}^{2}=1;\quad A_{i\alpha}=0\ \ \text{otherwise}.$$
(25)
The equilibrium and the quasi-equilibrium $g$-populations, $g_{i}^{\rm eq}$ and $g_{i}^{*}$ in (22), are defined with the help of Grad’s approximation [11],
$$\displaystyle g_{i}^{\rm eq}$$
$$\displaystyle=w_{i}\left(\rho E+\frac{\bm{q}^{\rm eq}\cdot\bm{c}_{i}}{\theta}+\frac{(\bm{R}^{\rm eq}-\rho E\theta\bm{I}):(\bm{c}_{i}\otimes\bm{c}_{i}-\theta\bm{I})}{2\theta^{2}}\right),$$
(26)
$$\displaystyle g_{i}^{*}$$
$$\displaystyle=w_{i}\left(\rho E+\frac{\bm{q}^{*}\cdot\bm{c}_{i}}{\theta}+\frac{(\bm{R}^{\rm eq}-\rho E\theta\bm{I}):(\bm{c}_{i}\otimes\bm{c}_{i}-\theta\bm{I})}{2\theta^{2}}\right),$$
(27)
Here, the weights $w_{i}=w_{c_{ix}}w_{c_{iy}}w_{c_{iz}}$ are the products of the one-dimensional weights $w_{0}=1-\theta$, $w_{1}=w_{-1}={\theta}/{2}$, and $\theta=1/3$ is the lattice reference temperature.
The equilibrium mixture energy flux $\bm{q}^{\rm eq}$ and the second-order moment tensor $\bm{R}^{\rm eq}$ in (26) and (27) are,
$$\displaystyle\bm{q}^{\rm eq}$$
$$\displaystyle=\sum_{i=0}^{Q-1}g_{i}^{\rm eq}\bm{c}_{i}=\left(H+\frac{u^{2}}{2}\right)\rho\bm{u},$$
(28)
$$\displaystyle\bm{R}^{\rm eq}$$
$$\displaystyle=\sum_{i=0}^{Q-1}g_{i}^{\rm eq}\bm{c}_{i}\otimes\bm{c}_{i}=\left(H+\frac{u^{2}}{2}\right)\bm{P}^{\rm eq}+P\bm{u}\otimes\bm{u},$$
(29)
where $H$ is the specific mixture enthalpy (17). The quasi-equilibrium energy flux $\bm{q}^{*}$ in (27) has the following form,
$$\displaystyle\bm{q}^{*}=\sum_{i=0}^{Q-1}g_{i}^{*}\bm{c}_{i}=\bm{q}-\bm{u}\cdot(\bm{P}-\bm{P}^{\rm eq})+\bm{q}^{\rm diff}+\bm{q}^{\rm corr}.$$
(30)
The two first terms in (30) include the energy flux $\bm{q}$ and the pressure tensor $\bm{P}$,
$$\displaystyle\bm{q}=\sum_{i=0}^{Q-1}g_{i}\bm{c}_{i},\quad\bm{P}=\sum_{i=0}^{Q-1}f_{i}\bm{c}_{i}\otimes\bm{c}_{i}.$$
(31)
Their contribution maintains a variable Prandtl number and is patterned from the single-component case [10].
The remaining two terms in the quasi-equilibrium energy flux (30), $\bm{q}^{\rm diff}$ and $\bm{q}^{\rm corr}$ pertain to the multicomponent case. The interdiffusion energy flux $\bm{q}^{\rm diff}$ is,
$$\displaystyle\bm{q}^{\rm diff}=\left(\frac{\omega_{1}}{\omega-\omega_{1}}\right)\rho\sum_{a=1}^{M}H_{a}Y_{a}\delta\bm{u}_{a},$$
(32)
where the diffusion velocities $\delta\bm{u}_{a}$ are defined according to Eq. (15).
The flux (32) contributes the enthalpy transport due to diffusion and hence it vanishes in the single-component case but is significant in reactive flows.
Finally, the correction flux $\bm{q}^{\rm corr}$, which also vanishes in the single-component case, is required in the two-population approach to the mixtures in order to recover the Fourier law of thermal conduction, see [8] for details,
$$\displaystyle\bm{q}^{\rm corr}=\frac{1}{2}\left(\frac{\omega_{1}-2}{\omega_{1}-\omega}\right){\delta t}P\sum_{a=1}^{M}H_{a}\nabla Y_{a}.$$
(33)
Prefactors featured in (32) and (33) were found in [8] based on the analysis of the hydrodynamic limit of the lattice Boltzmann system (21) and (22) and are not affected by the present reactive mixture case. Second-order accurate isotropic lattice operators proposed in [12] were used for the evaluation of spatial derivatives in the correction flux (33) as well as in the isotropy correction (24). Following [8], the continuity, the momentum and the energy equations for a reactive multicomponent mixture [13] are obtained as follows,
$$\displaystyle\partial_{t}\rho+\nabla\cdot(\rho\bm{u})=0,$$
(34)
$$\displaystyle\partial_{t}(\rho\bm{u})+\nabla\cdot({\rho\bm{u}\otimes\bm{u}})+\nabla\cdot\bm{\pi}=0,$$
(35)
$$\displaystyle\partial_{t}(\rho E)+\nabla\cdot(\rho E\bm{u})+\nabla\cdot\bm{q}+\nabla\cdot(\bm{\pi}\cdot\bm{u})=0.$$
(36)
The pressure tensor $\bm{\pi}$ in the momentum equation (35) reads,
$$\bm{\pi}=P\bm{I}-\mu\left(\nabla\bm{u}+\nabla\bm{u}^{\dagger}-\frac{2}{D}(\nabla\cdot\bm{u})\bm{I}\right)-\varsigma(\nabla\cdot\bm{u})\bm{I},$$
(37)
where the dynamic viscosity $\mu$ and the bulk viscosity $\varsigma$ are related to the relaxation parameter $\omega$,
$$\displaystyle\mu=\left(\frac{1}{\omega}-\frac{1}{2}\right)P{\delta t},\quad\varsigma=\left(\frac{1}{\omega}-\frac{1}{2}\right)\left(\frac{2}{D}-\frac{R}{C_{v}}\right)P{\delta t},$$
(38)
where $C_{v}=\sum_{a=1}^{M}Y_{a}C_{a,v}$ is the mixture specific heat at constant volume.
The heat flux $\bm{q}$ in the energy equation (36) reads,
$$\bm{q}=-\lambda\nabla T+\rho\sum_{a=1}^{M}H_{a}Y_{a}\delta\bm{u}_{a}.$$
(39)
The first term is the Fourier law of thermal conduction, with the thermal conductivity $\lambda$ related to the relaxation parameter $\omega_{1}$,
$$\lambda=\left(\frac{1}{\omega_{1}}-\frac{1}{2}\right)PC_{p}{\delta t},$$
(40)
where $C_{p}=C_{v}+R$ is the mixture specific heat at constant pressure. The second term in (39) is the interdiffusion energy flux.
The dynamic viscosity $\mu$ and the thermal conductivity $\lambda$ of the mixture are evaluated as a function of the local composition, temperature and pressure using the chemical kinetics solver Cantera [14], wherein a combination of methods involving interaction potential energy functions [15], hard sphere approximations and the methods described in [16] and [17] are employed to calculate the mixture transport coefficients.
Finally, in accord with a principle of strong coupling [8], the excess conservation laws arising due to a separated construction of the species diffusion model in sec. 3 and the two-population mixture model are eliminated by removing one set of species populations (here, the component $M$),
$$f_{Mi}=f_{i}-\sum_{a=1}^{M-1}f_{ai}.$$
(41)
Thus, the component $M$ is not an independent field any more but is slaved to the remaining $M-1$ species and the mixture $f$-populations.
Summarizing, the thermodynamically consistent framework of [8] allows for a straightforward extension to reactive mixtures provided the sensible energy and enthalpy are extended to include the energy and the enthalpy of formation.
5 Coupling between lattice Boltzmann and chemical kinetics
In this work, the lattice Boltzmann code is coupled to the open source code chemical kinetics solver Cantera [14]. The Cantera solver is supplied with the publicly accessible GRI-Mech 3.0 mechanism [18] as an input data file.
The communication between the lattice Boltzmann solver and the Cantera chemical kinetics solver is executed as follows:
1.
An input from the lattice Boltzmann solver to Cantera is provided during the collision step in terms of internal energy, specific volume and mass fractions.
2.
Cantera internally solves numerically the integral equation (20) and thus the temperature at that state is obtained.
3.
Cantera calculates the production rates of species $\dot{\rho}_{a}^{c}$ and the transport coefficients including dynamic viscosity, thermal conductivity and the Stefan–Maxwell diffusivities as a function of the current state.
4.
The temperature obtained from Cantera is used to evaluate the equilibrium and quasi-equilibrium moments and populations. The transport coefficients are used to calculate the corresponding relaxation times and thus the collision step is complete.
Other thermodynamic parameters necessary for the simulations such as the specific heats and molecular masses are also obtained through Cantera. The reference standard state temperature is $T_{0}=298.15K$ and the reference standard state pressure is $P_{0}=1\ atm$. The data required by the lattice Boltzmann solver during runtime is obtained by querying Cantera through its C++ API using the "IdealGasMix" and "Transport" classes.
6 Results
As a first validation, probing the basic validity of our model, we
compute the flame speed in a premixed hydrogen/air mixture with the reactive Stefan–Maxwell formulation in a wide range of equivalence ratios $\phi$.
Subsequently, in order to test the isotropy of the model, the problem of outward expanding circular flame [19, 20] is solved for the premixed hydrogen/air mixture. For both test cases, we use the detailed chemical kinetics mechanism [21] involving the following nine species: \ce N2, O2, H2, H, O, OH, H2O, HO2, H2O2. It is worthwhile to mention that the model is not restricted to the detailed mechanisms. Reduced mechanisms available in the literature such as the five-species propane mechanism has also been tested with this model. In this paper, we will restrict ourselves to the more interesting detailed hydrogen/air mechanism which forms sharper and faster propagating flames.
While this benchmark not only probes the model’s behaviour in two dimensions, it is also a stringent isotropy test where it is crucial that the circular shape of the flame is preserved and not contaminated or distorted by the errors of the discrete numerics on the underlying Cartesian grid.
Finally, the models ability to capture non-linear instabilities is probed by simulations of wrinkled flames, which form as a result of
polychromatic perturbations.
6.1 Laminar flame speed
In order to validate our model, we calculate the burning velocity of a hydrogen/air mixture in a one-dimensional setup. As illustrated in Fig. 1, the setup
consists of a one-dimensional tube initialized with unburnt mixture at $T_{\rm u}=300K$ throughout from the left end up to 80% of the domain towards the right. The remaining 20% of the domain are initialized with the adiabatic flame temperature $T_{\rm af}$ and with the equilibrium burnt composition at the respective equivalence ratio. The pressure is initialized uniformly at $P_{in}=1\,atm$. Zero gradient boundary conditions are used at both ends for all variables using equilibrium populations.
At the left end, the velocity is imposed to be zero so that the flame propagates from right to left into the stationary unburnt mixture. The setup is used to calculate the burning velocity of the premixed \ceH2, N2, O2 system. Nitrogen is considered as an inert gas and thus does not split or form any radicals like nitrous oxides. However, the heat capacity of the inert gas has a strong influence on the flame temperature and consequently on the burning velocity. This is naturally accounted for in the formulation. The burning velocity is measured for various equivalence ratios ranging from $\phi=0.5$ to $\phi=2.25$.
We use the laminar flame thickness $\delta_{\rm f}$ at $\phi=1$ for defining the reference length,
where $\delta_{\rm f}=(T_{\rm af}-T_{\rm u})/max(\left|{dT}/{dx}\right|)$.
In order to accurately calculate the burning velocity, we use a long domain of $N\approx 90\delta_{\rm f}$, which corresponds to $10^{4}$ lattice points. In order to avoid the effect of the boundaries and transients due to initial acceleration, the flame speed $S_{L}$ is measured when the flame front approaches the middle of the domain. The results are compared to the data provided by [22] from multiple experimental and computational sources in Fig. 2. It can be seen that flame speed computed by our model agrees well with the data available in the literature. Although there is considerable dispersion in the literature for the flame speed values for fuel-rich mixtures $\phi>1$, the location of the peak burning velocity between $\phi=1.5$ and $\phi=2.0$ has been correctly captured. This test case indicates that the present model is a promising candidate for simulating reactive flows with the lattice Boltzmann method.
6.2 Circular expanding premixed flame
After confirming the $1D$ behaviour of the model, we compute the $2D$ circular expanding flame in a premixed hydrogen/air mixture with detailed chemistry. Similarly to the study [19, 20], due to symmetry, only a quarter of the flame is solved. Symmetry boundary conditions are used on the left and bottom edges of the square domain while the characteristic based outlet boundary conditions [23, 24] are imposed at the right and top edges of the domain. The bottom left corner is initialized with a burnt quarter sector at the adiabatic flame temperature $T_{\rm af}=1844.27K$ corresponding to the equivalence ratio $\phi=0.6$. The rest of the domain is initialized with an unburnt mixture at the temperature $T_{\rm u}=298K$. The composition in the burnt section is set to the equilibrium composition and the pressure in the entire domain is initialized to a uniform pressure $P=5\,atm$.
For this premixed initial condition, the burning velocity is obtained as $S_{L}=38.11\text{ cm/s}$ from solving a $1D$ flame propagation setup in Cantera. The flame thickness at these initial conditions is obtained as $\delta_{\rm f}=8.8\times 10^{-3}$ cm. A square domain with the side $N\approx 51\delta_{f}$ was considered, which corresponds to $1200\times 1200$ lattice points. The radius of the region initialized with the burnt equilibrium conditions is $R_{\rm ig}\approx 8.5\delta_{\rm f}$.
The characteristic flame transit time is defined as $\tau={\delta_{f}}/{S_{L}}=2.31\times 10^{-4}\,s$ [20]. Contours of temperature, velocity and mole fractions of oxygen and the hydroxide radical are shown at $t=0.082\tau$ in Fig. 2(b). As can be verified from Fig. 2(b), the solution is not contaminated by numerical noise or anisotropies and the contours do not contain any other spurious features. The thin interface of the hydroxide radical at the flame front is captured correctly and the curvature of the flame is maintained. This is in contrast to, e.g., [20], where the errors of the underlying numerical discretization leading to a spurious behaviour were reported when using Cartesian grids.
Next, we study the response of this setup to a deterministic perturbation to validate the model with the Direct Numerical Simulation (DNS) of [20].
The initial circular profile of the flame is perturbed with a sinusoidal profile,
$$\displaystyle R(\theta)=R_{\rm ig}(1+A_{0}\cos(4n_{0}\theta)),$$
(42)
where $n_{0}=4$ corresponds to the number of modes of the perturbation per $\pi/2$ sector of the flame and $A_{0}=0.05$ is the amplitude of the perturbation. The evolution of the perturbation is shown in Fig. 4. The heat release rate, $\dot{h}^{c}=-\sum_{a=1}^{M}H_{a}\dot{\rho}_{a}^{c}$, is a measure of the reactivity of the mixture. As it is evident in Fig. 3(a), during the initial stages of the evolution, the perturbed modes are continuous and the heat release rate is uniform along the circumference of the flame. As explained in [20], the reactivity and therefore the heat release rate reduces at the crest due to diffusion and more consumption of the deficient reactant. This, along with the hydrodynamic instability due to the density ratio and the thermal-diffusive instability due to the heat and mass imbalance of the deficient reactant leads to splitting of the peak of the crests into smaller cells, as it is visible in Fig. 3(b). A snapshot of the temperature contours over time shown in Fig. 3(c) verifies that the splitting of the flame indeed occurs from crests. Therefore, the splitting stems from the deterministic perturbation as expected, and not because of numerical noise. The mean radius of the flame is calculated by integrating along the flame front circumference,
$$\displaystyle\bar{R}={A}^{-1}\int R\;dA.$$
(43)
Here $A$ is the circumferential length and $R$ is the distance of the mean temperature isoline from the centre. On fitting $\bar{R}=at^{\alpha}$, the growth exponent was found to be $\alpha=1.16$, in agreement with the results from DNS in the literature wherein the value of the exponent was found to be between almost linear [20] and $1.25$ [25].
The local displacement speed [19, 20] is calculated as,
$$\displaystyle S_{d}=\frac{1}{\rho C_{p}\lvert\nabla T\rvert}\left[-\sum_{a=1}^{M}H_{a}\dot{\rho}_{a}^{c}+\nabla\cdot(\lambda\nabla T)-\rho\left(\sum_{a=1}^{M}C_{a,p}Y_{a}\delta\bm{u}_{a}\right)\cdot\nabla T\right].$$
(44)
With the local flame normal $\bm{n}=-\nabla T/\lvert\nabla T\rvert$, the absolute propagation speed is calculated as $S_{a}=S_{d}+\bm{u}\cdot\bm{n}$. The density weighted displacement speed is defined as $\hat{S_{d}}=\rho S_{d}/\rho_{u}$, where $\rho_{u}$ is the density of the unburnt mixture. The flame speeds are calculated as a mean over the flame interface isoline of $T=3T_{\rm u}$ in a way similar to equation (43). After the initial transients, the absolute propagation speed was found to reach a value of $6.2S_{L}$ whereas the density weighted displacement speed was found to fluctuate about $1.3S_{L}$. The corresponding values from the DNS results [20] are $7S_{L}$ and $1.5S_{L}$ respectively. The difference could be attributed to a number of factors including the type of grid, resolution, type of diffusion model etc. Overall, the results agree well with the DNS [19, 20].
7 Conclusion
In this paper, we proposed a lattice Boltzmann framework to simulate reactive mixtures. The novelty of the model lies in the fact that temperature and energy changes due to chemical reaction are handled naturally without the need of additional ad-hoc modelling of the heat of reaction. This was possible because of the thermodynamic consistency of the underlying multi-component model [8], which was extended to compressible reactive mixtures.
The species interaction is modelled through the Stefan–Maxwell diffusion mechanism which has been extended in this work to accommodate for the creation and destruction rates of the species due to chemical reaction.
Computational efficiency has been achieved through reduced description of energy which makes it possible to describe the physical system by only $M+2$ kinetic equations instead of $2M$ kinetic equations while retaining necessary physics such as the inter-diffusion energy flux. The model has been realized on the standard $D3Q27$ lattice, which not only reduces the computational costs compared to multispeed approaches but also possesses a wide temperature range, which is crucial for combustion applications.
The proposed model was validated in one and two dimensions with the $9$-species $21$ steps detailed hydrogen-air reaction mechanism. The accuracy of the model was assessed by calculating the burning velocity of a premixed hydrogen-air mixture in $1D$. The calculated flame speed agrees well with the results in the literature. The ability of the model to capture complex physics was tested by simulating a $2D$ expanding circular flame. The circular flame simulation exhibited good isotropy and low numerical noise. The setup was then subjected to monochromatic perturbations in order to study the evolution of the perturbed flame.
Good agreement with DNS simulations demonstrates viability of the proposed LBM for complex reactive flows.
\aucontribute
N.S. implemented the model, ran the simulations and wrote the first draft of the manuscript. B.D. and I.V.K. supervised the project.
All authors contributed to conceptualization of the model as well as writing, reading and approving the paper.
\competing
The authors declare that they have no competing interests.
\funding
This work was supported by the European Research Council grant No. 834763-PonD.
\ack
Computational resources at the Swiss National Super Computing Center CSCS were provided under grant No. s897.
Authors thank Ch. Frouzakis at ETHZ for discussions about the circular expanding flame.
References
[1]
T. Krüger et al.
“The lattice Boltzmann method”
Springer International Publishing, 2017
[2]
Sauro Succi
“The Lattice Boltzmann Equation”
OUP Oxford, 2018
[3]
Qinjun Kang, Peter C. Lichtner and Dongxiao Zhang
“Lattice Boltzmann pore-scale model for multicomponent reactive transport in porous media”
In Journal of Geophysical Research: Solid Earth 111.B5, 2006
DOI: 10.1029/2005JB003951
[4]
E. Chiavazzo, I.V. Karlin, A.N. Gorban and K. Boulouchos
“Combustion simulation via lattice Boltzmann and reduced chemical kinetics”
In J. Stat. Mech., 2009, pp. P06013
DOI: 10.1088/1742-5468/2009/06/p06013
[5]
Y. Feng, M. Tayyab and P. Boivin
“A Lattice-Boltzmann model for low-Mach reactive flows”
In Combustion and Flame 196, 2018, pp. 249–254
[6]
S.A. Hosseini et al.
“Hybrid lattice Boltzmann - finite difference model for low Mach number combustion simulation”
In Combustion and Flame 209, 2019, pp. 394–404
[7]
M. Tayyab, S. Zhao, Y. Feng and P. Boivin
“Hybrid regularized Lattice-Boltzmann modelling of premixed and non-premixed combustion processes”
In Combustion and Flame 211, 2020, pp. 173–184
[8]
N. Sawant, B. Dorschner and I.V. Karlin
“Consistent lattice Boltzmann model for multicomponent mixtures”
In Journal of Fluid Mechanics 909, 2021, pp. A1
DOI: 10.1017/jfm.2020.853
[9]
I. Karlin and P. Asinari
“Factorization symmetry in the lattice Boltzmann method”
In Physica A: Statistical Mechanics and its Applications 389.8, 2010, pp. 1530–1548
[10]
M.H. Saadat, F. Bösch and I.V. Karlin
“Lattice Boltzmann model for compressible flows on standard lattices: Variable Prandtl number and adiabatic exponent”
In Phys. Rev. E 99.1, 2019, pp. 013306
DOI: 10.1103/PhysRevE.99.013306
[11]
H. Grad
“On the kinetic theory of rarefied gases”
In Comm. Pure Applied Math. 2.4, 1949, pp. 331–407
DOI: 10.1002/cpa.3160020403
[12]
S.P. Thampi, S. Ansumali, R. Adhikari and S. Succi
“Isotropic discrete Laplacian operators from lattice hydrodynamics”
In Journal of Computational Physics 234, 2013, pp. 1–7
DOI: https://doi.org/10.1016/j.jcp.2012.07.037
[13]
F.A. Williams
“Combustion theory: the fundamental theory of chemically reacting flow systems”
Redwood City, Calif.: Benjamin/Cummings Pub. Co., 1985
[14]
D.G. Goodwin, R.L. Speth, H.K. Moffat and B.W. Weber
“Cantera: An Object-oriented Software Toolkit for Chemical Kinetics, Thermodynamics, and Transport Processes”, 2018
DOI: 10.5281/zenodo.1174508
[15]
R.J. Kee, M.E. Coltrin and P. Glarborg
“Chemically Reacting Flow: Theory and Practice”
Hoboken, NJ: Wiley Pub. Co., 2003
[16]
C.R. Wilke
“A Viscosity Equation for Gas Mixtures”
In The Journal of Chemical Physics 18.4, 1950, pp. 517–519
DOI: 10.1063/1.1747673
[17]
S. Mathur, P.K. Tondon and S.C. Saxena
“Thermal conductivity of binary, ternary and quaternary mixtures of rare gases”
In Molecular Physics 12.6, 1967, pp. 569–579
DOI: 10.1080/00268976700100731
[18]
G.P. Smith et al.
“GRI-Mech 3.0”, 1999
URL: http://http://combustion.berkeley.edu/gri-mech/version30/text30.html
[19]
C. Altantzis, C.E. Frouzakis, A.G. Tomboulides and K. Boulouchos
“Numerical simulation of propagating circular and cylindrical lean premixed hydrogen/air flames”
In Proceedings of the Combustion Institute 34.1, 2013, pp. 1109–1115
DOI: https://doi.org/10.1016/j.proci.2012.07.072
[20]
Christos Altantzis, Christos E. Frouzakis, Ananias G. Tomboulides and Konstantinos Boulouchos
“Direct numerical simulation of circular expanding premixed flames in a lean quiescent hydrogen-air mixture: Phenomenology and detailed flame front analysis”
In Combustion and Flame 162.2, 2015, pp. 331–344
DOI: https://doi.org/10.1016/j.combustflame.2014.08.005
[21]
Juan Li, Zhenwei Zhao, Andrei Kazakov and Frederick L. Dryer
“An updated comprehensive kinetic model of hydrogen combustion”
In International Journal of Chemical Kinetics 36.10, 2004, pp. 566–575
DOI: 10.1002/kin.20026
[22]
A.E. Dahoe
“Laminar burning velocities of hydrogen–air mixtures from closed vessel gas explosions”
In Journal of Loss Prevention in the Process Industries 18.3, 2005, pp. 152–166
DOI: https://doi.org/10.1016/j.jlp.2005.03.007
[23]
T.J. Poinsot and S.K. Lelef
“Boundary conditions for direct simulations of compressible viscous flows”
In Journal of Computational Physics 101.1, 1992, pp. 104–129
DOI: 10.1016/0021-9991(92)90046-2
[24]
C Feuchter, O Wagner, A Stief and T Beisswenger
“Turbulent flow simulations around a surface-mounted finite cylinder using an entropic multi-relaxation lattice Boltzmann method”
In Fluid Dynamics Research 51.5, 2019, pp. 055509
DOI: 10.1088/1873-7005/ab3baf
[25]
Michael A. Liberman et al.
“Self-acceleration and fractal structure of outward freely propagating flames”
In Physics of Fluids 16.7, 2004, pp. 2476–2482
DOI: 10.1063/1.1729852 |
Abstract
The Galactic Center Region has been surveyed in 1997–1998
with the Narrow Field Instruments
on-board the BeppoSAX satellite (2–10 keV energy range).
The X–ray emission from Sgr A*, the putative supermassive black hole
at the center of our Galaxy, has been measured and its spectrum studied,
together with the X–ray emission from several
bright X–ray sources, Low Mass X–ray Binaries (LMXBs)
containing neutron stars and black holes located
within the region ($|l|<2^{\circ})\times(|b|<1^{\circ}$).
New point–like X–ray sources have also been discovered.
The results on the diffuse emission coming from the SgrA Complex are
summarized here: it displays
a two–temperature thermal spectrum (kT${}_{\rm 1}\sim$ 0.6 keV
and kT${}_{\rm 2}\sim$8 keV) and an energy–dependent morphology, with the
hard emission (5–10 keV) elongated along the
galactic plane and the soft one (2–5 keV) spatially correlated with
the radio halo of SgrA East.
The results of the study of the diffuse X–ray emission from other
three fields in the Galactic Center Region
containing the giant molecular clouds Sgr B2, Sgr C and Sgr D
are reported for the first time.
The diffuse emission from the direction of these
three molecular clouds
shows a double temperature nature as well,
with a prominent iron line at $\sim$6.7 keV, the intensity and equivalent width
of which depends on the galactic longitude.
Moreover, the Sgr B2 spectrum shows an intense
6.4 keV line of fluorescent origin.
While the hard component of the Galactic Center diffuse emission displays a similar
temperature with respect to the Galactic Ridge, the low temperature plasma
is significantly softer. This may be due to a net contribution
produced from the accretion of material onto old isolated neutron stars located
inside the giant molecular clouds.
THE BEPPOSAX VIEW OF THE GALACTIC CENTER REGION
SIDOLI LARA
SSD-ESA/ESTEC
Postbus 299, NL-2200 AG Noordwijk, The Netherlands
1 Introduction
The Galactic Center (hereafter GC) Region is still one of the most
enigmatic regions of our Galaxy,
despite the deep investigations performed in the entire
electromagnetic spectrum.
The extreme interstellar absorption severely affects its
observability, especially in the
optical/ultraviolet range.
The interstellar medium is transparent
for energies below 2$\mu$m and above $\sim$2 keV.
Thus, the X–ray band is one of the privileged
observing windows for the GC.
Since a large concentration of mass exists towards the GC
direction, the crowding of sources, at all energies,
is severe as well.
For this reason,
instruments with a good spatial resolution are needed.
Several peculiar
objects are present within a few parsec from the Center of our Galaxy,
including the SgrA Complex, a remarkable ensemble of structures
observed in the radio band.
It is composed by at least four distinct objects (see Fig. 1):
the SgrA East shell, the SgrA East halo,
the SgrA West HII region, with its mini–spiral shape, and the compact
non–thermal radio source Sgr A*.
The first two structures are non–thermal radio sources.
Their real nature is not clear from the radio observations alone.
They might be supernova remnants
or by–products of a past outburst of activity from the GC.
Sgr A*, located at the dynamical center of our Galaxy, is
thought to be a super–massive black hole of about 2.5$\times 10^{6}$
solar masses.
The presence of such a large dark mass towards it is supported by
the measurements of the mass distribution and
dynamics of infrared stars in the central parsec of the Galaxy (e.g., Ghez et al. 1998).
The overall spectrum of Sgr A* is still poorly determined.
Up to now there are no sicure counterparts of Sgr A*
at any other wavelengths, with the exception of
the recent detection with $Chandra$ (Baganoff et al. 1999).
The Center of our Galaxy is rather silent at high energies, emitting
at a level which is several orders of magnitude below the Eddington luminosity
for a super–massive black hole of its mass.
In fact, the hard X–ray images obtained with coded–mask detectors
have shown that the
high energy activity previously detected with non–imaging instruments
and associated with the GC,
is instead produced by two black hole candidates
of stellar origin, 1E 1740.7–2942 and GRS 1758–258,
unrelated to the GC.
Previous X–ray observations led to the discovery that
a high density of point–like sources
exists towards the GC.
These sources include several transients, i.e. objects that are
bright in X–rays only sporadically.
It is unclear whether this concentration follows
the stellar mass density or if there is
evidence of an additional population of X–ray sources.
The bright sources (L${}_{\rm X}>10^{36}$ erg s${}^{-1}$) are binaries
containing compact objects (neutron stars or black holes)
accreting matter from stellar companions (probably of low mass).
Some of these sources are particularly interesting, belonging to the
class of “micro–quasars”, with relativistic,
double-sided radio jets.
The GC region is also characterized by strong diffuse
X–ray emission, first observed with the $Einstein$ Observatory (Watson et al. 1981).
The $Ginga$ satellite revealed the presence of a 6.7 keV line
emission from the Galactic
Plane which was particularly intense towards the GC direction (Koyama et al. 1989).
The ASCA satellite confirmed the presence of this diffuse component, which
extends
symmetrically with
respect to the GC.
Its spectrum is well described by a thermal hot plasma
with a temperature of $\geq$7 keV, but there is also evidence
for a multi–temperature
plasma. In fact, several emission lines are present, with the K–lines from
iron and sulfur (at $\sim$2.4 keV) particularly bright (Koyama et al. 1996).
The nature of the diffuse emission is still unknown, especially since
its temperature is too high to allow the confinement of the emitting plasma
by the galactic gravity.
Part of it (especially that at the lower energy
region of the spectrum) could be due to the integrated
thermal emission from supernova remnants. Other emitting processes
have been proposed in order to reject the thermal nature
of this component or to decrease its temperature:
non–thermal emission from SNRs, inverse Compton
scattering by relativistic
electrons, emission lines of non–thermal origin
produced during capture of electrons by accelerated ions,
charge exchange by low energy
heavy ions with neutral gas, non–thermal emission from the
interaction of low energy cosmic ray electrons with the interstellar
medium (see, e.g., the recent results of Valinia et al. 2000).
Another diffuse component was discovered with ASCA: a 6.4 keV
iron line component of fluorescent origin, the extent of which well
correlates
with the distribution of the molecular clouds
in the GC region (Koyama et al. 1996).
This emission is thought to be the result of irradiation of the
dense molecular clouds by hard photons coming from bright X–ray sources,
located inside or outside the cloud.
However, this emission seems too intense to be due to
reprocessing of hard X–rays from
any known source in the GC region.
Thus, a possible explanation is that the illuminating source
could have been Sgr A*, during a past phase of high-energy activity (Koyama et al. 1996;
Churazov et al. 1996).
Thus, the study of the diffuse component can also cast light on the
past high energy activity from the GC itself.
I report here on the results of the survey (1997-1998) of the GC
region performed with the Narrow Field Instruments (NFI) on-board
the BeppoSAX satellite (Boella et al. 1997). Only data obtained
from the MECS instrument (2–10 keV energy range) are relevant here.
In particular, the main objects of this study are the following:
•
the spectral properties
of the point–like X–ray sources, low mass X-ray binaries (LMXBs)
containing neutron stars and black holes, both displaying
persistent and transient emission;
•
the X–ray counterpart of Sgr A*, its spectrum and its emission level;
•
the spectrum and spatial distribution of the diffuse emission from the
GC Region.
2 Spectroscopy of the Point–like Sources
The region of the sky covered by the BeppoSAX NFI
observations is shown in Fig. 2.
A pointing on the
black hole candidate GRS 1758–258, $\sim 4.5$ degrees away from
the GC, is not included in this mosaic
(see Sidoli et al. 1999 and Sidoli 2000 for
a detailed description of the analysis and results).
We discovered new X–ray sources: the X–ray emission of plerionic
origin coming from the radio core
of the composite supernova remnant G0.9+01
(Mereghetti et al. 1998; Sidoli et al. 2000) and weak X–ray emission from
a still unidentified object, discovered near the molecular cloud Sgr D
at the position R.A.=$17^{{\rm h}}~{}48^{{\rm m}}~{}16^{{\rm s}}$,
Dec.=$-28^{\circ}~{}08^{\prime}~{}13^{\prime\prime}$ (J2000, $\sim$1${}^{\prime}$ uncertainty). Its power-law
spectrum (photon index $\Gamma\sim$1.4, flux corrected for the absorption
F${}_{\rm X}$$\sim$1.2$\times 10^{-12}$ erg cm${}^{-2}$ s${}^{-1}$) displays a
prominent iron line (EW=$2.0\pm{0.3}$ keV; Sidoli 2000).
Several bright sources (L${}_{\rm X}\sim 10^{36}$erg s${}^{-1}$), all previously known,
have been studied (Sidoli et al. 1999):
1E 1743.1–2843 (Watson et al. 1981; Cremonesi et al. 1998),
the persistent black hole candidates 1E 1740.7–2942 (Hertz & Grindlay 1984)
and GRS 1758–258 (Sunyaev et al. 1991),
the X–ray bursters SLX 1744–299, SLX 1744–300, A 1742–294, KS 1741–293
(Skinner et al. 1990; Pavlinsky, Grebenev & Sunyaev 1994; Kawai et al. 1988),
and the source at the GC position
(Watson et al. 1981).
We also detected two X–ray transients discovered very recently:
the X–ray burster SAX J1747.0–2853 (in’t Zand et al. 1998; Sidoli et al. 1998)
and the new superluminal source
XTE J1748–288
(Smith, Levine & Wood 1998).
The results of the spectroscopy of all these bright sources, low
mass X–ray binaries containing neutron stars (indicated by type I
X–ray bursts)
or black holes (with spectra similar to the “low–hard state” of the
well know black hole candidate
Cyg X–1) are shown in Fig. 3, where a single power-law model has been
used to fit the 2–10 keV emission.
The large range in interstellar absorbing column density is also
shown in Fig. 4 in dependence
of the galactic latitude, that can be translated
into height from the galactic plane, since all these sources are likely located at the
GC distance.
3 The X–ray Counterpart to Sgr A*
The large concentration of mass towards the GC direction
severely complicates the analysis
of the X–ray emission from Sgr A*, especially with the MECS instrument,
the spatial resolution of which is at a level
of 1${}^{\prime}$ ($\sim$2.5 pc at the GC distance).
Both the presence of the diffuse emission, peaking at the GC, and
several
point–like sources (known from previous missions and displayed in the Fig. 5)
within few arcminutes from Sgr A*, allowed us to place an upper limit to the
X–ray flux contributed by Sgr A* in the 2–10 keV band.
With the first imaging of the GC region with the $Einstein$ Observatory
(Watson et al. 1981, 0.5–4.5 keV) the point–like source 1E 1742.5–2859,
positionally coincident with Sgr A*, was discovered.
A diffuse component around the GC was also detected.
The ROSAT satellite observed this same region
in the 0.1–2.4 keV energy range (Predehl & Trumper, 1994) and,
in addition to the diffuse emission,
showed the presence of three different
sources.
One of these sources
is highly absorbed and located within $10^{\prime\prime}$ from Sgr A*.
Other X–ray sources are located within few arcminutes from Sgr A*:
the transient A 1742–289,
discovered in outburst with Ariel V in 1975 (Branduardi et al. 1976) and the
LMXB AX J1745.6–2901 (Maeda et al. 1996).
The spectrum and flux to be ascribed to
the GC point source(s) have been estimated (using the MECS data only)
extracting counts from a circular region with $2^{\prime}$ radius
centered at the position of the X–ray peak flux (see Fig. 5).
A local background has been taken from an external annular region ($6^{\prime}-8^{\prime}$), in order
to subtract the contribution of the diffuse emission.
Several emission lines (especially the sulfur line at $\sim 2.4$ keV and
the iron line at 6.7 keV) are present in this spectrum, indicative
of a contamination from the diffuse emission,
that indeed peaks at the GC position.
A good fit has been obtained with a hot plasma model with a temperature
of $\sim 4$ keV, absorbed by a column density, N${}_{\rm H}$, of 7$\times 10^{22}$ cm${}^{-2}$.
No evidence for a 6.4 keV line of
fluorescent origin is present, contrary to what is found from the
surrounding diffuse emission (see next section).
The 2–10 keV flux corrected for the absorption is
$\sim 4\times 10^{-11}$ erg cm${}^{-2}$ s${}^{-1}$ and the
luminosity is $\sim 3\times 10^{35}$ erg s${}^{-1}$ (a distance of 8.5 kpc is assumed).
Since also the diffuse emission
contributes to this flux, only an upper limit to
the emission from Sgr A* can be placed, at a level of $10^{35}$ erg s${}^{-1}$ (2–10 keV).
This estimate has been obtained extrapolating the surface brightness of
the diffuse emission from the surrounding region up to the Sgr A* position,
and assuming that all the net flux is contributed only by Sgr A*.
Recently, a $Chandra$ observation of the GC detected one weak source,
probably the real counterpart to Sgr A*, with a luminosity,
L${}_{\rm X}$, of $4\times 10^{33}$ erg s${}^{-1}$ (0.1–10 keV) (Baganoff et al. 1999).
4 Diffuse X–ray emission
The diffuse X–ray emission from the innermost 4 degrees along the
galactic plane has been mapped with four MECS pointed observations.
The regions considered for the spectral analysis are indicated
in Fig. 6.
They are marked by circles (radii = 8${}^{\prime}$) representing
the inner part of the field of view of four MECS pointings.
The fields include Sgr D (l$\sim$1.06, b$\sim$–0.14; field n. 3),
Sgr B2 (l$\sim$0.65, b$\sim$–0.04; field n. 2),
Sgr A (GC, field n. 1) and Sgr C (l$\sim$359.4, b$\sim$–0.11; field n. 4),
the main GC molecular clouds complexes as derived with the
CO surveys (e.g. Oka et al. 1998).
These molecular clouds are part of a large layer of neutral gas
distributed along the galactic plane (width$\sim$50 pc, length $\sim$500 pc).
The MECS instrument is particularly suited for the study of
the diffuse sources, due to its simple and unstructured point spread function.
All the spectra were extracted from within 8${}^{\prime}$ from
the pointing direction in order to use the region of the detector
with the best spatial and spectral properties
(and also, in some cases, to avoid off–axis point sources).
Only the spectrum from the Sgr A pointing was extracted from an
annular corona with outer and inner radii of 8${}^{\prime}$ and 2${}^{\prime}$ respectively,
in order to avoid the central point source(s) and the
probable counterpart to Sgr A*.
4.1 Diffuse X–ray emission from the SgrA Complex
The overall spectrum from the SgrA Complex
displays several emission lines.
A fit with a thermal bremsstrahlung continuum
plus three gaussian lines centered at $\sim$1.8, 2.4 and 6.7 keV,
to account for the most prominent emission lines in
the spectrum (from Si, S and Fe respectively), resulted in a temperature
of $\sim$13 keV.
However this temperature
is too high to be consistent with the presence of the
low energy emission lines.
A possible explanation is a multi-temperature plasma.
Thus, we
fitted the spectrum with two thermal emission plasma models.
The best fit parameters are kT${}_{\rm 1}$$\sim$0.6 keV
and kT${}_{\rm 2}$$\sim$8 keV.
A gaussian line added at 6.4 keV gives an equivalent width EW$\sim$120 eV.
To study the spatial distribution of the diffuse
emission, images in two different
energy bands (2–5 keV and 5–10 keV) were extracted.
They show significantly different morphologies, with the hard emission
with an elliptical shape elongated in the direction of the Galactic plane,
while the soft component shows a triangular shape, spatially correlated with the
triangular halo of SgrA East.
The fact that also the spectral data could be well described
by a two–temperature
model (with kT${}_{\rm 1}$$\sim$0.6 and kT${}_{\rm 2}$$\sim$8 keV),
can suggest that the SgrA Complex diffuse emission
may be explained in terms of two plasma components at
different temperatures and with different spatial distributions.
In this scenario, it is remarkable that the lower temperature
plasma is well correlated with the
Sgr A East triangular radio halo, a likely evolved SNR (Pedlar et al. 1989).
From the spectral parameters of the soft component and the size of the
SgrA East halo (radius$\sim$10 pc),
an electron density n${}_{\rm e}$$\sim$3 cm${}^{-3}$
and a pressure P$\sim$3$\times 10^{-9}$ erg cm${}^{-3}$
can be calculated (Sidoli & Mereghetti 1999).
This value is consistent with the
pressure P${}_{\rm Sedov}$$\sim$4$\times 10^{-9}$ erg cm${}^{-3}$
derived for a SNR in a Sedov phase.
In conclusion, if we assume that this
thermal emission is mostly produced by the Sgr A East halo,
its X–ray luminosity, pressure, density,
temperature of the emitting gas (0.6 keV)
and size ($\sim$20 pc), match well
with a supernova remnant origin in which thermal line emission
is produced when the expanding supernova
explosion heats the ISM to X–ray temperatures.
4.2 Diffuse X–ray emission from the Molecular Clouds
Three fields located on the galactic plane (within about 200 pc from the GC), free
from bright X–ray sources, have been studied (Sidoli 2000).
The MECS pointings considered here were centered on three molecular clouds
complexes, Sgr B2, Sgr C and Sgr C (Fig. 6).
The X–ray emission from these fields is
harder than that coming from the Sgr A Complex, and
displays weaker emission lines at low energy.
On the other hand the Fe lines around 6–7 keV are
again particularly bright. An iron line at 6.4 keV is prominent
in the spectrum from field n. 2, due to fluorescent emission from the
molecular cloud Sgr B2.
The dependence on the galactic longitude
of the properties of the iron line diffuse emission are reported in Figs. 7–8.
For comparison, also the parameters for the central
pointing on Sgr A (Field n.1) are shown.
The intensity (Fig. 8) of the 6.7 keV iron line
(in the case of field n. 2 it is a blend of 6.7 and 6.4 keV lines)
shows a peak at the GC.
Field n. 2 displays an excess with respect to field n. 4,
located at the same galactocentric
distance, due to the iron emission from Sgr B2 molecular cloud itself.
The spectra from the three fields
have been fit with thermal plasma models, first with single
temperature, then with two–temperature, due to soft excesses
remaining when fitting with a single component model.
All the results are reported in Table 1.
The temperatures
of the soft and hard components from these three fields
are in the ranges 0.2–0.4 keV
and 7–9 keV respectively.
The fact that the GC diffuse emission can be well fit
with a two–temperature plasma model, suggests a comparison with
the properties of the diffuse emission coming from other regions of the Galactic plane.
In fact, the Galactic plane is permeated by a diffuse X–ray component (the so–called
Galactic Ridge emission) with a two–temperature as well,
kT${}_{\rm 1}$$\sim$0.8 keV and kT${}_{\rm 2}$$\sim$8 keV (ASCA observations of the
Scutum Arm region; Kaneda et al. 1997).
While the hard components in the Galactic Ridge and in the GC
display a similar temperature,
the soft component coming from the GC shows a significantly
lower temperature (0.2–0.4 keV) with
respect to the Galactic Ridge emission (0.8 keV).
This discrepancy, other than to different physical
conditions of the ISM in the GC region,
could be due to a higher interstellar absorption that makes
quite uncertain the estimate of the parameters of
the soft component in the GC direction. Indeed,
in the Scutum Arm both the less severe interstellar
absorption and the larger band
of the ASCA instruments (0.5–10 keV) resulted in a
more precise measurement of the temperature.
Another explanation can be a possible “contamination”
of the soft part of the spectra
by emission physically related with the molecular clouds themselves.
Molecular clouds can emit X–rays in different ways.
In several cases the emission is
produced in the star forming regions naturally located
inside the clouds (e.g. Koyama et al. 1996b).
Pre–main sequence stars are strong X–ray emitters
(up to 10${}^{30}-10^{31}$ erg s${}^{-1}$) with a large variety of behaviours,
both with persistent thermal emission and with hard flares.
Another possibility can be the emission from
old isolated neutron stars (ONS) accreting from the dense ISM
inside the molecular clouds (Treves & Colpi 1991; Zane, Turolla & Treves, 1996).
The spectrum emerging from such emission
is indeed thermal and rather soft and could explain
the lower soft temperature, with respect to the Galactic Ridge,
of the fields containing
molecular clouds.
In this case, the X–ray luminosity contributed by a single ONS
depends on the density n${}_{\rm cloud}$ of the molecular cloud and on
the relative velocity v of the neutron star with respect to the
accreting matter:
L${}_{\rm ONS}\sim 7\times 10^{31}$ n${}_{\rm cloud}$ v${}_{10}^{-3}$ erg s${}^{-1}$, where v${}_{10}$ is the
relative velocity in units of 10 km s${}^{-1}$.
Assuming v${}_{10}\sim$75–100 km s${}^{-1}$ and n${}_{\rm cloud}\sim 10^{4}$ cm${}^{-3}$ for
a typical molecular cloud in the GC region,
we get L${}_{\rm ONS}$$\sim$10${}^{32}-10^{33}$ erg s${}^{-1}$.
The number of neutron star expected to reside in a single molecular cloud
can be calculated from the stellar distribution
as n${}_{\rm ONS}\sim 4\times 10^{3}~{}r^{-1.8}$ pc${}^{-3}$, where r is the distance from the GC.
Assuming r$\sim$200 pc and a cloud volume V${}_{\rm cloud}\sim 500$ pc${}^{3}$,
n${}_{\rm ONS}\sim 150$ ns/cloud; thus each cloud could contribute at a level of
about $10^{34}-10^{35}$ erg s${}^{-1}$ to the luminosity of
the soft component of the diffuse emission
from the GC region.
Molecular clouds can also be the sites of
reprocessing, scattering and reflection of hard photons from X–ray sources
located inside or outside the clouds themselves.
The strong X–ray emission in the 6.4 keV line from
Sgr B2 has be explained by
Koyama et al. (1996) and by Sunyaev & Churazov (1996) with the reflection
of hard X–rays coming from the GC during a past outburst from Sgr A*.
Also our data require the addition of a 6.4 keV line
to better fit the spectrum from Sgr B2,
but we cannot claim the prevalence of the 6.4 keV fluorescent line
with respect to the 6.7 iron line as Murakami et al. (1999).
Observations with a higher spectral and spatial resolution
are needed in order to precisely define the properties of this
X–ray emitting cloud and to cast light on the possible link
with a past high–energy activity of the GC.
5 Acknowledgements
The BeppoSAX satellite is a joint Italian-Dutch programme.
I acknowledge an ESA Fellowship.
All the results reported here are part of my PhD Thesis,
carried out at the “G. Occhialini” Institute of Cosmic Physics (IFC/CNR)
of the C.N.R., Milano (Italy).
I would like to thank Sandro Mereghetti for his constant support
and careful supervision. Aldo Treves is acknowledged for many
suggestions and his guide during the thesis work.
Lucio Chiappetti, Gianluca Israel and Giorgio Matt provided very important
assistance in the SAX data analysis and it has been a pleasure
to collaborate with them.
I am very grateful to Silvano Molendi, especially for his help with the analysis
of the diffuse emission.
I am grateful to Arvind Parmar for his useful comments on this manuscript.
My special thanks go to Annamaria Borriello, for her important help in numerous occasions and
for many interesting and constructive discussions.
6 References
Baganoff, F., Angelini, L., Bautz, M., et al.: 1999, AAS 195, 6201
Boella, G., Butler, R.C., Perola, G.C., et al.: 1997, A&AS 122, 299
Churazov, E., Gilfanov, M. & Sunyaev, R.: 1996, ApJ 464, L71
Cremonesi, D.I., Mereghetti, S., Sidoli, L. & Israel, G.L.: 1999, A&A 345, 826
Ghez, A.M., Klein, B.L., Morris, M., Becklin, E.E.: 1998, ApJ 509, 678
Hertz, P. & Grindlay, J.E.: 1984, ApJ, 278, 137
in’t Zand, J.J. et al.: 1998b, IAU circ. n.6846
Kaneda, H. et al. 1997, ApJ 491, 638
Kawai, N., Fenimore, E.E., Middleditch, J.,et al.: 1988, ApJ 330, 130
Koyama, K., et al.: 1989, Nature 339, 603
Koyama, K., Maeda, Y., Sonobe, T., et al.: 1996, PASJ 48, 249
Koyama, K., et al.: 1996b, PASJ 48, L87
Maeda, Y., et al.: 1996, PASJ 48, 417
Menten, K.M., Reid, M.J., Eckart, A. & Genzel, R.: 1997, ApJ 475, L111
Mereghetti, S., Sidoli, L. & Israel, G.L.: 1998, A&A 336, L81
Murakami, H., Koyama, K., Sakano, M., et al., 2000, ApJ 534, 283
Oka, T., Hasegawa, T., Sato, F., et al., 1998, ApJ Suppl. Ser. 118, 455
Pavlinsky, M.N., Grebenev, S.A., & Sunyaev, R.A.: 1994, ApJ 425, 110
Pedlar, A., et al.: 1989, ApJ 342, 769
Predehl, P. & Trumper, J.: 1994, A&A 290, L29
Sidoli, L., Mereghetti, S., Israel, G.L., et al.: 1998, A&A 336, L81
Sidoli, L., Mereghetti, S., Israel, G.L., et al.: 1999, ApJ 525, 215
Sidoli, L. & Mereghetti, S.: 1999, A&A 349, L49
Sidoli, L.: 2000, PhD Thesis, University of Milan, Italy
Sidoli, L., Mereghetti, S., Israel, G.L., Bocchino, F.: 2000, A&A submitted
Skinner, G.K., Foster, A.J., Willmore, A.P., Eyles, C.J.: 1990, MNRAS 243, 72
Smith D.A., Levine, A. & Wood, A.: 1998, IAU Circ. n.6932
Sunyaev, R.A., Churazov, E., Gilfanov, M., et al.: 1991, A&A 247, L29
Treves, A. & Colpi, M.: 1991, A&A 241, 107
Valinia, A., et al.: 2000, ApJ in press
Watson, M.G., Willingale, R., Grindlay, J.E. & Hertz, P.: 1981, ApJ 250, 142
Zane, S., Turolla, R. & Treves A.: 1996, ApJ 471, 248 |
\SetPages
11
\SetVol672017
{Titlepage}\Title
The OGLE Collection of Variable Stars.
Classical, Type II, and Anomalous Cepheids
Toward the Galactic Center***Based on observations
obtained with the 1.3-m Warsaw telescope at the Las Campanas Observatory of the Carnegie Institution for Science.
\AuthorI. S o s z y ń s k i${}^{1}$,
A. U d a l s k i${}^{1}$,
M. K. S z y m a ń s k i${}^{1}$,
Ł. W y r z y k o w s k i${}^{1}$,
K. U l a c z y k${}^{2}$,
R. P o l e s k i${}^{3}$,
P. P i e t r u k o w i c z${}^{1}$,
S. K o z ł o w s k i${}^{1}$,
D. S k o w r o n${}^{1}$,
J. S k o w r o n${}^{1}$,
P. M r ó z${}^{1}$,
M. P a w l a k${}^{1}$,
K. R y b i c k i${}^{1}$,
and A. J a c y s z y n - D o b r z e n i e c k a${}^{1}$
${}^{1}$Warsaw University Observatory, Al. Ujazdowskie 4, 00-478 Warszawa, Poland
e-mail: soszynsk@astrouw.edu.pl
${}^{2}$Department of Physics, University of Warwick, Gibbet Hill Road, Coventry, CV4 7AL, UK
${}^{3}$Department of Astronomy, Ohio State University, 140 W. 18th Ave., Columbus, OH 43210, USA
\Received
\Abstract
We present a collection of classical, type II, and anomalous
Cepheids detected in the OGLE fields toward the Galactic center. The
sample contains 87 classical Cepheids pulsating in one, two or three
radial modes, 924 type II Cepheids divided into BL Her, W Vir,
peculiar W Vir, and RV Tau stars, and 20 anomalous Cepheids – first
such objects found in the Galactic bulge. Additionally, we upgrade the
OGLE Collection of RR Lyr stars in the Galactic bulge by adding 828
newly identified variables. For all Cepheids and RR Lyr stars, we
publish time-series VI photometry obtained during the OGLE-IV
project, from 2010 through 2017.
We discuss basic properties of our classical pulsators: their spatial
distribution, light curve morphology, period–luminosity relations,
and position in the Petersen diagram. We present the most interesting
individual objects in our collection: a type II Cepheid with
additional eclipsing modulation, W Vir stars with the period doubling
effect and the RVb phenomenon, a mode-switching RR Lyr star, and a
triple-mode anomalous RRd star.
Stars: variables: Cepheids – Stars: variables: RR Lyrae – Stars: oscillations (including pulsations) – Galaxy: center – Catalogs
\Section
Introduction
Cepheids and RR Lyrae stars, sometimes collectively called classical
pulsators, undergo radial oscillations driven by the
$\kappa$-mechanism in helium ionization zones. Cepheid variables can
be divided into several subclasses which exhibit markedly different
masses, ages, and evolutionary histories. The youngest ones are
classical (or type I) Cepheids which play a fundamental role in the
calibration of the extragalactic distance scale thanks to their
period–luminosity (PL) relations. On the contrary, type II Cepheids
belong to an old stellar population, but the exact stage of their
evolution depends on their pulsation periods. BL Her stars, with
periods ranging between 1 and 5 d, evolve away from the horizontal
branch toward the asymptotic giant branch. W Vir stars (periods from 5
to 20 d) likely undergo blueward loops from the asymptotic giant
branch due to helium shell flashes. RV Tau stars (periods longer than
20 d) evolve away from the asymptotic giant branch toward a white
dwarf domain. In turn, the evolutionary status of anomalous Cepheids
is under debate. The two most popular scenarios are the evolution of a
single, intermediate-age, metal-poor star with mass of 1–2$M_{\odot}$,
and the evolution of coalescent binary systems of old, low-mass
stars.
Classical pulsating stars in the central regions of the Milky Way have
been the subject of extensive research in recent years. On the one
hand, the Optical Gravitational Lensing Experiment (OGLE) published a
large catalog of Cepheids and RR Lyr stars in the Galactic bulge
(Soszyński \etal2011, 2013, 2014). On the other hand, near-infrared
surveys, like the VISTA Variables in the Vía Láctea (VVV;
Minniti \etal2010) or IRSF/SIRIUS (Nagashima \etal1999), led to the
discovery of classical pulsators in the highly-obscured regions of the
bulge, mostly within $\sim 1$ degree from the Galactic plane (\egDékány \etal2015, Matsunaga \etal2011, 2013, 2015, 2016).
The previous version of the OGLE catalog (Soszyński \etal2011, 2013)
consisted of 32 candidates for classical Cepheids and 357 type II
Cepheids detected in about 69 square degrees in the Galactic bulge
covered by the OGLE-II and OGLE-III fields. These samples have been
successfully used in several interesting studies on the stellar
pulsations itself and on the structure of our Galaxy. Smolec \etal(2012) reported the discovery of the first BL Her stars exhibiting a
period-doubling effect – a phenomenon theoretically predicted by
Buchler and Moskalik (1992) and noticed for the first time in the OGLE
catalog of type II Cepheids. Feast \etal(2014) used five
fundamental-mode classical Cepheids from the OGLE-III catalog to
discover flared outer disk of the Milky Way. Kovtyukh \etal(2016)
studied metallicity of double-mode classical Cepheids from the OGLE
Collection of Variable Stars (OCVS). Recently, Bhardwaj \etal(2017)
matched the OGLE type II Cepheids with the VVV near-infrared
photometry to determine the distance to the Galactic center and to
investigate the spatial distribution of the old stellar population in
the bulge.
In this paper, we extend the OGLE Collection of Cepheids in the
Galactic bulge by objects identified in the OGLE-IV fields. The new
version of our collection consists of classical Cepheids, type II
Cepheids and, for the first time, anomalous Cepheids in the central
regions of the Milky Way. We also supplement the OGLE catalog of
RR Lyr stars with 828 newly detected variables of this type and update
the time-series photometry of the previously published stars.
The rest of the paper is organized as follows. In Section 2, we
present the OGLE photometric data used in this study. Methods applied
for the variable star identification and classification are introduced
in Section 3. Section 4 describes the Cepheid collection itself. In
Section 5, we discuss some interesting features of the published
samples of pulsating stars. Finally, conclusions are presented in
Section 6.
\Section
Observations and Data Reduction
The OGLE-IV data set used in this investigation was obtained between
March 2010 and August 2017 with the 1.3-m Warsaw telescope located at
Las Campanas Observatory (operated by the Carnegie Institution for
Science), Chile. A mosaic camera composed of 32 CCD chips with
Johnson-Cousins VI filters was used, providing a field of view
of 1.4 square degrees with a pixel scale of 0.26 arcsec. Details of
the instrument can be found in Udalski \etal(2015).
In total, 121 OGLE-IV fields toward the Galactic center were searched
for classical pulsating stars. Together with the OGLE-II and OGLE-III
fields analyzed by Soszyński \etal(2011), we studied an area of about
182 square degrees. Owing to the OGLE observing strategy optimized to
detect and monitor gravitational microlensing events, the number of
collected data points varies significantly from field to field – from
about 40 to $15\;000$ epochs per star in the I passband (median
value: 773) and from 0 to 175 in the V band (median: 37). Some
of these light curves may be supplemented with observations obtained
during the previous phases of the OGLE project and published by
Soszyński \etal(2011). One should be aware that small offsets between
individual light curves obtained during different stages of the OGLE
project are possible, mostly because of contamination by neighboring
stars in the dense bulge region. This should be taken into account
when merging the photometry from different phases of the project.
Data reduction was carried out using the Difference Image Analysis
software (Alard and Lupton 1998, Woźniak 2000). Detailed descriptions
of the photometric reductions and astrometric calibrations of the
OGLE-IV data are provided by Udalski \etal(2015).
\Section
Selection and Classification of Cepheids
Each of the 400 million I-band light curves collected by OGLE in
the Galactic bulge was subjected to a period search procedure based on
the Fourier technique implemented by Z. Kołaczkowski in the Fnpeaks
code†††http://helas.astro.uni.wroc.pl/deliverables.php?lang=en&active=fnpeaks.
The probed frequency space ranged from 0 to 24 cycles per day, with a
step of 0.00005 cycles per day. For each star the primary period was
subtracted from the original light curve and the period search
procedure was repeated on the residual data. Both periods (primary and
secondary) were recorded with their amplitudes and signal-to-noise
ratios. The light curves with the primary periods between 0.2 and
100 d and amplitudes larger than 0.05 mag were fitted with a Fourier
cosine series and the low-order Fourier coefficients $R_{21}$,
$\phi_{21}$, $R_{31}$, $\phi_{31}$ (Simon and Lee 1981) were derived.
This dataset became a basis for our variable star detection and
classification procedure. We visually examined the light curves with the
Fourier parameters characteristic for classical pulsators. The final
decision about the classification of each star was made after careful
analysis of the light curve shape quantified by the Fourier
coefficients. In ambiguous cases, we also considered other available
parameters of the stars, like colors and period ratios (for multi-mode
pulsators). Additionally, some candidates for pulsating stars were
identified during the search for eclipsing binary systems in the Milky
Way bulge area (Soszyński \etal2016a).
Contrary to the fundamental-mode pulsators, the overtone classical
Cepheids and $\delta$ Sct stars (both single- and multi-mode)
constitute continuity and it is a matter of convention which period
will be adopted as a borderline between both types of variables. In
this study, we adopted the same discrimination period as in other
parts of the OCVS: pulsators with the first-overtone periods shorter
than 0.23 d were classified as $\delta$ Sct stars and will be
published elsewhere. The longer-period (Population I) pulsators were
classified as classical Cepheids.
The Fourier coefficients were crucial for the first unambiguous
identification of anomalous Cepheids in the Galactic bulge. The
complete samples of classical pulsators in the Magellanic Clouds
published by the OGLE project (Soszynski \etal2015, 2017) allowed us
to show that different types of Cepheids are well separated in the
period–$\phi_{21}$ and period–$\phi_{31}$ diagrams. Fig. 1 shows
these diagrams for the bulge variables overplotted on their
counterparts from the Large Magellanic Cloud (LMC). The distinction
between anomalous Cepheids and classical Cepheids (and also RR Lyr
stars) is very prominent for the fundamental-mode pulsators (shown in
the left panels of Fig. 1), while for the first-overtone pulsators
(right panels of Fig. 1) the distinction is not so clear, so our
classification is less certain for these objects. We firmly detected
20 anomalous Cepheids (19 fundamental-mode and 1 first-overtone) in
the Galactic bulge, eight of which were previously classified as
classical Cepheids (Soszyński \etal2011) and six as RR Lyr stars
(Soszyński \etal2014). Previous designations of the reclassified
variables are given in the remarks of the collection.
Type II Cepheids have traditionally been divided into BL Her, W Vir,
and RV Tau stars based upon their pulsation periods. We used the same
discrimination periods as in the OGLE-III Catalog of Variable Stars
(Soszyński \etal2011): 5 d to distinguish between BL Her and W Vir
stars‡‡‡Note that in the Magellanic Clouds (Soszyński \etal2008) we adopted a period of 4 d as a borderline between the BL Her
and W Vir classes. and 20 d to separate W Vir and RV Tau variables.
We also distinguished 30 candidates for peculiar W Vir stars. This
subclass of type II Cepheids was defined by Soszyński \etal(2008)
based on the analysis of stars in the LMC. Peculiar W Vir stars are on
average brighter and bluer than regular W Vir stars, and have
different light curve morphology, with the rising branch being steeper
than the declining one. This latter feature is reflected in the period
– Fourier coefficients diagrams (Fig. 2), where the peculiar W Vir
stars occupy a limited area, although with some overlap with the
regular W Vir stars, so our classification should be treated with
caution. In the Galactic bulge, we only found one peculiar W Vir star
exhibiting additional eclipsing modulation (see Section 5.2), while in
the Magellanic Clouds at least 30% of these pulsators show signs of
binarity: eclipsing or ellipsoidal variations.
Our analysis revealed 68 classical Cepheids, 574 type II Cepheids and
828 RR Lyr stars that were not included in the previous versions of
the OCVS. Some objects have been reclassified which is summarized in
Table 1. In addition to the aforementioned pulsators that were
reclassified as anomalous Cepheids, we removed from the collection
several objects which likely belong to other (usually unknown) types
of variable stars. We confirm that all the five stars in the flared
disk of the Milky Way studied by Feast \etal(2014) are indeed the
fundamental-mode classical Cepheids.
\Section
OGLE Collection of Cepheids in the Galactic Bulge
The newly detected classical pulsators have been added to the
previously published OGLE catalogs. In total, the OCVS now contains 87
classical Cepheids, 924 type II Cepheids, 20 anomalous Cepheids, and
$39\;074$ RR Lyr stars in the Galactic bulge. The entire collection
can be downloaded via the WWW interface or from the FTP site:
http://ogle.astrouw.edu.pl
ftp://ftp.astrouw.edu.pl/ogle/ogle4/OCVS/blg/cep/
ftp://ftp.astrouw.edu.pl/ogle/ogle4/OCVS/blg/t2cep/
ftp://ftp.astrouw.edu.pl/ogle/ogle4/OCVS/gal/acep/
ftp://ftp.astrouw.edu.pl/ogle/ogle4/OCVS/blg/rrlyr/
Each object in our collection received a unique identifier which, with
some exceptions, follows the scheme proposed in the previous parts of
the OCVS. For example, classical Cepheids are designated as
OGLE-BLG-CEP-NNN, where NNN is a three-digit number. In the OGLE-III
catalog (Soszyński \etal2011), we used two-digit numbers in the
identifiers of classical Cepheids, but in this work we reached number
OGLE-BLG-CEP-100, which forced us to slightly change the naming
scheme. Objects from OGLE-BLG-CEP-001 to OGLE-BLG-CEP-032 were
included in the OGLE-III Catalog of Variable Stars (Soszyński \etal2011), while objects from OGLE-BLG-CEP-033 to OGLE-BLG-CEP-100 are the
newly discovered classical Cepheids. These new pulsators are ordered
by increasing right ascension. The names of anomalous Cepheids –
OGLE-GAL-ACEP-NNN – follow the scheme proposed by Soszyński \etal(2017), who discovered seven Galactic anomalous Cepheids in front of
the Magellanic Clouds.
All tables on the FTP site are in ASCII format and contain basic
information about the stars: their pulsation modes, J2000 equatorial
coordinates, intensity mean magnitudes in the I- and V-bands, periods in days with their uncertainties (derived with the
Tatry code of Schwarzenberg-Czerny 1996), epochs of maximum
light, peak-to-peak amplitudes in the I-band, and Fourier
coefficients $R_{21}$, $\phi_{21}$, $R_{31}$, $\phi_{31}$ derived for
the I-band light curves. For the already known variables we
provide their identifiers from the International Variable Star Index
(Watson \etal2006).
\Section
Discussion
\SubsectionClassical Cepheids
Fig. 3 shows the sky distribution of the OGLE classical Cepheids in
the Galactic coordinates. Despite the fact that classical Cepheids are
young (<300 Myr), relatively massive (3.5–20$M_{\odot}$) stars, a large
fraction of our objects are located far from the Galactic plane, up to
the Galactic latitudes $|b|=5$ deg. Feast \etal(2014) measured
distances to five OGLE fundamental-mode Cepheids and showed that they
are located in the flared outer disk of the Milky Way, from 0.9 to
2.1 kpc from the Galactic plane. Our collection probably contains more
classical Cepheids in the flared disk, not necessarily only the
fundamental-mode pulsators.
The presence of classical Cepheids inside the Galactic bulge is
currently a matter of debate. The VVV survey reported the discovery of
35 classical Cepheids in the highly-obscured regions of the Galactic
bulge (Dékány \etal2015) and suggested that the young stellar
population form an inner thin disk surrounding the Galactic
center. Matsunaga \etal(2016) used the IRSF/SIRIUS near-infrared
observations to detect their own set of Cepheids in this area (partly
overlapping with the VVV Cepheids), but they concluded that almost all
of these variables are located behind the bulge and the Galactic
center lacks classical Cepheids. The only exception from this rule are
four classical Cepheids discovered by Matsunaga \etal(2011, 2015) in
the very center of the Milky Way, in the so called Nuclear Bulge or
Central Molecular Zone.
We cannot participate in this interesting discussion, because the
Nuclear Bulge is inaccessible to optical observations owing to
enormous interstellar extinction. However, the OGLE Collection of
Variable Stars also contains several classical Cepheids relatively
close ($|b|<1$ deg) to the plane of the Milky Way. These objects are
probably located in the Galactic disk in front of the Galactic
bulge. For most of these stars, the V-band data are not
available, usually because of the high reddening which shifts the V-band magnitudes below the OGLE detection threshold.
Our collection includes 16 double-mode and three triple-mode classical
Cepheids. Positions of these stars in the Petersen diagram
(shorter-to-longer period ratio \vsthe longer period) overplotted on
the LMC and SMC beat Cepheids are shown in Fig. 4. Soszyński \etal(2011) showed that period ratios of double-mode classical Cepheids in
the Galactic bulge are smaller than in the Magellanic Clouds, which is
probably caused by metallicity differences between these stellar
environments. Our extended sample confirms that indeed double-mode
Cepheids in the bulge, both F/1O and 1O/2O pulsators, have smaller
period ratios than their counterparts in the Magellanic Clouds, but
this rule is valid only for the short-period variables. The longest
period beat Cepheids (two F/1O and two 1O/2O pulsators) have period
ratios virtually the same as variables in the Magellanic Clouds.
Triple-mode Cepheids (marked with triangles in Fig. 4) have generally
short periods and also tend to have smaller period ratios compared to
the LMC and SMC variables.
\Subsection
Type II Cepheids
In this paper, we present the most numerous sample of type II Cepheids
detected in one stellar environment. Two-dimensional spatial
distribution of 924 type II Cepheids from our collection is presented
in Fig. 5. These stars generally show a strong concentration toward
the Galactic center, but they avoid regions around the Galactic
plane. This is of course caused by the large amount of interstellar
matter toward these regions which obscures stars in the optical
regime. Such a distribution suggests that the vast majority of our
type II Cepheids are members of the Galactic bulge.
The same conclusion can be drawn from the analysis of the
period–luminosity diagram (Fig. 6), where as the “luminosity” we
used the reddening-independent Wesenheit index in the Galactic bulge
defined by Pietrukowicz \etal(2015) as $W_{I}=I-1.14(V-I)$ (Fig. 6
includes only those variables, for which both, I- and V-band, mean magnitudes are available). The majority of type II
Cepheids populate a period–luminosity relation that is an extension
of the relation for RRab stars in the bulge, which indicates that both
classes are on average at the same distance from us. In turn, most of
the classical Cepheids are fainter than type II Cepheids with the same
periods, which suggests that the most classical Cepheids in our sample
are located far behind the Milky Way center, also in the flared disk
(Feast \etal2014). As already mentioned, a number of classical
Cepheids in our sample lie in front of the Galactic bulge, but most of
them do not have V-band measurements and they are not included
in the period – Wesenheit index diagram.
Our collection covers the full range of periods observed in type II
Cepheids: from 1 to 66 d§§§In the OCVS we provide “single”
periods, \ieintervals between successive minima, even if the
period-doubling effect is present. BL Her variables with the shortest
periods are adjacent to the longest-period RR Lyr stars and the
boundary period between these groups is not strictly defined. We
traditionally used a 1.0 day period as a borderline between RR Lyr and
BL Her stars.
In our collection, type II Cepheids with pulsation periods longer than
20 d are classified as RV Tau stars. The characteristic feature of
RV Tau variables – alternating deep and shallow minima (period
doubling) – seems not to be a distinctive classification
characteristic. On the one hand, the period doubling is a common
phenomenon also among long-period W Vir stars, down to a period of
about 16 d. Fig. 7 shows the light curve of OGLE-BLG-T2CEP-494
($P=16.736$ d) which exhibits alternations of minima and maxima. On
the other hand, our collection contains some long-period type II
Cepheids ($P>20$ d) with no clear alternations of minima, but in our
catalog they are formally categorized as RV Tau stars. Such stars may
also be classified as yellow semiregular variables (SRd stars).
At least 13 RV Tau variables in our collection belong to the RVb
class, which means that these stars experience long-term variations of
the mean brightness, with periods of 470–2800 d and amplitudes from
0.2 to 2.5 mag. Also two W Vir stars (with pulsation period shorter
than 20 d) exhibit such modulation (Fig. 7). The long-term changes are
commonly interpreted as being caused by periodic obscuration of a
binary system by a circumbinary dust disk (\egLloyd Evans 1985,
Pollard \etal1996, Van Winckel \etal1999). The long-term homogeneous
OGLE light curves in two standard filters may provide important
constraints on theoretical models of the RVb phenomenon. In some
objects, the depths of the RVb modulation significantly change from
cycle to cycle.
The OGLE catalog of type II Cepheids in the Magellanic Clouds
(Soszyński \etal2008, 2010) contains an exceptionally large fraction
of pulsators that are members of binary systems. Over a dozen type II
Cepheids show additional eclipsing or ellipsoidal modulation. Most of
these objects were classified as peculiar W Vir stars, which suggests
that the binarity might be related to the origin of this class of
pulsating stars (see the discussion in Pilecki \etal2017).
It appears that the fraction of eclipsing Cepheids in the Galactic
bulge is much lower than in the Magellanic Clouds. In our collection
we found only one type II Cepheid that exhibits additional eclipsing
modulation: OGLE-BLG-T2CEP-674. Its light curve is shown in the upper
panel of Fig. 8, while middle panels show disentangled pulsation and
eclipsing variabilities. The orbital period of this system is close to
2 years (714 d), which interferes with annual gaps in the OGLE
photometry, but luckily both, primary and secondary, eclipses are well
visible in the light curve.
A closer look at the photometric data of OGLE-BLG-T2CEP-674 reveals
yet another interesting feature. Based on the pulsation light curve we
constructed an observed-minus-calculated (O$-$C) diagram which is
presented in the lower panel of Fig. 8. This diagram shows a long-term
sinusoidal-like variations which may be caused by the light-travel
time effect in a binary system. Surprisingly, a possible period of the
sinusoidal variations visible in the O$-$C diagram (about 1900 d) is
much longer than the orbital period measured from the eclipsing
modulation (714 d), which suggest that OGLE-BLG-T2CEP-674 is a member
of at least a triple system with the third body on a $\sim$1900 d
orbit. However, we must stress here that this period is comparable to
the total time span of the OGLE-IV photometry (the star was not
observed during the previous phases of the OGLE survey) so we cannot
be sure that the variations in the O$-$C diagram are indeed periodic
ones.
\Subsection
Anomalous Cepheids
In this paper, we present the first bona fide anomalous Cepheids
detected in the Galactic bulge. Eight of these stars were published by
Soszyński \etal(2011) as classical Cepheids and six other objects
were previously classified as RR Lyr stars (Soszyński \etal2014),
because at that time there was no reliable method to distinguish
anomalous Cepheids from other types of classical pulsators based
solely on their light curve shape. This situation changed when
Soszyński \etal(2015, 2017) selected the richest known population of
anomalous Cepheids in the Magellanic Clouds and showed that the shape
of their light curves (quantitatively assessed by the Fourier
coefficients $\phi_{21}$ and $\phi_{31}$) is an efficient diagnostic
to separate these groups. Positions in the sky of the 20 anomalous
Cepheids in the Galactic bulge are shown in Fig. 9. Despite a small
number of objects, we may notice a lack of anomalous Cepheids close to
the Galactic plane due to high interstellar extinction in these
regions.
Fundamental-mode anomalous Cepheids are well separated from classical
Cepheids in the period–luminosity diagram (Fig. 6), but,
paradoxically, anomalous Cepheids have brighter apparent magnitudes
than their classical siblings. An analysis of pulsating stars in the
LMC (Soszyński \etal2015) shows that anomalous Cepheids are
intrinsically fainter by about 0.7 mag than classical Cepheids. On the
other hand, anomalous Cepheids are on average brighter than type II
Cepheids, just like in the Magellanic Clouds, so we can safely assume
that our sample of anomalous Cepheids belongs to the Galactic bulge.
\Subsection
RR Lyrae stars
The newly detected 828 RR Lyr stars represent about 2% of all
variables of this type discovered by OGLE in the central regions of
the Galaxy (Soszyński \etal2014). These new detections were
previously overlooked for various reasons. Most of the new RR Lyr
variables (72%) are first-overtone pulsators (RRc stars) which
usually show nearly sinusoidal light curves that can be confused with
other classes of variable stars, for example close binary
systems. Some variables were missed because of a small number of data
points, or very low amplitudes, or pulsation periods close to 1 or 2/3
of the sidereal day which affected the period determinations.
The time-series photometry of RR Lyr stars published by
Soszyński \etal(2014) have been supplemented with new observations
collected by the OGLE-IV survey up to August 2017. At present, the
time-span of the OGLE-IV light curves exceeds 7 years and for some
stars it can be increased to even 20 years by joining the OGLE-II and
OGLE-III data points published by Soszyński \etal(2011). These light
curves can be used for follow-up studies of all stationary and
non-stationary phenomena in pulsating stars: non-radial modes, Blazhko
effect, period changes, mode switching, light-time effect in binary
system hosting pulsating stars, etc.
Fig. 10 shows an example of such non-stationary behaviors: the light
curve of OGLE-BLG-RRLYR-17342 – a Blazhko RR Lyr star that switched
from a single-mode RRab star to a double-mode RRd star. Lower panels
of Fig. 10 display folded light curves of OGLE-BLG-RRLYR-17342
obtained by OGLE in selected seasons. The amplitude of the
fundamental-mode pulsation has continuously decreased over the eight
years of the monitoring and at present it is slightly above to the
detection limit of the OGLE photometry. In 2015, the first-overtone
mode appeared in the light curve and the star became a double-mode
pulsator. Three other mode-switching RR Lyr stars in the Galactic
bulge were reported by Soszyński \etal(2014).
We reclassified 24 double-mode RR Lyr stars from ordinary to anomalous
RRd stars. This latter class of pulsators was defined by
Soszyński \etal(2016b) who noticed a distinct group of double-mode
RR Lyr variables in the Magellanic Clouds. Anomalous RRd stars have
different period and amplitude ratios than typical RRd stars and most
of them show Blazhko modulation (Smolec \etal2015a). In the present
investigation, the main classification criterion was the position of a
given pulsator in the Petersen diagram (Fig. 11). All multi-mode
RR Lyr variables that are located outside the curved sequence in the
Petersen diagram (both, above and below, this sequence) were
classified as anomalous RRd stars. Thus, we have expanded the
definition of anomalous double-mode RR Lyr variables introduced by
Soszyński \etal(2016b) for the Magellanic Clouds members by adding
objects with the $P_{\textrm{1O}}/P_{\textrm{F}}$ period ratios higher
than observed for “classical” RRd stars. The majority of our
candidates for anomalous RRd stars in the Galactic bulge share the
features of their Magellanic Clouds counterparts: usually the
fundamental mode is the dominant one and most of these objects exhibit
the Blazhko effect. Currently, the OCVS contains 31 anomalous RRd
stars, both reclassified and newly discovered variables. This group
includes also two mode-switching RR Lyr stars: OGLE-BLG-RRLYR-13442
and OGLE-BLG-RRLYR-17342.
One of the new detections deserves special attention.
OGLE-BLG-RRLYR-38791 is a triple-mode star pulsating likely in the
fundamental, first-overtone, and second-overtone modes. Its
disentangled light curves corresponding to the three pulsation modes
are displayed in Fig. 12. The classification of this object is
unclear. We decided to list it among anomalous RRd stars for a few
reasons. First, its fundamental-mode and first-overtone periods place
this object among anomalous RRd pulsators in the Petersen diagram
(Fig. 11). Second, the fundamental-mode has the largest amplitude. And
third, the shape of the fundamental-mode light curve resembles those
in some anomalous RRd stars.
Another triple-mode RR Lyr star was found in the OCVS by Smolec \etal(2015b). OGLE-BLG-RRLYR-24137 (in the present work also classified as
an anomalous RRd star) exhibits two radial fundamental and
first-overtone modes and the third periodicity that may correspond to
the radial third-overtone mode or a non-radial mode.
\Section
Conclusions
We present the largest collection of classical, type II, and anomalous
Cepheids in and toward the central regions of the Milky Way. We
release the long-term time-series photometry obtained by the OGLE
project for all the stars. The OGLE samples of classical pulsators
provide us with a tool to test evolutionary and stellar pulsation
models, as well as to study the structure and star formation history
in the Galactic bulge region, which is crucial for our understanding
of the Milky Way evolution.
In the near future, the OCVS will be extended by classical pulsators
found within the OGLE Galaxy Variability Survey regularly observing an
area of about 2000 square degrees in the Galactic disk and outer
regions of the Galactic bulge. This sub-project of the OGLE survey has
been carried out since 2013 and currently it accumulated long-term
multi-epoch photometric data which can be used to perform an effective
search for variable stars with their reliable classification. This
dataset promises to be a powerful tool to improve our understanding of
the Galactic structure.
\Acknow
We would like to thank Profs. M. Kubiak and G. Pietrzyński,
former members of the OGLE team, for their contribution to the
collection of the OGLE photometric data over the past years. We are
grateful for discussions and constructive comments to R. Smolec.
We thank Z. Kołaczkowski and A. Schwarzenberg-Czerny for
providing software used in this study.
This work has been supported by the National Science Centre, Poland,
grant MAESTRO no. 2016/22/A/ST9/00009. The OGLE project has received
funding from the Polish National Science Centre grant MAESTRO no.
2014/14/A/ST9/00121.
References
\refitem
Alard, C., and Lupton, R.H.1998\ApJ503325
\refitemBhardwaj, A., \etal2017Å605A100
\refitemBuchler, J.R. and Moskalik, P.1992\ApJ391736
\refitemDékány, I., \etal2015\ApJ812L29
\refitemFeast, M.W., Menzies, J.W., Matsunaga, N., and Whitelock, P.A.2014Nature509342
\refitemKovtyukh, V., \etal2016\MNRAS4602077
\refitemEvans, T.L.1985\MNRAS217493
\refitemMatsunaga, N., \etal2011Nature477188
\refitemMatsunaga, N., \etal2013\MNRAS429385
\refitemMatsunaga, N., \etal2015\ApJ79946
\refitemMatsunaga, N., \etal2016\MNRAS462414
\refitemMinniti, D., \etal2010New Astronomy15433
\refitemNagashima C. \etal1999 in Nakamoto T., ed., Proc. Star Formation 1999. Nobeyama Radio Observatory, Nagano, p. 397
\refitemPietrukowicz, P., \etal2015\ApJ811113
\refitemPilecki, B., \etal2017\ApJ842110
\refitemPollard, K.R., Cottrell, P.L., Kilmartin, P.M., and Gilmore, A.C.1996\MNRAS279949
\refitemSchwarzenberg-Czerny, A.1996\ApJ460L107
\refitemSimon, N.R., and Lee, A.S.1981\ApJ248291
\refitemSmolec, R., \etal2012\MNRAS4192407
\refitemSmolec, R., \etal2015a\MNRAS4473756
\refitemSmolec, R., \etal2015b\MNRAS4473873
\refitemSoszyński, I., \etal2008\Acta58293
\refitemSoszyński, I., \etal2010\Acta6091
\refitemSoszyński, I., \etal2011\Acta61285
\refitemSoszyński, I., \etal2013\Acta6337
\refitemSoszyński, I., \etal2014\Acta64177
\refitemSoszyński, I., \etal2015\Acta65233
\refitemSoszyński, I., \etal2016a\Acta66405
\refitemSoszyński, I., \etal2016b\MNRAS4631332
\refitemSoszyński, I., \etal2017\Acta67103
\refitemUdalski, A., Szymański, M.K., and Szymański, G.2015\Acta651
\refitemVan Winckel, H., Waelkens, C., Fernie, J.D., and Waters, L.B.F.M.1999Å343202
\refitemWatson, C.L., Henden, A.A., and Price, A.2006 in the Society for Astronomical Sciences 25th Annual Symposium on Telescope Science, eds. B. Warner, \etal, Soc. Astron. Sci., Big Bear, California, 47
\refitemWoźniak, P.R.2000\Acta50421 |
Self-affine Asperity Model for earthquakes
V. De Rubeis${}^{1}$, R. Hallgass${}^{2}$, V. Loreto${}^{2}$,
G. Paladin${}^{3}$, L. Pietronero${}^{2}$ and P. Tosi${}^{1}$
${}^{1}$Istituto Nazionale di Geofisica
Via di Vigna Murata 605 I-00143 Roma, Italy
${}^{2}$Dipartimento di Fisica, Università di Roma
’La Sapienza’
P.le A.Moro 2 I-00185 Roma, Italy
${}^{3}$Dipartimento di Fisica, Università dell’Aquila
Via Vetoio I-67100 Coppito, L’Aquila, Italy
ABSTRACT
A model for fault dynamics consisting of two rough
and rigid brownian profiles that slide one over the other is introduced.
An earthquake occurs when there is an intersection between the
two profiles. The energy release is proportional to the overlap interval.
Our model exhibits some specific features which follow from the
fractal geometry of the fault:
(1) non-universality of the exponent of the Gutenberg-Richter law
for the magnitude distribution;
(2) presence of local stress accumulation before
a large seismic event; (3) non-trivial space-time clustering
of the epicenters. These properties are in good agreement with
various observations and lead to specific predictions that can be
experimentally tested.
PACS NUMBERS: 91.30.Px, 05.40.+j
Many forms of scaling invariance appear in seismic phenomena: the celebrated
Gutenberg-Richter law for the magnitude distribution [1],
the Omori law for the time correlations of aftershocks
[2], space-time clustering of the epicenters [3] are
a common mark of the earthquake statistics.
Unfortunately, the complexity of modelling the motion of a fault
system, even in rather well controlled situation such as the San
Andreas fault in California, is a highly difficult task and it is
still controversial what is the correct theoretical framework at
the very origin of scaling laws. It is thus important to
individuate models as simple as possible that are able to exhibit
the main qualitative features of the fault dynamics.
Their physical relevance stems from the specific predictions on the
real seismic activity which might be verified from experimental data.
One of the first attempt in this direction is due to Burridge and
Knopoff [4] who introduced a stick-slip model of coupled
oscillators to mimic the interaction of two fault surfaces. In practice
one considers blocks on a rough support connected to one another
by springs.
They are also connected by other springs to a driver which moves
at a very low constant velocity. The blocks stick until the
spring forces overwhelms the static friction
and then one or more blocks slide, releasing an ‘earthquake’ energy
proportional to the sum of the displacements.
A numerical integration of the Newton equations for
a one-dimensional chain with a large number of homogeneous blocks
have been showed to exhibit the Gutenberg-Richter law [5]
(see also [6] for the connection with the chaotic behaviour
of the system).
Moreover, it has been proposed that the qualitative aspects of
earthquakes (and of Burridge and Knopoff models) are captured by the
so-called
Sandpile models, which represent the paradigm of a large class of
Self-Organized Critical (SOC) systems [7], where the scaling is spontaneously
generated by the dynamics. In fact, there is a whole generation of SOC models
to explain the scale invariant properties of earthquakes [8,9].
These type of models suggest however that there is no stress accumulation
before a big earthquake and the exponent of the Gutenberg-Richter law is
expected (with some exceptions [10]) to be universal. In addition the
space-time distribution of the epicenters has no clear relation with
the experiments where non-trivial clustering and correlations are present.
In order to go beyond these limitations we propose here an alternative
approach where the critical behavior is
not self-organized but stems from the fractal geometry of the fault
that is supposed to arise as a consequence of geological processes on very long
time scales with respect to the seismic dynamics.
Looking at the system on the time scale of human records the fault structure
can be considered assigned and just slightly modified by earthquakes.
Many authors pointed out that natural rock surfaces
are represented by fractional brownian surfaces over a wide scale range
[11] and that also the topographic traces of the fault
surfaces exhibit scale invariance [12].
A fault can thus be regarded as a
statistically self-affine profile $h(x)$, whose height scales as
$|h(x+\ell)-h(x)|\sim\ell^{H}$. In $d=2$, such a profile
$h(x)$ can be generated by fractional brownian motion with exponent
$H$ and in $d=3$ by the standard generalization given by
brownian reliefs [13,14].
The exponent $0\leq H\leq 1$ controls the roughness of the fault
where the standard random walk profile corresponds to $H=1/2$,
and a differentiable curve corresponds to $H=1$.
The fractal dimension of the profile is well known to be $D_{F}=d-H$.
Let us now introduce the self-affine asperity model (SAM)
that is, in a certain sense, the limit of infinite rigidity
of the Burridge-Knopoff models and it represents an alternative limit
with respect to the SOC models.
The model is defined by the following dynamical rules:
(i) We consider two profiles, say $h_{1}(x)$ and $h_{2}(x)$,
on parallel supports of length $L$ at infinite distance.
The initial condition is obtained by putting them in contact in
the point where the height difference is minimal so that
$h_{1}-h_{2}\geq 0\,,\ \forall\,x\in[0,L]$ (see Fig. 1a);
(ii) The successive evolution is obtained by drifting
a profile in a parallel way with respect to the other one,
at a constant speed $v$, so that $h_{1}(x;t)=h_{1}(x-v\,t)$;
(iii) At each time step $t$, one controls whether there
are new contact points between the profiles, i.e.
whether $h_{1}(x;t)-h_{2}(x)<0$ for some $x$ value.
An intersection represents a single seismic event
and starts with the collision of two asperities of the profiles.
The energy released is assumed to be proportional to
the extension of the overlap between the two asperities
in contact, see Fig. 1b;
(iv) We do not allow the developing of new earthquakes
in a region where a seismic event is already taking place .
With these rules, the motion of the two profiles simulate the
slipping of the two walls of a single fault. The points of collision are
the points of the fault where the morphology prevents the free
slip: these are the points where there is an accumulation of stress
and, consequently, a raise of pressure.
When the local pressure exceeds a certain threshold, it happens
a breaking, an earthquake, which allows to relax the stress and
redistribute the energy, previously accumulated, all around.
Rule (iii) of the SAM stems from the fact that the magnitude of a
real earthquake is proportional to the log of the seismic moment
$M_{0}$, which, on its turn, is proportional to the average displacement
of the fault according to the standard geophysical definition.
For sake of simplicity, in the SAM, there is no real breaking of the
profiles as a consequence of an earthquake and the profiles maintain
their structures after a crash.
It is possible to introduce a more realistic breaking mechanism
where there is also a modification of the asperity form
after an earthquake. However, we have verified that the main
qualitative features remain unchanged. So we are in the opposite
perspective than SOC models. In our case
the earthquake dynamics has no effect on the structure of the profile.
Realistic situation could well correspond to intermediate cases of course.
It is worth to stress that the SAM exhibits a strong
non-locality since a collision in a point $x$, at the time $t$ can
trigger, at later time, a subsequent event also very far away.
One of the main advantage of the SAM consists in the possibility of
deriving various analytic results using the properties of brownian profiles.
The most impressive characteristic of the earthquake statistics
is the Gutenberg-Richter law. It states that the probability
$P(E)\;dE$ that an earthquake releases an energy in the interval
$[\,E\,,\,E+dE\,]$ scales according to a power law
$P(E)\sim E^{-\beta-1}$
with an exponent $\beta$ of order of the unity [10].
It is a controversial issue whether $\beta$ is universal or varies
in a narrow range according to the characteristics of the fault system.
In the framework of our model it is possible to relate the value of
the exponent $\beta$ to the geometrical properties of the faults.
In particular it can be showed that:
$$\beta=1-{H\over(d-1)}={D_{F}-1\over d-1}.$$
(1)1( 1 )
This relation accounts for the direct dependence of the $\beta$-exponent
on the roughness of the faults $H$.
In order to derive (1), consider the profile
$h_{1}(x;t)-h_{2}(x)$, which,
being given by the difference of two
brownian profiles is, on its turn, a brownian profile at any time $t$.
The statistics of the intersections between the two profiles
is then given by the statistics of the intersections of the brownian profile
difference with a straight line along the temporal axis.
Due to the invariance under temporal shifts of the profile, we
can assume that the statistics of the intersections obtained at any time
with a profile difference is given by the statistics of the intersections of
an infinite profile with a zero level straight line.
In this perspective, a seismic event releases an energy proportional
to the interval between two sub-sequent intersections between a
brownian profile and the zero level straight line.
It is well known that the set obtained by the intersection
between a fractional brownian profile or relief of dimension $d-H$
embedded in a $d$-dimensional space and a hyper-surface of
dimensionality $d-1$, is a fractal with dimension given by the law
of addition of the codimension [13], $(d-H)+(d-1)-d$,
so that the number of intersections in a hyper-surface of volume
$E\sim L^{d-1}$ scales as: $N(L)\sim L^{d-1-H}.$
Now, if we identify the energy released from an earthquake
with the size $E$ of an intersection, we can determine the exponent
$\beta$ by consistency requirements.
In our case the probability $P(E)$
is given by the probability of finding
an intersection, of size $E$,
between a $d$-dimensional surface and a $(d-1)$-dimensional
hyperplane.
As a consequence of the geometric properties of the model,
$P(E)$ follows a power law.
Let us consider the average value of the intersection size:
$${\langle E\rangle}\equiv\int_{0}^{L^{d-1}}P(E)\,E\,dE\sim L^{(d-1)(1-\beta)}.$$
(2)2( 2 )
While the typical length of a $d-1$-dimensional
interval is the total length $L^{d-1}$ of the support divided by
the number of intersection $N(L)$ so that:
${\langle E\rangle}=L^{d-1}/N(L)\sim L^{H}.$
Therefore, one gets $H=(d-1)(1-\beta)$
and $\beta=1-{H\over d-1}$ that leads to eq.(1).
It is interesting to notice that the value $\beta=1$ is an upper bound
reached when the roughness of the fault is maximal ($H=0$).
Moreover $\beta=1$ is also recovered for all $H$-values in the mean
field limit $d\to\infty$, while at $d=3$, $\beta$ can vary in the
range $[0.5\,,\,1]$.
We have performed numerical simulations by considering two brownian profiles,
one of which at rest and composed by $10^{4}$ points and the other,
slipping
over the first one, composed by $2\cdot 10^{4}$ points. In this way
each realization of the dynamics lasts a time $T=10^{4}$.
The probability distribution of earthquakes has been obtained by averaging
over many
realizations of the dynamics.
Fig. 2 shows the numerical results
in the case of $H=0.5$ and $d=2$. The exponent of the power law
in this case is $\beta=0.5$ in good agreement with our theoretical
prediction.
The Gutenberg Richter law is obtained by the cumulative distribution of the
frequency of earthquakes, i.e. the integral of the distribution showed in
figure.
Another interesting feature that can be studied in the framework
of the SAM is the phenomenology of the space-time correlations
of earthquakes. In particular we will focus on the problem of the spatial
clustering of epicenters [15] and we refer to [16] for a more
exhaustive treatment of this point, including the analysis of the
correlation functions and the temporal fractal distribution
of epicenters.
In our model the space location of an epicenter is defined
in correspondence of the first point of contact of the two profiles.
Numerical simulations, performed on the SAM in the cases with $H=0.3$,
$H=0.5$ and $H=0.7$seem to provide a clear evidence, see fig. 3,
of a spatial clustering of the epicenters on a set
with a fractal dimension smaller than $1$ ($D_{F}\simeq 0.78$
in the case with $H=0.5$).
However, this result is a non-trivial finite size effect,
since the set of epicenters tends to be compact.
In fact it can be proved, for $H=0.5$,
that the fractal dimension $D_{F}(L)$ of the epicenters set in
a fault of a linear size $L$ is:
$$D_{F}(L)\simeq 1-\gamma{\log\log L\over\log L},\ {\rm for\ large\ L}$$
(3)3( 3 )
Let us, indeed, consider two brownian profiles of length $L$ as in Fig. 1a.
The distance $h_{0}(L)$ between the barycentre of the two profiles
can be obtained from the Iterated Logarithm Theorem [17] which states
that, for a partial sum $S_{n}=\sum_{i=1}^{n}x_{i}$ of identically distributed
random variables $x_{i}$ with $<x_{i}>=0$ and $<x_{i}^{2}>=1\,\forall i\in 1,..,n$, it holds:
$$P\left(\limsup_{n\rightarrow\infty}{S_{n}\over\sqrt{2n\log\log n}}=1\right)=1.$$
(4)4( 4 )
That means that the maximum $M(L)$ of a brownian
profile scales as
$M(L)\sim\sqrt{2L\log\log L}.$
Now, the distance $h_{0}(L)$ is given exactly by the maximum value of a
brownian profile obtained by the difference of two brownian profiles,
that is $h_{0}(L)\sim M(L).$
On the basis of this result, it is possible to estimate how the number
of epicenters scales as a function of $L$.
Considering
the configuration where two brownian profiles are $h_{0}(L)$ apart,
the number of points of the lower profile at a certain
height $h$ with respect to its barycentre, is:
$$N_{down}\sim\sqrt{L}\exp{-\left(h^{2}\over 2\eta\,L\right)}$$
(5)5( 5 )
where $\eta$ is a constant depending on the value of $<x_{i}^{2}>$ [18].
We have now to integrate over all the possible values of $h$ that correspond
to the heights at which there could be an intersection of the two
profiles in order to obtain the number of events ($N_{ep}$).
The two integration extremes are given by the maximum value of the lower
profile and the minimum value of the upper one, that is:
$$N_{ep}(L)\sim\sqrt{L}\int_{h_{0}-\sqrt{2L\log\log L}}^{\sqrt{2L\log\log L}}%
\exp{-\left(h^{2}\over 2\eta\,L\right)}\,dh\sim{L\over(\log L)^{\gamma}}\sqrt{%
\log\log L},$$
(6)6( 6 )
where $\gamma=\alpha/\eta$ and $\alpha$
is an intermediate value between $\sqrt{2}-1$ and $1$.
Using the mass-length definition of fractal dimension,
$D_{F}(L)=\log N_{ep}(L)/\log L$,
relation (3) is proved.
The asymptotic value $D_{F}=1$ is reached very slowly at increasing $L$
and it cannot be detected but by huge simulations.
We have checked the validity of (5) for profiles
with a linear size $L$ varying in the range $10^{2}\,-\,10^{6}$.
Work is in progress to extend our results to the case of a generic roughness
index $H$ [18].
In summary, we have proposed a model of earthquakes where
the critical behavior is generated by a pre-existent fractal geometry
of the fault. The statistics of earthquakes is thus related to
the roughness of the fault via the scaling relation (1)
between critical indices.
This result suggests that the younger the fault system,
the larger the $\beta$ exponent, since the roughness
of a fault is expected to decrease in geological times.
The exponent $\beta$ therefore is non-universal.
The model exhibits complex space-time correlations between epicenters:
from the temporal point of view, there exists a fractal
clusterization [16], although the spatial
fractal distribution of the epicenters
turns out to be a finite size effect very difficult to be detected from
data analysis. Our model provides a possible explanation for the
highly irregular and non random distribution of epicenters that is
experimentally observed.
Moreover, the accumulation of pressure
is at the very origin of large seismic events in the SAM.
The presence of such an effect could be tested
also in real situations e.g. by piezo-electric measurements.
We are grateful for interesting discussions to E. Caglioti, O. Mazzella
and R. Scarpa.
Figure captions
Fig. 1 (a) Example of two brownian profiles modelling the fault
surfaces.
(b) Sketch for the definition of the energy released during an earthquake:
it is assumed to be proportional to the overlap interval
of the two fault surfaces during the slip.
Fig. 2 Probability density of the earthquakes releasing an energy $E$
vs. $E$
for roughness index
$H=0.5$.
Fig. 3 Box-counting analysis of the spatial distribution of epicenters
for roughness index
$H=0.5$ in a system with linear dimension $L=10^{4}$. The distribution
apparently
shows a fractal dimension $D_{F}=0.78$.
REFERENCES
[1] Gutenberg B. and Richter C.F. 1956 Ann. Geophys. 9, 1.
[2] Omori F. 1894, Rep. Earth. Inv. Comm., 2 , 103.
[3] Kagan Y.Y. and Knopoff L. 1980, Geophys. J. R. Astron. Soc.,
62, 303.
[4] Burridge R. and Knopoff L. 1967, Bull. Seismol. Soc. Am. 57, 341.
[5] Carlson J.M. and Langer J.S. 1989, Phys. Rev. Lett. 62,
2632;
Phys. Rev. A40, 6470
[6] Crisanti A., Jensen M.H., Vulpiani A. and Paladin G. 1993
Phys. Rev. A46 R7363
[7] Bak P., Tang C. and Wiesenfeld K. 1987, Phys. Rev. Lett. 59,
381;
1988 Phys. Rev. A 38, 364.
[8] Bak P. and Tang C. 1989, J. Geophys. Res. 94, 15635.
[9] Ito K. and Matsuzaki M. 1990, J. Geophys. Res. 95, 6853.
[10] Olami Z., Feder H. J. S. and Christensen 1992, Phys. Rev. Lett.
68 1244;
Christensen K. and Olami Z. 1992, J. Geophys. Res. 97, 8729.
[11] Brown S.R. and Scholz C.H. 1985, J. Geophys. Res., 90, 12575; Wu R.S. and Aki K. 1985, PAGEOPH 123, 805.
[12] Power W., Tullis T., Brown S., Boitnott G. and Scholz C.H.
1987, Geophys. Res. Lett. 14, 29.
[13] Mandelbrot B. 1983, The fractal geometry of nature,
Freeman and Co., New York, pp 256-258.
[14] Turcotte D. L. 1992, Fractals and chaos in geology and
geophysics, Cambridge Un. Press, Cambridge.
[15]
Smalley R.F. Jr., Chatelain J.L., Turcotte D.L. and Prévot R.: 1987 Bull.
Seis. Soc.
Am. 77, 1378; Hirata T.: 1989 J. Geophys. Res. 94, 7507;
Kagan Y.Y. and Jackson D.D.: 1991, Geophys. J. Int. 104, 117;
Main I. G. 1992, Geophys. J. Int. 111, 531;
De Rubeis V., Dimitriu P., Papadimitriou E. and Tosi P. 1993, Geophys. Res.
Lett.
20, 1911.
[16] V. De Rubeis, R. Hallgass, V. Loreto, L. Pietronero and P.Tosi:
1995
in preparation.
[17] Grimmett G. R., Stirzaker D. R. 1992, Probability and Random
Processes
second edition, Oxford Science Publications, New York.
[18]R. Hallgass, V. Loreto, O. Mazzella and G. Paladin: 1995 in
preparation. |
Large-scale Gadolinium-doped Water Čerenkov Detector for Non-Proliferation
M. Sweany
A. Bernstein
N.S. Bowden
S. Dazeley
G. Keefer
R. Svoboda
M. Tripathi
Lawrence Livermore National Laboratory, Livermore, CA 94550, USA
Department of Physics, University of California, Davis, CA 95616, USA
(January 12, 2021)
Abstract
Fission events from Special Nuclear Material (SNM), such as highly enriched uranium or plutonium, can produce simultaneous emission of multiple neutrons and high energy gamma-rays. The observation of time correlations between any of these particles is a significant indicator of the presence of fissionable material. Cosmogenic processes can also mimic these types of correlated signals. However, if the background is sufficiently low and fully characterized, significant changes in the correlated event rate in the presence of a target of interest constitutes a robust signature of the presence of SNM. Since fission emissions are isotropic, adequate sensitivity to these multiplicities requires a high efficiency detector with a large solid angle with respect to the target. Water Čerenkov detectors are a cost-effective choice when large solid angle coverage is required. In order to characterize the neutron detection performance of large-scale water Čerenkov detectors, we have designed and built a 3.5 kL water Čerenkov-based gamma-ray and neutron detector, and modeled the detector response in Geant4 geant4 . We report the position-dependent neutron detection efficiency and energy response of the detector, as well as the basic characteristics of the simulation.
††journal: Nuclear Instruments and Methods in Physics Research A
1 Introduction
Legitimate cross border trade involves the transport of an enormous number of cargo containers. In order to verify that these containers are not transporting SNM without impeding legitimate trade, there is a need for fast, highly efficient, and large detectors that are relatively inexpensive. Such detectors must produce consistent yet distinct responses to both SNM and background, so that their effectiveness is not reduced by false positive or negative detections. They also need to have limited sensitivity to background radiation, such as cosmic ray induced background or Naturally Occurring Radioactive Material (NORM) present in certain legitimate forms of cargo. Both of these may contribute to false positives or reduce sensitivity to real SNM.
SNM can either spontaneously fission or be induced to do so by an external source of gamma rays or neutrons. Since cargo containers are large, they can contain a significant amount of shielding. The fission emissions most likely to penetrate the container and interact with a detector are neutrons or high energy (greater than 3 MeV) gamma-rays. We propose that a water Čerenkov detector doped with a neutron capturing agent (such as GdCl${}_{3}$ salt) would be ideal for this application. Such a detector, sensitive to short timescale correlations between events, has a number of advantages; it is relatively inexpensive, nonflammable and noncombustible, environmentally safe, and easy to operate.
Čerenkov detectors do not produce correlated signals from single fast neutrons in the same way organic scintillator does: in scintillator, fast neutrons are capable of producing a correlated signal via proton recoil followed by neutron capture. Water Čerenkov detectors only use the thermal neutron capture and prompt gamma-ray signals.
Thermal neutron capture on natural Gadolinium has an extremely high cross section (49,000 barns). On capture, a gamma-ray shower with energies adding to approximately 8 MeV is produced, and Čerenkov radiation produced by the resulting Compton scatters is detectable by ordinary PMTs. SNO and Super-Kamiokande have shown that the Čerenkov process can generate enough photons in water to detect neutron captures or gamma-rays with an energy of approximately 3 MeV or greater, so long as the photocathode coverage is high ($\sim$40%) sno ; SK . Our group has since demonstrated the viability of this technique above ground, operating 250 liter water based neutron detector with a photocathode coverage of 10%; reflective detector walls made a lower photocathode coverage possible dazeley .
2 Detector Description
In order to measure the neutron detection efficiency and maximize the performance of a large-scale detector, we have constructed a 3.5 kL water Čerenkov-based neutron detector. The detector consists of a cylindrical polyethylene tank with two PMT arrays arranged on the bottom and top, each with twenty Hamamatsu R7081 10 inch PMTs: although the current generation of R7081 PMTs are high quantum efficiency, the model used here does not have that improvement. The total photocathode coverage is approximately 19%. The tank is approximately 1.5 meter high, with a 2 meter diameter. The detector sits atop a layer of lead bricks to shield against background gamma radiation from the floor. Figure 1 shows a schematic of the detector: the white cylindrical opening in the center, called the top hat, allows for deployment of calibration sources at several locations inside the detector through a 10 cm diameter portal in the lid. The calibration source sits at the end of a long arm (the calibration arm), shown in Figure 2 and discussed further below.
It has been demonstrated that GdCl${}_{3}$ doped water is not compatible with stainless steel, contact with which results in reduced water clarity coleman . All detector components in contact with the water are therefore constructed of either plastic or glass. The PMTs are mounted on a frame constructed out of acrylic and white polypropylene that sits inside the tank, shown in Figure 3a. Each detector quadrant contains ten PMTs, five each on the top and bottom, with the top PMT array supported by acrylic rods. Because the frame structure and PMTs are buoyant, stainless steel bars (sealed inside polypropylene bags) are used as ballast. Figure 3b, shows the bottom four quadrants with the support rods in place. To increase light collection, the detector wall in between the two PMT arrays is lined with UV reflecting Teflon.
The detector is filled from the bottom using a water purification system capable of obtaining ultra pure deionized water (resistivity greater than 17 M$\Omega\cdot$cm). Municipal water is passed through three large de-ionizing (DI) resin bottles and is then circulated through a purification system purchased from South Coast Water Inc., consisting of a UV sterilizer, 5 micron and 0.22 micron filters, and an additional DI unit. Each unit in the system can be by-passed through a series of valves. After purification, the DI unit is by-passed and the water is doped with GdCl${}_{3}$. Any particulate matter and remaining biological contamination from the GdCl${}_{3}$ is eliminated by continued re-circulation through the filters and UV sterilizer. Finally, the doped water is sent to the detector inlet.
2.1 Data Acquisition
The DAQ and trigger electronics are mostly commercial VME and NIM modules. The only modules designed and fabricated in-house are eight-channel signal pick-off modules. The pick-off modules are needed to separate the fast PMT signal from the high voltage bias, which is carried on the same cable coming out of the PMT.
From there, the PMT signals are passed through a Mini-Circuits 15542 BBLP-39+, 23 MHz low-pass filter in order to stretch out the signal, then amplified and digitized using CAEN V975 amplifiers and 200 MHz Struck SIS3320 digitizers. There are two triggers for the system: an internally generated trigger for physics events in the water and an external trigger used for calibration. The internal trigger is formed from a four-fold coincidence among a group of 16 bottom PMT channels. The PMT signals are fed to a CAEN V814 discriminator, set to trigger at $\sim$1 photo-electron. A CAEN V1495 FPGA registers the 4-fold coincidences and issues the trigger to the waveform digitizers.
3 Calibration and Detection Efficiency
Calibration of the detector is needed to equalize the PMT gains and to set an approximate energy scale for physics events. We have three artificial sources: an LED for PMT gain, a ${}^{252}$Cf fission source for neutrons, and a tagged neutron source to measure the neutron detection performance on an event-by-event basis. A polyethylene calibration arm was constructed in collaboration with a group at Harvey Mudd College to deploy the neutron sources at multiple positions inside the detector.
The tagged neutron source consists of an americium beryllium (AmBe) source and a Scionix type 51B51/2M-E1-BGO crystal and PMT deployed together inside the calibration arm. AmBe sources emit a neutron in coincidence with a 4.4 MeV gamma-ray. Detection of this gamma-ray by the BGO detector forms a ”tag”, indicating that a neutron has been emitted from the source. A rendering of the crystal, PMT, and AmBe source inside the polyethylene arm is shown in Figure 2. The arm was designed so that it can reach several positions inside the detector, making position dependent neutron detection efficiency measurements possible.
3.1 ${}^{252}$Cf Calibration
Three datasets were taken with a ${}^{252}$Cf source positioned at various locations outside the detector: seven inches from the detector wall, one meter from the detector wall, and two meters from the detector wall. The first position is used to calibrate the response from Monte Carlo and to establish the quality of event-level cuts on the data. The one and two meter positions are used to determine how the event-level cuts perform with distant sources.
Since high energy events, such as muons traversing the detector tend to saturate the response of the PMTs, we first screen our data to remove all events that contain at least one saturated PMT. Event level cuts are then applied to select neutrons in the detector. Each is determined by maximizing a Quality Factor, $Q$, defined as the significance after the cut is applied divided by the significance before the cut is applied. For a given parameter distribution divided into n independent bins, and for an analysis cut at the jth bin, the quality factor approaching from the left, or lower bound, is
$$Q^{\mathrm{L}}_{j}=\left(\frac{\sum\limits_{i=0}^{j}\mathrm{On}_{i}-\sum%
\limits_{i=0}^{j}\mathrm{Off}_{i}}{\sqrt{\sum\limits_{i=0}^{j}\mathrm{Off}_{i}%
}}\right)\left(\frac{\sqrt{\sum\limits_{i=0}^{n}\mathrm{Off}_{i}}}{\sum\limits%
_{i=0}^{n}\mathrm{On}_{i}-\sum\limits_{i=0}^{n}\mathrm{Off}_{i}}\right),$$
where $\mathrm{On}$ and $\mathrm{Off}$ represent the value of the parameter with and without the neutron source present, respectively. $Q$ is defined to have a value equal to 1 when no cut is applied, i.e. when j equals n. Once a lower bound cut value has been determined, we then apply the same formula in reverse, or from the right, to determine the best jth bin for an upper bound cut:
$$Q^{\mathrm{R}}_{j}=\left(\frac{\sum\limits_{i=n}^{j}\mathrm{On}_{i}-\sum%
\limits_{i=n}^{j}\mathrm{Off}_{i}}{\sqrt{\sum\limits_{i=n}^{j}\mathrm{Off}_{i}%
}}\right)\left(\frac{\sqrt{\sum\limits_{i=0}^{n}\mathrm{Off}_{i}}}{\sum\limits%
_{i=0}^{n}\mathrm{On}_{i}-\sum\limits_{i=0}^{n}\mathrm{Off}_{i}}\right).$$
The quality analysis for charge is shown in Figure 4. Because the fission source increases the number of uncorrelated gamma-rays interacting in the detector, using the on source uncorrelated background as the no source present histogram for the quality analysis provides better rejection of gamma energies. In Figure 4a, the correlated on source data is shown in red and the uncorrelated on source is shown in black. Figure 4b shows $Q$ for the left (pink) and right (blue) cuts. The resulting cuts are applied to both the current event and the previous event to establish neutron-neutron pair events in the detector.
Since the presence of a neutron source results in an increase in the uncorrelated trigger rate, a comparison of the inter-event time distribution with and without the source is not appropriate. Instead, the random trigger rate in the on source data is fit to an exponential, and a histogram filled randomly from the fit values serves as the no source present histogram. The results shown in Figure 5 with the on source inter-event time, as well as the quality factor.
In order to pick out neutron-neutron events, we cut on both the current event and the previous event charge. Finally, we reject events in which the time difference between the current event and the last muon is less than 46 $\mu$s, which acts as a muon veto; the value is chosen as the same maximum inter-event time value allowed. A muon veto cut of 46 $\mu$s rejects 9.9% of the correlated background and increases the dead-time of the detector by 2.5%. Although there is modest improvement in rejecting muon-induced backgrounds, the majority of cosmic backgrounds appear to result from muons in the vicinity of the detector but not traversing it. The final cut values for all four event-level cuts are shown in Table 1.
Table 2 shows the event rates averaged over 20 seconds for the data run with the ${}^{252}$Cf source seven inches from the detector edge compared to no source present. At this position, the solid angle coverage is 28% of 4$\pi$. The singles rate, or the rate of single neutron events in the detector, is the rate after both an energy cut on the current event and the muon veto cut has been applied. The doubles rate, or the rate of neutron-neutron pair events, is the rate after all cuts in Table 1 have been applied.
3.2 Neutron Detection Efficiency
The neutron detection efficiency is determined using the calibration arm described above. The AmBe gamma-ray spectrum from the BGO crystal in Figure 6 shows the primary 4.4 MeV gamma-ray peak. The smaller peaks at 3.9 MeV and 3.4 MeV are also due to 4.4 MeV gamma-rays, where pair production and subsequent positron annihilation results in either one or two 511 keV gamma-rays escaping from the crystal. To maximize the signal to background ratio of our tag, we select events in the range 3 MeV to 5 MeV, where the background gamma-ray rate in the crystal is very small. Figure 7 shows the timing distribution of delayed detector events after our selection of AmBe tags. This inter-event time distribution can be well parameterized by a sum of two exponentials, representing a correlated and uncorrelated set of events. The correlated set represents neutron capture events due to the AmBe source in our detector. The mean capture time is 35 $\mu$s. This is consistent with the expected capture time of thermal neutrons in water doped with $0.1\%$ gadolinium dazeley ; apollonio ; boehm ; piepke . The number of neutron captures detected can be estimated by subtracting the exponential fit to the un-correlated events. The efficiency is taken to be the integral of the correlated events, the black curve in Figure 7, divided by the total number of neutron tags (after accounting for the crystal’s background rate), or the red curve in Figure 7.
Three AmBe data sets were taken inside the detector to determine the efficiency and energy response at various radii. We assume that the detector is radially symmetric. One dataset was taken outside the detector, for which the efficiency is multiplied by the fraction of solid angle calculated from the source position. The position-dependent efficiency is shown in Figure 8; it ranges from 69.9% at the center of the detector to 31.3% outside the detector. The efficiency drop is expected near the detector edge, as neutrons can leave before being captured and many of the neutron capture gamma-rays escape the detector before interacting.
3.3 Monte Carlo Response
A full Monte Carlo was written in Geant4 to model our detector. Our objective is to utilize the Monte Carlo, tuned to reproduce the response of this detector, to determine the best design for a larger-scale detector.
The output of the Monte Carlo is the wavelength of photon hits on the PMT surfaces. In analysis, an energy-dependent quantum efficiency is applied, and a variable single photo-electron response is applied to each photo-electron. The trigger is modeled by requiring at least four PMT signals to be greater than a given threshold. The detector data is then scaled by the PE to ADC unit value determined from single photo-electron calibrations.
There are three optical properties that can be tuned to reproduce the response in data: the wall reflectivity, the attenuation length of water, and, to a limited extent, the PMT quantum efficiency. The attenuation length of water in our detector was likely adversely affected by UV stabilizers in the polyethylene tank. The attenuation length from quickenden ; sogandares ; pope scaled down to an approximately 10 meter maximum gave results consistent with our data. The quantum efficiency is taken from Hamamatsu specifications, and scaled down to account for losses in collection efficiency from stray magnetic fields: the total efficiency peaks at 22%. Finally, the wall reflectivity has a total reflectivity of 90% with a 5% specular component.
Good agreement between data and Monte Carlo has been obtained with both AmBe source data inside the detector (Figure 9) and the ${}^{252}$Cf source data outside the detector (Figure 10). For the AmBe comparison, the calibration arm is included in the Monte Carlo.
4 Discussion and Conclusions
Based on the average number of neutrons produced from our ${}^{252}$Cf source, as well as the average number of neutrons produced in reactor grade plutonium (RGP), we have determined that our current setup can detect approximately $150~{}$g equivalent of RGP over background at two meter standoff from the detector edge in 20 seconds. The solid angle presented by our detector at this distance is 2.4% of 4$\pi$. We have assumed 8% fraction of ${}^{240}$Pu in RGP (87,000 n/s/kg). Figure 11 is the signal over background for each cut in Table 1 as a function of radial distance from the detector. At one meter from the detector edge, or 6.4% of 4$\pi$, we have a signal to background ratio of 0.547 $\pm$ 0.120 after all cuts have been made. Before cuts, the signal over background was 0.101 $\pm$ 0.00810. At two meters, the signal over background before cuts is 0.00560 $\pm$ 0.00755 and increases to 0.113 $\pm$ 0.0927 after all cuts have been made. Since our detector is intended as a correlated event detector, it is expected that high solid angle coverage is required for efficient operation. However, with the proper event-level cuts our source is detectable even at two meters away: without requiring neutron-neutron pairs, the signal over background is 0.0702 $\pm$ 0.0154.
4.1 Next Generation Detector
There are many ways to improve the performance of this detection technology. Aside from optical and geometrical optimizations, decreasing the neutron capture window by increasing the concentration of GdCl${}_{3}$ will result in a decrease in the rate of accidental coincidences. An examination of this detector’s ${}^{252}$Cf calibration data shows that if the neutron capture window was decreased from 35 $\mu$s to 10 $\mu$s, and assuming the same number of neutrons are captured, the percentage of accidental coincidences drops from 53% of all events surviving the current and last event energy cuts to 23%. In other words, if all neutrons are captured within 10 $\mu$s, then for every 4.3 coincident events, one would be accidental. This is a significant improvement compared to our current concentration in which all neutrons are captured within 35$\mu$s: for every 1.9 coincident events, one is accidental. It has been shown that there is no measurable impact on water attenuation length at the level of 0.2% GdCl${}_{3}$ coleman . An increase in GdCl${}_{3}$ concentration to the 0.3% level is needed to obtain a characteristic capture time of $\sim$10 $\mu$s. Further study may be warranted in this area.
4.2 Conclusions
We have successfully operated a large-scale gadolinium-doped water Čerenkov detector and characterized its neutron detection performance for the purposes of SNM monitoring. Using a tagged americium beryllium source, the raw neutron detection efficiency has been measured in the center of the detector at 70% and outside the detector at 31%. Event-level cuts have been established to maximize the detection of correlated pairs of neutrons emitted simultaneously from a ${}^{252}$Cf fission source. We have demonstrated detection of approximately $150~{}$g equivalent of RGP over background within 20 seconds for a solid angle coverage of 2.4% of 4$\pi$; an optimized geometry is expected to perform even better.
The neutron capture response of the detector has been reproduced in Geant4 for two different source positions, using the water attenuation length, the PMT quantum efficiency, and the wall reflectivity as tuning parameters. The tuned value of each parameter is within the expected range for the water quality, PMTs, and Tyvek reflectivity. Future work will use this model to design an optimized radiation portal monitoring system for detection of correlated neutrons from undeclared fission sources within cargo containers.
Acknowledgements
The authors would like to thank Dennis Carr for assistance with detector design and construction, as well as Serge Ouedraogo for help with construction. The authors also wish to thank the DOE NA-22 for their support of this project.
This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344. Document release number LLNL-JRNL-479935.
References
(1)
J. Allison et. al. “Geant4 developments and applications” IEEE Transactions on Nuclear Science 53(1) (2006) p. 270.
(2)
The SNO Collaboration. “Electron energy spectra, fluxes, and day-night asymmetries of ${}^{8}$B solar neutrinos from measurements with NaCl dissolved in the heavy-water detector at the Sudbury Neutrino Observatory” Physical Review C 72 (2005) 055502.
(3)
The Super-Kamiokande Collaboration. “Solar neutrino measurements in Super-Kamiokande-I” Physical Review D 73 (2006) 112001.
(4)
W. Coleman, A. Bernstein, S. Dazeley and R. Svoboda. “Transparency of 0.2% GdCl3 doped water in a stainless steel test environment” Nuclear Instruments and Methods A 595 (2008) p. 339.
(5)
S. Dazeley, A. Bernstein, N.S. Bowden, and R. Svoboda. “Observation of neutrons with a Gadolinium doped water Cherenkov detector” Nuclear Instruments and Methods A 607 (2009) p. 616.
(6)
M. Apollonio et. al. “Search for neutrino oscillations on a long base-line at the CHOOZ nuclear power station” European Physical Journal C 27 (2003) p. 331.
(7)
F. Boehm et. al. “Final results from the Palo Verde neutrino oscillation experiment” Physical Review D 64 (2001) 112001.
(8)
A. B. Piepke, S.W. Moser, and V.M. Novikov. “Development of a Gd loaded liquid scintillator for electron anti-neutrino spectroscopy” Nuclear Instruments and Methods A 432 (1999) p. 392.
(9)
T. I. Quickenden and J. A. Irvin. “The ultraviolet absorption spectrum of liquid water” Journal of Chemical Physics 72 (8) (1980) p. 4416.
(10)
F. M. Sogandares and E. S. Fry. “Absorption spectrum (340 - 640 nm) of pure water. I. Photothermal measurements” Applied Optics 36 (1997) p. 8699.
(11)
R. M. Pope and E. S. Fry. “Absorption spectrum (380 - 700 nm) of pure water. II. Integrating cavity measurements” Applied Optics 36 (1997) p. 8710. |
Confotronic Dynamics of Tubular Lattices
Osman Kahraman${}^{1}$, Hervé Mohrbach${}^{1,2}$, Martin Michael Müller${}^{1,2}$, Igor M. Kulić${}^{2}$
${}^{1}$Equipe BioPhysStat, ICPMB-FR CNRS 2843, Université de Lorraine; 1
boulevard Arago, 57070 Metz, France
${}^{2}$CNRS, Institut Charles Sadron; 23 rue du Loess BP 84047, 67034
Strasbourg, France
(December 8, 2020)
Abstract
Tubular lattices are ubiquitous in nature and technology. Microtubules and nanotubes of all kinds act as important pillars of biological
cells and the man-made nano-world. We show that
when prestress is introduced in such structures, localized conformational
quasiparticles emerge and govern the collective shape dynamics of the lattice.
When coupled via cooperative interactions these quasiparticles form
larger-scale quasipolymer superstructures exhibiting collective dynamic modes
and giving rise to a hallmark behavior radically different from semiflexible beams.
pacs: 87.16.aj,82.35.Pq,87.15.-v
I Introduction
Tubular and cylindrical lattices are abundant in living nature and
give rise to important filamentous structures, such as bacterial
flagella and microtubules. Inspired by biology, nanotubes made
from various building blocks like carbon Iijima , DNA
mahtieu or amphiphilic molecules micelles ; micelles2
have been synthesized. Remarkably, in the presence of internal
prestress, almost all of these tubular objects can adopt
superhelical structures, i.e., tubes whose centerline
describes a large-scale helix in space. Superhelical carbon
nanotubes with micron-meter size pitch have been observed
Xang carbon and their elastic properties probed
Volodin . It is believed that their coiling results from the
periodic arrangement of defects (pairs of heptagon-pentagon rings)
in the hexagonal carbon lattice forming the wall of the tube
Dunlap carbon . DNA nanotubes Douglas DNA adsorbed on
a substrate resemble squeezed helices indicating that they assume
a three-dimensional superhelical structure under free conditions
Squeezed helix . Supramolecular chiral helical nanofibers
made of self-assembling lipids have been synthetized. It was shown
that the torsional stress, created by the steric interactions of
the chemical groups at the surface of the tube, causes the
nanofibers to coil into a superhelix that minimizes the internal
prestress Li micelles . In bacterial flagella and
microtubules the coexistence of several conformational states of
their individual constituents is in elastic conflict with their
lattice geometry, resulting in prestresses that can be minimized
by forming superhelical shapes Calladine ; mt2010 ; mt2012 . The
protein monomer units of bacterial flagella are arranged in eleven
protofilaments parallel to the centerline that can switch from a
short to a longer state and can slide down relative to their
lateral neighbors creating local twist and curvature
Calladine . In a similar manner, the wall of microtubules is
made of protofilaments built by the polymerization of tubulin
dimers. There are experimental indications that the tubulin dimers
can indeed switch between straight and curved conformational
states in the presence of taxol Amos and display an
allosteric cooperative interaction along their protofilaments’
axes ElieCaille2007 . The integrity of the tubular lattice
can be maintained either by forcing all protofilaments in their
straight state or by creating a mixed phase with a cluster of
curved protofilaments while the rest of the protofilaments stays
in their straight state. When this phase is energetically more
favorable, the microtubule bends in the direction of the block
formed by the curved protofilaments. Since microtubules have
internal twist Chretien (i.e., the protofilaments
are not parallel to the centerline of the tube but instead wind
around it), the resulting shape will be a superhelix whose pitch
is believed to be given by the internal twist
mt2010 ; mt2012 . Remarkably there is some evidence that, in
contrast to the bacterial flagella filament and all other
structures discussed,
microtubules are super-helices that are spontaneously and
permanently reshaping on experimental timescales: they change
their reference ground state due to thermal fluctuations. This
unusual collective movement was previously proposed and termed the
“wobbling mode” mt2010 ; mt2012 .
Notable phenomenological models for multistable helices have been
developed in the past in order to describe transformations of bacterial flagella
Powers ; WadaNetz1 ; Friedrich , coiled plant tendrils TendrillPerversion ,
or whole microorganisms WadaNetz2 ; BistableHelices .
The novel and rather unique feature of our present model, that we
will explore and illustrate here in depth, will be the
extraordinary dynamic behavior associated with the cooperative
“wobbling motion” of the filament.
Although biofilaments are usually studied in the framework of beam
elasticity, microtubules can also be modelled as cylindrically
wrapped membranes since they are hollow. Jánosi et
al., for example, modelled microtubule walls as elastic sheets in
order to analyze their elastic properties Janosi1998 .
Inspired by the polymorphic tube model previously proposed by some of
the authors mt2010 ; mt2012 ; mtsolvable we study a tubular system
that incorporates the idea of lattice confinement of bistable
units and cooperativity into an elastic sheet. Numerical
simulations and analytical models are combined here to provide an
in-depth intuitive understanding of this system. The prestress,
that will be built into our model, will give rise to a remarkable
phenomenon: Localized conformational deformations, that will behave
as quasiparticles, will emerge and govern the collective shape
dynamics of the lattice via elastically-mediated interactions.
When we switch on additional mechanical coupling terms in the
lattice, these quasiparticles will exhibit cooperative
interactions. The cooperativity will lead to the formation of
larger-scale “quasipolymer” superstructures that we will show to
exhibit rather unusual collective dynamic modes. The notion of
quasiparticles/-polymers is the most natural language to
quantitatively describe many new phenomena for which we will
collectively use the term “confotronic
dynamics”.
In real biofilament systems this dynamics is usually inaccessible
to direct observation. However, as the internal confotronic modes
also govern the behavior of the centerline of the tube as a whole,
their existence can be experimentally inferred from the
observation of anomalous behaviors of the tube’s centerline in
well chosen experiments. Among them, observing the dynamics of a
tube clamped at one end, turns out to be the experiment of choice.
We will see that a lot about the inner confotronic dynamics of
quasiparticles can be revealed from the external behavior of the
tube.
The paper starts with a description of the polymorphic tube model
in Sec. II. In Sec. III, the
notion of quasiparticles (called “confoplexes”) is presented in
detail. We will see that these particles can form ordered
conformational superstructures on the lattice
(called “confostacks”). This order is shown to be responsible
for the formation of the broken symmetry superhelical tube state,
very much akin to the superhelices observed in microtubules and
bacterial flagella. Finally, in Sec. IV the
confotronic dynamics of clamped polymorphic tubes is analyzed with
a particular focus on the remarkable collective mode (“wobbling
mode”) that emerges spontaneously in the system at finite
temperature.
II The Polymorphic Tube Model
We construct a hollow polymorphic tube by discretizing its surface
as a mesh of rectangular subunits (see Fig. 1).
Similarly to previous discrete models of (linear) elastic
membranes SeungNelson88 ; Janosi1998 , the elastic properties
of the tube will be enforced via stretching and bending energetic
penalties of the mesh. The polymorphic character of the tube is
implemented at the level of each subunit, decomposed into two
squares, which is multistable and can be either straight or curved
(see Fig. 1). Note that this choice of mesh is well
adapted to describe biofilaments such as microtubules. In this
picture, each rectangular, double-square subunit of the mesh
corresponds to a single tubulin dimer. The association of the
former along the vertical direction defines protofilaments (blue
in Fig. 1) which form the tube in a way similar to
protofilaments forming the wall of a microtubule. In this work we
always consider $13$ such protofilaments alluding again to
microtubules. To stay as generic as possible, however, we will
omit many system-specific details like the internal twist and
other particularities of microtubules like the “seam” Janosi1998 . Such generic
simplifications, including symmetry, are necessary in order to
extract the physical “gist” of such systems, as we will see from
the plethora of phenomena emerging already in this simplified
lattice geometry.
Each dimer consists of internal and external bonds. The horizontal
internal bond $B_{2}$ separates the subunit into two squares and
forms a hinge in the curved state of the dimer. The
internal bonds $B_{4}$, depicted by dashed lines in
Fig. 1, control the shearability of each square and
ensure their planarity. The external bonds are shared with
neighboring dimers: two bonds of type $B_{1}$ along the vertical
and four of type $B_{3}$ along the horizontal direction.
The elastic energy of the tube is a sum of contributions for
stretching and bending of the mesh, $E=E^{\text{S}}+E^{\text{B}}$.
Denoting $\mathbf{d}_{i}$ the vector associated to a bond of type
$B_{i}$, the total stretching energy of all bonds of type
$B_{i}$ is given by
$$E_{i}^{\text{S}}=\frac{1}{2}\mu_{si}\sum_{\left\{\mathbf{d}_{i}\right\}}\left(%
\left|\mathbf{d}_{i}\right|-d_{i}^{(0)}\right)^{2},\,\,i=1,2,3,4$$
(1)
in the harmonic approximation. The sum runs over all the edge
vectors $\mathbf{d}_{i}$ of the tube. In
Eq. (1), $d_{i}^{(0)}$ is the preferred
length of the bond $B_{i}$ when the tube is in its straight
cylindrical state (Fig. 1) and $\mu_{si}$ is the
stretching rigidity. The stretching energy of the
whole mesh is thus: $E^{\text{S}}=\sum_{i=1}^{4}E_{i}^{\text{S}}$.
Similarly, we associate a quadratic bending energy with all bonds of type
$B_{1}$, $B_{3}$, $B_{4}$:
$$E_{i}^{\text{BH}}=\frac{1}{2}\mu_{bi}\sum_{\left\{s_{i}\right\}}\left(s_{i}-s_%
{i}^{(0)}\right)^{2},\,\,i=1,3,4$$
(2)
where ${s}_{i}{=(\mathbf{n}^{a}\times\mathbf{n}^{b})\cdot\frac{\mathbf{d}_{i}}{|%
\mathbf{d}_{i}|}}$ is the sine of the angle between the outward normals
$\mathbf{n}$ of two adjacent triangles $a$ and $b$ at bond $\mathbf{d}_{i}$,
and $\mu_{bi}$ is the bending rigidity of this bond. The
constants $s_{i}^{(0)}$ are chosen such to enforce the straight
cylindrical state.
To allow the subunits to accommodate two stable conformations, we assign an
anharmonic bending potential $E_{2}^{\text{BAH}}$ for all bonds of type $B_{2}$:
$$E_{2}^{\text{BAH}}=\sum_{\left\{s_{2}\right\}}\left(As_{2}^{4}+Bs_{2}^{3}+Cs_{%
2}^{2}\right)\;,$$
(3)
where the coefficients $A$, $B$, and $C$ are all functions of the energy
difference $\Delta G$ and the barrier $\delta G$ and are chosen to favor the
curved state
(see Fig. 1). The total bending energy of the mesh is thus
$E^{\text{B}}=E_{1}^{\text{BH}}+E_{2}^{\text{BAH}}+E_{3}^{\text{BH}}+E_{4}^{%
\text{BH}}$.
In the following, the values of the stretching rigidities will be
expressed in units of $k_{B}T_{0}/d^{2}$ and the bending
rigidities in units of $k_{B}T_{0}$, where $T_{0}$ is the room
temperature and $d=4\,$nm the size of a monomer (see
appendix A for all the parameter values). The
temperature $T$ in the simulations will be measured in units of
$T_{0}$.
III Confoplexes, confostacks and spontaneous symmetry breaking
The rich conformational properties of the previously defined tube
model will now be analyzed with the help of numerical simulations
and phenomenological models. The large number of parameters of the
model is a serious limitation for exploring all possible
conformations in an attempt to build a complete phase diagram.
Instead, we reduce the number of independent parameters as much as
possible and look for interesting generic configurations and
behavior. In this spirit, we set
$\mu_{s}=\mu_{s1}=\mu_{s2}=\mu_{s3}$ and choose their value in the
typical range of the elastic constants of microtubules
Janosi1998 (see below). To facilitate the visual and
numerical detection of the characteristic behavior for the short
lattices practically accessible to our simulations, we have
deliberately chosen an intrinsic curvature of the protofilaments
much larger than for real microtubules. The other parameters are
adjusted to ensure the numerical stability of the mesh (see
appendix A).
We first discuss the results of the numerical simulations in which the
system is integrated in time with the Langevin Dynamics method (see again
appendix A). To understand the subtle competition between
the anharmonic potential (3) and the elasticity of the
lattice, our study is built up in a hierachical manner: first the different
conformations of a single section of a tube are explored, from which, in a
second step, we can understand the behavior of longer tubes. Finally, long
tubes with additional cooperative interactions along the protofilaments are studied.
III.1 Numerical simulations
III.1.1 A single section of the tube, the emergence of a confoplex
We first consider the ground state (zero temperature limit) of a
single section of the tube. By varying the elastic stretching
constant $\mu_{s}$ we observe three different types of
configurations (see Fig. 2): For sufficiently
large values of $\mu_{s}$ (at the order of $10^{5}$) the dimers
can not switch to their curved configuration and the ring
maintains its cylindrical form. However, when $\mu_{s}$ becomes
smaller ($2\times 10^{4}$), all subunits adopt a curved
conformation, forming what we call a “conformational complex”
or—more briefly—a confoplex. The ring is still
cylindrically symmetric but its shape is catenoid-like. Very
interestingly, for intermediate values of $\mu_{s}$ (around
$4\times 10^{4}$) a state with broken cylindrical symmetry
is the minimum-energy configuration. In this partial
confoplex conformation a cluster of neighboring dimers on one
side of the ring is switched to the curved state, thereby creating
a negatively curved scar which gradually decays towards the
opposite side of the tube. This causes the opposite wall to
bulge slightly outwards.
III.1.2 Interacting confoplexes
The emergence of full and partial confoplexes in the ring leads to
the interesting question of how single confoplexes interact with
each other in a larger lattice. In contrast to the single section,
which does not possess any longitudinal neighbors, the
bending around the bonds of type $B_{1}$ will now have an
important effect. For the case of full confoplexes successive
sections meet at positive curvatures at the $B_{1}$ bonds. Hence,
for non-zero $\mu_{b1}$, stacking one full confoplex on top of a
second one will cost some energy; when the value of $\mu_{b1}$ is
high enough to dominate the other terms, the tube goes back to the
straight configuration.
For values of $\mu_{s}$ for which the tube is composed of partial
confoplexes, the deflection of each section is expected to be in
almost independent directions if $\mu_{b1}$ is zero. (In fact,
they are not completely uncorrelated due to the stretching energy
of the bonds $B_{1}$.) This almost random mutual orientation has
been confirmed by simulations at finite temperature (not shown).
However, for non-zero values of $\mu_{b1}$, (e.g. for
$\mu_{b1}=300$), each section conserves its partial confoplex
configuration but neighboring confoplexes tend to align their
orientation opposite to each other, forming an “antiferromagnetic”-like zigzag pattern along the
tube (see Fig. 3 for an example at temperature T=3).
If we keep increasing $\mu_{b1}$, the confoplexes become gradually
smaller in size and finally vanish, similar to the
catenoid-to-cylinder transition found for the ring when $\mu_{s}$
is increasing.
III.1.3 Cooperativity: confostacks, helices and spontaneous symmetry breaking
The repulsive interaction of confoplexes implies that global
conformational superstructures of the tube are formed only if
neighboring confoplexes interact cooperatively via additional
interactions. Cooperative interactions among monomers in a
filament have been observed, for example, in single protofilaments
of microtubules in vitro ElieCaille2007 . Motivated
by this observation, we implement the coupling between neighboring
dimers on a protofilament by adding a cooperative interaction
between the angles associated to the anharmonic bonds $B_{2}$:
$$E^{\text{CP}}=\frac{\mu_{c}}{2}\sum_{\left\{s_{2}\right\}}\left(\left(s_{2}-s_%
{2,\mathrm{next}}\right)^{2}+\left(s_{2}-s_{2,\mathrm{prev}}\right)^{2}\right)\;,$$
(4)
where $\mu_{c}$ is the strength of the cooperative interaction
between a bond $B_{2}$ and its nearest neighbors (i.e.,
$B_{2}$-next and $B_{2}$-previous). In the presence of partial
confoplexes this term causes an attractive interaction between
them, opposing the repulsive one (due to the bending $\mu_{b1})$
and generating an interesting “frustrated” situation. The
attraction between neighboring partial confoplexes leads to a
stack of confoplexes forming a kind of “quasipolymer”
superstructure arrangement on the lattice. This particular
internal state of the lattice, being an ordered stack of
confoplexes, will from now on be called a
“confostack”. In a similar manner as the emergence of
the confoplex previously gave rise to the breaking of the cylindrical
symmetry on the single section scale, the formation of the
ordered confostack is responsible for the spontaneous breaking of
the large scale chiral symmetry of the tube
centerline in the three-dimensional space.
At finite temperature, thermal fluctuations tend to disorganize
the ordered confostack. In this case we observe the formation of
uncorrelated fluctuating domains of ordered confoplexes
distributed along the confostack. The characteristic size of such
a domain will be called the coherence length of the confostack and
denoted with $l_{c.}$ This coherence length shares some analogy to
the concept of persistence length of a semiflexible polymer. In
particular we can write $l_{c}=C/(k_{B}T)$ where $C$ (an unknown
function of the material parameters) can be seen as an effective
stiffness of the confostack. Note that an exact computation of
$l_{c}$ for a simplified model of microtubules can be found in
Ref. mtsolvable . We expect two extreme situations: For a tube of
length $L$ such that $L\ll l_{c},$ the confostack is fully ordered and
the entire tube breaks the cylindrical symmetry. For $L\gg l_{c},$
the confostack is made up of a juxtaposition of uncorrelated
fluctuating domains of ordered confoplexes leading to a
statistically straight tube on average.
What is the fundamental shape of a single domain of an
ordered stack of confoplexes? The answer comes from the numerical
simulations. Figs. 3-3 show a number
of snapshots of the typical configurations of the tube at finite
temperature for increasing values of the ratio
$\nu:=\mu_{c}/\mu_{b1}.$ In Fig. 3 where $\nu$ is
small, we see that the confostack is made of short coherent
helices of random handedness. This corresponds to the situation $L\gg l_{c},$ where the tube looks straight on long scales, since right- and
left-handed helices cancel each other out. The fundamental shape
of a domain of ordered confoplexes is thus a helix with a pitch
which depends on $\nu.$ The origin of this helicity will be
elucidated below in section III.2.2.
Increasing $\nu$ leads to a stronger alignment of the
partial confoplexes, i.e., an increase of the pitch of the
confostack. When the coherence length is larger than $L$ but the
pitch is still smaller than $L$, the switched dimers form a
coherent helical confostack, deflecting the centerline of the tube
into a superhelix in space as shown in Fig. 3. Note
that the superhelices of either handedness appear spontaneously
within a completely symmetric lattice. This can be seen as an
indication that real microtubules might form superhelices with
finite pitches by a similar spontaneous symmetry breaking
mechanism, even in the absence of an explicit internal lattice
twist mt2010 ; mt2012 . For very large $\nu$ the
cooperativity is strong enough for the pitch of the helical
confostack to become larger than $L$. Fig. 3 shows
such a situation with $L\ll l_{c}$. In this case the tube forms a
circular arc with an untwisted confostack living on it.
III.2 Phenomenological modelling
After outlining the empirical observations of various
interesting phenomena from the Langevin simulations in the
previous section, in this section we seek to analytically and
phenomenologically explain the observations. In the first
subsection we will explore how the very existence of full/partial
confoplexes can be qualitatively understood within a simple
analytical model. Following that, in the second subsection, we
outline a simple phenomenological model explaining the chiral
symmetry breaking and the formation of the left-/right-handed
superhelical confostacks.
III.2.1 Simple model for confoplex formation
In this section we revisit the transitions between the different
states of a single section (cylinder, partial and full confoplex)
and will explain them qualitatively. For all three states in
Fig. 2, we observe that the horizontal bonds
form polygons which are close to circles lying in three parallel
planes. For a full catenoid-like confoplex, we see that the circle
in the middle (called $C_{2}$ because it is associated to the
anharmonic bonds $B_{2}$) has the smallest radius, whereas for the
partial confoplex there is additionally a small shift of the
center of $C_{2}$. This shift is in the direction of the
non-switched region of the wall, which thus bulges slightly
outwards. In view of these observations, it is tempting
to approximate a section of the tube by three circles that will
interact elastically in a very simplified manner (thus the results
can only be qualitatively compared with the simulation). By fixing
the center position and the radius $R$ of the lower and upper
circles, the only variables are now the position and the radius of
the circle $C_{2}$ in the middle. This circle can shift its center
horizontally by an amount of $\Delta X$. It can also decrease its
radius $R$ by $\Delta R$ with an energy cost
$$E_{s}=\frac{\mu_{s}}{26}(2\pi\Delta R)^{2}\;,$$
(5)
where $\mu_{s}$ is the
stretching rigidity of $C_{2}$. For a confoplex there is an
additional cost of bending energy due to the angle $\alpha$
between the lines joining the three circles at an azimuthal
position $\varphi$ of $C_{2}$ (see Fig. 4).
This angle is chosen negative in the concave part of the confoplex and
given by $\sin\alpha\approx-2(\Delta R-\Delta X\cos(\varphi))$ to first order in $\Delta R$ and $\Delta X$.
Similar to the anharmonic bending energy (3) for the bonds $B_{2}$
used in the simulations, we write the bending energy of the simple
model as a double well potential and integrate it along
$C_{2}:$
$$E_{b}=\oint_{C_{2}}\left(A(\sin\alpha)^{4}+B(\sin\alpha)^{3}+C(\sin\alpha)^{2}%
\right)\text{d}\varphi\;.$$
(6)
By minimizing the total energy of this three circle model,
$E^{\text{3C}}=E_{s}+E_{b}$, with respect to $\Delta X$ and $\Delta R$, we observe the same three configurations as in the numerical
simulations (see Fig. 4 again): for
increasing $\mu_{s}$ the ground state shifts from a
full confoplex ($\Delta R>0$, $\Delta X=0)$ where $C_{2}$ can
easily stretch, to a partial one ($\Delta R>0$ and $\Delta X>0$)
which is the result of a compromise between the stretching
and the anharmonic bending energy. For an even larger stretching
constant we finally obtain a cylinder ($\Delta R=\Delta X=0)$.
This simplified model which only comprises the stretching energy
and the anharmonic bending potential of the bond $B_{2}$ gives a
simple explanation for the very existence of confoplexes. Of
course, this reduced model is not capable to predict the precise
transition values; nonetheless it provides a qualitative
explanation for the observed morphologies in the simulation. Due
to its simplicity, it is highly intuitive and also analytically
tractable.
III.2.2 Elastically-mediated
interactions between confoplexes and formation of confostacks
¿From what we learnt previously from the simulations, the shape of
the confostack results from the competition between the
cooperative attractive interaction and the elastic repulsion
between partial confoplexes (see Fig. 3). If both
interactions were short-ranged, one would observe the two
configurations depicted in Fig. 5 with equal
probability as they would have equal energies.
But the observed spontaneous symmetry breaking of the straight
tube, which leads to right- and left-handed superhelices, can only
be explained by the presence of an effective long-range
repulsion between the confoplexes. Intuitively, the confoplexes
can be seen as quasiparticles which
deform the lattice surrounding them, generating an extended
elastic field. As a consequence the confoplexes will interact with
each other through long-range elastically-mediated repulsive
forces, extending further away than to the nearest neighbors.
A full analytical treatment of this interaction in our concrete
case is complicated by the presence of various lattice parameters
in the simulation, but a qualitative understanding is possible if
we retain only the dominant contributions. In this spirit,
consider two confoplexes $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$
which azimuthally overlap with an angle $\delta$ (cf.
Fig. 5) and interact only elastically without
cooperativity (left image of Fig. 5). For
simplicity, let us assume that only the overlapping parts of
$\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ interact strongly with
each other. This seems reasonable, since the elastic interaction
between the parts of the confoplexes which do not sit on the same
protofilament will be more screened around the tube. If the
confoplexes are shifted by an angle $\Delta\varphi=\left|\varphi_{1}-\varphi_{2}\right|$ their interaction is then
proportional to the overlap angle $\delta$, and we can write
$$E_{\mathcal{C}_{1}-\mathcal{C}_{2}}\approx\frac{13\delta}{2\pi}E_{\mathcal{C}-%
\mathcal{C}}^{\text{single PF}}\;,$$
(7)
where $E_{\mathcal{C}-\mathcal{C}}^{\text{single PF}}$ is the
interaction energy of the parts of the two partial confoplexes
sitting on the same protofilament. For a small angular deviation
$\theta(u)$ with respect to the straight protofilament state this
energy can be approximated as
$$E_{\mathcal{C}-\mathcal{C}}^{\text{single PF}}=\frac{1}{2}\int\nolimits_{0}^{L%
}\left(B_{\text{eff}}\left(\frac{\text{d}\theta}{\text{d}u}\right)^{2}+F_{%
\text{eff}}\theta^{2}\right)du\;,$$
(8)
where $u$ is the arc length along the protofilament,
$B_{\text{eff}}\approx\mu_{b1}d$ is the bending rigidity of the
protofilament and $F_{\text{eff}}\approx{\mu}_{s}d$ is an intrinsic
effective tension due to the stretching rigidity of the bonds of
the protofilament.
Eq. (8) is formally equivalent to the interaction
energy of two proteins bound to the same side of a semiflexible
filament under an external pulling force. As shown in
RudnickBruinsma99 the elastic interaction is repulsive.
Similarly, cylindrical proteins bound to a locally flat membrane
are described with the same energy functional and repel as well
Weikl ; Martin1 ; Martin2 ; MkrtchyanIngChen ). The range of this
repulsion is given by an elastic screening length
$\lambda=\sqrt{B_{\text{eff}}/F_{\text{eff}}}$ (see
Fig. 5).
Despite the crude approximations considered here the basic physics of the interaction
is well captured.
In the presence of cooperativity, this long-range interaction leads to an interplay between nearest and
next-nearest neighbor repulsion between the quasiparticles; at short scale the
second configuration in Fig. 5 will be adopted. At a larger
scale the quasipolymer will thus form a helix on the lattice inducing a
superhelical structure of the tube in three-dimensional space.
Before embarking on the study of the dynamic properties of
polymorphic tubes, it is interesting to remark, that the
notion of quasiparticles, interacting by elastic fields in our
present case is rather similar to the notion of “twist-kinks”, the
quasiparticles formed in squeezed helices confined to two
dimensions Squeezed helix . In this context conformational
“twist-kink” quasiparticles appear as a natural concept as well,
indicating a broader relevance of this perspective for prestressed
filaments with (hidden) internal degrees of freedom.
IV Dynamics of clamped tubes: Confostacks diffuse through the lattice
The observation of microtubules clamped at one end has been a key
experiment revealing their anomalous behavior
pampaloni ; Taute and was the starting point for the
polymorphic tube model proposed in mt2010 ; mt2012 . In this
section we explore the dynamics of collective internal modes of
the lattice by simulating a tube of various lengths clamped at one
end, at finite temperature $T$. We focus on the simplest case in
which the tubes form circular arcs (helices with infinite pitch),
i.e., when they bear ideal untwisted confostacks with a
coherence length much larger than the tube’s length.
IV.1 Numerical simulations
IV.1.1 Observation of the wobbling motion
Even though the rigid attachment of the arc-shaped tube
seems to preclude large scale motion, the tube is not static in
shape. Instead, we clearly observe a random rotation of the tube around a fixed axis (see
Fig. 6) in agreement with the predictions of the
wobbling motion for polymorphic tubes mt2010 ; mt2012 . It
is intuitively clear that the symmetry-broken
circular arc state can in principle explore all its equivalent
sister states, with the arc pointing in one of 13 possible
directions. However, it seems not obvious a priori if the
switching between the states can practically occur, and if so, on
which timescale. To quantify the dynamic behavior of this collective
“wobbling mode”, we measure the time
evolution of the centerline at the free end, which diffuses inside
an annular strip in the $\rho\phi$ plane (see Fig. 6).
If the clamped tube was not polymorphic, but had the
shape of a static circular arc instead, its elastic
fluctuations could be simply decomposed into a radial and an
azimuthal elastic mode. In this case we would expect a classical
“wormlike chain” dynamic behavior, where the chain has merely an
intrinsic curvature. However, such an intrinsically curved elastic
filament would of course not rotate. In sharp contrast to the
wormlike chain case, the observed wobbling scenario allows an
efficient rotation (despite the fixed clamping of the end section)
in our system. This rotary motion, which is the very signature of a
polymorphic lattice with broken symmetry, is apparently caused by a
confostack that moves azimuthally around the lattice.
Note that we have a peculiarly interesting “polymer on a
polymer” motif here: A quasipolymeric entity (the confostack)
exists and moves on the surface of a tube, that itself
can be considered as a polymer on larger scales. The
experimentally observable motion of the tube is slaved to the
motion of the confostack. The latter might be practically hidden
from direct experimental detection but it affects the tube’s
centerline dynamics so severely that it becomes detectable by
tracing the tube’s centerline.
IV.1.2 Energy barrier
A closer look at the distribution of the azimuthal angle
$\phi$ of the tube’s end reveals that the distribution is centered
around discrete angular values $\frac{2\pi n}{13},$
$n\in\mathbb{Z}$, corresponding to the $13$ protofilaments (c.f.
the blobs in Fig. 6). This result
unveils the existence of a periodic potential $V(\phi)$—due to
the discrete lattice structure—in which both the
confostack and in turn also the free end of the tube diffuses.
Although the azimuthal rotation of the confostack can in principle
be continuous, it tends to visit the minimum of $V(\phi)$ much
more frequently. This potential is associated with an effective
energy barrier $\Delta E$ over which the confostack has to hop in
order to move azimuthally on the lattice from one minimum of
$V(\phi)$ to the next one. The consequences of $\Delta E$ on the
confostack movement will be revisited and explored later on.
Fig. 7 shows a histogram of the logarithm of the
angular distribution function of the end of the centerline which
equals $-V(\phi)/(k_{B}T)$ modulo $\frac{2\pi}{13}$. This yields a
first estimate of the barrier $\Delta E$ measured from the
difference between the maximum and the minimum of $V(\phi)$:
$\Delta E\approx 12.5$. As we will see in
Secs. IV.2.2 and
IV.2.3, the same value will be found
with two other methods and is independent of the length $L$ of the
tube for all the lengths considered in the simulations.
IV.1.3 Rotary diffusion of the clamped tube
¿From the simulations one can get a more quantitative
understanding of the wobbling mode kinetics by measuring the mean
square displacement of the azimuthal angle $\phi$ of the end point
of the centerline of the tube. These measurements have been
performed for tubes of various lengths $L=2Nd$ (with $N$ the
number of sections of the tube) and at different temperatures $T$.
The results display the typical behavior of a diffusive system
with a diffusion coefficient $D_{\phi}$: $\left\langle(\phi(t+t_{0})-\phi(t_{0}))^{2}\right\rangle=2D_{\phi}t$. In
Fig. 8 we observe that the diffusion
coefficients normalized by the corresponding temperatures (the
mobility) $\widetilde{D}_{\phi}=D_{\phi}/(k_{B}T)$ all scale with
the length as $N^{-5}$ but do not collapse to a single
temperature-independent curve.
The scaling of the diffusion constant with length
$\widetilde{D}_{\phi}\sim N^{-5}$ is typical for a rotating rigid
circular arc moving through a fluid. The normalized diffusion
coefficient in this case is given by (see
Appendix B) $\widetilde{D}_{\phi_{0}}=\frac{5}{8}\frac{N^{-5}}{\xi_{\perp}\kappa^{2}d^{5}}$, where $\kappa$ is the
curvature of the arc and $\xi_{\perp}$ the friction constant per
unit length.
The temperature dependence of $\widetilde{D}_{\phi}$ is
much more intriguing. Naïvely, we would expect that, like for a
rigid rotor moving through a fluid, $\widetilde{D}_{\phi}$ is
simply a constant with respect to the temperature. However, this
expectation is too simple as it considers only the friction of the
rotating arc through the external fluid medium and misses the
internal dynamics of the confostack, in particular the presence of
structural barriers, as discussed in the previous section. This
hidden internal dynamics nevertheless reflects itself in the
movement of the end point of the tube and leads to an additional
internal dissipation.
In summary, the simulations reveal that the tube’s
friction must result from a combination of the external friction
of the arc in the fluid medium and a yet to be characterized
internal dissipation mechanism. In the next section, we explore
the origin of this inner dissipation mechanism. In particular, we
take a closer look at the conformational mode responsible for
these inner losses via barrier crossing.
IV.1.4 Barrier crossing transition state: The confostack-kink
Until now, we have analyzed the wobbling motion by
tracking the time evolution of the end point of the centerline. In this
section we go a step further and take advantage of the simulation
to directly observe the shape of the confostack on the surface of
the lattice. At zero temperature, the tube assumes one of its
circular ground states oriented along one of the 13 possible
orientations. At any finite temperature, thermal excitations allow
for a continuous shape reorientation of the confostack between the
ground states by crossing the energy barrier $\Delta E$, while at
the same time reorienting the direction of the tube’s curvature.
What is the critical confostack mode, i.e., the
conformation of the confostack at the top of the energy barrier
$\Delta E$?
A naïve first look into the noisy simulation snapshots of the tube in
space does not allow to identify the critical conformational mode on the
barrier (at the transition) in a simple way. But if one rotates the tube in
its natural co-moving frame of reference first (see next section for details),
one can observe the behavior of the confostack on the surface of the tube in
detail. This frame is defined by the $xz$ plane formed by the attachment point
and the end of the centerline of the tube.
Therefore, the deflection perpendicular to the $xz$ plane, $y(s)$,
satisfies the condition $y(0)=y(L)$, where $s$ is the arc length of the centerline.
At zero temperature, the confostack lies in the $xz$ plane and the
centerline of the tube is a circular arc. In this case the
co-moving and the laboratory frames are identical. At finite
temperature and for the transition regions, we typically observe a
confostack forming a kink on the lattice which will propagate
until the whole structure has crossed the barrier (see
Fig. 9 for a snapshot). This confostack-kink
resembles a “polymorphic” dislocation that can be either left- or
right-handed, while reorienting the direction of the whole tube.
To capture and characterize the shape of the
confostack-kink under the conditions of strong thermal noise, we
have looked for statistical anomalies of the tube’s shape in the
co-moving coordinates. Since the centerline of the tube leaves
the $xz$ plane only slightly, we expect that $x(s)$ still
describes a circular arc approximately
($x(s)\approx\frac{1}{2}\kappa s^{2}$). This is confirmed by the
numerical data (not shown). What turns out to be more interesting
is the $y$ direction. The root mean square $\left\langle\sqrt{y^{2}(s)}\right\rangle_{\text{Tr}}$ of the deflection $y(s)$
of the centerline is computed over all transition states (red
circles in Fig. 6). They correspond to the critical confostack-kink
at the top of the energy barrier $\Delta E$. This kink can either move to
the next minimum of the periodic potential $V(\phi)$ or return to the original
minimum. The curve $\left\langle\sqrt{y^{2}(s)}\right\rangle_{\text{Tr}}$ is significantly
different from the measurement of the same quantity for all—transition
and non-transition—states of the simulation taken
together $\left\langle\sqrt{y^{2}(s)}\right\rangle_{\text{All}}$ as
shown in Fig. 9.
IV.2 Theory of the dynamics of the clamped tube
To understand the unusual dynamics of the clamped tube
observed in the simulations and to interpret the behavior of the
confostack-kink at the barrier, in this section we make some
theoretical developments.
IV.2.1 Modelling a single confostack-kink
We first define an external laboratory frame $(X,Y,Z)$. The tube
is clamped at the origin and is oriented in the $Z$ direction. For
small deviations around the $Z$ axis, the unit vector tangent to
the tube’s centerline is approximately given by
$\boldsymbol{t}\approx(\theta_{X},\theta_{Y},1)$ with
$\theta_{X/Y}$ the deflection angles of the centerline in the
$X/Y$ direction. This deflection can be decomposed as the sum
$\theta_{X/Y}=\theta_{\text{el},X/Y}+\theta_{\text{pol},X/Y}$ of
a purely elastic deviation and a polymorphic one mt2012 .
The polymorphic contribution can be expressed in terms of a
polymorphic phase $\varphi\left(s\right)$ which stands for the
confostack’s angular orientation with respect to the tube’s
material frame. Therefore, one has
$\theta_{\text{pol},X}(s)=\kappa\int\nolimits_{0}^{s}\cos{\varphi(s^{\prime})}%
\text{d}s^{\prime}$ and
$\theta_{\text{pol},Y}(s)=\kappa\int\nolimits_{0}^{s}\sin{\varphi(s^{\prime})}%
\text{d}s^{\prime}$. Neglecting the purely
elastic fluctuations, the lateral displacement of the tube in the
$(X,Y)$ plane can be written as
$$\displaystyle X(s)$$
$$\displaystyle=$$
$$\displaystyle\int\nolimits_{0}^{s}\sin\left[\theta_{\text{pol},X}(s^{\prime})%
\right]ds^{\prime}\;,\quad\text{and}$$
$$\displaystyle Y(s)$$
$$\displaystyle=$$
$$\displaystyle\int\nolimits_{0}^{s}\sin\left[\theta_{\text{pol},Y}(s^{\prime})%
\right]\text{d}s^{\prime}\;.$$
(9)
As defined previously, the co-moving frame ($x,y,z=Z)$ is given by the rotation
$$\displaystyle x(s)$$
$$\displaystyle=X(s)\cos\Phi+Y(s)\sin\Phi\;,$$
$$\displaystyle y(s)$$
$$\displaystyle=-X(s)\sin\Phi+Y(s)\cos\Phi$$
(10)
with $\Phi=\arctan(Y(L)/X(L))$. Once we know the polymorphic phase
$\varphi\left(s\right)$ of the
confostack configuration in the transition regime we can determine
the deflections $(x(s),y(s))$ of the centerline in the co-moving frame with the help of Eqs. (10)
and compare them with the experimental data.
To model the behavior of the confostack phenomenologically, we
assume that it moves along a tube of length $L=2Nd$ in a periodic
potential of amplitude $W$ with an effective polymorphic stiffness
$C_{p}$:
$$\Delta E(L)=\int_{-L/2}^{L/2}\text{d}s\;\left\{\frac{C_{p}}{2}\varphi^{\prime 2%
}+\frac{W}{2}[1+\cos{(13\varphi)}]\right\}\;,$$
(11)
where the interval of the arc length $s$ of the centerline has been shifted for
mathematical convenience $s\in\left[-L/2,L/2\right]$.
The transition state is found by minimizing $\Delta E(L)$ with the
natural boundary conditions $\varphi^{\prime}(\pm L/2)=0$, i.e.,
no torque at the confostack ends. As shown in Appendix C there are two
regimes:
a) For $L<\pi\ell$ with $\ell=\sqrt{2C_{p}/(13^{2}W)}$, the minimal energy
barrier-crossing configuration is $\varphi\left(s\right)=0$. This
corresponds to a uniform rotation of the confostack as a block over the
barrier. In this case the barrier energy grows linearly with the length,
$\Delta E(L)=WL$.
b) For $L>\pi\ell$, a nontrivial barrier crossing solution
minimizing $\Delta E(L)$ is:
$$\varphi(s)=\frac{2}{13}\arcsin\left(\frac{1}{\sqrt{m}}\operatorname{sn}\left[%
\frac{s}{\ell},1/m\right]\right)\;,$$
(12)
where $\operatorname{sn}$ is the Jacobi sine function of parameter
$m>1$ Abra . Eqn. (12) is a periodic
function in $s$. The solution which is monotonous and interpolates
between two successive minima of the periodic potential lies on a
finite interval given by the condition $L(m)=2\ell K[1/m]$, where
$K[1/m]$ is the complete elliptic integral of the first kind. In
the limit of large tube lengths $L$, we have $m\approx 1$ and the
transition state confostack Eq. (12) becomes a
kink:
$$\varphi(s)=(4\arctan{(\text{e}^{\frac{s-s_{0}}{\ell}})}-\pi)/13$$
(13)
where $s_{0}$ is the position of the center of the kink on the lattice.
To determine the typical size of the confostack at the transition,
we look at the energy of the solution (12)
computed in Appendix C (see Eq. (23)). In the
regime where the length $L\geq\pi\ell,$ the barrier $\Delta E(L)$
grows sublinearly and saturates at $\Delta E\approx 0.05C_{p}/\ell$
for large $L$. As already mentioned, the notable observation from
the computer simulation is that $\Delta E\approx 12.5$ for all the
lengths considered in the simulations, $L\geq 14d$. Therefore,
$\ell\ll L$ and Eq. (13) is a good approximation for
the typical confostack configuration at the transition. This
analysis of a single confostack-kink allows us to determine the
theoretical root mean square of $y(s)$ at the transition which can
then be compared to the simulations.
IV.2.2 Deflection of the tube at the transition
Assume that there is only a single confostack-kink of very short size $\ell\approx 0$ placed on a
random segment ($n=1,2,..N$) of the tube. Averaging over all positions of the kink with the
same probability we have computed $\left\langle\sqrt{y^{2}(s)}\right\rangle_{\text{model}}$
numerically for $N=20$ as shown in Fig. 9. We find that the model is close to the simulation
data of the transition state. This agreement is reassuring and validates the idea of
the confostack-kink as the main culprit for the crossing of the angular barrier.
In addition, from the ratio of $\left\langle\sqrt{y^{2}(s)}\right\rangle$ in this model over the simulation
data for all frames (transition and non-transition states) we can
infer the kink density in the simulation. Indeed
the probability to find the kink anywhere is $P\approx$ $\frac{\text{max}\left(\left\langle\sqrt{y^{2}(s)}\right\rangle_{\text{All}}%
\right)}{\text{max}\left(\left\langle\sqrt{y^{2}(s)}\right\rangle_{\text{model%
}}\right)}\approx 0.368$.
Therefore, the kinks are roughly three times more frequent in the transition
states as expected from a kink-mediated barrier crossing mechanism.
This result allows us to determine $\Delta E$ in another way:
The probability of finding a single kink on any segment of the tube ($n=1,2,..N$)
is given by:
$$P_{1}=\frac{\exp(-\frac{\Delta E}{k_{B}T})}{1+\exp(-\frac{\Delta E}{k_{B}T})}\;.$$
(14)
The probability to find at least one kink on the tube is then
given by $P=1-(1-P_{1})^{N}$. For a tube with $N=20$ segments at
$T=3$ we find that $P\approx 0.368$ leads to an energy barrier
$\Delta E\approx 12.5$ in agreement with the simulations. The
corresponding $P_{1}\approx 0.018$ shows that for $N=20$ the
density of kinks is $NP_{1}\approx 0.37$. Since the kinks are
roughly three times more frequent at the transition we deduce that
the density of kinks at the transition is about one which
justifies our original assumption.
IV.2.3 Rotor diffusion revisited
The theoretical modelling in the previous sections allowed us to deepen our understanding
of the confostack dynamics in the presence of a barrier. With this knowledge we can return
to the anomalous diffusion of the clamped tube.
Approximating the internal periodic barrier $V(\phi)$
by a sinusoidal of the form $V(\phi)=(\Delta E/2)\cos(13\phi)$, the azimuthal
diffusion coefficient $D_{\phi}$ of the free end of the centerline should obey
the relation Risken
$$D_{\phi}=D_{\phi_{0}}\left[I_{0}\left(\frac{\Delta E}{2k_{B}T}\right)\right]^{%
-2}\;,$$
(15)
where $I_{0}$ denotes the modified Bessel function of the first kind Abra . The
robustness of the scaling $\sim N^{-5}$ of $D_{\phi}$ confirms again that the energy
barrier $\Delta E$ has to be independent of $N$ for the lengths we considered.
To compare the diffusion at different temperatures, we introduce a
new scaled (temperature-independent) diffusion coefficient $\tilde{D}=\frac{D_{\phi}}{k_{B}T}\left[I_{0}\left(\frac{\Delta E}{2k_{B}T}%
\right)\right]^{2}$.
In Fig. 10 we see that the simulation results for
different temperatures collapse to a single curve for $\Delta E\approx 12.5$. This coincides again with the previous results. The
master curve is slightly below the curve for the perfect rotor
diffusion coefficient $\widetilde{D}_{\phi_{0}}$.
This small discrepancy can be understood as the
polymorphic tube is softer than the ideal rigid rotor. Indeed, it
has to sustain some additional deformations due to the presence
and migration of defects in the confostack. The reaction
coordinate between two angular orientations is therefore in
reality not straight as it would be for a rigid rotor, but instead
the system has to take a non-straight “detour” in the
configuration space. This leads to an apparent reduction of the
diffusion constant along the ideal (shortest path) rotary reaction
coordinate.
To summarize, we have seen that both the external friction of the
arc (through the external fluid medium) as well as an internal
dissipation mechanism (crossing an inner energetic barrier) of the
tube combine together into an effective friction $\xi_{\phi}=k_{B}T/D_{\phi}$. Note that contributions of an anomalous friction in
the short modes of microtubules were reported by Jansen &
Dogterom Jansen , Taute et al. Taute and Brangwynne
et al. Brangwynne . It was speculated by these researchers
that some form of internal dissipation mechanism was at work. In
this section we have seen how internal barriers and conformational
cooperativity give rise to such internal friction phenomena.
V Conclusion
The elastic and thermal properties of isotropic and anisotropic
macromolecular tubes have been the focus of scientific research
for decades. In this paper we have studied a new system consisting
of a tubular lattice whose individual elements can switch between
a flat and a curved configuration. This triggers the birth of
“confoplexes”, conformational
quasiparticles that interact via long-range repulsive
interactions mediated by the elasticity of the lattice. By
introducing structural cooperativity (as motivated by biological
systems) and in turn “polymerizing” a number of confoplexes on the
tubular lattice, a plethora of different phenomena have been
discovered: the tube spontaneously breaks its cylindrical symmetry
and forms superhelical structures in three-dimensional space.
Remarkably, at finite temperature, the movement of the
quasipolymer built out of confoplexes on the lattice constantly
reshapes the whole tube inducing a random rotation for a clamped
tube. This dynamics has been studied in detail by numerical
simulations and phenomenological theory. We found that the
quasipolymer—the confostack—has to cross a periodic energy
barrier to move azimuthally on the lattice. We observe that the
typical conformational mode for barrier crossing is a conformational
defect that we termed a confostack-kink. The associated kink-dynamics
of the confostack on the lattice was the clue to explain
the behavior of the diffusion coefficient of the clamped tube.
Looking back at what we have learned from the present tube model, we note one
interesting perspective crystallizing out. It is the idea of localized
conformational quasiparticles living on the lattice. We have seen how an
elementary quasiparticle—the confoplex—emerges and how it interacts with others of
the same kind via elastic lattice modes. Once an additional
cooperative interaction is introduced, the particle-like confoplexes are
forced together into an extended polymer-like conformational object—the
confostack. However, curiously the quasipolymeric confostack tends, once
again, to decompose into smaller discrete entities. This gives rise to another
discrete localized particle-like entity—the confostack-kink.
Exploring the implications of the quasiparticle point of view to study the
“confotronics” of a multitude of concrete biological (tubular or cylindrical)
monomer lattices, like flagellin, microtubules and actin, promises quite some
excitement ahead.
Acknowledgements.
The PMMS (Pôle Messin de Modélisation et de Simulation) is
acknowledged for providing the computer time. We thank Albert Johner,
Jean-François Joanny, Carlos Marques, Helmut Schiessel, René Messina
and Norbert Stoop for stimulating discussions.
Appendix A Details of the simulations
Each vertex of the lattice is treated as a bead subject to the
equation of motion:
$$\displaystyle m\ddot{\mathbf{r}}=\mathbf{f}-m\gamma\dot{\mathbf{r}}+\mathbf{%
\Gamma}\;,$$
(16)
where $m$ is the mass of the bead, and $\gamma$ is the damping
constant. $\mathbf{\Gamma}$ denotes the Gaussian white noise, and
$\mathbf{f}$ is the sum of the (elastic) forces acting on the
bead.
The corresponding discrete velocity Verlet algorithm reads
Schlick
$$\displaystyle\dot{\mathbf{r}}_{n+1/2}$$
$$\displaystyle=\dot{\mathbf{r}}_{n}+(\mathbf{f}(\vec{r}_{n})-m\gamma\dot{%
\mathbf{r}}_{n}+\mathbf{\Gamma}_{n})\frac{\Delta t}{2m}\;,$$
$$\displaystyle\mathbf{r}_{n+1}$$
$$\displaystyle=\mathbf{r}_{n}+\dot{\mathbf{r}}_{n+1/2}\Delta t\;,$$
(17)
$$\displaystyle\dot{\mathbf{r}}_{n+1}$$
$$\displaystyle=\dot{\mathbf{r}}_{n+1/2}+(\mathbf{f}(\vec{r}_{n+1})-m\gamma\dot{%
\mathbf{r}}_{n+1}+\mathbf{\Gamma}_{n+1})\frac{\Delta t}{2m}\;,$$
where the noise term is sampled with zero mean and a variance
$\langle\Gamma_{i}^{2}\rangle=2k_{B}T_{0}m\gamma/\Delta t$
for each component $i$ with room temperature $T_{0}$.
In the simulations, we set the Boltzmann constant, the mass and the
damping coefficient to unity. The integration time step is set to
$\delta t=0.001\,\tau$, where $\tau$ denotes the unit of time
in the simulations.
The elastic parameters in the simulations are chosen in the following manner:
The rest lengths are set to $d^{(0)}_{1}=d^{(0)}_{2}=d^{(0)}_{3}=d$ and
$d^{(0)}_{4}=\sqrt{2}\,d$, where $d=4\,$nm is the size of a monomer, chosen
as the unit length of the system.
The angular constants are given by $s^{(0)}_{1}=s^{(0)}_{4}$ and $s^{(0)}_{3}=\sin{\frac{2\pi}{13}}$.
These values ensure that the lattice is cylindrical in the absence of the anharmonic potential.
The value of the stretching rigidity $\mu_{s}$ is specified in the
main text. For the diagonal bonds, we apply $\mu_{s4}=1000\,k_{B}T_{0}/d^{2}$ which is sufficient to conserve the rectangular
nature of the subunits. A bending rigidity $\mu_{b4}=500\,\,k_{B}T_{0}$ is chosen for the diagonal bonds. The bending rigidity
of the bonds $B_{3}$ is set to $\mu_{b3}=1500\,\,k_{B}T_{0}$.
The coefficients of the anharmonic bending potential
$E_{2}^{\text{BAH}}$ of the bonds of type $B_{2}$, given in Eq. (3),
are set to $A=3350\,k_{B}T_{0}$, $B=2960\,k_{B}T_{0}$, $C=290\,k_{B}T_{0}$ to ensure that $\Delta G=200\,\,k_{B}T_{0}$ and
$\delta G=1\,\,k_{B}T_{0}$. The global minimum of
$E_{2}^{\text{BAH}}$ is situated at $\bar{s}_{2}=-0.588$. This
value leads to a preferred curved state with an angle $36^{\circ}$
of the free dimer. Note that, if we approximated the double well
potential by a harmonic one around $\bar{s}_{2}$, we could write
$E_{2}^{\text{BAH}}\approx\frac{\mu_{b2}}{2}\sum_{\{s_{2}\}}\left(s_{2}-\bar{s}%
_{2}\right)^{2}$ with an effective
bending rigidity $\mu_{b2}\approx 2000\,k_{B}T_{0}$, a value
slightly larger than $\mu_{b3}$.
Appendix B Rigid rotor dynamics in a surrounding fluid
To explain the temporal diffusion of the clamped polymorphic tube,
we need the diffusion coefficient of a rotating rigid circular arc
in a fluid. Assuming a constant friction per unit length for the
cross section of the arc, $\xi_{\perp}$, the Rayleigh dissipation
functional is given as
$P_{\text{diss}}=\frac{1}{2}\xi_{\perp}\int_{0}^{L}\dot{\rho}(s,t)^{2}\text{ds}%
=\frac{1}{2}\xi_{\perp}\int_{0}^{L}\rho^{2}\omega^{2}\text{ds}$ where $\omega$
is the angular velocity of the rotor. Since the deflection of the
arc is $\rho(s)=\frac{1}{2}\kappa s^{2}$, we obtain
$P_{\text{diss}}=\frac{1}{2}\xi_{\phi_{0}}\omega^{2}$ with an
effective rotational friction $\xi_{\phi_{0}}$ that can be
directly read off
$\xi_{\phi_{0}}=\frac{\xi_{\perp}\kappa^{2}L^{5}}{20}$. The
azimuthal diffusion coefficient
$D_{\phi_{0}}=\frac{k_{B}T}{\xi_{\phi_{0}}}$ of the free end of
the centerline is thus:
$$D_{\phi_{0}}=\frac{5}{8}\frac{k_{B}T}{\xi_{\perp}\kappa^{2}d^{5}}N^{-5}$$
(18)
where $N=L/(2d)$ (see Sec. IV.1.3).
Appendix C The emergence of a polymorphic confostack-kink
The wobbling mode of the clamped polymorphic tube is due to the
formation of a particular confostack configuration which allows
the tube to cross the angular energy barrier. The details of this
movement can be understood with the help of a phenomenological
model.
The energy of a confostack, Eq. (11), can be written
in terms of the transformed angle $\psi=13\varphi$ and the scaled
polymorphic stiffness $\widetilde{C}_{p}=13^{-2}C_{p}$ as
$$\Delta E(L)=\int_{-L/2}^{L/2}\text{d}s\;\left[\frac{\widetilde{C}_{p}}{2}\psi^%
{\prime 2}+\frac{W}{2}(1+\cos{\psi})\right]\;.$$
(19)
This energy has to be minimized with the boundary conditions
$\psi^{\prime}(\pm L/2)=0$ (no torque) to find the
barrier-crossing configuration of the confostack. This gives the
Euler-Lagrange equation
$$\ell^{2}\psi^{\prime\prime}=-\sin\psi$$
(20)
with $\ell=\sqrt{2\widetilde{C}_{p}/W}$. There is always the
trivial configuration $\psi\left(s\right)=0$ with energy
$\Delta E(L)=WL$ and a non-trival antisymmetric solution with
$\psi\left(0\right)=0$:
$$\psi(s)=2\operatorname{am}\left(\frac{s}{\ell\sqrt{m}},m\right)\text{ \ with }%
m=\frac{4}{2+C}$$
(21)
with $\operatorname{am}(s,m)$ the Jacobi amplitude function of
parameter $m\in\left[0,1\right]$ Abra and $C$ a constant
of integration. The solution describes a revolving pendulum wich
is a multi-kink solution. A single kink can be defined on a finite
region of size $L$, given implicitly by the
relation $\sqrt{m}K[m]=L/2\ell$. However, the boundary conditions
$\psi^{\prime}\left(\pm L/2\right)=0$ can never be satisfied in
this case, except for $L\rightarrow\infty$ $(m\to 1)$.
Nevertheless, a physical solution for a finite
length of the confostack can be obtained by analytic continuation of Eq. (21) choosing $m>1:$
$$\psi(s)=2\arcsin\left(\frac{1}{\sqrt{m}}\operatorname{sn}\big{[}\frac{s}{\ell}%
,1/m\big{]}\right)\text{
\ \ with }m>1\;.$$
(22)
This solution is a periodic function. It is monotonous on a finite
interval given by $L(m)=2\ell K[1/m]$. The associated energy is
$$\Delta E(m)=\frac{\widetilde{C}_{p}}{\ell}\left(8\operatorname{E}[1/m]-\frac{4%
(m-1)}{m}\operatorname{K}[1/m]\right)\;.$$
(23)
with $\operatorname{K}$ and $\operatorname{E}$ being the complete
elliptic integrals of the first and the second kind.
For $m\simeq 1$, and thus $L$ very large,
$\psi(s)=4\arctan{(\text{e}^{\frac{s}{\ell}})}-\pi$ and the
barrier energy is a constant $\Delta E\approx 8\widetilde{C}_{p}/\ell$. Increasing $m$ decreases $L,$
and the energy stays close to its plateau value
$8\widetilde{C}_{p}/\ell$. When $L$ approaches
$L_{c}=\pi\ell$ , $\Delta E$ decreases sublinearly and reaches $\Delta E\approx 2\widetilde{C}_{p}\pi/\ell$ for $L=L_{c}$ $(m=\infty)$. For $L<L_{c}$, $\psi\left(s\right)=0$ is the only solution, and the barrier scales
linearly with $L$ in this regime. A comparison with the results of the numerical
simulations is presented in the main text.
References
(1)
S. Iijima, Helical microtubules of grafitic carbon,
Nature 354, 56
(1991).
(2)
F. Mathieu et al., Six-Helix Bundles Designed from
DNA, Nano Letters 5, 661
(2005).
(3)
T. Shimizu, M. Masuda, H. Minamikawa,
Supramolecular Nanotube Architectures Based on Amphiphilic Molecules,
Chem. Rev. 105, 1401 (2005).
(4)
Y. Yamamoto et al., Photoconductive Coaxial
Nanotubes of Molecularly Connected Electron Donor and Acceptor Layers,
Science 314, 1761 (2006).
(5)
X. B. Zhang et al., The Texture of Catalytically
Grown Coil-Shaped Carbon Nanotubules, Europhys. Lett. 27, 141 (1994).
(6)
A. Volodin, M. Ahlskog, E. Seynaeve, C. van Haesendonck,
Imaging the Elastic Properties of Coiled Carbon Nanotubes with Atomic
Force Microscopy, Phys. Rev. Lett. 84, 3342 (2000).
(7)
B. I. Dunlap, Constraints on small graphitic
helices, Phys. Rev. B 50, 8134 (1994).
(8)
S. M. Douglas, J. J. Chou, W. M. Shih,
DNA-nanotube-induced alignment of membrane proteins for NMR structure
determination, Proc. Natl. Acad. Sci. U.S.A. 104, 6644 (2007).
(9)
G.-M. Nam et al., Helices at interfaces,
Europhys. Lett. 100, 28001 (2012).
(10)
L.-S. Li et al., A Torsional Strain Mechanism To
Tune Pitch in Supramolecular Helices, Angew. Chem. Int. Ed. 119, 5977
(2007).
(11)
C. R. Calladine, Construction of bacterial
flagella, Nature 255, 121
(1975).
(12)
H. Mohrbach, A. Johner, I. M. Kulić, Tubulin
Bistability and Polymorphic Dynamics of Microtubules, Phys. Rev. Lett.
105, 268102 (2010).
(13)
H. Mohrbach, A. Johner, I. M. Kulić, Cooperative
lattice dynamics and anomalous fluctuations of microtubules Eur.
Biophys. J. 41, 217 (2012)
(14)
L. A. Amos, W. B. Amos, The bending of sliding
microtubules imaged by confocal light microscopy and negative stain electron
microscopy, J. Cell Sci. Suppl. 14, 95
(1991).
(15)
C. Elie-Caille et al., Straight GDP-tubulin
protofilaments form in the presence of taxol, Curr. Biol. 17, 1765
(2007).
(16)
D. Chretien, R. H. Wade, New data on the microtubule
surface lattice, Biol. Cell. 71, 161
(1991).
(17)
S. V. Srigiriraju, T. R. Powers,
Continuum Model for Polymorphism of Bacterial Flagella,
Phys. Rev. Lett. 94, 248101 (2005).
(18)
H. Wada, R. R. Netz,
Discrete elastic model for stretching-induced flagellar
polymorphs, Europhys. Lett. 82, 28001 (2008).
(19)
B. Friedrich, A mesoscopic model for helical
bacterial flagella, J. Math. Biol. 53, 162
(2006).
(20)
A. Goriely, M. Tabor, Spontaneous Helix Hand Reversal and Tendril
Perversion in Climbing Plant, Phys. Rev. Lett. 80, 1564
(1998).
(21)
H. Wada, R. R. Netz,
Model for Self-Propulsive Helical Filaments: Kink-Pair
Propagation, Phys. Rev. Lett. 99, 108102 (2007).
(22)
R. E. Goldstein, A. Goriely, G. Huber, C. W. Wolgemuth,
Bistable helices, Phys. Rev. Lett. 84, 1631
(2000).
(23)
I. M. Jánosi, D. Chrétien, H. Flyvbjerg,
Modeling elastic properties of microtubule tips and walls, Eur.
Biophys. J. 27, 501
(1998).
(24)
H. Mohrbach, I. M. Kulić, Solvable model for
polymorphic biofilaments, Phys. Rev. E 85, 031903 (2012).
(25)
H. S. Seung, D. R. Nelson, Defects in flexible
membranes with crystalline order, Phys. Rev. A 38, 1005
(1988).
(26)
J. Rudnick, R. Bruinsma, DNA-Protein
Cooperative Binding through Variable-Range Elastic Coupling, Biophys. J.
76, 1725
(1999).
(27)
T. R. Weikl, Indirect interactions of membrane-adsorbed cylinders,
Eur. Phys. J. E 12, 265 (2003).
(28)
M. M. Müller, M. Deserno, J. Guven, Geometry of
surface-mediated interactions, Europhys. Lett. 69, 482 (2005).
(29)
M. M. Müller, M. Deserno, J. Guven,
Interface-mediated interactions between particles: A geometrical
approach, Phys. Rev. E 72, 061407 (2005).
(30)
S. Mkrtchyan, C. Ing, and J. Z. Y. Chen,
Adhesion of cylindrical colloids to the surface of a membrane,
Phys. Rev. E 81, 011904 (2010).
(31)
F. Pampaloni et al., Thermal fluctuations of
grafted microtubules provide evidence of a length-dependent persistence
length, Proc. Natl. Acad. Sci. U.S.A. 103, 10248 (2006).
(32)
K. M. Taute, F. Pampaloni, E. Frey, E.-L. Florin,
Microtubule dynamics depart from the wormlike chain model, Phys. Rev.
Lett. 100, 028102 (2008).
(33)
M. Abramowitz, I. Stegun, Handbook of Mathematical
Functions, Dover, New York (1972).
(34)
H. Risken, The Fokker-Planck Equation: Methods of
Solution and Applications, Springer (1996).
(35)
M. E. Janson, M. Dogterom, A bending mode analysis for
growing microtubules: evidence for a velocity-dependent rigidity, Biophys. J.
87, 2723
(2004).
(36)
C. Brangwynne, G. Koenderink, E. Barry, Z. Dogic, F.
MacKintosh, D. Weitz, Bending dynamics of fluctuating biopolymers
probed by automated high-resolution filament tracking, Biophys. J.
93, 346
(2007).
(37)
T. Schlick, Molecular Modeling and Simulation: An
Interdisciplinary Guide, Springer (2010). |
Preventing eternality in phantom inflation
Chao-Jun Feng
fengcj@shnu.edu.cn
Shanghai United Center for
Astrophysics (SUCA),
Shanghai Normal University,
100 Guilin Road, Shanghai 200234, People’s Republic of China
Xin-Zhou Li
kychz@shnu.edu.cn
Shanghai United Center for
Astrophysics (SUCA),
Shanghai Normal University,
100 Guilin Road, Shanghai 200234, People’s Republic of China
Emmanuel N. Saridakis
msaridak@phys.uoa.gr
College of Mathematics
and Physics,
Chongqing University of Posts and
Telecommunications, Chongqing, 400065, People’s Republic of China
Abstract
We have investigated the necessary conditions that prevent phantom inflation from being eternal. Allowing additionally
for a nonminimal coupling between the phantom field and gravity, we present the slow-climb requirements, perform an
analysis of the fluctuations, and finally we extract the overall conditions that are necessary in order to prevent
eternality. Furthermore, we verify our results by solving explicitly the cosmological equations in a simple example of
an exponential potential, formulating the classical motion plus the stochastic effect of the fluctuations through
Langevin equations. Our analysis shows that phantom inflation can be finite without the need of additional exotic
mechanisms.
pacs: 98.80.Cq, 98.80.-k
I Introduction
After almost three decades of extensive research, inflation is now
considered to be a crucial part of the cosmological history of the
Universe inflation , having affected indelibly its
observational features. Introducing a scalar field, the inflaton,
and a suitable potential, one can make various scenarios of
inflation realization in conventional, as well as in higher-dimensional frameworks Lindebook ; LidseyRMP ; Baumann:2009ni .
Additionally, one could generalize the aforementioned paradigm,
allowing for a nonminimal interaction of the scalar field with
gravity nonminimal0 , since nonminimal inflation could
improve the obtained perturbation spectrum
miaoLi ; recent nonminimal .
One important subject that has to be addressed in this paradigm is that of the exit from the inflationary epoch, that
is to examine whether inflation can be eternal or not. In particular, in the new inflation scenario it was shown that
the procedure could be eternal since the “false” vacuum (in which the field lies during inflation) is never dominated
by the “true” one (the approach of which causes the end of inflation) steinhardt-nuffield ; Vilenkin:1983xq .
Additionally, even in advanced scenarios, such as the chaotic inflation, where there is no false vacuum state,
slow-roll eternality is also possible Linde:1986fd due to a different mechanism. In particular, in this
model-subclass the inflaton is classically rolling down its potential slope, however the quantum fluctuations can
conditionally drive it upwards and thus inflation will never end recent pro eternal ; Linde:2007fr ; Guth:2007ng .
Thus, one must in general examine the conditions for the realization of eternality Guth:2000ka .
An interesting class of inflation scenarios
Piao:2004tq ; Lidsey:2004xd ; Nojiri:2005pu ; Capozziello:2005tf ; GonzalezDiaz:2004df ; Wu:2006wu ; Piao:2007ne ; Elizalde:2008yf
is achieved through the use of phantom fields phant0 , inspired by the wide use of such fields to explain the
late-time universe acceleration phant1 . The simplest realization of phantom fields is the use of a negative
kinetic term in the Lagrangian, but this could lead their quantum theory to be problematic, due to the causality and
stability problems and the possible spontaneous breakdown of the vacuum into phantoms and conventional particles
CHT ; Cline:2003gs . However, one could consider that the phantom fields arise through an effective description of
a nonphantom fundamental (probably higher-dimensional) underlying theory, consistently with the basic requirements of
quantum field theory quantumphantom0 . Indeed actions with phantomlike behavior may arise in supergravity
N , scalar tensor gravity BEPS , higher derivative gravity P , braneworld SS , k-field
ADM , stringy Frampton:2002tu and others scenarios Chimento:2003qy ; Stefancic:2003bj .
The peculiar nature of phantom fields requires the inflation
paradigm to be suitably redesigned. In particular, since phantoms
behave inversely in potential slopes, climbing up along them, in
order to avoid an early-time Big Rip singularity BigRip , one
must use potentials with maxima instead of minima, and the slow-roll parameters are replaced by the “slow-climb” ones
Sami:2003xv . However, even with potentials bounded from
above, the problem of eternal inflation still exists, and one
should examine in detail the possible exits from the inflationary
epoch
Piao:2004tq ; GonzalezDiaz:2004df ; Wu:2006wu ; Piao:2007ne ; Elizalde:2008yf .
In the present work we are interested in investigating the
necessary conditions that prevent phantom inflation from being
eternal, going beyond the basic requirements of bounded-from-above
potentials. In particular, we examine whether the quantum
fluctuations could affect the classical motion towards the
potential maximum, preventing inflation to the end. Furthermore, in
order to be general we allow for a nonminimal coupling of the
phantom field with gravity, since this interaction could also
affect the eternality conditions, similarly to the canonical case
Feng:2009kb .
The plan of this work is the following: In Sec.
II we present the phantom-inflation scenario
under the conditions of slow-climb. In Sec. III we
perform a fluctuation analysis and we extract the conditions for
preventing eternality, while in Sec. IV we verify
our results by solving explicitly the Langevin equations for the
cosmological evolution in a simple example. Finally, Sec.
V is devoted to the summary of the obtained
results.
II Phantom slow-climb inflation
Let us present briefly the cosmological scenario of nonminimal
phantom inflation Elizalde:2008yf , focusing on the
conditions required for its long-time, efficient duration. The
action of a universe constituted by a phantom scalar field
$\varphi$ is given by
$$S=\int d^{4}x\sqrt{-g}\left[\frac{R}{2}+\frac{1}{2}g^{\mu\nu}\partial_{\mu}%
\varphi\partial_{\nu}\varphi-V(\varphi)-\frac{1}{2}f(R)\varphi^{2}\right],$$
(1)
with $V(\varphi)$ the corresponding potential, and where for
simplicity we have set $8\pi G=M_{p}^{-2}=1$. As usual $R$ is
the Ricci scalar, and $f(R)$ is the function describing the
coupling of the phantom field to gravity. Throughout this work we
consider a flat Friedmann-Robertson-Walker geometry with the
unperturbed metric
$$ds^{2}=-dt^{2}+a^{2}(t)\left(dx^{2}+dy^{2}+dz^{2}\right),$$
(2)
with $a(t)$ the scale factor and $t$ the comoving time. Thus,
defining the Hubble parameter as $H\equiv\dot{a}/a$ the
scalar curvature reads:
$$R=6\left(\dot{H}+2H^{2}\right),$$
(3)
where the dot denotes the derivative with respect to $t$.
Variation of the action (1) leads to the two Friedmann
equations
$$3H^{2}=-\frac{1}{2}\dot{\varphi}^{2}+V+\frac{1}{2}f\varphi^{2}+3H^{2}\left[%
\frac{d}{dt}\left(\frac{f^{\prime}\varphi^{2}}{H}\right)-f^{\prime}\varphi^{2}\right]$$
(4)
$$-2\dot{H}=-\dot{\varphi}^{2}+H^{3}\frac{d}{dt}\left(\frac{f^{\prime}\varphi^{2%
}}{H^{2}}\right)-\frac{d^{2}}{dt^{2}}\bigg{(}f^{\prime}\varphi^{2}\bigg{)},$$
(5)
where the prime denotes differentiation with respect to the
corresponding argument, that is $f^{\prime}=df/dR$ and $V^{\prime}=dV/d\varphi$. Additionally, the evolution equation for the scalar
field writes:
$$\ddot{\varphi}+3H\dot{\varphi}-V^{\prime}-f\varphi=0.$$
(6)
As we discussed in the Introduction, the phantom fields require
potentials bounded from above, since they climb upwards the
potential slopes. Therefore, in order to acquire a long-time
inflation in phantom cosmology we impose the following slow-climb
conditions Sami:2003xv
$$|\dot{H}|\ll H^{2}\,,\quad|\ddot{\varphi}|\ll 3H|\dot{\varphi}|,$$
(7)
which corresponds to the slow-roll conditions of canonical
inflation Lindebook . After some algebra, and assuming
potential domination ($|-\frac{\dot{\varphi}^{2}}{2}|\ll V$) to
simplify the calculations, the slow-climb conditions write as
$$|f^{\prime}\varphi^{2}\dot{H}|\ll V\,,\quad\bigg{|}V^{\prime\prime}+f-\frac{3%
Hf^{\prime}\dot{R}}{f}\frac{f\varphi}{V^{\prime}+f\varphi}\bigg{|}\ll 9H^{2}.$$
(8)
Furthermore, in the usual case where $f$ is a monomial of
$R$, for instance $f\sim R^{n}$, we obtain $3H(\log f)^{\prime}\dot{R}\approx 6n\dot{H}$. Therefore, if $|f\varphi|\ll|V^{\prime}|$ or
$|f\varphi|\gg|V^{\prime}|$, the third term on the left hand side of the
second equation in (8) can be neglected. Thus, the
aforementioned expressions are simplified to
$$|f^{\prime}\varphi^{2}\dot{H}|\ll V\,,\quad\bigg{|}V^{\prime\prime}+f\bigg{|}%
\ll 9H^{2}\,.$$
(9)
In summary, under these slow-climb conditions, the first Friedmann
equation (4) becomes
$$3H^{2}=\frac{1}{2}\left(f-6f^{\prime}H^{2}\right)\varphi^{2}+V\,,\\
$$
(10)
while the phantom field equation of motion, for the two examined
limiting cases, is simplified as
$$\displaystyle 3H\dot{\varphi}$$
$$\displaystyle=$$
$$\displaystyle V^{\prime}\,,\quad|f\varphi|\ll|V^{\prime}|\quad(\text{Case I}~{%
})\,,$$
(11)
$$\displaystyle 3H\dot{\varphi}$$
$$\displaystyle=$$
$$\displaystyle f\varphi\,,\quad|f\varphi|\gg|V^{\prime}|\quad(\text{Case II})\,.$$
(12)
At this stage we introduce the standard
dimensionless slow-climb parameters as Piao:2004tq ; Wu:2006wu
$$\epsilon=-\frac{\dot{H}}{H^{2}}\,,\quad\eta=M_{p}^{2}\frac{V^{\prime\prime}}{V%
}\,,$$
(13)
and following Feng:2009kb we define a new dimensionless
slow-climb parameter
$$\Delta\equiv M_{p}^{2}\frac{f}{V}$$
(14)
to account for the nonminimal coupling, where we have recovered
the Planck mass to indicate that these parameters are indeed
dimensionless. Using these parameters, the slow-climb conditions
(9) become
$$\epsilon\Delta\varphi^{2}\ll 1\,,\quad\eta+\Delta\ll 1,$$
(15)
having also used for simplicity $f^{\prime}\sim f/R$ although this is not
necessary. Therefore, if $\epsilon,\eta,\Delta\ll 1$, the
slow-climb conditions (9) are indeed satisfied.
In order to continue, we consider explicitly the usual ansatz for
$f(R)$ of the literature nonminimal0 ; recent nonminimal ,
namely $f=\xi R$, with $\xi$ the coupling parameter. Thus, the
Friedmann equation (10) becomes
$$3H^{2}=\frac{V}{1-\xi\varphi^{2}}\,,$$
(16)
and the slow-climb parameter $\Delta$ reads
$$\Delta=\frac{2\xi(2-\epsilon)}{(1-\xi\varphi^{2})}.$$
(17)
As we see, $\Delta\ll 1$ requires $\xi\ll 1$ in the model at hand, a
condition which is usually satisfied in all nonminimal scenarios.
Finally, differentiating the Friedmann equation (10),
we deduce that in the case $|f\varphi|\ll|V^{\prime}|$ (Case I), that is
$\Delta\varphi^{2}\ll 1$, the slow-climb parameter $\epsilon$
becomes
$$\epsilon=-\frac{V^{\prime 2}}{2V^{2}}\bigg{[}1-\left(1-\frac{2V}{V^{\prime}%
\varphi}\right)\xi\varphi^{2}\bigg{]}\,,$$
(18)
while in the case of $|f\varphi|\gg|V^{\prime}|$ (Case II), i.e. $\Delta\varphi^{2}\gg 1$, it becomes
$$\epsilon=-\frac{f\varphi V^{\prime}}{2V^{2}}\bigg{[}1-\left(1-\frac{2V}{V^{%
\prime}\varphi}\right)\xi\varphi^{2}\bigg{]}\,.$$
(19)
Note that in the latter case the condition $\Delta\varphi^{2}\gg 1$ requires the field values to be large
and therefore, without loss of generality, in the following we
consider large-field inflation.
III Fluctuations and conditions for preventing eternality
In the previous section we extracted the basic conditions for an efficient long time, but noneternal phantom
inflation. However, as we discussed in the Introduction, even if one manages to stop inflation at the classical level
using suitable potentials, the backreaction of the metric plus the inflaton’s quantum fluctuations on the background
space-time could make the inflaton field follow a Brownian motion in which half of the time the inflaton field in a
given domain will jump downwards, instead of drifting up to the potential. Thus, the necessary conditions for preventing
eternality in phantom inflation will arise through examination of the overall effects of the classical behavior plus
the fluctuations.
In order to calculate the quantum fluctuation of the inflaton, we
expand the action (1) to second order, since the action
approach guarantees the correct normalization for the quantization
of fluctuations. It is convenient to work in the Arnowitt-Deser-Misner (ADM) formalism and
write the metric as
$$ds^{2}=-N^{2}dt^{2}+h_{ij}(dx^{i}+N^{i}dt)(dx^{j}+N^{j}dt)\,,$$
(20)
where $N$ is the lapse function and $N^{i}$ is the shift vector.
Note that such perturbations have been studied in a different
framework, for the minimal case, in Piao:2004tq ; Wu:2006wu .
The action (1) becomes
$$\displaystyle S=\frac{1}{2}\int dtdx^{3}\sqrt{h}\left[NR^{(3)}+N^{-1}\left(E_{%
ij}E^{ij}-E^{2}\right)\ \ \ \ \ \ \ \ \ \ \ \right.$$
$$\displaystyle\left.-N^{-1}\left(\dot{\varphi}-N^{i}\partial_{i}\varphi\right)^%
{2}+Nh^{ij}\partial_{i}\varphi\partial_{j}\varphi-N\left(2V+f\varphi^{2}\right%
)\right],$$
(21)
where $h=\det h_{ij}$ and the symmetric tensor $E_{ij}$ is defined
as
$$E_{ij}=\frac{1}{2}\bigg{(}\dot{h}_{ij}-\nabla_{i}N_{j}-\nabla_{j}N_{i}\bigg{)}%
\,,\quad E=E^{i}_{~{}i}\,.$$
(22)
In (III) $R^{(3)}$ is the three-dimensional Ricci
curvature, which is computed from the metric $h_{ij}$, and
$K_{ij}=E_{ij}/N$ is the extrinsic curvature. In the following we
work in the spatially-flat gauge and we neglect the tensor
perturbations. Thus, we write
$$\varphi(t,x)=\bar{\varphi}(t)+\delta\varphi(t,x)\,,\quad h_{ij}=a^{2}\delta_{%
ij}\,,$$
(23)
where $\bar{\varphi}(t)$ is the background value of the scalar field
and $\delta\varphi$ is a small fluctuation around the background
value.
In the ADM formalism one can consider $N$ and $N^{i}$ as Lagrange
multipliers, and in order to obtain the action for $\xi$ one needs
to solve the constraint equations for $N$ and $N^{i}$ and substitute
the result back in the action. The equations of motion for $N^{i}$
and $N$ are the momentum and Hamiltonian constraints
$$\displaystyle\nabla_{i}\bigg{[}(1-f^{\prime}\varphi^{2})N^{-1}\left(E^{i}_{j}-%
\delta^{i}_{j}E\right)\bigg{]}$$
$$\displaystyle\ \ \ \ \ \ \ +N^{-1}\left(\dot{\varphi}-N^{i}\partial_{i}\varphi%
\right)\partial_{j}\varphi=0$$
(24)
and
$$\displaystyle R^{(3)}-(1-2f^{\prime}\varphi^{2})N^{-2}\left(E_{ij}E^{ij}-E^{2}\right)$$
$$\displaystyle+N^{-2}\left(\dot{\varphi}-N^{i}\partial_{i}\varphi\right)^{2}-2V%
-f\varphi^{2}+h^{ij}\partial_{i}\varphi\partial_{j}\varphi=0.$$
(25)
We now decompose $N^{i}$ into
$$\displaystyle N^{i}=\partial^{i}\psi+N^{i}_{T}$$
(26)
with
$\partial_{i}N^{i}_{T}=0$, and we define
$$\displaystyle N_{1}\equiv N-1,$$
(27)
where $N_{1},N^{i}_{T},\psi\sim\mathcal{O}(\delta\varphi)$. Thus, inserting these expansions into
(24) and (25), we can obtain the
solutions up the first order in $\xi$. In particular, in the usual
case $f=\xi R$, we can derive the first order solutions
similarly to the Appendix of Feng:2009kb . Simplifying the
notation using $\varphi$ to denote the background value
$\bar{\varphi}$, we finally acquire:
$$N_{1}=-\frac{\delta\varphi}{1-\xi\varphi^{2}}\left(\frac{\dot{\varphi}}{2H}+2%
\xi\varphi\right)\,,\quad N^{i}_{T}=0\,,$$
(28)
and
$$\displaystyle(1-\xi\varphi^{2})\partial^{2}\psi=-N_{1}\frac{\dot{\varphi}^{2}}%
{2H}+\frac{\dot{\varphi}}{2H}\delta\dot{\varphi}$$
$$\displaystyle+\left(\frac{3}{2}\frac{\dot{\varphi}}{H}+6\xi\varphi-\frac{V^{%
\prime}}{2H^{2}}\right)H\delta\varphi,$$
(29)
with suitable boundary conditions. Furthermore, we obtain the
exact background dynamical equation
$$3H^{2}(1-\xi\varphi^{2})=-\frac{1}{2}\dot{\varphi}^{2}+V,$$
(30)
which coincides with expression (16) in the
slow-climb limit.
Now, in order to find the quadratic action for $\delta\varphi$, we
need to insert relations (28) and (III) in the
action (III) and expand it up to second order. However,
as we can see these expressions for $N$ and $N^{i}$ are subleading
in the slow-climb limit ($\dot{\varphi}^{2}\ll H^{2}$) and large-field
inflation ($\varphi^{2}\gg 1$) , comparing to $\delta\varphi$ (on
the other hand, if the momentum of the inflaton was comparable
with its energy density, namely $|\dot{\varphi}|\sim H$, the quantum
fluctuation of the background would become significant and could
cause instabilities on the background). Therefore, it is adequate
to consider just the action (1) for $\delta\phi$ in the
de Sitter background, resulting in the second-order action
Feng:2009kb
$$S_{2}=\frac{1}{2}\int d^{4}xa^{3}\bigg{[}-\delta\dot{\varphi}^{2}+(\nabla{%
\delta\varphi})^{2}-V^{\prime\prime}\delta\varphi^{2}-12\xi H^{2}\delta\varphi%
^{2}\bigg{]}.$$
(31)
Moreover, introducing the Fourier transform of $\delta\varphi$
through $\delta\varphi_{k}$, the perturbation equation writes
$$\delta\ddot{\varphi}_{k}+3H\delta\dot{\varphi}_{k}+\frac{k^{2}}{a^{2}}\delta%
\varphi_{k}=0\,,$$
(32)
where we have used $\eta\ll 1$ and $\Delta\ll 1$. Therefore, as we
observe, the quantum fluctuations in a Hubble time have the same
value as in the canonical case Guth:2007ng ; Feng:2009kb
$$\delta_{q}\varphi\approx\frac{H}{2\pi}.$$
(33)
Expression (33) provides the quantum fluctuations of
the inflaton in one Hubble time. On the other hand, it is known
that usually the classical motion of the inflaton during one
Hubble time is given by Lindebook ; LidseyRMP
$$|\delta_{c}\varphi|\approx|\dot{\varphi}H^{-1}|\sim\frac{|V^{\prime}|}{3H^{2}}%
\bigg{(}1+\Delta\varphi^{2}\bigg{)}.$$
(34)
Thus, we deduce that if the quantum fluctuations are larger than
the classical ones, namely $\delta_{q}\varphi>|\delta_{c}\varphi|$,
then inflation will be eternal. Therefore, the necessary
conditions for exiting phantom inflation is to use the suitably
defined and bounded-from-above potentials of the phantom-inflation
literature
Piao:2004tq ; Lidsey:2004xd ; GonzalezDiaz:2004df ; Nojiri:2005pu ; Capozziello:2005tf ; Wu:2006wu ; Piao:2007ne ; Elizalde:2008yf ,
plus the condition $\delta_{q}\varphi<|\delta_{c}\varphi|$.
Thus, since slow-climb always requires $\Delta\varphi^{2}\ll 1$ and
the validity of (16), the condition that prevent
eternality reads
$$\left|\frac{dV(\varphi)}{d\varphi}\right|\gtrsim\left|V(\varphi)\right|^{3/2}%
\left(1+\frac{3\xi}{2}\varphi^{2}\right).$$
(35)
This condition restricts the potential-forms that can give rise to
a finite inflation, or inversely, for a given potential it
determines the bounds inside which the field can move, in order to
avoid eternality. Finally, in the limit $\xi\rightarrow 0$ the
above relation provides the corresponding condition for the
minimal phantom inflation.
IV Langevin analysis for the nonminimal slow-climb phantom scenario
In the previous section we extracted the general condition that prevents eternality in phantom inflation, estimating
separately the effects of the classical motion and of the quantum fluctuations. In this section we will try to verify
the aforementioned results, solving explicitly the cosmological equations, formulating the classical motion plus the
stochastic effect of the quantum fluctuations through a Langevin analysis Chen:2006hs . In order to be able to
provide analytical results we will use the toy example of the exponential potential $V(\varphi)=V_{0}e^{\lambda\varphi}$, with $\lambda>0$, which satisfies the basic requirements for phantom inflation.
The overall evolution of the phantom field, including quantum fluctuations, is modeled through a random walk, and
therefore it can be described by the following Langevin equation Chen:2006hs ,
$$3H\dot{\varphi}-V^{\prime}(\varphi)-12\xi H^{2}\varphi=\frac{3}{2\pi}H^{5/2}n(%
t)\,,$$
(36)
where $n(t)$ is a Gaussian white noise normalized as
$$\langle n(t)\rangle=0\,,\qquad\langle n(t)n(t^{\prime})\rangle=\delta(t-t^{%
\prime})\,.$$
(37)
As can be seen, $n(t)$ has dimensions of mass to the power of one half. Using the exponential potential, and taking the
approximation $3H^{2}\approx V_{0}$ during inflation, which means we do not consider the backreaction from the space-time
to the classical evolution of the inflaton and focus on the quantum fluctuation of the inflaton itself which is modeled
by the stochastic process, then we get
$$\dot{\varphi}-\bigg{(}\lambda e^{\lambda\varphi}+4\xi\varphi\bigg{)}\sqrt{%
\frac{V_{0}}{3}}=qn(t),$$
(38)
where we have defined $q\equiv\frac{H^{3/2}}{2\pi}$.
In Eq. (38), if the term on the right-hand side
is absent we recover the usual slow-climb equation of motion and
the inflaton will follow a classical trajectory $\varphi_{c}(t)$.
Therefore, we expand the field $\varphi(t)$ around its classical
value $\varphi_{c}(t)$ up to order $\mathcal{O}(q^{2})$, namely
$$\varphi(t)=\varphi_{c}(t)+q\varphi_{1}(t)+q^{2}\varphi_{2}(t)+\mathcal{O}(q^{3%
})\,.$$
(39)
Substituting this expansion into (38) and setting
the coefficients of the $q$-powers to zero, we acquire the
equations
$$\displaystyle\dot{\varphi}_{c}$$
$$\displaystyle=$$
$$\displaystyle\sqrt{V_{0}/3}\big{(}\lambda e^{\lambda\varphi_{c}}+4\xi\varphi_{%
c}\big{)}$$
(40)
$$\displaystyle\dot{\varphi}_{1}$$
$$\displaystyle=$$
$$\displaystyle\sqrt{V_{0}/3}\big{(}\lambda^{2}e^{\lambda\varphi_{c}}+4\xi\big{)%
}\varphi_{1}+n(t)$$
(41)
$$\displaystyle\dot{\varphi}_{2}$$
$$\displaystyle=$$
$$\displaystyle\sqrt{V_{0}/3}\bigg{[}\frac{1}{2}\lambda^{3}e^{\lambda\varphi_{c}%
}\varphi_{1}^{2}+\big{(}\lambda^{2}e^{\lambda\varphi_{c}}+4\xi\big{)}\varphi_{%
2}\bigg{]}.$$
(42)
These three equations can be solved analytically in Case I and
Case II of (11),(12), namely for $|f\varphi|\ll|V^{\prime}|$ and $|f\varphi|\gg|V^{\prime}|$ respectively. The
explicit solutions are presented in the Appendix.
For case I, the condition for the Hubble parameter not to be
changed significantly by the quantum noise (see Appendix
A.1) reads
$$\varphi_{0}\lesssim\lambda^{-1}\ln V_{0}^{-1}\,,$$
(43)
while for Case II the corresponding condition (see Appendix
A.2) reads
$$\varphi_{0}\lesssim\lambda^{-1}\ln V_{0}^{-1}+\lambda^{-1}\ln(\sqrt{\xi}/%
\lambda).$$
(44)
In other words, if these conditions are satisfied, that is if the
inflaton remains smaller than these critical values, then
inflation will not be eternal.
Let us now compare these expressions with the condition
(35) derived in the previous section. Applying
(35) in the case of the exponential potential of the
present section, and keeping up to zeroth order in terms of $\xi$
(since otherwise we obtain transcendental equations), we acquire
$$\varphi_{0}\lesssim\lambda^{-1}\ln V_{0}^{-1}+2\lambda^{-1}\ln\lambda.$$
(45)
Clearly, this expression is consistent with both (43) and (44), and the slight differences
arise from the performed assumptions that were necessary in order
to solve the Langevin equation. Additionally, going to first order
in $\xi$ in (35), one can numerically show the
agreement too. Therefore, we conclude that the results of the
previous sections are indeed reliable.
V Conclusions
In this work we investigated the necessary conditions that prevent
phantom inflation to be eternal, going beyond the basic conditions
of slow-climb behavior. In particular, even using potentials
bounded from above and with suitable slopes, which give rise to
slow climbing, quantum fluctuations could still lead inflation to
be eternal. Thus, after presenting the slow-climb conditions, we
performed an analysis of the fluctuations, extracting the overall
conditions that are necessary for preventing eternality. Finally,
in order to be general, we moreover allowed for a nonminimal
coupling of the phantom field with gravity.
Our main result is expression (35), which is the
condition restricting the potential-forms that can give rise to a
finite inflation, or inversely the condition determining the
bounds inside which the field can move in a given, slow-climb
potential, in order to avoid eternality. Note that in our analysis
we did not need any additional mechanism in order to exit eternal
phantom inflation, such as the use of an extra scalar
Piao:2004tq , the imposition of strong backreaction
Wu:2006wu , the consideration of multiuniverses
GonzalezDiaz:2004df , or the use of specially-designed
braneworld models with brane/flux annihilation
Piao:2007ne .
Furthermore, in order to verify the obtained results, we solved
explicitly the cosmological system in a simple example of an
exponential potential, formulating the classical motion plus the
stochastic effect of the quantum fluctuations through Langevin
equations. Requiring finite parameters in the inflation we resulted to similar
conditions with those obtained by the above fluctuation-analysis
procedure.
Let us make a comment here, on the limits of applicability of our
analysis. First of all, as we have mentioned, the phantom field
must be smaller than the Planck scale, thus its backreaction will
be small and not capable of bringing inflation to eternality (in
Langevin-equation terms, this means that the expansion around the
classical trajectory (39) is valid). However, in an
inflating universe, even if the examined region satisfies these
conditions, its neighboring regions can have very high densities,
and thus one could ask whether this behavior could bring about
strong quantum effects in the examined region too. Therefore, we
have to make an additional assumption, namely that the initially
low-density, slow-roll-inflating region has been already causally
disconnected from its possible high-density neighboring regions,
and the possible interactions lie outside the horizon. In such a
case, the inflation of the observable universe will not be led to
eternality.
Phantom fields could have interesting implications either in
inflation or in describing the late-time acceleration of the
Universe. Although their quantum behavior could be problematic at
first, one can consider the phantoms to arise through an effective
description of a nonphantom, fundamental, higher-dimensional,
underlying theory, consistently with the basic requirements of
quantum field theory. Therefore, the examination of their
cosmological implications is valuable and can improve our
understanding of nature. In these lines, the fact that phantom
inflation can be noneternal makes the scenario at hand a
candidate for the description of the early universe.
Acknowledgements.
This work is supported by National Education Foundation of China grant No. 2009312711004 and Shanghai Natural Science
Foundation, China grant No. 10ZR1422000.
Appendix A Solution of the Langevin equations
Since we are dealing with stochastic variables, we perform the
average of any physical quantity by defining the statistical
measure. In particular, we use the Fokker-Planck approach and
define the measure to be the physical volume of the Hubble patch,
and thus the average is defined as
$$\langle H(t)\rangle_{p}=\frac{\langle H(t)e^{3N(t)}\rangle}{\langle e^{3N(t)}%
\rangle}\,,\quad N(t)=\int_{0}^{t}H(t^{\prime})dt^{\prime}\,.$$
(46)
Since the Hubble patch that is eternally inflating will have an exponentially larger physical volume, taking the
largest weight in the average at late times, the physical volume can be a good measure to characterize eternal
inflation. Therefore, the average $\langle H(t)\rangle_{p}$ could be significantly changed by quantum fluctuations if
eternal inflation is realized. Furthermore, we shall use the functional technique developed in Chen:2006hs and
define a generating functional
$$W_{t}[\mu]=\ln\langle e^{M_{t}[\mu]}\rangle\,,\quad M_{t}[\mu]=\int_{0}^{t}\mu%
(t^{\prime})H(t^{\prime})dt^{\prime}\,.$$
(47)
Thus, $\langle H(t)\rangle_{p}$ can be evaluated by functionally
differentiating $W_{t}[\mu]$ with respect to $\mu$ and setting
$\mu=3$, resulting to the following equations up to
$\mathcal{O}(q^{2})$:
$$\displaystyle\langle H(t)\rangle_{p}$$
$$\displaystyle=$$
$$\displaystyle\frac{\delta W_{t}[\mu]}{\delta\mu(t)}\bigg{|}_{\mu(t)=3}$$
(48)
$$\displaystyle=$$
$$\displaystyle\langle H(t)\rangle+3\int_{0}^{t}\langle\langle H(t)H(t^{\prime})%
\rangle\rangle dt^{\prime},$$
$$\displaystyle\langle\langle H(t)H(t^{\prime})\rangle\rangle$$
$$\displaystyle=$$
$$\displaystyle\langle H(t)H(t^{\prime})\rangle-\langle H(t)\rangle_{p}\langle H%
(t^{\prime})\rangle_{p}\,.$$
(49)
After these definitions we can proceed to the solution of the
Langevin equations.
A.1 Case I: $|f\varphi|\ll|V^{\prime}|$
In this case, the phantom field can be regarded as minimally
coupled to gravity and the solution to (40) writes:
$$\displaystyle\varphi(t)$$
$$\displaystyle=$$
$$\displaystyle\varphi_{c}(t)+q\,e^{\lambda\varphi_{c}(t)}\Xi(t)+q^{2}\,e^{%
\lambda\varphi_{c}(t)}\Pi(t^{\prime})$$
(50)
with
$$\displaystyle\varphi_{c}(t)$$
$$\displaystyle=$$
$$\displaystyle-\lambda^{-1}\ln\bigg{[}e^{-\lambda\varphi_{0}}-\lambda^{2}t\sqrt%
{V_{0}/3}\bigg{]}\,,$$
(51)
where the subscript $0$ denotes the initial value of the field (at
$t=0$). In (50) we have defined the quantities
$$\displaystyle\Xi(t)$$
$$\displaystyle=$$
$$\displaystyle\int_{0}^{t}n(t^{\prime})e^{-\lambda\varphi_{c}(t^{\prime})}dt^{\prime}$$
(52)
$$\displaystyle=$$
$$\displaystyle\int_{0}^{t}n(t^{\prime})\bigg{[}e^{-\lambda\varphi_{0}}-\lambda^%
{2}t^{\prime}\sqrt{V_{0}/3}\bigg{]}dt^{\prime}$$
and
$$\displaystyle\Pi(t)$$
$$\displaystyle=$$
$$\displaystyle\frac{\lambda^{3}}{2}\sqrt{\frac{V_{0}}{3}}\int_{0}^{t}e^{2%
\lambda\varphi_{c}(t^{\prime})}\Xi^{2}(t^{\prime})dt^{\prime},$$
(53)
where $\Xi(t)$ is a new stochastic variable normalized as
$$\displaystyle\langle\Xi(t)\rangle=0,$$
(54)
$$\displaystyle\langle\Xi(t)\Xi(t^{\prime})\rangle$$
$$\displaystyle=\frac{e^{-3\lambda\varphi_{0}}}{\lambda^{2}\sqrt{3V_{0}}}\bigg{%
\{}1-\left[1-\lambda^{2}e^{\lambda\varphi_{0}}\sqrt{V_{0}/3}\text{min}(t,t^{%
\prime})\right]^{3}\bigg{\}}.$$
(55)
The Hubble parameter reads
$$\displaystyle H(t)=H_{c}(t)+q\,\sqrt{\frac{V_{0}}{3}}\frac{\lambda}{2}e^{3%
\lambda\varphi_{c}(t)/2}\Xi(t)$$
$$\displaystyle+q^{2}\,\sqrt{\frac{V_{0}}{3}}\frac{\lambda}{8}e^{3\lambda\varphi%
_{c}(t)/2}\left[\lambda e^{\lambda\varphi_{c}(t)}\Xi^{2}(t)+4\Pi(t)\right],$$
(56)
where $H_{c}(t)=\sqrt{V(\varphi_{c}(t))/3}=e^{\lambda\varphi_{c}/2}\sqrt{V_{0}/3}$.
Using (54), (55) and (A.1), we can
further obtain
$$\langle H(t)\rangle=\langle H_{c}(t)\rangle+q^{2}\,\frac{e^{-2\lambda\varphi_{%
0}}}{8}e^{3\lambda\varphi_{c}(t)/2}\bigg{(}e^{\lambda[\varphi_{c}(t)-\varphi_{%
0}]}-1\bigg{)}$$
(57)
and
$$\displaystyle 3\int_{0}^{t}\langle\langle H(t)H(t^{\prime})\rangle\rangle=%
\frac{q^{2}e^{-\lambda\varphi_{c}(t)}}{10\lambda^{2}}\left\{5e^{3\lambda[%
\varphi_{c}(t)-\varphi_{0}]}\right.$$
$$\displaystyle\left.+1-6e^{5\lambda[\varphi_{c}(t)-\varphi_{0}]/2}\right\}.$$
(58)
Now, in the limit $t\ll t_{0}\equiv\lambda^{-2}e^{-\lambda\varphi_{0}}\sqrt{3/V_{0}}$ we can
acquire the leading order behavior of $\langle H(t)\rangle_{p}$ in
terms of $t$ as
$$\displaystyle\langle H(t)\rangle_{p}=\langle H(t=0)\rangle_{p}$$
$$\displaystyle+\frac{e^{\lambda\varphi_{0}/2}}{2}\sqrt{\frac{V_{0}}{3}}\left(%
\frac{t}{t_{0}}\right)+\frac{q^{2}e^{-\lambda\varphi_{0}/2}}{8}\left(\frac{t}{%
t_{0}}\right),$$
(59)
where the second term arises from expanding the classical motion
$H_{c}(t)$, while the last term comes from the quantum correction
(57). Note that the contribution from
(58) is of the order of $(t/t_{0})^{2}$.
Requiring the Hubble parameter not to be changed significantly by
the quantum noise, we need to impose
$$\frac{e^{\lambda\varphi_{0}/2}}{2}\sqrt{\frac{V_{0}}{3}}\lesssim\frac{q^{2}e^{%
-\lambda\varphi_{0}/2}}{8},$$
(60)
which provides the bound when $(t/t_{0})\ll 1$ as:
$$\varphi_{0}\lesssim\lambda^{-1}\ln V_{0}^{-1}.$$
(61)
A.2 Case II: $|f\varphi|\gg|V^{\prime}|$
In this case, the solution to (40) reads
$$\displaystyle\varphi(t)$$
$$\displaystyle=$$
$$\displaystyle\varphi_{c}(t)+q\,\varphi_{c}(t)\Xi(t)+q^{2}\,\varphi_{c}(t)\frac%
{\varphi_{20}}{\varphi_{0}}$$
(62)
with
$$\displaystyle\varphi_{c}(t)$$
$$\displaystyle=$$
$$\displaystyle\varphi_{0}\exp\bigg{(}4\xi t\sqrt{V_{0}/3}\bigg{)},$$
(63)
where $\varphi_{20}\equiv\varphi_{2}(t=0)$ and similarly to the
previous subsection we have defined
$$\displaystyle\Xi(t)$$
$$\displaystyle=$$
$$\displaystyle\int_{0}^{t}n(t^{\prime})\varphi_{c}^{-1}(t^{\prime})dt^{\prime}$$
(64)
$$\displaystyle=$$
$$\displaystyle\varphi_{0}^{-1}\int_{0}^{t}n(t^{\prime})\exp\bigg{(}-4\xi t\sqrt%
{V_{0}/3}\bigg{)}dt^{\prime},$$
normalized as
$$\displaystyle\langle\Xi(t)\rangle=0,$$
(65)
$$\displaystyle\langle\Xi(t)\Xi(t^{\prime})\rangle$$
$$\displaystyle=\frac{\varphi_{0}^{-2}}{8\xi\sqrt{V_{0}/3}}\left\{1-\exp\left[-8%
\xi\sqrt{V_{0}/3}\text{min}(t,t^{\prime})\right]\right\}.$$
(66)
The Hubble parameter is
$$\displaystyle H(t)=H_{c}(t)+q\,\sqrt{\frac{V_{0}}{3}}\frac{\lambda}{2}e^{%
\lambda\varphi_{c}(t)/2}\varphi_{c}(t)\Xi(t)$$
$$\displaystyle+q^{2}\,\sqrt{\frac{V_{0}}{3}}\frac{\lambda}{8}e^{3\lambda\varphi%
_{c}(t)/2}\left[\lambda\varphi_{c}^{2}(t)\Xi^{2}(t)+4\varphi_{c}(t)\frac{%
\varphi_{20}}{\varphi_{0}}\right],$$
(67)
where $H_{c}(t)=e^{\lambda\varphi_{c}/2}\sqrt{V_{0}/3}$. Moreover we
obtain
$$\displaystyle\langle H(t)\rangle=\langle H_{c}(t)\rangle+q^{2}\,\sqrt{\frac{V_%
{0}}{3}}\frac{\lambda}{8}e^{3\lambda\varphi_{c}(t)/2}\left\{4\varphi_{c}(t)%
\frac{\varphi_{20}}{\varphi_{0}}\right.$$
$$\displaystyle\left.+\frac{\lambda}{8\xi\sqrt{V_{0}/3}}\left[\left(\frac{%
\varphi_{c}(t)}{\varphi_{0}}\right)^{2}-1\right]\right\}$$
(68)
and
$$\displaystyle 3\int_{0}^{t}\langle\langle H(t)H(t^{\prime})\rangle\rangle=$$
$$\displaystyle\ \ \ \ \ \ =\frac{3q^{2}\lambda^{2}e^{\frac{\lambda\varphi_{c}}{%
2}}}{128\xi^{2}}\left(\frac{\varphi_{c}}{\varphi_{0}}\right)\left\{\frac{2}{%
\lambda\varphi_{0}}\left(e^{\frac{\lambda\varphi_{c}}{2}}-e^{\frac{\lambda%
\varphi_{0}}{2}}\right)\right.$$
$$\displaystyle\ \ \ \ \ \ +\left(\frac{\varphi_{0}}{\varphi_{c}}e^{\frac{%
\lambda\varphi_{c}}{2}}-e^{\frac{\lambda\varphi_{0}}{2}}\right)$$
$$\displaystyle\ \ \ \ \ \ \left.+\frac{\lambda\varphi_{0}}{2}\left[\text{Ei}%
\left(-\frac{\lambda\varphi_{c}}{2}\right)-\text{Ei}\left(-\frac{\lambda%
\varphi_{0}}{2}\right)\right]\right\},$$
(69)
where Ei is the exponential integral function.
In the limit $t\ll t_{0}\equiv\sqrt{3/V_{0}}/(4\xi)$ we can obtain
the leading order behavior of $\langle H(t)\rangle_{p}$ in terms of
$t$ as
$$\displaystyle\langle H(t)\rangle_{p}=\langle H(t=0)\rangle_{p}+\frac{\lambda%
\varphi_{0}e^{\lambda\varphi_{0}/2}}{2}\sqrt{\frac{V_{0}}{3}}\left(\frac{t}{t_%
{0}}\right)$$
$$\displaystyle+q^{2}\sqrt{\frac{V_{0}}{3}}\frac{\lambda}{8}e^{3\lambda\varphi_{%
0}/2}\bigg{[}\frac{\lambda}{4\xi\sqrt{V_{0}/3}}+4\varphi_{20}+6\lambda\varphi_%
{0}\varphi_{20}\bigg{]}\left(\frac{t}{t_{0}}\right),$$
where the second term arises from expanding the classical motion
$H_{c}(t)$ and the last term comes from the quantum correction
(68). Note that the contribution from
(69) is of the order of $(t/t_{0})^{2}$.
Requiring the Hubble parameter not to be changed significantly by
the quantum noise, we impose
$$4\varphi_{0}\lesssim\frac{q^{2}\,\lambda e^{\lambda\varphi_{0}}}{4\xi\sqrt{V_{%
0}/3}}\,,$$
(70)
where we have used that $\xi\ll 1$ and
$\varphi_{0}\gg\Delta^{-1/2}\sim\xi^{-1/2}$. Thus, we conclude that
at $(t/t_{0})\ll 1$:
$$\varphi_{0}\lesssim\lambda^{-1}\ln V_{0}^{-1}+\lambda^{-1}\ln(\sqrt{\xi}/%
\lambda).$$
(71)
References
(1)
A. H. Guth,
Phys. Rev. D 23, 347 (1981);
A. D. Linde,
Phys. Lett. B 108, 389 (1982);
A. J. Albrecht and P. J. Steinhardt,
Phys. Rev. Lett. 48, 1220 (1982).
(2)
A. D. Linde,
Particle Physics and Inflationary Cosmology,
CRC Press, Boca Raton, USA (1990).
(3)
D. Baumann and L. McAllister,
Ann. Rev. Nucl. Part. Sci. 59, 67 (2009)
[arXiv:0901.0265 [hep-th]].
(4)
J. E. Lidsey et. al., Rev. Mod. Phys. 69, 373 (1997).
(5)
V. Sahni and S. Habib,
Phys. Rev. Lett. 81, 1766 (1998);
J. P. Uzan, Phys. Rev. D 59, 123510 (1999);
F. Perrotta, C. Baccigalupi and S. Matarrese,
Phys. Rev. D 61, 023507 (2000);
V. Faraoni, Phys.
Rev. D 68, 063508 (2003); E. Elizalde, S. Nojiri and S.
Odintsov, Phys. Rev. D 70, 043539 (2004);
S. Nojiri, S. D. Odintsov
and M. Sami, Phys. Rev. D 74, 046004 (2006);
M. R. Setare and E. N. Saridakis,
JCAP 0903, 002 (2009);
E. N. Saridakis and S. V. Sushkov,
Phys. Rev. D 81, 083510 (2010).
(6)
V. Faraoni,
Phys. Rev. D 62, 023504 (2000);
M. Li,
JCAP 0610, 003 (2006)
[arXiv:astro-ph/0607525];
B. Chen, M. Li, T. Wang and Y. Wang,
Mod. Phys. Lett. A 22, 1987 (2007)
[arXiv:astro-ph/0610514].
(7)
Y. S. Piao, Q. G. Huang, X. m. Zhang and Y. Z. Zhang,
Phys. Lett. B 570, 1 (2003)
[arXiv:hep-ph/0212219];
X. Z. Li and X. H. Zhai,
Phys. Rev. D 67, 067501 (2003)
[arXiv:hep-ph/0301063];
D. J. Liu and X. Z. Li,
Phys. Rev. D 70, 123504 (2004)
[arXiv:astro-ph/0402063];
K. Nozari and B. Fazlpour,
JCAP 0711, 006 (2007)
[arXiv:0708.1916 [hep-th]];
S. C. Park and S. Yamaguchi,
JCAP 0808, 009 (2008)
[arXiv:0801.1722 [hep-ph]];
F. Bauer and D. A. Demir,
Phys. Lett. B 665, 222 (2008)
[arXiv:0803.2664 [hep-ph]];
G. Gupta, E. N. Saridakis and A. A. Sen,
Phys. Rev. D 79, 123013 (2009).
(8)
Steinhardt, P J 1983 ”Natural inflation” in The Very Early
Universe, Proceedings of the Nuffield Workshop, Cambridge, 21
June – 9 July, 1982, eds: Gibbons, G W, Hawking, S W and Siklos,
S T C (Cambridge: Cambridge University Press), pp. 251–66.
(9)
A. Vilenkin,
Phys. Rev. D 27, 2848 (1983).
(10)
A. D. Linde,
Phys. Lett. B 175, 395 (1986).
(11)
Y. F. Cai and Y. Wang,
JCAP 0706, 022 (2007)
[arXiv:0706.0572 [hep-th]];
M. Li and Y. Wang,
JCAP 0708, 007 (2007)
[arXiv:0706.1691 [hep-th]];
Q. G. Huang, M. Li and Y. Wang,
JCAP 0709, 013 (2007)
[arXiv:0707.3471 [hep-th]];
Y. F. Cai and Y. Wang,
JCAP 0801, 001 (2008)
[arXiv:0711.4423 [gr-qc]];
S. Winitzki,
Phys. Rev. D 78, 043501 (2008)
[arXiv:0803.1300 [gr-qc]];
D. I. Podolsky,
Grav. Cosmol. 15, 69 (2009)
[arXiv:0809.2453 [gr-qc]].
(12)
A. D. Linde,
Lect. Notes Phys. 738, 1 (2008)
[arXiv:0705.0164 [hep-th]].
(13)
A. H. Guth,
J. Phys. A 40, 6811 (2007)
[arXiv:hep-th/0702178].
(14)
A. H. Guth,
Phys. Rept. 333, 555 (2000)
[arXiv:astro-ph/0002156].
(15)
J. E. Lidsey,
Phys. Rev. D 70, 041302 (2004)
[arXiv:gr-qc/0405055].
(16)
S. Nojiri and S. D. Odintsov,
Gen. Rel. Grav. 38, 1285 (2006)
[arXiv:hep-th/0506212].
(17)
S. Capozziello, S. Nojiri and S. D. Odintsov,
Phys. Lett. B 632, 597 (2006)
[arXiv:hep-th/0507182].
(18)
P. F. Gonzalez-Diaz and J. A. Jimenez-Madrid,
Phys. Lett. B 596, 16 (2004)
[arXiv:hep-th/0406261].
(19)
Y. S. Piao,
Phys. Rev. D 78, 023518 (2008)
[arXiv:0712.3328 [gr-qc]].
(20)
E. Elizalde, S. Nojiri, S. D. Odintsov, D. Saez-Gomez and V. Faraoni,
Phys. Rev. D 77, 106005 (2008)
[arXiv:0803.1311 [hep-th]].
(21)
Y. S. Piao and Y. Z. Zhang,
Phys. Rev. D 70, 063513 (2004)
[arXiv:astro-ph/0401231].
(22)
P. Wu and H. W. Yu,
JCAP 0605, 008 (2006)
[arXiv:gr-qc/0604117].
(23)
R. R. Caldwell, Phys. Lett. B 545, 23 (2002); R. R. Caldwell, M. Kamionkowski and N. N. Weinberg, Phys. Rev.
Lett. 91, 071301 (2003);
M. Baldi, F. Finelli and S. Matarrese,
Phys. Rev. D 72, 083504 (2005)
[arXiv:astro-ph/0505552].
(24)
J. G. Hao and X. Z. Li,
Phys. Rev. D 67, 107303 (2003)
[arXiv:gr-qc/0302100];
J. G. Hao and X. Z. Li,
Phys. Rev. D 68, 043501 (2003);
X. Z. Li and J. G. Hao,
Phys. Rev. D 69, 107303 (2004);
P. F. Gonzalez-Diaz and C. L. Siguenza,
Nucl. Phys. B 697, 363 (2004);
J. G. Hao and X. Z. Li,
Phys. Rev. D 70, 043529 (2004);
S. Nojiri and S. D. Odintsov, Phys. Rev. D 72, 023003
(2005); H. Garcia-Compean, G. Garcia-Jimenez, O. Obregon, and C.
Ramirez, JCAP 0807, 016 (2008);
E. N. Saridakis, P. F. Gonzalez-Diaz and C. L. Siguenza,
Class. Quant. Grav. 26, 165003 (2009);
X. M. Chen, Y. G. Gong and E. N. Saridakis,
JCAP 0904, 001 (2009);
S. Dutta and R. J. Scherrer,
Phys. Lett. B 676, 12 (2009);
E. N. Saridakis,
Nucl. Phys. B 819, 116 (2009).
(25)
S.M. Carroll, M. Hoffman and M. Trodden, Phys. Rev. D 68
023509 (2003).
(26)
J. M. Cline, S. Jeon and G. D. Moore,
Phys. Rev. D 70, 043543 (2004)
[arXiv:hep-ph/0311312].
(27)
S. Nojiri and S. D. Odintsov,
Phys. Lett. B 562, 147 (2003);
S. Nojiri and S. D. Odintsov,
Phys. Lett. B 571, 1 (2003).
(28)
H.P. Nilles, Phys. Rept. 110 1 (1984).
(29)
B. Boisscau, G. Esposito-Farese, d.
Polarski and A.A. Starobinsky, Phys. Rev. Lett. 85
2236(2000).
(30)
M.D. Pollock, Phys. Lett. B 215, 635 (1988).
(31)
V. Sahni and Y. Shtanov,
JCAP 0311, 014 (2003)
[arXiv:astro-ph/0202346];
D. J. Liu and X. Z. Li,
Phys. Rev. D 68, 067301 (2003)
[arXiv:hep-th/0307239].
(32)
C. Armendariz-Picon, T. Damour and V. F. Mukhanov,
Phys. Lett. B 458, 209 (1999)
[arXiv:hep-th/9904075];
T. Chiba, T. Okabe and M. Yamaguchi,
Phys. Rev. D 62, 023511 (2000)
[arXiv:astro-ph/9912463];
P. F. Gonzalez-Diaz,
Phys. Lett. B 586, 1 (2004)
[arXiv:astro-ph/0312579].
(33)
P. H. Frampton,
Phys. Lett. B 555, 139 (2003)
[arXiv:astro-ph/0209037].
(34)
L. P. Chimento and R. Lazkoz,
Phys. Rev. Lett. 91, 211301 (2003)
[arXiv:gr-qc/0307111];
J. G. Hao and X. Z. Li,
Phys. Lett. B 606, 7 (2005)
[arXiv:astro-ph/0404154].
(35)
H. Stefancic,
Eur. Phys. J. C 36, 523 (2004)
[arXiv:astro-ph/0312484].
(36)
R. Kallosh, J. Kratochvil, A. Linde, E. Linder and M. Shmakova,
JCAP 0310, 015 (2003);
P. F. Gonzalez-Diaz, Phys. Rev. D
68, 021303 (2003);
S. Nojiri, S. D. Odintsov and S. Tsujikawa,
Phys. Rev. D 71, 063004 (2005).
(37)
M. Sami and A. Toporensky,
Mod. Phys. Lett. A 19, 1509 (2004)
[arXiv:gr-qc/0312009];
P. Singh, M. Sami and N. Dadhich, Phys. Rev. D 68, 023522
(2003).
(38)
C. J. Feng and X. Z. Li,
arXiv:0911.3994 [astro-ph.CO].
(39)
X. Chen, S. Sarangi, S. H. Henry Tye and J. Xu,
JCAP 0611, 015 (2006)
[arXiv:hep-th/0608082];
S. Gratton and N. Turok,
Phys. Rev. D 72, 043507 (2005)
[arXiv:hep-th/0503063];
A. D. Linde, D. A. Linde and A. Mezhlumian,
Phys. Rev. D 49, 1783 (1994)
[arXiv:gr-qc/9306035]. |
3D-2D Stokes-Darcy coupling for the modelling of seepage with
an application to fluid-structure interaction with contact
Erik Burman
Department of Mathematics, University College London, London, UK–WC1E 6BT, United Kingdom
Miguel A. Fernández
Inria, 75012 Paris,
Sorbonne Université & CNRS, UMR 7598 LJLL, 75005 Paris.
Stefan Frei
Department of Mathematics & Statistics, University of Konstanz, Germany.
Fannie M. Gerosa
Inria, 75012 Paris,
Sorbonne Université & CNRS, UMR 7598 LJLL, 75005 Paris.
()
Abstract
In this note we introduce a mixed dimensional Stokes-Darcy coupling where a
$d$ dimensional Stokes’ flow is coupled to a Darcy model on the
$d-1$ dimensional boundary of the domain. The porous layer introduces
tangential creeping flow along the boundary and allows for the modelling
of boundary flow due to surface roughness. This leads to a new
model of flow in fracture networks with reservoirs in an
impenetrable bulk matrix. Exploiting this modelling capability, we then formulate a
fluid-structure interaction method with contact, where the porous layer allows
for mechanically consistent contact and release. Physical seepage in the contact zone due
to rough surfaces is modelled by the porous layer. Some numerical
examples are reported, both on the Stokes’-Darcy coupling alone and on
the fluid-structure interaction with contact in the porous boundary layer.
1 Introduction
In numerous environmental or biomedical applications there is a need to model the coupling between
a flow in a reservoir and flow in a surrounding porous
medium. This is particularly challenging if the porous medium is
fractured and the bulk matrix has very low permeability.
Typically the fractures are modelled as $d-1$ dimensional
manifolds, embedded in a $d$ dimensional porous bulk matrix.
For the modelling of the fractured porous medium we refer to
[3]. Observe however that if the bulk
permeability is negligible the fluid in the reservoir can not penetrate
into the fractures since the $d-1$ dimensional manifolds have an
intersection of the reservoir boundary of $d-1$ measure zero. This
means that such a model can not be used for the fluid flow between two
reservoirs connected by a fracture in an impenetrable medium. Here we
propose to introduce a Darcy equation for the tangential flow on the
boundary of the reservoir. Since this equation is set on a $d-1$
dimensional manifold it can provide an interface allowing for flow
from the reservoir to the cracks. The flow on the boundary
communicates with the flow in the cracks through continuity of
pressure and conservation expressed by Kirchhoff’s law. This gives a cheap and flexible
model for flow in reservoirs connected by fractures.
Our original motivation for this model is the particular case of fluid
structure interaction with contact where the situation described above
occurs when two boundaries enter in contact provoking a change of
topology of the fluid domain. It has recently been observed by several
authors [1, 4] that the consistent modelling of fluid-structure
interaction with contact requires a fluid model, in particular a pressure, also in the contact
zone. Indeed, some seepage is expected to occur due to permeability of
the contacting bodies or their surface roughness. Otherwise there is no continuous mechanism for the release of
contact and non-physical voids can occur. For instance, it was argued
in [1] that a consistent modelling of FSI with contact requires a
complete modelling of the FSI-poroelastic coupling. Similar ideas were
introduced in [4], but for computational
reasons. Indeed, in the latter reference an elastic body immersed in
a fluid enters in contact with a rigid wall and to allow for a
consistent numerical modelling the permeability of the wall is
relaxed. This motivates the introduction of an artificial porous medium
whose permeability goes to zero with the mesh-size. Both approaches allow for the
seepage that appears to be necessary for physical contact and
release.
However, in case the contacting solids are (modelled as) impenetrable, this seepage
must be due to porous media flow in a thin layer in the contact zone due to surface roughness.
The complete modelling of the poroelastic interaction of [1] or the
bulk porous medium flow of [4] then appears
artificial and unnecessarily expensive. For such situations the mixed
dimensional modelling suggested above can offer an attractive
compromise between model detail and computational cost.
In this note, we will focus exclusively on the modelling aspect. The
coupled Stokes-Darcy model is introduced in section
2. Then, in section 3, we show how the ideas of
[4] can be used to model FSI with contact
together with the mixed-dimensional fluid system. Finally, we
illustrate the two model situations numerically in section
4. First, the Stokes’-Darcy reservoir coupling (section
4.1) and then the
full FSI with contact (section 4.2). In the latter
case, we also give comparisons with the results from
[4].
The numerical analysis of the resulting methods will be the subject of
future work.
2 The coupled Stokes-Darcy system
We consider the coupling of a Darcy system in a thin-walled domain $\Omega_{l}=\Sigma_{l}\times(-\frac{\epsilon}{2},\frac{\epsilon}{2})\in\mathbb{%
R}^{d}$ for $d=2,3$
with a Stokes equation in the bulk domain $\Omega_{f}$. The Darcy problem on $\Omega_{l}$ writes
$$\left\{\begin{aligned} \displaystyle u_{l}+K\nabla p_{l}=0&\\
\displaystyle\nabla\cdot u_{l}=0&\\
\end{aligned}\right.\quad\mbox{in}\quad\Omega_{l},$$
(1)
where $u_{l}$ denotes the Darcy velocity, $p_{l}$ the Darcy pressure and $K$ is a $d\times d$ matrix that allows for the decomposition
$$\displaystyle K\nabla p_{l}=K_{\tau}\nabla_{\tau}p_{l}+K_{n}\partial_{n}p_{l}.$$
We denote the upper boundary of $\Omega_{l}$ which couples to $\Omega_{f}$ by $\gamma_{f}$ and
the outer boundary by $\gamma_{o}$. The normal vector $n$ of the middle surface $\Sigma_{l}$ of $\Omega_{l}$ is chosen in such a way that it points towards $\gamma_{o}$.
By averaging across the thickness $\epsilon$, Martin, Jaffré and Roberts derived in [3] an effective equation
for the averaged pressure across the thickness
$$P_{l}:=\frac{1}{\epsilon}\int_{-\frac{\epsilon}{2}}^{\frac{\epsilon}{2}}p_{l}.$$
Under the modelling assumption that the average pressure is equal to the mean of the pressures on the upper and lower boundary
$$P_{l}=\frac{1}{2}\left(p_{l}|{\gamma_{f}}+p_{l}|{\gamma_{o}}\right)\quad\mbox{%
in}\quad\Sigma_{l},$$
(2)
the authors derived the system
$$\left\{\begin{aligned} \displaystyle-\nabla_{\tau}\cdot\left(\epsilon K_{\tau}%
\nabla_{\tau}P_{l}\right)&\displaystyle=u_{l,n}|{\gamma_{f}}-u_{l,n}|_{\gamma_%
{o}}\\
\displaystyle p_{l}|{\gamma_{f}}&\displaystyle=P_{l}+\frac{\epsilon K_{n}^{-1}%
}{4}\left(u_{l,n}|{\gamma_{o}}+u_{l,n}|{\gamma_{f}}\right)\end{aligned}\right.%
\quad\mbox{in}\quad\Sigma_{l}.$$
(3)
Here, $u_{l,n}=u_{l}\cdot n$ denotes the normal component of the velocity and $\tau$ is a tangential vector of $\Sigma_{l}$.
We will couple (3) to Stokes flow in $\Omega_{f}$
$$\left\{\begin{aligned} \displaystyle\rho_{f}\partial_{t}u_{f}-\nabla\cdot%
\sigma_{f}(u_{f},p_{f})=0&\\
\displaystyle\nabla\cdot u_{f}=0&\end{aligned}\right.\quad\mbox{in}\quad\Omega%
_{f},$$
(4)
where $u_{f}$ denotes the fluid velocity, $p_{f}$ the pressure, $\rho^{\rm f}$ the fluid density,
$$\sigma_{f}(u_{f},p_{f}):=\mu(\nabla u_{f}+\nabla u_{f}^{T})-p_{f}I,$$
the fluid Cauchy stress tensor and $\mu$ the dynamic viscosity.
We
assume that the coupling to the Darcy system (1) on $\gamma_{f}$ takes place
via the interface conditions
$$\left\{\begin{aligned} \displaystyle\sigma_{f,nn}=-p_{l}&\\
\displaystyle\tau^{T}\sigma_{f}n=0&\\
\displaystyle u_{f,n}=u_{l,n}&\\
\end{aligned}\right.\quad\mbox{on}\quad\gamma_{f},$$
(5)
where $\sigma_{f}=\nabla u_{f}-p_{f}I$ and $\sigma_{f,nn}=n^{T}\sigma_{f}n$.
In the lower porous wall $\gamma_{o}$ we assume for simplicity that $u_{l,n}=0$.
Then, the relations (3) can be written as
$$\left\{\begin{aligned} \displaystyle-\nabla_{\tau}\cdot\left(\epsilon K_{\tau}%
\nabla_{\tau}P_{l}\right)=u_{f,n}&\\
\displaystyle\sigma_{f,nn}=-P_{l}-\frac{\epsilon K_{n}^{-1}}{4}u_{f,n}&\end{%
aligned}\right.\quad\mbox{in}\quad\Sigma_{l}.$$
Note that the only remaining porous medium variable is the averaged pressure $P_{l}$.
In the limit of permeability $K_{n}\to 0$, the system converges to a pure Stokes system with slip conditions on $\gamma_{f}$
with an extension of the fluid forces into the porous medium pressure $P_{l}$.
We have the following coupled variational problem for $(u_{f},p_{f},P_{l})$:
$$\left\{\begin{aligned} \displaystyle\rho_{f}(\partial_{t}u_{f},v_{f})_{\Omega_%
{f}}+(\sigma_{f}(u_{f},p_{f}),\nabla v_{f})_{\Omega_{f}}+(q_{f},\nabla\cdot u_%
{f})_{\Omega_{f}}&\\
\displaystyle+\big{(}P_{l},v_{f,n}\big{)}_{\Sigma_{l}}+\frac{\epsilon K_{n}^{-%
1}}{4}\big{(}u_{f,n},v_{f,n}\big{)}_{\Sigma_{l}}&\displaystyle=0,\\
\displaystyle(\epsilon K_{\tau}\nabla_{\tau}P_{l},\nabla_{\tau}q_{l})_{\Sigma_%
{l}}-(u_{f,n},q_{l}\big{)}_{\Sigma_{l}}&\displaystyle=0,\end{aligned}\right.$$
(6)
for all $v_{f},q_{f},q_{l}$, where $n=n_{f}$ is the outer normal of the fluid domain $\Omega_{f}$.
3 The fluid-structure-poroelastic-contact interaction system
Now, we consider a fluid-structure-contact interaction system with a thin porous layer on the part of the exterior boundary,
where contact might take place.
The moving boundary of the solid is denoted by $\Sigma(t)$ and the porous layer by $\Sigma_{l}$. In absence of contact, we have
the following system of equations
$$\displaystyle\left\{\begin{aligned} \displaystyle\rho_{f}\partial_{t}u_{f}-%
\nabla\cdot\sigma_{f}(u_{f},p_{f})=0&\\
\displaystyle\nabla\cdot u_{f}=0&\end{aligned}\right.\quad\mbox{in}\quad\Omega%
_{f}(t),$$
$$\displaystyle\rho_{s}\partial_{t}\dot{d}-\nabla\cdot\sigma_{s}(d)=0\quad\mbox{%
in}\quad\Omega_{s}(t),$$
$$\displaystyle u_{f}=\dot{d},\quad\sigma_{s}n=\sigma_{f}n\quad\mbox{in}\quad%
\Sigma(t),$$
$$\left\{\begin{aligned} \displaystyle-\nabla_{\tau}\cdot\left(\epsilon K_{\tau}%
\nabla_{\tau}P_{l}\right)=u_{l,n}|_{\gamma_{f}}\\
\displaystyle\sigma_{f,nn}=\underbrace{-P_{l}-\frac{\epsilon K_{n}^{-1}}{4}u_{%
l,n}|_{\gamma_{f}}}_{\sigma_{p}}\\
\displaystyle\tau^{T}\sigma_{f}n=0&\end{aligned}\right.\quad\mbox{in}\quad%
\Sigma_{l},$$
(7)
where, in addition to the quantities introduced above, $\rho_{s}$ denotes the solid density,
$d$ stands for the solid displacement and $\sigma_{s}$ denotes the tensor of linear elasticity
$$\displaystyle\sigma_{s}=\frac{\lambda_{s}}{2}\left(\nabla d+\nabla d^{T}\right%
)+\frac{\mu_{s}}{2}\text{tr}\left(\nabla d+\nabla d^{T}\right).$$
In addition, we impose that the solid $\Omega_{s}$ can not penetrate into the porous medium $\Sigma_{l}$
$$d_{n}-g\leq 0,\quad\lambda\leq 0,\quad\lambda(d_{n}-g)=0\quad\text{ on }\Sigma%
(t).$$
(8)
Here, $g$ denotes the gap function to $\Sigma_{l}$ and $\lambda$ is a Lagrange multiplier for the no-penetration condition defined by
$$\displaystyle\lambda$$
$$\displaystyle=\sigma_{s,nn}-\sigma_{f,nn}\qquad\text{ on }\Sigma(t)\setminus%
\Sigma_{l},$$
$$\displaystyle\lambda$$
$$\displaystyle=\sigma_{s,nn}-\sigma_{p}\qquad\quad\text{ on }\Sigma(t)\cap%
\Sigma_{l}.$$
The “switch” on the right-hand side occurs, as the solid on one side of $\Sigma(t)$ couples either to the fluid $\Omega_{f}$ or the porous
medium $\Sigma_{l}$ on the other side of $\Sigma(t)$. The conditions (8) can equivalently be written as
$$\displaystyle\lambda=\gamma_{C}\big{[}\underbrace{d_{n}-g-\gamma_{C}^{-1}%
\lambda}_{P_{\gamma}}\big{]}_{+}\quad\text{ on }\Sigma(t)$$
for arbitrary $\gamma_{C}>0$. Using this notation, we can characterise the zone of “active” contact as follows
$$\Sigma_{c}(t)=\left\{x\in\Sigma_{(}t)\,|\,P_{\gamma}>0\right\}.$$
To summarise, we have the following interface conditions:
•
Contact condition on $\Sigma(t)$:
$$d_{n}-g\leq 0,\quad\lambda\leq 0,\quad\lambda(d_{n}-g)=0\quad\text{on}\quad%
\Sigma(t).$$
•
Kinematic coupling on $\Sigma_{fsi}(t)=\Sigma(t)\backslash\Sigma_{l}$
$$u_{f}=\dot{d}\quad\mbox{on}\quad\Sigma_{fsi}(t).$$
•
Dynamic coupling on $\Sigma(t)$:
$$\displaystyle\sigma_{s}n$$
$$\displaystyle=\lambda n-\sigma_{p}n=\gamma_{C}[P_{\gamma}]_{+}n-\sigma_{p}n%
\quad\text{on }\Sigma(t)\cap\Sigma_{l},$$
$$\displaystyle\sigma_{s}n$$
$$\displaystyle=\lambda n-\sigma_{f}n=\gamma_{C}[P_{\gamma}]_{+}n-\sigma_{f}n%
\quad\text{on }\Sigma(t)\setminus\Sigma_{l}.$$
We have the following Nitsche-based variational formulation: Find $u_{f}\in{\mathcal{V}}_{f},p_{f}\in{\mathcal{L}}_{f},d\in{\mathcal{V}}_{s},P_{l%
}\in{\mathcal{V}}_{l}$ such that
$$\displaystyle(\partial_{t}u_{f},v)_{\Omega_{f}}+(\partial_{t}\dot{d},w)_{%
\Omega_{s}}+a_{f}\big{(}u_{f},p_{f};v,q\big{)}+a_{s}(d,w)\\
\displaystyle-(\sigma_{f}n,v-w)_{\Sigma(t)\setminus\Sigma_{l}}-(u_{f}-\dot{d},%
\sigma_{f}(v,-q))_{\Sigma(t)\setminus\Sigma_{l}}+\frac{\gamma_{\text{fsi}}}{h}%
(u_{f}-\dot{d},v-w)_{\Sigma(t)\setminus\Sigma_{l}}\\
\displaystyle-(\sigma_{p},v\cdot n)_{\Sigma_{l}\setminus\Sigma(t)}-(\sigma_{p}%
,w\cdot n)_{\Sigma_{l}\cap\Sigma(t)}+\big{(}[P_{\gamma}]_{+},w\cdot n\big{)}_{%
\Sigma(t)}\\
\displaystyle+(\epsilon K_{\tau}\nabla_{\tau}P_{l},\nabla_{\tau}q_{l})_{\Sigma%
_{l}}-\big{(}u_{f,n},q_{l}\big{)}_{\Sigma_{l}\setminus\Sigma(t)}-\big{(}\dot{d%
}_{n},q_{l}\big{)}_{\Sigma_{l}\cap\Sigma(t)}=0\\
\displaystyle\forall v\in{\mathcal{V}}_{f},q\in{\mathcal{L}}_{f},w\in{\mathcal%
{V}}_{s},q_{l}\in{\mathcal{V}}_{l}.$$
The porous stress $\sigma_{p}$ is given by
$$\displaystyle\sigma_{p}=-P_{l}+\frac{\epsilon K_{n}^{-1}}{4}u_{l,n}|_{\gamma_{%
f}}=\begin{cases}&-P_{l}+\frac{\epsilon K_{n}^{-1}}{4}u_{f,n}\quad\text{ on }%
\Sigma_{l}\setminus\Sigma(t)\\
&-P_{l}+\frac{\epsilon K_{n}^{-1}}{4}\dot{d}_{n}\quad\text{ on }\Sigma_{l}\cap%
\Sigma(t).\end{cases}$$
(9)
4 Numerical experiments
Here we will report on some numerical experiments using the above
models. First we consider the mixed dimensional Stokes’-Darcy system
and then the fluid-structure interaction system with contact and porous layer
in the contact zone.
4.1 Stokes-Darcy example
In this example, we consider two disconnected fluid reservoirs, the domain $\Omega_{f}$,
connected through a thin-walled porous media located on the bottom wall $\Sigma_{l}$, as shown in
Figure 1. The physical parameters are $\mu=0.03$, $\rho_{f}=1$, $\epsilon=0.01$ and $K_{\tau}=K_{n}=1$.
We impose a pressure drop across the two parts of the boundary $\Gamma_{f}^{N}$. The purpose of this example is to illustrate how the
porous model is able to connect the fluid flow between the two containers. This can be clearly inferred from the results reported in
Figure 2, which respectively show a snapshot of the fluid velocity, the elevation of the fluid pressure and the associated porous pressure.
4.2 Fluid-structure interaction with contact
To test the FSI-contact model, we consider flow in a 2-dimensional pipe, where the upper wall is elastic, see Figure 3. Due to the application of a
large pressure $\overline{P}$ on the left and right boundary, the
upper wall is deflected downwards until it reaches the bottom. Note
that when contact occurs, the configuration is topologically
equivalent to the situation in section 4.1.
Shortly before the time of impact we set $\overline{P}$ to zero, such that contact is realeased again after a certain time. This model
problem is taken from [4], where further details on the configuration and the discretisation can be found. To deal with the topology change in the fluid domain at the impact,
we apply a Fully Eulerian approach for the FSI problem [2].
In order to obtain a continuous and physically relevant transition from FSI to solid-solid contact, we use the FSI-contact model derived in section 3
and place a thin porous domain $\Sigma_{l}$ on the lower boundary.
In Figure 4 we compare this model for different parameters $K=K_{\tau}=K_{n}$ and $\epsilon$ with the approaches for FSI-contact problems
introduced in [4] in terms of the
minimal distance of the solid to $\Sigma_{p}$ over time. In [4] two approaches
were presented in order to extend the fluid stresses to the contact region during solid-solid contact, namely a so-called relaxed and an
artificial fluid approach. It was observed that for the artificial fluid approach contact happens earlier, as penetration of the
fluid flow into the artificial region is prevented only asymptotically, i.e. $u_{f,n}\to 0\,(h\to 0)$ on $\Sigma_{p}$, in contrast to $u_{f,n}=0$
for the relaxed approach. In the model presented here, we have
similarly from (7) and $u_{l,n}=u_{f,n}$ on $\Sigma_{p}$
$$\displaystyle u_{f,n}=-\nabla_{\tau}\cdot(\epsilon K_{\tau}\partial_{\tau}P_{l%
})\to 0\quad(\epsilon K_{\tau}\to 0).$$
For this reason we observe in Figure 4 that the impact happens earlier for a larger value of $\epsilon K_{\tau}$.
The time of the release seems to depend also
on $\epsilon K_{n}^{-1}$, which appears in the definition of $\sigma_{p}$ (9). A detailed investigation of this dependence and the
investigation of stability and convergence of the numerical method are subject to future work.
Acknowledgments
Erik Burman was partially supported by the EPSRC grant: EP/P01576X/1. Stefan Frei acknowledges support by the DFG Research Scholarship FR3935/1-1.
References
[1]
C Ager, B Schott, AT Vuong, A Popp, and WA Wall.
A consistent approach for fluid-structure-contact interaction based
on a porous flow model for rough surface contact.
Int J Numer Methods Eng, 119(13):1345–1378, 2019.
[2]
S Frei.
Eulerian finite element methods for interface problems and
fluid-structure interactions.
PhD thesis, Heidelberg University, 2016.
http://www.ub.uni-heidelberg.de/archiv/21590.
[3]
V Martin, J Jaffré, and JE Roberts.
Modeling fractures and barriers as interfaces for flow in porous
media.
SIAM J. Sci. Comput., 26(5):1667–1691, 2005.
[4]
E Burman S Frei and MA Fernández.
Nitsche-based formulation for fluid-structure interactions with
contact.
ESAIM: M2AN (published online).
https://doi.org/10.1051/m2an/2019072. |
Solvable model for the standard folklore of the glassy state
Theo M. Nieuwenhuizen
Department of Physics and Astronomy, University of Amsterdam
Valckenierstraat 65, 1018 XE Amsterdam, The Netherlands
(Revised, September 28, 1999)
Abstract
A model system with fast and slow processes is introduced.
After integrating out the fast ones,
the considered dynamics of the slow variables is exactly solvable.
In statics the system undergoes a Kauzmann transition to a glassy state.
The relaxation time obeys a generalized
Vogel-Fulcher-Tammann-Hesse law.
The aging dynamics on the approach to and below the Kauzmann temperature
is analyzed; it has logarithmic behavior.
The structure of the results could be general, as they satisfy
laws of thermodynamics far from equilibrium.
The original VFTH law is on the border-line between the regime
where only the effective temperature of the slow modes is needed,
and the regime where also an effective field occurs.
The production rates of entropy and heat are calculated.
pacs: 64.70.Pf, 75.10Nr,75.40Cx,75.50Lk
Half a century ago Kauzmann pointed at the
“entropy crisis” in glass forming liquids:
The naive extrapolation of the entropy of an undercooled liquid
goes below the entropy of the crystalline state, which is
physically incorrect [1].
It is often assumed that a thermodynamic phase transition takes place
around the “Kauzmann temperature” $T_{K}$, characterized by a
vanishing configurational entropy.
This prediction is very difficult to test experimentally,
since the relaxation time may be very long.
Experimental data for glass forming liquids
are often fit to a Vogel-Fulcher-Tammann-Hesse (VFTH) [2]
behavior $\tau_{\rm eq}$ $\sim$
$\exp[A^{\gamma}/(T-T_{0})^{\gamma}]$. An entropic argument for the
standard choice, $\gamma=1$, was given by Adam and
Gibbs [3]. A field-theoretic approach by
Parisi [4] leads to $\gamma>1$.
These approaches share the commonly accepted point that $T_{0}$ and $T_{K}$
coincide. In practice exponents $\gamma\neq 1$ are compatible
with data; this merely affects the width of the fitting interval.
Here we shall investigate universal aspects of this standard picture.
Very little is known about dynamics below the Kauzmann temperature.
One simple, intriguing idea comes to the mind. Consider a glassy
system that has aged a long time $t$. Can one say that all processes
with equilibration time much less than $t$ are in equilibrium,
while the ones with timescale much larger than $t$ are still
quenched, leaving the processes with timescale of order $t$ as the only
interesting ones? Such an approach would be too naive for continuous
phase transitions, where the timescale diverges
algebraically at the critical point.
However, in glasses there is an exponential divergence,
and this might induce an asymptotic decoupling of the time-decades.
Indeed, it occurs in the glassy regime of
models with an Arrhenius (rather than VFTH) law,
like the ferromagnetic Ising chain [5],
harmonic oscillator and spherical spin models [6].
This asymptotic decoupling could be the basic ground for
a generalization of equilibrium thermodynamics to systems
far from equilibrium [6].
That approach involved systems where one extra variable was
needed to describe the non-equilibrium physics, namely the
effective temperature. In the present work we shall
encounter situations, where also
an effective field is needed to describe the
two observable quantities, energy and magnetization out of equilibrium.
That case is of principle interest for glass forming
liquids, as it was found experimentally that at given $T,p$ the same
volume can be reached via different histories.
As they then lead to different futures,
the glassy state of these systems
cannot be coded using one extra variable [7].
In order to gain insight in properties of the standard picture,
we introduce a simple model, with solvable dynamics. In the
long time aging regime it leads to first order differential
equations. We thus can study aging
on the approach to and below the glass transition.
Even though the physics of the model is quite simple, we shall
set out to discover general aspects of the results by
phrasing them in a thermodynamic language.
The model contains a set of $N$ spins $S_{i}$, coupled to a slowly
changing background field, that we model by a set of $N$ harmonic
oscillators $x_{i}$. All variables are only coupled locally,
as is described by the Hamiltonian
$${\cal H}=\frac{1}{2}K\sum_{i}x_{i}^{2}-H\sum_{i}x_{i}-J\sum_{i}x_{i}S_{i}-L%
\sum_{i}S_{i}$$
(1)
The spins have no fixed length, but satisfy the
spherical constraint $\sum_{i}S_{i}^{2}=N$.
In order to model the $\beta$-processes of a glass,
we assume that the spins $S_{i}$ move fast, and we
shall consider only the time regime where they
are at thermodynamic equilibrium.
To model $\alpha$-processes, we shall
consider a dynamics where the $x_{i}$ may fall out of equilibrium.
Integrating out the $S_{i}$ replaces the spin-part of the energy (1)
by its free energy, which will act as an effective
Hamiltonian for the slowly moving variables $x_{i}$,
$${\cal H}_{\rm eff}=U-TS_{\rm ep}$$
(2)
Here $U=Nu$ is the internal energy given by
$$u=\frac{U}{N}=\frac{1}{2}Km_{2}-Hm_{1}-w+\frac{T}{2},$$
(3)
It depends on the two lowest moments of the $x_{i}$,
$m_{k}=(1/N)M_{k}=(1/N)\sum_{i}x_{i}^{k}$,
and $w$ is the short hand
$$w=\sqrt{J^{2}m_{2}+2JLm_{1}+L^{2}+{\small\frac{1}{4}}T^{2}},$$
(4)
$S_{\rm ep}$ is the entropy of the equilibrium processes,
i.e., of the spins,
$$S_{\rm ep}=\frac{1}{2}N-\frac{1}{2}N\ln[\beta(w+\frac{1}{2}T)]$$
(5)
Being interested in the long-time aging behavior,
we may restrict ourselves to the time-regime
where the dynamics of the $x_{i}$ can be
described by a Monte Carlo dynamics.
We must assume parallel updating,
as sequential updating would not produce a long timescale in
our system without two-particle interactions. We are thus considering
a simple example of models typically studied numerically.
Since it can even be solved analytically, we shall gain qualitative
insight in the long time regime.
The rules for the MC dynamics are standard, and lead to
evolution equations for $m_{1,2}$ that have the form
$$\displaystyle\dot{m}_{1}$$
$$\displaystyle=$$
$$\displaystyle\int_{-\infty}^{\infty}{\rm d}x\,y_{1}(x)p(x|m_{1,2})W(\beta x)$$
(6)
$$\displaystyle\dot{m}_{2}$$
$$\displaystyle=$$
$$\displaystyle 2\int_{-\infty}^{\infty}{\rm d}x\,\frac{x+{\tilde{H}}y_{1}(x)}{{%
\tilde{K}}}p(x|m_{1,2})W(\beta x)$$
(7)
Let us explain how this comes about.
For one MC step one shifts
all $x_{i}$ by an amount $r_{i}/\sqrt{N}$
with the $r_{i}$ drawn from a symmetric
Gaussian with variance $\Delta^{2}$.
This updates $M_{1,2}^{\prime}=M_{1,2}+\delta M_{1,2}$.
According to the Metropolis algorithm,
the move is accepted with probability
$W(\beta x)=\exp(-\beta x)$ if
the change in effective energy, $x={\cal H}_{\rm eff}^{\prime}-{\cal H}_{\rm eff}$,
is positive; if $x$ is negative
the move is always accepted ($W(\beta x)=1$). Such dynamical
problems are well understood [8] [9].
The essential ingredient is the thermally averaged transition
probability for a move characterized by $x$ and $\delta M_{1}$.
It is the product of two factors:
the first is a Gaussian in $\delta M_{1}$ with center
$y_{1}(x)$ and width $\Delta_{y}$;
the second factor, $p(x|m_{1,2})$ occurring in eqs.
(6) and (7), is a Gaussian in $x$ with center
$x_{0}$ and width $\Delta_{x}$. Defining $\mu_{1}={{\tilde{H}}}/{{\tilde{K}}}-m_{1}$ with
$$\displaystyle{\tilde{H}}=H+\frac{JL}{w+T/2};\qquad{\tilde{K}}=K-\frac{J^{2}}{w%
+T/2},$$
(8)
the parameters of these Gaussians take the form
$$\displaystyle x_{0}=\Delta^{2}{\tilde{K}}/2,$$
$$\displaystyle\Delta_{x}=\Delta^{2}{\tilde{K}}T_{e},$$
(9)
$$\displaystyle y_{1}=\beta_{e}\mu_{1}\,(x_{0}-x),$$
$$\displaystyle\Delta_{y}=\Delta^{2}\,(1-\beta_{e}{\tilde{K}}\mu_{1}^{2}).$$
(10)
Here $T_{e}=1/\beta_{e}$ is a short hand
for $T_{e}={\tilde{K}}(m_{2}-m_{1}^{2})$; later we shall interpret
it as an effective temperature.
We are still free to choose the variance $\Delta^{2}$ of the
Monte Carlo moves. To model glassy behavior we shall consider a
dynamics that constrains the system’s phase space to
$m_{2}-m_{1}^{2}\geq m_{0}$ for some fixed $m_{0}$.
This mimics a glass that has no crystalline state,
as occurs in atactic polymer melts.
For some fixed $m_{0}$ and $B$, we thus let $\Delta$
depend on the instantaneous values of $m_{1,2}(t)$ according to
$$\Delta^{2}(t)=\frac{8T_{e}(t)}{{\tilde{K}}(t)}\,\Gamma(t)$$
(11)
where
$$\Gamma(t)\equiv\left(\frac{B}{\mu_{2}(t)}\right)^{\gamma};\quad\mu_{2}(t)=m_{2%
}(t)-m_{1}^{2}(t)-m_{0}$$
(12)
The power $\gamma$ remains a free parameter, because we will
not make a connection to an underlying microscopy.
The fact that $\Delta^{2}$ becomes large near $\mu_{2}=0$ implies that
there the trial moves are typically large, and thus unfavorable.
Most of them will be rejected,
implying, as shown below, a VFTH-type equilibrium relaxation time
$$\tau_{eq}(T)\sim\exp\Gamma_{eq}\sim\exp\left(\frac{A(T_{0})}{T-T_{0}}\right)^{\gamma}$$
(13)
while $\tau_{eq}=\infty$ for $T<T_{0}$.
Though the physics of our model (1) is simple,
the choice (11), (12) for the timescale
allows us to derive aspects of the long time aging regime below $T_{K}$,
which could well be model independent.
Let us first look at the static regime, that has $\mu_{1}=0$.
For large enough $T$, the variable $\mu_{2}$ remains positive
and the effective temperature is trivial: $T_{e}^{\rm eq}=T$.
A Kauzmann transition occurs at $T_{K}$ set by $\mu_{2}=0$.
Since states with $\mu_{2}<0$ cannot be reached, it is evident
that the generally expected equality $T_{0}=T_{K}$ holds.
In statics $\mu_{2}$ vanishes for all $T<T_{0}$, implying that
$T_{e}^{\rm eq}$ follows from
$$\frac{Km_{0}-T_{e}^{\rm eq}+T}{Km_{0}-T_{e}^{\rm eq}}+\frac{D^{2}m_{0}}{(JT_{e%
}^{\rm eq})^{2}}=\frac{J^{2}m_{0}}{(Km_{0}-T_{e}^{\rm eq})^{2}}$$
(14)
where $D=HJ+KL$. The Kauzmann temperature is the solution of
$T_{e}^{\rm eq}(T_{0})=T_{0}$. Below $T_{0}$, $T_{e}^{\rm eq}$ exceeds $T$, and
it remains finite for $T\to 0$.
All static observables can now be expressed in $T_{e}^{\rm eq}$.
Since $w=[{J^{2}m_{0}+(DJ/T_{e}^{\rm eq})^{2}+T^{2}/4}]^{1/2}$,
we have the result for $m_{1}={\tilde{H}}/{\tilde{K}}$ via (8),
and using $m_{2}=m_{1}^{2}+m_{0}$, we can find the energy from (3).
Since ${\rm d}T_{e}^{\rm eq}/{\rm d}T<1$ at $T_{0}^{-}$, the specific heat
makes an upward jump as function of $T$; this happens also
for $-\partial m_{1}/\partial T$. The very same features occur in
the mean field glass model, the $p$-spin model,
where $T_{e}^{\rm eq}=x(T)/T$, with $x$ the Parisi
breakpoint [10].
We shall restrict ourselves to aging experiments at fixed $T,H$,
where the system evolves from some initial state. An example is
fast quenching from an equilibrium state
at large temperature, for which initially $\mu_{1}=0$, $0<\mu_{2}\ll 1$.
In the long time regime eqs. (6), (7)
lead to first order differential equations for $\mu_{1,2}$
$$\displaystyle\dot{\mu}_{1}$$
$$\displaystyle=$$
$$\displaystyle 4[aJrT_{e}-(1+aD)(\Gamma+r)\mu_{1}]I$$
(15)
$$\displaystyle\dot{\mu}_{2}$$
$$\displaystyle=$$
$$\displaystyle-8[\frac{rT_{e}}{{\tilde{K}}}-(\Gamma+r)\mu_{1}^{2}]I$$
(16)
where we denote $r=(T_{e}-T)/(2T_{e}-T)$,
$I=\int{\rm d}x\,p\,W$, and we introduce the shorthands
$$\displaystyle a=\frac{DJ^{2}}{{\tilde{K}}^{3}w(w+T/2)^{2}},\qquad b=\frac{J^{4%
}T_{e}}{2{\tilde{K}}^{2}w(w+T/2)^{2}}$$
(17)
$\dot{\mu}_{2}$ in eq. (16) is dominated by the small
factor $I\sim\exp(-\Gamma)$.
This is the dynamic imprint of the static VFTH law (13),
which may be more general. The solution follows readily,
since the integral of an exponential is basically the same exponential.
It is this property that leads to the asymptotic decoupling
of decades, discussed in the introduction.
In the aging regime one gets $t\sim t_{0}\exp\Gamma$, equivalent to
$$\mu_{2}=\frac{B}{\left[{\ln(t/t_{0})}\right]^{1/\gamma}}$$
(18)
where $t_{0}$ depends logarithmically on $t$. Such logarithmic
behavior is supposed to be the universal finger print for the
glassy state; notice, however, that there can be a non-trivial
exponent. For $T>T_{0}$ $\mu_{2}$
will stick at $\mu_{2}^{\rm eq}\sim T-T_{0}$, whereas for $T<T_{0}$ it will
ultimately vanish.
To solve for $\mu_{1}(t)$ we
divide eq. (15) by (16), which
allows to express $\mu_{1}$ in terms of $\mu_{2}$ of eq. (18),
$$\displaystyle\frac{{\rm d}\mu_{1}}{{\rm d}\mu_{2}}=\frac{-aJ{\tilde{K}}rT_{e}+%
(1+aD)(\Gamma+r){\tilde{K}}\mu_{1}}{2rT_{e}-2(\Gamma+r){\tilde{K}}\mu_{1}^{2}}$$
(19)
There exist several ranges of solutions:
$\bullet$ Regime 0: $T>T_{0}$, all $\gamma$:
Here $r\sim T_{e}-T\sim\mu_{2}$. The solution is obtained by
equating the numerator of the right hand side to zero,
$$\mu_{1}=\frac{aJ{\tilde{K}}(1+b)}{1+aD}\,\frac{\mu_{2}}{\Gamma}$$
(20)
Since $\mu_{1}\ll\mu_{2}$, it can be neglected to leading order.
$\bullet$ Regime 1: $T<T_{0}$, $\gamma>1$:
As in Regime 0, but now $r$ is of order unity,
$$\mu_{1}=\frac{aJT_{e}}{1+aD}\,\frac{r}{\Gamma}$$
(21)
Also in this case $\mu_{1}\ll\mu_{2}$.
$\bullet$ Regime 2: $T<T_{0}$, $\gamma=1$:
This is the case with a true VTFH-law. Quite surprisingly,
eq. (19) exhibits several subregimes with
different behavior: $\gamma=1$ is the most difficult situation,
too intricate to explain here in detail.
$\bullet$ Regime 3: $T<T_{0}$, $\gamma<1$
Now the denominator of (19) becomes subleading, of order
$\Gamma\mu_{2}=B^{\gamma}\mu_{2}^{1-\gamma}\ll 1$, but it remains
positive, thus assuring that $\mu_{2}$ finally vanishes. It follows that
$$\mu_{1}=\sqrt{\frac{rT_{e}}{{\tilde{K}}\,\Gamma}}(1-\frac{(1+aD){\tilde{K}}}{2%
\gamma rT_{e}}\Gamma\mu_{2})\sim\mu_{2}^{\gamma/2},$$
(22)
so in this regime $\mu_{2}\ll\mu_{1}$ can be neglected.
The magnetization approaches equilibrium as
$$m_{1}(t)=\frac{{\tilde{H}}^{\rm eq}}{{\tilde{K}}^{\rm eq}}-\frac{1}{1+aD}\mu_{%
1}(t)-\frac{aJ{\tilde{K}}}{2(1+aD)}\mu_{2}(t)$$
(23)
In regimes 0 and 1 its time-dependence is dominated by
$\mu_{2}\sim(\ln t)^{-1/\gamma}$, while in regime 3 the dominant
term is $\mu_{1}\sim(\ln t)^{-1/2}$.
In all cases both $m_{2}=m_{0}+m_{1}^{2}+\mu_{2}$ and the energy,
see eq. (3), behave similar to $m_{1}$.
It has long been a challenge to formulate glassy behavior in
terms of the laws of thermodynamics, because that could
point at universal behavior. Attempts to do so
remained unfruitful till recently [7][11].
In a recent series of papers we have shown that it can be done in
model systems where only one extra variable, the effective
temperature $T_{e}$, is needed to describe the physics [12]
[5][9]. The crucial step was to explain
the paradoxes concerning the Ehrenfest relations and the
Prigogine-Defay ratio [12].
In the present model there are two independent observables,
$m_{1,2}(t)$ or, equivalently, $m_{1}(t)$ and $u(t)$.
They result from the dynamics, but they can also
be described by a thermodynamics
at an effective temperature $T_{e}(t)$ and an effective field $H_{e}(t)$.
To do so, let us calculate in a quasi-stationary approach
the partition sum over macroscopically equivalent states
$$Z_{e}=\int Dx\,e^{-\beta_{e}[{\cal H}_{\rm eff}+HM_{1}-H_{e}M_{1}]}\delta(M_{1%
,2}-Nm_{1,2}(t))$$
(24)
By the extra terms
in the exponent we eliminate $H$ in favor of $H_{e}$.
We find for the free energy $F=-T_{e}\ln Z_{e}$
$$\displaystyle F=U+(H-H_{e})M_{1}-TS_{\rm ep}-T_{e}{\cal I}$$
(25)
where $U$ and $S_{\rm ep}$ were defined in (3) and (5).
The new term ${\cal I}=N[1+\ln(m_{2}-m_{1}^{2})]/2$ is the
entropy of the configurations with given $m_{1,2}$. Though our model is
classical, we can impose ${\cal I}=0$ at $T=0$ by adding a constant,
yielding
${\cal I}=(N/2)\ln(1+\mu_{2}/m_{0})$.
In his seminal paper Kauzmann pointed out that
the equilibrium configurational entropy vanishes already at $T_{K}$.
We see that this is indeed the case here.
We can also check another aspect of the standard folklore.
Comparison with eq. (13) yields
$\ln\tau_{eq}$$\sim$ $(N/{\cal I})^{\gamma}$. This is the generalization
of the Adam-Gibbs relation [3] to cases with $\gamma\neq 1$.
$T_{e}$ and $H_{e}$ are required to satisfy the saddle point equations that
follow from varying $m_{1,2}$ in eq. (25), implying
$$\displaystyle T_{e}$$
$$\displaystyle=$$
$$\displaystyle{\tilde{K}}(m_{2}-m_{1}^{2})={\tilde{K}}(m_{0}+\mu_{2})$$
(26)
$$\displaystyle H_{e}$$
$$\displaystyle=$$
$$\displaystyle Km_{1}-\frac{J(Jm_{1}+L)}{w+{T/2}}=H-{\tilde{K}}\mu_{1}$$
(27)
They only depend on $H$ through $m_{1,2}=m_{1,2}(t;T,H)$.
In eqs. (9)-(11) it was already seen
that the expression (26) for $T_{e}$ was useful
to shorten the dynamical equations.
It can now be checked that $T_{e}-T_{e}^{\rm eq}\sim+\mu_{2}-\mu_{1}$, so it has no definite sign, whereas
$H_{e}-H\sim-\mu_{1}$ is negative.
It is now seen that the “simple” Regimes 0 and 1 have a natural
interpretation: one then has $\mu_{1}\ll\mu_{2}$,
so the effective field $H_{e}\approx H$ is not needed:
the glassy state is coded by the effective temperature alone.
Such a situation was encountered before in model glasses with $T_{0}=0$
[6][9], and in the $p$-spin model
[12][13].
In the “complicated” Regimes 2 and 3
one ultimately has $T_{e}-T_{e}^{\rm eq}\sim H_{e}-H$. Both parameters
are thus needed to describe the evolution, a result
that should be valid beyond our model.
Systems with a true VFTH-law ($\gamma=1$) are at the borderline,
and thus more subtle.
We now consider which processes are involved at large time $t$.
The change of energy may be cast in the form
$$\dot{u}(t)=-\frac{u(t)}{\tau(T,T_{e}(t))}$$
(28)
For $T>T_{0}$ the behavior $I\sim\exp(-\Gamma)$
reproduces at equilibrium eq. (13);
Outside equilibrium it follows that $\tau(T,T_{e})\approx\tau_{eq}(T_{e})$,
as was observed in systems with an Arrhenius law [6].
For $T<T_{0}$, $\gamma>1$ (Regime 1)
the timescale of a system characterized by $T$ and $T_{e}$
obeys a generalization of the VFTH-law
$$\tau(T,T_{e})\sim\exp\Gamma=\exp\left(\frac{A(T)}{T_{e}-T_{e}^{\rm eq}(T)}%
\right)^{\gamma}.$$
(29)
saying that the equilibrium VFTH law has a definite fingerprint on the
aging dynamics below $T_{0}$.
Here, as well as in eq. (13), $A(T)=B{\tilde{K}}(1+b/(1+aD))$.
In the cases $\gamma\leq 1$ there are various behaviors,
that may not be universal.
In all these cases it holds that for $t\gg t_{0}$
the active processes are indeed those with timescale
$\tau(T,T_{e}(t))\sim t$, as is consistent with the logarithmic
time dependence: $u(t)=\tilde{u}(\ln t)$ implies
$\dot{u}=-u/\tau$ $\to$ $\tau=(-u/\tilde{u}^{\prime})\,t$.
Let us now consider the results from a more general thermodynamic
point of view. The first law says that the change in energy
is equal to the heat added to the system and the work done on the
system, ${\rm d}U={\mathchar 22\mskip-11.0mu {\rm d}}Q-M_{1}{\rm d}H$. It is satisfied with
$$\displaystyle{\mathchar 22\mskip-11.0mu {\rm d}}Q$$
$$\displaystyle=$$
$$\displaystyle T{\rm d}S_{\rm ep}\,+\,T_{e}{\rm d}{\cal I}+(H_{e}-H){\rm d}M_{1}$$
(30)
This holds because of the presence of a free energy,
eq. (25), satisfying ${\rm d}F=-S_{\rm ep}{\rm d}T-{\cal I}{\rm d}T_{e}-M_{1}{\rm d}H_{e}$.
It does not invoke the relation between $m_{1}(t)$ and $m_{2}(t)$,
which differs in the various regimes.
The two entropic terms in (30) are related to the
fast and slow processes, respectively [5][12].
The last term is new; it results from irreversible internal work.
In the aging process
the system will go to lower energy by producing heat at
a rate $|{\mathchar 22\mskip-11.0mu {\rm d}}Q/{\rm d}t|$, that follows from (30).
It holds that $|{\mathchar 22\mskip-11.0mu {\rm d}}Q/{\rm d}t|\sim|\dot{\mu}_{1}|+|\dot{\mu}_%
{2}|$ .
The dominant term is $\dot{S}_{\rm ep}+\dot{\cal I}\sim\dot{\mu}_{2}$ in regimes 0 and
1, while it is solely due to $\dot{S}_{\rm ep}\sim\dot{\mu}_{1}$ in regime 3.
In thermodynamics close to equilibrium a central role is played by
the irreversible entropy production. For systems far from
equilibrium, this is also the case.
We now show that it can be derived explicitly for systems
such as the present one. We write the change in total entropy $S=S_{\rm ep}+{\cal I}$
as ${\rm d}S={\rm d}_{e}S+{\rm d}_{i}S$,
where ${\rm d}_{e}S={\mathchar 22\mskip-11.0mu {\rm d}}Q/T$ is the externally supplied entropy,
and ${\rm d}_{i}S$ the internally produced entropy.
This gives the general result
$$\frac{{\rm d}S_{i}}{{\rm d}t}=\frac{T-T_{e}}{T}\,\frac{{\rm d}{\cal I}}{{\rm d%
}t}+\frac{H-H_{e}}{T}\,\frac{{\rm d}M_{1}}{{\rm d}t}$$
(31)
It can be verified explicitly in our model that both contributions
are non-negative, in accordance with the second law.
The first term of (31) is of order $r\dot{\mu}_{2}$, and is
dominant in regimes 0 and 1; The last term is
of order $\mu_{1}|\dot{\mu}_{1}|$, and it dominates in regime 3.
In conclusion, we introduce an exactly solvable model glass
with a set of fast variables (spherical spins) and a set of slow
variables (harmonic oscillators). The fast modes are summed out.
For the slow modes we assume a dynamics that obeys a
generalized VFTH law, $\tau_{eq}\sim\exp\,A^{\gamma}(T-T_{0})^{-\gamma}$.
In statics there occurs a Kauzmann transition.
We solve the aging dynamics, and
verify a generalized Adam-Gibbs relation between the
relaxation time and the configurational entropy.
Below $T_{0}$ the relaxation time for a system described by its effective
temperature may have a VFTH form, or a different one.
The dynamically active processes have timescale $\sim t$.
The aging solution can be described by a quasi-static
thermodynamic approach.
Above the Kauzmann temperature $T_{0}$
one extra variable is needed, the effective temperature, that depends
logarithmically on time. Below $T_{0}$ this persists in some cases,
while in other cases also an effective field is needed.
Systems with the original VFTH-law are at the borderline.
We finally derive the production rates of entropy and heat for systems
described by an effective temperature and field.
They are non-vanishing whenever the effective
temperature deviates from the bath temperature, and/or
when the effective field deviates from the external field.
We thank A. Allahverdyan, A. Crisanti, L. Leuzzi, L.S. Suttorp
and G.H. Wegdam for discussion.
References
[1]
W. Kauzmann, Chem. Rev. 43 (1948) 219
[2]
H. Vogel, Physik. Z. 22 (1921) 645;
G.S. Fulcher, J. Am. Ceram. Soc. 8 (1925) 339;
G. Tammann and G. Hesse, Z. Anorg. Allgem. Chem. 156 (1926) 245
[3]
G. Adam and J.H. Gibbs,
J. Chem. Phys. 43 (1965) 139
[4]
G. Parisi,
in The Oscar Klein Centenary, U. Lindström ed.,
(World Scientific, Singapore, 1995)
[5]
Th.M. Nieuwenhuizen, J. Phys. A 31
(1998) L201
[6]
Th.M. Nieuwenhuizen,
Phys. Rev. Lett. 80 (1998) 5580
[7]
G.B. McKenna, in Comprehensive Polymer Science
2: Polymer Properties, C. Booth and C. Price, eds.
(Pergamon, Oxford, 1989), pp 311
[8]
L.L. Bonilla, F.G. Padilla, and F. Ritort,
Physica A 250 (1998) 315
[9]
Th.M. Nieuwenhuizen,
Phys. Rev. E, in press; cond-mat/9807161
[10]
A. Crisanti and H.J. Sommers, Z. Phys. B87 341
(1992).
[11]
C.A. Angell, Science 267 (1995) 1924
[12]
Th.M. Nieuwenhuizen, Phys. Rev. Lett. 79(1997) 1317
[13]
J.A. Hertz, D. Sherrington, and Th.M. Nieuwenhuizen,
Phys. Rev. E 60 (1999) 2460 |
SI-HEP-2006-01
[0.2cm]
January 8, 2007
Testing the left-handedness of the $b\to c$ transition
B. M. Dassinger, R. Feger, T. Mannel
[0.1cm]
Theoretische Physik 1, Fachbereich Physik,
Universität Siegen
D-57068 Siegen, Germany
We analyse the spin structure of inclusive semileptonic $b\to c$ transitions
and the effects of non-standard model couplings on the rates and the spectra.
The calculation includes the ${\cal O}(\alpha_{s})$ corrections as well as the
leading non-perturbative ones.
1 Introduction
The enormous amount of data produced at the $B$ factories and the perspectives concerning future experiments with $b$ quarks will allow us to test the flavour non-diagonal sector of the standard model in a stringent way. However, aside from enough data on flavour changing decays this requires also a sufficient theoretical control in the calculation of the standard model predictions.
The sensitivity to “new physics effects” is expected to be largest in channels which do not have a big standard model contribution. These are in particular the loop induced flavour changing neutral current processes such as $b\to s$ transitions, which are suppressed by the GIM mechanism. Still there is also room for “new physics effects” in charged current processes, e.g. through the exchange of a heavy charged Higgs particle.
The theoretical description of semi-leptonic decays has become very precise over the last couple of years and seems to be under good control [1]. In combination with the very large data sets from the $B$ factories this has led to precise determinations of of the CKM matrix elements $V_{cb}$ and $V_{ub}$, assuming that the standard model is correct [2, 3].
In turn, these data may also serve to test the standard model. The charged currents will allow us to test aspects of the standard model different from the ones governed by loop induced flavour changing neutral current processes. In particular, the detailed data on semileptonic $B$ decays are sensitive to the helicity structure of the underlying quark transition, such as right-handed, scalar or tensor charged currents.
The type of analysis we are suggesting here is well known in the context of leptonic muon and $\tau$ decays and is usually expressed as values or limits for the “Michel parameters” [4, 5]. In the present paper we analyse the $b\to c$ transition current along the same lines and discuss the sensitivity which can be obtained on new couplings on the basis of the precise data on spectral moments in inclusive semileptonic $B$ decays.
The theoretical methods available for inclusive processes are in a mature state [6] and we shall compute the corresponding differential rates using the $1/m_{b}$ expansion [7, 8]. The state-of-the-art analysis within the standard model involves terms up to and including the terms of the order $1/m_{b}^{3}$ [9], the ones of order $1/m_{b}^{4}$ [10] and the next-to-leading QED and QCD radiative corrections to the leading (partonic) term [13]. In the present paper we perform the analysis for general currents at the appropriate level of precision.
In the next section we discuss the relevant operators using an effective field theory language. This allows us to identify the structure of the operators and will give us a qualitative argument on the sizes of the different contributions. In section 3 the OPE for the general interaction is performed, and in section 4 we discuss the radiative corrections to the leading term which is the partonic calculation. Finally we perform a numerical analysis of the effects of non-standard model interactions on the moments of semileptonic decay spectra and conclude.
2 Effective Field Theory Description of “New Physics”
Since the standard model is the most general renormalizable theory with the observed particle spectrum and interactions, possible effects from physics beyond the standard model will show up as $SU(3)\times SU(2)\times U(1)$ invariant operators of higher dimension, which are suppressed by powers of the new-physics’ scale. Focussing on the quark sector the symmetries of the standard model enforce that the leading operators are of dimension 6.
The list of relevant operators has been given in [14] and we shall use the notations of [15]. The quark fields are grouped into
$$\displaystyle Q_{L}$$
$$\displaystyle=$$
$$\displaystyle\left(\begin{array}[]{c}u_{L}\\
d_{L}\end{array}\right),\quad\left(\begin{array}[]{c}c_{L}\\
s_{L}\end{array}\right),\quad\left(\begin{array}[]{c}t_{L}\\
b_{L}\end{array}\right)\mbox{ for the left handed quarks and}$$
(1)
$$\displaystyle q_{R}$$
$$\displaystyle=$$
$$\displaystyle\left(\begin{array}[]{c}u_{R}\\
d_{R}\end{array}\right),\quad\left(\begin{array}[]{c}c_{R}\\
s_{R}\end{array}\right),\quad\left(\begin{array}[]{c}t_{R}\\
b_{R}\end{array}\right)\mbox{ for the right handed quarks}$$
(2)
where $Q_{L}$ are doublets under $SU(2)_{L}$ and $q_{R}$ are doublets under a (explicitly broken) $SU(2)_{R}$. The Higgs field and its charge conjugate are written as a $2\times 2$ matrix
$$H=1/\sqrt{2}\left(\begin{array}[]{cc}\phi_{0}-i\chi_{0}&\sqrt{2}\phi_{+}\\
-\sqrt{2}\phi_{-}&\phi_{0}+i\chi\end{array}\right)$$
(3)
transforming under $SU(2)_{L}\times SU(2)_{R}$. The potential of the Higgs fields leads to a vacuum expectation value (VEV) for the field $\phi_{0}$.
The $SU(2)_{L}\times SU(2)_{R}$ is broken by the Yukawa couplings and the weak hypercharge down to the standard model symmetry $SU(2)_{L}\times U(1)_{Y}$; however, keeping the notion of the larger $SU(2)_{L}\times SU(2)_{R}$ symmetry is useful for bookkeeping reasons.
The standard model is the most general renormalizable Lagrangian with an $SU(2)_{L}\otimes U(1)_{Y}$ symmetry and the phenomenologically correct particle content.This means that any effect from a high scale $\Lambda$ is suppressed by inverse powers of $\Lambda$, and — in the sense of an effective field theory — we are led to consider operators of dimension higher than 4. Hence, at scales of the order of the weak scale we may write the Lagrangian as an expansion in inverse powers of $\Lambda$
$$\mathcal{L}=\mathcal{L}_{4D}+\frac{1}{\Lambda}\mathcal{L}_{5D}+\frac{1}{%
\Lambda^{2}}\mathcal{L}_{6D}+...$$
(4)
where the leading term is the standard model Lagrangian $\mathcal{L}_{SM}=\mathcal{L}_{4D}$. Furthermore, $\mathcal{L}_{5D}$, $\mathcal{L}_{6D}$ … have to be invariant under the standard model symmetry $SU(2)_{L}\times U(1)_{Y}$.
The next-to-leading terms in the $1/\Lambda$ expansion involve only dim-6 or higher operators, since for quarks there is no $SU(2)_{L}\otimes U(1)_{Y}$ invariant dim-5 operator. The list of possible operators is sizable and can be found e.g. in [14]. However, in the present paper we shall concentrate on semileptonic $B$ decays which restricts us to a smaller subgroup. It contains operators involving two quarks and gauge fields as well as four fermion operators, consisting of two quarks and two leptons. Since we know from semileptonic $\tau$ decays that a possible contribution from a four fermion operator at the weak scale is likely to be small, we shall focus on the operators with two quarks and gauge fields.
These two quark operators of dimension 6 may be classified according to the helicities of the quark fields, left-left (LL), right-right (RR) and left-right (LR). A full list of the relevant two quark operators can be found in [15] where the following basis of operators has been suggested
$$\displaystyle O^{(1)}_{LL}$$
$$\displaystyle=$$
$$\displaystyle\bar{Q}_{A}\,\mathchoice{\ooalign{$\hfil\displaystyle\mskip 0.0mu%
/\hfil$\cr$\displaystyle L$}}{\ooalign{$\hfil\textstyle\mskip 0.0mu /\hfil$%
\cr$\textstyle L$}}{\ooalign{$\hfil\scriptstyle\mskip 0.0mu /\hfil$\cr$%
\scriptstyle L$}}{\ooalign{$\hfil\scriptscriptstyle\mskip 0.0mu /\hfil$\cr$%
\scriptscriptstyle L$}}G_{AB}^{(1)}\,Q_{B},$$
(5)
$$\displaystyle O^{(2)}_{LL}$$
$$\displaystyle=$$
$$\displaystyle\bar{Q}_{A}\,\mathchoice{\ooalign{$\hfil\displaystyle\mskip 0.0mu%
/\hfil$\cr$\displaystyle L$}}{\ooalign{$\hfil\textstyle\mskip 0.0mu /\hfil$%
\cr$\textstyle L$}}{\ooalign{$\hfil\scriptstyle\mskip 0.0mu /\hfil$\cr$%
\scriptstyle L$}}{\ooalign{$\hfil\scriptscriptstyle\mskip 0.0mu /\hfil$\cr$%
\scriptscriptstyle L$}}_{3}G_{AB}^{(2)}\,Q_{B}$$
(6)
with
$$\displaystyle L^{\mu}$$
$$\displaystyle=$$
$$\displaystyle H\left(iD^{\mu}H\right)^{\dagger}+\left(iD^{\mu}H\right)H^{%
\dagger},$$
(7)
$$\displaystyle L^{\mu}_{3}$$
$$\displaystyle=$$
$$\displaystyle H\tau_{3}\left(iD^{\mu}H\right)^{\dagger}+\left(iD^{\mu}H\right)%
\tau_{3}H^{\dagger}$$
(8)
and all coupling matrices $G^{(i)}_{AB}$ being Hermitian. The terms proportional to $\tau_{3}$ have once again been included to break the custodial symmetry explicitly.
In the same spirit we define RR-operators
$$\displaystyle O^{(1)}_{RR}$$
$$\displaystyle=$$
$$\displaystyle\bar{q}_{A}\,\mathchoice{\ooalign{$\hfil\displaystyle\mskip 0.0mu%
/\hfil$\cr$\displaystyle R$}}{\ooalign{$\hfil\textstyle\mskip 0.0mu /\hfil$%
\cr$\textstyle R$}}{\ooalign{$\hfil\scriptstyle\mskip 0.0mu /\hfil$\cr$%
\scriptstyle R$}}{\ooalign{$\hfil\scriptscriptstyle\mskip 0.0mu /\hfil$\cr$%
\scriptscriptstyle R$}}F_{AB}^{(1)}\,q_{B}$$
(9)
$$\displaystyle O^{(2)}_{RR}$$
$$\displaystyle=$$
$$\displaystyle\bar{q}_{A}\,\left\{\tau_{3},\mathchoice{\ooalign{$\hfil%
\displaystyle\mskip 0.0mu /\hfil$\cr$\displaystyle R$}}{\ooalign{$\hfil%
\textstyle\mskip 0.0mu /\hfil$\cr$\textstyle R$}}{\ooalign{$\hfil\scriptstyle%
\mskip 0.0mu /\hfil$\cr$\scriptstyle R$}}{\ooalign{$\hfil\scriptscriptstyle%
\mskip 0.0mu /\hfil$\cr$\scriptscriptstyle R$}}\right\}F_{AB}^{(2)}\,q_{B}$$
(10)
$$\displaystyle O^{(3)}_{RR}$$
$$\displaystyle=$$
$$\displaystyle i\bar{q}_{A}\,\left[\tau_{3},\mathchoice{\ooalign{$\hfil%
\displaystyle\mskip 0.0mu /\hfil$\cr$\displaystyle R$}}{\ooalign{$\hfil%
\textstyle\mskip 0.0mu /\hfil$\cr$\textstyle R$}}{\ooalign{$\hfil\scriptstyle%
\mskip 0.0mu /\hfil$\cr$\scriptstyle R$}}{\ooalign{$\hfil\scriptscriptstyle%
\mskip 0.0mu /\hfil$\cr$\scriptscriptstyle R$}}\right]F_{AB}^{(3)}\,q_{B}$$
(11)
$$\displaystyle O^{(4)}_{RR}$$
$$\displaystyle=$$
$$\displaystyle\bar{q}_{A}\,\tau_{3}\mathchoice{\ooalign{$\hfil\displaystyle%
\mskip 0.0mu /\hfil$\cr$\displaystyle R$}}{\ooalign{$\hfil\textstyle\mskip 0.0%
mu /\hfil$\cr$\textstyle R$}}{\ooalign{$\hfil\scriptstyle\mskip 0.0mu /\hfil$%
\cr$\scriptstyle R$}}{\ooalign{$\hfil\scriptscriptstyle\mskip 0.0mu /\hfil$\cr%
$\scriptscriptstyle R$}}\tau_{3}F_{AB}^{(4)}\,q_{B}$$
(12)
with
$$\displaystyle R^{\mu}$$
$$\displaystyle=$$
$$\displaystyle H^{\dagger}\left(iD^{\mu}H\right)+\left(iD^{\mu}H\right)^{%
\dagger}H$$
(13)
and again Hermitian coupling matrices $F_{AB}^{(i)}$.
Using an odd number of Higgs fields we can construct invariant LR operators
$$\displaystyle O^{(1)}_{LR}$$
$$\displaystyle=$$
$$\displaystyle\bar{Q}_{A}\,HH^{\dagger}H\widehat{K}_{AB}^{(1)}\,q_{B}+h.c.$$
(14)
$$\displaystyle O^{(2)}_{LR}$$
$$\displaystyle=$$
$$\displaystyle\bar{Q}_{A}\,\left(\sigma_{\mu\nu}B^{\mu\nu}\right)H\widehat{K}_{%
AB}^{(2)}\,q_{B}\,+h.c.$$
(15)
$$\displaystyle O^{(3)}_{LR}$$
$$\displaystyle=$$
$$\displaystyle\bar{Q}_{A}\,\left(\sigma_{\mu\nu}W^{\mu\nu}\right)H\widehat{K}_{%
AB}^{(3)}\,q_{B}+h.c.$$
(16)
$$\displaystyle O^{(4)}_{LR}$$
$$\displaystyle=$$
$$\displaystyle\bar{Q}_{A}\,\left(iD_{\mu}H\right)iD^{\mu}\widehat{K}_{AB}^{(4)}%
\,q_{B}+h.c.$$
(17)
with the coupling matrices
$$\widehat{K}_{AB}^{(i)}={K}^{(i)}_{AB}+\tau_{3}{K}_{AB}^{(i)\prime}\,.$$
(18)
After spontaneous symmetry breaking we want to consider the contributions of these operators to the charged current $b\to c$ transitions. The relevant interactions are
$$\displaystyle O^{(1)}_{LL}$$
$$\displaystyle=$$
$$\displaystyle\frac{v^{2}g}{\sqrt{2}}G_{cb}^{(1)}V_{cb}\,\bar{c}\mathchoice{%
\ooalign{$\hfil\displaystyle\mskip 0.0mu /\hfil$\cr$\displaystyle W$}}{%
\ooalign{$\hfil\textstyle\mskip 0.0mu /\hfil$\cr$\textstyle W$}}{\ooalign{$%
\hfil\scriptstyle\mskip 0.0mu /\hfil$\cr$\scriptstyle W$}}{\ooalign{$\hfil%
\scriptscriptstyle\mskip 0.0mu /\hfil$\cr$\scriptscriptstyle W$}}^{+}P_{-}b$$
(19)
$$\displaystyle O^{(1)}_{RR}$$
$$\displaystyle=$$
$$\displaystyle\frac{v^{2}g}{\sqrt{2}}F_{cb}^{(1)}V_{cb}\,\bar{c}\mathchoice{%
\ooalign{$\hfil\displaystyle\mskip 0.0mu /\hfil$\cr$\displaystyle W$}}{%
\ooalign{$\hfil\textstyle\mskip 0.0mu /\hfil$\cr$\textstyle W$}}{\ooalign{$%
\hfil\scriptstyle\mskip 0.0mu /\hfil$\cr$\scriptstyle W$}}{\ooalign{$\hfil%
\scriptscriptstyle\mskip 0.0mu /\hfil$\cr$\scriptscriptstyle W$}}^{+}P_{+}b$$
(20)
$$\displaystyle O^{(3)}_{RR}$$
$$\displaystyle=$$
$$\displaystyle\frac{v^{2}g}{\sqrt{2}}2iF_{cb}^{(3)}V_{cb}\,\bar{c}\mathchoice{%
\ooalign{$\hfil\displaystyle\mskip 0.0mu /\hfil$\cr$\displaystyle W$}}{%
\ooalign{$\hfil\textstyle\mskip 0.0mu /\hfil$\cr$\textstyle W$}}{\ooalign{$%
\hfil\scriptstyle\mskip 0.0mu /\hfil$\cr$\scriptstyle W$}}{\ooalign{$\hfil%
\scriptscriptstyle\mskip 0.0mu /\hfil$\cr$\scriptscriptstyle W$}}^{+}P_{+}b$$
(21)
$$\displaystyle O^{(4)}_{RR}$$
$$\displaystyle=$$
$$\displaystyle-\frac{v^{2}g}{\sqrt{2}}F_{cb}^{(4)}V_{cb}\,\bar{c}\mathchoice{%
\ooalign{$\hfil\displaystyle\mskip 0.0mu /\hfil$\cr$\displaystyle W$}}{%
\ooalign{$\hfil\textstyle\mskip 0.0mu /\hfil$\cr$\textstyle W$}}{\ooalign{$%
\hfil\scriptstyle\mskip 0.0mu /\hfil$\cr$\scriptstyle W$}}{\ooalign{$\hfil%
\scriptscriptstyle\mskip 0.0mu /\hfil$\cr$\scriptscriptstyle W$}}^{+}P_{+}b$$
(22)
$$\displaystyle O^{(3)}_{LR}$$
$$\displaystyle=$$
$$\displaystyle\frac{vg}{2}V_{cb}\,\bar{c}\sigma^{\mu\nu}\left\{\partial_{\mu}W_%
{\nu}^{+}\widetilde{K}_{cb}^{(3)}P_{+}+\partial_{\nu}W_{\mu}^{+}\widetilde{K}_%
{cb}^{\dagger(3)}P_{-}\right\}b$$
(23)
$$\displaystyle O^{(4)}_{LR}$$
$$\displaystyle=$$
$$\displaystyle\frac{vg}{2}V_{cb}\,\bar{c}\left\{W_{\mu}^{+}i\partial^{\mu}%
\widetilde{K}_{cb}^{(4)}P_{+}+W_{\mu}^{+}i\partial^{\mu}\widetilde{K}_{cb}^{%
\dagger(4)}P_{-}\right\}b$$
(24)
containing the couplings $\widetilde{K}_{cb}^{(i)}=K_{cb}^{(i)}-K_{cb}^{\prime(i)}$ and the projectors $P_{\pm}=(1\pm\gamma_{5})/2$.
Including also the contribution from the standard model
$$\mathcal{L}_{SM}=\frac{g}{\sqrt{2}}V_{cb}\,\bar{c}\mathchoice{\ooalign{$\hfil%
\displaystyle\mskip 0.0mu /\hfil$\cr$\displaystyle W$}}{\ooalign{$\hfil%
\textstyle\mskip 0.0mu /\hfil$\cr$\textstyle W$}}{\ooalign{$\hfil\scriptstyle%
\mskip 0.0mu /\hfil$\cr$\scriptstyle W$}}{\ooalign{$\hfil\scriptscriptstyle%
\mskip 0.0mu /\hfil$\cr$\scriptscriptstyle W$}}^{+}P_{-}b\,,$$
(25)
we get the generalized interaction as
$$\begin{split}\displaystyle\mathcal{L}&\displaystyle=\frac{g}{\sqrt{2}}\left(1+%
\frac{v^{2}}{\Lambda^{2}}G_{cb}^{(1)}\right)V_{cb}\,\bar{c}\mathchoice{%
\ooalign{$\hfil\displaystyle\mskip 0.0mu /\hfil$\cr$\displaystyle W$}}{%
\ooalign{$\hfil\textstyle\mskip 0.0mu /\hfil$\cr$\textstyle W$}}{\ooalign{$%
\hfil\scriptstyle\mskip 0.0mu /\hfil$\cr$\scriptstyle W$}}{\ooalign{$\hfil%
\scriptscriptstyle\mskip 0.0mu /\hfil$\cr$\scriptscriptstyle W$}}^{+}P_{-}b\\
&\displaystyle+\frac{v^{2}g}{\sqrt{2}}\frac{1}{\Lambda^{2}}\left(F_{cb}^{(1)}-%
F_{cb}^{(4)}+2iF_{cb}^{(3)}\right)V_{cb}\,\bar{c}\mathchoice{\ooalign{$\hfil%
\displaystyle\mskip 0.0mu /\hfil$\cr$\displaystyle W$}}{\ooalign{$\hfil%
\textstyle\mskip 0.0mu /\hfil$\cr$\textstyle W$}}{\ooalign{$\hfil\scriptstyle%
\mskip 0.0mu /\hfil$\cr$\scriptstyle W$}}{\ooalign{$\hfil\scriptscriptstyle%
\mskip 0.0mu /\hfil$\cr$\scriptscriptstyle W$}}^{+}P_{+}b\\
&\displaystyle+\frac{vg}{2}\frac{1}{\Lambda^{2}}V_{cb}\,\bar{c}\left\{W_{\mu}^%
{+}i\partial^{\mu}\widetilde{K}_{cb}^{(4)}P_{+}+W_{\mu}^{+}i\partial^{\mu}%
\widetilde{K}_{cb}^{\dagger(4)}P_{-}\right\}b\\
&\displaystyle+\frac{vg}{2}\frac{1}{\Lambda^{2}}V_{cb}\,\bar{c}\sigma^{\mu\nu}%
\left\{\partial_{\mu}W_{\nu}^{+}\widetilde{K}_{cb}^{(3)}P_{+}+\partial_{\nu}W_%
{\mu}^{+}\widetilde{K}_{cb}^{\dagger(3)}P_{-}\right\}b.\end{split}$$
(26)
Going to hadronic energy scales we also integrate out the weak bosons. Since possible new physics contributions to the leptonic sector are strongly constrained, we shall use here only the standard model piece. Integrating out the weak bosons amounts to the replacement
$$W_{\mu}^{-}=\frac{4G_{F}}{\sqrt{2}}\bar{e}\gamma_{\mu}P_{-}\nu_{e}.$$
(27)
from which we obtain the effective Hamiltonian
$$\mathcal{H}_{eff}=\frac{4G_{F}V_{cb}}{\sqrt{2}}J_{q,\mu}J_{l}^{\mu},$$
(28)
where $J_{l}^{\mu}=\bar{e}\,\gamma^{\mu}P_{-}\,\nu_{e}$ is the usual leptonic current and $J_{h,\mu}$ is the generalized hadronic $b\to c$ current which is given by
$$\displaystyle J_{h,\mu}$$
$$\displaystyle=$$
$$\displaystyle c_{L}\,\,\bar{c}\gamma_{\mu}P_{-}b+c_{R}\,\,\bar{c}\gamma_{\mu}P%
_{+}b+g_{L}\,\,\bar{c}\frac{iD_{\mu}}{m_{b}}P_{-}b+g_{R}\,\,\bar{c}\frac{iD_{%
\mu}}{m_{b}}P_{+}b$$
$$\displaystyle+d_{L}\,\,\frac{i\partial^{\nu}}{m_{b}}(\bar{c}i\sigma_{\mu\nu}P_%
{-}b)+d_{R}\,\,\frac{i\partial^{\nu}}{m_{b}}(\bar{c}i\sigma_{\mu\nu}P_{+}b)\,,$$
where we have renamed the coupling constants to streamline the notation, $P_{\pm}$ denotes the projector on positive/negative chirality and $D_{\mu}$ is the QCD covariant derivative. Note that the term proportional to $c_{L}$ contains the standard model contribution.
In principle one could have written (2) from the start, but the derivation via an effective field theory approach allows us to consider the expected orders of magnitudes of the different contributions. Note that the derivatives acting on the fields in (26) will yield momenta of the particles involved in the decay (see (2)), and hence the typical scale which has to be assigned to these derivatives is the mass of the $b$ quark. Furthermore, we have already included the corresponding inverse powers of $m_{b}$ in the definitions of the couplings $g_{L/R}$ and $d_{L/R}$.
Comparing (2) with (26) we get an estimate for the orders of magnitudes
$$c_{L}\propto 1;\hskip 28.452756ptc_{R}\propto\frac{v^{2}}{\Lambda^{2}};\hskip 2%
8.452756ptd_{R/L}\propto\frac{v\,m_{b}}{\Lambda^{2}};\hskip 28.452756ptg_{R/L}%
\propto\frac{v\,m_{b}}{\Lambda^{2}};$$
(30)
where a possible overall loop factor $1/(16\pi^{2})$ has been omitted.
We note further that in minimal flavour violating scenarios [16] any occurrence of a right handed quark is related to a helicity flip and hence mass factors occur. While the above estimates remain the same for $g_{R/L}$ and $d_{R/L}$, the estimate for $c_{R}$ contains a strong additional suppression factor of $m_{b}m_{c}/v^{2}$.
The additional factor $m_{b}/\Lambda$ in $g_{L/R}$ and $d_{L/R}$ reflects the fact that in order to obtain a helicity flip one has to have an additional Yukawa coupling, making these contributions small compared to the helicity-conserving ones.
In the (differential) rate the relevant contributions are the interference terms with the standard model piece. A coupling between left and right handed contributions requires to flip the helicity of the final state $c$ quark and hence an additional factor $m_{c}/m_{b}$ will appear in these contributions.
Thus for the rates we obtain additional contributions of the orders
$c_{L}c_{R}\sim(m_{c}/m_{b})v^{2}/\Lambda^{2}$,
$c_{L}d_{L}\sim c_{L}g_{L}\sim(m_{b}v)/\Lambda^{2}$ and
$c_{L}d_{R}\sim c_{L}g_{R}\sim(m_{c}/m_{b})(m_{b}v)/\Lambda^{2}$,
while all other contributions will be too small to be relevant here.
3 OPE for the Differential Inclusive Rate
The calculation of the differential rate proceeds along the same lines as the corresponding calculation in the standard model. The starting point is a correlator of the two hadronic currents
$$T_{\mu\nu}=\int d^{4}x\,e^{-ix(m_{b}v-q)}\langle B(p)|\bar{b}_{v}(x)\Gamma_{%
\mu}c(x)\,\bar{c}(0)\Gamma_{\nu}^{\dagger}b_{v}(0)|B(p)\rangle,$$
(31)
where $\Gamma$ is the Dirac matrix corresponding to (2), $v=p/M_{B}$ is the four velocity of the decaying $B$ meson and $q$ is the momentum transferred to the leptons.
This correlator may be decomposed into scalar form factors according to
$$T_{\mu\nu}=-g_{\mu\nu}T_{1}+v_{\mu}v_{\nu}T_{2}-i\epsilon_{\mu\nu\alpha\beta}v%
^{\alpha}q^{\beta}T_{3}+q_{\mu}q_{\nu}T_{4}+(q_{\mu}v_{\nu}+v_{\mu}q_{\nu})T_{%
5}.$$
(32)
Contracting the imaginary part of $T_{\mu\nu}$ with the tensor $L_{\mu\nu}$
obtained from the leptonic currents one obtains the differential decay rate
$$d\Gamma=\frac{G_{F}^{2}}{4M_{B}}{\rm Im}T_{\mu\nu}L^{\mu\nu}d\phi_{\rm PS}$$
(33)
where $d\phi_{\rm PS}$ is the corresponding phase space element.
As discussed above, we shall use the standard model expression for the leptonic side of the process. Using the charged lepton energy $E_{\ell}$, the neutrino energy $T$ and the leptonic invariant mass $S$ as independent variables one obtains for the triply differential rate
$$\frac{d^{3}\Gamma}{dE_{\ell}\,dTdS^{2}}=\frac{G_{F}^{2}m_{b}}{4\pi^{3}}\left[2%
m_{b}S^{2}\,{\rm Im}T_{1}-m_{b}(S^{2}-4E_{\ell}T)\,{\rm Im}T_{2}-2S^{2}m_{b}(T%
-E_{\ell})\,{\rm Im}T_{3}\right]$$
(34)
In the following calculation we shall adopt the notations of [13].
3.1 Tree level expansion
The tree-level expansion in $1/m_{b}$ is most easily set up along the lines described in [10]. In the Feynman diagram shown in fig. 1, the double line denotes the propagator of a charm quark which propagating in the background field of the soft gluons of the $B$ meson
$$S_{\rm BGF}=\frac{-i}{\mathchoice{\ooalign{$\hfil\displaystyle\mskip 0.0mu /%
\hfil$\cr$\displaystyle Q$}}{\ooalign{$\hfil\textstyle\mskip 0.0mu /\hfil$\cr$%
\textstyle Q$}}{\ooalign{$\hfil\scriptstyle\mskip 0.0mu /\hfil$\cr$%
\scriptstyle Q$}}{\ooalign{$\hfil\scriptscriptstyle\mskip 0.0mu /\hfil$\cr$%
\scriptscriptstyle Q$}}+i\mathchoice{\ooalign{$\hfil\displaystyle\mskip 0.0mu %
/\hfil$\cr$\displaystyle D$}}{\ooalign{$\hfil\textstyle\mskip 0.0mu /\hfil$\cr%
$\textstyle D$}}{\ooalign{$\hfil\scriptstyle\mskip 0.0mu /\hfil$\cr$%
\scriptstyle D$}}{\ooalign{$\hfil\scriptscriptstyle\mskip 0.0mu /\hfil$\cr$%
\scriptscriptstyle D$}}-m_{c}}$$
(35)
where $Q=m_{b}v-q$ and $D$ denotes the covariant derivative with respect to the background gluon field.
The OPE of the forward scattering amplitude is obtained by multiplying (35) by the appropriate Dirac structures for the currents (see (2) and by expanding (35) to third order in $iD$
$$iS_{\rm BGF}=\frac{1}{\mathchoice{\ooalign{$\hfil\displaystyle\mskip 0.0mu /%
\hfil$\cr$\displaystyle Q$}}{\ooalign{$\hfil\textstyle\mskip 0.0mu /\hfil$\cr$%
\textstyle Q$}}{\ooalign{$\hfil\scriptstyle\mskip 0.0mu /\hfil$\cr$%
\scriptstyle Q$}}{\ooalign{$\hfil\scriptscriptstyle\mskip 0.0mu /\hfil$\cr$%
\scriptscriptstyle Q$}}-m_{c}}-\frac{1}{\mathchoice{\ooalign{$\hfil%
\displaystyle\mskip 0.0mu /\hfil$\cr$\displaystyle Q$}}{\ooalign{$\hfil%
\textstyle\mskip 0.0mu /\hfil$\cr$\textstyle Q$}}{\ooalign{$\hfil\scriptstyle%
\mskip 0.0mu /\hfil$\cr$\scriptstyle Q$}}{\ooalign{$\hfil\scriptscriptstyle%
\mskip 0.0mu /\hfil$\cr$\scriptscriptstyle Q$}}-m_{c}}(i\mathchoice{\ooalign{$%
\hfil\displaystyle\mskip 0.0mu /\hfil$\cr$\displaystyle D$}}{\ooalign{$\hfil%
\textstyle\mskip 0.0mu /\hfil$\cr$\textstyle D$}}{\ooalign{$\hfil\scriptstyle%
\mskip 0.0mu /\hfil$\cr$\scriptstyle D$}}{\ooalign{$\hfil\scriptscriptstyle%
\mskip 0.0mu /\hfil$\cr$\scriptscriptstyle D$}})\frac{1}{\mathchoice{\ooalign{%
$\hfil\displaystyle\mskip 0.0mu /\hfil$\cr$\displaystyle Q$}}{\ooalign{$\hfil%
\textstyle\mskip 0.0mu /\hfil$\cr$\textstyle Q$}}{\ooalign{$\hfil\scriptstyle%
\mskip 0.0mu /\hfil$\cr$\scriptstyle Q$}}{\ooalign{$\hfil\scriptscriptstyle%
\mskip 0.0mu /\hfil$\cr$\scriptscriptstyle Q$}}-m_{c}}+\frac{1}{\mathchoice{%
\ooalign{$\hfil\displaystyle\mskip 0.0mu /\hfil$\cr$\displaystyle Q$}}{%
\ooalign{$\hfil\textstyle\mskip 0.0mu /\hfil$\cr$\textstyle Q$}}{\ooalign{$%
\hfil\scriptstyle\mskip 0.0mu /\hfil$\cr$\scriptstyle Q$}}{\ooalign{$\hfil%
\scriptscriptstyle\mskip 0.0mu /\hfil$\cr$\scriptscriptstyle Q$}}-m_{c}}(i%
\mathchoice{\ooalign{$\hfil\displaystyle\mskip 0.0mu /\hfil$\cr$\displaystyle D%
$}}{\ooalign{$\hfil\textstyle\mskip 0.0mu /\hfil$\cr$\textstyle D$}}{\ooalign{%
$\hfil\scriptstyle\mskip 0.0mu /\hfil$\cr$\scriptstyle D$}}{\ooalign{$\hfil%
\scriptscriptstyle\mskip 0.0mu /\hfil$\cr$\scriptscriptstyle D$}})\frac{1}{%
\mathchoice{\ooalign{$\hfil\displaystyle\mskip 0.0mu /\hfil$\cr$\displaystyle Q%
$}}{\ooalign{$\hfil\textstyle\mskip 0.0mu /\hfil$\cr$\textstyle Q$}}{\ooalign{%
$\hfil\scriptstyle\mskip 0.0mu /\hfil$\cr$\scriptstyle Q$}}{\ooalign{$\hfil%
\scriptscriptstyle\mskip 0.0mu /\hfil$\cr$\scriptscriptstyle Q$}}-m_{c}}(i%
\mathchoice{\ooalign{$\hfil\displaystyle\mskip 0.0mu /\hfil$\cr$\displaystyle D%
$}}{\ooalign{$\hfil\textstyle\mskip 0.0mu /\hfil$\cr$\textstyle D$}}{\ooalign{%
$\hfil\scriptstyle\mskip 0.0mu /\hfil$\cr$\scriptstyle D$}}{\ooalign{$\hfil%
\scriptscriptstyle\mskip 0.0mu /\hfil$\cr$\scriptscriptstyle D$}})\frac{1}{%
\mathchoice{\ooalign{$\hfil\displaystyle\mskip 0.0mu /\hfil$\cr$\displaystyle Q%
$}}{\ooalign{$\hfil\textstyle\mskip 0.0mu /\hfil$\cr$\textstyle Q$}}{\ooalign{%
$\hfil\scriptstyle\mskip 0.0mu /\hfil$\cr$\scriptstyle Q$}}{\ooalign{$\hfil%
\scriptscriptstyle\mskip 0.0mu /\hfil$\cr$\scriptscriptstyle Q$}}-m_{c}}+\cdots$$
(36)
keeping track of the ordering of the covariant derivatives.
The remaining task is to evaluate the matrix elements of operators of the form
$$b_{v}(iD_{\mu_{1}})....(iD_{\mu_{n}})b_{v}$$
where $b_{v}$ is the full QCD field. This is done most conveniently in a recursive fashion as described in [10].
Expanding to $1/m_{b}^{2}$ requires the two well known matrix elements of dimension 5, which are the kinetic energy parameter $\mu_{\pi}$ and the chromomagnetic moment $\mu_{G}$
$$\displaystyle 2M_{B}\mu_{\pi}^{2}$$
$$\displaystyle=$$
$$\displaystyle-\langle B(p)|\bar{b}_{v}(iD)^{2}b_{v}|B(p)\rangle$$
(37)
$$\displaystyle 2M_{B}\mu_{G}^{2}$$
$$\displaystyle=$$
$$\displaystyle\langle B(p)|\bar{b}_{v}(iD_{\mu})(iD_{\nu})(-i\sigma^{\mu\nu}b_{%
v}|B(p)\rangle$$
(38)
The dim-4 matrix elements contain only a single derivative, while the dim-3 contributions have no derivative. They can be expressed in terms of $\mu_{\pi}$, $\mu_{G}$ up to ${\cal O}(1/m_{b}^{2})$ accuracy. The relevant general matrix elements are
[10]
$$\displaystyle\langle B(p)\,|\bar{b}_{v}(iD^{\rho})(iD^{\sigma})b_{v}\,|\,B(p)%
\,\rangle=-2M_{B}\left[\frac{1}{6}P_{+}\left(g^{\rho\sigma}-v^{\rho}v^{\sigma}%
\right)\mu_{\pi}^{2}-\frac{1}{12}P_{+}(-i\sigma^{\rho\sigma})P_{+}\mu_{G}^{2}%
\right],$$
$$\displaystyle\langle B(p)|\bar{b}_{v}(iD^{\rho})b_{v}|B(p)\rangle=-\frac{M_{B}%
}{2m_{b}}P_{+}\bigg{\{}v^{\rho}(\mu^{2}_{G}-\mu_{\pi}^{2})\bigg{\}}+\frac{M_{B%
}}{6m_{b}}\bigg{\{}(\gamma^{\rho}-v^{\rho}\mathchoice{\ooalign{$\hfil%
\displaystyle\mskip 0.0mu /\hfil$\cr$\displaystyle v$}}{\ooalign{$\hfil%
\textstyle\mskip 0.0mu /\hfil$\cr$\textstyle v$}}{\ooalign{$\hfil\scriptstyle%
\mskip 0.0mu /\hfil$\cr$\scriptstyle v$}}{\ooalign{$\hfil\scriptscriptstyle%
\mskip 0.0mu /\hfil$\cr$\scriptscriptstyle v$}})(\mu^{2}_{G}-\mu_{\pi}^{2})%
\bigg{\}},$$
$$\displaystyle\langle B(p)|\bar{b}_{v}b_{v}|B(p)\rangle=P_{+}\,\,M_{B}+\frac{M_%
{B}}{4m_{b}^{2}}(\mu^{2}_{G}-\mu_{\pi}^{2}),$$
(39)
and the result to order $1/m_{b}^{2}$ is obtained using the expansion (36) and the general vertices (2).
In this way we evaluate the scalar components of the correlator $T_{\mu\nu}$ given in (31). Although it is straightforward to compute all contributions we shall give here only the ones which can be expected to be sizable, these are the interferences with the standard model contribution. We write
$$T_{i}=T_{i}^{\rm c_{L}c_{L}}+T_{i}^{\rm c_{L}c_{R}}+T_{i}^{\rm c_{L}g_{L}}+T_{%
i}^{\rm c_{L}g_{R}}+T_{i}^{\rm c_{L}d_{L}}+T_{i}^{\rm c_{L}d_{R}},$$
(40)
where all scalar components have an expansion in $1/m_{b}$ according to
$$T_{i}^{\rm c_{L}c_{L}}=T_{i,0}^{\rm c_{L}c_{L}}+T_{i,2}^{\rm c_{L}c_{L}}+T_{i,%
3}^{\rm c_{L}c_{L}}+...$$
(41)
For the standard model contribution this expansion has be performed already to $1/m_{b}^{4}$ [10], while for the new physics contributions it is sufficient to retain the tree level terms, since the nonperturbative corrections at least to the low moments are tiny [11, 12]. More relevant are the QCD radiative corrections which we shall consider in the next section.
3.2 QCD Radiative Corrections
The main corrections to the non-standard model contributions is due to QCD radiative effects, since at the current level of precision the non-perturbative corrections are too small to be relevant. Hence we consider now the order $\alpha_{s}$ radiative corrections, which are computed by evaluating the Feynman diagrams shown in fig. 2. Note that the scalar current involves also an effective quark-quark-gluon-boson vertex in order to maintain QCD gauge invariance. Taking the imaginary part at the end implements the various cuts of the diagrams in fig. 2 corresponding to real and virtual radiation.
The standard model contribution involves only left handed currents and has been computed already long ago in [17, 18]. The left handed current is a dim-3 operator and has a vanishing anomalous dimension; hence the standard model calculation for the sum of the real and the virtual radiation yields an ultraviolet finite result.
As discussed above, the non-standard model operators correspond to dim-6 operators, but after spontaneous symmetry breaking the operators may be treated as dim-3 or dim-4-operators as far as the QCD corrections are concerned.
For the right handed current the same argument holds as for the left handed current. It is also a dim-3 operator and is conserved; thus the anomalous dimension vanishes and the sum of the real and the virtual QCD corrections is ultraviolet finite.
The tensor and the scalar operators connect different helicities and thus they contain only a single power of the weak VEV and in addition one derivative. Consequently, from the QCD point-of-view they are dim-4 operators which renormalize in QCD.
However, we are interested only in the QCD effects on the shape of the spectra and hence the virtual corrections do not play a role, since they only renormalize the tree level result. In particular, the partonic mass moments $\langle(p_{X}^{2}-m_{c}^{2})^{n}\rangle$ (with $n>0$) do not have a tree level contribution (and hence no contribution from virtual QCD corrections), since the tree level rate is proportional to $\delta(p_{X}^{2}-m_{c}^{2})$. Hence we need to compute only the contributions to the real radiation and can ignore - at least for this purpose - the question of the renormalization of the helicity-changing operators.
4 Results and Discussion
The new physics contributions will have an impact on the shapes of the spectra of the hadronic and leptonic energies as well as on the spectrum of the hadronic invariant mass.
It is instructive to first take a look at the tree level results for the charged lepton energy spectrum. Using the variable $y=2E_{\ell}/m_{b}$ we write the different contributions as
$$\frac{\text{d}\Gamma}{\text{d}y}=\sum_{i}\frac{\text{d}\Gamma^{(i)}}{\text{d}y%
}\quad,\,i=c_{L}c_{L},\,c_{L}c_{R},\,c_{L}d_{L},\,c_{L}d_{R},\,c_{L}g_{L},\,c_%
{L}g_{R}$$
(42)
where the superscript denotes the different contributions according to the
coupling constants as defined in (2). The term proportional
to $c_{L}c_{L}$ contains the well known standard-model contribution
$$\frac{\text{d}\Gamma^{c_{L}c_{L}}}{\text{d}y}=\frac{G_{F}^{2}|V_{cb}|^{2}m_{b}%
^{5}}{192\pi^{3}}\left[2y^{2}(3-2y)-6y^{2}\rho-\frac{6y^{2}\rho^{2}}{(1-y)^{2}%
}+\frac{2y^{2}(3-y)\rho^{3}}{(1-y)^{3}}\right],$$
(43)
while the contribution from a right handed current is given by
$$\frac{\text{d}\Gamma^{c_{L}c_{R}}}{\text{d}y}=-\frac{G_{F}^{2}|V_{cb}|^{2}m_{b%
}^{3}}{192\,\pi^{3}}\sqrt{\rho}\left[12y^{2}-\frac{24y^{2}\rho}{1-y}+\frac{12y%
^{2}\rho^{2}}{(1-y)^{2}}\right]\,.$$
(44)
The tree level results involving the tensor currents is
$$\frac{\text{d}\Gamma^{c_{L}d_{R}}}{\text{d}y}=-\frac{G_{F}^{2}|V_{cb}|^{2}m_{b%
}^{5}}{192\pi^{3}}\left[4y^{3}-\frac{12y^{3}\rho^{2}}{(1-y)^{2}}+\frac{8y^{3}%
\rho^{3}}{(1-y)^{3}}\right]$$
(45)
and
$$\frac{\text{d}\Gamma^{c_{L}d_{L}}}{\text{d}y}=-\frac{G_{F}^{2}|V_{cb}|^{2}m_{b%
}^{5}}{192\pi^{3}}\sqrt{\rho}\left[\frac{4y^{2}(3-y)(y-1+\rho)^{3}}{(1-y)^{3}}\right]$$
(46)
Likewise, the tree contributions involving the scalar current are
$$\frac{\text{d}\Gamma^{c_{L}g_{L}}}{\text{d}y}=\frac{G_{F}^{2}|V_{cb}|^{2}m_{b}%
^{5}}{192\pi^{3}}\sqrt{\rho}\left[\frac{6y^{2}(y-1+\rho)^{2}}{1-y}\right]=%
\sqrt{\rho}\,\,\frac{\text{d}\Gamma^{c_{L}g_{R}}}{\text{d}y}.$$
(47)
In fig. 3 we show the effect of the additional contributions in the hadronic current at tree level. This plot already shows that a contribution of a different helicity has a large impact on the shape of the spectrum.
It is well known that the moments are sufficiently inclusive and can be calculated reliably in the $1/m_{b}$ expansion, even if a (not too high) cut on the lepton energy is applied. In the following tables we give the results for various moments without an energy cut for the charged lepton energy $E_{l}$ and with a cut of 1 GeV for this quantity. In the tables we list the results for
$$L_{n}=\frac{1}{\Gamma_{0}}\int_{E_{\rm cut}}dE_{\ell}\,E_{\ell}^{n}\,\frac{d%
\Gamma}{dE_{\ell}}$$
(48)
and
$$H_{ij}=\frac{1}{\Gamma_{0}}\int_{E_{\rm cut}}dE_{\ell}\,\int dE_{\rm had}\,dM_%
{\rm had}^{2}\,(M_{\rm had}^{2}-m_{c}^{2})^{i}\,E_{\rm had}^{j}\,\frac{d^{3}%
\Gamma}{dE_{\ell}\,dE_{\rm had}dM_{\rm had}^{2}}$$
(49)
with
$$\Gamma_{0}=\frac{G_{F}^{2}|V_{cb}|^{2}m_{b}^{5}}{192\pi^{3}}\left[1-8\rho-12%
\rho^{2}\ln\rho+8\rho^{3}-\rho^{4}\right]$$
(50)
The entries in the tables contain the coefficients corresponding to the expansion (42) of the differential rate
$$\displaystyle L_{n}$$
$$\displaystyle=$$
$$\displaystyle c_{L}^{2}L_{n}^{(c_{L}c_{L})}+c_{L}c_{R}L_{n}^{(c_{L}c_{R})}+c_{%
L}d_{L}L_{n}^{(c_{L}d_{L})}+c_{L}d_{R}L_{n}^{(c_{L}d_{R})}+c_{L}g_{L}L_{n}^{(c%
_{L}g_{L})}+c_{L}g_{R}L_{n}^{(c_{L}g_{R})}$$
$$\displaystyle H_{ij}$$
$$\displaystyle=$$
$$\displaystyle c_{L}^{2}H_{ij}^{(c_{L}c_{L})}+c_{L}c_{R}H_{ij}^{(c_{L}c_{R})}+c%
_{L}d_{L}H_{ij}^{(c_{L}d_{L})}+c_{L}d_{R}H_{ij}^{(c_{L}d_{R})}+c_{L}g_{L}H_{ij%
}^{(c_{L}g_{L})}+c_{L}g_{R}H_{ij}^{(c_{L}g_{R})}$$
For the following numerical estimates we use $m_{b}=4.6$ GeV and $m_{c}=1.15$ GeV, resulting in $\rho=m_{c}^{2}/m_{b}^{2}=0.0625$.
Table 1 contains the results for the tree level contributions for the moments of the lepton energy spectrum without cut, table 2 contains the same quantities including a lepton energy cut.
Radiative corrections will give additional contributions proportional to $\alpha_{s}/\pi$ according to the expansion
$$L_{n}=L_{n,0}+\frac{\alpha_{s}}{\pi}L_{n,1}\qquad\mbox{and}\qquad H_{ij}=H_{ij%
,0}+\frac{\alpha_{s}}{\pi}H_{ij,1}$$
(51)
which is again split into the various contributions of the different coupling constants as defined in (42). Table 3 contains the numerical results for the coefficients $L_{n}$, which we give according to the discussion of the last section only for the $c_{L}c_{R}$ contribution. The radiative corrections are sizable as expected from the ones for the total rate, but not abnormally large.
The tree level result for the hadronic moments $H_{ij}$ is proportional to $\delta(M_{\rm had}^{2}-m_{c}^{2})$ which means that the tree level contribution vanishes for $i>0$:
$$H_{ij,0}=0\,\,\mbox{for}\,\,i>0.$$
(52)
As discussed above, only the $\alpha_{s}$ corrections will yield a nontrivial shape of the spectrum. Thus we list in table 6 the $\alpha_{s}/\pi$ coefficients of the hadronic moments without a lepton-energy cut, and in table 7 the same quantity including a cut of $E_{l}>1$ GeV.
The tables show that the various moments have a strong sensitivity to the non-standard couplings. With very few exceptions the coefficients of the new contributions are of the same order as the $c_{L}^{2}$ term, which contains the standard model. Thus if the moments can be determined at the level of a few percent this would result in a determination of the non-standard couplings at a similar level. However, a precise statement concerning the precision of an extraction of a non-standard model coupling is difficult, since a full analysis requires a combined fit of all the parameters, including the heavy quark parameters and heavy quark masses, which is beyond the scope of the present paper.
5 Conclusions
In this paper we have computed the effect of a non-standard coupling for the $b\to c$ transition in a semileptonic decay, assuming that the leptonic current is purely left handed. The latter assumption is well justified by the data on purely leptonic processes, in particular the muon decay.
Although the calculation of nonperturbative effects for these non-standard couplings is straightforward, the main corrections are the perturbative QCD effects, which are sizable and have to be taken into account. Due to the vanishing anomalous dimension of the left and the right handed currents the QCD effects are finite for these currents; however, additional work is required to compute the virtual corrections to the scalar and tensor currents, which renormalize under QCD. Fortunately, these virtual corrections do not change the shape of the spectra and thus statements about moments are still possible.
On general grounds one would not expect new physics to show up in a charged current interaction, but this may as well be a false prejudice. At least from the generic point of view the most general parametrizations in terms of dimension six operators allow new physics effects in charged currents. As an example for a model one can consider a multi-Higgs Model, where a charged Higgs boson would induce an effect in a charged current already at tree level.
The effect of non-standard couplings on the moments can be sizable and thus this method opens a road to constrain possible new physics effects in charged current interactions. However, a detailed analysis needs a combined fit of all parameters in the heavy quark expansion, including the quark masses and the heavy quark expansion parameters, and the additional results given in this paper will allow us to perform such a fit.
Note added:
When this paper was almost completed a parallel computation was communicated to us by I. Bigi with similar results. We thank I. Bigi and G. Song for communicating the results of their work prior to publication.
Acknowledgements
We acknowledge useful discussions with O. Buchmüller and J. Kühn. Parts of the calculations were done with FeynArts [19], FormCalc and LoopTools [20].
This work was partially supported by
the German Research Foundation (DFG) under contract No.
MA1187/10-1, and by the German Minister of Research (BMBF), contract No. 05HT6PSA.
References
[1]
A Textbook presentation is given in: A. V. Manohar and M. B. Wise,
Camb. Monogr. Part. Phys. Nucl. Phys. Cosmol. 10 (2000) 1.
[2]
O. L. Buchmüller, H. U. Flächer,
Phys. Rev. D 73, 073008 (2006)
[3]
A. Kuzmin [Belle Collaboration], arXiv:hep-ex/0611047.
[4]
L. Michel,
Proc. Phys. Soc. A 63, 514 (1950).
[5]
C. Bouchiat and L. Michel, Phys. Rev. 106, 170 (1957).
[6]
I. I. Y. Bigi, M. A. Shifman, N. G. Uraltsev and A. I. Vainshtein,
Phys. Rev. Lett. 71, 496 (1993)
[arXiv:hep-ph/9304225].
[7]
A. V. Manohar and M. B. Wise,
Phys. Rev. D 49, 1310 (1994)
[arXiv:hep-ph/9308246].
[8]
T. Mannel,
Nucl. Phys. B 413, 396 (1994)
[arXiv:hep-ph/9308262].
[9]
M. Gremm and A. Kapustin,
Phys. Rev. D 55, 6924 (1997)
[arXiv:hep-ph/9603448].
[10]
B. M. Dassinger, T. Mannel and S. Turczyk,
arXiv:hep-ph/0611168.
[11]
B. M. Dassinger,
Diploma thesis,
University of Siegen,
2006.
[12]
R. P. Feger,
Diploma thesis,
University of Siegen,
2005.
[13]
D. Benson, I. I. Bigi, T. Mannel and N. Uraltsev,
Nucl. Phys. B 665, 367 (2003)
[arXiv:hep-ph/0302262].
[14]
W. Buchmüller and D. Wyler,
Nucl. Phys. B 268, 621 (1986).
[15]
T. Hansmann and T. Mannel,
Phys. Rev. D 68, 095002 (2003)
[arXiv:hep-ph/0306043].
[16]
G. D’Ambrosio, G. F. Giudice, G. Isidori and A. Strumia,
Nucl. Phys. B 645, 155 (2002)
[arXiv:hep-ph/0207036].
[17]
G. Altarelli, N. Cabibbo, G. Corbo, L. Maiani and G. Martinelli,
Nucl. Phys. B 208, 365 (1982).
[18]
M. Jezabek and J. H. Kühn,
Nucl. Phys. B 320, 20 (1989).
[19]
T. Hahn,
Comput. Phys. Commun. 140, 418 (2001)
[arXiv:hep-ph/0012260].
[20]
T. Hahn and M. Perez-Victoria,
Comput. Phys. Commun. 118, 153 (1999)
[arXiv:hep-ph/9807565]. |
Discreteness Effects in Simulations of Hot/Warm Dark Matter
Jie Wang
$\thanks{Email: wangjie@mpa-garching.mpg.de},SimonD.~{}M.~{}White$
${}^{1}$Max–Planck–Institut für Astrophysik,
Karl–Schwarzschild–Str. 1, D-85748 Garching, Germany
Email: wangjie@mpa-garching.mpg.de
(Accepted 2007 May 25.
Received 2006 May 25;
in original form 2007 February 21)
Abstract
In Hot or Warm Dark Matter universes the density fluctuations at early times
contain very little power below a characteristic wavelength related
inversely to the particle mass. We study how discreteness noise influences
the growth of nonlinear structures smaller than this coherence scale in
$N$-body simulations of cosmic structure formation. It has been known for 20
years that HDM simulations in which the initial uniform particle load is a
cubic lattice exhibit artifacts related to this lattice. In particular, the
filaments which form in such simulations break up into regularly spaced
clumps which reflect the initial grid pattern. We demonstrate that a similar
artifact is present even when the initial uniform particle load is not a
lattice, but rather a glass with no preferred directions and no long-range
coherence. Such regular fragmentation also occurs in simulations of the
collapse of idealised, uniform filaments, although not in simulations of the
collapse of infinite uniform sheets. In HDM or WDM simulations all
self-bound nonlinear structures with masses much smaller than the free
streaming mass appear to originate through spurious fragmentation of
filaments. These artificial fragments form below a characteristic mass which
scales as $m_{p}^{1/3}k_{peak}^{-2}$, where $m_{p}$ is the $N$-body particle
mass and $k_{peak}$ is the wavenumber at the maximum of $k^{3}P(k)$
($P(k)$ is the power spectrum). This has the unfortunate
consequence that the effective mass resolution of such simulations
improves only as the cube root of the number of particles employed.
keywords:
methods: N-body simulations –methods: numerical – dark matter:
massive neutrinos
††pagerange: Discreteness Effects in Simulations of Hot/Warm Dark Matter–A††pubyear: 2007
1 Introduction
In the absence of a full analytic understanding of nonlinear structure growth,
numerical simulations provide a critical link between the weak density
fluctuations measured in the cosmic microwave background and the strong
inhomogeneities observed on all but the very largest scales in the present
Universe. Indeed, numerical simulations played a decisive role in excluding
massive neutrinos as a dark matter candidate (White
et al., 1983) and
in establishing the $\Lambda$CDM model as the leading and now standard
paradigm for the formation of all structure (Davis et al., 1985; White et al., 1987; Cen et al., 1994; Navarro
et al., 1996).
With the development of
more powerful computer hardware, of more accurate numerical algorithms, and of
methods to follow additional physical processes, the importance of simulations
as a tool to interpret observations of observed structure continues to
increase dramatically. In this paper we are concerned with one aspect of the
simplest kind of cosmological structure formation simulation, namely how
discreteness effects can drive the growth of spurious small-scale structure in
N-body simulations of evolution from initial conditions containing no such
structure.
To create initial conditions for a cosmological simulation, a uniform particle
distribution is needed. This can be perturbed by a random realisation of the
linear fluctuation field associated with the specific structure formation
model to be simulated (e.g. $\Lambda$CDM). A uniform Poisson distribution of
particle positions is not suitable for this purpose, because stochastic
“root-$N$” fluctuations can exceed the density fluctuations predicted by the
desired model over a wide range of scales. To avoid this problem, most early
simulations chose a regular cubic lattice as the initial uniform load.
Symmetry then assures that there can be no growth of structure in the absence
of imposed perturbations (Efstathiou et al., 1985). The preferred
directions and the large-scale coherence of the lattice may, however, be a
disadvantage, since they can give rise to numerical artifacts. As an
alternative, White (1996) suggested using a glass-like initial particle load
created by carrying out a cosmological simulation from Poisson initial
conditions but with the sign of the peculiar gravitational accelerations
reversed, so that each particle is repelled by all the others. When such a
system reaches quasi-equilibrium, the total force on each particle vanishes,
as for a grid, but there are no preferred directions and no long-range
order. The power spectrum on scales much larger than the mean interparticle
spacing approaches a power-law $P(k)\sim k^{n}$ with $n=4$ (Baugh et
al. 1995), where $n=4$ is the minimal large-scale power expected for a
discrete stochastic system (Peebles, 1980, section 28).
Normally, grid and glass initial loads are considered equivalent.
Nevertheless, artifacts due the initial lattice are obvious in early images of
the sheets, filaments and “voids” formed in Hot Dark Matter (HDM)
simulations (e.g. Centrella &
Melott, 1983; Frenk
et al., 1984; Efstathiou et al., 1985; Centrella et al., 1988). Baugh
et al. (1995) and White (1996)
showed that low density regions appear very different in simulations
with a grid initial load than in simulations started from a glass, although
Baugh et al. found that this does not show up as a difference in their power
spectra. Nevertheless, the regularly spaced clumps seen along filaments in HDM
simulations are clearly related to the initial particle grid, and so seem
unlikely to reflect a true physical instability. Despite this, Bode
et al. (2001)
and Knebe et al. (2003) interpreted analogous structures in their Warm Dark Matter
(WDM) simulations (which were set up using a grid initial load) as the result
of the physical fragmentation of filaments. The nonlinear formation
of such small-scale structure could have important consequences in models like
HDM or WDM where power on small scales is strongly suppressed in the linear
initial conditions. It is thus important to establish which simulated
structures are real and which are artifacts, as well as to understand whether
the simulations can be improved by, for example, choosing a glass initial load
in place of a grid.
Götz &
Sommer-Larsen (2002, 2003) carried out WDM simulations using both
grid and glass initial loads and reported significant differences. With a grid
they found spurious low-mass halos evenly spaced along filaments, exactly as
in earlier HDM experiments. The spacing is simply that of the initial grid,
stretched or compressed by the large-scale distortion field. They emphasised,
however, that such unphysical halos were less evident in their simulations
starting from a glass. This conclusion disagrees with our own work below,
where we find spurious halos also in simulations from glass initial conditions
and with a frequency very similar to that found in the grid case. Curiously,
in the glass case also we find the spurious halos to be regularly spaced along
filaments even though the initial condition is not regular over the
relevant scales.
In this paper, we wish clarify this issue by isolating the numerical
artifact, by exhibiting it in idealised filament formation
simulations, by exploring its dependence on the nature of the uniform
particle load, and by establishing the dependence of its
characteristic scale on the discreteness scale of the simulation and
the coherence scale of the WDM/HDM initial conditions. We carry out
cosmological simulations of an HDM universe at a wide range of
resolutions and with both grid and glass initial loads. In addition,
we simulate the collapse of an infinite straight uniform density
filament from glass initial conditions, showing that it fragments into
regularly spaced clumps. To gain additional insight, we also consider
the collapse of a glass to a uniform sheet, and the growth of
structure in a uniform, space-filling, but anisotropically compressed
glass. Rapid fragmentation on small scales occurs only in the filament
case. Our tests also demonstrate that considerable care is needed to
produce an initial glass load for which the growth of small-scale
structure in filaments is optimally suppressed. We propose a
randomisation technique which successfully washes out most
code-dependent periodic signals in the initial load.
The remainder of our paper is organised as follows. In $\lx@sectionsign 2$ we first discuss
the aspects of our simulation code which are relevant to the problem at hand,
in particular, how it estimates gravitational accelerations and how it is
modified in order to create a uniform glass distribution. We then describe
the way in which initial conditions are created for the simulations presented
in the rest of the paper. $\lx@sectionsign 3$ presents results from our Hot Dark Matter
simulations, showing that all small-scale collapsed structures appear to form
initially as regularly spaced clumps along filaments, and that these are
similar for grid and for glass initial loads. Results for our studies of
idealised structure formation from anisotropically compressed glasses are
presented in $\lx@sectionsign 4$. Rapid fragmentation on small scales occurs only in the
filament case. $\lx@sectionsign 5$ examines this filament fragmentation in more detail,
showing that its characteristic scale is related to the interparticle
separation for a well-constructed glass, but that scales related to the
Poisson-solver of the glass-construction code can play an important role if
their influence is not carefully controlled. Finally, $\lx@sectionsign 6$ summarises the
implications of our results for simulations of structure formation. In
particular, we show that for nonlinear structures the effective mass
resolution of simulations of HDM or WDM universes improves only as the cube
root of the number of simulation particles employed. This is much more
pessimistic than the direct proportionality to $N$ which might naively have
been expected.
2 Simulation methods and initial conditions
All simulations in this paper were performed using the massively parallel
N-body code L-Gadget2. This is a lean version of Gadget2 (Springel, 2005)
with
the SPH part excluded and with the memory requirements minimised. It was
originally written in order to carry out the Millennium Simulation
(Springel
et al., 2005).
The computation of gravitational forces is the most critical and
time-consuming element of any cosmological $N$-body code. Gadget2 uses a
hybrid tree-PM method where the long-range force is calculated at low
resolution using a particle-mesh scheme, and is supplemented by a
high-resolution but short-range correction calculated using a tree algorithm.
The short-range correction is assembled in real space by collecting
contributions from all neighbouring particles. The long-range force is
calculated by assigning the particles to a regular cubic mesh, by using
Fourier methods to obtain the corresponding potential, and by numerically
differencing the result. For a single particle this scheme introduces a
maximum force error of 1-2 percent near the split scale. Choosing a suitable
split scale (typically several times the mean interparticle separation)
results in force errors for smooth distributions of particles which are almost
everywhere far below 1 percent. Gadget2 uses a space-filling fractal, the
Peano-Hilbert curve, to control the domain decomposition associated with
parallelisation. Because there is a good correspondence between the spatial
decomposition obtained from this self-similar curve and the hierarchical tree
used to compute forces, it is possible to ensure that the tree decomposition
used by the code is independent of the platform, in particular of the number
of processors on which it is run. In addition, the “round-off” errors in
the forces induced when summing contributions from all processors are
explicitly considered. As a result, the forces are independent of
the number of processors and the domain cuts that are made. We believe that
all code-related numerical effects relating to the calculation of
gravitational accelerations are well under control in Gadget2.
Glass construction is embedded in Gadget2 by using some compile options. A
preset number of particles is initially distributed at random within the
cubic computational volume and the standard scheme is used to obtain the
gravitational acceleration on every particle. After reversing the signs of the
accelerations, all particles are advanced for a suitably chosen timestep. The
velocities are then reset to zero and the whole procedure is repeated. After
about a hundred steps the acceleration of each particle approaches zero.
Notice that this is the acceleration as obtained by the code, including the
effects of force anisotropy, domain decomposition, etc. Because the glass is
made in a periodic cube, we can get a large glass file cheaply by tiling a big
box with many replications of the original glass. However, the accelerations
calculated by the code may no longer vanish exactly for this larger glass
because the force anisotropies and inaccuracies now occur on a different scale
than when the glass was created.
The FFT calculation and the Barnes & Hut tree used in Gadget2 are
both based on static grids. This spatially fixed decomposition
introduces weak periodic signals in the force calculation, and these
are reflected in the particle distribution at the end of the
glass-making procedure. We will see below that that this can introduce
measurable spikes in the 1-D power spectrum of the final glass. To
reduce such effects we randomly offset the particle distribution in
all three coordinates with respect to the computational box before
carrying out each force computation during glass-making. This
suppresses the induced signals quite effectively but does not fully
eliminate them. A glass constructed in this fashion is referred to as
a “good” glass in the following, while a glass constructed without
the random offset technique and showing significant high spikes is
referred to as a “poor” glass. In the rest of this paper, we use
“good” glasses for our initial conditions except where explicitly
noted. These issues are further discussed in section 5.
Below we consider grid initial loads in addition to glasses in order to
compare their performance and to check the results of previous work
(Bode
et al., 2001; Götz &
Sommer-Larsen, 2002, 2003). A third quasi-uniform particle
distribution, the Quaquaversal distribution, has recently been suggested by
Hansen et al. (2007) as a non-periodic uniform initial load, a possible
alternative to a glass. We have created such a Quaquaversal load with
$2\times 8^{7}$ particles using the code provided by Hansen et al at their web
site, and we compare its performance to our grid and glass initial loads in
an Appendix. It produces significantly worse discreteness artifacts than
either grids or glasses.
Two types of simulation are considered below. The first is a series of
cosmological simulations of evolution from Hot Dark Matter (HDM)
initial conditions. Most of these are for a single realisation of the
HDM density field within a 100$h^{-1}$Mpc cube, but with different
kinds of initial load and with varying mass resolution. One considers
a 200$h^{-1}$Mpc cube in order to better constrain the abundance of
large objects. For these simulations, we choose an Einstein-de Sitter
universe dominated by a single massive neutrino. We take $H_{0}=76.5{\rm kms^{-1}Mpc^{-1}}$ which implies a neutrino mass of $\sim 55{\rm eV}$ and a corresponding free-streaming scale $\lambda=22.2~{}{\rm Mpc}$ (Bond &
Szalay, 1983) below which initial fluctuations are
exponentially suppressed relative to an assumed $P(k)\propto k$
primordial power spectrum. The power spectrum we actually use to
impose fluctuations on our initial loads is based on the theoretical
predictions of Bardeen et al. (1986) and agrees with numerical estimates
from the Boltzmann solver CMBFAST (Seljak &
Zaldarriaga, 1996). Since the same
realisation is used for all our 100$h^{-1}$Mpc simulations, they
should all produce identical structures. We start integrating at
redshift $z=20$ and evolve structure to a formal present-day amplitude
of $\sigma_{8}=2$. This corresponds to the collapse of the first
nonlinear structure in the simulation at $z\sim 6$. As a check of our
starting redshift we reran one simulation glass128 starting from
$z=100$. At $z=15$ the power spectrum of this simulation differed from
that of the original run by few percent or less on all scales.
Our simulations are listed with their parameters in Table 1: pre-IC, L, $m_{p}$,
$\epsilon$,and $n_{p}$ denote the initial load, the box size, the particle mass,
the softening length, and the particle number respectively.
Our second type of simulation is designed specifically to study the
discreteness artifacts which show up in the filamentary structures
within our HDM simulations. These simulations follow evolution from a variety
of highly idealised initial conditions, all based on uniformly but
anisotropically compressed glasses. We consider three different cases:
Anisotropic glass: The glass initial load is compressed by a factor of
2 along one axis, by a factor of 3 along a second axis, and is unaltered along
the third axis. Six replications of this configuration are then used to tile
the computational cube to produce an initial condition which is uniform and
glass-like on large scales but where the forces are no longer balanced on the
scale of the interparticle separation.
Sheet: The glass initial load is compressed along one dimension by a
factor of 2, so that it fills half of the computational volume. The other half
remains empty. This configuration collapses to form an infinite uniform
sheet.
Filament: Our glass initial load is compressed by a factor of 2 along
two of its periodic directions while the third remains unchanged. The
particles then fill a quarter of the computational volume, the rest remaining
empty. This configuration collapses to form a uniform straight filament.
All the simulations carried out from these initial conditions assume an
Einstein-de Sitter background universe. We define the expansion factor $a$ to
be unity at the initial time.
We identify collapsed “halos” in our HDM simulations using a
Friends-of-Friends (FOF) algorithm with linking length 0.2 times the mean
interparticle spacing (Davis et al 1985). In the following we will only
consider FOF halos with 32 or more particles. Subhalos within these halos were
identified using the SUBFIND algorithm (Springel et al., 2001) with parameters
set to retain all overdense self-bound regions with at least 20 particles.
Based on these subhalo catalogues we use the techniques of Springel
et al. (2005)
to construct merging trees which allow us to follow the formation
and evolution of all halos and subhalos.
3 Filament Fragmentation in HDM Simulations
The original motivation for this paper came from an unexpected phenomenon in
our HDM simulations. Even when we use a glass initial load, we find that the
filaments in these simulations break up into regularly spaced clumps, just as
in early simulations based on grid initial loads. We illustrate this in Fig. 1
which shows a slice through simulation glass128. All FOF haloes with $32\leq N_{fof}<300$ are indicated by red points. It is obvious that many of these
low-mass haloes lie in the filaments, and that they are surprisingly regularly
spaced along them.
We find similar behaviour in all our HDM simulations, independent of the
initial load and the mass resolution. In Fig. 2 we focus on a small cubic
subregion containing a filament. (This region is indicated by a blue square in
Fig. 1). In all four simulations, small clumps are visible at regularly spaced
positions along the filament. Their spacing is very similar in the two $128^{3}$
simulations, even though one started from a glass and the other from a grid.
The spacing is reduced by about a factor of two in the $256^{3}$ simulation and
by about another factor of two in the $512^{3}$ simulation. The clumps are
difficult to see in this last case, but this is merely a consequence of the
resolution of the image (compare Fig. 3 below). There are strong indications
that these clumps are a numerical artifact, not least because in the grid128
case they line up with the distorted but still recognisable pattern of the
initial grid, just as in early HDM simulations (e.g. Centrella & Melott 1983)
or in the WDM simulations of Götz &
Sommer-Larsen (2003). Götz &
Sommer-Larsen (2003) reported
that the effect is absent when starting from a glass, although
their Fig. 1 appears to show it in the filament at the lower right
corner of their image. Bode
et al. (2001) also noticed that many low-mass
haloes formed in the filaments of their WDM simulation, but they
interpreted this as a result of physical pancake fragmentation.
The tight relation between clump scale and mass resolution
manifest in Fig. 2 makes it clear, however, that this is actually a reflection
of N-body discreteness effects. Its surprising aspect is that the regular clump
spacing persists in the glass case where the initial load has no large-scale
coherence. We consider this issue further below.
Another consequence of this artifact is illustrated in Fig. 3. It appears that
almost all the subhaloes within the massive objects present at $z=0$ actually
originated through spurious filament fragmentation. Fig. 3 focusses on a
small subregion of the glass512 simulation which contains the largest FOF
halo, an object with several linked density centres at $z=0$. The lower right
panel shows the final mass distribution in this cubic region. All subhaloes
with SUBFIND particle count greater than 20 are indicated by blue circles. The
other three panels use our merging trees to trace all progenitors of
these subhaloes back to earlier times. All of them appear to form initially as
evenly spaced “beads” strung along filaments. They later fall into the large
halo where they are seen at $z=0$. The artificial regularity of their
formation is illustrated by the two zooms in the upper panels. We conclude
that the first generations of haloes in pure HDM or WDM universes should
contain no dark matter subhaloes of smaller mass scale.
4 Structure Growth in Idealised Glass Collapses
In our HDM simulations there is very little power in the imposed HDM power
spectrum below the free-streaming scale. The regular fragmentation of the
filaments is clearly related to the mesh for a grid initial load, but the
origin of the equally regular fragmentation in the glass case is less
obvious. In Fig.4 we analyse the structure of a “good” $160^{3}$ particle
glass to search for signs of unexpected periodic behaviour. The irregular
blue line in this figure is the dimensionless 3-D power per unit $\ln k$,
$\Delta^{2}(k)=k^{3}P_{3}(k)$ where $P_{3}(k)$ is the 3-D power spectrum. For
comparison, the straight blue line gives the expectation for a Poisson
distribution with the same number of particles ($P(k)=1/N$). On large scales
the power in the glass is far below this white noise level, with
$\Delta^{2}(k)\propto k^{7}$ rather than $\Delta^{2}(k)\propto k^{3}$. For a relatively
narrow frequency band near $k=160$, however, the power is noticeably above the
Poisson expectation. This corresponds approximately to the
separation of the clumps which form on the filaments so there may be some
connection to this artifact in our HDM simulations.
Fig. 4 also shows the power per unit $\ln k$ in 2-D and 1-D projections of
this same glass, $\Delta^{2}(k)=k^{2}P_{2}(k)$ and $\Delta^{2}(k)=kP_{1}(k)$
respectively. Again the measured results are compared to the expectation for a
Poisson distribution with the same number of particles. Both cases show
features directly analogous to those seen in the 3-D power spectrum. On large
scales (small $k$) the power is strongly suppressed relative to the white
noise level, with a spectrum which is steeper than white noise by four powers
of $k$. Near $k=160$ there is a narrow range of wavenumbers where the power
rises significantly above the white noise level. Again this feature could be
related to the break up of sheets or filaments into clumps spaced regularly at
about the mean interparticle separation.
Whether these features can indeed explain the unexpected fragmentation
of filaments in our HDM simulations depends, of course, on how they
are amplified as the particle distribution evolves. For the first
structures to collapse in an HDM universe, this evolution can be
idealised as a succession of three phases (Zel’dovich, 1970). During
early nonlinear growth, the tidal field causes a locally anisotropic
flow which first reverses along a single preferred direction while
continuing to expand (although at different rates) along the two
orthogonal directions. In the second phase, collapse along the
preferred axis gives rise to a quasi-two-dimensional sheet-like
structure, a “pancake”. Collapse along one of the other two axes
then produces a filament. Finally, material flows along filaments to
produce dark matter haloes at their intersections. These features are
all clearly visible in Fig. 1, although in practice the different
phases overlap and interact significantly. In Fig. 5 we show the
results from a set of idealised simulations of anisotropic collapse
designed to explore how discreteness noise grows for a glass initial
load during these various phases.
To illustrate structure growth in the first of the above phases, the top panel
of Fig. 5 shows the evolution of the total 3-D power per $\ln k$ for evolution
from an anisotropically compressed, but space-filling and otherwise
unperturbed glass. (It is easier for us to simulate the isotropic expansion
of an anisotropically distorted glass than the anisotropic expansion of an
initially isotropic glass.) The $160^{3}$ “good” glass of Fig. 4 was here
compressed by a factor of 2 along the $x$-axis and a factor of 3 along the
$y$-axis, then replicated 6 times in order to tile the full simulation cube.
The exact periodicities introduced by this procedure are responsible for the
regular gaps in power visible at low $k$ in the initial power histogram
(the black curve in Fig. 5). It is interesting that the “bump” in power at
the discreteness scale is broader than in Fig. 4 and now stretchess from near
$k=160$, the interparticle separation of the original glass past $k\sim 300$, the interparticle separation of the compressed glass. Clumping during
expansion from this initial condition is extremely slow. The green curve shows
the power distribution after expansion by a factor of 1000. By this time the
matter has aggregated into small dense knots which typically contain 50
particles, but on larger scales the distribution remains almost uniform. At
long wavelengths the power has grown by about six orders of magnitude, just as
predicted by linear theory. Thus the characteristic mass of the clumps grows
as $M_{*}\propto a^{6/7}$. The amplification of discreteness noise is
very weak during the first phase of anisotropic evolution from glass initial
conditions.
The second panel of Fig. 5 studies the growth of discrete noise during and
after collapse to a sheet. We compress our $160^{3}$ glass by a factor of 2
along one axis, leaving the other half of the simulation cube empty. This
initial condition collapses to a thin uniform sheet and thereafter remains
thin with the particles oscillating about the symmetry plane. The figure shows
the total 2-D power per $\ln k$ in the projection of particle distribution
onto this plane at four different times: the initial time, the moment when the
mass first collapses to a thin configuration ($a=3.1$), the moment the sheet
reaches minimum thickness for the second time ($a=12.5$) and a substantially
later time ($a=100$). In this case the power on large scales grows much faster
than in the previous case with $\Delta^{2}$ increasing approximately as $a^{6}$
rather than as $a^{2}$. At the time of first collapse, the power in the
discreteness peak has grown rather little, even though the power
on larger scales has already amplified substantially. By the time of second
collapse many nonlinear clumps are already evident in the projected mass
distribution and the feature at the scale of the initial interparticle
separation is no longer visible in the power spectrum. The characteristic
nonlinear scale is determined by the point where the amplified long wavelength
$\Delta^{2}\propto k^{6}$ tail crosses $\Delta^{2}\sim 1$. This scale increases
rapidly with time, approximately as $M_{*}\propto a^{3}$. Structure in a pancake
thus grows by an accelerated version of the standard hierarchical aggregation
mechanism illustrated in a more familiar context in the top panel of
Fig. 5. Discreteness effects do not appear to play a role other than by
setting the initial amplitude of the long wavelength tail of $\Delta^{2}(k)$.
The lowest panel of Fig. 5 shows similar data for an idealised simulation of
collapse to a filament. We compress our $160^{3}$ glass by a factor of 2 along
two orthogonal axes, leaving the remaining three quarters of the simulation
cube empty. This bar-like initial condition collapses to a thin, straight
filament. The figure shows the 1-D power per $\ln k$ for the projection of the
particle distribution onto the axis of the filament at four different times:
the initial time, the time of first collapse ($a=2.2$), a time shortly
thereafter ($a=2.4$) and a significantly later time ($a=5$). The power spectra
here are considerably noisier here than in the top two panels because there
are far fewer modes per bin in $\ln k$. By first collapse the large-scale
power has grown substantially but there is rather little amplification near
the discreteness peak at $k\sim 160$. This is similar to the sheet
case. Shortly after first collapse, however, the power in the discreteness
peak has grown by a large factor, reaching nonlinear levels. This shows up as
regular clumping along the filament with a periodicity close to $k=160$. It
differs from the behaviour in the sheet case and is apparently analogous to
the filament fragmentation we saw in our HDM simulations. We investigate it
further in the next section. After the filament collapses the large-scale tail
of the power distribution amplifies extremely rapidly, roughly as
$\Delta^{2}\propto a^{20}$. At the last time plotted this growth is again
setting the nonlinear scale, as in the upper two panels, and there is no
obvious feature near $k=160$.
In all three of these tests the scale of nonlinearity at late times reflects
the amplified small-$k$ tail of the initial power spectrum of the glass. This
tail grows much more rapidly in a sheet than in a uniform 3-D distribution and
much more rapidly in a filament than in a sheet. The initial glass, if well
made, does exhibit the theoretical minimum power on large scales
$P(k)\propto k^{4}$ (Zel’dovich 1965; Peebles 1980), so that no better
suppression of discreteness effects can be hoped for. In the filament case,
however, the first nonlinear structures are clearly different in nature
and are related to the interparticle separation scale of the uncompressed glass. We now consider this instability more closely.
5 Fragmentation of filaments
In Fig. 6 we illustrate the evolution of the collapsing filament discussed in
the last section. The first and second columns show projections perpendicular
to and along the filament, while the third shows its 1-dimensional projected
density. Only a tenth of the full length of the filament is plotted in order
to make its structure more visible. Shortly after $a=2$ the filament collapses
to minimum thickness and at almost the same time it breaks up into regularly
spaced clumps. The clump spacing is very nearly equal to the mean
interparticle separation in the unperturbed glass; we find $\sim 160$ clumps
along the full length of the filament. The number
of clumps is independent of the FFT grid used. For $128^{3}$,
$243^{3}$, $400^{3}$ FFT grids, we find the total number of lumps to be
always around $\sim 160$.
Furthermore, we have repeated this fragmentation experiment with
different compression factors in the initial condition. This changes the time
of first collapse but it changes neither the fact that the filament breaks up
just after first collapse, nor the spacing of the clumps. The same is true
even if we adopt different compression factors along the two axes (provided
both are well above unity) or if we impose an initial perturbation which is
axially symmetric and has no sharp edges. Using an initial glass with a
different number of particles produces a change in the interclump separation
which scales as the cube-root of $N$. Clearly then, the break up is associated
with a feature of the unperturbed glass.
The regular spacing of these artifacts indicates that modes with
$k\sim N^{1/3}$ dominate at least the early nonlinear evolution of
structure along the filament. This is visible in the left panels of
Fig. 7, which repeat the power spectra at the initial time and at
$a=2.4$ (just after collapse) from Fig. 5. At the initial time the
power around $k\sim 160$ is more than three orders of magnitude below
the threshold for nonlinearity, but by $a=2.4$ it is already approaching
unity and is well above the power on all the other scales
plotted. This is the reflection in Fourier space of the remarkable
regularity seen in Fig. 6. The lower left panel of Fig. 7 plots growth
factors for individual modes between the two times. The fastest
growing modes have $k$ somewhat smaller than 100, but their growth is
insufficient for them to overtake the initial power peak near $k\sim 160$. The power in all the modes in this peak grows by a similar
amount so the peak remains relatively narrow. This causes the
regular spacing of clumps along the filament. Tests with a $270^{3}$
particle glass show identical behaviour but with the peak shifted to
$k\sim 270$. We conclude that the regularly spaced clumps which form
on the filaments of our HDM simulations are produced by a narrow peak
in power near the mean interparticle separation of our initial glass
load. This peak is amplified to nonlinearity by the remarkably rapid
growth of structure which occurs once a filament has collapsed.
Careful examination of Fig. 7 shows that there are particular modes
for which the growth appears anomalously strong, notably those with
$k=64,128,256,384...$. This is very likely a consequence of
anisotropies in Gadget’s Poisson solver which is based on a binary
decomposition of the computational volume. For the “good” glass used
here these modes do not grow enough to overtake the power in the peak
associated with the interparticle separation, so it is the latter
which determines the initial fragmentation scale of the filament.
We now show that this is not always the case.
Up to this point all our results have been based on such a “good”
initial glass for which the offset technique discussed in §2 was
used to minimise features due to anisotropies in the force
calculations. Nevertheless, artifacts due to force anisotropies are
still visible in some of our plots. For example, spikes can be seen in
Fig. 5 at $k=128$ and 256 in the $a=3.1$ power spectrum of the sheet
(these spikes are solely due to modes with wave-vector parallel to the
fundamental axes of the computational cube) and at $k=64$ and 256 in
the $a=2.4$ power spectrum of the filament (see also the left panels
of Fig. 7). The growth factors in Fig. 7 show that these “special”
modes grow substantially more rapidly than neighboring modes. If we do
not use random offsets to reduce the impact of algorithmic boundaries
in the force calculation, then features of this kind can be strong
enough in the initial glass to significantly affect later
evolution. An example is shown in the right panels of Fig. 7 and also
in Fig. 8. Here we have carried out exactly the same filament collapse
test as before, but using a “poor” initial glass constructed without
using the offset technique. Spikes are now visible at $k=128,256$
and $384$ in the initial 1-D power spectrum. These are amplified
by the evolution and at $a=2.4$ the power is dominated by the
amplified spike at $k=128$ rather than by modes in the neighborhood of
the discreteness peak at $k\sim 160$. Spikes at $k=256$ and 384 are
also very strong and several other spikes are clearly visible. As
Fig. 8 shows, these spikes cause the filament to break up initially
into $\sim 128$ rather than $\sim 160$ clumps. Subsequent aggregation
into larger objects is similar in the two cases, however, with
large-scale effects overwhelming the initial differences.
Eliminating these troublesome power spikes from the initial conditions and
from subsequent evolution is not easy. Changing the nominal accuracy of the
force calculation affects the amplitude of the spikes but does not remove
them. We were surprised to find that similar spikes are present in the initial
glass used for the GIF simulations (Kauffmann et al., 1999) even though this
was created using a different code based on a $\rm{P^{3}M}$ Poisson
solver. (For the GIF simulations the artifact was of no consequence, because
of the substantial small-scale power imposed in the CDM initial conditions.)
The large-scale PM force calculation in both codes imposes a regular “power
of 2” spatial structure, and for Gadget-2 this is reinforced by the static
Barnes-Hut oct-tree which underlies the calculation of the short range
forces. The unexpected spikes appear to reflect these structural properties of
the force construction algorithms. To test this, we projected the initial
glass onto periodic directions which are not aligned with the axes of
the computational box. The corresponding 1-D power spectra do not show any
sharp spikes.
The tests in this section demonstrate that even with our random offset
procedure artifacts due to our Poisson solver are not entirely eliminated. On
the other hand, these tests are extremely sensitive to such artifacts because
of the very high growth rates which occur in the idealised straight filaments
we have been studying. While it is clearly important to be aware of the
possibility of such numerical effects when simulating WDM or HDM universes,
our results here show that for a carefully constructed glass the effects due
to the Poisson solver remain subdominant with respect to effects caused by the
discreteness of the particle distribution. The latter cannot be eliminated for
any choice of initial particle load. They set the fundamental lower limit to
the effective resolution of such simulations
6 Discussion and Conclusions
In this paper we have studied how discreteness effects limit the effective
mass resolution of $N$-body simulations of cosmogonies like WDM or HDM where
structure on small scales is suppressed in the linear initial conditions.
Filaments occur in such models as part of the natural development of the
cosmic web, but in simulations they fragment into regularly spaced clumps with
a separation which reflects the mean interparticle distance in the initial
load. These spurious clumps are responsible for all the low-mass substructures
we have been able to identify at late times in collapsed halos. Thus, it
appears that in an idealised WDM or HDM universe the first generations of dark
haloes are predicted to contain no self-bound substructures of
significantly smaller scale. Our tests on idealised systems show this
fragmentation to occur because 1-D projections of a 3-D quasi-uniform
particle distribution retain substantial power on the scale of the
3-D interparticle separation, and this power amplifies very
rapidly as the effectively 1-D system evolves.
We find (in disagreement with Götz &
Sommer-Larsen (2002, 2003)) that spurious
fragmentation of filaments occurs in almost identical fashion whether the
initial particle load is a glass or a grid. Indeed, as we illustrate in
Fig. 9, the effect appears slightly worse for a glass than for a grid. This
plot gives mass functions for 9 simulations from HDM initial conditions which
use different particle numbers and different initial loads. In each case there
is a sharp upturn in abundance at small masses which reflects the clumps
visible within filaments in Figures 1 to 3. For initial loads of a given type
this upturn shifts to smaller masses by a factor of 2 for each factor of 8
increase in the number of particles. For a given number of particles the
upturn clearly occurs at somewhat larger masses in the glass case than in the
grid case. Notice also that the upturn for the $N=256^{3}$ glass simulation in a
$200h^{-1}{\rm Mpc}$ box agrees very well with that for the $N=128^{3}$ glass
simulation in a $100h^{-1}{\rm Mpc}$ box. This confirms that it is the mean
interparticle separation which sets the mass scale, rather than properties of
the simulation code or of the particular HDM realisation simulated.
If we take the effective lower resolution limit of our HDM simulations
to be given by the dashed vertical lines in the lower panel of Fig. 9,
we find that it can be expressed as $M_{lim}=10.1\times\bar{\rho}~{}d~{}k_{peak}^{-2}$, where $\bar{\rho}$ is the mean density of
the universe, $k_{peak}$ is the wavenumber at the maximum of
$\Delta^{2}(k)$, the dimensionless power per $\ln k$ in the linear
initial conditions, $d=N^{-1/3}L$ is the mean interparticle
separation, $N$ is the number of simulation particles, and $L$ is the
side of the computational box. For our HDM initial conditions
$k_{peak}=4.23\times\lambda_{fs}^{-1}=0.1\times(m_{\nu}/30eV){\rm Mpc}^{-1}$. The coefficient in our expression for $M_{\rm lim}$
is estimated directly from our HDM results. It may depend
significantly on the shape of the primordial power spectrum and so
need modification for WDM initial conditions. The scaling
$M_{\rm lim}\propto N^{-1/3}$ should still hold
in this case, however. Comparing our formula without modification to
the numerical results of Bode
et al. (2001) using $k_{peak}=1.0$ and
$0.5~{}{\rm Mpc}^{-1}$, as appropriate for their two WDM models, gives
$M_{\rm lim}=3\times 10^{10}$ and $1.2\times 10^{11}h^{-1}{\rm M_{\odot}}$. These values agree well with the upturns
in the mass functions which they plot in their Fig. 9.
We can also compare with the WDM simulations presented by
Knebe et al. (2002, 2003). These authors followed Bode
et al. (2001) in
arguing that small mass haloes form along filaments by top-down
fragmentation. However, if we compare the mass functions they present
for three different simulations in Fig. 4 of Knebe et al. (2002) and
Fig. 3 of Knebe et al. (2003), we find the upturn at low mass for the two
simulations with the same numerical resolution but different WDM
particle mass to occur at masses which differ by about a factor of 4,
while the upturn for the two runs with different numerical resolution
but the same WDM particle mass also occur at masses differing by a
factor of about 4. This is in roughly agreement with the scaling we
predict for $M_{\rm lim}$ and is unexpected for a physical (rather
than numerical) feature. We conclude that these results, as well as
those of Bode
et al. (2001) are consistent with a spurious numerical
origin for the low mass halos in filaments similar to that we find in
our HDM simulations. Furthermore, with our parametrisation of the
characteristic scale based on the wavenumber at the peak of
$\Delta^{2}(k)$, the dependence of the characteristic mass of the effect
on the overall shape of the power spectrum appears to be weak.
This effective resolution limit is unfortunate news for simulations of HDM and
WDM universes. In our highest resolution HDM model, for example, the $N=512^{3}$
glass simulation of a $100{\rm Mpc/h}$ box, the resolution limit is $M_{lim}=8.8\times 10^{12}h^{-1}{\rm M_{\odot}}$, which corresponds to a clump of $4300$
simulation particles. Thus only halos with $5000$ particles or more can be
considered reliable. This is two or three orders of magnitude below the
masses of typical big halos in the simulation. Contrast this with simulations
of CDM universes where the positions, velocities and masses of haloes are
reasonably well reproduced even for objects with about 100 simulation
particles, giving a logarithmic dynamic range which is about twice as
large. Furthermore the effective dynamic range in halo mass increases in
proportion to $N$ for CDM simulations, but only in proportion to $N^{1/3}$ in
HDM or WDM simulations.
These results are interesting for the question of whether WDM models can
reproduce the observed properties of dwarf satellite galaxies in the Milky
Way. Available kinematic data for dwarf spheroidals suggest that they are
sitting in dark matter halos with maximum circular velocities of order 30 km/s
(e.g. Stoehr et al., 2002; Kazantzidis et al., 2004) corresponding to masses (for
an isolated object) of about $10^{10}{\rm M_{\odot}}$. After discounting
the spurious low-mass halos, the mass functions shown in Fig. 9 of Bode
et al. (2001)
demonstrate that halos of such small mass are not expected for a
WDM particle mass of 175 eV and are still strongly suppressed relative to
$\Lambda$CDM for a mass of 350 eV. We infer that WDM particle masses well in
excess of 500 eV will be necessary to produce “Milky Way” halos with
sufficient substructure to host the observed satellites. This is, however,
less stringent by a factor of several than constraints based on structure in
the Lyman $\alpha$ forest (Viel et al., 2006). It will be interesting to
carry out simulations of sufficient resolution to test whether the internal
structure of subhalos in a WDM universe is consistent with that inferred for
the halos of Milky Way dwarfs. The resolution limitations we have explored in
this paper imply that, although possible, this will be a major computational
challenge.
Acknowledgements
We thank Volker Springel for help in devising a glass-making scheme
which suppresses Poisson-solver-induced power spikes. We thank Adrian
Jenkins for the suggestion to consider idealised bar collapses. We
thank both of them and also Liang Gao for a number of very useful
discussions of the material presented in this paper.
References
Bardeen et al. (1986)
Bardeen J. M., Bond J. R., Kaiser N., Szalay A. S., 1986, ApJ,
304, 15
Baugh
et al. (1995)
Baugh C. M., Gaztanaga E., Efstathiou G., 1995, MNRAS, 274, 1049
Bode
et al. (2001)
Bode P., Ostriker J. P., Turok N., 2001, ApJ, 556, 93
Bond &
Szalay (1983)
Bond J. R., Szalay A. S., 1983, ApJ, 274, 443
Cen et al. (1994)
Cen R., Miralda-Escudé J., Ostriker J. P., Rauch M., 1994,
ApJ, 437, L9
Centrella &
Melott (1983)
Centrella J., Melott A. L., 1983, Nature, 305, 196
Centrella et al. (1988)
Centrella J. M., Gallagher III J. S., Melott A. L., Bushouse
H. A., 1988, ApJ, 333, 24
Davis et al. (1985)
Davis M., Efstathiou G., Frenk C. S., White S. D. M., 1985, ApJ,
292, 371
Efstathiou et al. (1985)
Efstathiou G., Davis M., White S. D. M., Frenk C. S., 1985,
ApJS, 57, 241
Frenk
et al. (1984)
Frenk C. S., White S. D. M., Davis M., 1984, in Setti G., Van
Hove L., eds, Large-Scale Structure of the Universe Elementary Particles
and the Large-Scale Structure of the Universe - Short Contributions.
pp 257–+
Götz &
Sommer-Larsen (2002)
Götz M., Sommer-Larsen J., 2002, Ap&SS, 281, 415
Götz &
Sommer-Larsen (2003)
Götz M., Sommer-Larsen J., 2003, Ap&SS, 284, 341
Hansen et al. (2007)
Hansen S. H., Agertz O., Joyce M., Stadel J., Moore B.,
Potter D., 2007, ApJ, 656, 631
Kauffmann et al. (1999)
Kauffmann G., Colberg J. M., Diaferio A., White S. D. M., 1999,
MNRAS, 303, 188
Kazantzidis et al. (2004)
Kazantzidis S., Mayer L., Mastropietro C., Diemand J., Stadel J.,
Moore B., 2004, ApJ, 608, 663
Knebe et al. (2003)
Knebe A., Devriendt J. E. G., Gibson B. K., Silk J., 2003,
MNRAS, 345, 1285
Knebe et al. (2002)
Knebe A., Devriendt J. E. G., Mahmood A., Silk J., 2002, MNRAS,
329, 813
Navarro
et al. (1996)
Navarro J. F., Frenk C. S., White S. D. M., 1996, ApJ, 462, 563
Peebles (1980)
Peebles P. J. E., 1980, The large-scale structure of the universe.
Princeton, N.J., Princeton University Press, 1980. 435 p.
Seljak &
Zaldarriaga (1996)
Seljak U., Zaldarriaga M., 1996, ApJ, 469, 437
Springel (2005)
Springel V., 2005, MNRAS, 364, 1105
Springel
et al. (2005)
Springel V., White S. D. M., Jenkins A., Frenk C. S., Yoshida N.,
Gao L., Navarro J., Thacker R., Croton D., Helly J.,
Peacock J. A., Cole S., Thomas P., Couchman H., Evrard A.,
Colberg J., Pearce F., 2005, Nature, 435, 629
Springel et al. (2001)
Springel V., White S. D. M., Tormen G., Kauffmann G., 2001,
MNRAS, 328, 726
Stoehr et al. (2002)
Stoehr F., White S. D. M., Tormen G., Springel V., 2002, MNRAS,
335, L84
Viel et al. (2006)
Viel M., Lesgourgues J., Haehnelt M. G., Matarrese S., Riotto
A., 2006, Physical Review Letters, 97, 071301
White (1996)
White S. D. M., 1996, in Schaeffer R., Silk J., Spiro M.,
Zinn-Justin J., eds, Cosmology and Large Scale Structure Formation and
Evolution of Galaxies.
pp 349–+
White
et al. (1983)
White S. D. M., Frenk C. S., Davis M., 1983, ApJ, 274, L1
White et al. (1987)
White S. D. M., Frenk C. S., Davis M., Efstathiou G., 1987, ApJ,
313, 505
Zel’dovich (1965)
Zel’dovich Ya. B., 1965, Adv Astron Astrophys., 3,241
Zel’dovich (1970)
Zel’dovich Ya. B., 1970, A&A, 5, 84
Appendix A The Quaquaversal distribution
Hansen et al. (2007) have suggested using an initial particle load constructed
through a simple “Quaquaversal” tiling of space (sometimes known as a
Q-set). In particular, they suggested using this initial load for WDM
simulations. In this appendix, we show what happens if this distribution is
used instead of a grid or glass initial load in a number of the tests we have
studied in our paper.
A Q-set “lives” in a rectangular box with side ratio $1:1:\sqrt{3}$, and
requires a total particle number of the form $N=2\times 8^{n}$ with $n$ an
integer. Periodic boundary conditions can be assumed on opposite faces of the
box to represent an infinite cosmological distribution. Fig. A1 compares the
3-D power spectrum of such a distribution with that of our “good” $160^{3}$
glass. This Q-set has $n=7$, resulting in a total of $N=4194304$ particles,
and was set up using the codes provided by Hansen et al. (2007). In order to
facilitate the comparison we shift $P(k)$ for the Q-set so that its white
noise level is at $160^{-3}$ and the mode number $k=1$ corresponds to a
wavelength 160 times the mean interparticle spacing. At all scales
significantly larger than the mean interparticle spacing, the power in the
Q-set lies well above that in the glass. The mean power declines approximately
as $k^{3.4}$ for small $k$ rather than as $k^{4}$, and there is substantial power
in a series of additional narrow peaks separated by factors of 2 in $k$.
We have used this Q-set as the initial particle load for our standard
$100h^{-1}{\rm Mpc}$ box HDM simulation. Since this simulation is carried out
in a cubic region we need to chop off the long end of the Q-set rectangular
box leaving a total of $2424140\sim 134^{3}$ particles in the simulation. This
truncation results in a violation of periodicity in the initial load for one
pair of faces of the cubic volume. As a result, spurious small clumps form
along this interface during later evolution, but these effects do not
propagate into the rest of the simulation. They are not visible in Fig. A2,
which is a slice through the simulation to be compared directly with
Fig. 1. We exclude a thin slice of the simulation which contains this
discontinuity when we calculate the halo mass function at $z=0$. This is
displayed in Fig. A3 and compared to our earlier results.
A comparison of Fig. A2 with Fig. 1 shows that the regularities of the Q-set
are much more visually apparent than those of the glass, particularly in low
density regions. The small halos indicated by red points are again found along
filaments and on the outskirts of massive objects. Figure A3 shows that the
Q-set reproduces the halo mass function of our other simulations at high
mass, but that the turn-up due to spurious low-mass halos occurs at
significantly higher mass than for glass or grid initial loads. The effective
mass resolution of the Q-set simulation is about a factor of three worse than
in either of the other cases.
We have also carried out our standard idealised filament simulation
starting from a compressed Q-set rather than a compressed glass. The
particle distribution along a section of the filament is shown just
after collapse in Fig. A4 for comparison with the central left-hand
panel of Fig. 6. Substantial and regular clumping is seen, although
the regularity is considerably more complex (and also stronger) than
in the glass case. As in the glass case, the clumps rapidly aggregate
into a small number of massive objects during later evolution. The
different initial power spectra of the glass and Q-set cases
(Fig. A1), result in different rates of growth of structure along the
filament at later times. Structures are always more massive are
arranged in a more complex pattern along the filament for Q-set than
in the glass case.
Our conclusion from these tests is that for given particle number the Q-set
performs significantly worse as an initial load than either a grid
or a glass. Since the visual regularities it induces are very similar to
those seen for a grid and are much stronger than those found with a glass,
there does not seem to be any obvious situation where a Q-set would be the
preferred choice for an initial quasi-uniform particle load. |
Polarization Diffusion from Spacetime Uncertainty
Carlo R. Contaldi, Fay Dowker and Lydia Philpott
Blackett Laboratory,
Imperial College, London SW7 2AZ, U.K.
Abstract
A model of Lorentz invariant random fluctuations in photon
polarization is presented. The
effects are frequency dependent and affect the polarization of photons
as they propagate through space. We test for this effect by
confronting the model with the latest measurements of polarization of
Cosmic Microwave Background (CMB) photons.
All approaches to the problem of quantum gravity
predict that the spacetime itself will suffer from quantum
uncertainty
at Planckian scales. This basic idea is
the inspiration behind many attempts to
formulate phenomenological models of quantum gravitational
effects with
potentially observable consequences.
To date, much of the effort spent on modeling the effect of
spacetime fluctuations has produced Lorentz
symmetry violating models.
However, with constraints on violations of Lorentz symmetry becoming
tighter all the time
(see e.g. Collaborations:2009zq)
it is more important than ever to discover quantum gravity
phenomenology that respects Lorentz symmetry.
That such models can exist has been demonstrated in Dowker:2003hb and
Philpott:2008vd. That work was motivated by the causal set
approach to quantum gravity Bombelli:1987aa; Sorkin:2003bx; Henson:2006kf
but the scheme is quite general and does not depend on
any details of the underlying theory except that it
should be Lorentz invariant.
The basic idea is that
certain dynamical quantities such as particle trajectories
are subject to minute, quasi-local, random fluctuations due to
the uncertainty in spacetime structure at the Planck scale.
The model-building strategy is
straightforward: identify a space of states for the
system, work out how Lorentz transformations act
and hence deduce the most general
Lorentz invariant diffusion process on that space.
In the case of a massive point particle, the outcome of this
strategy is an Ornstein-Uhlenbeck-type process in which the
momentum of the particle undergoes Brownian motion on the
mass shell in proper time Dowker:2003hb.
For massless particles, the Lorentz symmetry
restricts the process to be a one dimensional diffusion in
energy – the particles always travel on the light cone – but a
second independent parameter enters which governs a
drift in energy Philpott:2008vd.
In this paper we will
apply the
strategy described above to polarization degrees of freedom
as suggested in Sorkin:2007qi.
We model a photon classically as a point particle with
a spacetime position $x^{\mu}$, null momentum $k^{\mu}=(k^{0},\vec{k})$ and
a polarization state to be identified.
A more realistic description would use wave packets
and an even better model would take account of the quantal nature of
photons.
For now we assume that this classical state is
a good approximate description of each of the free streaming
photons produced by astrophysical and cosmological
sources which reach our detectors.
The state space for a classical photon is therefore
$\mathbb{M}^{4}\times{\cal{H}}_{0}^{3}\times\cal{B}$
where $\mathbb{M}^{4}$ is 4 dimensional Minkowski spacetime, ${\cal{H}}_{0}^{3}$
is the 3 dimensional “cone” of future-pointing null 4-vectors, and
$\cal{B}$ is the space of polarization states which we
will see is the Bloch sphere.
The polarization state of a massless particle of momentum
$k^{\mu}$ can be given by a complex 4-vector $a^{\mu}$ such that
$k^{\mu}a_{\mu}=0$ and $a^{\mu*}a_{\mu}=1$.
The vector ${a^{\prime}}^{\mu}=a^{\mu}+\lambda k^{\mu}$, for any complex number $\lambda$, will
describe the same state. To eliminate this gauge freedom,
we can consider the polarization state to be given by the complex
two form, $P=k\wedge a$, whose components, $P_{\mu\nu}=k_{\mu}a_{\nu}-a_{\mu}k_{\nu}$
satisfy the Lorentz invariant conditions
$$\displaystyle P^{\mu\nu}k_{\nu}$$
$$\displaystyle=0\,,$$
(1)
$$\displaystyle P^{\mu\nu}P_{\mu\nu}$$
$$\displaystyle=0\,,$$
(2)
$$\displaystyle P^{\mu\nu*}P_{\mu\nu}$$
$$\displaystyle=0\,,$$
(3)
$$\displaystyle P^{\mu\nu*}P_{\mu\sigma}$$
$$\displaystyle=k^{\mu}k_{\sigma}\,.$$
(4)
If $k^{\mu}=s^{\mu}$ where $s^{\mu}:=(1,0,0,1)$, $P$ has the
following components
$$P_{\mu\nu}=\begin{pmatrix}0&-a_{1}&-a_{2}&0\\
a_{1}&0&0&-a_{1}\\
a_{2}&0&0&-a_{2}\\
0&a_{1}&a_{2}&0\end{pmatrix}\,,$$
(5)
where $a_{1}$ and $a_{2}$ are complex
numbers such that $|a_{1}|^{2}+|a_{2}|^{2}=1$.
This corresponds to a polarization vector $a_{\mu}=(0,a_{1},a_{2},0)$.
The phase of the 2-d complex unit vector $(a_{1},a_{2})$
is not relevant for the polarization
state of a single photon and so the
polarization state space has two real dimensions: it is the
Bloch sphere, ${\cal{B}}\cong\mathbb{C}{\mathbb{P}}^{1}$.
Let $\alpha$ and $\beta$ be, respectively,
the usual polar and azimuthal
angles on $\cal{B}$, then they are related to the
components of $P_{\mu\nu}$ by
$$\displaystyle a_{1}$$
$$\displaystyle=\frac{e^{i\gamma}}{\sqrt{2}}\left(\cos{\frac{\alpha}{2}}+e^{i%
\beta}\sin{\frac{\alpha}{2}}\right)\,,$$
(6)
$$\displaystyle a_{2}$$
$$\displaystyle=i\frac{e^{i\gamma}}{\sqrt{2}}\left(\cos{\frac{\alpha}{2}}-e^{i%
\beta}\sin{\frac{\alpha}{2}}\right)\,,$$
(7)
where $\gamma$ is an irrelevant phase.
The
north and south poles,
$\alpha=0,\pi$, are
the circularly polarized states and the equator, $\alpha=\pi/2$,
consists of the linearly polarized states.
Now consider a general photon state $(k^{\mu},P_{\mu\nu})$.
For a general $k^{\mu}$, the polarization 2-form $P_{\mu\nu}$ must be
transformed by a Lorentz transformation that takes $k^{\mu}$ to $s^{\mu}$
in order for it to be compared to the standard polarization basis and
its coordinates on $\cal{B}$ determined.
This can be done using a standard Lorentz transformation
defined for example in Weinberg:2005.
If $P(k)$ is the polarization 2-form thus transformed, then
it will have components of the form (5) and
$$P(k)_{\mu\nu}=s_{\mu}a_{\nu}-a_{\mu}s_{\nu}\,,$$
(8)
where $a_{\mu}=(0,P(k)_{10},P(k)_{20},0)$.
The $(\alpha,\beta)$ coordinates of the polarization state
on $\cal{B}$ are then obtained from
(6) with $a_{1}=P(k)_{10}$ and $a_{2}=P(k)_{20}$ .
In this way, every photon state is specified by coordinates $(x^{\mu},k^{\mu},\alpha,\beta)$
on ${\mathbb{M}}^{4}\times{\cal{H}}_{0}^{3}\times\cal{B}$.
Under a Lorentz transformation
the photon state $(k^{\mu},P_{\mu\nu})$ transforms in
the usual way as a vector and 2-tensor and
it can be shown that this translates into a
polar rotation on $\cal{B}$, a rotation around
the north-south polar axis
generated by $\frac{\partial}{\partial\beta}$.
Details of these derivations will appear elsewhere.
The Stokes parameters (see e.g. Kamionkowski:1996ks) are a
convenient way to parameterize the polarization of a beam of
electromagnetic radiation.
A monochromatic beam
with
Stokes parameters $(I,Q,U,V)$ can be modeled as a
bunch of photons with the same momentum $k^{\mu}$ and
polarization states distributed over $\cal{B}$. $I$ is the
intensity of the beam and since our process
preserves particle number $I$ is fixed.
If a beam consists of photons of momentum $k^{\mu}$ which are all in
the same polarization state $(\alpha,\beta)\in\cal{B}$ then
the Stokes parameters of this perfectly polarized beam are
$Q=I\sin\alpha\cos\beta$,
$U=I\sin\alpha\sin\beta$ and
$V=I\cos\alpha$.
If the photons
have a distribution of polarizations
the Stokes parameters are weighted by the
probability density $\rho(\alpha,\beta)$
on $\cal{B}$ e.g.
$Q=I\int_{\cal{B}}\,\sin\alpha\cos\beta\,\,\rho(\alpha,\phi)\,d\alpha d\beta$ and similarly for $U$ and $V$.
There are many
distributions that will model a given set of Stokes
parameters. For example, an
unpolarized beam, $Q=U=V=0$,
could be modeled by a uniform distribution
of linearly polarized states, or a uniform distribution on the
two circularly polarized states alone.
In general, the more spread out the distribution on $\cal{B}$, the
smaller the polarization fraction,
${\cal{P}}:=\sqrt{Q^{2}+U^{2}+V^{2}}/I$.
Having identified the state space of the photon
as $\mathbb{M}^{4}\times{\cal{H}}_{0}^{3}\times{\cal{B}}$
we can deduce
the most general Lorentz invariant
diffusion process on this space. As described in Philpott:2008vd
the trajectory in spacetime is simple: the photon moves along
null lines according to $\frac{dx^{\mu}}{d\lambda}=k^{\mu}$
where $\lambda$ is affine time.
Moreover the process on the momentum
space ${\cal{H}}_{0}^{3}$ must not disturb the
blackbody nature of the CMBR spectrum over the age of the
universe. This means that we can neglect this effect for the
purposes of this paper: we assume that the photon’s frequency
is constant along its worldline. We are left with the
task of deducing the Lorentz invariant
diffusion equation on $\cal{B}$.
We refer to
coordinates on $\cal{B}$ as $X^{A}=(\alpha,\beta)$.
Then, following Sorkin:1986, the most general
diffusion equation on $\cal{B}$ is
$$\frac{\partial\rho}{\partial\lambda}=\partial_{A}\left(K^{AB}\,n\,\partial_{B}%
\left(\frac{\rho}{n}\right)-u^{A}\rho\right)\,,$$
(9)
where $\lambda$ is affine time,
$K^{AB}$ is a symmetric, positive semi-definite 2-tensor,
$u^{A}$ is a vector and $n$ is a scalar density (“density of
states”) on $\cal{B}$. These
geometric quantities are
the phenomenological parameters of the
model and must be Lorentz invariant.
There is an embarrassment of choice of parameters
because Lorentz transformations act as polar rotations
only. Any tensor, vector or scalar density
that does not depend on the azimuthal angle, $\beta$,
is Lorentz invariant. Free parameters that are whole functions do not
make for powerful phenomenology. If, however, we restrict attention
to the linear polarization states alone (corresponding
to setting Stokes parameter $V$ to zero), the
model recovers its predictive power.
The space of linearly polarized states is the unit circle, the
equator of the Bloch sphere, and the Lorentz transformations
act as rotations of the circle. The coordinate around the circle
is $\beta$ and there is, up to a constant factor, one
Lorentz invariant vector, $\partial/\partial\beta$.
A Lorentz invariant density $n$ must be constant on
the circle. We deduce a simple
diffusion-cum-drift on the circle for the
distribution $\rho=\rho(\alpha,\beta)$:
$$\frac{\partial\rho}{\partial\lambda}=c\frac{\partial^{2}}{\partial\beta^{2}}%
\rho-d\frac{\partial}{\partial\beta}\rho\,,$$
(10)
where $c>0$ and $d$ are constants.
Transforming from affine time to “cosmic time”,
i.e. time in the observatory frame, $t=h\nu\lambda$
where $h$ is Planck’s constant and $\nu$ is the
frequency of the photon, we have
$$\frac{\partial\rho}{\partial t}=\frac{c}{\nu}\frac{\partial^{2}}{\partial\beta%
^{2}}\rho-\frac{d}{\nu}\frac{\partial}{\partial\beta}\rho\,.$$
(11)
We absorbed the $h$ into the free parameters
governing the diffusion and drift.
We see that the rates of diffusion and drift in polarization angle
are frequency dependent.
Note that the Lorentz symmetry we have assumed is only
invariance under the proper orthochronous component of the
full Lorentz Group. The drift term explicitly breaks parity
invariance.
Our model has assumed that spacetime is Minkowski spacetime.
Following Philpott:2008vd we can model the effect of
an expanding universe by setting the frequency
to depend on time, $\nu=\nu(t)$, such that $a(t)\nu(t)=a_{0}\nu_{0}$
where $a(t)$ is the scale factor of the universe, $a_{0}$ is the
current value of $a$ and $\nu_{0}$ is the current (observed)
value of the frequency of the photon. If we define a new
time coordinate $t^{\prime}$ by $dt^{\prime}/dt=a(t)/a_{0}$ then our
diffusion equation keeps the same form as (11)
$$\frac{\partial\rho}{\partial t^{\prime}}=\frac{c}{\nu_{0}}\frac{\partial^{2}}{%
\partial\beta^{2}}\rho-\frac{d}{\nu_{0}}\frac{\partial}{\partial\beta}\rho\,.$$
(12)
For a matter dominated universe $a\sim t^{2/3}$ and
a range of
$t$ of $10^{60}$ Planck times becomes a range of $t^{\prime}$ of
$3/5\times 10^{60}$ Planck times. We drop the subscript
$0$ from $\nu$ and the prime from
t in what follows.
Consider a beam of photons of frequency $\nu$ whose polarization is
initially described by Stokes parameters $(U,Q)$.
The drift will result, after a time $t$ in a beam whose
polarization angle, $\Phi$, has rotated by $\chi:=td/\nu$:
$$\Phi=\tan^{-1}\left(\frac{U}{Q}\right)\rightarrow\Phi^{\prime}=\Phi+\chi\,.$$
(13)
The diffusion will result in a decrease in the
magnitude of polarization by a factor $\exp(-\mu)$:
$${\cal{P}}:=\sqrt{U^{2}+Q^{2}}\rightarrow{\cal{P}}^{\prime}=e^{-\mu}{\cal{P}}\,.$$
(14)
where $\mu$ can be calculated to be $4tc/\nu$.
Clearly, such an effect would be most pronounced in photons that have
propagated over long distances. Thus CMB photons, whose polarization
is correlated over incoming directions and have traveled over
cosmological distances without rescattering, offer the best chance to
constrain the parameters in the model. The polarization of the photons
is imprinted during the last stages of recombination as the photons
decouple from baryons and enter the free streaming regime. The
correlation in the polarization of photons arriving from different
directions is encoded in a set of angular power spectra $C_{\ell}^{\rm XY}$ where $\rm XY$ are the different spectral cross-correlation
in total intensity $T$ and grad-type and curl-like components $E$ and
$B$ respectively. The spectra are calculated by solving the full
Einstein-Boltzmann system describing the evolution of perturbed fluids
in a particular cosmological model Bond:1984. The rotation and
suppression of polarization along the trajectory will modify the
angular power spectra of the observed CMB photons $C_{\ell}^{\rm XY}\rightarrow\tilde{C}_{\ell}^{\rm XY}$. The mapping for each spectrum is
given by Lue:1998mq; Feng:2004mq
$$\displaystyle\tilde{C}_{\ell}^{EE}$$
$$\displaystyle=$$
$$\displaystyle e^{-2\mu}\,C_{\ell}^{EE}\,\cos^{2}(2\chi)\,,$$
$$\displaystyle\tilde{C}_{\ell}^{BB}$$
$$\displaystyle=$$
$$\displaystyle e^{-2\mu}\,C_{\ell}^{EE}\,\sin^{2}(2\chi)\,,$$
$$\displaystyle\tilde{C}_{\ell}^{TE}$$
$$\displaystyle=$$
$$\displaystyle e^{-\mu}\,C_{\ell}^{TE}\,\cos(2\chi)\,,$$
(15)
$$\displaystyle\tilde{C}_{\ell}^{TB}$$
$$\displaystyle=$$
$$\displaystyle e^{-\mu}\,C_{\ell}^{TB}\,\sin(2\chi)\,,$$
$$\displaystyle\tilde{C}_{\ell}^{EE}$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{2}e^{-2\mu}\,C_{\ell}^{EE}\,\sin(4\chi)\,,$$
where we have assumed no $BB$ contribution to the original spectra (no
primordial gravitational waves). The $TT$ spectrum is not modified as
it is not sensitive to the polarization of the photons.
A number of assumptions are implicit in the simple mapping given in
(Polarization Diffusion from Spacetime Uncertainty). Firstly it assumes that the picture of recombination
(when the polarization is imprinted on the CMB) and reionisation (a
further source of polarization) is not altered in this model. It
assumes the background cosmological evolution is the same for a given
set of cosmological parameters. In fact, polarization on large scales
is also generated after the universe is reionized and this could
introduce a mild scale dependence of the diffusion-rotation effect.
To constrain this scenario with available CMB data we modify the CosmoMC111http://cosmologist.info/cosmomc/ Monte Carlo
Markov Chain (MCMC) package to fit for standard $\Lambda CDM$ model
parameters together with polarization rotation $\chi$ and polarization
depth $\mu$. The standard parameters are: cold dark matter and
baryonic matter physical densities $\Omega_{c}h^{2}$ and $\Omega_{b}h^{2}$,
angular diameter distance measure $\theta$, optical depth to
reionisation $\tau$, and primordial scalar perturbation amplitude
$A_{s}$ and spectral index $n_{s}$. We assume a uniform prior of
sufficient range in each parameter and do not include any primordial
tensor contributions. However we do fit to $TB$, $EB$, and $BB$
data since these are not expected to vanish any longer in the modified
model.
We fit to a combination of data which includes all polarization
sensitive experiments which have reported a detection of the $EE$
power. These are the DASI results Leitch:2005, the final
CBIpol results Sievers:2005, the Boomerang 2003 flight results
Montroy:2006, the WMAP 5-year results Nolta:2009 and the
latest BICEP Chiang:2009 and QUaD quad results. Of
these, the last two contain the highest signal-to-noise determination
of the polarization spectra on scales below a degree. Both BICEP and
QUaD have published $TB$ and $EB$ data which are crucial in
constraining polarization rotation effects
Chiang:2009; Xia:2009. We include the published $TB$ and $EB$
band powers with band power window functions mimicking the published
$TE$ and $EE$ ones. The frequency dependence is accounted for by
scaling the effect for each experiment to a reference frequency of 150
GHz.
The MCMC chains sample the posterior density in the 8-dimensional
parameter space. Once the sampling has converged we integrate the
densities over the standard parameters which gives the marginalized
posterior in $\chi_{150}$ and $\mu_{150}$. We show the result of the
marginalization in the $\chi_{150}$ $\mu_{150}$ plane in
Fig. 1. We plot the 68% and 95% density contours for two
data combinations. The tightest constraints are obtained by the
combination of all polarized data and are consistent with no rotation
and vanishing polarization depth. We also include the results for the
case where the QUaD data is excluded. Although the ‘no QUaD’ result,
driven mainly by the BICEP and WMAP measurements, prefers a non-zero
rotation angle, the indication is weaker than that reported in
Xia:2009. However we do recover their result when not
accounting for the frequency dependence. This may be an indication of
a frequency dependent effect in the data whereby the lower frequency
WMAP data tend favour less rotation given its effect is roughly
twice as large compared to the reference frequency of 150GHz. Future
observations at multiple frequencies will easily determine whether
this is the case.
Table 1 shows the 1-d marginalized constraints on
$\chi_{150}$ and $\mu_{150}$ showing the marginal bias towards a
negative rotation driven mainly by the BICEP and WMAP data. The “All
polarized” result is consistent with no effect.
The model building strategy employed here is based on the assumption
that the fluctuations in dynamical variables due to spacetime
uncertainty are small enough that they result, in the hydrodynamic
approximation, in a continuous, Brownian motion through the state
space. If spacetime uncertainty causes more violent, discontinuous
jumps in the variables, this will have to be modeled by a Boltzmann
equation rather than a diffusion equation. For this reason and also
as data begins to constrain our phenomenology, the need for
microscopic models that can give us a handle on the parameters from
more fundamental physics becomes more acute. For example, an
improvement on point particle models would be to treat particles as
wave packets of a scalar field on a causal set using the discrete
D’Alembertian operators described in
Henson:2006kf; Sorkin:2007qi; Benincasa:2010ac. Assuming
continuity, however, the power of our model is its robustness: if
photons can be modeled as classical particles with polarization
degrees of freedom, then any Lorentz invariant effect of
underlying spacetime uncertainty – whether due to discreteness,
fluctuations, fuzziness, foaminess or whatever – will be of this
form.
Acknowledgements.
We thank Rafael Sorkin for helpful discussions. FD’s research was
supported by EC grant MRTN-CT-2004-005616,
and Royal Society Grant IJP 2006/R2.
LP acknowledges the support of a TEC doctoral scholarship.
FD is grateful to the
Perimeter Institute for Theoretical Physics, Waterloo, Canada, for hospitality
during work on this paper. |
A State-Based Regression Formulation for
Domains with Sensing Actions and Incomplete Information
Le-Chi Tuan\rsupera
\lsupera
GCAS Incorporated, 1531 Grand Avenue, San Marcos, CA 92078, USA
lctuan@gcas.net
,
Chitta Baral\rsuperb
\lsuperb
Computer Science and Engineering,
Arizona State University, Tempe, AZ 85287, USA
chitta@asu.edu
and
Tran Cao Son\rsuperc
\lsuperc
Computer Science Department,
New Mexico State University,
Las Cruces, NM 88003, USA
tson@cs.nmsu.edu
Abstract.
We present a state-based regression function for planning
domains where an agent does not have complete information and may
have sensing actions. We consider binary domains and employ a
three-valued characterization of domains with sensing actions to
define the regression function. We prove the soundness and
completeness of our regression formulation with respect to the
definition of progression. More specifically, we show that (i) a
plan obtained through regression for a planning problem is indeed
a progression solution of that planning problem, and that (ii) for
each plan found through progression, using regression one obtains
that plan or an equivalent one.
Key words and phrases:
Reasoning about Action and Change, Regression, Sensing Actions,
Conditional Planning
1991 Mathematics Subject Classification: I.2.4, I.2.8
\lmcsheading
2 (4:2) 2006
1–34
Jan. 13, 2006
Oct. 01, 2006
1. Introduction and Motivation
An important aspect in reasoning about actions and characterizing
the semantics of action description languages is to define a
transition function that encodes the transition between states due
to actions. This transition function is often viewed as a progression function in that it denotes the progression of the
world by the execution of actions. The ‘opposite’ or ‘inverse’ of
progression is referred to as regression.
Even for a simple case where we have only non-sensing actions
and the progression transition function is deterministic, there are
various formulations of regression. For example, let us consider the
following. Let $\Phi$ be the progression transition function from
actions and states to states. I.e., intuitively, $\Phi(a,s)=s^{\prime}$
means that if the action $a$ is executed in the state $s$ then the
resulting state will be $s^{\prime}$. One way to define a regression
function $\Psi_{1}$ is to define it with respect to states. In that
case $s\in\Psi_{1}(a,s^{\prime})$ will mean that the state $s^{\prime}$ is reached
if $a$ is executed in $s$. Another way to define regression is with
respect to formulas. In that case $\Psi_{2}(a,f)=g$, where $f$ and
$g$ are formulas, means that if $a$ is executed in a state
satisfying $g$ then a state satisfying $f$ will be reached.
For planning using heuristic search, often a different
formulation of regression is given. Typically, a planning problem is
specified by a set of actions, an initial state, and a goal state,
which is a conjunction of literals. As such, regression is often
defined with respect to a set of literals and an action. In that
case the conjunction of literals (the goal) denotes a set of states,
one of which needs to be reached. This regression is slightly
different from $\Psi_{2}$ as the intention is to regress to another
set of literals (not an arbitrary formula), denoting a sub-goal.
With respect to the planning language STRIPS [9], where each
action $a$ has an add list $Add(a)$, a delete list $Del(a)$, and a
precondition list $Prec(a)$, the progression function is defined
as $Progress(s,a)=s+Add(a)-Del(a)$; and the regression
function is defined as $Regress(conj,a)=conj+Prec(a)-Add(a)$, where $conj$ is a set of atoms. Intuitively,
$Regress(conj,a)$ represents a minimal requirement on states
from which the execution of $a$ leads to states satisfying
$conj$. The relation between
these two, formally proven in [16], shows the correctness
of regression based planners; which, through use of
heuristics (e.g. [4, 14]), have done
well in planning competitions. However, the focus of these papers
has been the regression function in domains where agents have
complete knowledge about the world. The following example
shows that this property does not always holds.
Example 1.1.
Consider the following do-or-die story111
This story was brought to us by a participant of
a Texas Action Group (TAG) meeting at Lubbock in 2002 during a discussion
on the need of sensing actions in reasoning about actions
and changes and planning.
:
A wannabe prince faces the last task in
his endeavor. He stands in front of two rooms. In one room is a
tiger and in the other is the princess, whom he wants to marry. Opening
the room with the tiger will result in him being eaten. Otherwise,
he will be able to rescue the princess and will get to marry her. He
does not know exactly in which room the princess is. However, he can
use a specialized222Because the rooms are too close to each
other, the natural smelling ability of a human is not quite
accurate. smell sensor that can precisely tell him where the tiger is.
The story can be formalized as follows. Let us denote the rooms by
$1$ and $2$. $in(t,R)$ (resp. $in(p,R)$) denotes that the tiger
(resp. the princess) is in room $R$. Initially, the agent (i.e., the
want-to-be prince) is $alive$; he does not know what is behind the
door of each room (i.e., the truth value of $in(t,R)$ and $in(p,R)$
is unknown to him) but he knows that the tiger and the princess are
in different rooms (i.e., if $in(t,1)$ is true then $in(p,2)$ is
true, etc.); he can execute $open(1)$ and $open(2)$. Executing the
action $open(R)$, when $in(t,R)$ is true, causes him to be death
($\neg alive$); otherwise, the princess gets rescued. The agent can
determine (by smelling) the truth value of $in(t,1)$ and
$in(t,2)$. If he dies, he can not execute any action.
It is easy to see that the only possible way for the agent to achieve
his goal is to begin by determining where the tiger is (by executing
the action smell); after that, depending on where the tiger is,
he can open the other room to rescue the princess. Observe that this
plan involves the action $smell$ whose execution does not change the
world but changes the knowledge of the agent. Furthermore, the second
action of the plan depends on the knowledge of the agent after the
execution of the first action. We say that the agent needs a conditional plan with sensing actions to achieve his goal.
Reasoning about the effects of actions and changes in the presence
of sensing actions and incomplete information has been the topic of
intensive research (e.g.,
[10, 12, 13, 18, 20] and the discussion in
these papers). In general, the progression function for action
theories with sensing actions and incomplete information is defined
as a mapping from pairs of actions and belief states to belief
states, where each belief state is a set of possible states.
Intuitively, a belief state represents the set of possible states an
agent thinks he might be in given his knowledge about the world. For
example, the initial belief state of the want-to-be-prince in
Example 1.1 consists of every possible state of the world;
and, after the execution of the action smell, his belief state
consists of a single state in which he is alive and knows the
location of the tiger and the princess.
It has been also recognized that the planning problem in domains
with sensing actions and incomplete information has a higher
complexity than the planning problem in domains with complete
information [1]. Furthermore, plans for
achieving a goal in these domains will sometime require sensing
actions and conditionals [10, 20]. It should be noted that
there is an alternative approach to planning in the presence of
incomplete information, called conformant planning, where no sensing
action is used and a plan is a sequence of actions leading to the
goal from every possible initial situation. Example 1.1
indicates that this is inadequate for many planning problems. In the
past, several planners capable of generating conditional plans have
been developed (e.g., [5, 11, 21]) in which some
form of the progression function has been used.
In this regards, two natural questions arise:
$\bullet$
How to define a regression function in the presence of
incomplete information and sensing actions?
$\bullet$
What should be the result of the regression of a
state or a formula over a conditional plan? and, how can it be computed?
In the literature, we can find several proposals addressing the
first question [19, 18, 6, 20], among them
only the proposal in [20] partly discusses the second
one. Moreover, all previous regression functions with respect to
domains with incomplete information and sensing actions are about
regression of formulas.
In this paper we are concerned with domains where the agent
does not have complete information about the world, and may have
sensing actions. For such domains, we define a regression function
with respect to states. We then formally relate our definition of
regression with the earlier notion of progression and show that
planning using our regression function will not only give us correct
plans but also will not miss plans. In summary the main
contributions of our paper are:
$\bullet$
A state-based regression function
for STRIPS domains with sensing actions and incomplete
initial state;
$\bullet$
An extended regression function that allows for
the regression from a (goal) state over a conditional plan;
and
$\bullet$
A formal result showing the soundness
and completeness of our regression function with respect to the
progression function.
The rest of this paper is organized as follows. First, we review
the necessary background information for understanding the
technical details of the paper. We then present the regression
formulation (Section 3)
and prove its soundness and completeness with respect
to the progression function (Section 4).
We relate
our work to other work in regression in Section 5 and conclude in
Section 6.
2. Background
In this section, we present our action and plan representation
and its semantics.
2.1. Action and Plan Representation
We employ a STRIPS-like action representation [9] and
represent a planning problem by a tuple $P=\langle A,O,I,G\rangle$ where $A$ is a finite set of fluents, $O$ is a finite set
of actions, and $I$ and $G$ are sets of fluent literals
where a fluent literal is either a fluent $f\in A$ (a.k.a.
positive fluent literal) or its negation $\neg f$ (a.k.a.
negative fluent literal).
Intuitively, $I$ encodes what is known about the initial state and
$G$ encodes what is desired of a goal state.
An action $a\in O$ is either a non-sensing action or a
sensing action and is defined as follows:
$\bullet$
A non-sensing action $a$ is specified by an expression of the form
$$\textnormal{action }a\hskip 14.454pt\textnormal{:Pre }Pre_{a}\hskip 14.454pt%
\textnormal{:Add }Add_{a}\hskip 14.454pt\textnormal{:Del }Del_{a}$$
where $Pre_{a}$ is a set of fluent literals representing the
precondition for $a$’s execution, $Add_{a}$ and
$Del_{a}$ are two disjoint sets of fluents representing
the positive and negative effects of $a$, respectively; and
$\bullet$
A sensing action $a$ is specified by an expression of the form
$$\textnormal{action }a\hskip 14.454pt\textnormal{:Pre }Pre_{a}\hskip 14.454pt%
\textnormal{:Sense }Sens_{a}$$
where $Pre_{a}$ is a set of fluent literals and $Sens_{a}$ is a subset of
the fluents that do not appear in $Pre_{a}$, .i.e.,
$Pre_{a}\cap(\{\neg f\mid f\in Sens_{a}\}\cup Sens_{a})=\emptyset$.
As with non-sensing actions, a sensing action might only be executed
under certain condition, which is represented by $Pre_{a}$.
Intuitively, $Pre_{a}$ is the condition under which $a$
can be executed, and hence, needs to be known to be true
before the execution
of $a$. On the other hand, $Sens_{a}$ is the set of fluents that
are unknown at the time of execution. For this reason, we require
that none of the fluents in $Sens_{a}$ appear in $Pre_{a}$.
As an example of a sensing action with precondition,
consider the action of looking into
the refrigerator to determine whether there is some beer or not.
This action requires that the refrigerator door is open and
can be represented by the action $look$ with the condition
$Pre_{look}=\{door\_open\}$ and $Sens_{look}=\{beer\_in\_fridge\}$.
The next example shows a simple domain in our representation.
Example 2.1.
Figure (1) shows the actions of
the “Getting to Evanston” domain from [17]
in our representation.
The first four rows of the tables describe different non-sensing
actions (driving actions) with their corresponding preconditions and
add- and delete-effects. Each action can be executed when the agent
is at certain locations (the second column) and changes the location
of the agent after its completion. For instance,
goto-western-at-belmont can be executed if the agent is
at-start; its effect is that the agent will be on-western and
on-belmont (third column) and no longer at-start (fourth
column).
The last two rows represent two sensing actions, neither requires a
precondition; one allows the agent to check for traffic condition (
check-traffic) and the other one ( check-on-western) allows for
the agent to check whether it is ( on-belmont) or not.
The notion of a plan in the presence of incomplete information and
sensing actions has been extensively discussed in the
literature [10, 12, 19, 20]. In this paper,
we consider conditional plans that are formally defined as
follows.
Definition 2.2 (Conditional Plan).
$\bullet$
An empty sequence of actions, denoted by $[\ ]$, is a conditional plan.
$\bullet$
If $a$ is a non-sensing action, then $a$ is a conditional plan.
$\bullet$
If $a$ is a sensing action and
$\varphi_{1},\ldots,\varphi_{n}$ are mutually exclusive
conjunctions of fluent literals
and $c_{1},\ldots,c_{n}$ are conditional plans, then
$$a;case(\varphi_{1}\rightarrow c_{1},\ldots,\varphi_{n}\rightarrow c_{n})$$
is a conditional plan 333We often refer to this type of conditional plans as case
plans
..
$\bullet$
If
$a$ is a non-sensing action and
$c$ is a conditional plan, then $a;c$ is a
conditional plan.
$\bullet$
Nothing else is a conditional plan.
Intuitively, to execute a plan $a;case(\varphi_{1}\rightarrow c_{1},\ldots,\varphi_{n}\rightarrow c_{n})$, first $a$ is executed,
$\varphi_{i}$’s are then evaluated. If one of $\varphi_{i}$ is true
then $c_{i}$ is executed. If none of $\varphi_{i}$ is true then the
plan fails. To execute a plan $a;c$, first $a$ is executed
then $c$ is executed.
Example 2.3 (Getting to Evanston).
The following is a conditional plan:
$\mathit{check\mbox{-}traffic};\\
\hskip 21.681ptcase(\ \\
\hskip 36.135pt\mathit{traffic\mbox{-}bad}\rightarrow\\
\hskip 50.589ptgoto\mbox{-}western\mbox{-}at\mbox{-}belmont;\\
\hskip 50.589pttake\mbox{-}belmont;\\
\hskip 50.589pttake\mbox{-}ashland\\
\hskip 36.135pt\neg\mathit{traffic\mbox{-}bad}\rightarrow\\
\hskip 50.589ptgoto\mbox{-}western\mbox{-}at\mbox{-}belmont;\\
\hskip 50.589pttake\mbox{-}western\\
\hskip 21.681pt)$
2.2. The Progression Function
In the presence of incomplete information, the knowledge of an agent
can be approximately captured by three disjoint sets of fluents: the
set of fluents known to be true, false, and unknown to him,
respectively. Thus, we represent the knowledge of an agent by a pair
$\langle T,F\rangle$, called an approximate state (or
a-state), where $T{\subseteq}A$ and $F{\subseteq}A$ are two
disjoint sets of fluents. Intuitively, $\langle T,F\rangle$
represents the knowledge of an agent
who knows that
fluents in $T$ (resp. $F$) are true (resp. false) and
does not have any knowledge about fluents in $A\setminus(T\cup F)$.
It can also be considered as the
intersection of all belief states satisfying
$T\cup\{\neg f\mid f\in F\}$.
Given a fluent $f$, we
say that $f$ is true (resp. false) in $\sigma$ if $f\in T$ (resp. $f\in F$). $f$ (resp. $\neg f$) holds in $\sigma$
if $f$ is true (resp. false) in $\sigma$. $f$ is known (resp.
unknown) in $\sigma$ if $f\in(T\cup F)$ (resp. $f\not\in(T\cup F)$). A set $L$ of fluent literals holds in an a-state
$\sigma=\langle T,F\rangle$ if every member of $L$ holds in
$\sigma$. A set $X$ of fluents is known in $\sigma$ if every
fluent in $X$ is known in $\sigma$. An action $a$ is executable in $\sigma$ if $Pre_{a}$ holds in $\sigma$.
Furthermore, for two a-states $\sigma_{1}{=}\langle T_{1},F_{1}\rangle$ and $\sigma_{2}{=}\langle T_{2},F_{2}\rangle$:
(1)
We call $\sigma_{1}{\cap}\sigma_{2}{=}\langle T_{1}{\cap}T_{2},F_{1}{\cap}F_{2}\rangle$
the intersection of $\sigma_{1}$ and $\sigma_{2}$.
(2)
We say $\sigma_{1}$ extends $\sigma_{2}$, denoted by $\sigma_{2}{\preceq}\sigma_{1}$ if $T_{2}{\subseteq}T_{1}$ and $F_{2}{\subseteq}F_{1}$.
$\sigma_{1}{\setminus}\sigma_{2}$ denotes the set $(T_{1}{\setminus}T_{2}){\cup}(F_{1}{\setminus}F_{2})$.
(3)
For a set of fluents $X$, we
write $X{\setminus}\langle T,F\rangle$ to denote $X{\setminus}(T{\cup}F)$. To simplify the presentation, for a set of literals
$L$, by $L^{+}$ and $L^{-}$ we denote the set of fluents $\{f\mid f{\in}L,\;f$ is a fluent $\}$ and $\{f\mid\neg f{\in}L,\;f$
is a fluent $\}$.
The transition function (for progression) is defined next.
Definition 2.4 (Transition Function).
For an a-state $\sigma=\langle T,F\rangle$ and an action $a$,
$\Phi(a,\sigma)$ is defined as follows:
$\bullet$
if $a$ is not executable in $\sigma$ then $\Phi(a,\sigma)=\{\bot\}$;
$\bullet$
if $a$ is executable in $\sigma$ and $a$
is a non-sensing action then
$$\Phi(a,\sigma)=\{\langle(T\setminus Del_{a})\cup Add_{a},(F\setminus Add_{a})%
\cup Del_{a}\rangle\};$$
$\bullet$
if $a$ is executable in $\sigma$ and $a$ is a sensing action then
$$\Phi(a,\sigma)=\{\sigma^{\prime}|\sigma\preceq\sigma^{\prime}\textnormal{ and %
}Sens_{a}\setminus\sigma=\sigma^{\prime}\setminus\sigma\}.$$
Here, $\bot$ denotes the “error state.”
$\Phi(a,\sigma)=\{\bot\}$ indicates that the action $a$ cannot be
executed in the a-state $\sigma$.
The next example illustrates the above definition.
Example 2.5 (Getting to Evanston).
Consider the a-state
$$\sigma=\langle\{at\hbox{-}start\},\{on\hbox{-}western,on\hbox{-}belmont,on%
\hbox{-}ashland,at\hbox{-}evanston\}\rangle.$$
We have that $\mathit{check\mbox{-}traffic}$ is executable in $\sigma$
and
$$\Phi(\mathit{check\mbox{-}traffic},\sigma)=\{\sigma_{1},\sigma_{2}\}$$
where:
$\sigma_{1}=\langle\{at\hbox{-}start,\mathit{traffic\hbox{-}bad}\},\{on\hbox{-}%
western,on\hbox{-}belmont,on\hbox{-}ashland,at\hbox{-}evanston\}\rangle$,
$\sigma_{2}=\langle\{at\hbox{-}start\},\{\mathit{traffic\hbox{-}bad},on\hbox{-}%
western,on\hbox{-}belmont,on\hbox{-}ashland,at\hbox{-}evanston\}\rangle$.
Similarly,
$$\Phi(goto\mbox{-}western\mbox{-}at\mbox{-}belmont,\sigma)=\{\sigma_{3}\}$$
where: $\sigma_{3}=\langle\{$on-western,on-belmont$\},\{$at-start, on-ashland,
at-evanston $\}\rangle$.
The function $\Phi$ can be extended to define the function
$\Phi^{*}$ that maps each pair of a conditional plan $p$ and
a-states $\sigma$ into a set of a-states, denoted by
$\Phi^{*}(p,\sigma)$. $\Phi^{*}$ is defined similarly to the extended function
$\hat{\Phi}$ in [20].
Definition 2.6 (Extended Transition Function).
For an a-state $\sigma$,
$\bullet$
if $c=[\ ]$, then $\Phi^{*}([\ ],\sigma)=\{\sigma\}$;
$\bullet$
if $c=a$ and
$a$ is a non-sensing action,
then $\Phi^{*}(c,\sigma)=\Phi(a,\sigma)$;
$\bullet$
if $c=a;case(\varphi_{1}\rightarrow p_{1},\ldots,\varphi_{n}\rightarrow p_{n})$ is a case plan,
then
$$\Phi^{*}(c,\sigma)=\bigcup_{\sigma^{\prime}\in\Phi(a,\sigma)}E(case(\varphi_{1%
}\rightarrow p_{1},\ldots,\varphi_{n}\rightarrow p_{n}),\sigma^{\prime})$$
where
$$E(case(\varphi_{1}\rightarrow p_{1},\ldots,\varphi_{n}\rightarrow p_{n}),%
\gamma)=\left\{\begin{array}[]{ll}\Phi^{*}(p_{j},\gamma),&\hbox{if $\varphi_{j%
}$ holds in $\gamma$ ($1\leq j\leq n$);}\\
\{\bot\},&\hbox{if none of $\varphi_{1},\ldots,\varphi_{n}$ holds in $\gamma$.%
}\\
\end{array}\right.$$
$\bullet$
if $c$ is a conditional plan and $a$ is a non-sensing action, then
$$\Phi^{*}(a;c,\sigma)=\bigcup_{\sigma^{\prime}\in\Phi(a,\sigma)}\Phi^{*}(c,%
\sigma^{\prime}).$$
Furthermore, $\Phi^{*}(c,\bot)=\{\bot\}$ for any conditional plan $c$.
Intuitively, $\Phi^{*}(c,\sigma)$ is the set of a-states
resulting from the execution of the plan $c$ in $\sigma$.
Given a planning problem $P=\langle A,O,I,G\rangle$, the
a-state representing $I$ is defined by $\sigma_{I}=\langle I^{+},I^{-}\rangle$. $\Sigma_{G}=\{\sigma\mid\sigma_{G}\preceq\sigma\}$, where $\sigma_{G}=\langle G^{+},G^{-}\rangle$, is the set
of a-states satisfying the goal $G$. We define a progression solution as
follows.
Definition 2.7 (Progression Solution).
A progression solution
to the planning problem $P$ is a conditional
plan $c$ such that
$\Phi^{*}(c,\sigma_{I})\subseteq\Sigma_{G}$.
Note that, since $\bot$
is not a member of $\Sigma_{G}$, we have that $\bot\not\in\Phi^{*}(c,\sigma_{I})$ if $c$ is a progression solution to $P$.
In other words, the execution of $c$ will not fail if $c$
is a progression solution of $P$.
Example 2.8 (Getting to Evanston - cont’d).
Let $P=\langle A,O,I,G\rangle$
where $A$ and $O$ are given in Figure (1) and
$I=\{at$-$start,\neg on$-$western,\neg on$-$belmont,\neg on$-$ashland,\neg at$-$evanston\}$;
$G=\{at$-$evanston\}$,
respectively. We can easily check that the conditional plan
in Example 2.3 is a progression solution of $P$.
2.3. Some Properties of the Progression Function $\Phi$
There have been several proposals on defining a progression function
for domains with sensing actions and incomplete information
[10, 12, 13, 18, 20]. We will show next that
for domains considered in this paper, the function $\Phi$
(Definition 2.4) is equivalent to the transition function
defined in [20]. By virtue of the equivalent results
between various formalisms, in [20], we can conclude
that $\Phi$ is equivalent to the progression functions defined in
several other formalisms as well. First, let us review the
definition of the function in [20], which will be
denoted by $\Phi_{c}$. We need the following notations. For an action
theory given by a set of fluents $A$, a set of operators $O$, and an
initial state $I$, a state $s$ is a set of fluents. A combined
state (or c-state) of an agent is a pair $\langle s,\Sigma\rangle$ where $s$ is a state and $\Sigma$ is a set of states.
Intuitively, the state $s$ in a c-state $\langle s,\Sigma\rangle$
represents the real state of the world whereas $\Sigma$ is the set
of possible states which an agent believes it might be in. A c-state
$\omega=\langle s,\Sigma\rangle$ is grounded if $s\in\Sigma$.
A fluent $f$ is true (resp. false) in $s$ iff $f\in s$ (resp. $f\not\in s$). $f$ is known to be true (resp. false)
in a c-state $\langle s,\Sigma\rangle$ iff $f$ is true (resp. false) in every state $s^{\prime}\in\Sigma$; and $f$ is known in $\langle s,\Sigma\rangle$, if
$f$ is known to be true or known to be false in $\langle s,\Sigma\rangle$.
For an action $a$ and a state $s$, $a$ is executable in $s$
if $Pre_{a}^{+}\subseteq s$ and $Pre_{a}^{-}\cap s=\emptyset$. The state resulting from executing $a$
in $s$, denoted by $Res(a,s)$, is defined by
$Res(a,s)=(s\setminus Del_{a})\cup Add_{a}$.
The function $\Phi_{c}$ is a mapping from pairs of
actions and c-states into c-states and is defined as
follows. For a c-state $\omega=\langle s,\Sigma\rangle$
and action $a$,
(1)
if $a$ is not
executable in $s$ then $\Phi_{c}(a,\omega)$ is undefined,
denoted by $\Phi_{c}(a,\omega)=\bot$;
(2)
if $a$ is executable in $s$ and $a$ is a non-sensing action,
then
$$\Phi_{c}(a,\omega)=\langle Res(a,s),\{s^{\prime}\ |\ s^{\prime}=Res(a,s^{%
\prime\prime}),\ \exists s^{\prime\prime}\in\Sigma\mbox{ s.t. }a\mbox{ is %
executable in }s^{\prime\prime}\}\rangle;$$
and
(3)
if $a$ is executable in $s$ and $a$ is a
sensing action
then
$$\Phi_{c}(a,\omega)=\langle s,\{s^{\prime}\ |\ s^{\prime}\in\Sigma\mbox{ s.t. }%
Sens_{a}\setminus s=Sens_{a}\setminus s^{\prime},\mbox{ and }a\mbox{ is %
executable in }s^{\prime}\}\rangle.$$
The set of initial states of a planning problem $P=\langle A,O,I,G\rangle$ is $\Sigma_{0}=\{s\mid I^{+}\subseteq s,\textnormal{ and }I^{-}\cap s=\emptyset\}$; and the set of initial c-states of $P$,
denoted by $\Omega_{I}$, is given by $\Omega_{I}=\{\langle s_{0},\Sigma_{0}\rangle\mid s_{0}\in\Sigma_{0}\}$. The function $\Phi_{c}$ can
be extended to define an extended progression function $\Phi^{*}_{c}$
over conditional plans and c-states, similar to the extended
function $\Phi^{*}$ in Definition 2.6. The notion of a
progression solution can then be defined accordingly. The following
theorem states the equivalence between $\Phi$ and $\Phi_{c}$ for
domains representable by the action representation language given in the
previous subsection.
Proposition 2.9.
For a planning problem $\langle A,O,I,G\rangle$, a conditional
plan $c$ is a progression solution with respect to $\Phi$ iff it is
a progression solution with respect to $\Phi_{c}$.
Proof.
For an
a-state $\sigma=\langle T,F\rangle$, let $\Sigma_{\sigma}=\{s\mid T\subseteq s\subseteq A\textnormal{ and }F\cap s=\emptyset\}$, $\Delta_{\sigma}=\{\langle T^{\prime},F^{\prime}\rangle\mid T\subseteq T^{%
\prime}\subseteq A,\>F\subseteq F^{\prime}\subseteq A,\textnormal{ and }T^{%
\prime}\cap F^{\prime}=\emptyset\}$, and $\Omega_{\sigma}=\{\langle s,\Sigma_{\sigma})\rangle\mid s\in\Sigma_{\sigma}\}$.
Furthermore, for an action $a$ and a set of c-states $\Omega$,
let $\widehat{\Phi_{c}}(a,\Omega)=\{\Phi_{c}(a,\omega)\mid\omega\in\Omega\}$. From
the definition of $\Phi$ and $\Phi_{c}$, we can easily verify that the
following properties hold:
(1)
An action $a$ is executable in $\sigma$ iff $a$ is executable in
every c-state belonging to $\Omega_{\sigma}$.
(2)
If a non-sensing action $a$ is executable in $\sigma$ and
$\Phi(a,\sigma)=\{\langle T^{\prime},F^{\prime}\rangle\}$ then
$$T^{\prime}=\bigcap_{u\in\Sigma,\langle s,\Sigma\rangle\in\widehat{\Phi_{c}}(a,%
\Omega_{\sigma})}u$$
and
$$F^{\prime}=\bigcap_{u\in\Sigma,\langle s,\Sigma\rangle\in\widehat{\Phi_{c}}(a,%
\Omega_{\sigma})}(A\setminus u)$$
(3)
If a sensing action $a$ is executable in $\sigma$ then
$$\widehat{\Phi_{c}}(a,\Omega_{\sigma})=\bigcup_{\sigma^{\prime}\in\Delta_{%
\sigma},\>\sigma^{\prime\prime}\in\Phi(a,\sigma^{\prime})}\Sigma_{\sigma^{%
\prime\prime}}$$
The conclusion of the proposition can be verified using induction on
the structure of a plan and the fact that for a planning problem
$P$, $\Omega_{I}=\Omega_{\sigma_{I}}$.∎
Remark 2.10.
For the discussion on the complexity of planning using the
progression function $\Phi$, it will be useful to note that $\Phi$
is also equivalent to the 0-approximation $\Phi_{0}$ defined in
[20]. Indeed, this can be easily verified from the
definitions of $\Phi_{0}$ and $\Phi$. In the notation of this paper,
$\Phi_{0}$ is also a mapping from pairs of actions and a-states into
a-states and for an a-state $\sigma=\langle T,F\rangle$ and an
action $a$, $\Phi_{0}(a,\sigma)$ is defined as follows:
$\bullet$
if $a$ is not executable in $\sigma$ then $\Phi_{0}(a,\sigma)=\{\bot\}$;
$\bullet$
if $a$ is executable in $\sigma$ and $a$
is a non-sensing action then
$$\Phi_{0}(a,\sigma)=\{\langle(T\setminus Del_{a})\cup Add_{a},(F\setminus Add_{%
a})\cup Del_{a}\rangle\};$$
$\bullet$
if $a$ is executable in $\sigma$ and $a$ is a sensing action then
$$\Phi_{0}(a,\sigma)=\{\sigma^{\prime}|\sigma\preceq\sigma^{\prime}\textnormal{ %
and }Sens_{a}\setminus\sigma=\sigma^{\prime}\setminus\sigma\}.$$
This implies that $\Phi$ is identical to $\Phi_{0}$.
3. A State-Based Regression Formulation
In this section, we present our formalization of a regression
function, denoted by ${\mathcal{R}}$, and prove that it is both sound
and complete with respect to the progression function $\Phi$.
${\mathcal{R}}$ is a state-based regression function that maps each pair of
an action and a set of a-states into an a-state.
Observe that our progression formulation states that a plan $p$
achieves the goal $G$ from an a-state $\sigma$ if $G$ holds in all a-states belonging to $\Phi^{*}(p,\sigma)$, i.e., $G$ holds in
$\cap_{\sigma^{\prime}\in\Phi^{*}(p,\sigma)}\sigma^{\prime}$. In addition, similar
to [4], we will also define regression with
respect to the goal. These suggest us to introduce the notion of a
partial knowledge state (or p-state) as a pair $[T,F]$ where
$T\subseteq A$ and $F\subseteq A$ are two disjoint sets of
fluents. Intuitively, a p-state $\delta=[T,F]$ represents a
collection of a-states which extend the a-state $\langle T,F\rangle$. We denote this set by $ext(\delta)$ and call it the
extension set of $\delta$. Formally, $ext(\delta)=\{\langle T^{\prime},F^{\prime}\rangle\mid T\subseteq T^{\prime}%
\subseteq A,F\subseteq F^{\prime}\subseteq A,T^{\prime}\cap F^{\prime}=\emptyset\}$. Any a-state $\sigma^{\prime}\in ext(\delta)$
is called an extension of $\delta$. Given a p-state $\delta{=}[T,F]$, we say a partial state $\delta^{\prime}=[T^{\prime},F^{\prime}]$ is a partial
extension of $\delta$ if $T\subseteq T^{\prime},F\subseteq F^{\prime}$. For a set
of p-states $\Delta=\{\delta_{1},\ldots,\delta_{n}\}$, $\Delta^{\prime}=\{\delta^{\prime}_{1},\ldots,\delta^{\prime}_{n}\}$ is said to be an extension of
$\Delta$, written as $\Delta\sqsubseteq\Delta^{\prime}$ if $\delta^{\prime}_{i}$ is
a partial extension of $\delta_{i}$ for every $i=1,\ldots,n$. For a
fluent $f$, we say that $f$ is true (resp. false, known, unknown) in
$\delta$ if $f\in T$ ($f\in F$, $f\in T\cup F$, $f\not\in T\cup F$). A set of fluents $S$ is said to be true (resp. false,
known, unknown) in $\delta$ if every fluent $f$ in $S$ is true
(resp. false, known, unknown) in $\delta$.
The regression function ${\mathcal{R}}$ will be defined separately for
non-sensing actions and sensing actions to take into consideration
the fact that the execution of a non-sensing action
(resp. sensing action) in an a-state results in a single
a-state (resp. set of a-states). Thereafter, ${\mathcal{R}}$
is extended
to define regression over conditional plans.
The key requirement on $\mathcal{R}$ is that it
should be sound (i.e., plans obtained through
regression must be plans based on the progression function) and complete
(i.e., for each plan based on progression, using regression one
should obtain that plan or a simpler plan with the same effects)
with respect to
progression.
We will also need to characterize the conditions
under which an action should not be used for regression.
Following [4], we refer to this condition
as “the applicability
condition.”
We begin with non-sensing actions.
3.1. Regression Over Non-Sensing Actions
We begin with the
applicability condition of non-sensing actions and then give the
definition of the function ${\mathcal{R}}$ for non-sensing actions.
Definition 3.1 (Regression Applicability Condition – Non-Sensing Action).
Given a non-sensing
action $a$ and a p-state $\delta=[T,F]$. We say that $a$ is
applicable in $\delta$ if
(i)
$Add_{a}\cap T\neq\emptyset$
or $Del_{a}\cap F\neq\emptyset$, and
(ii)
$Add_{a}\cap F=\emptyset$, $Del_{a}\cap T=\emptyset$,
$Pre^{+}_{a}\cap F\subseteq Del_{a}$, and $Pre^{-}_{a}\cap T\subseteq Add_{a}$.
Intuitively, the aforementioned applicability condition requires
that $a$ is relevant (item (i)) and consistent (item
(ii)) in $\delta$. Item (i) is considered “relevant” as it makes
sure that the effects of $a$ will contribute to $\delta$ after
its execution. Item (ii) is considered “consistent” as it makes sure
that the situation obtained by progressing $a$, from a situation
yielded by regressing from $\delta$ through $a$, will be consistent with
$\delta$. Observe also that this definition will exclude the conventional
operator no-op from consideration for regression as it is never
applicable.
Since the application of
a non-sensing action in an a-state results in a single a-state,
the regression of a p-state over a non-sensing action should
result in a p-state.
This is defined next.
Definition 3.2 (Regression – Non-Sensing Action).
Given
a non-sensing action $a$ and a p-state $\delta=[T,F]$,
$\bullet$
if $a$ is not applicable in
$\delta$ then ${\mathcal{R}}(a,\delta)=\bot$;
$\bullet$
if $a$ is applicable in $\delta$ then ${\mathcal{R}}(a,\delta)=[(T\setminus Add_{a})\cup Pre_{a}^{+},(F\setminus Del_%
{a})\cup Pre_{a}^{-}]$.
Like in the progression function, the symbol $\bot$
indicates a “failure.” In other words, ${\mathcal{R}}(a,\delta)=\bot$
means that $\delta$ cannot be regressed on $a$ (or the regression
from $\delta$ over $a$ fails).
For later use, we extend the regression function ${\mathcal{R}}$ for
non-sensing actions over a set of p-states and define
$${\mathcal{R}}(a,\{\delta_{1},\ldots,\delta_{n}\})=\{{\mathcal{R}}(a,\delta_{1}%
),\ldots,{\mathcal{R}}(a,\delta_{n})\}$$
where $\delta_{1},\ldots,\delta_{n}$ are p-states and $a$ is a
non-sensing action.
Example 3.3 (Getting to Evanston - con’t).
The actions $take\mbox{-}western$ and $take\mbox{-}ashland$ are applicable in
$\delta=[\{$at-evanston$\},\{\}]$.
${\mathcal{R}}(take\mbox{-}western,\delta)=[\{on\mbox{-}western\},\{\mathit{%
traffic\mbox{-}bad}\}]$, and
${\mathcal{R}}(take\mbox{-}ashland,\delta)=[\{$on-ashland$\},\{\}]$.
3.2. Regression Over Sensing Actions
Let $a$ be a sensing action and $\sigma$ be an a-state. The
definition of the progression function $\Phi$ states that the
execution of $a$ in $\sigma$ results in a set of a-states
$\Phi(a,\sigma)$. Furthermore, if $a$ is executable in $\sigma$ then
every member of $\Phi(a,\sigma)$ extends $\sigma$ by a set of
fluents $s_{a}\subseteq Sens_{a}$ and every $f\in Sens_{a}\setminus s_{a}$ is known in $\sigma$. As such, the regression over a sensing
action should be with respect to a set of p-states and result in a
p-state. Moreover, our definition for the applicability condition of
a sensing action must account for the fact that the set of p-states,
from which the regression is done, satisfies the two properties:
(i) $Sens_{a}$ is known in each of its members; and (ii)
the difference between two of its members is exactly $Sens_{a}$. This
leads to the following definition.
Definition 3.4 (Sensed Set of Fluents).
Let $\Delta=\{\delta_{1},\ldots,\delta_{n}\}$ be a set of p-states and $a$ be a
sensing action. A sensed set of fluents of $\Delta$ with
respect to $a$, denoted by $p(a,\Delta)$, is a non-empty subset of
$Sens_{a}$ satisfying the following properties:
$\bullet$
$Sens_{a}$ is known in $\Delta$;
$\bullet$
$n=2^{|p(a,\Delta)|}$;
$\bullet$
for every partition444
For a set of fluents $X$, a partition of $X$ is a pair of sets
of fluents $(P,Q)$ where $P\cap Q=\emptyset$ and $P\cup Q=X$.
$(P,Q)$ of $p(a,\Delta)$, there exists only one $\delta_{i}\in\Delta$ ($1\leq i\leq n$) such that $\delta_{i}.T\cap p(a,\Delta)=P,\ \delta_{i}.F\cap p(a,\Delta)=Q$; and
$\bullet$
$\delta_{i}.T\setminus p(a,\Delta)=\delta_{j}.T\setminus p(a,\Delta)$ and $\delta_{i}.F\setminus p(a,\Delta)=\delta_{j}.F\setminus p(a,\Delta)$
for every pair of $i$ and $j$, $1\leq i\leq n$
and $1\leq j\leq n$.
It can be seen from Definition 2.4 (Case 2) that
when a sensing action $a$ is executed in an a-state $\sigma$, the result
is a set of a-states $\Phi(a,\sigma)$ where for
each $\sigma^{\prime}\in\Phi(a,\sigma)$, $\sigma^{\prime}\setminus\sigma=Sens_{a}\setminus\sigma$.
The above definition captures the inverse of the progression process.
Intuitively,
$p(a,\Delta)$ is the set of fluents which are unknown
before the execution of $a$ and are known after its execution;
for example, if $\Delta=\Phi(a,\sigma)$, then $p(a,\Delta)$
should encode the set $Sens_{a}\setminus\sigma$.
It is easy to see that the second condition on $p(a,\Delta)$
warrants that it is a maximal subset of $Sens_{a}$ satisfying the
four stated conditions.
Observe also that due to the second condition, $\Delta$ must be
a non-empty set.
The next lemma proves that the sensed set of a set of p-states
with respect to an action is unique.
Lemma 3.5.
For every sensing action $a$ and set of p-states $\Delta$,
$p(a,\Delta)$ is unique if it exists.
Proof.
Abusing the notation, we write $p(a,\Delta)=\bot$ whenever $p(a,\Delta)$ does not exist. Clearly, the lemma
holds if $\Delta=\emptyset$ as $p(a,\Delta)=\bot$ for every $a$.
So, we need to prove it for the case $\Delta\neq\emptyset$.
Assume that $p(a,\Delta)$ exists but it is not unique, i.e, we can
find different sensed sets of $\Delta$ with respect to $a$, say $X$
and $X^{\prime}$. By Definition 3.4, we have that $X\neq\emptyset$ and $X^{\prime}\neq\emptyset$.
Since $X\neq\emptyset$, let us consider a fluent $f\in X$. By
Definition 3.4, for two partitions $(\{f\},X\setminus\{f\})$ and $(X\setminus\{f\},\{f\})$ of $X$, there
exist $\delta\in\Delta$ and $\delta^{\prime}\in\Delta$ such that $\{f\}=\delta.T\cap X$ and $\{f\}=\delta^{\prime}.F\cap X$, i.e. $f$ is true
in $\delta$ and false in $\delta^{\prime}$. [*].
Suppose that $f\not\in X^{\prime}$. By Item 4, Definition
3.4, either $f\in\delta.T\setminus X^{\prime}$ or
$f\in\delta.F\setminus X^{\prime}$ for $\delta\in\Delta$, i.e $f$ is
either true or false in every $\delta\in\Delta$. In either case,
this contradicts with [*]. Therefore, $f\in X^{\prime}$.
Symmetrically, we can argue that, if $f\in X^{\prime}$ then $f\in X$.
Thus, $f\in X$ iff $f\in X^{\prime}$, i.e. $X=X^{\prime}$.∎
Definition 3.6 (Properness).
A set of p-states $\Delta$
is proper with respect to a sensing action $a$
if $p(a,\Delta)\neq\bot$.
For convenience, we sometime write $p(a,\Delta)=\bot$ to indicate that
$\Delta$ is not proper with respect to $a$.
Example 3.7 (Getting to Evanston – Cond’t).
Consider a
set $\Delta_{1}=\{\delta_{1},\delta_{2}\}$ where $\delta_{1}=[\{at\mbox{-}start,\mathit{traffic\mbox{-}bad}\},\{$on-western, on-belmont,
on-ashland, at-evanston$\}]$ and
$$\delta_{2}=[\{at\mbox{-}start\},\{\mathit{traffic\mbox{-}bad},on\mbox{-}%
western,on\mbox{-}belmont,on\mbox{-}ashland,at\mbox{-}evanston\}].$$
We can easily check that
$$p(\mathit{check\mbox{-}traffic},\Delta_{1})=\{\mathit{traffic\mbox{-}bad}\}.$$
So, $\Delta_{1}$ is proper with respect to
$check\mbox{-}\mathit{traffic}$.
On the other hand
$$p(\mathit{check\mbox{-}traffic},\Delta_{2})=\bot.$$
where $\Delta_{2}=\{\delta_{1},\delta_{3}\}$ and $\delta_{3}=[\{at\mbox{-}start\},\{\mathit{traffic\mbox{-}bad},at\mbox{-}%
evanston\}]$.
This is because the fourth condition (Definition 3.4)
cannot be satisfied for any non-empty subset of $\mathit{traffic\mbox{-}bad}$.
So, $\Delta_{2}$ is not proper with respect to
$check$-$\mathit{traffic}$.
We are now ready to define the applicability condition for
sensing actions. The definition is given in two steps.
First, we define the strong applicability condition as follows.
Definition 3.8 (Strong Regression Applicability Condition – Sensing Action).
Let $a$ be a sensing action and
$\Delta$ be a set of p-states. We
say that $a$ is strongly applicable in $\Delta$ if
(i)
$p(a,\Delta)\neq\bot$; and
(ii)
$Pre^{+}_{a}\cap\delta.F=\emptyset$ and $Pre^{-}_{a}\cap\delta.T=\emptyset$
for every $\delta\in\Delta$.
In the above definition, (i) and (ii) correspond to the “relevancy”
and “consistency” requirement for non-sensing actions (Definition
3.1) respectively.
(i) corresponds to the fact that
executing a sensing action $a$ in an a-state $\sigma$ results in a
set of $2^{|p(a,\Delta)|}$ a-states, each of which extends $\sigma$
by $p(a,\Delta)$ and (ii) guarantees that $a$ must be executable
prior to its execution.
It is easy to see that if a sensing action $a$ is strongly
applicable in $\Delta$, then for every a-state $\sigma^{\prime}$ extending
the p-state
$$\delta^{\prime}=[((\bigcup_{\delta\in\Delta}\delta.T)\setminus p(a,\Delta))%
\cup Pre^{+}_{a},((\bigcup_{\delta\in\Delta}\delta.F)\setminus p(a,\Delta))%
\cup Pre^{-}_{a}]$$
it holds that every member of $\Phi(a,\sigma^{\prime})$ belongs to the
extension of some $\delta_{i}\in\Delta$. As such, $\delta^{\prime}$ could
be viewed as the result of the regression from $\Delta$ through $a$.
Unfortunately, the conditions in Definition
3.8 are sometime unnecessarily strong as the
following example demonstrates.
Example 3.9 (Strong Regression Applicability Condition).
Let
$$P=\langle\{f,g,h\},\{sense_{f},a_{1},a_{2}\},\{h\},\{g\}\rangle$$
be a planning problem, where $sense_{f}$ is a sensing action with
$$Pre_{sense_{f}}=\{h\},\;Sens_{sense_{f}}=\{f\};$$
$a_{1}$ and
$a_{2}$ are two non-sensing actions with
$$Pre_{a_{1}}=\{h,f\},\;Add_{a_{1}}=\{g\},Del_{a_{1}}=\emptyset,$$
$$Pre_{a_{2}}=\{\neg f\},\;Add_{a_{2}}=\{g\},\textnormal{
and }Del_{a_{2}}=\emptyset.$$
Clearly,
$$c=sense_{f};case(f\rightarrow a_{1},\neg f\rightarrow a_{2})$$
is a progression solution to $P$. Thus, it is reasonable to expect
that if we regress from the goal $\delta=[\{g\},\emptyset]$ on
$c$ — step-by-step — we will receive a p-state $\delta^{\prime}$ such that
$\langle\{h\},\emptyset\rangle\in ext(\delta^{\prime})$.
This process begins with the
regression on $a_{1}$ and $a_{2}$ from $\delta$. Thereafter, we
receive a set of p-states from which the regression
on $sense_{f}$ can be done. It is easy to see that ${\mathcal{R}}(a_{1},\delta)=[\{h,f\},\emptyset]=\delta_{1}$ and ${\mathcal{R}}(a_{2},\delta)=[\emptyset,\{f\}]=\delta_{2}$. It is also easy to see that the
strong applicability condition implies that $sense_{f}$ is not
applicable in $\{\delta_{1},\delta_{2}\}$ because $p(sense_{f},\{\delta_{1},\delta_{2}\})=\bot$. This means that, we cannot
regress on $c$ from the goal state.
Notice that the problem in the above example lies in
the fact that $\{\delta_{1},\delta_{2}\}$ violates the properness
definition in that $\delta_{1}$ and $\delta_{2}$
do not have the same values
on the set of fluents that do not belong to $Sens_{sense_{f}}$.
To overcome the problem posed by the strong applicability condition,
we relax this condition.
Definition 3.10 (Regression Applicability Condition – Sensing Action).
Let $a$ be a sensing action and $\Delta$ be a set of p-states. $a$ is
applicable in $\Delta$ if
(i)
there exists a set of p-states $\Delta^{\prime}$ such that
$\Delta\sqsubseteq\Delta^{\prime}$ and $a$
is strongly applicable in $\Delta^{\prime}$; and
(ii)
$Sens_{a}$ is known in $\Delta$.
Example 3.11 (Continuation of Example 3.9).
It is easy to see that $sense_{f}$ is applicable in $\{\delta_{1},\delta_{2}\}$
since it is strongly applicable in $\{\delta_{1},\delta_{2}^{\prime}\}$
where $\delta_{2}^{\prime}=[\{h\},\{f\}]$ and
$\{\delta_{1},\delta_{2}\}\sqsubseteq\{\delta_{1},\delta_{2}^{\prime}\}$.
Definition 3.12 (Sensed Set).
Let $\Delta$ be a set of p-states and $a$ be a sensing action
such that $a$ is applicable in $\Delta$. We say that $X$ is
a sensed set of fluents of $\Delta$ with respect to $a$,
denoted by $S_{a,\Delta}$,
if there exists a set of p-states $\Delta^{\prime}$ such that
$\Delta\sqsubseteq\Delta^{\prime}$,
$a$ is strongly applicable in $\Delta^{\prime}$, and
$X=p(a,\Delta^{\prime})$.
Again, we write $S_{a,\Delta}=\bot$ to say that
the sensed set of fluents of $\Delta$ with respect to $a$
does not exist.
The next lemma states that $S_{a,\Delta}$ is unique.
Lemma 3.13.
For every sensing action $a$ and set of p-states $\Delta$,
$S_{a,\Delta}$ is unique if it exists.
Proof.
Obviously, the lemma holds for
$\Delta=\emptyset$. So, we need to prove it for the case $\Delta\neq\emptyset$.
Assume the contrary, $S_{a,\Delta}$ is not unique. This implies that
there exists $\Delta^{\prime}$ and $\Delta^{\prime\prime}$ such that $\Delta\sqsubseteq\Delta^{\prime}$ and $\Delta\sqsubseteq\Delta^{\prime\prime}$, $p(a,\Delta^{\prime})\neq\bot$, $p(a,\Delta^{\prime\prime})\neq\bot$, and $p(a,\Delta^{\prime})\neq p(a,\Delta^{\prime\prime})$. Again, by Definition 3.4 we
can conclude that $p(a,\Delta^{\prime})\neq\emptyset$ and $p(a,\Delta^{\prime\prime})\neq\emptyset$. Without loss of generality, we conclude that there
exists some $f\in p(a,\Delta^{\prime})\setminus p(a,\Delta^{\prime\prime})$.
Since $p(a,\Delta^{\prime\prime})\neq\bot$, $f\in Sens_{a}$, and $f$ is known in
$\Delta^{\prime\prime}$, by Definition 3.4, we must have
two cases.
(1)
$f\in{\delta}^{\prime\prime}.T\setminus p(a,\Delta^{\prime\prime})$ for every $\delta^{\prime\prime}\in\Delta^{\prime\prime}$. Since $f\in p(a,\Delta^{\prime})$, by Definition
3.4, for the partition $(p(a,\Delta^{\prime})\setminus\{f\},\{f\})$ of $p(a,\Delta^{\prime})$, there exists $\delta^{\prime}\in\Delta^{\prime}$ such that $\delta^{\prime}.F\cap p(a,\Delta^{\prime})=\{f\}$, i.e.
$f$ is false in $\delta^{\prime}$.
Because $\Delta\sqsubseteq\Delta^{\prime}$, there exists some $\delta\in\Delta$ such that $\delta^{\prime}$ is a partial extension of $\delta$. So,
we have that $\delta.F\subseteq\delta^{\prime}.F$.
Also, as $Sens_{a}$ is known in $\delta$, we must have that $f\in\delta.F$.
Since $\Delta\sqsubseteq\Delta^{\prime\prime}$, we know that there exists some
${\delta}^{\prime\prime}\in\Delta^{\prime\prime}$ which is a partial extension of $\delta$.
This implies that $\delta.F\subseteq{\delta}^{\prime\prime}.F$, i.e., $f\in{\delta}^{\prime\prime}.F$. This contradicts with the fact that $f\in{\delta}^{\prime\prime}.T$.
(2)
$f\in{\delta}^{\prime\prime}.F\setminus p(a,\Delta^{\prime\prime})$ for
every $\delta^{\prime\prime}\in\Delta^{\prime\prime}$. Similarly to the first case, we can
derive a contradiction.
The above two cases show that if $f\in p(a,\Delta^{\prime})$ then $f\in p(a,\Delta^{\prime\prime})$.
This shows that
$p(a,\Delta^{\prime})=p(a,\Delta^{\prime\prime})$.∎
We illustrate the above definition in the next example.
Example 3.14 (Getting to Evanston - con’t).
Consider the
set $\Delta_{2}$ and the sensing action $\mathit{check\mbox{-}traffic}$ in Example
3.7. We have that
(i)
$\mathit{check\mbox{-}traffic}$ is not strongly applicable in $\Delta_{2}$
(because $p(\mathit{check\mbox{-}traffic},\Delta_{2})=\bot$, Example 3.7);
however,
(ii)
$\mathit{check\mbox{-}traffic}$ is applicable in $\Delta_{2}$.
This is because $\Delta_{1}$ (Example 3.7) consists of partial
extensions of p-states in $\Delta_{2}$, and
$\mathit{check\mbox{-}traffic}$ is
strongly applicable in $\Delta_{1}$.
We are now ready to define the regression function for sensing actions.
Definition 3.15 (Regression – Sensing Action).
Let $a$ be a sensing action and $\Delta$ be a set of p-states.
$\bullet$
if $a$ is not applicable in
$\Delta$ then ${\mathcal{R}}(a,\Delta)=\bot$; and
$\bullet$
if $a$ is applicable in
$\Delta$
$${\mathcal{R}}(a,\Delta)=[((\bigcup_{\delta\in\Delta}\delta.T)\setminus S_{a,%
\Delta})\cup Pre^{+}_{a},((\bigcup_{\delta\in\Delta}\delta.F)\setminus S_{a,%
\Delta})\cup Pre^{-}_{a}].$$
Example 3.16 (Getting to Evanston – Cond’t).
The action $check\mbox{-}\textnormal{\em traffic}$ is applicable in
$\Delta_{2}$ with respect to $\{\textnormal{\em traffic}\mbox{-}bad\}$
(see Example 3.14) and we have
${\mathcal{R}}(check\mbox{-}\textnormal{\em traffic},\Delta_{2})=$
$[\{at\mbox{-}start\},\{on\mbox{-}western,on\mbox{-}belmont,on\mbox{-}ashland,%
at\mbox{-}evanston\}].$
3.3. Regression Over Conditional Plans
We now extend
${\mathcal{R}}$ to define ${\mathcal{R}}^{*}$ that allows us to perform
regression over conditional plans. For a conjunction of fluent
literals, by $\varphi^{+}$ and
$\varphi^{-}$ we denote the sets of fluents occurring positively
and negatively in $\varphi$, respectively.
Definition 3.17 (Extended Regression Function).
Let $\delta$ be a p-state. The extended transition function
${\mathcal{R}}^{*}$ is defined as follows:
$\bullet$
${\mathcal{R}}^{*}([\ ],\delta)=\delta$.
$\bullet$
For a non-sensing action $a$, ${\mathcal{R}}^{*}(a,\delta)$ = ${\mathcal{R}}(a,\delta)$.
$\bullet$
For a conditional plan $p=a;case(\varphi_{1}{\rightarrow}c_{1},\ldots,\varphi_{n}{\rightarrow}c_{n})$
where $a$ is a sensing action and $c_{i}$’s are conditional
plans,
–
if ${\mathcal{R}}^{*}(c_{i},\delta){=}\bot$ for some $i$, ${\mathcal{R}}^{*}(p,\delta)=\bot$;
–
if ${\mathcal{R}}^{*}(c_{i},\delta){=}[T_{i},F_{i}]$
for $i=1,\ldots,n$, then
$${\mathcal{R}}^{*}(p,\delta)={\mathcal{R}}(a,\{R(\varphi_{1}\rightarrow c_{1},%
\delta),\ldots,R(\varphi_{n}\rightarrow c_{n},\delta)\})$$
where $R(\varphi_{i}\rightarrow c_{i},\delta)=[T_{i}\cup\varphi_{i}^{+},F_{i}\cup%
\varphi_{i}^{-}]$ if $\varphi_{i}^{+}\cap F_{i}=\emptyset$ and $\varphi_{i}^{-}\cap T_{i}=\emptyset$; otherwise,
$R(\varphi_{i}\rightarrow c_{i},\delta)=\bot$.
$\bullet$
For $p=a;c$, where $a$ is a non-sensing action and $c$ is
a conditional plan,
$${\mathcal{R}}^{*}(p,\delta)={\mathcal{R}}(a,{\mathcal{R}}^{*}(c,\delta));$$
$\bullet$
${\mathcal{R}}^{*}(p,\perp)=\perp$ for
every plan $p$.
The notion of a regression solution is defined as follows.
Definition 3.18 (Regression Solution).
A conditional plan $c$ is a
regression solution to the planning problem $P=\langle A,O,I,G\rangle$ if ${\mathcal{R}}^{*}(c,\delta_{G})\neq\bot$ and
$\sigma_{I}\in ext({\mathcal{R}}^{*}(c,\delta_{G}))$ where $\delta_{G}=[G^{+},G^{-}]$ and $\sigma_{I}=\langle I^{+},I^{-}\rangle$.
The above definition is a generalization of the notion of a plan
obtained by regression in domains without sensing actions and
with complete information about the initial state to domains with
sensing actions and incomplete information. An important property
that any regression function needs to satisfy is its soundness with
respect to the corresponding progression function. Here, we would
like to guarantee that $\mathcal{R}$ and $\mathcal{R}^{*}$ are sound
with respect to the progression function $\Phi$ and $\Phi^{*}$, respectively. As
such, we require that a regression solution $c$ to a planning
problem $P=\langle A,O,I,G\rangle$ be a plan achieving the goal
$G$ from $I$. This property is proved in Theorem
4.8.
As the soundness of the regression function with respect to the
progression function is guaranteed, it will be interesting to
investigate its completeness. In this paper, we opt for a definition
that — when used in planning — will give us optimal solutions in
the sense that regression solutions do not contain redundant
actions. This is evident from Definitions 3.1 and
3.10 in which we require that the action, over which
the regression is done, must add new information to the regressed
state. We will elaborate in more detail on this point in the next
section.
4. Soundness and Completeness Results
In this section, we show that our regression function ${\mathcal{R}}^{*}$
is sound and complete with respect to the progression function
$\Phi$.
4.1. Soundness Result
As with its definition, the soundness of ${\mathcal{R}}^{*}$
is proved in two steps. First, we prove the soundness of $\mathcal{R}$,
separately for non-sensing and sensing actions.
Second, we extend this result to regression solutions.
To establish the soundness of ${\mathcal{R}}$
on non-sensing actions, we need the following lemma.
Lemma 4.1.
Let $\delta$ be a p-state. An a-state
$\sigma$ is an extension of $\delta$ (i.e. $\sigma\in ext(\delta)$)
iff $\sigma$ is an a-state of the form $\langle\delta.T\cup X,\delta.F\cup Y\rangle$ where $X,Y$ are two disjoint sets of
fluents and $X\cap\delta.F=\emptyset,\ Y\cap\delta.T=\emptyset$.
Proof.
$\bullet$
Case “$\Rightarrow$”:
Let $\sigma\in ext(\delta)$ be an extension of $\delta$. By
the definition of an extension, $\sigma$ is an a-state where $\delta.T\subseteq\sigma.T$ and $\delta.F\subseteq\sigma.F$. Denote $X=\sigma.T\setminus\delta.T$ and $Y=\sigma.F\setminus\delta.F$. Clearly, $X$
and $Y$ are two set of fluents where $X\cap Y=\emptyset$ and
$X\cap\delta.F=\emptyset,\ Y\cap\delta.T=\emptyset$.
$\bullet$
Case “$\Leftarrow$”:
Let $\sigma$ be an a-state of the form $\langle\delta.T\cup X,\delta.F\cup Y\rangle$ where $X,Y$ are two disjoint
sets of fluents and $X\cap\delta.F=\emptyset,\ Y\cap\delta.T=\emptyset$.
It’s easy to see that $\sigma.T\cap\sigma.F=\emptyset$,
i.e. $\sigma$ is consistent. Furthermore, $\delta.T\subseteq\sigma.T$ and $\delta.F\subseteq\sigma.F$, i.e. by definition
of an extension, $\sigma$ is an extension of $\delta$.∎
Intuitively, the soundness of ${\mathcal{R}}$ for a non-sensing action
states that the regression over a
non-sensing action from a p-state yields another p-state such that
the execution of the action in any extension of the latter results
in a subset of a-states belonging to the extension set of the
former. This is illustrated in Figure
2.
Theorem 4.2 (Non-sensing Action).
Let $\delta$ be a p-state and $a$ be a non-sensing action. If
${\mathcal{R}}(a,\delta)=\delta^{\prime}$ and $\delta^{\prime}\neq\bot$, then for
every $\sigma^{\prime\prime}\in ext(\delta^{\prime})$ we have that $\Phi(a,\sigma^{\prime\prime})\subseteq ext(\delta)$.
Proof.
Let $\delta=[T,F]$. From the fact that ${\mathcal{R}}(a,\delta)=\delta^{\prime}\neq\bot$, we have that $a$ is applicable in $\delta$.
By Definition 3.2,
$$\delta^{\prime}={\mathcal{R}}(\delta,a)=[(T\setminus Add_{a})\cup Pre^{+}_{a},%
(F\setminus Del_{a})\cup Pre^{-}_{a}].$$
Let $\sigma^{\prime\prime}\in ext(\delta^{\prime})$. It follows from Lemma
4.1 that
$$\sigma^{\prime\prime}=\langle(T\setminus Add_{a})\cup Pre^{+}_{a}\cup X,(F%
\setminus Del_{a})\cup Pre^{-}_{a}\cup Y\rangle,$$
where $X$ and $Y$ are two sets of fluents such that $\sigma^{\prime\prime}.T\cap\sigma^{\prime\prime}.F=\emptyset$. We now prove that (i) $a$ is executable in
$\sigma^{\prime\prime}$ and (ii) $\Phi(a,\sigma^{\prime\prime})\subseteq ext(\delta)$.
$\bullet$
Proof of (i):
Since $Pre^{+}_{a}\subseteq\sigma^{\prime\prime}.T$ and $Pre^{-}_{a}\subseteq\sigma^{\prime\prime}.F$, we conclude that ${lem1-maintext}a$ is executable in $\sigma^{\prime\prime}$.
$\bullet$
Proof of (ii):
By definition of the transition function $\Phi$, we have that
$$\Phi(a,\sigma^{\prime\prime})=\{\langle(((T\setminus Add_{a})\cup Pre^{+}_{a}%
\cup X)\setminus Del_{a})\cup Add_{a},(((F\setminus Del_{a})\cup Pre^{-}_{a}%
\cup Y)\setminus Add_{a})\cup Del_{a}\rangle\}$$
Since $a$ is applicable in $\delta$, we have that $T\cap Del_{a}=\emptyset$, $F\cap Add_{a}=\emptyset$. Furthermore, $Del_{a}\cap Add_{a}=\emptyset$. Therefore, we have that $(((T\setminus Add_{a})\cup Pre^{+}_{a}\cup X)\setminus Del_{a})\cup Add_{a}=((%
T\setminus Add_{a})\cup((Pre^{+}_{a}\cup X)\setminus Del_{a}))\cup Add_{a}%
\supseteq T\cup((Pre^{+}_{a}\cup X)\setminus Del_{a})\supseteq T$. This
concludes that $T\subseteq\Phi(a,\sigma^{\prime\prime}).T$. Similarly, we have
that $F\subseteq\Phi(a,\sigma^{\prime\prime}).F$. This shows that $\Phi(a,\sigma^{\prime\prime})\subseteq ext(\delta)$.∎
Observe that the conclusion of the theorem indicates that
$\bot\not\in\Phi(a,\sigma^{\prime\prime})$, i.e., $a$ is executable in
$\sigma^{\prime\prime}$. This shows that ${\mathcal{R}}$ can be “reversed” for
non-sensing actions.
We will next establish a result similar to Theorem 4.2
for sensing actions. Intuitively, the result should state that
the regression over a
sensing action from a set of p-states yields a p-state such that the
execution of the action in any extension of the latter results in
a set of a-states belonging to the union of the extension sets
of the former, i.e., it should allow us to conclude that ${\mathcal{R}}$ can be “reversed”
for sensing actions. Figure 3 illustrates this idea.
We need the following lemma.
Lemma 4.3.
Let $\sigma^{\prime}$ be an a-state and $a$ be a
sensing action executable in $\sigma^{\prime}$. For any $S_{a}\subseteq Sens_{a}$ and $\sigma\in\Phi(a,\sigma^{\prime})$, let $\sigma.T\cap S_{a}=S^{+}_{\sigma}$ and $\sigma.F\cap S_{a}=S^{-}_{\sigma}$, we have that
$S^{+}_{\sigma}\cup S^{-}_{\sigma}=S_{a}$ and $S^{+}_{\sigma}\cap S^{-}_{\sigma}=\emptyset$.
Proof.
It is easy to see that the lemma is correct
for the case $S_{a}=\emptyset$. Let us consider the case $S_{a}\neq\emptyset$. Since $S^{+}_{\sigma}\subseteq\sigma.T$ and $S^{-}_{\sigma}\subseteq\sigma.F$, we have that $S^{+}_{\sigma}\cap S^{-}_{\sigma}=\emptyset$.
. This also shows
Consider $f\in S^{+}_{\sigma}\cup S^{-}_{\sigma}$, we have that $f\in S^{+}_{\sigma}$ or $f\in S^{-}_{\sigma}$. In both cases, we have $f\in S_{a}$.
Consider $f\in S_{a}$. Since $S_{a}\subseteq Sens_{a}$, we have that $f\in Sens_{a}$. By the definition of $\Phi$, we have that $f\in\sigma.T$ or $f\in\sigma.F$. From this fact, it’s easy to see
that $f\in S^{+}_{\sigma}$ or $f\in S^{-}_{\sigma}$.∎
With the help of the above lemma, we can prove the following theorem.
Theorem 4.4 (Sensing action).
Let
$\Delta$ be a set of p-states and $a$ be a sensing action. If
${\mathcal{R}}(a,\Delta)=\delta^{\prime}$ and $\delta^{\prime}\neq\bot$, then for
every $\sigma^{\prime\prime}\in ext(\delta^{\prime})$, we have that $\Phi(a,\sigma^{\prime\prime})\subseteq\bigcup_{\delta\in\Delta}ext(\delta)$.
Proof.
From the fact
that ${\mathcal{R}}(a,\Delta)=\delta^{\prime}\neq\bot$, we have that $a$
is applicable in $\Delta$ with respect to some set $S_{a,\Delta}\subseteq Sens_{a}$ ($S_{a,\Delta}\neq\emptyset$). By Definition
3.15 we have:
$$\delta^{\prime}={\mathcal{R}}(a,\Delta)=[(\bigcup_{\delta\in\Delta}\delta.T%
\setminus S_{a,\Delta})\cup Pre^{+}_{a},(\bigcup_{\delta\in\Delta}\delta.F%
\setminus S_{a,\Delta})\cup Pre^{-}_{a}].$$
Let $\sigma^{\prime\prime}\in ext(\delta^{\prime})$ be an arbitrary extension of
$\delta^{\prime}$. We will now prove (i) $a$ is executable in $\sigma^{\prime\prime}$ and
(ii) $\Phi(a,\sigma^{\prime\prime})\subseteq\bigcup_{\delta\in\Delta}ext(\delta)$.
$\bullet$
Proof of (i):
It follows from Lemma 4.1 that
$$\sigma^{\prime\prime}=\langle(\bigcup_{\delta\in\Delta}\delta.T\setminus S_{a,%
\Delta})\cup Pre^{+}_{a}\cup X,(\bigcup_{\delta\in\Delta}\delta.F\setminus S_{%
a,\Delta})\cup Pre^{-}_{a}\cup Y\rangle$$
where $X$ and $Y$ are two sets of fluents such that $\sigma^{\prime\prime}.T\cap\sigma^{\prime\prime}.F=\emptyset$.
From the fact that $Pre^{+}_{a}\subseteq\sigma^{\prime\prime}.T$ and
$Pre^{-}_{a}\subseteq\sigma^{\prime\prime}.F$, we conclude that $a$ is executable in
$\sigma^{\prime\prime}$. [*]
$\bullet$
Proof of (ii):
Consider an arbitrary $\sigma\in\Phi(a,\sigma^{\prime\prime})$. We need to
prove that there exists some $\delta\in\Delta$ such that $\sigma\in ext(\delta)$.
Since $a$ is applicable in $\Delta$, by Definition
3.10, there exists $\Delta^{\prime}$ such that $\Delta\sqsubseteq\Delta^{\prime}$ and $a$ is strongly applicable in $\Delta^{\prime}$ and
$S_{a,\Delta}=p(a,\Delta^{\prime})$.
Let $S^{+}_{\sigma}=\sigma.T\cap S_{a,\Delta}$ and $S^{-}_{\sigma}=\sigma.F\cap S_{a,\Delta}$. By Lemma 4.3, we have that
$S^{+}_{\sigma}\cup S^{-}_{\sigma}=S_{a,\Delta}$ and $S^{+}_{\sigma}\cap S^{-}_{\sigma}=\emptyset$.
By Definition 3.4, there exists $\delta^{\prime}\in\Delta^{\prime}$ such that $\delta^{\prime}.T\cap S_{a,\Delta}=S^{+}_{\sigma}$ and
$\delta^{\prime}.F\cap S_{a,\Delta}=S^{-}_{\sigma}$. Because $\Delta\sqsubseteq\Delta^{\prime}$, there exists some $\delta\in\Delta$ such
that $\delta^{\prime}$ is a partial extension of $\delta$.
We will show that $\sigma\in ext(\delta)$, i.e., $\delta.T\subseteq\sigma.T$ and $\delta.F\subseteq\sigma.F$.
Since $\delta.T\subseteq\delta^{\prime}.T$, we have that $\delta.T\cap S_{a,\Delta}\subseteq\delta^{\prime}.T\cap S_{a,\Delta}=S^{+}_{\sigma}$.
Therefore:
$$\begin{array}[]{lll}\delta.T&=&(\delta.T\setminus(\delta.T\cap S_{a,\Delta}))%
\cup(\delta.T\cap S_{a,\Delta})\\
&=&(\delta.T\setminus S_{a,\Delta})\cup(\delta.T\cap S_{a,\Delta})\\
&\subseteq&(\delta.T\setminus S_{a,\Delta})\cup S^{+}_{\sigma}.\end{array}$$
Similarly, we can show that $\delta.F\subseteq(\delta.F\setminus S_{a,\Delta})\cup S^{-}_{\sigma}$.
Since $\sigma\in\Phi(a,\sigma^{\prime\prime})$, by the definition of $\Phi$,
we have that $\sigma^{\prime\prime}.T\subseteq\sigma.T$. Let $\sigma.T\setminus\sigma^{\prime\prime}.T=\omega$, we have that
$$\sigma.T=\sigma^{\prime\prime}.T\cup\omega=((\bigcup_{\delta\in\Delta}\delta.T%
)\setminus S_{a,\Delta})\cup Pre^{+}_{a}\cup X\cup\omega.$$
Since $\sigma.T\cap S_{a,\Delta}=S^{+}_{\sigma}$ and
$(((\bigcup_{\delta\in\Delta}\delta.T)\setminus S_{a,\Delta})\cup Pre^{+}_{a})%
\cap S_{a,\Delta}=\emptyset$ (because $Sens_{a}\cap Pre^{+}_{a}=\emptyset$), we must have that $(X\cup\omega)\cap S_{a,\Delta}=S^{+}_{\sigma}$, i.e. $S^{+}_{\sigma}\subseteq X\cup\omega$. From the fact that $\delta.T\subseteq(\delta.T\setminus S_{a,\Delta})\cup S^{+}_{\sigma}$ and $S^{+}_{\sigma}\subseteq X\cup\omega$, it is easy to see that $\delta.T\subseteq\sigma.T$.
Similarly, we can show that $\delta.F\subseteq\sigma.F$. From this
fact, we conclude that $\sigma\in ext(\delta)$. [**]
From [*] and [**] the theorem is proved.∎
To prove the final result about the correctness of ${\mathcal{R}}$
(and Theorem 4.27 in the next section) we need a number of
additional notations and definitions.
Definition 4.5 (Branching Count).
Let $c$ be a conditional plan,
we define the number of case plans of $c$, denoted by $count(c)$,
inductively as follows:
(1)
if $c=[\ ]$ then $count(c)=0$;
(2)
if $c=a$, $a$ is a non-sensing action, then $count(c)=0$;
(3)
if $a$ is a non-sensing action and $c$ is a conditional plan
then $count(a;c)=count(c);$
(4)
if $c$ is a case plan of the form
a; case($\varphi_{1}\rightarrow c_{1},\ldots,\varphi_{n}\rightarrow c_{n}$)
where $a$ is a sensing action, then
$count(c)=1+\sum^{n}_{i=1}count(c_{i}).$
Lemma 4.6 (Sequence of Non-sensing Action).
For p-states
$\delta$ and $\delta^{\prime}$, and a sequence of non-sensing actions $c=a_{1};\ldots;a_{n}$ $(n\geq 0)$, ${\mathcal{R}}^{*}(c,\delta)=\delta^{\prime}\neq\bot$ implies that $\Phi^{*}(c,\sigma^{\prime\prime})\subseteq ext(\delta)$
for every $\sigma^{\prime\prime}\in ext(\delta^{\prime})$.
Proof.
By induction on $n$.
$\bullet$
Base Case: $n=1$.
This means that $c$ has only one action $a$. Using
Theorem 4.2, and Definition 3.17 – item 2 – the base case is
proved. Notice that for the case $n=0$, i.e. $c=[\ ]$,
the lemma follows directly from Definitions 3.17
and 2.6.
$\bullet$
Inductive Step:
Assume that the lemma is shown for $1\leq n\leq k$.
We now prove the lemma for $n=k+1$.
Let $c^{\prime}=a_{2};\ldots;a_{k+1}$ and ${\mathcal{R}}^{*}(c^{\prime},\delta)=\delta^{*}$.
By Definition 3.17
$${\mathcal{R}}^{*}(c,\delta)={\mathcal{R}}(a_{1},{\mathcal{R}}^{*}(c^{\prime},%
\delta))=\delta^{\prime}.$$
Since ${\mathcal{R}}(a_{1},\delta^{*})=\delta^{\prime}\neq\bot$,
we have that $\delta^{*}\neq\bot$.
Let $\sigma^{\prime\prime}\in ext(\delta^{\prime})$.
By Theorem 4.2, we have
that
$\Phi(a_{1},\sigma^{\prime\prime})=\{\sigma\}\subseteq ext(\delta^{*})$, i.e., $\sigma\in ext(\delta^{*})$.
By the definition of $\Phi^{*}$, we also have
that $\Phi^{*}(c,\sigma^{\prime\prime})=\Phi^{*}(c^{\prime},\Phi^{*}(a_{1},\sigma^{%
\prime\prime}))$.
Using the induction hypothesis for $c^{\prime}$,
where ${\mathcal{R}}^{*}(c^{\prime},\delta)=\delta^{*}$ and $\sigma\in ext(\delta^{*})$, we have:
$$\Phi^{*}(c^{\prime},\Phi^{*}(a_{1},\sigma^{\prime\prime}))=\Phi^{*}(c^{\prime}%
,\sigma)\subseteq ext(\delta).$$
Therefore,
$\Phi^{*}(c,\sigma^{\prime\prime})\subseteq ext(\delta)$.∎
Lemma 4.7.
Let $\delta$ be a p-state and $c$ be a
conditional plan. If ${\mathcal{R}}^{*}(c,\delta)=\delta^{\prime}$ and
$\delta^{\prime}\neq\bot$, then for every $\sigma\in ext(\delta^{\prime})$,
$\Phi^{*}(c,\sigma)\subseteq ext(\delta)$.
Proof.
By induction on $count(c)$, the number of
case plans in $c$.
$\bullet$
Base Case: $count(c)=0$. Then $c$ is a sequence of
non-sensing actions. The base case follows from Lemma
4.6.
$\bullet$
Inductive Step: Assume that we have proved the
lemma for $count(c)\leq k$ ($k\geq 0$). We need to prove the lemma for
$count(c)=k+1$. By the definition of a conditional plan, we
have two cases:
(1)
$c=a;p$ is a case plan where $a$ is a sensing action and
$p=case\ (\varphi_{1}\rightarrow p_{1},\ldots,\varphi_{m}\rightarrow p_{m})$.
Since
$count(c)=1+\sum^{m}_{j=1}count(p_{j})\leq k+1$, we have that $count(p_{i})\leq k$ for $i=1,\ldots,m$.
By Definition 3.17,
$$\bot\neq\delta^{\prime}={\mathcal{R}}^{*}(c,\delta)={\mathcal{R}}(a,\{R(%
\varphi_{1}\rightarrow p_{1},\delta),\ldots,R(\varphi_{m}\rightarrow p_{m},%
\delta)\}).$$
Let us denote $R(\varphi_{i}\rightarrow p_{i},\delta)$
by $\delta_{i}$ ($1\leq i\leq m$) and $\Delta=\{\delta_{1},\ldots,\delta_{m}\}$.
We have that $\delta_{i}\models\varphi_{i}$ for $1\leq i\leq m$, and $a$ is applicable in $\Delta$.
From Theorem 4.4, we
have that
$$\Phi(a,\sigma)\subseteq\bigcup_{\delta^{\prime\prime}\in\Delta}ext(\delta^{%
\prime\prime})$$
for every $\sigma\in ext(\delta^{\prime})$.
Consider an arbitrary $\sigma^{\prime}\in\Phi(a,\sigma)$.
Because of the above relation, we can conclude that there exists
some $i$, $1\leq i\leq m$, such that $\sigma^{\prime}\in ext(\delta_{i})$.
Because ${\mathcal{R}}^{*}(p_{i},\delta).T\subseteq R(\varphi_{i}\rightarrow p_{i},%
\delta).T$ and
${\mathcal{R}}^{*}(p_{i},\delta).F\subseteq R(\varphi_{i}\rightarrow p_{i},%
\delta).F$,
${\sigma}^{\prime}\in ext(\delta_{i})$ implies
${\sigma}^{\prime}\in ext({\mathcal{R}}^{*}(p_{i},\delta))$.
Using inductive hypothesis for $count(p_{i})\leq k$, we have that
$\Phi^{*}(p_{i},{\sigma}^{\prime})\subseteq ext(\delta)$.
Since this holds for every $\sigma^{\prime}\in\Phi(a,\sigma)$,
from Definition 2.6,
we conclude that
$\Phi^{*}(c,\sigma)\subseteq ext(\delta)$.
(2)
$c=a;p$ where $a$ is a non-sensing action and
$p$ is a conditional plan. Because $count(c)>0$, from Definition 2.2 we conclude
that there exists a sequence of non-sensing actions $b_{1},\ldots,b_{t}$
and a case plan $q$ such that $c=b_{1};\ldots;b_{t};q$.
Let $c^{\prime}=b_{1};\ldots;b_{t}$ and ${\mathcal{R}}^{*}(q,\delta)=\delta^{*}$.
Using the first case, we can show that for every
$\sigma^{\prime}\in ext(\delta^{*})$,
$\Phi^{*}(q,\sigma^{\prime})\subseteq ext(\delta)$.
Furthermore, because
$$\delta^{\prime}={\mathcal{R}}^{*}(c,\delta)={\mathcal{R}}^{*}(c^{\prime},{%
\mathcal{R}}^{*}(q,\delta))$$
and Lemma 4.6, we can show that
for every
$\sigma\in ext(\delta)$,
$\Phi^{*}(c,\sigma)\subseteq ext(\delta)$.
From cases 1 and 2, the lemma is proved.∎
We are now ready to prove the soundness of the extended
regression function ${\mathcal{R}}^{*}$ with respect to the extended
progression transition function $\Phi^{*}$, which is illustrated in the next
figure.
Theorem 4.8 (Soundness of Regression).
Let $P=\langle A,O,I,G\rangle$ be a planning
problem and $c$ be a regression solution of $P$. Then, $c$ is also a
progression solution of $P$, i.e., $\Phi^{*}(c,\sigma_{I})\subseteq ext(\delta_{G})$.
Proof.
Let $\delta^{\prime}={\mathcal{R}}^{*}(c,\delta_{G})$.
Since $\delta^{\prime}\neq\bot$ and $\sigma_{I}\in ext(\delta^{\prime})$
(Definition 3.18), the conclusion of the theorem follows
immediately from Lemma
4.7.∎
4.2. Completeness Result
We now proceed towards a completeness result. Ideally, one
would like to have a completeness result that expresses that for a
given planning problem, any progression solution can also be found
by regression. In our formulation, however, the definition of the
progression function allows an action $a$ to execute in any a-state
$\sigma$ if $a$ is executable in $\sigma$, regardless whether or not
$a$ would add “new” information to $\sigma$. In contrast, our
definition of the regression function requires that an action $a$
can only be applied in a p-state (or a set of p-states) if $a$
contributes something to the applied p-state(s) 555Note that
this condition is also applied for regression planning systems such
as [4] and [14].. Thus,
given a planning problem $P=\langle A,O,I,G\rangle$, a
progression solution $c$ of $P$ may contain redundant actions or
extra branches. As a result, we may not obtain $c$ via our
regression, i.e. ${\mathcal{R}}^{*}(c,\delta_{G})=\bot$. To illustrate
this point, let us consider the following two examples.
Example 4.9 (Redundancy).
Let $P=\langle\{f,g\},\{b,c\},\{f\},\{g\}\rangle$ be a planning
problem where $c$ is a non-sensing action with $Pre_{c}=\{f\}$,
$Add_{c}=\{g\}$, and $Del_{c}=\emptyset$; $b$ is also a
non-sensing action with $Pre_{b}=\{g\}$, $Add_{b}=\{f\}$, and
$Del_{b}=\emptyset$. Clearly
$$\begin{array}[]{l}p_{1}=c\>\>\>\>\>p_{2}=c;b\>\>\>\>\>p_{3}=c;c\end{array}$$
are three progression solutions of $P$. Plan $p_{1}$ indicates that
$b$ (in $p_{2}$) and a copy (a.k.a. an instance) of $c$ (in $p_{3}$) are
redundant.
It is easy to check that
$${\mathcal{R}}^{*}(p_{2},[\{g\},\emptyset])={\mathcal{R}}^{*}(p_{3},[\{g\},%
\emptyset])=\bot$$
whereas
$${\mathcal{R}}^{*}(c,[\{g\},\emptyset])=[\{f\},\emptyset].$$
Example 4.10 (Redundancy).
Let $P=\langle\{f,g\},\{b,c\},\{f\},\{g\}\rangle$ be a
planning problem. Let $c$ be a sensing action where $Pre_{c}=\emptyset$, $Sens_{c}=\{f,g\}$; $b$ is a non-sensing action
where $Pre_{b}=\{f,\neg g\}$, $Add_{b}=\{g\}$, and
$Del_{b}=\emptyset$. A plan achieving $g$ is:
$$p=c;case(f\wedge\neg g\rightarrow b,f\wedge g\rightarrow[\ ],\neg f\wedge\neg g%
\rightarrow[\ ],\neg f\wedge g\rightarrow[\ ]).$$
Notice that, the
conditions $\neg f\wedge\neg g$ and $\neg f\wedge g$
are always evaluated to false after the execution of $c$ because
$f$ is true before the execution of $c$.
Thus, the two last branches of $p$ are
never used to achieve $g$.
We have that
$$\begin{array}[]{l}{\mathcal{R}}^{*}([],[\{g\},\emptyset])=[\{g\},\emptyset]\\
{\mathcal{R}}^{*}(b,[\{g\},\emptyset])={\mathcal{R}}(b,[\{g\},\emptyset])=[\{f%
\},\{g\}]\\
\end{array}$$
We can also verify that
$$\begin{array}[]{llllllll}R(f\wedge\neg g\rightarrow b,[\{g\},\emptyset])=[\{f%
\},\{g\}]&&R(f\wedge g\rightarrow[],[\{g\},\emptyset])=[\{f,g\},\emptyset]\\
R(\neg f\wedge\neg g\rightarrow[],[\{g\},\emptyset])=\bot&&R(\neg f\wedge g%
\rightarrow[],[\{g\},\emptyset])=[\{g\},\{f\}]\end{array}$$
This implies that
$${\mathcal{R}}^{*}(p,[\{g\},\emptyset])={\mathcal{R}}(c,\{[\{f\},\{g\}],[\{f,g%
\},\emptyset],\bot\})=\bot.$$
Let $p^{\prime}$ be the conditional plan obtained from $p$ by removing
the last two branches of $p$, i.e.,
$$p^{\prime}=c;case(f\wedge\neg g\rightarrow b,f\wedge g\rightarrow[\ ]).$$
We can easily check that
${\mathcal{R}}^{*}(p^{\prime},[\{g\},\emptyset])=[\{f\},\emptyset]\neq\bot$.
The above discussion suggests us the following completeness
result: if a conditional plan can be found through progression we
can find an equivalent conditional plan through regression. The plan
found through regression does not have redundancies, both in terms
of extra actions and extra branches. We refer to these notions as
“redundancy” and “plan equivalence”. We now formalize these
notions.
Definition 4.11 (Subplan).
Let $c$ be a conditional plan. A conditional plan $c^{\prime}$ is a subplan of $c$ if
$\bullet$
$c^{\prime}$ can be obtained from $c$ by
(i)
removing an instance of a non-sensing action from
$c$; or
(ii)
removing a case plan or a branch
$\varphi_{i}\rightarrow c_{i}$ from a case plan in $c$; or
(iii)
replacing a case plan
$a;case(\varphi_{1}\rightarrow p_{1};c_{n}\ldots,\varphi_{m}\rightarrow p_{m})$
in $c$ with one of its branches $p_{i}$ for some $i$, $1\leq i\leq m$; or
$\bullet$
$c^{\prime}$ is a subplan of $c^{\prime\prime}$ where $c^{\prime\prime}$
is a subplan of $c$.
The above definition allows us to define redundant plans as follows.
Definition 4.12 (Redundancy).
Let $c$ be a conditional
plan, $\sigma$ be an a-state, and $\delta$ be a p-state. We say
that $c$ contains redundancy (or is redundant) with respect to
$(\sigma,\delta)$ if
(i)
$\Phi^{*}(c,\sigma)\subseteq ext(\delta)$; and
(ii)
there exists a subplan $c^{\prime}$ of $c$
with respect to $\sigma$ such that
$\Phi^{*}(c^{\prime},\sigma)\subseteq ext(\delta)$.
Note that, if $c^{\prime}$ is a subplan of a conditional plan $c$ then
$c^{\prime}\neq c$. The equivalence of two conditional plans is defined
formally as follows.
Definition 4.13 (Equivalent Plan).
Let $\sigma$ be an a-state, $\delta$ be a p-state, and $c$ be a
conditional plan such that
and
$\Phi^{*}(c,\sigma)\subseteq ext(\delta)$. A conditional
plan $c^{\prime}$ is equivalent to $c$ with respect to $(\sigma,\delta)$
if $\Phi^{*}(c^{\prime},\sigma)\subseteq ext(\delta)$.
Example 4.14 (Equivalence).
Consider the
plans in Example 4.9, we have
that $p_{1}$ is a subplan of $p_{3}$ which is equivalent to $p_{3}$ with
respect to $(\langle\{f\},\emptyset\rangle,[\{g\},\emptyset])$.
Similarly, for planning problem in Example
4.10, $p^{\prime}$ is a subplan of $p$ and is
equivalent to $p$ with respect to $(\langle\{f\},\emptyset\rangle,[\{g\},\emptyset])$.
It is easy to see that if $c^{\prime}$
and $c^{\prime\prime}$ are equivalent to $c$ with respect to
$(\sigma,\delta)$ then $c^{\prime}$ and $c^{\prime\prime}$ are equivalent with respect
to $(\sigma,\delta)$. To prove the completeness result of our
regression formulation, we will need to introduce a few
more definitions and notations. Recall that our purpose is to use
regression to find an equivalent conditional plan for a given
progression solution. To do that, we will provide conditions
characterizing when a conditional plan is regressable, i.e. when the
${\mathcal{R}}^{*}$ function can be applied on it to produce a p-state.
We refer to conditional plans satisfying such conditions as
regressable conditional plans. We will later show that, for a
given progression solution of a planning problem $P$ there exists an
equivalent, regressable conditional plan that is also a regression
solution of $P$.
To define a regressable conditional plan, we begin with some
additional notations. For a non-empty set of fluents $S=\{f_{1},...,f_{k}\}$, a binary representation of $S$ is a formula of the
form $l_{1}\wedge\ldots\wedge l_{k}$ where $l_{i}\in\{f_{i},\neg f_{i}\}$ for $i=1,\ldots,k$.
For a non-empty set of fluents $S$, let $BIN(S)$ denote the set of
all different binary representations of $S$. We say a conjunction
$\phi$ of literals is consistent if there exists no fluent $f$ such
that both $f$ and $\neg f$ appear in $\phi$. A set of consistent
conjunctions of literals $\chi=\{\varphi_{1},\ldots,\varphi_{n}\}$
is said to span over some set of fluents $S$ if there exists a
consistent conjunction of literals $\varphi\not\in\chi$, such
that:
(1)
$S\cap(\varphi^{+}\cup\varphi^{-})=\emptyset$ where
$\varphi^{+}$ and $\varphi^{-}$ denote the sets of fluents occurring positive and negative in
$\varphi$, respectively;
(2)
$\varphi_{i}=\varphi\wedge\psi_{i}$ where $BIN(S)=\{\psi_{1},\ldots,\psi_{n}\}$.
Notice that for a non-empty set $S$, we can easily check whether
the set $\chi=\{\varphi_{1},\ldots,\varphi_{n}\}$ spans over S. We
say that a set $\chi=\{\varphi_{1},\ldots,\varphi_{n}\}$ is
factorable if it spans over some non-empty set of fluents $S$.
Example 4.15 (Getting to Evanston – Cond’t).
Consider a set $S=\{\mathit{traffic}$-$bad\}$, a conjunction
$\varphi=on$-$ashland$ and a set of conjunctions $\chi=\{on$-$ashland\wedge\mathit{traffic}$-$bad,on$-$ashland\wedge\neg\mathit{traffic}$-$bad\}$.
We have that $BIN(S)=\{\mathit{traffic}$-$bad,\neg\mathit{traffic}$-$bad\}$
and $\chi$ spans over $S$.
We can show that for a non-empty set of consistent conjunctions of
literals $\chi=\{\varphi_{1},\ldots,\varphi_{n}\}$ be a non-empty
set if $\chi$ is factorable, then there exists a unique non-empty
set of fluents $S$ such that $\chi$ spans over $S$. This allows us
to define the notion of regressable plans as follows.
Definition 4.16 (Potentially Regressable Case Plan).
A case plan
$$p=a;case(\varphi_{1}\rightarrow c_{1},\ldots,\varphi_{n}\rightarrow c_{n})$$
is potentially
regressable if
(i)
there exists a non-empty set $\emptyset\neq S_{a}\subseteq Sens_{a}$ such that $\{\varphi_{1},\ldots,\varphi_{n}\}$ spans
over $S_{a}$, and
(ii)
for $1\leq i\leq n$,
$Sens_{a}\subseteq(\varphi^{+}_{i}\cup\varphi^{-}_{i})$.
Definition 4.17 (Regressable Conditional Plan).
Let $c$ be a conditional plan, $\sigma$ be an a-state, and $\delta$
be a p-state. We say $c$ is regressable with respect to
$(\sigma,\delta)$ if
(i)
every case plan occurring in $c$ is
potentially regressable,
(ii)
$\Phi^{*}(c,\sigma)\subseteq ext(\delta)$, and
(iii)
$c$ is not redundant with respect to
$(\sigma,\delta)$.
We will now prove a series of lemmae that will be used in the proof of
the completeness of ${\mathcal{R}}^{*}$. Lemma 4.18 is about the uniqueness of
a set of literals over which a factorable set of conjunctions spans.
Lemmae 4.19-4.20 state that the regressable property
of a sequence of non-sensing actions
is maintained by the function ${\mathcal{R}}^{*}$.
Lemma 4.21-4.23 extend this result to regressable
conditional plans. Lemmae 4.24-4.26 show that for each
progression solution there exists an equivalent regressable plan which
can be found through regression.
Lemma 4.18.
Let $\chi=\{\varphi_{1},\ldots,\varphi_{n}\}$ be a non-empty set of consistent conjunctions of
literals. If $\chi$ is factorable, then there exists a unique
non-empty set of fluents $S$ such that $\chi$ spans over $S$.
Proof.
Since $\chi$ is factorable, there exists a
non-empty set of fluents $S$ such that $\chi$ spans over $S$, i.e.
there exists $\varphi$ such that $\varphi_{i}=\varphi\wedge\psi_{i}$
where $\psi_{i}\in BIN(S)$ for $i=1,\ldots,n$ and $BIN(S)=\{\psi_{1},\ldots,\psi_{n}\}$. Assume that $S$ is not unique. This means
that there exists a non-empty set $S^{\prime}\neq S$ such that $\chi$ spans
over $S^{\prime}$, i.e. there exists $\varphi^{\prime}$ such that $\varphi_{i}=\varphi^{\prime}\wedge\psi^{\prime}_{i}$ where $\psi^{\prime}_{i}\in BIN(S^{\prime})$ for $i=1,\ldots,n$.
Consider $f\in S\setminus S^{\prime}$. For every $1\leq i\leq n$, we
have that $\varphi_{i}=\varphi^{\prime}\wedge\psi^{\prime}_{i}$. Since $f\not\in S^{\prime}$ and $\varphi_{i}$ is consistent ($1\leq i\leq n$), $f$ must
occur either positively or negatively in $\varphi^{\prime}$. This means that
$f$ occurs either positively or negatively in all $\varphi_{i}$ for $1\leq i\leq n$.
Consider the case that $f$ occurs positively in all $\varphi_{i}$
for $1\leq i\leq n$ [*]. Since $f\in S$, there exists a binary
representation $\psi_{j}\in BIN(S)$ ($1\leq j\leq n$) such that $f$
appears negatively in $\psi_{j}$ i.e. $f$ appears negatively in
$\varphi_{j}$. This contradicts with [*]. Similarly we can show a
contradiction in the case that $f$ occurs negatively in all
$\varphi_{i}$ for $1\leq i\leq n$. We conclude that $S$ is
unique.∎
Lemma 4.19.
Let $\sigma$ be an a-state, $\delta$ be a
p-state, and $c=a_{1};\ldots;a_{n}$ ($n\geq 1$) be a sequence of
non-sensing actions. Assume that $c$ is regressable with respect to
$(\sigma,\delta)$. Then, ${\mathcal{R}}^{*}(a_{n},\delta)=\delta^{\prime}$,
$\delta^{\prime}\neq\bot$, and $c^{\prime}=a_{1};\ldots;a_{n-1}$ is regressable
with respect to $(\sigma,\delta^{\prime})$.
Proof.
By induction on $n$.
$\bullet$
Base Case: $n=1$. Similar to the inductive step,
we can show that $a_{1}$ is applicable in $\delta$. Let
$\delta^{\prime}={\mathcal{R}}(a_{1},\delta)$ and $\Phi(a_{1},\sigma)=\{\sigma^{\prime}\}$. We have that, $\delta^{\prime}.T=(\delta.T\setminus Add_{a_{1}})\cup Pre_{a_{1}}^{+}$ and $\sigma^{\prime}.T=(\sigma.T\setminus Del_{a_{1}})\cup Add_{a_{1}}$. Using the facts $\sigma^{\prime}\in ext(\delta)$, $Add_{a_{1}}\cap Del_{a_{1}}=\emptyset$, and the above
equations, we can show that $\delta^{\prime}.T\subseteq\sigma.T$.
Similarly, $\delta^{\prime}.F\subseteq\sigma.F$. Since $[\ ]$ is not
redundant with respect to $(\sigma,\delta^{\prime})$, we have that $[\ ]$ is
a plan that is regressable with respect to $(\sigma,\delta^{\prime})$.
$\bullet$
Inductive Step: Assume that we have proved the lemma
for $0<n\leq k$. We need to prove the lemma for $n=k+1$.
Let $\Phi^{*}(a_{1};\ldots;a_{k},\sigma)=\{\sigma_{k}\}$, we have
that
$$\Phi^{*}(c,\sigma)=\Phi(a_{k+1},\sigma_{k})=\{\sigma^{\prime}\}\subseteq ext(%
\delta).$$
We will prove that (1) $a_{k+1}$ is applicable in $\delta$,
(2) ${\mathcal{R}}(a_{k+1},\delta)=\delta^{*}\neq\bot$ and
$\sigma_{k}\in ext(\delta^{*})$, and (3) $c^{\prime}=a_{1};\ldots;a_{k}$ is
regressable with respect to $(\sigma,\delta^{*})$.
–
Proof of (1):
We first show that
$Add_{a_{k+1}}\cap\delta.T\neq\emptyset$ or
$Del_{a_{k+1}}\cap\delta.F\neq\emptyset$.
Assume the contrary, $Add_{a_{k+1}}\cap\delta.T=\emptyset$ and
$Del_{a_{k+1}}\cap\delta.F=\emptyset$.
By Definition 2.4, we have that
$$\sigma^{\prime}.T=(\sigma_{k}.T\setminus Del_{a_{k+1}})\cup Add_{a_{k+1}}$$
and
$$\sigma^{\prime}.F=(\sigma_{k}.F\setminus Add_{a_{k+1}})\cup Del_{a_{k+1}}.$$
Since $\sigma^{\prime}\in ext(\delta)$, we have $\delta.T\subseteq\sigma^{\prime}.T$. By our assumption, $Add_{a_{k+1}}\cap\delta.T=\emptyset$,
we must have that $\delta.T=\delta.T\setminus Add_{a_{k+1}}\subseteq\sigma^{\prime}.T\setminus Add%
_{a_{k+1}}$. Because for arbitrary sets $X,Y$,
$(X\cup Y)\setminus Y=X\setminus(X\cap Y)$, we have that
$$\sigma^{\prime}.T\setminus Add_{a_{k+1}}=((\sigma_{k}.T\setminus Del_{a_{k+1}}%
)\cup Add_{a_{k+1}})\setminus Add_{a_{k+1}}=$$
$$(\sigma_{k}.T\setminus Del_{a_{k+1}})\setminus((\sigma_{k}.T\setminus Del_{a_{%
k+1}})\cap Add_{a_{k+1}})\subseteq\sigma_{k}.T,$$
i.e. $\delta.T\subseteq\sigma_{k}.T\setminus Del_{a_{k+1}}$.
This shows that $\delta.T\subseteq\sigma_{k}.T$. Similarly,
we can show that $\delta.F\subseteq\sigma_{k}.F$. We conclude that $\sigma_{k}\in ext(\delta)$, i.e.
$c$ is redundant with respect to $(\sigma,\delta)$.
This is a contradiction. Therefore, $Add_{a_{k+1}}\cap\delta.T\neq\emptyset$ or
$Del_{a_{k+1}}\cap\delta.F\neq\emptyset$.
(i)
Since $\sigma^{\prime}\in ext(\delta)$, we have $\delta.T\subseteq\sigma^{\prime}.T$ and $\delta.F\subseteq\sigma^{\prime}.F$. As $a_{k+1}$ is executable in $\sigma_{k}$, we have
$Add_{a_{k+1}}\cap\sigma^{\prime}.F=\emptyset$ and $Del_{a_{k+1}}\cap\sigma^{\prime}.T=\emptyset$. This concludes that $Add_{a_{k+1}}\cap\delta.F=\emptyset$ and $Del_{a_{k+1}}\cap\delta.T=\emptyset$.
(ii)
Now, assume that there exists $f\in Pre^{+}_{a_{k+1}}\cap\delta.F$ and $f\not\in Del_{a_{k+1}}$. By Definition
2.4, it’s easy to see
that $f\in\sigma^{\prime}.T$ and $f\in\sigma^{\prime}.F$. This is a
contradiction, therefore $Pre^{+}_{a_{k+1}}\cap\delta.F\subseteq Del_{a_{k+1}}$. Similarly, we can show that $Pre^{-}_{a_{k+1}}\cap\delta.T\subseteq Add_{a_{k+1}}$.
(iii).
From (i), (ii), and (iii) we conclude
that $a_{k+1}$ is applicable in $\delta$.
–
Proof of (2):
Because $a_{k+1}$ is applicable in $\delta$, we have that
${\mathcal{R}}(a_{k+1},\delta)=\delta^{*}$ for some
partial state $\delta^{*}\neq\bot$.
We will show that $\sigma_{k}\in ext(\delta^{*})$.
From the fact $\sigma^{\prime}\in ext(\delta)$, by Definition 2.4,
we have
$$\delta.T\subseteq\sigma^{\prime}.T=(\sigma_{k}.T\setminus Del_{a_{k+1}})\cup
Add%
_{a_{k+1}}$$
and
$$\delta.F\subseteq\sigma^{\prime}.F=(\sigma_{k}.F\setminus Add_{a_{k+1}})\cup
Del%
_{a_{k+1}}.$$
By Definition 3.2 we have
$$\delta^{*}.T=(\delta.T\setminus Add_{a_{k+1}})\cup Pre^{+}_{a_{k+1}}$$
and
$$\delta^{*}.F=(\delta.F\setminus Del_{a_{k+1}})\cup Pre^{-}_{a_{k+1}}.$$
Since $a_{k+1}$ is executable in $\sigma_{k}$, we have that
$Pre^{+}_{a_{k+1}}\subseteq\sigma_{k}.T$
and $Pre^{-}_{a_{k+1}}\subseteq\sigma_{k}.F$. Therefore, to prove
that $\delta^{*}.T=(\delta.T\setminus Add_{a_{k+1}})\cup Pre^{+}_{a_{k+1}}\subseteq%
\sigma_{k}.T$, we only need to show that
$\delta.T\setminus Add_{a_{k+1}}\subseteq\sigma_{k}.T$. As
$\delta.T\subseteq(\sigma_{k}.T\setminus Del_{a_{k+1}})\cup Add_{a_{k+1}}$, we have
$$\delta.T\setminus Add_{a_{k+1}}\subseteq((\sigma_{k}.T\setminus Del_{a_{k+1}})%
\cup Add_{a_{k+1}})\setminus Add_{a_{k+1}}.$$
From the proof of item (1), we have that $((\sigma_{k}.T\setminus Del_{a_{k+1}})\cup Add_{a_{k+1}})\setminus Add_{a_{k+1%
}}\subseteq\sigma_{k}.T$. This concludes that $\delta.T\setminus Add_{a_{k+1}}\subseteq\sigma_{k}.T$. Similarly, we can show that $\delta.F\setminus Del_{a_{k+1}}\subseteq\sigma_{k}.F$, i.e., $\sigma_{k}\in ext(\delta^{*})$ or $\{\sigma_{k}\}\subseteq ext(\delta^{*})$.
–
Proof of (3):
Suppose that $c^{\prime}$ is redundant with respect to $(\sigma,\delta^{*})$.
By Definition 4.12, there exists a subplan $c^{\prime\prime}$ of $c$
such that
$\Phi^{*}(c^{\prime\prime},\sigma)=\{\sigma^{\prime\prime}\}\subseteq ext(%
\delta^{*})$.
By Theorem 4.2, we have that $\Phi(a_{k+1},\sigma^{\prime\prime})\subseteq ext(\delta)$. Since
$$\Phi^{*}(c^{\prime\prime};a_{k+1},\sigma)=\Phi(a_{k+1},\sigma^{\prime\prime})%
\subseteq ext(\delta),$$
we have that $c$ is redundant with respect to
$(\sigma,\delta)$. This
contradicts with the assumption that $c$ is not redundant
with respect to $(\sigma,\delta)$. Since $c^{\prime}$ has no case
plan, this concludes that $c^{\prime}$ is not
redundant with respect to $(\sigma,\delta^{*})$. Since
$\Phi^{*}(a_{1};\ldots;a_{k},\sigma)=\{\sigma_{k}\}\subseteq ext(\delta^{*})$
we have that $c^{\prime}$ is regressable with respect to $(\sigma,\delta^{*})$.∎
Lemma 4.20.
Let $\sigma$ be an a-state and $\delta$ be a
p-state. Let $c=a_{1};\ldots;a_{n}$ be a sequence of non-sensing
actions that is regressable with respect to $(\sigma,\delta)$. Then,
there exists some p-state $\delta^{*}\neq\bot$ such that ${\mathcal{R}}^{*}(c,\delta)=\delta^{*}$ and $\sigma\in ext(\delta^{*})$.
Proof.
By induction on $n$.
$\bullet$
Base Case: $n=0$. Then $c$ is an empty sequence of
non-sensing actions. The base case follows from Definition 3.17
(with $\delta^{*}=\delta$ and $[\ ]$ is not redundant with respect
to $(\sigma,\delta)$).
$\bullet$
Inductive Step: Assume that we have proved the
lemma for $0\leq n\leq k$. We need to prove the lemma for
$n=k+1$.
It follows from Lemma 4.19
that $\delta^{\prime}={\mathcal{R}}(a_{k+1},\delta)$,
$\delta^{\prime}\neq\bot$, and
$c^{\prime}=a_{1};\ldots;a_{k}$ is a plan that is regressable
with respect to $(\sigma,\delta^{\prime})$.
By inductive hypothesis, we have that
${\mathcal{R}}^{*}(c^{\prime},\delta^{\prime})=\delta^{*}\neq\bot$
and $\sigma\in ext(\delta^{*})$.
The inductive step follows from this and the fact
${\mathcal{R}}^{*}(c,\delta)={\mathcal{R}}^{*}(c^{\prime},{\mathcal{R}}(a_{k+1}%
,\delta))$.∎
Lemma 4.21.
Let $\sigma$ be an a-state, $a$ be a sensing
action which is executable in $\sigma$. Let $S_{a}=Sens_{a}\setminus\sigma$. Then, we have that
(1)
$\Phi(a,\sigma)=\{\sigma_{1},\ldots,\sigma_{m}\}$ where $m=2^{|S_{a}|}$,
(2)
$a$ is strongly
applicable in $\Delta=\{\delta_{1},\ldots,\delta_{m}\}$ where
$\delta_{i}=[\sigma_{i}.T,\sigma_{i}.F]$, $i=1,\ldots,m$, and
(3)
${\mathcal{R}}(a,\Delta)=[\sigma.T,\sigma.F]$.
Proof.
(1)
From Definition 2.4, we have that
$$\bot\not\in\Phi(a,\sigma)=\{\sigma^{\prime}|Sens_{a}\setminus\sigma=\sigma^{%
\prime}\setminus\sigma\}$$
and, for every $\sigma^{\prime}\in\Phi(a,\sigma)$, $\sigma^{\prime}\setminus\sigma=(\sigma^{\prime}.T\setminus\sigma.T)\cup(\sigma%
^{\prime}.F\setminus\sigma.F)$. Denote $\sigma^{\prime}.T\setminus\sigma.T$ by $P$ and
$\sigma^{\prime}.F\setminus\sigma.F$ by $Q$, we have that $(P,Q)$ is a
partition of $S_{a}$. Since there are $2^{|S_{a}|}$ partitions of $S_{a}$,
we have that $m\leq 2^{|S_{a}|}$. Furthermore, for a partition
$(P,Q)$ of $S_{a}$ there exists an a-state $\sigma^{\prime}=\langle P\cup\sigma.T,Q\cup\sigma.F\rangle\in\Phi(a,\sigma)$ because
$\sigma^{\prime}\setminus\sigma=P\cup Q$. Therefore $2^{|S_{a}|}\leq m$.
We conclude that $m=2^{|S_{a}|}$.
(2)
We first show that $\Delta$ is proper with respect
to $S_{a}$, i.e. $S_{a}$ is a sensed set of $\Delta$ with respect to
$a$. Indeed, by Definition 2.4 and the proof of (1), we
have that the first three conditions of Definition 3.6
are satisfied. The fourth condition of Definition 3.6
is satisfied because we have that $\delta_{i}.T\setminus S_{a}=\sigma_{i}.T\setminus S_{a}=\sigma.T$ and $\delta_{i}.F\setminus S_{a}=\sigma_{i}.F\setminus S_{a}=\sigma.F$ ($1\leq i\leq m$). Therefore,
we conclude that $p(a,\Delta)=S_{a}$.
Since $a$ is an action that is executable in $\sigma$ we have
that $(Pre^{+}_{a}\cup Pre^{-}_{a})\cap Sens_{a}=\emptyset$ and $Pre^{+}_{a}\cap\sigma.F=\emptyset$, $Pre^{-}_{a}\cap\sigma.T=\emptyset$,
therefore $Pre^{+}_{a}\cap\delta_{i}.F=\emptyset$, $Pre^{-}_{a}\cap\delta_{i}.T=\emptyset$ ($1\leq i\leq m$). By Definition
3.8, we conclude that $a$ is strongly
applicable in $\Delta$.
(3)
Since $a$ is executable in $\sigma$, we have that
$Pre^{+}_{a}\subseteq\sigma.T$ and $Pre^{-}_{a}\subseteq\sigma.F$. From
the proof of (2), $\delta_{i}.T\setminus S_{a}=\sigma.T$ and
$\delta_{i}.F\setminus S_{a}=\sigma.F$ ($1\leq i\leq m$). The proof
follows from Definition 3.15.∎
Lemma 4.22.
Let $\sigma$ be an a-state, $\delta$ be a
p-state, and $c=\alpha;c^{\prime}$ is a conditional plan where $\alpha$
is a non-empty sequence of non-sensing actions and $c^{\prime}=a;case(\varphi_{1}\rightarrow p_{1},\ldots,\varphi_{m}\rightarrow p%
_{m})$.
If $c$ is regressable with respect to $(\sigma,\delta)$, then
(1)
there exists some a-state $\sigma_{1}\neq\bot$ such that
$\Phi^{*}(\alpha,\sigma)=\{\sigma_{1}\}$;
(2)
$m=2^{|S_{a}|}$ where $S_{a}=Sens_{a}\setminus\sigma_{1}$;
(3)
$\{\varphi_{1},\ldots,\varphi_{m}\}$ spans over
$S_{a}$;
(4)
For each $i$, $1\leq i\leq m$, there exists a
unique a-state $\sigma^{\prime}\in\Phi(a,\sigma_{1})$ such that $p_{i}$ is
regressable with respect to $(\sigma^{\prime},\delta)$.
Proof.
(1)
By Definition 2.6, we have that
$$\Phi^{*}(c,\sigma)=\bigcup_{\sigma^{\prime}\in\Phi^{*}(\alpha,\sigma)}\Phi^{*}%
(c^{\prime},\sigma^{\prime}).$$
Since $c$ is regressable with respect to $(\sigma,\delta)$ we have
that $\bot\not\in\Phi^{*}(c,\sigma)$. This implies that $\bot\not\in\Phi^{*}(\alpha,\sigma)$. Furthermore, because $\alpha$ is a
sequence of non-sensing actions, we conclude that there exists some
a-state $\sigma_{1}\neq\bot$. such that $\Phi^{*}(\alpha,\sigma)=\{\sigma_{1}\}$.
(2)
By definition of $S_{a}$ we conclude that $S_{a}$ is
the set of fluents that belong to $Sens_{a}$ which are unknown in
$\sigma_{1}$. By Definition 2.4, we conclude that $\Phi(a,\sigma_{1})$ consists of $2^{|S_{a}|}$ elements where for each $\sigma^{\prime}\in\Phi(a,\sigma_{1})$, $\sigma^{\prime}\setminus\sigma_{1}=S_{a}$. Because
$$\bot\not\in\Phi^{*}(c,\sigma)=\bigcup_{\sigma^{\prime}\in\Phi(a,\sigma_{1})}E(%
case(\varphi_{1}\rightarrow p_{1},\ldots,\varphi_{m}\rightarrow p_{m}),\sigma^%
{\prime})$$
we conclude that for each $\sigma^{\prime}\in\Phi(a,\sigma_{1})$ there
exists one $j$, $1\leq j\leq m$, such that $\varphi_{j}$ is satisfied
in $\sigma^{\prime}$. Since $\varphi$’s are mutual exclusive we conclude
that for each $j$, $1\leq j\leq m$, there exists at most one
$\sigma^{\prime}\in\Phi(a,\sigma_{1})$ such that $\varphi_{j}$ is satisfied
in $\sigma^{\prime}$. This implies that $m\geq 2^{|S_{a}|}$. The
non-redundancy property of $c$ implies that $m\geq 2^{|S_{a}|}$. Thus,
$m=2^{|S_{a}|}$.
(3)
Since $c$ is regressable with respect to
$(\sigma,\delta)$ we have that $a;(case(\varphi_{1}\rightarrow p_{1},\ldots,\varphi_{m}\rightarrow p_{m})$ is potentially regressable.
This implies that $\{\varphi_{1},\ldots,\varphi_{m}\}$ spans over a set
of fluents $S\subseteq Sens_{a}$ and there exists a $\varphi$ such
that for every $i$, $\varphi_{i}=\psi_{i}\wedge\varphi$ where
$\psi_{i}\in BIN(S)$ and $S\cap(\varphi^{+}\cup\varphi^{-})=\emptyset$. From Lemma 4.18 we know that $S$ is unique.
We will show now that $S=S_{a}$. Assume the contrary, $S\neq S_{a}$.
We consider two cases:
$\bullet$
$S\setminus S_{a}\neq\emptyset$. Consider a fluent $f\in S\setminus S_{a}$. Because $\{\varphi_{1},\ldots,\varphi_{m}\}$ spans over
$S$, there exists some $i$ such that $f$ occurs positively in
$\varphi_{i}$. From the proof of the previous item and the fact that
$f\not\in S_{a}$, we conclude that $f$ must be true in $\sigma_{1}$
(otherwise, we have that the subplan $c^{\prime}$ of $c$, obtained by
removing the branch $\varphi_{i}\rightarrow p_{i}$, satisfies $\bot\not\in\Phi^{*}(c^{\prime},\sigma)\subseteq ext(\delta)$, which implies that
$c$ is redundant with respect to $(\sigma,\delta)$). Similarly,
there exists some $j$ such that $f$ occurs negatively in
$\varphi_{j}$, and hence, $f$ must be false in $\sigma_{1}$. This is a
contradiction. Thus, this case cannot happen.
$\bullet$
$S_{a}\setminus S\neq\emptyset$. Consider a fluent $f\in S_{a}\setminus S$. Again, from the fact that $c$ is regressable with
respect to $(\sigma,\delta)$, we conclude that $f$ occurs either
positively or negatively in $\varphi_{i}$. Because $f\not\in S$, we
have that $f$ occurs in $\varphi$, and hence, $f$ occurs positively
or negatively in all $\varphi_{i}$. In other words, $f$ is true or
false in every $\sigma^{\prime}\in\Phi(a,\sigma_{1})$. Thus, $f$ is true or
false in $\sigma_{1}$. This contradicts the fact that $f\in S_{a}=Sens_{a}\setminus\sigma_{1}$. Thus, this case cannot happen too.
The above two cases imply that $S_{a}=S$. This means that
$\{\varphi_{1},\ldots,\varphi_{m}\}$ spans over $S_{a}$.
(4)
Consider an arbitrary $i$, $1\leq i\leq m$. From
the proof of the second item, we know that there exists a unique
$\sigma^{\prime}\in\Phi(a,\sigma_{1})$ such that $\varphi_{i}$ is satisfied by
$\sigma^{\prime}$. We will show now that $p_{i}$ is regressable with respect
to $(\sigma^{\prime},\delta)$. From the fact that $c$ is regressable, we
conclude that every case plan in $p_{i}$ is potentially regressable.
Furthermore, because $\Phi^{*}(p_{i},\sigma^{\prime})\subseteq\Phi^{*}(c,\sigma)$, we have that $\bot\not\in\Phi^{*}(p_{i},\sigma^{\prime})\subseteq ext(\delta)$. Thus, to complete the proof, we need to show that
$p_{i}$ is not redundant with respect to $(\sigma^{\prime},\delta)$. Assume
the contrary, there exists a subplan $p^{\prime}$ of $p_{i}$ such that $\bot\not\in\Phi^{*}(p^{\prime},\sigma^{\prime})\subseteq ext(\delta)$. This implies
that the subplan $c^{\prime}$ of $c$, obtained by replacing $p_{i}$ with $p^{\prime}$,
will satisfy that $\bot\not\in\Phi^{*}(c^{\prime},\sigma)\subseteq ext(\delta)$, i.e., $c$ is redundant with respect to $(\sigma,\delta)$. This contradicts the condition of the lemma, i.e., our
assumption is incorrect. Thus, $p_{i}$ is not redundant with respect
to $(\sigma^{\prime},\delta)$, and hence, $p_{i}$ is regressable with respect
to $(\sigma^{\prime},\delta)$.∎
Lemma 4.23.
Let $\sigma$ be an a-state, $\delta$ be a
p-state, and $c$ is a conditional plan that is regressable with
respect to $(\sigma,\delta)$. Then, there exists some p-state
$\delta^{\prime}\neq\bot$ such that ${\mathcal{R}}^{*}(c,\delta)=\delta^{\prime}$
and $\sigma\in ext(\delta^{\prime})$.
Proof.
By induction on $count(c)$, the number of
case plans in $c$.
$\bullet$
Base Case: $count(c)=0$. Then $c$ is a sequence
of non-sensing actions. The base case follows from Lemma
4.20.
$\bullet$
Inductive Step:
Assume that we have proved the lemma
for $count(c)\leq k$. We need to prove the lemma for
$count(c)=k+1$.
Since $c$ is a conditional plan, we have that $c=\alpha;c^{\prime}$ where
$\alpha$ is a sequence of non-sensing actions and $c^{\prime}=a;p$ and
$p=case\ (\varphi_{1}\rightarrow p_{1},\ldots,\varphi_{m}\rightarrow p_{m})$.
Because $\alpha$ is a sequence of non-sensing actions we have that
$\Phi^{*}(\alpha,\sigma)$ is a singleton. Let $\Phi^{*}(\alpha,\sigma)=\{\sigma_{1}\}$.
Let $S_{a}=Sens_{a}\setminus\sigma_{1}$. Since $c$ is not
redundant with respect to $(\sigma,\delta)$ we conclude that $S_{a}\neq\emptyset$.
It follows from the fact that $c$ is regressable with respect
to $(\sigma,\delta)$ and Lemma 4.22 that $\{\varphi_{1},\ldots,\varphi_{m}\}$ spans over $S_{a}$ and for every $i$, $1\leq i\leq m$, there exists a unique $\sigma^{\prime}\in\Phi(a,\sigma_{1})$ such
that $p_{i}$ is regressable with respect to $(\sigma^{\prime},\delta)$. By
inductive hypothesis for $p_{i}$, we conclude that ${\mathcal{R}}^{*}(p_{i},\delta)=\delta_{i}\neq\bot$ and $\sigma^{\prime}\in ext(\delta_{i})$. Because $\varphi_{i}$ is satisfied by $\sigma^{\prime}$ we
have that $R(\varphi_{i}\rightarrow p_{i},\delta)=[\delta_{i}.T\cup\varphi_{i}^{+},\delta%
_{i}.F\cup\varphi_{i}^{-}]$ is consistent and hence
$R(\varphi_{i}\rightarrow p_{i},\delta)\neq\bot$. This also implies
that $\sigma^{\prime}\in ext(R(\varphi_{i}\rightarrow p_{i},\delta))$ and
$R(\varphi_{i}\rightarrow p_{i},\delta)\neq R(\varphi_{j}\rightarrow p_{j},\delta)$ for $i\neq j$.
Let $\Delta=\{R(\varphi_{i}\rightarrow p_{i},\delta)\mid i=1,\ldots,m\}$. We will show next that $a$ is applicable in
$\Delta$. Consider $\Delta^{\prime}=\Phi(a,\sigma_{1})$, we have that for
each $i$, $1\leq i\leq m$, there exists one $\sigma^{\prime}\in\Delta^{\prime}$
and $\sigma^{\prime}\in ext(R(\varphi_{i}\rightarrow p_{i},\delta))$. It
follows from Lemma 4.21 that $a$ is strongly applicable in
$\Delta^{\prime}$. Thus, $a$ is applicable in $\Delta$.
By definition of ${\mathcal{R}}$, we have that
$$\begin{array}[]{ll}{\mathcal{R}}(a,\Delta)=[((\bigcup_{i=1}^{m}R(\varphi_{i}%
\rightarrow p_{i},\delta).T)\setminus S_{a})\cup Pre^{+}_{a},\\
\hskip 72.27pt((\bigcup_{i=1}^{m}R(\varphi_{i}\rightarrow p_{i},\delta).F)%
\setminus S_{a})\cup Pre^{-}_{a}]=\delta^{*}\neq\bot.\end{array}$$
Since $a$ is executable in $\sigma_{1}$, from Lemma 4.21, and
the fact that for each $\sigma^{\prime}\in\Phi(a,\sigma_{1})$ there exists
an $i$ such that $\sigma^{\prime}\in ext(R(\varphi_{i}\rightarrow p_{i},\delta))$, we can conclude $\sigma_{1}\in ext(\delta^{*})$.
To continue our proof, we will now show that $q=\alpha$ is not
redundant with respect to $(\sigma,\delta^{*})$. Assume the contrary,
there exists a subplan $q^{\prime}$ of $q$ such that $\Phi^{*}(q^{\prime},\sigma)\subseteq ext(\delta^{*})$. This, together with the fact that
${\mathcal{R}}^{*}(c^{\prime},\delta)=\delta^{*}$ and Theorem 4.8
implies that $\Phi^{*}(c^{\prime\prime},\sigma)\subseteq ext(\delta)$ for $c^{\prime\prime}=q^{\prime};c^{\prime}$, i.e., $c$ is redundant with respect to $(\sigma,\delta)$.
This contradicts the assumption of the lemma, i.e., we have proved
that $q$ is not redundant with respect to $(\sigma,\delta^{*})$.
Applying the inductive hypothesis for the plan $q$ and
$(\sigma,\delta^{*})$, we have that ${\mathcal{R}}^{*}(q,\delta^{*})=\delta^{\prime}\neq\bot$ and $\sigma\in ext(\delta^{\prime})$. The inductive
hypothesis is proved because ${\mathcal{R}}^{*}(c,\delta)={\mathcal{R}}^{*}(q,\delta^{*})$.∎
Lemma 4.24.
Let $\sigma$ be an a-state, $\delta$ be
a p-state, and $c$ be a sequence of non-sensing actions such that
$\Phi^{*}(c,\sigma)\subseteq ext(\delta)$. Then, there exists a
subplan $c^{\prime}$ of $c$ that is not redundant with respect to
$(\sigma,\delta)$ and $c^{\prime}$ is equivalent to $c$ with respect to
$(\sigma,\delta)$.
Proof.
Notice that the length of $c$ is
finite666
By this we mean that $c$ is given and hence its length (the number of
actions in $c$) is finite.
. Consider two cases:
$\bullet$
Case (i): $c$ is not redundant with respect to $(\sigma,\delta)$.
It’s easy to see that $c^{\prime}=c$ satisfies the condition of the lemma.
$\bullet$
Case (ii): $c$ is redundant with respect to $(\sigma,\delta)$.
By definition of redundancy, there exists a subplan of $c$ which
are equivalent to $c$ with respect to $(\sigma,\delta)$.
Let $c^{\prime}$ be a subplan of $c$ which is equivalent to $c$
with respect to $(\sigma,\delta)$ whose length is minimal among all subplans
which is equivalent to $c$ with respect to $(\sigma,\delta)$. To prove the lemma,
it suffices to show that $c^{\prime}$ is not redundant with respect to $(\sigma,\delta)$.
Assume the contrary, there exists a subplan $c^{\prime\prime}$ of $c^{\prime}$ which
is equivalent to $c$ with respect to $(\sigma,\delta)$. Trivially,
the number of actions in $c^{\prime\prime}$ is smaller than the number
of actions in $c^{\prime}$. By definition, we have that $c^{\prime\prime}$ is also a subplan of $c$
which is equivalent
to $c$ with respect to $(\sigma,\delta)$. This contradicts the fact that
$c^{\prime}$ has the minimal length among all subplans of $c$ which are equivalent to $c$.
So, we conclude that
$c^{\prime}$ is not redundant with respect to $(\sigma,\delta)$. The lemma is
proved.∎
Lemma 4.25.
Let $\sigma$ be an a-state and $c=a;case\ (\varphi_{1}\rightarrow p_{1},\ldots,\varphi_{m}\rightarrow p_{m})$
be a case plan such that $\bot\not\in\Phi^{*}(c,\sigma)$. Then, if
$Sens_{a}\setminus\sigma\neq\emptyset$, there exists a potentially
regressable plan $c^{\prime}=a;case\ (\varphi_{1}^{\prime}\rightarrow p^{\prime}_{1},\ldots,%
\varphi_{n}^{\prime}\rightarrow p^{\prime}_{n})$
such that $\Phi^{*}(c,\sigma)=\Phi^{*}(c^{\prime},\sigma)$.
Proof.
We prove the lemma by constructing $c^{\prime}$. Let
$S=\{\varphi_{1},\ldots,\varphi_{m}\}$ and $S_{a}=Sens_{a}\setminus\sigma$. Let $L=\{f\mid f\in S_{a}\}\cup\{\neg f\mid f\in S_{a}\}$. First, observe that because of $\bot\not\in\Phi^{*}(c,\sigma)$ we have that $a$ is executable in $\sigma$.
Furthermore, for each $\sigma^{\prime}\in\Phi(a,\sigma)$ there exists one
$\varphi_{i}\in S$ such that $\varphi_{i}$ is satisfied in $\sigma^{\prime}$.
Without loss of generality, we can assume that for each $\varphi_{i}\in S$, there exists (at least) one $\sigma^{\prime}\in\Phi(a,\sigma)$
such that $\varphi_{i}$ is satisfied in $\sigma^{\prime}$.
It is easy to see that for each $i$, we can write $\varphi_{i}=\psi_{i}\wedge\chi_{i}$ where $\psi_{i}$ is the conjunction of literals
occurring in $\varphi_{i}$ and belonging to $L$ and $\chi_{i}$ is the
conjunction of literals that do not belong to $L$. From the above
observation, we have that $\chi_{i}$ is satisfied by $\sigma$. So,
$\varphi=\wedge_{i=1}^{m}\chi_{i}$ holds in $\sigma$. Thus, the
conditional plan $c_{1}=a;case\ (\varphi^{\prime}_{1}\rightarrow p_{1},\ldots,\varphi^{\prime}_{%
m}\rightarrow p_{m})$ where $\varphi^{\prime}_{i}=\psi_{i}\wedge\varphi$ satisfies that
$\Phi^{*}(c,\sigma)=\Phi^{*}(c_{1},\sigma)$.
Since $\psi_{i}$ is a consistent conjunction of literals from $L$
and $\psi_{i}$’s are mutual exclusive, there exists a partition
$(S_{1},\ldots,S_{m})$ of $BIN(S_{a})$ such that for every $\eta\in S_{i}$,
$\eta=\psi_{i}\wedge\eta^{\prime}$. Let
$$\begin{array}[]{rll}c_{2}=a;case(&\\
&&\gamma^{1}_{1}\rightarrow p_{1},\ldots,\gamma^{|S_{1}|}_{1}\rightarrow p_{1}%
,\\
&&\gamma^{1}_{2}\rightarrow p_{1},\ldots,\gamma^{|S_{2}|}_{2}\rightarrow p_{2}%
,\\
&&\ldots\\
&&\gamma^{1}_{m}\rightarrow p_{1},\ldots,\gamma^{|S_{m}|}_{m}\rightarrow p_{m}%
,\\
)&&\\
\end{array}$$
where
$${\hbox{}\,\vbox{\openup 3.0pt\halign{\cr}$\displaystyle{\gamma^{j}_{i}}$&$%
\displaystyle{{}=\eta^{j}_{i}\wedge\varphi\wedge\gamma}$\cr$\displaystyle{S_{i%
}}$&$\displaystyle{{}=\{\eta^{1}_{i},\ldots,\eta^{|S_{i}|}_{i}\}\quad\hbox{for%
$i=1,\ldots,m$,
and}}$\cr$\displaystyle{\gamma}$&$\displaystyle{{}=\bigwedge{}_{f\in Sens_{a}%
\cap\sigma.T}\,f\wedge\bigwedge{}_{f\in Sens_{a}\cap\sigma.F}\,\neg f\ .}$}}\,$$
We have that $\Phi^{*}(c,\sigma)=\Phi^{*}(c_{2},\sigma)$. It is easy to
see that the set $\{\gamma^{1}_{1},\ldots,\gamma^{|S_{m}|}_{m}\}$ spans over
$S_{a}$ and $Sens_{a}\subseteq(\gamma^{j}_{i})^{+}\cup(\gamma^{j}_{i})^{-}$.
Thus, $c_{2}$ is potentially regressable. The lemma is proved with $c^{\prime}=c_{2}$.∎
Lemma 4.26.
Let $\sigma$ be an a-state, let $\delta$
be a p-state, and let $c$ be a conditional plan such that
$\Phi^{*}(c,\sigma)\subseteq ext(\delta)$. There exists a plan $c^{\prime}$
such that $c^{\prime}$ is regressable with respect to $(\sigma,\delta)$ and
$c^{\prime}$ is equivalent to $c$ with respect to $(\sigma,\delta)$.
Proof.
By induction on $count(c)$, the number of case plans in $c$.
$\bullet$
Base case: $count(c)=0$
This follows from Lemma 4.24.
$\bullet$
Inductive Step: Assume that we have proved the
lemma for $count(c)\leq k$. We need to prove the lemma for
$count(c)=k+1$.
By construction of $c$, we have two cases
(1)
$c=a;p$ where
$p=case\ (\varphi_{1}\rightarrow p_{1},\ldots,\varphi_{m}\rightarrow p_{m})$.
Here, we have two cases.
(a)
$Sens_{a}\setminus\sigma=\emptyset$. In this case, we have
that there exists some $j$ such that $\varphi_{j}$ is satisfied by $\sigma$
and $\Phi^{*}(c,\sigma)=\Phi^{*}(p_{j},\sigma)$.
Thus, $c$ is equivalent to $p_{j}$ with respect to $(\sigma,\delta)$.
Since $count(p_{j})<count(c)$, by the inductive hypothesis and
transitivity of the equivalence relation, we conclude that
there exists a plan $c^{\prime}$ such that
$c^{\prime}$ is regressable with respect to $(\sigma,\delta)$ and $c^{\prime}$
is equivalent to $c$ with respect to $(\sigma,\delta)$.
(b)
$Sens_{a}\setminus\sigma\neq\emptyset$.
Without loss of generality, we can assume that for each $\varphi_{i}\in S$, there exists (at least) one $\sigma^{\prime}\in\Phi(a,\sigma)$
such that $\varphi_{i}$ is satisfied in $\sigma^{\prime}$.
Using Lemma 4.25, we can construct a plan
$c_{1}=a;case\ (\varphi_{1}^{\prime}\rightarrow p_{1}^{\prime},\ldots,\varphi^{%
\prime}_{n}\rightarrow p^{\prime}_{n})$
which is potentially regressable and
$\Phi^{*}(c_{1},\sigma)=\Phi^{*}(c,\sigma)$.
From the construction of $c_{1}$, we know that for each
$\sigma^{\prime}\in\Phi(a,\sigma)$ there exists one and only one
$j$, $1\leq j\leq n$, such that $\varphi_{j}^{\prime}$ is satisfied
in $\sigma^{\prime}$. Applying the inductive hypothesis
for $(\sigma^{\prime},\delta)$ and the plan $p_{i}^{\prime}$,
we know that there exists a regressable plan $q_{i}$
which is equivalent to $p_{i}^{\prime}$ with respect to $(\sigma^{\prime},\delta)$.
This implies that
$c^{\prime}=a;case\ (\varphi_{1}^{\prime}\rightarrow q_{1},\ldots,\varphi^{%
\prime}_{n}\rightarrow q_{n})$
is equivalent to $c$ with respect to $(\sigma,\delta)$.
Furthermore, every case plan in $c^{\prime}$ is potentially regressable
and each $q_{i}$ is regressable with respect to $(\sigma^{\prime},\delta)$.
To complete the proof, we will show that $c^{\prime}$ is not
redundant with respect to $(\sigma,\delta)$.
Because
for each $\sigma^{\prime}\in\Phi(a,\sigma)$ there exists
at most one $j$ such that $\varphi_{j}^{\prime}$ is satisfied in
$\sigma^{\prime}$, none of the branches can be removed.
Since $Sens_{a}\setminus\sigma\neq\emptyset$
there are more than one a-state in
$\Phi(a,\sigma)$.
Therefore, we cannot replace $c^{\prime}$ by one of its branches.
This,
together with the fact that $q_{j}$ is not redundant with respect to
$(\sigma^{\prime},\delta)$, implies
that $c^{\prime}$ is not redundant with respect to $(\sigma,\delta)$.
The inductive hypothesis is proved for this case as well.
(2)
$c=\alpha;c_{1}$ where $\alpha$ is a sequence of
non-sensing actions and $c_{1}$ is a case plan.
Let $P_{\alpha}=\{\alpha^{\prime}\mid\alpha^{\prime}$ is a subplan
of $\alpha$ and there exists some $c_{1}^{\prime\prime}$ such that
$\alpha^{\prime};c_{1}^{\prime\prime}$ is equivalent to $c$ with respect to $(\sigma,\delta)\}$.
Let $\beta$ be a member of $P_{\alpha}$
such that $|\beta|=\min\{|\alpha^{\prime}|\mid\alpha^{\prime}\in P_{\alpha}\}$777
For a sequence of actions $\gamma$, $|\gamma|$
denotes the length of $\gamma$.
.
Since $P_{\alpha}\neq\emptyset$, $\beta$ exists.
We have that $\beta$ is a sequence of non-sensing actions,
and so, $\Phi^{*}(\beta,\sigma)=\{\sigma_{1}\}$. It follows from the above
case and the inductive hypothesis that
there exists a regressable plan $c_{1}^{\prime}$ which
is equivalent to $c_{1}$ with respect to $(\sigma_{1},\delta)$.
Consider the plan $c^{\prime}=\beta;c_{1}^{\prime}$.
We have that $c^{\prime}$ is a potentially regressable
conditional plan. To complete the proof, we will show that
$c^{\prime}$ is not redundant with respect to $(\sigma,\delta)$.
Assume the contrary, we will have three cases:
(a)
There exists a subplan $\beta^{\prime}$ of $\beta$
such that $q=\beta^{\prime};c_{1}^{\prime}$ is equivalent to $c^{\prime}$
with respect to $(\sigma,\delta)$. This implies that
$\beta^{\prime};c_{1}$ is equivalent to $c^{\prime}$
with respect to $(\sigma,\delta)$ which contradicts the
construction of $\beta$.
(b)
There exists a subplan $c^{\prime\prime}$ of $c_{1}^{\prime}$
such that $q=\beta;c^{\prime\prime}$ is equivalent to $c^{\prime}$
with respect to $(\sigma,\delta)$. This implies that
$c^{\prime\prime}$ is equivalent to $c_{1}^{\prime}$
with respect to $(\sigma_{1},\delta)$ which contradicts the
construction of $c_{1}^{\prime}$.
(c)
There exists a subplan $\beta^{\prime}$ of $\beta$ and
a subplan $c^{\prime\prime}$ of $c_{1}^{\prime}$
such that $q=\beta^{\prime};c^{\prime\prime}$ is equivalent to $c^{\prime}$
with respect to $(\sigma,\delta)$. This implies that
$\beta^{\prime}\in P_{\alpha}$ and $|\beta^{\prime}|<|\beta|$, which is
a contradiction on the construction of $\beta$.
Thus this
case cannot happen as well.
This shows that $c^{\prime}$ is not redundant with respect to $(\sigma,\delta)$.
So, we have proved that $c^{\prime}$ is regressable
and equivalent to $c$
with respect to $(\sigma,\delta)$.
The inductive step is proved for this case.∎
We are now ready to prove the completeness of our regression formulation,
which is illustrated by Figure 5.
Theorem 4.27 (Completeness of Regression).
Given a planning problem $P=\langle A,O,I,G\rangle$ and a
progression solution $c$ of $P$, there exists a regression solution
$c^{\prime}$ of $P$ such that $c^{\prime}$ is not redundant and is equivalent to $c$
with respect to $(\sigma_{I},\delta_{G})$.
Proof.
Lemma
4.26 implies that there exists a regressable plan $c^{\prime}$ with
respect to $(\sigma_{I},\delta_{G})$ which is equivalent to $c$ with
respect to $(\sigma_{I},\delta_{G})$. The non-redundancy of $c^{\prime}$
follows from the fact that it is a regressable plan. The conclusion
of the theorem follows directly from Lemma 4.23 and Theorem
4.8.∎
5. Related Work
Waldinger [25] is probably the first to discuss
regression in Artificial Intelligence. In his paper, Waldinger uses
the concept of regression in plan modification. To plan for
several goals simultaneously, say $P$ and $Q$, his strategy was to
first find a plan to achieve $P$, then modify that plan to achieve
$Q$. In order to achieve $Q$, regression is used to make sure that
any action added to the existing plan will not interfere with $P$.
Waldinger’s regression is based on the idea of “weakest
precondition” proposed by Dijkstra in 1975 [8](see
also, e.g., [2, 7]). Intuitively, regression from
a logical sentence that is represented by a conjunction of goals,
$conj$, via an action, $A$, yields another logical sentence that
encodes what must be true before $A$ is performed to make $conj$
true immediately afterwards. This is computed by the formula
$$S^{\prime}=Prec(A)\cup(S\setminus Add(A)),$$
where $S$ denotes the set of
goals in the conjunction $conj$, $S^{\prime}$ denotes subgoals in the
regressed conjunction, $Pre(A)$ denotes the set of preconditions of
$A$, and $Add(A)$ denotes the set of add conditions of $A$;
something similar to what is proposed in [26]. Following
Waldinger, Nilsson [15] discusses regression with respect
to partially grounded actions and proposes a regression algorithm
for plan generation.
Another early effort in formulating regression over simple
(non-sensing) actions is due to Pednault [16]. In his Ph.D.
thesis [16], Pednault proposed the language $ADL$ (Action
Description Language) that extends STRIPS and allows, amongst other
things, conditional effects. In addition, Pednault also presents
sound and complete formula-based regression operators for $ADL$
actions. Addressing a similar problem, Reiter [18] also
presents a sound and complete formula-based regression formulation
over simple actions within the Situation Calculus framework. It
reduces reasoning about future situations to reasoning about the
initial situation using first-order theorem proving. Regression
operators are provided for the formulae, with and without functional
fluents.
Scherl and Levesque [19] were probably the first to
extend the regression formulation for simple actions to include
sensing actions. They directly formalize regression in first order
logic, within the framework of Situation Calculus. Their
formula-based regression operator is defined with respect to a set
of successor state axioms which was based on Moore’s formulation of
accessible worlds [13]. They show that, for any plan $P$
expressed by a ground situation term $s_{gr}$ (a ground situation
term is built on the initial situation by repeatedly applying the
function $do$ on it), the axiomatization $F$ of a domain including
the successor state axioms $F_{ss}$, G is an arbitrary sentence then
$$F\models G(s_{gr})\ \;\Leftrightarrow\;\ F\setminus F_{ss}\models R^{*}[G(s_{%
gr})],$$
where $R^{*}(\varphi)$ indicates that the regression
operator is repeatedly applied until the regressed formulae is
unchanged. Intuitively, this shows that the regression is sound and
complete. However, Scherl and Levesque do not define regression over
conditional plans. Later, Reiter adapts the work of Scherl and
Levesque in his book [18]. He does not, however, consider
regression on conditional plans. De Giacomo and Levesque
[6] consider a generalized action theory where successor
state axioms and sensing information are conditionally applicable.
For example the following conditional successor state axiom
[6] expresses that if a robot is alone in a building, then
the status of the door is only defined by the robot’s actions $open$
and $close$.
$Alone(s)\supset$
$DoorOpen(x,do(a,s)\equiv$
$a=open(x)\vee(a\neq close(x)\wedge DoorOpen(x,s)).$
Here the sensor fluent formula $Alone(s)$ expresses the
condition that the robot is alone in the building in a situation
$s$, $DoorOpen(x,do(a,s)$ expresses the fact that a door $x$ is
open in the situation after the robot performs an action in the
situation $s$, and $DoorOpen(x,s)$ expresses the fact the door $x$
is open in the situation $s$. Similarly, the following conditional
sensed fluent axiom [6] expresses the condition that if
the robot is outdoors, then its on-board thermometer always measures
the temperature around the robot.
$Outdoor(s)\supset$
$OutDoorTemperature(n,s)\equiv thermometer(s)=n$.
Their formula-based regression is then defined over
histories. A history is defined as a sequence
$(\overrightarrow{v_{0}}).(A_{1},\overrightarrow{v_{1}}),\ldots,(A_{n},%
\overrightarrow{v_{n}})$ where each $A_{i}$ is an action,
$\overrightarrow{v_{i}}$ represents a vector of the values $\langle v_{i,1},\ldots,v_{i,m}\rangle$ and $v_{i,j}$ represents the
reading value of $j^{th}$ sensor after the $i^{th}$ action.
However, they showed that the regression although sound, does
not guarantee completeness in some circumstances. They also did
not consider regression on conditional plans.
In another direction, Son and Baral [20] study
regression over sensing actions using the high-level action language
${\mathcal{A}}_{K}$. In this work, they provide a state-based transition
function and a formula-based regression function with respect to the
full semantics. Different from the work of [6, 18], Son
and Baral define regression over conditional plans. They also prove
that their regression formulation is both sound and complete with
respect to the transition function. However in [20],
Son and Baral do not consider precondition of actions in their
regression formulation.
The regression formalism presented in this paper differs from
earlier notion of regression for action theories with sensing
actions in [18, 19, 20] in that our definition
is a state-based regression formalism while the earlier definitions
are formula-based. With regards to regression on conditional plans,
we are not aware of any other work except [20]. For
regression on non-sensing actions, our definition is close to the
formula used in [4].
6. Conclusion, Discussion, and Future Work
In this paper, we developed a state-based regression function
in domains with sensing actions, incomplete information, and actions without
conditional effects.
We also extended the regression function to allow for the
regression over conditional plans.
We proved the soundness of the extended regression function
with respect to the definition of the progression function and
developed a relaxed notion of completeness for the regression
function.
It is interesting to note that for planning problems described in
this paper, the progression function developed in this paper is
equivalent to the full semantics for domains with sensing actions
and incomplete information and to the 0-approximation developed in
[20]. This implies that the regression function
$\mathcal{R}$ (and hence ${\mathcal{R}}^{*}$) is also complete with
respect to the full semantics for planning problems as defined in
Section 2.1. Since the complexity of (conditional) planning
with respect to the 0-approximation is lower than that with respect
to the full semantics of sensing actions, this means that the
conditional planning problem for domains presented in this paper has
a lower complexity than it is in general. In other words, the
complexity of the conditional planning problem presented in this
paper in NP-complete, whereas the complexity of the
conditional planning problem for action theories with conditional
effects is $\Sigma^{2}_{P}$-complete [1]. We
observe that this complexity results are somewhat different than the
complexity results in [3], as the planning problems in
[3] do not contain sensing actions and are complete.
It should be noted that the notion of a conditional plan in this paper
is not as general as in [20]. For example, we
do not consider plans of the form $c_{1};c_{2}$ where $c_{1}$ and $c_{2}$ are
case plans. This is done to make the presentation of
the proofs easier to follow. Indeed, in [22], we proved that
all of the theorems in this paper are valid with respect to
conditional plans defined in [20].
Finally, we would like to mention that we have developed
a regression-based planner, called CPR, using the regression formulation
proposed in this paper [23]. The planner
employs the best first search strategy with a heuristic function
similar to the HSP-r heuristic function [4].
Due to the fact that most of the available benchmarks in planning
with sensing actions allow disjunction in the initial state and
conditional effects, an experimental evaluation of CPR
against other planners could not be done with respect to the benchmarks.
We have therefore developed our own domains to test CPR.
Our initial experimental result shows that CPR performs reasonably
well [23]. The code of CPR and the domains are
available at http://www.cs.nmsu.edu/~tson/CPR.
Our main goal in the near future is to extend the regression
formalism proposed in this paper to allow conditional effects
and disjunctive initial states. This will allow us to extend CPR
to deal with conditional effects and to evaluate the planning approach
based on regression against forward chaining approaches.
Acknowledgment
We are grateful to the anonymous referees whose useful comments
helped us improve this paper. We would also like to thank Tu Phan for his
comments on an ealier version of this paper which help us establish
Proposition 1.
A preliminary version of this paper appeared in [24].
Le-Chi Tuan and Chitta Baral were partially supported by NSF grants
0070463 and 0412000 and a ARDA/DTO contract.
Tran Cao Son was also partially supported by NSF grants CNS-0454066
EIA-0220590 and HRD-0420407.
References
[1]
C. Baral, V. Kreinovich, and R. Trejo.
Computational complexity of planning and approximate planning in the
presence of incompleteness.
Artificial Intelligence, 122:241–267, 2000.
[2]
E. Best.
Semantics of Sequential and Parallel Programs.
Prentice Hall, 1996.
[3]
T. Bylander.
The Computational Complexity of Propositional STRIPS Planning.
Artificial Intelligence, 69:165–204, 1994.
Computational complexity of planning and approximate planning in the
presence of incompleteness.
Artificial Intelligence, 122:241–267, 2000.
[4]
B. Bonet and H. Geffner.
Planning as Heuristic Search.
Artificial Intelligence, 129(1–2):5–33, 2001.
[5]
D. Bryce, S. Kambhampati, and D. Smith.
Planning Graph Heuristics for Belief Space Search.
Technical report, Arizona State University, Computer Science and
Engineering, 2004.
http://www.public.asu.edu/~danbryce/papers/.
[6]
G. De Giacomo and H. Levesque.
Projection using regression and sensors.
In Proc. of the Sixteen International Joint Conference on
Artificial Intelligence (IJCAI’99), pages 160–165, 1999.
[7]
J. W. De Bakker.
Mathematical theory of program correctness.
Englewood Cliffs, N.J, Prentice-Hall International, 1980.
[8]
E. W. Dijkstra.
Guarded commands, nondeterminacy and formal derivation of programs.
In Communications of the ACM, 18(8):453457, August 1975.
[9]
R. Fikes and N. Nilson.
STRIPS: A new approach to the application of theorem
proving to problem solving.
Artificial Intelligence, 2(3–4):189–208, 1971.
[10]
H. Levesque.
What is planning in the presence of sensing?
In Proceedings of the 14th Conference on Artificial
Intelligence, pages 1139–1146. AAAI Press, 1996.
[11]
J. Lobo.
COPLAS: a COnditional PLAnner with Sensing actions.
Technical Report FS-98-02, AAAI, 1998.
[12]
J. Lobo, S. Taylor, and G. Mendez.
Adding knowledge to the action description language ${\mathcal{A}}$.
In AAAI 97, pages 454–459, 1997.
[13]
R. Moore.
A formal theory of knowledge and action.
In J. Hobbs and R. Moore, editors, Formal theories of the
commonsense world. Ablex, Norwood, NJ, 1985.
[14]
X.L Nguyen, S. Kambhampati, and R. Nigenda.
Planning graph as the basis for deriving heuristics for plan
synthesis by state space and CSP search.
Artificial Intelligence, 135(1-2):73–123, 2002.
[15]
N. Nilson.
Principles of Artificial Intelligence.
Tioga publishing company, 1980.
[16]
E. Pednault.
Toward a Mathematical Theory of Plan Synthesis.
PhD thesis, Stanford University, 1986.
[17]
L. Pryor and G. Collins.
Planning for contingencies: A decision-based approach.
Journal of Artificial Intelligence Research, 4:287–339, 1996.
[18]
R. Reiter.
KNOWLEDGE IN ACTION: Logical Foundations for Describing and
Implementing Dynamical Systems.
MIT Press, 2001.
[19]
R. Scherl and H. Levesque.
Knowledge, action, and the frame problem.
Artificial Intelligence, 144(1-2), 2003.
[20]
T.C. Son and C. Baral.
Formalizing sensing actions - a transition function based approach.
Artificial Intelligence, 125(1-2):19–91, January 2001.
[21]
P.H. Tu, T.C. Son, and C. Baral.
Reasoning and Planning with Sensing Actions, Incomplete
Information, and Static Causal Laws using Answer Set Programming.
Theory and Practice of Logic Programming, 2006.
[22]
L.C. Tuan.
Regression in the Presence of Incomplete Information and Sensing
Actions, and its Application to Conditional Planning.
PhD thesis, Arizona State University, 2004.
[23]
L.C. Tuan, C. Baral, and T.C. Son.
Regression-based Conditional Planning in the Presence of Sensing
Actions and Uncertainty in the Initial State.
Technical report, Computer Science Department, New Mexico State
University, 2005.
http://www.cs.nmsu.edu/TechReports/2005/004.ps.
[24]
L.C. Tuan, C. Baral, X. Zhang, and T.C. Son.
Regression With Respect to Sensing Actions and Partial States.
In Proceedings of the Nineteenth National Conference on
Artificial Intelligence (AAAI’04), pages 556–561. AAAI Press, 2004.
[25]
R. Waldinger.
Achieving several goals simultaneously.
Machine Intelligence, pages 94–136, 1977.
[26]
D. Weld.
An introduction to least commitment planning.
AI Magazine, 15(4):27–61, winter 1994.
[27]
D. Weld, C. Anderson, and D. Smith.
Extending graphplan to handle uncertainty and sensing actions.
In Proceedings of the Fifteenth National Conference on
Artificial Intelligence. AAAI Press, 1998. |
Addendum to my paper
“The Lebesgue summability of trigonometric integrals”
(J. Math. Anal. Appl. 390 (2012), 188-196)
Ferenc Móricz
We observed that statement (2.9) in Theorem 2 remains valid if condition (2.7) is replaced
by the weaker condition that
$$f(t)\in L^{1}(-T,T)\quad{\rm for\ all}\quad T>0.$$
(2.7′)superscript2.7′( 2.7 start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )
Furthermore, Theorem 3 also remains valid if condition (2.7) is replaced by (2.7${}^{\prime}$).
To be more precise, the following Theorems 2${}^{\prime}$ and 3${}^{\prime}$ can be proved in the same way as
Theorems 2 and 3 are proved, while using Lemmas 2${}^{\prime}$ and 3${}^{\prime}$ below instead of Lemmas 2 and 3.
Theorem 2${}^{\prime}$. If $f:{\tenopen R}\to{\tenopen C}$ is such that conditions (2.7${}^{\prime}$)
and
$$\lim_{T\to\infty}{1\over T}\int_{|t|<T}|tf(t)|dt=0$$
(2.8)2.8( 2.8 )
are satisfied, then we have uniformly in $x\in{\tenopen R}$ that
$$\lim_{h\downarrow 0}\Big{\{}{\Delta{\cal L}(x;h)\over 2h}-I_{1/h}(x)\Big{\}}=0.$$
(2.9)2.9( 2.9 )
Theorem 3${}^{\prime}$. Suppose $f:{\tenopen R}\to{\tenopen C}$ is such that conditions
(2.7${}^{\prime}$) and
$${1\over T}\int_{|t|<T}|tf(t)|dt\leq B\quad for\ all\quad T>T_{1},$$
(2.10)2.10( 2.10 )
are satisfied, where $B$ and $T_{1}$ are constants. If the finite limit
$$\lim_{T\to\infty}I_{T}(x):=\lim_{T\to\infty}\int_{|t|<T}f(t)e^{itx}dt=\ell$$
(2.3)2.3( 2.3 )
exists at some point $x\in{\tenopen R}$, then (2.9) holds at this $x$.
We emphasize that in this addendum, the definition
$$\int_{\tenopen R}f(t){e^{itx}\over it}dx=:{\cal L}(x),\quad x\in{\tenopen R},$$
(2.4)2.4( 2.4 )
is interpreted only formally; that is, the integral in (2.4) may not exist in
Lebesgue’s sense. However, under the conditions in Theorems 2${}^{\prime}$ and 3${}^{\prime}$, the integral in the
representation
$${\Delta{\cal L}(x;h)\over 2h}:=\int_{\tenopen R}f(t)e^{itx}{\sin th\over th}dt%
,\quad h>0,$$
(2.6)2.6( 2.6 )
does exist in Lebesgue’s sense.
As we have mentioned above, the proofs of Theorems 2${}^{\prime}$ and 3${}^{\prime}$ hinge on the following Lemmas 2${}^{\prime}$ and 3${}^{\prime}$.
We note that we essentially use only Part (i)
in our earlier Lemmas 2 and 3, while we substitute condition (2.7${}^{\prime}$) for (2.7).
Lemma 2${}^{\prime}$. If $f:{\tenopen R}\to{\tenopen C}$ is such that condition (2.7${}^{\prime}$) and (2.8)
are satisfied, then
$$\lim_{T\to\infty}T\int_{|t|>T}\Big{|}{f(t)\over t}\Big{|}dt=0.$$
(3.2)3.2( 3.2 )
Lemma 3${}^{\prime}$. If $f:{\tenopen R}\to{\tenopen C}$ is such that conditions (2.7${}^{\prime}$) and (2.10) are satisfied, then
$$T\int_{|t|>T}\Big{|}{f(t)\over t}\Big{|}dt\leq 4B\quad for\ all\quad T>T_{1}.$$
(3.8)3.8( 3.8 )
We note that the converse implications (3.2) $\Rightarrow$ (2.8) in Lemma 2${}^{\prime}$ and (3.8) $\Rightarrow$
(2.10) in Lemma 3${}^{\prime}$ do hold under the supplementary condition (2.7). But we do not need these converse implications in
the proofs of Theorems 2${}^{\prime}$ and 3${}^{\prime}$. |
Frequency Resolved Measurement of Longitudinal Impedances Using
Transient Beam Diagnostics††thanks: This work was supported by DOE contract
No. DE-AC03-76SF00515
D. Teytelman
dim@slac.stanford.edu
J. Fox
S. Prabhakar
Stanford Linear Accelerator Center
Stanford
CA
J.Byrd
Lawrence Berkeley National Laboratory
One Cyclotron Road
Berkeley
CA
Abstract
In this paper we present several techniques for characterizing
longitudinal impedances based on transient measurements of the growth rates
and tune shifts of unstable coupled-bunch modes. These techniques are
applicable to measurement of both fundamental and higher-order mode
impedances and allow characterization of shunt impedances and quality
factors of the HOMs. Methods presented here are complementary to lab bench
measurements of RF cavities, in that the beam based measurements directly
sense the physical impedance in the installed configuration. In contrast to
a single-bunch integrated impedance measurement these techniques resolve the
impedances in the frequency domain. These methods allow determination of the
impedance’s unaliased frequency by analyzing synchronous phase
transients. Experimental results from ALS and BESSY-II are presented showing
the use of these techniques to measure complex impedances.
SLAC–PUB–8884
June 2001
Frequency Resolved Measurement of Longitudinal Impedances Using
Transient Beam Diagnostics111Work supported by
Department of Energy contract DE–AC03–76SF00515.
D. Teytelman, J. Fox, S. Prabhakar
Stanford Linear Accelerator Center, Stanford University,
Stanford, CA 94309
J. Byrd
Lawrence Berkeley National Laboratory,
1 Cyclotron Road, Berkeley, CA 94563
Abstract
In this paper we present several techniques for characterizing
longitudinal impedances based on transient measurements of the growth rates
and tune shifts of unstable coupled-bunch modes. These techniques are
applicable to measurement of both fundamental and higher-order mode
impedances and allow characterization of shunt impedances and quality
factors of the HOMs. Methods presented here are complementary to lab bench
measurements of RF cavities, in that the beam based measurements directly
sense the physical impedance in the installed configuration. In contrast to
a single-bunch integrated impedance measurement these techniques resolve the
impedances in the frequency domain. These methods allow determination of the
impedance’s unaliased frequency by analyzing synchronous phase
transients. Experimental results from ALS and BESSY-II are presented showing
the use of these techniques to measure complex impedances.
Presented at IEEE Particle Accelerator Conference (PAC 2001), Chicago,
Illinois, 18-22 Jun 2001
1 Introduction
The interaction of charged particles in a storage ring or circular
accelerator with the ring impedance determines many important accelerator
dynamics parameters. Single and multi-bunch instabilities are the result of
interactions of the bunches with the impedance of the machine, and achieving
high stored currents requires knowledge and control of the ring components
which produce the dominant narrow-band impedances. There are several
laboratory techniques to measure impedances of physical components
[1, 2]. Beam-based impedance measurement
techniques exist as well. Frequency-resolved information about the coupling
impedance can be extracted from a measurement of the beam transfer function
(BTF) [3]. However such a measurement can only be
performed below the instability threshold. In addition network analyzer
sweeps have to be repeated for each unstable mode making BTF approach slow
and cumbersome.
This paper presents several beam-based longitudinal impedance measurement
techniques. These fast transient multi-bunch techniques measure the aliased
longitudinal impedance as a function of frequency in a sampling bandwidth up
to 1/2 the RF frequency. Consequently various higher-order mode resonators
can be identified and their complex impedance (and parameters such as center
frequency and Q) measured.
2 Longitudinal impedances and coupled-bunch instabilities
Bunches of charged particles passing through the vacuum chamber of a storage
ring leave behind electromagnetic fields. These fields (wake fields) affect
the energy of the following bunches providing a bunch-to-bunch coupling
mechanism. At high beam currents such coupling can cause instabilities.
The bunch motion in a storage ring can be projected onto the orthonormal
basis of the even fill eigenmodes (EFEMs).
Eigenvalue of mode $l$ is given by [4]
$$\displaystyle\Lambda_{l}=-d_{r}+j\omega_{s}+\frac{\pi\alpha ef_{rf}^{2}I_{0}}{%
E_{0}h\omega_{s}}Z^{\parallel eff}(l\omega_{0}+\omega_{s})$$
(1)
$$\displaystyle Z^{\parallel eff}(\omega)=\frac{1}{\omega_{rf}}\sum_{p=-\infty}^%
{\infty}(ph\omega_{0}+\omega)Z^{\parallel}(ph\omega_{0}+\omega)$$
(2)
where $d_{r}$ is the radiation damping rate, $\omega_{s}$ is the synchrotron
frequency, $\alpha$ is the momentum compaction factor, $e$ is the charge of
the electron, $f_{rf}$ is the frequency in the accelerating cavities, $I_{0}$
is the beam current, $E_{0}$ is the beam energy, $h$ is the ring harmonic
number, $\omega_{0}$ is the revolution frequency, and $Z^{\parallel}(\omega)$
is the total longitudinal impedance.
In order to measure modal eigenvalues $\Lambda$ we use the capabilities of a
programmable longitudinal feedback system [5]. The system is
able to measure the unique synchronous phase and centroid motion of every
bunch in a storage ring, and uses digital memory to record time sequences of
the bunch motion. In a transient grow/damp measurement feedback loop is
opened under software control for a predetermined period of time and then
closed. In the open-loop conditions unstable modes grow exponentially due to
noise and feedback system records the motion of the bunches during the
transient. The motion is then projected on the EFEM basis and modal
exponential growth and damping rates as well as oscillation frequencies are
extracted [6]. Once the eigenvalues are measured it is
possible to extract the aliased impedance according to Eq. 1.
The aliased beam-derived impedance, combined with knowledge about the
impedances from bench measurements of ring components may be properly
assigned as an unaliased impedance.
3 Synchronous phase transients
For the cases when ring fill pattern is uneven additional information about
the impedance can be obtained from analyzing the dependence of
synchronous phases on bunch currents. Previous work by Prabhakar
[7] presents the relationship between the bunch
currents, impedances, and synchronous phases. This work is applicable to
fill patterns where all buckets are populated, however unevenly. For empty
buckets synchronous phase is not measurable. Extending the analysis to fills
with empty buckets (gaps) we get
$$\displaystyle\vec{\phi}_{U}=\frac{-N}{|V_{c}cos(\phi_{s}^{0})|}{\bf A}^{UV}%
\vec{Z}^{\dagger}_{V}$$
(3)
$$\displaystyle Z^{\dagger}_{n}=\sum_{m=-\infty}^{\infty}Z^{\parallel}((mN+n)%
\omega_{0})$$
where $\vec{\phi}$ is the vector of bunch phases, $U$ is the set that
includes all non-empty buckets, $V_{c}$ is the peak RF cavity voltage and
$\phi_{s}^{0}$ is the synchronous phase in absence of wake fields. Matrix
${\bf A}^{UV}$ is computed using inverse DFT (Discrete Fourier Transform)
matrix and a DFT of the vector of individual bunch currents. Set $V$
includes revolution harmonics excited by the DFT of bunch currents. By
solving an overdetermined linear system of equations described by
Eq. 3 in the least-squares sense we obtain
$\vec{Z}^{\dagger}_{V}$.
4 ALS measurements
The goal of the first measurement is to quantify the HOM impedances of the
two 500 MHz main RF cavities installed at the ALS. Past measurements have
determined that there are two dominant EFEMs, modes 205 and 233, excited by
the impedances in the main RF cavities [8]. Using the
measurements made on the spare cavity identical to the ones installed in the
ring mode 205 had been identified as driven by the $TM_{011}$ longitudinal
mode at 812 MHz. Mode 233 has two potential driving HOMs, at 2.353 GHz and
2.853 GHz [2].
Due to technical limitations it is only possible to fill 320 RF
buckets at the ALS. All of the transient measurements described here were
taken with 320 buckets maximally equally filled leaving a gap of 8 RF
buckets. Since the gap is small we assume that eigenmodes of the fill are
very close to those of an even-fill.
In order to characterize the frequency dependence of the impedance we
shifted the center frequencies of the cavity HOM resonances by changing the
temperature of the cavity. At each
point the temperature was allowed to stabilize and the open-loop eigenvalues
of the unstable modes were measured using the transient grow-damp technique.
In Fig. 1 the growth rates and oscillation frequencies of
modes 205 and 233 are plotted versus temperature of cavity 2.
These measurements agree well with the expected effect of the HOM resonators.
However these measurements do not provide a way to distinguish between the
two possible HOMs at 2.353 and 2.853 GHz as the source of the aliased
impedance. To resolve this ambiguity the ring was filled with
a single bunch while a cavity probe signal was monitored on a spectrum
analyzer. We observed that change of cavity temperature had very small
effect on the magnitude of the revolution harmonics excited within the
2.353 GHz resonance while signal at 2.853 GHz scaled with temperature in
agreement with the growth rate measurements.
Thus the resonance measured in the temperature scan is at 2.853 GHz.
In addition the impedance presented by the 2.353 GHz HOM can be considered
constant.
In order to quantify impedance parameters $R_{s}$ and $Q$ we convert cavity
temperatures to center frequencies of the resonance. Conversion factor is
determined by matching cavity probe signal levels between two temperatures and two
RF frequency settings. Using nonlinear least-squares estimation we extract
parameter values. In Table 1 results for both cavities are
summarized. Note that characteristics of the 2.853 GHz resonances in two
cavities differ significantly. The cavities have RF windows of different
designs which can cause variations in the $R/Q$ values. Additionally, the
mode in question is close to the beam pipe cut-off frequency and is strongly
affected by the field leakage.
Using growth rates vs. RF cavity temperature results it is possible to
optimize operating temperatures of the main RF cavities. Since temperatures
affect the transverse impedances as well as longitudinal impedances, mapping
growth rates in horizontal and vertical planes is necessary for a full
understanding of the tradeoff.
5 BESSY-II measurements
These measurements were aimed at quantifying longitudinal impedances at
BESSY-II. The machine was filled with 350 consecutive
bunches out of 400 to a current of 165 mA. A series of 15 transient
grow/damp experiments was conducted over a period of 10 minutes during which
the machine configuration remained unchanged. There are three unstable EFEMs
seen in the data: 281, 396, and 397. Using Eq. 1 we extract
complex longitudinal impedances from the measured growth rates and
oscillation frequencies.
$$\displaystyle Z^{\parallel eff}_{281}$$
$$\displaystyle=$$
$$\displaystyle(63.2\pm 8.1)+(0\pm 94)j\ \textrm{k}\Omega$$
$$\displaystyle Z^{\parallel eff}_{396}$$
$$\displaystyle=$$
$$\displaystyle(59.0\pm 3.3)+(1115\pm 53)j\ \textrm{k}\Omega$$
$$\displaystyle Z^{\parallel eff}_{397}$$
$$\displaystyle=$$
$$\displaystyle(59.6\pm 3.7)-(726\pm 36)j\ \textrm{k}\Omega$$
Impedance measurement for modes 396 and 397 correlates well with the
impedance of four third harmonic cavities parked between 3 and 4 revolution
harmonics below $3f_{rf}$.
As described in Sec. 3 we can estimate the impedance by
analyzing the synchronous phase transient. In Fig. 2
synchronous phase transient in BESSY-II is presented with
350 consecutive buckets filled nearly equally.
Periodic pulse excitation of the fill pattern generates oscillatory behavior
of the synchronous phases. Solving
Eq. 3 in the least-squares sense we obtain aliased
impedances. Least-squares estimate of the synchronous phases is also shown
in Fig. 2 for comparison with experimental data. Using
15 BESSY transient measurements described above we get
$Z^{\dagger}_{396}=(35\pm 22)+(344\pm 14)j$ k$\Omega$ and
$Z^{\dagger}_{397}=(22\pm 6)-(233\pm 15)j$ k$\Omega$.
The two methods of measuring the impedance can be used together in order to
determine unaliased frequencies. This is possible due to the fact that
during aliasing into $Z^{\parallel eff}$ impedance is scaled by resonant
frequency, while in $Z^{\dagger}$ it is unscaled. From Eq. 2 we
have
$$\displaystyle|Z^{\parallel eff}_{396}|$$
$$\displaystyle=$$
$$\displaystyle\frac{(ph+396)\omega_{0}}{h\omega_{0}}|Z^{\dagger}_{396}|$$
(4)
$$\displaystyle p_{exp}$$
$$\displaystyle=$$
$$\displaystyle\frac{|Z^{\parallel eff}_{396}|}{|Z^{\dagger}_{396}|}-\frac{396%
\omega_{0}}{400\omega_{0}}=2.2$$
Since $p$ in Eq. 4 is an integer by definition, comparison
above indicates that the physical impedance is at $2\omega_{rf}+396\omega_{0}=3\omega_{rf}-4\omega_{0}$. This conclusion agrees perfectly
with the expected position of the parked third-harmonic cavities.
6 Summary
We have demonstrated several methods for measuring the impedance of
accelerator components using transient diagnostic capabilities of the
DSP-based longitudinal feedback systems. The methods extend the capabilities
of laboratory bench measurements by quantifying the physical impedances as
installed in the accelerator. Dependence of the impedances on operating
conditions such as temperature or tuner position can be extracted and used
to select optimal working points. By comparing information obtained from
growth transients with the analysis of the synchronous phase transients for
uneven fills it is possible to determine the spectral position of the
driving impedance.
7 Acknowledgments
Authors would like to thank Jorn Jacob of ESRF and Greg Stover of LBNL for
help in setting up and conducting ALS measurements. We also thank Shaukat
Khan and Tom Knuth for setting up and taking BESSY-II transient measurements.
References
[1]
L. Palumbo and V. G. Vaccaro, in Frontiers of Particle Beams:
Observation, Diagnosis and Correction (Springer-Verlag, Berlin, 1989), pp. 312–354.
[2]
J. N. Corlett and J. M. Byrd, in 1993 IEEE Particle Accelerator
Conference: proceedings (IEEE, Piscataway, NJ, USA, 1994), pp. 3408–3410.
[3]
A. Hofmann and B. Zotter, IEEE Trans. Nucl. Sci. 24, 1487 (1977).
[4]
S. Prabhakar, Ph.D. thesis, Stanford University, 2000, SLAC-R-554.
[5]
J. Fox et al., in Proceedings of the 1999 Particle Accelerator
Conference (IEEE, Piscataway, NJ, USA, 1999), pp. 636–640.
[6]
S. Prabhakar, J. D. Fox, D. Teytelman, and A. Young, Phys. Rev. ST Accel. Beams
2, 084401 (1999).
[7]
S. Prabhakar et al., in Proceedings of the Sixth European Particle
Accelerator Conference (IOP Publishing, Bristol, 1998), pp. 996–998.
[8]
S. Prabhakar et al., Part. Accel. 57, 175 (1997). |
Spectroscopy of phase transitions for multiagent systems
Niccolò Zagli
n.zagli18@imperial.ac.uk
Department of Mathematics, Imperial College London, London, SW7 2AZ, UK
Department of Mathematics and Statistics, University of Reading, Reading, RG6 6AX, UK
Centre for the Mathematics of Planet Earth, University of Reading, Reading, RG6 6AX, UK
Valerio Lucarini
Department of Mathematics and Statistics, University of Reading, Reading, RG6 6AX, UK
Centre for the Mathematics of Planet Earth, University of Reading, Reading, RG6 6AX, UK
Grigorios A. Pavliotis
Department of Mathematics, Imperial College London, London, SW7 2AZ, UK
Abstract
In this paper we study phase transitions for weakly interacting multiagent systems. By investigating the linear response of a system composed of a finite number of agents, we are able to probe the emergence in the thermodynamic limit of a singular behaviour of the susceptibility. We find clear evidence of the loss of analyticity due to a pole crossing the real axis of frequencies. Such behaviour has a degree of universality, as it does not depend on either the applied forcing nor on the considered observable. We present results relevant for both equilibrium and nonequilibrium phase transitions by studying the Desai-Zwanzig and Bonilla-Casado-Morillo models.
pacs: 05.40.-a, 05.45.-a, 05.45.Xt, 05.70.Fh, 64.60.-i
Multiagent models feature in a very vast range of applications in natural sciences, social sciences, and engineering. We study here the Desai-Zwanzig and Bonilla-Casado-Morillo models, which are paradigmatic for equilibrium and nonequilibrium conditions, respectively. Phase transitions result from the coordination between the individual agents, and are associated with the divergence of the linear response of the system. The occurrence of phase transitions is universal: it does not depend on the acting forcing, and can be detected by looking at virtually any observable of the system. We showcase here how response theory is capable of providing a useful angle for understanding the universal properties of phase transitions in complex systems.
I Introduction
Agent based models are regularly employed to model various phenomena in the natural sciences, social sciences and engineering Naldi, Pareschi, and Toscani (2010); Pareschi and Toscani (2013).
Multiagent systems are used to model diverse phenomena such as cooperation Dawson (1983), synchronisation Acebrón et al. (2005), systemic risk Garnier, Papanicolaou, and Yang (2013) and consensus formation Wang et al. (2017); Garnier, Papanicolaou, and Yang (2017). They are fundamental in developing algorithms for sampling and optimization Garbuno-Inigo, Nüsken, and Reich (2020) and they have also been used for the management of natural hazard Simmonds, Gómez, and Ledezma (2019) and climate change impact Geisendorf (2018).
Multiagent systems can often exhibit abrupt changes in their behaviour, often corresponding to critical transitions that occur when a parameter, e.g. interaction strength or temperature, passes a certain threshold. Such transitions are often associated to cataclysmic events such climate change, market crashes etc Scheffer (2009); Sornette (0006). The importance of developing tools for predicting critical transitions has long been recognized. One of the main tools used in order to develop early warning signals for critical transitions is that of linear response theory.
Following the seminal contribution by Kubo Kubo (1966), linear response theory represents a very powerful framework for studying the properties of statistical mechanical systems by investigating how they respond to external perturbations Marconi et al. (2008a); Baiesi and Maes (2013); Sarracino and Vulpiani (2019). Linear response theory has been successfully applied to classic problems of solid-state physics and optics Lucarini et al. (2005) as well as plasma physics and stellar dynamics (Binney and Tremaine, 2008, Ch.5); see some examples of application of the theory in both equilibrium and nonequilibrium systems Leith (1975); North, Bell, and Hardin (1993); Öttinger (2005); Lucarini, Ragone, and Lunkeit (2017); Cessac (2019); Gottwald (2020). Rigorous mathematical foundations for linear response theory have been provided for the case of Axiom A systems Ruelle (1998, 2009) (see e.g. Baladi (2014) for further developments in the context of deterministic systems) and for diffusion processes, both in finite and in infinite dimensions Dembo and Deuschel (2010); Hairer and Majda (2010); see also the interesting contributions Wormell and Gottwald (2019) that bridges the deterministic and the stochastic viewpoints.
Critical transitions arise when the spectral gap of the transfer operator of the unperturbed system shrinks to zero Liverani and Gouëzel (2006); Chekroun et al. (2014); Lucarini (2016); Tantet, Lucarini, and Dijkstra (2018) as the Ruelle-Pollicott poles Pollicott (1985); Ruelle (1986), touch the real axis. Near criticality, the negative feedbacks of the system become increasingly ineffective, resulting in arbitrarily large, usually non-Gaussian, fluctuations and a divergence of correlation properties of the system Dawson (1983); Shiino (1987); Delgadino, Gvalani, and Pavliotis (2021).
In the thermodynamic limit, multiagent system can also undergo a qualitative change of their properties through a different mathematical mechanism, namely phase transitions Dawson (1983); Shiino (1987), defined as exchange of stability of nonunique stationary distributions as the parameters of the systems vary; see a detailed analysis in Carrillo et al. (2020).
In a previous paper Lucarini, Pavliotis, and Zagli (2020) we derived linear response formulas for a system of weakly interacting diffusions described by an $N-$particle Fokker-Planck equation and have explicitly identified two qualitatively different scenarios for the breakdown of the linear response, associated with the previously mentioned critical transitions and phase transitions. We focus here on the latter case.
Phase transitions are a genuine thermodynamic phenomenon, where the divergence of the response stems from the coordination taking place, in suitable conditions, because of the coupling between the infinite number of agents composing the total system. The coupling among the subsystems results in a memory effect that leads to obtaining the macroscopic response function of the system as a renormalised version of its microscopic counterpart Lucarini, Pavliotis, and Zagli (2020), with formal similarities with the well-known Clausius-Mossotti relation Lucarini et al. (2005); Jackson (1975); Talebian and Talebian (2013). The role of memory in determining criticality due to endogenous processes has been emphasised in Sornette and Helmstetter (2003); Sornette (0006). The link between phase transitions and slow decay of correlations for interacting particle systems is well established, see. e.g. Yoshida (2003).
I.1 This Paper
In this paper we focus in much greater detail on the relationship between the occurrence of phase transitions and the non-analyticity of the susceptibility of the system describing the frequency-dependent response of an observable to a given perturbation in the upper complex frequency plane. The singularity manifests itself as a pole that crosses the real axis of the frequency variable. We use a formalism that mirrors spectroscopic techniques that are used for investigating the frequency dependence of the optical properties of materials Lucarini et al. (2005). By studying how the real and imaginary part of the susceptibility of the systems depend on the number of agents, we are able to predict the position of the pole and the associated residue, which describe the emergence of the singularity in the thermodynamic limit. We verify that the position of the pole depends on the considered model, but, instead, that for a given model the loss of analyticity depends neither on the choice of the observable, nor on the applied perturbation, and is, in this sense, an universal feature of the system.Our numerical investigations are performed on the Desai-Zwanzig (DZ) Desai and Zwanzig (1978) and the Bonilla-Casado-Morillo (BCM) Bonilla, Casado, and Morillo (1987) models. The DZ model exhibits a paradigmatic example of an equilibrium order-disorder phase transition, analogous to the Ising ferromagnetic transition Shiino (1987); Dawson (1983), while the BCM model describes an out-of-equilibrium synchronisation transition of an infinite collection of coupled nonlinear oscillators. As the transition point is crossed, the order parameter (magnetization)
acquires a non vanishing constant value for the DZ model and is periodically oscillating for the BCM model.
II The general framework
We investigate a system composed of $N$ exchangeable interacting $M$ dimensional sub-systems whose dynamics is determined by the following Itô stochastic differential equations (SDEs)
$$\mathrm{d}\mathbf{x}^{k}=\mathbf{F}_{\alpha}(\mathbf{x}^{k})\mathrm{d}t-\frac{\theta}{N}\sum_{l=1}^{N}\mathbf{\nabla}\mathcal{U}\left(\mathbf{x}^{k}-\mathbf{x}^{l}\right)\mathrm{d}t+\sigma\mathbf{S}(\{\mathbf{x}^{k}\})\mathrm{d}\mathbf{W},$$
(1)
where $\mathbf{x}^{k}\in\mathbb{R}^{M}$ and $k=1,\ldots,N$.
$\mathbf{F_{\alpha}}:\mathbb{R}^{M}\rightarrow\mathbb{R}^{M}$ is a smooth vector field, possibly depending on a parameter $\alpha$, and $\mathbf{W}$ denotes a standard $P-$dimensional Brownian motion; $\mathbf{S}:\mathbb{R}^{M}\rightarrow\mathbb{R}^{M\times P}$ is the volatility matrix and the parameter $\sigma>0$ controls the strength of the stochastic forcing, i.e. plays the role of the temperature.
We consider a fully coupled system given by the quadratic (Curie-Weiss) interaction potential $\mathcal{U}\left(\mathbf{y}\right)=\frac{|\mathbf{y}|^{2}}{2}$. In this case, the order parameter is known and it is given by the first moment/magnetization.
The coefficient $\theta$ modulates the intensity of the coupling, which attempts at synchronising all systems by attracting them to the center of mass. In the thermodynamic limit $N\rightarrow\infty$ the one-particle distribution function converges to the distribution $\rho(\mathbf{x},t)$ that satisfies a nonlinear and nonlocal Fokker-Planck equation Dawson (1983); Sznitman (1989); Oelschlager (1984); Dawson and Gärtner (1987)
$$\begin{split}\partial_{t}\rho(\mathbf{x},t)=&-\nabla\cdot\left[\rho(\mathbf{x},t)\left(\mathbf{F}_{\alpha}(\mathbf{x})+\theta\left(\langle\mathbf{x}\rangle(t)-\mathbf{x}\right)\right)\right]\\
&+\frac{\sigma^{2}}{2}\nabla\cdot\nabla\cdot(\mathbf{D}\rho(\mathbf{x},t)),\end{split}$$
(2)
where $\mathbf{D}=\mathbf{S}\mathbf{S}^{T}$. This mean field Partial Differential Equation might support multiple coexisting stationary measures at low temperatures/large interaction strengths. In particular, in a conservative system described by a confining potential $\mathbf{F_{\alpha}}(\mathbf{y})=-\nabla V_{\alpha}(\mathbf{y})$ with additive noise such that $\mathbf{S}$ is the identity matrix, stationary solutions of Eqn. 2 correspond to local minima of $V_{\alpha}(\mathbf{y})$. In this case, the thermodynamic limit (2) can be written in the standard form as $\lim_{N\rightarrow+\infty}-\frac{1}{N}\log Z_{N}=\inf F(\rho)$, where $Z_{N}$ denotes the partition function of the $N-$particle system and $F(\rho)$ denotes the free energy of mean field system Helffer (2002). A stationary state is characterised by the order parameter $\langle\mathbf{x}\rangle_{0}$ and the associated stationary distribution $\rho_{0}(\mathbf{x})$.
We now perturb the stationary state by setting $\mathbf{F}_{\alpha}(\mathbf{x})\rightarrow\mathbf{F}_{\alpha}(\mathbf{x})+\varepsilon\mathbf{X}(\mathbf{x})T(t)$ and we study the response of the system by expanding the distribution function as $\rho(\mathbf{x},t)=\rho_{0}(\mathbf{x})+\varepsilon\rho_{1}(\mathbf{x},t)+O(\varepsilon^{2})$. Following a tedious calculation reported in Lucarini, Pavliotis, and Zagli (2020), the response of the order parameter in the frequency domain is written in terms of a macroscopic (or renormalised) susceptibility $\tilde{\Gamma}_{i}(\omega)$ as (Repeated indices are summed):
$$\langle x_{i}\rangle_{1}(\omega)=P_{ij}^{-1}(\omega)\Gamma_{j}(\omega)T(\omega)\equiv\tilde{\Gamma}_{i}(\omega)T(\omega)$$
(3)
where $P_{ij}(\omega)=\delta_{ij}-\theta\mathcal{Y}_{ij}(\omega)$ and the susceptibilities $\Gamma_{i}(\omega)$and $\mathcal{Y}_{ij}(\omega)$ are respectively the Fourier Transform of the microscopic response functions that can be written as correlation functions in the unperturbed state as Lucarini, Pavliotis, and Zagli (2020)
$$\displaystyle G_{i}(\tau)$$
$$\displaystyle=-\Theta(\tau)\left\langle\frac{\nabla\cdot\left(\rho_{0}(\mathbf{y})\mathbf{X}(\mathbf{y})\right)}{\rho_{0}(\mathbf{y})}\exp\left(\mathcal{L}^{+}_{\langle\mathbf{x}\rangle_{0}}\tau\right)y_{i}\right\rangle_{0}$$
(4)
$$\displaystyle Y_{ij}(\tau)$$
$$\displaystyle=-\Theta(\tau)\left\langle\partial_{y_{j}}\log\rho_{0}(\mathbf{y})\exp\left(\mathcal{L}^{+}_{\langle\mathbf{x}\rangle_{0}}\tau\right)y_{i}\right\rangle_{0}$$
(5)
where $\mathcal{L}^{+}_{\langle\mathbf{x}\rangle_{0}}$represents the adjoint of $\mathcal{L}_{\langle\mathbf{x}\rangle_{0}}$ and $\langle\cdot\rangle_{0}$ is the expectation value on the unperturbed state $\rho_{0}(\mathbf{x})$. The Fokker-Planck operator $\mathcal{L}_{\langle\mathbf{x}\rangle_{0}}$ appears on the right hand side of (2) (evaluated at the stationary state $\rho_{0}(\mathbf{x})$) and its adjoint $\mathcal{L}^{+}_{\langle\mathbf{x}\rangle_{0}}$ can be interpreted as the generator of the Koopman operator of the stationary dynamics Frank (2004); Lucarini, Pavliotis, and Zagli (2020). As such, correlation properties of the system are related to the spectrum of $\mathcal{L}^{+}_{\langle\mathbf{x}\rangle_{0}}$ Chekroun et al. (2020, 2014); Lasota and Mackey (1994). The renormalisation of the susceptibility derives from the coupling among the subsystems; note that $\tilde{\Gamma}_{i}(\omega)$ inherits the poles of both $\Gamma_{i}(\omega)$ and of the matrix $P^{-1}_{ij}(\omega)$. Away from criticality, both the microscopic and macroscopic susceptibilities are analytic in the upper half of the $\omega$ complex plane.
As discussed in Lucarini, Pavliotis, and Zagli (2020), the critical behaviour of this class of multiagent systems, signified by the singular behaviour of the susceptibility, originates from two distinct physical phenomena that are associated to either the poles of $\Gamma_{i}(\omega)$ or $P_{ij}^{-1}(\omega)$. The case where $\Gamma_{i}(\omega)$ diverges pertains to the occurrence of critical transitions.
It is of interest here the case where poles appear in the real $\omega$ axis for $\tilde{\Gamma}_{i}(\omega)$ because the matrix $P^{-1}_{ij}(\omega)$ becomes singular.
This corresponds to phase transitions originated by the coupling and do not show a divergence of correlation properties, because $\Gamma_{i}(\omega)$ is, instead, analytic in the upper complex $\omega$ plane. Equivalently, the spectral gap of $\mathcal{L}^{+}_{\langle x\rangle_{0}}$ remains finite at a phase transition.
However, at the transition point, the usual dispersion relations need to be modified Lucarini, Pavliotis, and Zagli (2020); Lucarini et al. (2005). The conditions underpinning the breakdown of linear response theory do not depend on the perturbation field $\mathbf{X}(\mathbf{x})$ nor on the choice of the observable ($\mathbf{x}$, in our case) and can be related to the spectral properties of a modified transfer operator Lucarini, Pavliotis, and Zagli (2020).
III Numerical results
Below, we present results for the Desai-Zwanzig (DZ) Desai and Zwanzig (1978) and the Bonilla-Casado-Morillo (BCM) Bonilla, Casado, and Morillo (1987) models. These are composed by $N$ interacting agents evolving according to Eq. 1. Each individual agent of the DZ (BCM) model evolves in $\mathbb{R}$ ($\mathbb{R}^{2}$). A detailed description of the two models is presented in the Appendix. We repeat our experiments for various choices of $N$, in order to detect the emergence of singularities for the combination of the parameters corresponding to phase transitions. Here, we keep fixed the values of the internal parameter $\alpha$. Both models undergo a phase transition at the transition line $\tilde{\sigma}=\sigma(\theta;\alpha)$ in the parameter space $(\sigma,\theta)$, see Appendix for the analytical evaluation of the transition line. Since one of the two parameters is redundant, we fix the coupling intensity $\theta$ and we vary, instead, the noise strength $\sigma$.
Following Marconi et al. (2008b), we perform $n$ simulations where the initial conditions are chosen according to the unperturbed invariant measure $\rho_{0}(\mathbf{x})$ and where at time $\tau=0$ we apply a perturbation proportional to a Dirac $\delta$ function. The average of the response for the observable $\mathbf{x}$ over the $n$ simulations gives an estimate $\tilde{G}_{i}(\tau;N)$ of the renormalised response function . Details on the numerical simulations are also reported in the Appendix. Figure 1 shows the response functions $\tilde{G}_{i}(\tau;N)$ for an additive perturbation $\mathbf{X}(\mathbf{x})=1$ for the DZ model (left panel) and $\mathbf{X}(\mathbf{x})=(1,0)$ for the BCM model (right panel). The two response functions are qualitatively different because, by and large, the one for the DZ model describes a monotonic decay, whereby the system relaxes towards the unperturbed state, while the one for the BCM combines the decay with an oscillatory behaviour taking place at the natural frequency $\tilde{\omega}=1$.
In the DZ model, the response functions initially undergo a fast and substantial decay, both far from and at the phase transition, associated with a time scale of order $1$. However, at the phase transition, a new, much longer, timescale appears. This timescale increases monotonically with $N$. The same is observed in the case of the BCM model if one considers the envelope of the response function rather than the response function itself: at the transition the decay of the oscillations becomes slower and slower as $N$ increases.
The origin of the new timescales resides in the appearance of simple pole at $\omega=\omega_{0}$ in the susceptibility $\tilde{\Gamma}_{i}(\omega;N)$, the Fourier transform of the response function. The pole is located at $\omega_{0}=0$ for the DZ model and at $\omega_{0}=\tilde{\omega}=1$ for the BCM model. When considering finite values of $N$, the susceptibilities describing the response of (virtually) any observable to (virtually) any external perturbation have a contribution of the form $\frac{\kappa}{\omega-\omega_{0}+i\gamma(N)}$, where $\gamma(N)\rightarrow 0^{+}$ as $N\rightarrow+\infty$ and $\kappa$ represents the residue of the pole, because $\lim_{N\rightarrow\infty}\frac{\kappa}{\omega-\omega_{0}+i\gamma(N)}=-i\pi\kappa\delta(\omega-\omega_{0})+\kappa\mathcal{P}(1/(\omega-\omega_{0}))$. The quantity $\kappa\in\mathbb{C}$ depends on the choice of observable and of the perturbation. We remark that the asymptotic property does not depend on how fast the function $\gamma(N)$ vanishes for increasing values of $N$. Following Delgadino, Gvalani, and Pavliotis (2021), one might conjecture that for the Desai-Zwanzig model and related models the function $\gamma$ would scale as $1/N$. Instead, we have observed here that the behaviour of $\gamma$ is different. This is an issue of fundamental importance that we will explore in future work, also in the case of nonequilibrium systems. Note also that, in the case of equibrium systems, the mean field limit $N\rightarrow\infty$ and the limit $T\rightarrow T_{c}$, where $T_{c}$ is the critical temperature, do not commute Chavanis (2014).
We next investigate the phase transitions by looking at the properties of the susceptibilities, see Fig. 2. When $\sigma\neq\tilde{\sigma}$, the susceptibilities do not show any singularity nor any remarkable dependence on $N$, thus indicating that the thermodynamic limit has been reached to a good approximation.
As $N$ increases, for both the DZ model (left panel) and the BCM model (right panel) the resonance at $\omega=\omega_{0}$ of the real part of the susceptibility approaches the limiting Dirac function $\pi k\delta(\omega-\omega_{0})$ with coefficient $k>0$. This singular behaviour is clear from the plot of the primitive function of the real part of the susceptibility (bottom inset) that tends to step function. For both models $\kappa=\mathrm{i}k$ is an imaginary number. Indeed, the imaginary part of the susceptibility behaves exactly as the Cauchy principal value distribution and can be used to get easily a quantitative estimate of $k$. The top insets of Figure 2 shows the function $(\omega-\omega_{0})\mathbf{Im}\{\tilde{\Gamma}_{i}(\omega)\}$. As $N\rightarrow\infty$, this function converges to $k$ everywhere except for $\omega=\omega_{0}$.
An explicit expression for $k$ is known Lucarini, Pavliotis, and Zagli (2020) in the case of the DZ model 111There is a typo in the formula given in Lucarini, Pavliotis, and Zagli (2020). Furthermore, the convention for the Fourier transform we use here has the opposite sign, ditto the residue.
$k=\frac{\langle x^{2}\rangle_{0}}{\theta\int_{0}^{+\infty}\mathrm{d}t\langle x(t)x(0)\rangle_{0}}$. Using the statistics of the unperturbed runs we obtain $k\approx 0.89$, which agrees within 2% with the one obtained from the limiting behaviour of the susceptibility, thus validating our results. In the case of the BCM model, our procedure allows one to derive a direct estimate $k\approx 0.44$; in this case no expression for the residue is available in the literature and, following Bonilla, Casado, and Morillo (1987); Lucarini, Pavliotis, and Zagli (2020), its evaluation seems cumbersome.
We here observe that, by evaluating the susceptibility for finite values of $N$, we are able to predict the residue of the pole at $\omega=\omega_{0}$, which appears, instead, only in the thermodynamic limit. The residue plays the role of a latent heat of phase change in classical thermodynamics. Our results, though, allow one to deal with the case of a dynamical latent heat, that is observed for perturbations occurring a non-vanishing frequency.
As discussed earlier, the singular behaviour of the susceptibility has some degree of universality. By this we mean that while for a given model the value of the residue is forcing- and observable-dependent, its position is a fundamental property of the model itself;
see the Appendix for an additional examples.
IV Conclusions
The study of how a large network of identical agents respond to exogenous perturbations is of the uttermost importance in different areas of science. One might be interested not only in the smooth response of the system, where its properties change ever so slightly, but also in the critical, nonsmooth, regime, where small perturbations can lead to large and possibly undesired changes. Multiagent systems modeled as weakly interacting Itô diffusions represent a rich class of models exhibiting such critical behaviour, and for which rigorous analysis and careful numerical investigations can be carried out. Usually these phenomena are accompanied by a large spatial (among sub-systems) and temporal restructuring of the system where correlations get highly magnified. The critical behaviour due to the emergence of a phase transition is a genuine thermodynamic phenomenon arising from the complex interactions among the infinite number of agents. Nevertheless, we have shown in this paper that linear response theory provides a powerful framework for detecting and anticipating phase transitions by investigating the response of a finite particle system to external perturbations.We have been able to predict the appearance of poles in the susceptibility, which describes the frequency-dependent response of the system, as well as to obtain a correct estimate of critical thermodynamic properties, such as the residue of the poles, based on the knowledge of the response for the finite particle system in two paradigmatic models describing equilibrium and nonequilibrium phase transitions. This is an encouraging starting point for improving our ability to understand and predict transitions in more complex multiagent systems.
Data Availability
The data that support the findings of this study are openly available in "Spectroscopy of Phase Transitions" at https://figshare.com/projects/Spectroscopy_of_phase_transitions/101846, reference number Zagli (2021).
Acknowledgements.
VL acknowledges the support received by the European Union’s Horizon 2020 program through the project TiPES (Grant Agreement No. 820970) and from the EPSRC project EP/T018178/1. The work of GP was partially funded by the EPSRC, grant number EP/P031587/1, and by J.P. Morgan Chase & Co. Any views or opinions expressed herein are solely those of the authors listed, and may differ from the views and opinions expressed by J.P. Morgan Chase & Co. or its affiliates. This material is not a product of the Research Department of J.P. Morgan Securities LLC. This material does not constitute a solicitation or offer in any jurisdiction. NZ has been supported by an EPSRC studentship as part of the Centre for Doctoral Training in Mathematics of Planet Earth (grant number EP/L016613/1).
Appendix A The models
Weakly interacting diffusions represent a rich class of agent based models, describing a network of interacting subsystems. The local dynamics of each subsystem is determined by a smooth vector field $\mathbf{F}_{\alpha}(\mathbf{x})$. The local force is in general non conservative, leading to irreversible and dissipative processes that can exhibit complex behaviours, such as deterministic chaos, see Figure 3.
An all-to-all coupling between the subsystems is given by a matrix $L_{ij}=\mathbf{\nabla}\mathcal{U}(\mathbf{x}^{i}-\mathbf{x}^{j})$ where $\mathcal{U}(\mathbf{x})=\mathcal{U}(-\mathbf{x})$ represents the interaction potential and $\mathbf{x}^{i}$ is the state vector of the $i$-th subsystem. Weakly interacting diffusions are characterised by a coupling strength which is inversely proportional to the number of subsystems $N$. As $N$ increases, the interaction structure gets more and more intricate, while the intensity becomes weaker and weaker, see Figure 3.
As mentioned in the main text, the DZ model has been introduced, and thereafter commonly used, as a paradigmatic example of an equilibrium continuous phase transition reminiscent of the Ising-like ferromagnetic transitions in spin systems Dawson (1983); Shiino (1987). The DZ model describes a network of one-dimensional subsystems $x^{k}$ whose dynamics is prescribed by the following equations (see main text for notation)
$$\mathrm{d}x^{k}=-V^{\prime}_{\alpha}(x^{k})\mathrm{d}t-\frac{\theta}{N}\sum_{l=1}^{N}\left(x^{k}-x^{l}\right)\mathrm{d}t+\sigma\mathrm{d}W^{k}$$
(6)
where $k=1,\dots,N$ and the confining potential $V_{\alpha}(x)=-\frac{\alpha}{2}x^{2}+\frac{x^{4}}{4}$ has a double well shape for $\alpha>0$. Without loss of generality, we here consider $\alpha=1$. In the absence of coupling, $\theta=0$, the above equations describe the simple motion of a particle in a double well potential, subject to additive noise. The presence of the coupling allows for a long range coordination of the system that in the thermodynamic limit $N\rightarrow+\infty$ results in a proper phase transition. In this regime, by varying the parameters $(\alpha,\theta)$, the order parameter $\langle x\rangle$ undergoes a continuous order-disorder transition, similar to the pitchfork bifurcation diagram for the Ising model. It is possible to show Dawson (1983) that the critical line is given by $\frac{D_{-3/2}\left(\frac{\theta-1}{\sigma}\right)}{D_{-1/2}\left(\frac{\theta-1}{\sigma}\right)}=\frac{\sigma}{\theta}$
where $D_{\nu}(z)$ is a parabolic cylinder function. Here the coupling $\theta$ is kept fixed ($\theta=0.55$) and we vary $\sigma$ to approach the transition point.
The BCM model describes an ensemble of bi-dimensional nonlinear oscillators undergoing an out of equilibrium self-synchronisation transition. The time evolution of the network of oscillators is given by the following equations
$$\mathrm{d}\mathbf{x}^{k}=\mathbf{F}_{\alpha}(\mathbf{x}^{k})\mathrm{d}t-\frac{\theta}{N}\sum_{l=1}^{N}\left(\mathbf{x}^{k}-\mathbf{x}^{l}\right)\mathrm{d}t+\sigma\mathrm{d}\mathbf{W}^{k},$$
(7)
where the local force is not conservative, giving rise to the non equilibrium features of the system, and reads $\mathbf{F}_{\alpha}(\mathbf{x})=\left(\alpha-|\mathbf{x}|^{2}\right)\mathbf{x}+\mathbf{x}^{+}$where $\mathbf{x}^{+}=(-x_{2},x_{1})$. The latter term corresponds to a rotation and makes the stationary state a non equilibrium one. The parameter $\alpha>0$ controls the amplitude of the oscillations of the individual non linear oscillators. In fact, when $\theta=\sigma=0$, each subsystem oscillates as $\mathbf{x}^{j}(t)=\sqrt{\alpha}\left(\cos(t+\beta_{j}),\sin(t+\beta_{j})\right)$ where $\beta_{j}=\tan(x^{j}_{2}(0)/x^{j}_{1}(0))$. The coupling tries to synchronise the subsystems by attracting them towards the center of mass $\frac{1}{N}\sum_{j=1}^{N}\mathbf{x}^{j}$. In the thermodynamic limit and for sufficiently low values of the noise, the order parameter $\langle\mathbf{x}\rangle=\langle\mathbf{x}\rangle(t)$ exhibits a periodic time evolution, resulting from the subsystem oscillating in a coherent way. On the other hand, high values of the noise correspond to a non synchronised state where the order parameter vanishes. In particular, the transition happen at the surface of the $(\alpha,\theta,\sigma)$ parametric space defined by Bonilla, Casado, and Morillo (1987)
$$A=\frac{\delta^{2}}{2}\left[1-\frac{1}{\delta}\exp\left(-\frac{A^{2}}{\delta^{2}}\right)\left[\int_{-\frac{A}{\delta}}^{\infty}e^{-r^{2}}dr\right]^{-1}\right]$$
(8)
where $A=\frac{\alpha}{\theta}-1$ and $\delta=\frac{\sqrt{2\sigma^{2}}}{\theta}$.
In the following we have $\theta=\alpha=2$. The colour code for the figures is given by:
•
non critical black : DZ $\sigma\approx 1$ , BCM $\sigma\approx 2$.
•
non critical blue : DZ $\sigma\approx 0.87$ , BCM $\sigma\approx 1.8$
•
critical red : DZ $\tilde{\sigma}\approx 0.75$ , BCM $\tilde{\sigma}\approx 1.59$.
A.1 Numerical linear response experiments
As mentioned in the main text, we perform $n$ simulations of the response given by Eqs. 6 and 7 where the initial conditions are chosen according to the respective unperturbed invariant measure $\rho_{0}(\mathbf{x})$ and where at time $\tau=0$ we apply a perturbation proportional to a Dirac’s $\delta(\tau=0)$. The average of the response for the observable $\mathbf{x}$ over the $n$ simulations gives an estimate of $\tilde{G}_{i}(\tau;N)$. The response functions away from the transitions are estimated on an ensemble of $n=10^{5}$ simulations, while the critical response functions with $n=10^{6}$ for DZ and $n=7\times 10^{6}$ for BCM. Furthermore we investigate the response up to time $\tau=5\times 10^{3}$. The corresponding susceptibility $\tilde{\Gamma}_{i}(\omega;N)$ is simply defined as the Fourier Transform of $\tilde{G}_{i}(\tau;N)$.
In the main text, we show the results for an additive perturbation $\mathbf{X}(\mathbf{x})\equiv\mathbf{X}$. However, the critical behaviour of the response does not depend on the type of perturbation, modulo a potential degenerate class of perturbations that have zero projection on the invariant measure $\rho_{0}(\mathbf{x})$. We have thus decided to investigate the response of the DZ model for a spatially dependent perturbation $X(x)=x^{2}$, see Figure 4. The response function $\tilde{G}_{i}(\tau;N)$, both away and at the phase transition, has a rapid initial decay with a timescale that is different from the response function shown in the main text. As a matter of fact, the timescale associated to the dominant mode of the response function for $\tau\rightarrow 0$ does in general depend on the applied perturbation Lucarini, Pavliotis, and Zagli (2020). As expected, the response function at the phase transition develops a much longer timescale that increases as the number of particle increases. A more accurate comparison with the result shown in the main text can only be performed in the frequency domain. Figure 4 (right panel) shows that, away from the transition, the susceptibilities have a smooth behaviour and no evident dependence on $N$. At the phase transition, the susceptibility develops the expected singular behaviour $\frac{\kappa}{\omega-\omega_{0}+i\gamma(N)}$, where $\gamma(N)\rightarrow 0^{+}$ as $N\rightarrow+\infty$ due to the appearance of a simple pole $\omega_{0}=0$. The residue $\kappa$ is purely imaginary and its magnitude $k$ can be inferred by visual inspection of the top inset representing the function$(\omega-\omega_{0})\mathbf{Im}\{\tilde{\Gamma}(\omega)\}$ to be just less than 0.29. As mentioned in the main text, the residue depends both on the observable and on the perturbation $\mathbf{X}(\mathbf{x})$.
References
Naldi, Pareschi, and Toscani (2010)
G. Naldi, L. Pareschi, and G. Toscani, Mathematical
Modeling of Collective Behavior in Socio-Economic and Life Sciences (Birkhäuser Basel, 2010).
Pareschi and Toscani (2013)
L. Pareschi and G. Toscani, “Interacting
multiagent systems: kinetic equations and monte carlo methods,” (2013).
Dawson (1983)
D. A. Dawson, “Critical
dynamics and fluctuations for a mean-field model of cooperative behavior,” J. Stat. Phys. 31, 29–85 (1983).
Acebrón et al. (2005)
J. A. Acebrón, L. L. Bonilla, C. J. Pérez Vicente, F. Ritort, and R. Spigler, “The kuramoto
model: A simple paradigm for synchronization phenomena,” Rev.
Mod. Phys. 77, 137–185
(2005).
Garnier, Papanicolaou, and Yang (2013)
J. Garnier, G. Papanicolaou, and T. Yang, “Large deviations
for a mean field model of systemic risk,” SIAM Journal
on Financial Mathematics 4, 151–184 (2013), https://doi.org/10.1137/12087387X .
Wang et al. (2017)
C. Wang, Q. Li, W. E, and B. Chazelle, “Noisy hegselmann-krause systems: Phase transition and the
2r-conjecture,” J. Stat. Phys. 166, 1209–1225 (2017).
Garnier, Papanicolaou, and Yang (2017)
J. Garnier, G. Papanicolaou, and T. Yang, “Consensus
convergence with stochastic effects,” Vietnam Journal of Mathematics 45, 51–75 (2017).
Garbuno-Inigo, Nüsken, and Reich (2020)
A. Garbuno-Inigo, N. Nüsken, and S. Reich, “Affine invariant
interacting Langevin dynamics for Bayesian inference,” SIAM J. Appl.
Dyn. Syst. 19, 1633–1658 (2020).
Simmonds, Gómez, and Ledezma (2019)
J. Simmonds, J. A. Gómez, and A. Ledezma, “The role of
agent-based modeling and multi-agent systems in flood-based hydrological
problems: a brief review,” Journal of Water and Climate
Change (2019), 10.2166/wcc.2019.108, jwc2019108, https://iwaponline.com/jwcc/article-pdf/doi/10.2166/wcc.2019.108/625177/jwc2019108.pdf
.
Geisendorf (2018)
S. Geisendorf, “Evolutionary
climate-change modelling: A multi-agent climate-economic model,” Computational Economics 52, 921–951 (2018).
Scheffer (2009)
M. Scheffer, Critical Transitions in Nature and Society, Princeton Studies in Complexity (Princeton University
Press, Princeton, 2009).
Sornette (0006)
D. Sornette, “Endogenous
versus exogenous origins of crises,” in Extreme Events in Nature and Society, edited by K. H. Albeverio S., Jentsch V. (,
Berlin, Heidelberg, 20006) pp. 95–119.
Kubo (1966)
R. Kubo, “The
fluctuation-dissipation theorem,” Reports on Progress in Physics 29, 255–284 (1966).
Marconi et al. (2008a)
U. M. B. Marconi, A. Puglisi, L. Rondoni, and A. Vulpiani, “Fluctuation–dissipation:
Response theory in statistical physics,” Physics Reports 461, 111 – 195 (2008a).
Baiesi and Maes (2013)
M. Baiesi and C. Maes, “An update on the
nonequilibrium linear response,” New Journal of Physics 15, 013004 (2013).
Sarracino and Vulpiani (2019)
A. Sarracino and A. Vulpiani, “On the
fluctuation-dissipation relation in non-equilibrium and non-hamiltonian
systems,” Chaos: An Interdisciplinary Journal of Nonlinear
Science 29, 083132
(2019), https://doi.org/10.1063/1.5110262 .
Lucarini et al. (2005)
V. Lucarini, J. J. Saarinen, K.-E. Peiponen, and E. M. Vartiainen, Kramers-Kronig
relations in Optical Materials Research (Springer, New York, 2005).
Binney and Tremaine (2008)
J. Binney and S. Tremaine, Galactic Dynamics, 2nd ed. (Princeton University
Press, Princeton, 2008).
Leith (1975)
C. E. Leith, “Climate response
and fluctuation dissipation,” J. Atmos. Sci. 32, 2022 (1975).
North, Bell, and Hardin (1993)
G. North, R. Bell, and J. Hardin, “Fluctuation dissipation in a general
circulation model,” Clim. Dyn. 8, 259 (1993).
Öttinger (2005)
H. Öttinger, Beyond Equilibrium Thermodynamics (Wiley, Hoboken, 2005).
Lucarini, Ragone, and Lunkeit (2017)
V. Lucarini, F. Ragone, and F. Lunkeit, “Predicting climate change
using response theory: Global averages and spatial patterns,” J. Stat. Phys. 166, 1036–1064 (2017).
Cessac (2019)
B. Cessac, “Linear response
in neuronal networks: From neurons dynamics to collective response,” Chaos 29, 103105 (2019).
Gottwald (2020)
G. A. Gottwald, “Introduction
to focus issue: Linear response theory: Potentials and limits,” Chaos: An Interdisciplinary Journal of Nonlinear Science 30, 020401 (2020), https://doi.org/10.1063/5.0003135 .
Ruelle (1998)
D. Ruelle, “Nonequilibrium
statistical mechanics near equilibrium: computing higher-order terms,” Nonlinearity 11, 5–18 (1998).
Ruelle (2009)
D. Ruelle, “A review of
linear response theory for general differentiable dynamical systems,” Nonlinearity 22, 855–870 (2009).
Baladi (2014)
V. Baladi, “Linear response,
or else,” in ICM Seoul 2014,
Proceedings, Vol. III (2014) p. 525–545.
Dembo and Deuschel (2010)
A. Dembo and J.-D. Deuschel, “Markovian
perturbation, response and fluctuation dissipation theorem,” Ann. Inst. Henri Poincaré Probab. Stat. 46, 822–852 (2010).
Hairer and Majda (2010)
M. Hairer and A. J. Majda, “A simple
framework to justify linear response theory,” Nonlinearity 23, 909–922 (2010).
Wormell and Gottwald (2019)
C. L. Wormell and G. A. Gottwald, “Linear
response for macroscopic observables in high-dimensional systems,” Chaos: An Interdisciplinary Journal of Nonlinear Science 29, 113127 (2019).
Liverani and Gouëzel (2006)
C. Liverani and S. Gouëzel, “Banach
spaces adapted to Anosov systems,” Ergodic Theory and Dynamical Systems 26, 189–217 (2006).
Chekroun et al. (2014)
M. D. Chekroun, J. D. Neelin, D. Kondrashov,
J. C. McWilliams, and M. Ghil, “Rough parameter dependence in climate
models and the role of Ruelle-Pollicott resonances,” Proceedings of the National Academy of Sciences 111, 1684–1690 (2014).
Lucarini (2016)
V. Lucarini, “Response
operators for Markov processes in a finite state space: Radius of
convergence and link to the response theory for Axiom A systems,” J. Stat. Phys. 162, 312–333 (2016).
Tantet, Lucarini, and Dijkstra (2018)
A. Tantet, V. Lucarini, and H. A. Dijkstra, “Resonances in a Chaotic
Attractor Crisis of the Lorenz Flow,” J.
Stat. Phys. 170, 584–616 (2018), arXiv:1705.08178 .
Pollicott (1985)
M. Pollicott, “On the rate
of mixing of Axiom A flows,” Inventiones Mathematicae 81, 413–426 (1985).
Ruelle (1986)
D. Ruelle, “Resonances of
chaotic dynamical systems,” Physical Review Letters 56, 405–407 (1986).
Shiino (1987)
M. Shiino, “Dynamical
behavior of stochastic systems of infinitely many coupled nonlinear
oscillators exhibiting phase transitions of mean-field type: H theorem on
asymptotic approach to equilibrium and critical slowing down of
order-parameter fluctuations,” Phys. Rev. A 36, 2393–2412 (1987).
Delgadino, Gvalani, and Pavliotis (2021)
M. G. Delgadino, R. S. Gvalani, and G. A. Pavliotis, “On the
diffusive-mean field limit for weakly interacting diffusions exhibiting phase
transitions,” Archive for Rational Mechanics and Analysis (2021), 10.1007/s00205-021-01648-1.
Carrillo et al. (2020)
J. A. Carrillo, R. S. Gvalani, G. A. Pavliotis, and A. Schlichting, “Long-time
behaviour and phase transitions for the McKean-Vlasov equation on the
torus,” Arch. Ration. Mech. Anal. 235, 635–690 (2020).
Lucarini, Pavliotis, and Zagli (2020)
V. Lucarini, G. A. Pavliotis, and N. Zagli, “Response theory
and phase transitions for the thermodynamic limit of interacting identical
systems,” Proc. R. Soc. A. 476 (2020), https://doi.org/10.1098/rspa.2020.0688 .
Jackson (1975)
J. D. Jackson, Classical electrodynamics; 2nd ed. (Wiley, New York, NY, 1975).
Talebian and Talebian (2013)
E. Talebian and M. Talebian, “A general
review on the derivation of clausius-mossotti relation,” Optik 124, 2324 – 2326 (2013).
Sornette and Helmstetter (2003)
D. Sornette and A. Helmstetter, “Endogenous
versus exogenous shocks in systems with memory,” Physica A: Statistical Mechanics and its Applications 318, 577 – 591 (2003).
Yoshida (2003)
N. Yoshida, “Phase
transition from the viewpoint of relaxation phenomena,” Rev.
Math. Phys. 15, 765–788
(2003).
Desai and Zwanzig (1978)
R. C. Desai and R. Zwanzig, “Statistical
mechanics of a nonlinear stochastic model,” J. Stat.
Phys. 19, 1–24
(1978).
Bonilla, Casado, and Morillo (1987)
L. L. Bonilla, J. Casado, and M. Morillo, “Self-synchronization of
populations of nonlinear oscillators in the thermodynamic limit,” J.
Stat. Phys. 48, 571–591
(1987).
Sznitman (1989)
A. Sznitman, Topics in propagation
of chaos., Hennequin PL. (eds) Ecole d’Eté de
Probabilités de Saint-Flour XIX — 1989. Lecture Notes in Mathematics,
Vol. 1464 (Springer, Berlin,
Heidelberg, 1989).
Oelschlager (1984)
K. Oelschlager, “A
martingale approach to the law of large numbers for weakly interacting
stochastic processes,” Ann. Probab. 12, 458–479 (1984).
Dawson and Gärtner (1987)
D. A. Dawson and J. Gärtner, “Large
deviations from the McKean-Vlasov limit for weakly interacting
diffusions,” Stochastics 20, 247–308 (1987).
Helffer (2002)
B. Helffer, Semiclassical analysis, Witten Laplacians, and statistical
mechanics, Series in Partial Differential Equations
and Applications, Vol. 1 (World
Scientific Publishing Co., Inc., River Edge, NJ, 2002) pp. x+179.
Frank (2004)
T. Frank, “Fluctuation–dissipation theorems for nonlinear Fokker–Planck equations
of the Desai–Zwanzig type and Vlasov–Fokker–Planck equations,” Physics Letters A 329, 475 – 485 (2004).
Chekroun et al. (2020)
M. D. Chekroun, A. Tantet,
H. A. Dijkstra, and J. D. Neelin, “Ruelle–pollicott resonances of
stochastic systems in reduced state space. part i: Theory,” J. Stat. Phys. (2020), 10.1007/s10955-020-02535-x.
Lasota and Mackey (1994)
A. Lasota and M. C. Mackey, Chaos, fractals, and
noise, 2nd ed., Applied
Mathematical Sciences, Vol. 97 (Springer-Verlag, New York, 1994) pp. xiv+472.
Marconi et al. (2008b)
U. M. B. Marconi, A. Puglisi, L. Rondoni, and A. Vulpiani, “Fluctuation-dissipation:
Response theory in statistical physics,” Phys. Rep. 461, 111 (2008b).
Chavanis (2014)
P.-H. Chavanis, “The brownian
mean field model,” The European Physical Journal B 87, 120 (2014).
Note (1)
There is a typo in the formula given in Lucarini, Pavliotis, and Zagli (2020).
Furthermore, the convention for the Fourier transform we use here has the
opposite sign, ditto the residue.
Zagli (2021)
N. Zagli, “Spectroscopy of
phase transitions,” Figshare (2021). |
Scaled tree fractals do not strictly self-assemble
Kimberly Barth
Department of Computer Science, University of Wisconsin–Oshkosh, Oshkosh, WI 54901, USA,
barthk63@uwosh.edu.
David Furcy
Department of Computer Science, University of Wisconsin–Oshkosh, Oshkosh, WI 54901, USA,
furcyd@uwosh.edu.
Scott M. Summers
Department of Computer Science, University of Wisconsin–Oshkosh, Oshkosh, WI 54901, USA,
summerss@uwosh.edu.
Paul Totzke
Department of Computer Science, University of Wisconsin–Oshkosh, Oshkosh, WI 54901, USA,
totzkp00@uwosh.edu.
()
Abstract
In this paper, we show that any scaled-up version of any discrete self-similar tree fractal does not strictly self-assemble, at any temperature, in Winfree’s abstract Tile Assembly Model.
1 Introduction
The stunning, often mysterious complexities of the natural world, from nanoscale crystalline structures to unthinkably massive galaxies, all arise from the same elemental process known as self-assembly. In the absence of a mathematically rigorous definition, self-assembly is colloquially thought of as the process through which simple, unorganized components spontaneously combine, according to local interaction rules, to form some kind of organized final structure. A major objective of nanotechnology is to harness the power of self-assembly, perhaps for the purpose of engineering atomically precise medical, digital and mechanical components at the nanoscale. One strategy for doing so, developed by Nadrian Seeman, is DNA tile self-assembly [8, 9].
In DNA tile self-assembly, the fundamental components are “tiles”, which are comprised of interconnected DNA strands. Remarkably, these DNA tiles can be “programmed”, via the careful configuration of their constituent DNA strands, to automatically coalesce into a desired target structure, the characteristics of which are completely determined by the “programming” of the DNA tiles. In order to fully realize the power of DNA tile self-assembly, we must study the algorithmic and mathematical underpinnings of tile self-assembly.
Perhaps the simplest mathematical model of algorithmic tile self-assembly is Erik Winfree’s abstract Tile Assembly Model (aTAM) [11]. The aTAM is a deliberately over-simplified, combinatorial model of nanoscale (DNA) tile self-assembly that “effectivizes” classical Wang tiling [10] in the sense that the former augments the latter with a mechanism for sequential “growth” of a tile assembly. Very briefly, in the aTAM, the fundamental components are un-rotatable, translatable square “tile types” whose sides are labeled with (alpha-numeric) glue “colors” and (integer) “strengths”. Two tiles that are placed next to each other interact if the glue colors on their abutting sides match, and they bind if the strengths on their abutting sides match and sum to at least a certain (integer) “temperature”. Self-assembly starts from a “seed” tile type, typically assumed to be placed at the origin, and proceeds nondeterministically and asynchronously as tiles bind to the seed-containing assembly one at a time.
Despite its deliberate over-simplification, the aTAM is a computationally expressive model. For example, Winfree [11] proved that the model is Turing universal, which means that, in principle, the process of self-assembly can be directed by any algorithm. In this paper, we will specifically study the extent to which tile sets in the aTAM can be algorithmically directed to “strictly” self-assemble (place tiles at and only at locations that belong to) shapes that are discrete self-similar tree fractals.
There are examples of prior results related to the self-assembly of fractals in the aTAM, in general [6, 2, 1], as well as the strict self-assembly of tree fractals in the aTAM, specifically [3, 4]. In fact, a notable example of the latter is [3], Theorem 3.2 of which bounds from below the size of the smallest tile set in which an arbitrary shape $X$ strictly self-assembles by the depth of $X$’s largest finite sub-tree. Although not stated explicitly, an immediate corollary of Theorem 3.2 of [3] is that no tree fractals strictly self-assemble in the aTAM.
While the strict self-assembly of tree fractals in the aTAM is well-understood (via Theorem 3.2 of [3]), nothing is known about the strict self-assembly of “scaled-up” versions of tree fractals (“scaled-up” meaning each point in the original shape is replaced by a $c\times c$ block of points). After all, the scaled-up version of any shape – tree or otherwise – is not a tree in the sense of the “full connectivity graph” of the shape, i.e., each point in the shape is represented by one vertex and edges exist between vertices that represent adjacent points in the shape. This means that prior proof techniques, which exploit the intricate geometry of tree fractals (e.g., [3, 4]), simply cannot be applied to scaled-up versions of tree fractals. Thus, in this paper, we ask if it is possible for a scaled-up version of a tree fractal to strictly self-assemble in the aTAM.
The main contribution of this paper provides an answer to the previous question, perhaps not too surprisingly to readers familiar with the aTAM, in the negative: we prove that there is no tree fractal that strictly self-assembles in the aTAM – at any positive scale factor. Thus, our main result generalizes Theorem 3.4 of [3], which says that the Sierpinski triangle, perhaps the most famous, well-studied example of a tree fractal, does not strictly self-assemble at scale factor $1$. Our proof makes crucial use of a recent technical lemma developed by Meunier, Patitz, Summers, Theyssier, Winslow and Woods [5], known as the “Window Movie Lemma” (WML), which gives a sufficient condition for taking any pair of tile assemblies, at any temperature, and “splicing” them together to create a new valid tile assembly. The WML is, in some sense, a pumping lemma for self-assembly that mitigates the need to use overly-complicated, convoluted case-analyses that typically arise when doing impossibility proofs in self-assembly.
What follows is a list of the main technical contributions presented in – and the general outline of – this paper:
•
In Section 2.2, we exhibit a natural
characterization of a tree fractal in terms of a few simple,
easily checkable geometric properties of its generator. While
perhaps well-known, this type of characterization, to the best
of our knowledge, has yet to be explicitly documented or proved
in the literature.
•
In Section 2.3, we develop a
modified version of the general WML. Our version of the WML,
which we call the “Closed Window Movie Lemma”, allows us to
replace one portion of a tile assembly with another, assuming
certain extra “containment” conditions are met. Moreover,
unlike in the original WML that lacks the extra containment
assumptions, the replacement of one tile assembly with another
in our Closed WML only goes “one way”, i.e., the part
of the tile assembly being used to replace another part cannot
itself be replaced by the part of the tile assembly it is
replacing.
•
In Section 3, we use our closed WML to prove that any scaled-up version of any tree fractal does not strictly self-assemble in the aTAM at any temperature. Our main result generalizes the claim that every tree fractal, at scale factor 1, does not strictly self-assemble in the aTAM (an implicit corollary to the main negative result of [3]).
2 Definitions
In this section, we give a formal definition of Erik Winfree’s abstract Tile Assembly Model (aTAM), define and characterize tree fractals and develop a “Closed” Window Movie Lemma.
2.1 Formal description of the abstract Tile Assembly Model
This section gives a formal definition of the abstract Tile Assembly Model (aTAM) [11]. For readers unfamiliar with the aTAM, [7] gives an excellent introduction to the model.
Fix an alphabet $\Sigma$.
$\Sigma^{*}$ is the set of finite strings over $\Sigma$. Let $\mathbb{Z}$, $\mathbb{Z}^{+}$, and $\mathbb{N}$ denote the set of integers, positive integers, and nonnegative integers, respectively. Given $V\subseteq\mathbb{Z}^{2}$, the full grid graph of $V$ is the undirected graph $G^{\mathrm{f}}_{V}=(V,E)$,
such that, for all $\vec{x},\vec{y}\in V$, $\left\{\vec{x},\vec{y}\right\}\in E\iff\|\vec{x}-\vec{y}\|=1$, i.e., if and only if $\vec{x}$ and $\vec{y}$ are adjacent in the $2$-dimensional integer Cartesian space.
A tile type is a tuple $t\in(\Sigma^{*}\times\mathbb{N})^{4}$, e.g., a unit square, with four sides, listed in some standardized order, and each side having a glue $g\in\Sigma^{*}\times\mathbb{N}$ consisting of a finite string label and a nonnegative integer strength.
We assume a finite set of tile types, but an infinite number of copies of each tile type, each copy referred to as a tile. A tile set is a set of tile types and is usually denoted as $T$.
A configuration is a (possibly empty) arrangement of tiles on the integer lattice $\mathbb{Z}^{2}$, i.e., a partial function $\alpha:\mathbb{Z}^{2}\dashrightarrow T$.
Two adjacent tiles in a configuration interact, or are attached, if the glues on their abutting sides are equal (in both label and strength) and have positive strength.
Each configuration $\alpha$ induces a binding graph $G^{\mathrm{b}}_{\alpha}$, a grid graph whose vertices are positions occupied by tiles, according to $\alpha$, with an edge between two vertices if the tiles at those vertices interact. An assembly is a connected, non-empty configuration, i.e., a partial function $\alpha:\mathbb{Z}^{2}\dashrightarrow T$ such that $G^{\mathrm{f}}_{{\rm dom}\;\alpha}$ is connected and ${\rm dom}\;\alpha\neq\varnothing$.
Given $\tau\in\mathbb{Z}^{+}$, $\alpha$ is $\tau$-stable if every cut-set
of $G^{\mathrm{b}}_{\alpha}$ has weight at least $\tau$, where the weight
of an edge is the strength of the glue it represents.111A
cut-set is a subset of edges in a graph which, when removed from
the graph, produces two or more disconnected subgraphs. The
weight of a cut-set is the sum of the weights of all of the edges
in the cut-set. When $\tau$ is clear from context, we say $\alpha$ is
stable. Given two assemblies $\alpha,\beta$, we say $\alpha$
is a subassembly of $\beta$, and we write $\alpha\sqsubseteq\beta$, if ${\rm dom}\;\alpha\subseteq{\rm dom}\;\beta$ and, for all points $p\in{\rm dom}\;\alpha$, $\alpha(p)=\beta(p)$. For two non-overlapping
assemblies $\alpha$ and $\beta$, $\alpha\cup\beta$ is defined as the
unique assembly $\gamma$ satisfying, for all $\vec{x}\in{\rm dom}\;{\alpha}$, $\gamma(\vec{x})=\alpha(\vec{x})$, for all $\vec{x}\in{\rm dom}\;{\beta}$, $\gamma(\vec{x})=\beta(\vec{x})$, and
$\gamma(\vec{x})$ is undefined at any point $\vec{x}\in\mathbb{Z}^{2}\backslash\left({\rm dom}\;{\alpha}\cup{\rm dom}\;{%
\beta}\right)$.
A tile assembly system (TAS) is a triple $\mathcal{T}=(T,\sigma,\tau)$, where $T$ is a tile set, $\sigma:\mathbb{Z}^{2}\dashrightarrow T$ is the finite, $\tau$-stable, seed assembly, and $\tau\in\mathbb{Z}^{+}$ is the temperature.
Given two $\tau$-stable assemblies $\alpha,\beta$, we write $\alpha\to_{1}^{\mathcal{T}}\beta$ if $\alpha\sqsubseteq\beta$ and $|{\rm dom}\;\beta\setminus{\rm dom}\;\alpha|=1$. In this case we say $\alpha$ $\mathcal{T}$-produces $\beta$ in one step. If $\alpha\to_{1}^{\mathcal{T}}\beta$, ${\rm dom}\;\beta\setminus{\rm dom}\;\alpha=\{p\}$, and $t=\beta(p)$, we write $\beta=\alpha+(p\mapsto t)$.
The $\mathcal{T}$-frontier of $\alpha$ is the set $\partial^{\mathcal{T}}\alpha=\bigcup_{\alpha\to_{1}^{\mathcal{T}}\beta}({\rm
dom%
}\;\beta\setminus{\rm dom}\;\alpha$), the set of empty locations at which a tile could stably attach to $\alpha$. The $t$-frontier $\partial^{\mathcal{T}}_{t}\alpha\subseteq\partial^{\mathcal{T}}\alpha$ of $\alpha$ is the set $\left\{\ p\in\partial^{\mathcal{T}}\alpha\ \left|\ \alpha\to_{1}^{\mathcal{T}}%
\beta\text{ and }\beta(p)=t\right.\ \right\}.$
Let $\mathcal{A}^{T}$ denote the set of all assemblies of tiles from $T$, and let $\mathcal{A}^{T}_{<\infty}$ denote the set of finite assemblies of tiles from $T$.
A sequence of $k\in\mathbb{Z}^{+}\cup\{\infty\}$ assemblies $\alpha_{0},\alpha_{1},\ldots$ over $\mathcal{A}^{T}$ is a $\mathcal{T}$-assembly sequence if, for all $1\leq i<k$, $\alpha_{i-1}\to_{1}^{\mathcal{T}}\alpha_{i}$.
The result of an assembly sequence $\vec{\alpha}$, denoted as $\textmd{res}(\vec{\alpha})$, is the unique limiting assembly (for a finite sequence, this is the final assembly in the sequence).
We write $\alpha\to^{\mathcal{T}}\beta$, and we say $\alpha$ $\mathcal{T}$-produces $\beta$ (in 0 or more steps) if there is a $\mathcal{T}$-assembly sequence $\alpha_{0},\alpha_{1},\ldots$ of length $k=|{\rm dom}\;\beta\setminus{\rm dom}\;\alpha|+1$ such that
(1) $\alpha=\alpha_{0}$,
(2) ${\rm dom}\;\beta=\bigcup_{0\leq i<k}{\rm dom}\;{\alpha_{i}}$, and
(3) for all $0\leq i<k$, $\alpha_{i}\sqsubseteq\beta$.
If $k$ is finite then it is routine to verify that $\beta=\alpha_{k-1}$.
We say $\alpha$ is $\mathcal{T}$-producible if $\sigma\to^{\mathcal{T}}\alpha$, and we write $\mathcal{A}[\mathcal{\mathcal{T}}]$ to denote the set of $\mathcal{T}$-producible assemblies. The relation $\to^{\mathcal{T}}$ is a partial order on $\mathcal{A}[\mathcal{\mathcal{T}}]$ [7, 3].
An assembly $\alpha$ is $\mathcal{T}$-terminal if $\alpha$ is $\tau$-stable and $\partial^{\mathcal{T}}\alpha=\varnothing$.
We write $\mathcal{A}_{\Box}[\mathcal{\mathcal{T}}]\subseteq\mathcal{A}[\mathcal{%
\mathcal{T}}]$ to denote the set of $\mathcal{T}$-producible, $\mathcal{T}$-terminal assemblies. If $|\mathcal{A}_{\Box}[\mathcal{\mathcal{T}}]|=1$ then $\mathcal{T}$ is said to be directed.
We say that a TAS $\mathcal{T}$ strictly (a.k.a. uniquely) self-assembles $X\subseteq\mathbb{Z}^{2}$ if, for all $\alpha\in\mathcal{A}_{\Box}[\mathcal{\mathcal{T}}]$, ${\rm dom}\;\alpha=X$; i.e., if every terminal assembly produced by $\mathcal{T}$ places tiles on – and only on – points in the set $X$.
In this paper, we consider scaled-up versions of subsets of $\mathbb{Z}^{2}$. Formally, if $X$ is a subset of $\mathbb{Z}^{2}$ and $c\in\mathbb{Z}^{+}$, then a $c$-scaling of $X$ is defined as the set $X^{c}=\left\{(x,y)\in\mathbb{Z}^{2}\;\left|\;\left(\left\lfloor\frac{x}{c}%
\right\rfloor,\left\lfloor\frac{y}{c}\right\rfloor\right)\in X\right.\right\}$. Intuitively, $X^{c}$ is the subset of $\mathbb{Z}^{2}$ obtained by replacing each point in $X$ with a $c\times c$ block of points. We refer to the natural number $c$ as the scaling factor or resolution loss.
2.2 Discrete self-similar tree fractals
In this section, we introduce a new formal characterization of
discrete self-similar tree fractals. The proof of
Theorem 2.1 below is omitted from this version of the
paper due to lack of space.
Notation 1
We use $\mathbb{N}_{g}$ to denote the subset $\{0,\ldots,g-1\}$ of
$\mathbb{N}$.
Notation 2
If $A$ and $B$ are subsets of $\mathbb{N}^{2}$ and $k\in\mathbb{N}$, then $A+kB=\{\vec{m}+k\vec{n}~{}|~{}\vec{m}\in A$ and $\vec{n}\in B\}$.
The following definition is adapted from [6].
Definition 1
Let $1<g\in\mathbb{N}$ and $\mathbf{X}\subset\mathbb{N}^{2}$. We say that
$\mathbf{X}$ is a $g$-discrete self-similar fractal (or $g$-dssf
for short), if there is a set $\{(0,0)\}\subset G\subset\mathbb{N}_{g}^{2}$ with at least one point in every row and column, such
that $\displaystyle\mathbf{X}=\bigcup_{i=1}^{\infty}X_{i}$, where $X_{i}$, the
$i^{th}$ stage of $\mathbf{X}$, is defined by $X_{1}=G$ and
$X_{i+1}=X_{i}\ +\ g^{i}G$. We say that $G$ is the
generator of $\mathbf{X}$.
Intuitively, a $g$-dssf is built as follows. Start by selecting points
in $\mathbb{N}_{g}^{2}$ satisfying the constraints listed in
Definition 1. This first stage of the fractal is the
generator. Then, each subsequent stage of the fractal is obtained by
adding a full copy of the previous stage for every point in the
generator and translating these copies so that their relative
positions are identical to the relative positions of the individual
points in the gnerator.
In this paper, we focus on tree fractals, that is, fractals whose
underlying graph is a tree. We introduce terminology and notation that
will help us in formulating a complete characterization of tree
fractals in terms of geometric properties of their generator.
Definition 2
Let $S$ be any finite subset of $\mathbb{Z}^{2}$. Let $l$, $r$, $b$, and
$t$ denote the following integers:
$\displaystyle l_{S}=\min_{(x,y)\in S}x\qquad r_{S}=\max_{(x,y)\in S}x\qquad b_%
{S}=\min_{(x,y)\in S}y\qquad t_{S}=\max_{(x,y)\in S}y$
An h-bridge of $S$ is any subset of $S$ of the form
$hb_{S}(y)=\{(l_{S},y),(r_{S},y)\}$. Similarly, a v-bridge of $S$ is any
subset of $S$ of the form $vb_{S}(x)=\{(x,b_{S}),(x,t_{S})\}$. We say that a bridge is connected if there is a simple path in $S$ connecting the two bridge points.
Notation 3
Let $S$ be any finite subset of $\mathbb{Z}^{2}$. We will denote by
$nhb_{S}$ and $nvb_{S}$, respectively, the number of h-bridges and
the number of v-bridges of $S$.
The following theorem is a new characterization of tree fractals in
terms of simple connectivity properties of their generator.
Theorem 2.1
$\displaystyle\mathbf{T}=\bigcup_{i=1}^{\infty}{T_{i}}$ is a $g$-discrete self-similar tree fractal, for some $g>1$, with generator $G$ if and only if
a. $G$ is a tree, and
b. $nhb_{G}=nvb_{G}=1$
Notation 4
The directions $\mathcal{D}=\{N,E,S,W\}$ will be used as functions
from $\mathbb{Z}^{2}$ to $\mathbb{Z}^{2}$ such that $N(x,y)=(x,y+1)$,
$E(x,y)=(x+1,y)$, $S(x,y)=(x,y-1)$ and $W(x,y)=(x-1,y)$. Note
that $N^{-1}=S$ and $W^{-1}=E$.
Notation 5
Let $X\subseteq\mathbb{Z}^{2}$. We say that a point $(x,y)\in X$ is
$D$-free in $X$, for some direction $D$, if $D(x,y)\not\in X$.
Definition 3
Let $G$ be the generator of any $g$-discrete self-similar fractal. A
pier is a point in $G$ that is $D$-free for exactly three of
the four directions in $\mathcal{D}$. We say that a pier $(p,q)$ is
$D$-pointing if $D^{-1}(p,q)\in G$. Note that a pier always points
in exactly one direction
Finally, the following observation follows from the
fact that a tree with more than one vertex must contain at least two
leaf nodes.
Observation 1
If $G$ is the generator of any discrete self-similar fractal and $G$ is a tree, then it must
contain at least two piers.
2.3 The Closed Window Movie Lemma
In this subsection, we develop a more accommodating (modified) version
of the general Window Movie Lemma (WML) [5]. Our
version of the WML, which we call the “Closed Window Movie Lemma”,
allows us to replace one portion of a tile assembly with another,
assuming certain extra “containment” conditions are met. Moreover,
unlike in the original WML that lacks the extra containment
assumptions, the replacement of one tile assembly with another in our
Closed WML only goes “one way”, i.e., the part of the tile assembly
being used to replace another part cannot itself be replaced by the
part of the tile assembly it is replacing. We must first define some
notation that we will use in our closed Window Movie Lemma.
A window $w$ is a set of edges forming a cut-set of the full grid
graph of $\mathbb{Z}^{2}$. For the purposes of this paper, we say that a
closed window $w$ induces a cut222A cut is
a partition of the vertices of a graph into two disjoint subsets
that are joined by at least one edge.
of the full grid graph of
$\mathbb{Z}^{2}$, written as $C_{w}=(C_{<\infty},C_{\infty})$, where
$C_{\infty}$ is infinite, $C_{<\infty}$ is finite and for all pairs of
points $\vec{x},\vec{y}\in C_{<\infty}$, every simple path connecting
$\vec{x}$ and $\vec{y}$ in the full grid graph of $C_{<\infty}$ does
not cross the cut $C_{w}$. We call the set of vertices that make up
$C_{<\infty}$ the inside of the window $w$, and write
$inside(w)=C_{<\infty}$ and $outside(w)=\mathbb{Z}^{2}\backslash\,inside(w)=C_{\infty}$. We say that a window $w$ is enclosed in another
window $w^{\prime}$ if $inside(w)\subseteq inside(w^{\prime})$.
Given a window $w$ and an assembly $\alpha$, a window that intersects $\alpha$ is a partitioning of $\alpha$ into two
configurations (i.e., after being split into two parts, each part may
or may not be disconnected). In this case we say that the window $w$
cuts the assembly $\alpha$ into two configurations $\alpha_{L}$
and $\alpha_{R}$, where $\alpha=\alpha_{L}\cup\alpha_{R}$. For
notational convenience, if $w$ is a closed window, we write $\alpha_{I}$
for the assembly inside $w$ and $\alpha_{O}$ for the assembly outside
$w$. Given a window $w$, its translation by a vector $\vec{c}$,
written $w+\vec{c}$ is simply the translation of each of $w$’s
elements (edges) by $\vec{c}$.
For a window $w$ and an assembly sequence $\vec{\alpha}$, we define a window movie $M$ to be the order of placement, position and glue type for each glue that appears along the window $w$ in $\vec{\alpha}$. Given an assembly sequence $\vec{\alpha}$ and a window $w$, the associated window movie is the maximal sequence $M_{\vec{\alpha},w}=(v_{0},g_{0}),(v_{1},g_{1}),(v_{2},g_{2}),\ldots$ of pairs of grid graph vertices $v_{i}$ and glues $g_{i}$, given by the order of the appearance of the glues along window $w$ in the assembly sequence $\vec{\alpha}$.
Furthermore, if $k$ glues appear along $w$ at the same instant (this happens upon placement of a tile that has multiple sides touching $w$) then these $k$ glues appear contiguously and are listed in lexicographical order of the unit vectors describing their orientation in $M_{\vec{\alpha},w}$.
Let $w$ be a window and $\vec{\alpha}$ be an assembly sequence and $M=M_{\vec{\alpha},w}$. We use the notation $\mathcal{B}\left(M\right)$ to denote the bond-forming submovie of $M$, i.e., a restricted form of $M$ that consists of only those steps of $M$ that place glues that eventually form positive-strength bonds in the assembly $\alpha=\textrm{res}(\vec{\alpha})$. Note that every window movie has a unique bond-forming submovie.
Lemma 1 (Closed Window Movie Lemma)
Let $\vec{\alpha}~{}=~{}(\alpha_{i}~{}|~{}0\leq~{}i<~{}l)$, with $l\in\mathbb{Z}^{+}\cup\{\infty\}$, be an assembly sequence in some TAS $\mathcal{T}$
with result $\alpha$. Let $w$ be a closed window that partitions
$\alpha$ into $\alpha_{I}$ and $\alpha_{O}$, and $w^{\prime}$ be a closed window
that partitions $\alpha$ into $\alpha_{I}^{\prime}$ and $\alpha_{O}^{\prime}$. If
$\mathcal{B}(M_{\vec{\alpha},w})+\vec{c}=\mathcal{B}(M_{\vec{\alpha},w^{\prime}})$ for some $\vec{c}\neq(0,0)$ and
the window $w+\vec{c}$ is enclosed in $w^{\prime}$, then the assembly $\alpha^{\prime}_{O}\cup(\alpha_{I}+\vec{c})$ is
in $\mathcal{A[\mathcal{T}]}$.
Proof
Before we proceed with the proof, the next paragraph introduces some
notation taken directly from [5].
For an assembly sequence $\vec{\alpha}=(\alpha_{i}\mid 0\leq i<l)$, we write $\left|\vec{\alpha}\right|=l$ (note that if $\vec{\alpha}$ is infinite, then $l=\infty$). We write $\vec{\alpha}[i]$ to denote $\vec{x}\mapsto t$, where $\vec{x}$ and $t$ are such that $\alpha_{i+1}=\alpha_{i}+\left(\vec{x}\mapsto t\right)$, i.e., $\vec{\alpha}[i]$ is the placement of tile type $t$ at position $\vec{x}$, assuming that $\vec{x}\in\partial_{t}\alpha_{i}$. We write $\vec{\alpha}[i]+\vec{c}$, for some vector $\vec{c}$, to denote $\left(\vec{x}+\vec{c}\right)\mapsto t$. We define $\vec{\alpha}=\vec{\alpha}+\left(\vec{x}\mapsto t\right)=(\alpha_{i}\mid 0\leq i%
<k+1)$, where $\alpha_{k}=\alpha_{k-1}+\left(\vec{x}\mapsto t\right)$ if $\vec{x}\in\partial_{t}\alpha_{k-1}$ and undefined otherwise, assuming $\left|\vec{\alpha}\right|>0$. Otherwise, if $\left|\vec{\alpha}\right|=0$, then $\vec{\alpha}=\vec{\alpha}+\left(\vec{x}\mapsto t\right)=(\alpha_{0})$, where $\alpha_{0}$ is the assembly such that $\alpha_{0}\left(\vec{x}\right)=t$ and is undefined at all other positions. This is our notation for appending steps to the assembly sequence $\vec{\alpha}$: to do so, we must specify a tile type $t$ to be placed at a given location $\vec{x}\in\partial_{t}\alpha_{i}$. If $\alpha_{i+1}=\alpha_{i}+\left(\vec{x}\mapsto t\right)$, then we write $Pos\left(\vec{\alpha}[i]\right)=\vec{x}$ and $Tile\left(\vec{\alpha}[i]\right)=t$. For a window movie $M=(v_{0},g_{0}),(v_{1},g_{1}),\ldots$, we write $M[k]$ to be the pair $\left(v_{k},g_{k}\right)$ in the enumeration of $M$ and $Pos\left(M[k]\right)=v_{k}$, where $v_{k}$ is a vertex of a grid graph.
We now proceed with the proof, throughout which we will assume that $M=\mathcal{B}\left(M_{\vec{\alpha},w}\right)$ and $M^{\prime}=\mathcal{B}\left(M_{\vec{\alpha},w^{\prime}}\right)$. Since $M+\vec{c}=M^{\prime}$ for some $\vec{c}\neq(0,0)$ and $w$ and $w^{\prime}$ are both closed windows, it must be the case that the seed tile of $\alpha$ is in ${\rm dom}\;{\alpha_{O}}\cap{\rm dom}\;{\alpha^{\prime}_{O}}$ or in ${\rm dom}\;{\alpha_{I}}\cap{\rm dom}\;{\alpha^{\prime}_{I}}$. In other words, the seed tile cannot be in ${\rm dom}\;{\alpha_{I}}\backslash\,{\rm dom}\;{\alpha^{\prime}_{I}}$ nor in ${\rm dom}\;{\alpha^{\prime}_{I}}\backslash\,{\rm dom}\;{\alpha_{I}}$. Therefore, assume without loss of generality that the seed tile is in ${\rm dom}\;{\alpha_{O}}\cap{\rm dom}\;{\alpha^{\prime}_{O}}$.
The algorithm in Figure 1 describes how to produce a new valid assembly sequence $\vec{\gamma}$.
If we assume that the assembly sequence $\vec{\gamma}$ ultimately produced by the algorithm is valid, then the result of $\vec{\gamma}$ is indeed $\alpha^{\prime}_{O}\cup\left(\alpha_{I}+\vec{c}\right)$. Observe that $\alpha_{I}$ must be finite, which implies that $M$ is finite. If $|\vec{\alpha}|<\infty$, then all loops will terminate. If $|\vec{\alpha}|=\infty$, then $|\alpha^{\prime}_{O}|=\infty$ and the first two loops will terminate and the last loop will run forever. In either case, for every tile in $\alpha^{\prime}_{O}$ and $\alpha_{I}+\vec{c}$, the algorithm adds a step to the sequence $\vec{\gamma}$ involving the addition of this tile to the assembly. However, we need to prove that the assembly sequence $\vec{\gamma}$ is valid. It may be the case that either: 1. there is insufficient bond strength between the tile to be placed and the existing neighboring tiles, or 2. a tile is already present at this location.
Case 1:
In this case, we claim the following: at each step of the algorithm, the current version of $\vec{\gamma}$ is a valid assembly sequence whose result is a producible subassembly of $\alpha^{\prime}_{O}\cup\left(\alpha_{I}+\vec{c}\right)$. Note that three loops in the algorithm iterate through all steps of $\vec{\alpha}$, such that at any time when adding $\vec{\alpha}[i]$ (or $\vec{\alpha}[j]+\vec{c}$) to $\vec{\gamma}$, all steps of the window movie occurring before $\vec{\alpha}[i]$ (or $\vec{\alpha}[j]$) in $\vec{\alpha}$ have occurred. Similarly, all tiles in $\alpha^{\prime}_{O}$ (or $\alpha_{I}+\vec{c}$) added to $\alpha$ before step $i$ in the assembly sequence have occurred.
So, if the tile $Tile\left(\vec{\alpha}[i]\right)$ that is added to the subassembly of $\alpha$ produced after $i-1$ steps can bond at a location in $\alpha^{\prime}_{O}$ to form a $\tau$-stable assembly, then the same tile added to the producible assembly of $\vec{\gamma}$ must also bond to the same location in $\vec{\gamma}$, as the neighboring glues consist of (i) an identical set of glues from tiles in the subassembly of $\alpha^{\prime}_{O}$ and (ii) glues on the side of the window movie containing $\alpha_{I}+\vec{c}$. Similarly, the tiles of $\alpha_{I}+\vec{c}$ must also be able to bind.
Case 2: Since we only assume that $\mathcal{B}\left(M_{\vec{\alpha},w}\right)+\vec{c}=\mathcal{B}\left(M_{\vec{%
\alpha},w^{\prime}}\right)$, as opposed to the stronger condition $\mathcal{B}\left(M_{\vec{\alpha},w+\vec{c}}\right)=\mathcal{B}\left(M_{\vec{%
\alpha},w^{\prime}}\right)$, which is assumed in the original WML, we must show that ${\rm dom}\;{\left(\alpha_{I}+\vec{c}\right)}\cap{\rm dom}\;{\alpha^{\prime}_{O%
}}=\varnothing$. To see this, observe that, by assumption, $w+\vec{c}$ is enclosed in $w^{\prime}$, which, by definition, means that $inside\left(w+\vec{c}\right)\subseteq inside(w^{\prime})$. Then we have $\vec{x}\in{\rm dom}\;{\alpha^{\prime}_{O}}\Rightarrow\vec{x}\in outside(w^{%
\prime})\Rightarrow\vec{x}\not\in inside\left(w^{\prime}\right)\Rightarrow\vec%
{x}\not\in inside\left(w+\vec{c}\right)\Rightarrow\vec{x}\not\in{\rm dom}\;{%
\left(\alpha_{I}+\vec{c}\right)}$. Thus, locations in $\alpha_{I}+\vec{c}$ only have tiles from $\alpha_{I}$ placed in them, and locations in $\alpha^{\prime}_{O}$ only have tiles from $\alpha^{\prime}_{O}$ placed in them.
So the assembly sequence of $\vec{\gamma}$ is valid, i.e., every addition to $\vec{\gamma}$ adds a tile to the assembly to form a new producible assembly. Since we have a valid assembly sequence, as argued above, the finished producible assembly is $\alpha^{\prime}_{O}\cup\left(\alpha_{I}+\vec{c}\right)$.
∎
3 Main result: scaled tree fractals do not strictly self-assemble
In this section, we first define some notation and then prove our main
result.
3.1 Notation
Recall that each stage $X_{s}$ ($s>1$) of a $g$-dssf (scaled by a factor
$c$) is made up of copies of the previous stage $X_{s-1}$, each of
which is a square of size $cg^{s-1}$.
In the proof of our main result, we will need to refer to one of
the squares of size $cg^{s-2}$ inside the copies of stage $X_{s-1}$, leading to
the following notation.
Notation 6
Let $c\in\mathbb{Z}^{+}$, $1<s\in\mathbb{N}$ and $1<g\in\mathbb{N}$. Let $e,f,p,q\in\mathbb{N}_{g}$. We use
$S_{s}^{c}(e,f,p,q)$ to denote $\{0,1,\ldots,cg^{s-2}-1\}^{2}+cg^{s-1}(e,f)+cg^{s-2}(p,q)$ and
$W_{s}^{c}(e,f,p,q)$ to denote the square-shaped, closed window whose inside
is $S_{s}^{c}(e,f,p,q)$.
In Figure 2 below, the small and large red
windows are $W_{2}^{1}(0,1,3,2)$ and $W_{3}^{1}(0,1,3,2)$, respectively.
Next, we will need to translate a small window to a position inside a
larger window. These two windows will correspond to squares at the
same relative position in different stages $i$ and $j$ of a $g$-dssf.
Notation 7
Let $c\in\mathbb{Z}^{+}$, $i,j\in\mathbb{N}\,\backslash\{0,1\}$, with $i<j$, and $e,f,p,q\in\mathbb{N}_{g}$.
We use $\vec{t}^{c}_{i\rightarrow j}(e,f,p,q)$ to denote
the vector joining the southwest corner of $W_{i}^{c}(e,f,p,q)$ to the
southwest corner of $W_{j}^{c}(e,f,p,q)$. In other words, $\vec{t}^{c}_{i\rightarrow j}(e,f,p,q)=\left(c\left(g^{j-1}-g^{i-1}\right)e+c%
\left(g^{j-2}-g^{i-2}\right)p,c\left(g^{j-1}-g^{i-1}\right)f+c\left(g^{j-2}-g^%
{i-2}\right)q\right)$.
For example,
in Figure 2 below,
$\vec{t}_{2\rightarrow 3}^{1}(0,1,3,2)=(9,18)$.
Finally, to apply Lemma 1, we will need the bond-forming
submovies to line up. Therefore, once the two square windows share
their southwest corner after using the translation defined above, we
will need to further translate the smallest one either up or to the
right, or both, depending on which side of the windows contains the
bond-forming glues, which, in the case of scaled tree fractals, always
form a straight (vertical or horizontal) line of length $c$. We will
compute the coordinates of this second translation in our main
proof. For now, we establish an upper bound on these coordinates that
will ensure that the translated window will remain enclosed in the
larger window.
Lemma 2
Let $c\in\mathbb{Z}^{+}$, $i,j\in\mathbb{N}\,\backslash\{0,1\}$, with $i<j$,
$e,f,p,q\in\mathbb{N}_{g}$, and $x,y\in\mathbb{N}$. Let $m=c(g^{j-2}-g^{i-2})$. If $x\leq m$ and $y\leq m$, then the window
$W^{c}_{i}(e,f,p,q)+\vec{t}^{c}_{i\rightarrow j}(e,f,p,q)+(x,y)$ is
enclosed in the window $W^{c}_{j}(e,f,p,q)$.
Finally, the following lemma establishes that any scaled tree fractal
$\mathbf{T}^{c}$ contains an infinite number of closed windows that all
cut the fractal along a single line of glues.
Lemma 3
Let $\mathbf{T}$ be any tree fractal with generator $G$. If $c\in\mathbb{Z}^{+}$, then it is always possible to pick one pier $(p,q)$
and one point $(e,f)$, both in $G$, such that, for $1<s\in\mathbb{N}$, $W^{c}_{s}(e,f,p,q)$ encloses a configuration that is connected
to $\mathbf{T}^{c}$ via a single connected (horizontal or vertical) line
of glues of length $c$.
The proofs of the lemmas in this sub-section are omitted from this version of the paper due to lack of space.
3.2 Application to scaled tree fractals
The main contribution of this paper is the
following theorem.
Theorem 3.1
Let $\mathbf{T}$ be any tree fractal. If $c\in\mathbb{Z}^{+}$, then
$\mathbf{T}^{c}$ does not strictly self-assemble in the aTAM.
Proof
Let $\mathbf{T}$ be any tree fractal with a $g\times g$ generator $G$,
where $1<g\in\mathbb{N}$. Let $c$ be any positive integer.
For the sake of obtaining a contradiction, assume that $\mathbf{T}^{c}$
does strictly self-assemble in some TAS $\mathcal{T}=(T,\sigma,\tau)$. Further assume that $\vec{\alpha}$ is some assembly
sequence in $\mathcal{T}$ whose result is $\alpha$, such that ${\rm dom}\;\alpha=\mathbf{T}^{c}$.
According to Lemma 3, we can always pick one pier
$(p,q)$ and a point $(e,f)$, both in $G$, such that, for $1<s\in\mathbb{N}$,
the window $W^{c}_{s}(e,f,p,q)$, which we will abbreviate $w_{s}$, encloses
a configuration that is connected to $\mathbf{T}^{c}$ via a single line
of glues of length $c$.333Without loss of generality, we will
assume that this line of glues is positioned on the western side of
the windows and is thus vertical (see the orange circles in
Figure 2, where $s=2$ and $s=3$ for the small
and large red windows, respectively, and $(p,q)=(3,2)$ and
$(e,f)=(0,1)$), because the chosen pier in our example points
east. A similar reasoning holds for piers pointing north, south or
west. The maximum number of distinct combinations and orderings of
glue positionings along this line of glues is finite.444This
number is $(T_{glue})^{2c}\cdot(2c)!$, where $T_{glue}$ is the total
number of distinct glue types in $T$. By the generalized
pigeonhole principle, since
$\left|\left\{w_{s}~{}\left|~{}1<s\in\mathbb{N}\right.\right\}\right|$ is infinite,
there must be at least one bond-forming submovie such that an infinite
number of windows generate this submovie (up to translation). Let us
pick two such windows, say, $w_{i}$ and $w_{j}$ with $i<j$, such that
$\mathcal{B}(M_{\vec{\alpha},w_{i}})$ and
$\mathcal{B}(M_{\vec{\alpha},w_{j}})$ are equal (up to translation). We
must pick these windows carefully, since as stated in the proof of
Lemma 1, the seed of $\alpha$ must be either in both
windows or in neither. This condition can always be satisfied. The
only case where the seed is in more than one window is when it is at
position $(0,0)$ and $e=f=p=q=0$, which implies that all windows
include the origin. In all other cases, none of the windows
overlap. So, if the seed belongs to one of them, say $w_{k}$, then we
can pick any $i$ greater than $k$ (and $j>i$). Finally, if the seed
does not belong to any windows, then any choice of $i$ and $j>i$ will
do.
We will now prove that $w_{i}$ and $w_{j}$ satisfy the two
conditions of Lemma 1.
First, we compute $\vec{c}$ such that
$\mathcal{B}(M_{\vec{\alpha},w_{i}})+\vec{c}=\mathcal{B}(M_{\vec{\alpha},w_{j}})$. We know that $w_{i}+\vec{t}^{c}_{i\rightarrow j}(e,f,p,q)$ and $w_{j}$ share their southwest
corner. We need to perform one more translation to align the
bond-forming glues of $w_{i}$ and $w_{j}$. We use $(a,b)$ to denote the
position of the western point in the horizontal bridge of the
generator. In our example (east-pointing pier), $a=0$ and $b$ is a
variable with domain $\mathbb{N}_{g}$ ($b=2$ in
Figure 2). To align the bond-forming glues of
$w_{i}$ and $w_{j}$, we must translate $w_{i}+\vec{t}^{c}_{i\rightarrow j}(e,f,p,q)$ by $(x,y)=\left(0,bc\sum_{k=i-2}^{j-3}g^{k}\right)$. The general computation for
this translation is illustrated in Figure 3. Since
$x\leq m$ (as defined in Lemma 2) and
$bc\sum_{k=i-2}^{j-3}g^{k}\leq(g-1)c\sum_{k=i-2}^{j-3}g^{k}=c\left(\sum_{k=i-1}%
^{j-2}g^{k}-\sum_{k=i-2}^{j-3}g^{k}\right)=c\left(g^{j-2}-g^{i-2}\right)=m$, we can apply
Lemma 2 to infer that, with $\vec{c}=\vec{t}^{c}_{i\rightarrow j}(e,f,p,q)+(x,y)$, $w_{i}+\vec{c}$ is
enclosed in $w_{j}$. Therefore, the second condition of
Lemma 1 holds.
Second, by construction, $\mathcal{B}(M_{\vec{\alpha},w_{i}})+\vec{c}=\mathcal{B}(M_{\vec{\alpha},w_{j}})$. Therefore, the first condition of
Lemma 1 holds.
In conclusion, the two
conditions of Lemma 1 are satisfied, with $\alpha_{I}$ and
$\alpha_{O}^{\prime}$ defined as the intersection of $\mathbf{T}^{c}$ with the inside
of $w_{i}$ and the outside of $w_{j}$, respectively. We can thus conclude
that the assembly $\alpha_{I}\cup(\alpha_{O}^{\prime}-\vec{c})$ is producible
in $\mathcal{T}$. Note that this assembly is identical (up to
translation) to $\mathbf{T}^{c}$, except that the interior of the large
window $w_{j}$ is replaced by the interior of the small window
$w_{i}$. Since the configurations in these two windows cannot be
identical, we have proved that $\mathcal{T}$ does not strictly
self-assemble $\mathbf{T}^{c}$, which is a contradiction. ∎
4 Conclusion
In this paper, we made three contributions. First, we gave a new characterization of tree fractals in terms of simple geometric properties of
their generator. Second, we proved a new variant of the Window Movie
Lemma in [5], which we call the “Closed Window
Movie Lemma.” Third, we proved that no scaled-up version of any
discrete self-similar tree fractal strictly self-assembles in the
aTAM. In future work, we plan to extend this result to larger classes of
non-tree fractals similar to the class of pinch-point
fractals in [6].
References
[1]
Steven M. Kautz and James I. Lathrop, Self-assembly of the discrete
Sierpinski carpet and related fractals, DNA, 2009, pp. 78–87.
[2]
Steven M. Kautz and Brad Shutters, Self-assembling rulers for
approximating generalized sierpinski carpets, Algorithmica 67
(2013), no. 2, 207–233.
[3]
James I. Lathrop, Jack H. Lutz, and Scott M. Summers, Strict
self-assembly of discrete Sierpinski triangles, Theoretical Computer
Science 410 (2009), 384–405.
[4]
Jack H. Lutz and Brad Shutters, Approximate self-assembly of the
sierpinski triangle, Theory Comput. Syst. 51 (2012), no. 3,
372–400.
[5]
P.-E. Meunier, M. J. Patitz, S. M. Summers, G. Theyssier, A. Winslow, and
D. Woods, Intrinsic universality in tile self-assembly requires
cooperation, Proceedings of the 25th Annual ACM-SIAM Symposium on Discrete
Algorithms (SODA), 2014, pp. 752–771.
[6]
Matthew J. Patitz and Scott M. Summers, Self-assembly of discrete
self-similar fractals, Natural Computing 1 (2010), 135–172.
[7]
Paul W. K. Rothemund, Theory and experiments in algorithmic
self-assembly, Ph.D. thesis, University of Southern California, December
2001.
[8]
Nadrian C. Seeman, Nucleic-acid junctions and lattices, Journal of
Theoretical Biology 99 (1982), 237–247.
[9]
, De novo design of sequences for nucleic acid structural
engineering, Journal of Biomolecular Structural Dynamics 8 (1990),
573–581.
[10]
Hao Wang, Proving theorems by pattern recognition – II, The Bell
System Technical Journal XL (1961), no. 1, 1–41.
[11]
Erik Winfree, Algorithmic self-assembly of DNA, Ph.D. thesis,
California Institute of Technology, June 1998. |
11institutetext: Observatorio Astronómico Nacional (OAN-IGN), Alfonso XII 3, 28014, Madrid, Spain
11email: i.gallardocava@oan.es
22institutetext: Observatorio Astronómico Nacional (OAN-IGN), Apartado 112, 28803, Alcalá de Henares, Madrid, Spain
33institutetext: Centro de Desarrollos Tecnológicos, Observatorio de Yebes (IGN), 19141, Yebes, Guadalajara, Spain
44institutetext: Institut de Radioastronomie Millimétrique, 300 rue de la Piscine, 38406 Saint-Martin-d’Hères, France
Abstract
Context:There is a class of binary post-asymptotic giant branch (post-AGB) stars that exhibit remarkable near-infrared (NIR) excess.
These stars are surrounded by disks with Keplerian or quasi-Keplerian dynamics and outflows composed of gas escaping from the rotating disk.
Depending on the dominance of these components, there are two subclasses of binary post-AGB stars: disk-dominated and outflow-dominated.
Aims:We aim to properly study the hourglass-like structure that surrounds the Keplerian disk around 89 Her.
Methods:We present total-power on-the-fly maps of ${}^{12}$CO and ${}^{13}$CO $J=$ 2 – 1 emission lines in 89 Her. Previous studies are known to suffer from flux losses in the most extended components.
We merge these total-power maps with previous NOEMA maps.
The resulting combined maps are expected to detect the whole nebula extent of the source.
Results:Our new combined maps contain the entirety of the detectable flux of the source and at the same time are of high spatial resolution thanks to the interferometric observations.
We find that the hourglass-like extended outflow around the rotating disk is larger and more massive than suggested by previous works. The total nebular mass of this very extended nebula is 1.8 $\times$ 10${}^{-2}$ M${}_{\odot}$, of which $\sim$ 65% comes from the outflow.
The observational data and model results lead us to classify the envelope around 89 Her as an outflow-dominated nebula, together with R Sct and IRAS 19125+0343 (and very probably AI CMi, IRAS 20056+1834, and IRAS 18123+0511).
The updated statistics on the masses of the two post-AGB main components reveal that there are two distinct subclasses of nebulae around binary post-AGB stars depending on which component is the dominant one. We speculate that the absence of an intermediate subclass of sources is due to the different initial conditions of the stellar system and not because both subclasses are in different stages of the post-AGB evolution.
Conclusions:
The nebula around the binary post-AGB star 89 Herculis ††thanks: Based on observations with the 30 m IRAM telescope and NOEMA. IRAM is supported by INSU/CNRS (France), MPG (Germany), and IGN (Spain). ††thanks: Final datacubes are available at the CDS via anonymous FTP
to cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/gcat?J/A+A/
I. Gallardo Cava
11
J. Alcolea
11
V. Bujarrabal
22
M. Gómez-Garrido
1133
and A. Castro-Carrizo
44
(Received 5 July 2022 / Accepted 5 January 2023)
Key Words.:
stars: AGB and post-AGB $-$ binaries: general $-$ circumstellar matter $-$ stars: individual: 89 Herculis $-$ radio lines: stars $-$ ISM: planetary nebulae: general
1 Introduction
Circumstellar envelopes around AGB stars (CSE-AGBs) are often found to be spherical and in slow expansion, while their descendants, that is pre-planetary and planetary nebulae (pPNe and PNe), present different morphologies: pPNe present strongly aspherical (often axisymmetrical) shapes resulting from the interaction of axial fast winds with the CSE formed in the AGB stage (Ueta et al., 2000; Sahai et al., 2007; Castro-Carrizo et al., 2010);
PNe tend to present (quasi-)spherical, axisymmetrical, or irregular shapes (Sahai et al., 2011; Stanghellini et al., 2016).
This evolution is very rapid and takes place in a very short time, namely $\sim$ 1 000 years
(see Balick & Frank, 2002; Van Winckel, 2003).
This spectacular transformation is thought to be due to magnetocentrifugal launching of outflows from rotating disks, which implies the presence of a stellar companion to provide the necessary amount of angular momentum (see e.g., Frank & Blackman, 2004; Bujarrabal et al., 2016).
There is a class of binary post-AGB stars (binary systems including a post-AGB star) that systematically present circumbinary disks with Keplerian dynamics.
These sources tend to present remarkable near-infrared (NIR) excess and narrow CO line profiles characteristic of rotating disks (Van Winckel, 2003; Bujarrabal et al., 2013, and references therein).
The IR data of these sources reveal the presence of highly processed dust grains, which implies that these disks must be stable structures (Jura, 2003; Sahai et al., 2011; Gielen et al., 2011).
In the four sources in which the disk rotation has been well observed (the Red Rectangle, AC Herculis, IW Carinae, IRAS 08544$-$4431), the disk contains most of the mass, but there is also gas in expansion, a disk wind that is extracted from the disk and represents $\stackrel{{\scriptstyle\sf<}}{{\scriptstyle\sf\sim}}$ 15% of the total nebular mass (Bujarrabal et al., 2016, 2017, 2018; Gallardo Cava et al., 2021).
In contrast, there is a subclass of these binary post-AGB stars whose nebulae are dominated by the expanding component instead of the Keplerian disk: the outflow-dominated subclass (see Gallardo Cava et al., 2021).
In this latter class of sources, the nebula emission is quite different from that of the other well-studied cases (the disk-dominated nebulae), because their millimeter(mm)-wave interferometric data confirm the presence and dominance of an extended and expanding component that contains most of the detected nebular material.
This is the case for R Scuti and IRAS 19125+0343, where $\sim$ 75% of the total nebular mass corresponds to the extended and expanding component (Gallardo Cava et al., 2021). AI CMi, IRAS 20056+1834, and IRAS 18123+0511 also very probably belong to this subclass.
As we show in this paper, the last member of this (outflow-dominated) subclass is 89 Herculis.
89 Herculis is a binary post-AGB star with warm dust located in a stable structure and large dust grains formed and settled onto the midplane (see Shenton et al., 1995; de Ruyter et al., 2006; Hillen et al., 2014).
This source has been thoroughly studied using single-dish observations. Studies reveal narrow CO line profiles similar to those of the Red Rectangle, but with prominent wings, suggesting a significant contribution of the extended component (Bujarrabal et al., 2013).
Observations at 1.3, 2, and 3 mm reveal the presence of C${}^{18}$O, C${}^{17}$O, CS, SiS, and HCN and a very detailed analysis reveals that the nebula around 89 Her is C-rich (see Gallardo Cava et al., 2022).
According to Gallardo Cava et al. (2021), the nebula around 89 Her is composed of a Keplerian disk and an hourglass-shaped structure (see also Alcolea & Bujarrabal, 1995; Fong et al., 2006; Bujarrabal et al., 2007).
The mass of the nebula was found to be 1.4 $\times$ 10${}^{-2}$ M${}_{\odot}$, of which 6.4 $\times$ 10${}^{-3}$ M${}_{\odot}$ corresponds to the Keplerian disk ($\sim$ 50%). The findings of previous studies suggest that 89 Her is the only source of our sample in which there is the same mass in the disk as in the disk wind.
However, the emission of the outflow was found to be underestimated because the NOEMA maps presented a significantly increased amount of filtered-out flux in the line wings in comparison to the single-dish 30 m IRAM profile taken towards the center of the source.
Furthermore, the NOEMA maps show that the total extent of the outflow has a size comparable to the Half Power Beam Width (HPBW) of the 30 m IRAM at the frequency of the CO $J=$ 2 – 1transitions.
Therefore, it is also possible that the lost flux problem of the interferometric observation is even more severe, as the 30 m IRAM single-dish data may not contain all the flux emitted by the source, as NOEMA maps demonstrate that the source is larger than the beam. This implies that the mass derived for the outflow could be largely underestimated.
To overcome this problem and properly derive the mass of the extended outflow component in a definitive way, here we present new single-dish total-power maps of 89 Her. We merge these maps, which probe the full extent of 89 Her, with the previous interferometric data, resulting in high-resolution maps of 89 Her containing the total flux for all the nebular components detected in molecular gas.
2 Observations and observational results
As mentioned above, to solve the problem of the underestimation of the mass of the extended outflow component, it is necessary to construct maps of the source including the large spatial scales and not only those provided by the interferometric observations, which only probe structures of 5${}^{\prime\prime}$ or smaller. To do this, we built new total-power single-dish maps of 92${}^{\prime\prime}$ in diameter, and later combined these with the previously obtained interferometric maps already presented by Gallardo Cava et al. (2021).
2.1 Total-power data
These new single-dish 89 Her observations were performed using the 30 m IRAM telescope at Pico Veleta (Granada, Spain).
The observations were carried out between June 2 and 8, 2021, for a total of 65 hours of telescope time (project 061-21).
We focus our observations on the ${}^{12}$CO and ${}^{13}$CO $J=$ 2 – 1 emission lines (230.5 and 220.4 GHz), but we also simultaneously observed the ${}^{12}$CO and ${}^{13}$CO $J=$ 1 – 0 lines. However, the signal-to-noise ratio (S/N) attained for the $J=$ 1 – 0 lines did not result in their detection, and therefore these lines are not discussed in this paper.
The HPBW of the telescope is around 11.${}^{\prime\prime}$3 at 230.5 and 220.4 GHz.
During the observing time the weather conditions were good, with typical 225 GHz zenital opacities of between 0.1 and 0.3.
We performed 92${}^{\prime\prime}$ $\times$ 92${}^{\prime\prime}$ maps every 20 min. The scanning of these maps was performed in six different directions, at paralactic angles (PAs) of 0°, 30°, 60°, 90°, 120°, and 150°, in order to minimize interleaving and weaving patterns in the resulting final averaged map.
The observations were performed in on-the-fly (OTF) mode, with 23 parallel scans per map, and a separation between scans of 4${}^{\prime\prime}$, a scanning velocity of 4${}^{\prime\prime}$s${}^{-1}$, and a dump rate of 1 s, resulting in a 4${}^{\prime\prime}$ $\times$ 4${}^{\prime\prime}$ grid (to be compared with the HPBW of 11.${}^{\prime\prime}$3, i.e., a factor 3 oversampling).
The observations were performed in on-off mode, observing the reference position (600${}^{\prime\prime}$ away in the R.A. direction) every three scans ($\sim$ 70 s). A calibration was performed at the beginning of each individual map using the chopper-wheel method, observing the sky and both hot and cold loads. Observations of NGC 7027 were also performed to check the consistency of the intensity scale: As we did not find calibration changes larger that 20%, no re-scale has been applied to the data.
We connected the Fast Fourier Transform Spectrometer (FTS) units to the EMIR receiver with a spectral velocity resolution of 0.25 km s${}^{-1}$ ($\sim$ 200 kHz) per channel. We obtained spectra for vertical and horizontal linear polarization receivers; as no significant changes were found between the two polarizations, the two maps were averaged.
We applied the standard data-reduction procedure, which consists of the removal of baselines, calculating a polarization average, and resampling data to the desired final spectral resolution. Individual OTF maps were inspected for consistency of the pointing: we found no significant differences in the position of the central peak, and so all maps were averaged without applying corrections.
The OTF maps for the $J=2-1$ lines are shown in Figs. 9 and 10.
As we can see in Fig. 9, the extent of the nebula is restricted to a region of 20${}^{\prime\prime}$ in diameter. This is fully compatible with the extent of the nebula in the interferometric maps considering the resolution of the OTF data. This relatively large extent also justifies the flux loss in the NOEMA data. Moreover, this extent is larger than the HPBW of the 30 m IRAM, which explains the larger total flux in the new OTF observations in comparison with the old single-pointing 30 m data.
On the contrary, this size is smaller than the HPBW of the primary beam of the individual NOEMA antennas at 1.3 mm ($\sim$ 22${}^{\prime\prime}$). Nevertheless, the primary beam shape attenuation is taken into account, because it is corrected before the process of merging with the total-power data, following the standard procedure of the GILDAS software111GILDAS is a software package designed to reduce and analyze mainly mm observations from single-dish and interferometric telescopes. It is developed and maintained by IRAM, LAOG/Université de Grenoble, LAB/Observatoire de Bordeaux, and LERMA/Observatoire de Paris. See https://www.iram.fr/IRAMFR/GILDAS.
2.2 Interferometric data
Interferometric observations of the ${}^{12}$CO and ${}^{13}$CO $J=$ 2 – 1 rotational transitions were carried out toward 89 Her with the IRAM NOEMA interferometer at Plateau de Bure (Grenoble, France).
${}^{12}$CO $J=$ 2 – 1 observations were performed under project name P05E, while ${}^{13}$CO $J=$ 2 – 1 observations were performed under project names X073 and W14BT.
These maps were published by Gallardo Cava et al. (2021), where the complete technical description can be found.
2.3 Combined maps
In this work, we present NOEMA maps of ${}^{12}$CO and ${}^{13}$CO $J=2-1$ emission lines, which include short-spacing pseudovisibilities obtained from the total-power maps with the 30 m IRAM through OTF mode. Therefore, our combined maps contain all detectable flux because (a) they recover the lost flux of the extended component filtered out by the interferometer and
(b) include large areas that single-dish single-pointing observations cannot detect because of the limitations of the beam of the telescope; see Figs. 1 and 2.
To include the large scales probed by the OTF maps into the interferometric data, these total-power maps must first be converted into pseudo-visibility data cubes with exactly the same velocity configuration (same number of channels, velocity spacing, and velocity value for the reference channel) as the NOEMA data. We did this by resampling the OTF maps into a spectral resolution of 0.5 km s${}^{-1}$ and adopting a LSR velocity of $-$7.94 km s${}^{-1}$ for the central channel. Then, data cubes with a pixel size of 2${}^{\prime\prime}$ $\times$ 2${}^{\prime\prime}$ were built from the OTF scans, which are Fourier transformed for obtaining the corresponding pseudo-visibilities to be merged with the interferometric visibilities from the corresponding NOEMA observations. Before the merging, these interferometric data are corrected for the NOEMA primary beam attenuation following the standard procedure of GILDAS.
After the merging of the $uv$ data sets, we verified the compatibility of the calibration of the two instruments, 30 m IRAM and NOEMA. The new maps were then produced by first mapping and then cleaning the merged uv data sets.
For the ${}^{12}$CO $J=$ 2 – 1 maps, we used robust weighting and a $uv$ tapper of 200 m, resulting in a HPBW synthetic clean beam of 1.${}^{\prime\prime}$95 $\times$ 1.${}^{\prime\prime}$45.
For the case of ${}^{13}$CO $J=$ 2 – 1, as the interferometric data were of better quality, we used robust weighting and no tappering.
This resulted in a HPBW synthetic clean beam of 0.${}^{\prime\prime}$76 $\times$ 0.${}^{\prime\prime}$59.
The resulting new merged maps for ${}^{12}$CO $J=$ 2 – 1 and ${}^{13}$CO $J=$ 2 – 1 are presented in Figs. 3 and 4, respectively.
We verified that the new maps are compatible with the old interferometric data and that the total flux is the same as in the new OTF maps, that is, no flux is missing in the new data.
We therefore conclude that our new combined maps contain all the detectable flux, because they are not affected by any interferometric losses and include the full extent of the source. This new total flux is larger than that obtained from the interferometric observation alone and that provided by single-dish, single-pointing observations; see Figs. 1 and 2.
In addition, position–velocity (PV) diagrams for PA $=150\degree$ are shown in Figs. 5 and 6.
These velocity maps and equatorial PV diagrams reveal an intense central clump and an expanding component.
On the one hand, the central clump corresponds to an unresolved rotating disk very probably with Keplerian dynamics.
On the other hand, we also see an expanding component that very nicely shows hourglass-like features.
Following the same reasoning as in our previous work (Gallardo Cava et al., 2021), we find that PV diagrams along a PA of 150° are the most efficient at revealing the presence of the rotating gas (see Figs. 5 and 6). Therefore, the PV diagrams with PA = 150°suggest that a rotating disk with moderate velocity dispersion is responsible for the compact and central clump in 89 Her.
Thanks to our new combined maps, we confirm the presence of asymmetrical velocities in both maps, because more emission is present at positive velocities than at negative ones. This phenomenon is a result of self-absorption by cold gas in expansion located in front of the rotating component
and is often found to be present in the disk-containing sources that we have observed in the past, such as the Red Rectangle or R Sct, a disk- and an outflow-dominated source, respectively (see Bujarrabal et al., 2016; Gallardo Cava et al., 2021).
Our combined maps, with all possible detected flux, show an hourglass outflow whose symmetry axis is almost aligned with our line of sight. These maps also reveal that the size of this outflowing component is larger than we found it to be in a previous study, both in the height of the hourglass and in the width of its expanding walls. This recovered flux has a relevant impact on the mass of the outflow that surrounds the disk (see Sect. 3).
Apart from the rotating disk and the very large hourglass-like expanding component, we do not detect any other structures (such as a halo).
We verified this by performing radial averages of the emission in the OTF maps. The resulting intensity-versus-offset profiles obtained for both ${}^{12}$CO and ${}^{13}$CO $J=$ 2 – 1 maps are compatible with the 89 Her nebula consisting only in the two structures mentioned above.
Therefore, we conclude that the molecular envelope of 89 Her comprises an hourglass-shaped outflow and a rotating disk with Keplerian dynamics.
We note that the S/N is higher in the new combined maps, which allows detection of the outflow at slightly larger expansion velocities. The only new feature worth mentioning is the tentative detection of a spiral-like pattern seen at some receding velocities ($-$5.95 km s${}^{-1}$) in the ${}^{12}$CO $J=$ 2 – 1 maps.
If real, this structure cannot be explained as a result of the orbital motions in the binary system, because this would imply a period much larger (1 800 years for an expansion velocity of $\sim$ 10 km s${}^{-1}$ and a distance of 1 000 pc) than that derived from the radial velocity curve of the primary (289.1 days; see Oomen et al., 2018).
This structure could instead be due to the effect of a precessing jet launched from the compact companion (see Raga et al., 2009; Velázquez et al., 2011). However, this should be investigated more thoroughly using more sensitive maps.
This is similar to the case of the pattern seen in the Red Rectangle, which also suggests a periodicity much larger than the orbital period in the system (see Cohen et al., 2004; Waelkens et al., 1996).
3 Model fitting: Mass of the nebula
To derive the physical parameters of the nebula from the new data presented here, we adopted a nebular model based on a rotating disk with Keplerian dynamics surrounded by an extended and expanding hourglass-shaped structure. We use the code described in Gallardo Cava et al. (2021), where a complete description can be found. As in the previous work, we adopt a distance of 1 000 pc.
We assume LTE populations for the $J=$ 2 – 1 CO lines, which is a reasonable assumption for low-$J$ rotational levels, simplifies the calculations, and provides a more comprehensible interpretation of the fitting parameters.
We adopt the same relative abundance values as in Gallardo Cava et al. (2021): X(${}^{13}$CO) = 2 $\times$ 10${}^{-5}$ and an abundance ratio of X(${}^{12}$CO)/X(${}^{13}$CO) = 10. These abundance values are usually found in nebulae around binary post-AGB stars.
We assume the presence of a rotating disk in the innermost region of the nebula and a large and wide hourglass-like extended and expanding component.
We confirm the inclination of the nebular symmetry axis with respect to the line of sight, together with PA = 150° for the equatorial direction.
We also corroborate the self-absorption effects at low negative expansion velocities.
The main parameters of our new best model can be seen in Table 1 and can be described as follows. As in the first work, we find the same reliable results for density and temperature laws with high slope values.
The density of the rotating disk is assumed to vary with the distance to the binary system following a potential law, $n\propto r^{-2}$, with a value of 2.0 $\times$ 10${}^{7}$ cm${}^{-3}$ at 2.5 $\times$ 10${}^{15}$ cm.
The temperature varies as $T\propto r^{-2.5}$, with a temperature of 425 K at 5 $\times$ 10${}^{15}$ cm (the disk radius).
Moreover, we assume Keplerian rotation in the disk, with 1.5 km s${}^{-1}$ at 10${}^{16}$ cm, which is compatible with a central total stellar mass of 1.7 M${}_{\odot}$.
In the case of the outflow/disk wind, we assume a radial velocity with a modulus that increases linearly with the distance to the center, and we find moderate velocities of $\sim$ 10 km s${}^{-1}$.
We assume an hourglass 67% larger with walls 10% wider compared to the previous work (see Fig. 7).
Therefore, the total height of the hourglass is 2.4 $\times$ 10${}^{17}$ cm and its walls are $\sim$ 8 $\times$ 10${}^{15}$ cm wide.
The density law varies as $n\propto r^{-2}$, with values in between 10${}^{5}$ cm${}^{-3}$ in the zones closest to the rotating disk and $\leq$ 10${}^{3}$ cm${}^{-3}$ in the most external regions of the outflow.
The temperature of the outflow varies as $T\propto r^{-0.2}$, with values $\leq$ 10 K in the external regions of the outflow. We therefore find a very extended outflow composed of cold gas, as in the case of R Sct or IRAS 19125+0343, which are similar objects (see Gallardo Cava et al., 2021).
We show the predictions from our best model in our synthetic velocity maps (Figs. 11 and 12) and synthetic PV diagrams (Figs. 13 and 14) and our new nebula model for 89 Her in Fig. 7.
We also checked that our model satisfactorily reproduces previous interferometric maps of the ${}^{12}$CO $J=$ 1 – 0 emission line (data taken from Fong et al., 2006, whose maps also show no flux loss).
The model reproduces the observational data and yields a total mass for the nebula of 1.8 $\times$ 10${}^{-2}$ M${}_{\odot}$, of which 1.2 $\times$ 10${}^{-2}$ M${}_{\odot}$ corresponds to the outflow and 6.4 $\times$ 10${}^{-3}$ M${}_{\odot}$ to the rotating disk. This means that 89 Her presents a Keplerian disk surrounded by a very extended and expanding outflow that represents $\sim$ 65% of the total mass of the nebula.
Some of the nebula properties are not accurately determined because of the relatively low resolution of the interferometric data, such as the height of the Keplerian disk. On the contrary, the outflow structure is well determined, because the hourglass-like shape is clear in our maps. See Gallardo Cava et al. (2021), for further details on the uncertainties of our model.
4 Disk-mass ratio in binary post-AGB stars
In our previous work, we classified 89 Her as an intermediate-subclass nebula, that is, intermediate between the disk- and the
outflow-dominated subclasses, because both the disk and its outflow were found to present very similar masses. However, a significant amount of flux was found to be filtered out in those maps (see Sect. 1).
According to our new merged maps and the new model, the pPN around 89 Her is dominated by its disk wind, because $\sim$ 65% of the total mass is located in the very large hourglass-like expanding component.
After this update on the masses of both outflow and disk in 89 Her, this source is more in line with what is found in those classified as outflow-dominated. In fact, if we only consider sources for which these masses are relatively well determined (see Table 2 and Fig. 8), it seems that there is not an intermediate case in between the disk- and the outflow-dominated subclass. However, before drawing conclusions, it is better to revise any problem in the estimation of the masses of the two components, such as the impact of and hypothetical loss of flux (as in the case of 89 Her, which we review immediately above), the impact of other nebular components —in particular the neutral and ionized atomic gas—, or the contribution of the layers where molecules have been photo-dissociated by the interstellar UV field.
We reiterate that the main goal of this work is to study the circumbinary disk and the outflow or disk wind that is escaping from the rotating component during the post-AGB phase.
Other nebular components, such as collimated high-velocity stellar winds (launched by the companion) or larger haloes (formed during the previous AGB phase and the detection of which could be quite difficult), are not part of this work.
In any case, we discuss the relevance of these structures and their (possible) contribution to the total nebular mass.
4.1 The effects of the interferometric flux losses
Seven sources have been thoroughly studied through interferometric observations: AC Her, the Red Rectangle, IRAS 08544$-$4431, IW Car, 89 Her, IRAS 19125+0343, and R Sct.
Here, we discuss the flux loss present in the maps used to estimate the molecular mass in each source of our sample and update the values of the mass of the outflow and consequently the disk-to-outflow mass ratio. In the following, we assume that the filtered-out flux is due to the outflow, as no other large-scale structures have been detected in these kinds of objects.
4.1.1 AC Herculis
The ${}^{12}$CO $J=$ 2 – 1 mm-wave interferometric observations of this source do not present a significant amount of filtered-out flux. In addition, its angular size is around 2${}^{\prime\prime}$, which is too small compared with the beam size. Therefore, we can confidently discard the presence of a very extended and undetected component.
According to this, the total molecular nebular mass is 8.3 $\times$ 10${}^{-4}$ M${}_{\odot}$ and the derived disk-mass ratio of AC Her of $\sim$ 95% is well determined. See Gallardo Cava et al. (2021) for further details.
4.1.2 The Red Rectangle
This source is the best studied of our sample. The nebular mass was derived from model fitting of maps of lines ${}^{12}$CO $J=3-2$ and $J=6-5$, ${}^{13}$CO $J=3-2$, C${}^{17}$O $J=6-5$, and H${}^{13}$CN $J=4-3$. The total nebular mass is 1.4 $\times$ 10${}^{-2}$ M${}_{\odot}$, of which $\sim$ 9% corresponds to the outflow that surrounds the Keplerian disk.
Bujarrabal et al. (2016) mentioned that they found 25% flux loss in the line wings. If we assume that this filtered-out flux corresponds to the outflow, the contribution of the outflow to the total molecular mass of the nebula is up to 11%. Therefore, the Red Rectangle remains as a disk-dominated source.
4.1.3 IRAS 08544$-$4431
The nebular mass of this source is 2.0 $\times$ 10${}^{-2}$ M${}_{\odot}$ (Bujarrabal et al., 2018). According to the authors, the mass of the outflow is 10%. The ${}^{12}$CO $J=3-2$ maps present a small fraction of flux loss ($<$ 20%), and the ${}^{13}$CO $J=3-2$ maps, with a less extended brightness distribution, do not show a significant amount of filtered-out flux.
Assuming that this flux loss is due to the outflow, its contribution to the total molecular mass will be $\leq$ 12%.
Additionally, the angular size of this source is around 4${}^{\prime\prime}$, which is considerably smaller than the beam size.
4.1.4 IW Carinae
This source, of 4${}^{\prime\prime}$ in angular size, presents a total nebular mass of 4 $\times$ 10${}^{-3}$ M${}_{\odot}$, of which 11% is located in the expanding component (see Bujarrabal et al., 2017, for details).
The authors find a small fraction of filtered-out flux (25%) in the ${}^{12}$CO $J=3-2$ maps, which could be higher in the line wings.
If this flux loss corresponds to the expanding component, it mass contribution is $\leq$ 14% of the total nebular mass. Additionally, Bujarrabal et al. (2013), via single-dish observations, found very narrow CO line profiles characteristic of rotating disk and weak wings that correspond to the outflow. This source would still be a disk-dominated source even in in the least favorable flux-loss scenario.
4.1.5 89 Herculis
In this work, we present interferometric maps merged with large-scale single-dish maps. These combined maps do not present filtered-out flux (see Sect. 2).
Our new combined maps and model allow us to estimate the mass of the nebula that surrounds 89 Her. We find a total molecular mass of 1.8 $\times$ 10${}^{-2}$ M${}_{\odot}$, in which the hourglass-shaped outflow represents 65% of the mass (see Sect. 3).
4.1.6 IRAS 19125+0343
This nebula presents a total mass of 1.1 $\times$ 10${}^{-2}$ M${}_{\odot}$ and 71% corresponds to the outflow (see Gallardo Cava et al., 2021, for details). The interferometric visibilities were merged with zero-spacing data obtained with the 30 m IRAM telescope. Therefore, there is no filtered-out flux in the ${}^{12}$CO $J=$ 2 – 1 final maps, and so this source is clearly an outflow-dominated source.
4.1.7 R Scuti
The nebular mass is 3.2 $\times$ 10${}^{-2}$ M${}_{\odot}$, of which 73% is located in the extended outflow that surrounds the Keplerian disk (see Gallardo Cava et al., 2021, for complete description). The ${}^{12}$CO $J=$ 2 – 1 maps do not present filtered-out flux, because they were merged with short-spacing pseudo-visibilities obtained from on-the-fly maps with the 30 m IRAM telescope to compensate for the extended component filtered out by the interferometer, as this source presents a large angular size. This case is similar to that of 89 Her.
4.2 Atomic mass
Some sources of our sample have been studied in the C ii line (157.14 $\mu$m).
The flux of this line can be used to measure the mass of the low-excitation atomic component in pPNe, because this transition is useful for estimating the total mass of photodissociation regions (PDRs).
PDRs almost perfectly coincide with the cold atomic gas regions, because they are very clearly delimited zones in between the molecular and the H ii regions. In addition, most of our sources show relatively low stellar temperatures, of namely $<$ 10 000 K, and so H ii regions are not expected in them. We note that in Gallardo Cava et al. (2022), no radio recombination lines were detected in these spectral surveys.
Castro-Carrizo et al. (2001) and Fong et al. (2001) found that the PDR masses for AC Her, the Red Rectangle, 89 Her, and R Sct (rescaled to our distances; see Table 2) are $<$ 10${}^{-2}$, $<$ 10${}^{-2}$, $<$ 2 $\times$ 10${}^{-3}$, and $<$ 3 $\times$ 10${}^{-2}$ M${}_{\odot}$.
Bujarrabal et al. (2016) detected C ii in the Red Rectangle and, using the method described in Castro-Carrizo et al. (2001), the mass of the PDR derived from the measured flux would represent $<$ 10${}^{-3}$ M${}_{\odot}$ which would mostly be located in the rotating disk.
We conclude that there are no indications of a significant contribution from PDRs or H ii regions to the masses of the disk-containing nebulae around binary post-AGB stars.
Therefore, there is no need to consider these small contributions in the disk-mass ratio analysis.
4.3 Post-AGB stellar winds
Systematic high-resolution radial velocity studies reveal that stellar jets or winds are a common phenomenon in binary post-AGB stars (see Bollen et al., 2021, and references therein). A collimated low-density and high-velocity jet launched by the companion often operates during the late AGB and early post-AGB stages (see Bollen et al., 2022, and their Fig. 1).
We note that this is a nebular component not studied in this work (focused on rotating disks and disk winds). However, we argue that the mass of this component is small.
This kind of stellar wind is present in the best studied source of our sample, the Red Rectangle, and can be clearly seen in the optical image (Cohen et al., 2004). At first sight, the H$\alpha$ emission of the famous X-shaped structure seems to dominate the whole nebula.
Nevertheless, according to the high velocities and mass-loss found in this stellar wind ($\sim$ 200 km s${}^{-1}$ and 10${}^{-6}$ M${}_{\odot}$ a${}^{-1}$, respectively222We follow the recommendations for units of the IAU Style Manual (Wilkins, 1990, 1995). Therefore, we use the term annus, abbreviated to “a”, for year., see Thomas et al., 2013; Bollen et al., 2019), we deduce that this jet is $\stackrel{{\scriptstyle\sf<}}{{\scriptstyle\sf\sim}}$ 1 000 years old and its mass is $\stackrel{{\scriptstyle\sf<}}{{\scriptstyle\sf\sim}}$ 10${}^{-3}$ M${}_{\odot}$.
Although this stellar wind is not part of our study, its mass is small compared with the mass derived in this work for the rotating disk, and is slightly lower than that of the molecular outflow.
In addition, we highlight that the “X”-shaped structure seen in Cohen et al. (2004) only shows the gas that is closest to the axis, while CO interferometric observations reveal that the optical X-shaped structure represents the inner layer of the outflowing biconical component (Bujarrabal et al., 2016).
Moreover, Men’shchikov et al. (2002), from a detailed model of this nebula in the polar direction, found that the gas and dust density values sharply decay from 2${}^{\prime\prime}$. This is roughly consistent with the strong decrease in brightness seen in Cohen et al. (2004) at $\pm$ 2${}^{\prime\prime}$, as the optical brightness of the outermost regions of the Red Rectangle ($\pm$ 10${}^{\prime\prime}$) is more than 50 times weaker than that at $\pm$ 2${}^{\prime\prime}$.
In view of what is stated in Sect. 4.1.2, the rotating disk dominates the Red Rectangle nebula. As for the expanding gas, we clearly differentiate between an outflow composed of low-velocity expanding gas escaping from the disk, and these collimated high-velocity stellar winds that do not contribute notably to the total nebular mass (and their study is not part of this work).
Therefore, we prefer not to take this contribution into account, as it does not significantly affect our results.
We expect a similar or even more favorable situation for all the sources in our study. For instance, the size of 89 Her via CO data seems to be compatible with that in the optical or IR image.
4.4 Possible presence of haloes
It is thought that $\stackrel{{\scriptstyle\sf<}}{{\scriptstyle\sf\sim}}$ 1 M${}_{\odot}$ of mass-loss from the AGB phase is necessary to reach the post-AGB stage. However, the masses detected in this study are small compared to what is predicted from theoretical works. One might wonder whether or not there is a large component surrounding the material that we detect.
The origin of these large quasi-spherical fossil shells requires successive ejecta of stellar material when the star reached the tip of the AGB phase (Guerrero & Manchado, 1999).
This structure can be difficult to detect and its mass can vary between 10${}^{-4}$ and $\sim$ 0.1 M${}_{\odot}$; and its contribution to the total mass may not be negligible
(Chu et al., 1991; Bujarrabal et al., 2001; Guerrero et al., 2003; Cox et al., 2012; Van de Steene et al., 2015).
Models of dust emission from our sources and standard pre-planetary nebulae show significant differences, with no hint in our disk-containing objects of extended shells, which are characteristic of standard pPNe; see e.g., Gezer et al. (2015); Hrivnak et al. (1989) (as in general detected in CO lines).
In particular, from far-infrared (FIR) data in Gezer et al. (2015), in particular the emission excess at 20 $-$ 100 $\mu$m in their Fig. 2, one finds that the mass of the undetected outer-shell in 89 Her must be more than 100 times lower than that of the well-known pPN HD 161796. We reiterate the fact that the Gaia distance of HD 161796 is $\sim$ 2 kpc and that its nebula is supposed to be placed at 1 $-$ 3 $\times$ 10${}^{17}$cm, just beyond the nebula we detect in 89 Her. As the mass of the HD 161796 nebula is around 0.1 M${}_{\odot}$ (from CO data in Bujarrabal et al., 2001, and again accounting for the new Gaia distance), we derive a lower limit for the mass for any very outer shell around 89 Her, $\stackrel{{\scriptstyle\sf<}}{{\scriptstyle\sf\sim}}$ 10${}^{-3}$ M${}_{\odot}$, which is a small value compared with the mass values derived in this paper for the rotating disk and disk wind.
The objects of our sample could present the same well-known missing mass problem as the other evolved sources (see e.g., Santander-García et al., 2021, and references therein). This missing mass could reside in a hard-to-detect halo.
Future ultrasensitive observations and theoretical works will be needed to answer this question.
In any case, these haloes, which by definition arise from mass loss during the AGB stage, are unrelated to the post-AGB structures studied in this work (rotating disks and outflows).
4.5 Contribution of outer layers of the disk wind
We remind the reader that, according to our previous discussion, the contribution of very wide components is not significant, in particular in the case of 89 Her (see also Hrivnak et al., 1989; Gezer et al., 2015).
In our analysis, the contribution of very outer layers, where CO can be photodissociated by interstellar UV, has not been included. This is a very general problem in the study of circumstellar envelopes, because these cool, dissociated layers are very difficult to detect and analyze. We think that the contribution of such outer haloes is likely negligible. General studies of FIR dust emission (see e.g., Gezer et al., 2015; Vickers et al., 2015), which is known to be useful for identifying very outer layers beyond the CO photodissociation radius, do not reveal the presence of sizable very extended components in binary post-AGB stars.
Additionally, Men’shchikov et al. (2002) modeled the dust emission from the Red Rectangle in regions far from the poles in great detail, including an extended component in expansion, and found that the density decreases very steeply from a radius of 1 000 to 2 000 AU, which is very similar to what was derived from CO maps (Bujarrabal et al., 2016).
Seemingly, the contribution of outer photodissociated shells to the total mass budget should be negligible in our sources.
Moreover, from the CO maps, we know that the kinetic time of the CO outflows in binary post-AGB stars is of a few thousand years. According to the evolutionary models, this timescale is comparable to the total expected time spent by a star in its transition from the AGB to the PN phase (see Bloecker & Schoenberner, 1990; Miller Bertolami, 2016). High-mass Keplerian disks are a typical post-AGB phenomenon, being very rare or absent in AGB stars, and the ejection of the observed outflow from the disk must also be a post-AGB phenomenon.
In conclusion, our results so far suggest that we cannot expect any greatly extended outflows in the objects we have studied. Nevertheless, we recognize that the analysis of these difficult-to-detect layers is uncertain.
Even in the hypothetical case where the mass of outer and undetected layers of the outflow is comparable to that of the detected ones (i.e. a factor of 2), the contribution of the disk wind to the total nebular mass would always be $<$ 25% in the case of the disk-dominated sources. On the contrary, this outflow mass ratio would be $\stackrel{{\scriptstyle\sf>}}{{\scriptstyle\sf\sim}}$ 80% in the case of the outflow-dominated ones.
Therefore, even in this hypothetical scenario, our main conclusions of the disk-to-mass ratio fraction would not change significantly.
4.6 Bimodal distribution of the wind contribution
Following this very detailed analysis of each object, we update the total mass and contribution of the outflow (i.e., the disk wind) for the sample. These updated values can be seen in Table 2 and Fig. 8.
In particular, the figure shows that there is a clear bimodal distribution of the contribution of the disk wind to the total nebular mass around binary post-AGB stars. There are two types of sources; disk-dominated and outflow-dominated sources.
We note that the segregation between the two subclasses is very clear. The disk-dominated sources always present disk-mass percentages above 15%, whereas in the outflow-dominated ones, this value is always below 40%.
More precisely, the disk-dominated sources present an outflow contribution of 10 $\pm$ 5%, while in the outflow-dominated ones this figure is 70 $\pm$ 10%.
We note that there are no intermediate sources in between the two subclasses; see Table 2 and Fig. 8.
The absence of intermediate sources indicates that the two subclasses of sources are probably not related via an evolutionary link; otherwise one would expect to find intermediate sources because of the similar outflow and post-AGB lifetimes (Sect. 4.5). Therefore, we suggest that the existence of these two subclasses is related to the initial stellar properties and configuration of the binary system, resulting in the ability to launch a more or a less massive wind from the rotating disk. Of course, such reasoning would have to be revised if the post-AGB life scales of our sources are demonstrated to be much longer than for standard pPNe, at least by an order of magnitude, which cannot be excluded; to explain the bimodal distribution, we would then have to assume that the mass loss takes place in a critically small fraction of the post-AGB evolution.
5 Conclusions
The new combined maps of 89 Her with no flux loss lead us to study the very large hourglass-like extended component that surrounds the Keplerian disk. We find a total nebular mass of 1.8 $\times$ 10${}^{-2}$ M${}_{\odot}$, of which 65% is located in the outflow. According to these new mass estimations, 89 Her has been re-classified as an outflow-dominated source.
After a very detailed analysis of each object of our sample of nebulae around binary post-AGB stars, we clearly find two subclasses: the disk- and outflow-dominated sources.
Fig. 8 shows a histogram that indicates the mass percentage of the rotating disk and outflow components for our sample, showing that 85% $-$ 95%
of the mass in disk-dominated sources is located in the rotating component, while the rotating component of the outflow-dominated sources only contains 25% $-$ 35% of the total mass (HD 52961 and IRAS 19157$-$0247 have not been considered in this work because they were classified as intermediate sources with a high level of uncertainty, and they could belong to either subclass).
The nebulae around binary post-AGB stars present a bimodal distribution (see Fig. 8). The existence of these two very different subclasses does not support an evolutionary relationship between them, as the timescale of the post-AGB evolution is believed to be very short, and comparable to the time required to form these disk winds. On the contrary, the existence of these two subclasses could be due to a different initial configuration of the binary system or different initial masses. We propose that the outflow-dominated sources could result from the presence of a very efficient mass loss from the disk in certain objects, the cause of which remains unknown. It is possible that the outflow would be particularly conspicuous in objects that have spent a relatively long time in the post-AGB phase and have then had time to form very extended,
high-mass outflows.
Acknowledgements.
We are grateful to the anonymous referee for the relevant recommendations and comments on the paper.
This work is based on observations of IRAM telescopes. IRAM is supported by INSU/CNRS (France), MPG (Germany), and IGN (Spain).
This work is part of the AxiN and EVENTs / NEBULAE WEB research programs supported by Spanish AEI grants AYA 2016-78994-P and PID2019-105203GB-C21, respectively.
IGC acknowledges Spanish MICIN the funding support of BES2017-080616.
References
Alcolea & Bujarrabal (1995)
Alcolea, J. & Bujarrabal, V. 1995, A&A, 303, L21
Balick & Frank (2002)
Balick, B. & Frank, A. 2002, ARA&A, 40, 439
Bloecker & Schoenberner (1990)
Bloecker, T. & Schoenberner, D. 1990, A&A, 240, L11
Bollen et al. (2019)
Bollen, D., Kamath, D., Van Winckel, H., & De Marco, O. 2019, A&A,
631, A53
Bollen et al. (2022)
Bollen, D., Kamath, D., Van Winckel, H., et al. 2022, A&A, 666, A40
Bollen et al. (2021)
Bollen, D., Kamath, D., Van Winckel, H., De Marco, O., & Wardle, M.
2021, MNRAS, 502, 445
Bujarrabal et al. (2013)
Bujarrabal, V., Alcolea, J., Van Winckel, H., Santander-García,
M., & Castro-Carrizo, A. 2013, A&A, 557, A104
Bujarrabal et al. (2001)
Bujarrabal, V., Castro-Carrizo, A., Alcolea, J., & Sánchez
Contreras, C. 2001, A&A, 377, 868
Bujarrabal et al. (2016)
Bujarrabal, V., Castro-Carrizo, A., Alcolea, J., et al. 2016, A&A,
593, A92
Bujarrabal et al. (2017)
Bujarrabal, V., Castro-Carrizo, A., Alcolea, J., et al. 2017, A&A,
597, L5
Bujarrabal et al. (2018)
Bujarrabal, V., Castro-Carrizo, A., Van Winckel, H., et al. 2018, A&A,
614, A58
Bujarrabal et al. (2007)
Bujarrabal, V., van Winckel, H., Neri, R., et al. 2007, A&A, 468, L45
Castro-Carrizo et al. (2001)
Castro-Carrizo, A., Bujarrabal, V., Fong, D., et al. 2001, Astronomy &
Astrophysics, 367, 674
Castro-Carrizo et al. (2010)
Castro-Carrizo, A., Quintana-Lacaci, G., Neri, R., et al. 2010, A&A,
523, A59
Chu et al. (1991)
Chu, Y.-H., Manchado, A., Jacoby, G. H., & Kwitter, K. B. 1991, ApJ,
376, 150
Cohen et al. (2004)
Cohen, M., Van Winckel, H., Bond, H. E., & Gull, T. R. 2004, AJ, 127,
2362
Cox et al. (2012)
Cox, N. L. J., Kerschbaum, F., van Marle, A. J., et al. 2012, A&A,
537, A35
de Ruyter et al. (2006)
de Ruyter, S., van Winckel, H., Maas, T., et al. 2006, A&A, 448, 641
Fong et al. (2001)
Fong, D., Meixner, M., Castro-Carrizo, A., et al. 2001, A&A, 367, 652
Fong et al. (2006)
Fong, D., Meixner, M., Sutton, E. C., Zalucha, A., & Welch, W. J.
2006, ApJ, 652, 1626
Frank & Blackman (2004)
Frank, A. & Blackman, E. G. 2004, ApJ, 614, 737
Gallardo Cava et al. (2022)
Gallardo Cava, I., Bujarrabal, V., Alcolea, J., Gómez-Garrido, M.,
& Santander-García, M. 2022, A&A, 659, A134
Gallardo Cava et al. (2021)
Gallardo Cava, I., Gómez-Garrido, M., Bujarrabal, V., et al. 2021,
A&A, 648, A93
Gezer et al. (2015)
Gezer, I., Van Winckel, H., Bozkurt, Z., et al. 2015, MNRAS, 453, 133
Gielen et al. (2011)
Gielen, C., Bouwman, J., van Winckel, H., et al. 2011, A&A, 533, A99
Guerrero et al. (2003)
Guerrero, M. A., Chu, Y.-H., Manchado, A., & Kwitter, K. B. 2003, AJ,
125, 3213
Guerrero & Manchado (1999)
Guerrero, M. A. & Manchado, A. 1999, ApJ, 522, 378
Hillen et al. (2014)
Hillen, M., Menu, J., Van Winckel, H., et al. 2014, A&A, 568, A12
Hrivnak et al. (1989)
Hrivnak, B. J., Kwok, S., & Volk, K. M. 1989, ApJ, 346, 265
Jura (2003)
Jura, M. 2003, ApJ, 582, 1032
Men’shchikov et al. (2002)
Men’shchikov, A. B., Schertl, D., Tuthill, P. G., Weigelt, G., &
Yungelson, L. R. 2002, A&A, 393, 867
Miller Bertolami (2016)
Miller Bertolami, M. M. 2016, A&A, 588, A25
Oomen et al. (2018)
Oomen, G.-M., Van Winckel, H., Pols, O., et al. 2018, A&A, 620, A85
Raga et al. (2009)
Raga, A. C., Esquivel, A., Velázquez, P. F., et al. 2009, ApJ,
707, L6
Sahai et al. (2011)
Sahai, R., Claussen, M. J., Schnee, S., Morris, M. R., & Sánchez
Contreras, C. 2011, ApJ, 739, L3
Sahai et al. (2007)
Sahai, R., Morris, M., Sánchez Contreras, C., & Claussen, M. 2007,
AJ, 134, 2200
Santander-García et al. (2021)
Santander-García, M., Jones, D., Alcolea, J., Bujarrabal, V., &
Wesson, R. 2021, arXiv e-prints, arXiv:2110.15261
Shenton et al. (1995)
Shenton, M., Evans, A., & Williams, P. M. 1995, MNRAS, 273, 906
Stanghellini et al. (2016)
Stanghellini, L., Shaw, R. A., & Villaver, E. 2016, ApJ, 830, 33
Thomas et al. (2013)
Thomas, J. D., Witt, A. N., Aufdenberg, J. P., et al. 2013, MNRAS,
430, 1230
Ueta et al. (2000)
Ueta, T., Meixner, M., & Bobrowsky, M. 2000, ApJ, 528, 861
Van de Steene et al. (2015)
Van de Steene, G. C., van Hoof, P. A. M., Exter, K. M., et al. 2015,
A&A, 574, A134
Van Winckel (2003)
Van Winckel, H. 2003, ARA&A, 41, 391
Velázquez et al. (2011)
Velázquez, P. F., Steffen, W., Raga, A. C., et al. 2011, ApJ, 734,
57
Vickers et al. (2015)
Vickers, S. B., Frew, D. J., Parker, Q. A., & Bojičić, I. S.
2015, MNRAS, 447, 1673
Waelkens et al. (1996)
Waelkens, C., Van Winckel, H., Waters, L. B. F. M., & Bakker, E. J.
1996, A&A, 314, L17
Wilkins (1990)
Wilkins, G. A. 1990, Transactions of the International Astronomical Union,
Series B, 20, Siii
Wilkins (1995)
Wilkins, G. A. 1995, Vistas in Astronomy, 39, 277
Appendix A ${}^{12}$CO and ${}^{13}$CO $J=$ 2 – 1 total-power maps
Here we present total-power maps for 89 Her of the ${}^{12}$CO and ${}^{13}$CO $J=$ 2 – 1 emission lines; see Figs. 9 and 10. These observations were performed in OTF mode using the 30 m IRAM telescope.
We detect emission at angular distances of around 20${}^{\prime\prime}$ in diameter. Our new total-power maps present expansion velocities, which reveal a larger expanding component. See Sect. 2.1 for a complete description.
Appendix B Model results
In this Appendix, we show the synthetic velocity maps of 89 Her for ${}^{12}$CO and ${}^{13}$CO $J=$ 2 – 1 emission for our best-fit model (see Figs. 11 and 12).
The large hourglass-like structure is present in our synthetic velocity maps.
We also show the synthetic PV diagrams along the equatorial direction for ${}^{12}$CO and ${}^{13}$CO $J=$ 2 – 1 line emission (see Figs. 13 and 14).
See Sect. 3 for further details. |
DoS & DDoS in Named-Data Networking
Paolo Gasti
Gene Tsudik
Ersin Uzun
Lixia Zhang
Abstract
With the growing realization that current Internet protocols are
reaching the limits of their senescence, a number of on-going
research efforts aim to design potential next-generation Internet
architectures. Although they vary in maturity and scope, in order to
avoid past pitfalls, these efforts seek to treat security and privacy
as fundamental requirements. Resilience to Denial-of-Service (DoS)
attacks that plague today’s Internet is a major issue for any new
architecture and deserves full attention.
In this paper, we focus on DoS in a specific candidate next-generation
Internet architecture called Named-Data
Networking (NDN) – an instantiation of Information-Centric Networking approach. By stressing content
dissemination, NDN appears to be attractive and viable approach to
many types of current and emerging communication models. It also
incorporates some basic security features that mitigate certain
attacks. However, NDN’s resilience to DoS attacks has not been
analyzed to-date. This paper represents the first step towards
assessment and possible mitigation of DoS in NDN. After identifying
and analyzing several new types of attacks, it
investigates their variations, effects and counter-measures.
This paper also sheds some light on the long-standing debate
about relative virtues of self-certifying, as opposed to human-readable,
names.
Keywords: Future Internet Architectures;
Content-Centric Networks; Information-Centric Networks, Named-data
Networking; Security; Denial-of-Service; Distributed
Denial-of-Service.
I Introduction
The Internet clearly represents an overwhelming and unique global success story.
Billions of people worldwide use it to perform a wide range of
everyday tasks. It hosts a large number of information-intensive
services and interconnects many millions of wired, wireless,
fixed and mobile computing devices.
The Internet also serves as a means of disseminating
enormous (and ever-increasing) amounts of digital content. Since
its inception, the amount of data exchanged over the Internet has
witnessed exponential growth. Recently, this growth intensified due
to increases in: (1) distribution of multimedia content, (2)
popularity of social networks and (3) amount of user-generated
content. Unfortunately, the same usage model that fostered Internet’s
success is also exposing its limitations. Core ideas of today’s
Internet were developed in the 1970-s, when telephony – exemplified by a
point-to-point conversation between two entities – was the only
successful example of effective global communication
technology. Moreover, original Internet applications were few and
modest in terms of bandwidth and throughput requirements, e.g.,
store-and-forward email and remote computer access.
The way people access and utilize the Internet has changed
dramatically since the 1970-s and today the Internet has to
continuously accommodate new services and applications as well as
different usage models. To keep pace with changes and move the
Internet into the future, a number of research efforts to design new
Internet architectures have been initiated in recent years.
Named-Data Networking (NDN) [29] is one such effort. NDN is an
on-going research project that aims to develop a candidate
next-generation Internet architecture. It instantiates the so-called
Content-Centric (CCN) or Information-Centric (ICN) approach
[19, 22, 24] to
networking. NDN explicitly names content instead of physical locations
(i.e., hosts or network interfaces) and thus transforms content into
a first-class entity. NDN also stipulates that each piece of named
content must be digitally signed by its producer. This allows
decoupling of trust in content from trust in the entity that might
store and/or disseminate that content. These NDN features facilitate
automatic caching of content to optimize bandwidth use and enable
effective simultaneous utilization of multiple network interfaces.
NDN has been demonstrated as a viable and attractive architecture for content
distribution [22] as well as real-time
[21] and anonymous communication
[12].
A number of other NDN-related research efforts are also underway.
I-A DoS and DDoS
In recent years, denial of service (DoS) and distributed denial of
service (DDoS) attacks have become more and more common and notorious. In the
latter, the adversary exploits a large number of compromised hosts
(zombies), that surgically aim their attacks at specific target(s).
Although DDoS attacks are generally easy to instantiate and require
little technical sophistication on the part of the adversary, they
are often very effective and difficult to mitigate. (NOTE: hereafter, we
use the term DoS to include both single-source and multi-source
– i.e., DDoS – denial-of-service attacks.)
We believe that any new Internet architecture should: (1) be
resilient to existing DoS attacks, or at least limit their
effectiveness, (2) anticipate new attacks that take advantage of its
idiosyncrasies, and (3) incorporate basic defenses in its design.
To the best of our knowledge, there has been no scientific and
systematic assessment of how NDN fares with respect to DoS attacks.
We believe that such assessment is both timely and very important. While NDN
appears to be quite efficient in terms of content distribution between
well-behaved (honest) entities, it is unclear how it would cope with
malicious parties. This paper tries to address these issues by
analyzing the impact of current DoS attacks on NDN, identifying new
attacks that rely on NDN features, and proposing some
countermeasures. We emphasize that this paper should be seen not as a
comprehensive treatment of security (or even DoS) in NDN. Instead, it
represents a first step towards identifying DoS attacks as well as their impact
and securing NDN against them.
I-B Anticipated Contributions
The main goal of this paper is to explore, and evaluate
effects of DoS attacks against NDN. We believe that only by better understanding the
effects and repercussions of these attacks we can begin to develop
sensible and effective countermeasures. Anticipated contributions are as follows:
•
We show that DoS attacks effective in the current IP-based Internet
are largely ineffective against NDN. This is not really
surprising since current attacks are very much tailored to
TCP/IP, current routing protocols, HTTP, DNS, etc.
•
We identify and describe two new and major types of NDN-specific
DoS attacks, based on: (1) interest flooding, and (2)
content/cache poisoning. We argue that both types of attack require
long-term exploration.
•
We then analyze the impact of several flavors of both attack types and
propose a set of potential counter-measures.
Organization: The rest of the paper is structured as follows: Section II provides an
overview of NDN. We discuss how NDN copes with existing DoS/DDoS attacks in
Section III, and examine new – NDN-specific – ones in Sections IV and
V. Section VI overviews related work. Finally, we conclude in Section
VII
II NDN Overview
As mentioned in Section I,
NDN [29, 22] is a network architecture based on named content.
Rather than addressing content by its location, NDN refers to it by
name. A content name is composed of one or more variable-length
components that are opaque to the network. Component boundaries are
explicitly delimited by “/” in the usual representation.
For example, the name of a CNN news content for May 20, 2012 might
look like: /ndn/cnn/news/2012may20. Large pieces of content
can be split into fragments with predictable names: fragment $137$ of
Alice’s YouTube video could be named:
/ndn/youtube/alice/video-749.avi/137. Since NDN’s main
abstraction is content, there is no explicit notion of “hosts”,
albeit, their existence is assumed.
Communication adheres to the pull model: content is delivered
to consumers only upon explicit request. A consumer requests content
by sending an interest packet. If an entity (a router or a host)
can “satisfy” a given interest, it returns the corresponding data packet.
Interest and content are the only types of packets in NDN. A
content packet with name X is never forwarded or routed
unless it is preceded by an interest for name X. (Strictly speaking,
content named X${}^{\prime}\neq$ X can be delivered in response to an interest
for X, but only if X is a prefix of X${}^{\prime}$.)
NDN routers must include of the following components:
•
Content Store (CS),
used for content caching and retrieval;
•
Forwarding Interest Base (FIB), that
contains a table of name prefixes and corresponding outgoing interfaces
(to route interests);
•
Pending Interest Table (PIT) – a table containing a set of currently
unsatisfied interests and their corresponding incoming interfaces;
When a router receives an interest for name X and there are no
pending interests for the same name in its PIT, it forwards the
interest to the next hop, according to its FIB. For each forwarded
interest, a router stores some amount of state information, including
the name in the interest and the interface on which it arrived.
However, if an interest for X arrives while there is already an entry
for the same name in the PIT, the router collapses the present
interest (and any subsequent ones for X) storing only the interface
on which it was received. When content is returned, the router
forwards it out on all interfaces from which an interest for X has
arrived and flushes the corresponding PIT entry. Since no additional
information is needed to deliver content, an interest does not carry
a “source address”.
Any NDN router can provide content caching through its CS. The
size of CS is limited only by resource availability.
Consequently, content might be fetched from any number of network
caches, rather than from its original producer. Hence, NDN has no
notion of “destination addresses”.
NDN deals with content authenticity and integrity by making digital
signatures mandatory for all content. A signature binds content with
its name, and provides origin authentication no matter how, when or
from where it is retrieved. Public keys are treated as regular
content: since all content is signed, each public key content is
effectively a simple “certificate”. NDN does not mandate any
particular certification infrastructure, relegating trust management
to individual applications.
Content objects are named data packets.111In the rest of the paper, we use the terms content object and data packet interchangeably.
Fields of a data packet include [7]:
•
Signature: public key signature (e.g., RSA or DSA)
computed over the entire data packet, including its name.
•
Keylocator: references the key
needed to verify the content signature.
This field can contain one of the following: (1) verification (public) key;
(2) certificate containing verification key; or (3) NDN name referencing verification key.
•
PublisherPublicKeyDigest: hash of the data packet producer’s public key.
In addition to the name of requested content, an interest packet carries
several fields [8]. In this paper, we are interested in the following
(others are omitted for clarity):
•
PublisherPublicKeyDigest: this optional field contains the
hash of the producer’s public key for the requested piece of data.
•
Exclude: an optional field that embodies a description of
name components that should not appear in the data packet in response to the interest.
•
AnswerOriginKind: encodes determines whether the answer
to an interest can retrieved from a CS or must be generated by the producer.
•
Scope: limits where the Interest may propagate; Scope 0 and 1 limit
propagation to the originating host; Scope 2 limits propagation to no further than the next host.
III NDN and Current DoS Attacks
We now consider how NDN fares against DoS attacks. We initially
focus on current DoS types often encountered on the Internet.
Next, we analyze the impact of these attacks on NDN and discuss our
findings. Then, in the next two sections, we identify some new DoS attacks that take advantage
of NDN features, and discuss tentative countermeasures.
There is a wide variety of DoS attacks targeting different network and host
resources, protocol layers as well as specific software. A typical attack is
composed of three elements:
(1) a set of zombies under control of a master node or nodes,
(2) master node(s) controlling the zombies, and (3) a set of victim hosts and/or routers.
A common way to obtain zombies to remotely exploit software vulnerabilities
and inject malware into a set of unpatched hosts. NDN makes this somewhat harder
– yet far from impossible – because hosts are not directly addressable.
However, there is no indication that software vulnerabilities will
disappear or even decrease significantly over time. Thus,
we assume that adversaries will continue to have access to large
numbers of zombies by means such as exploitation of un-patched vulnerabilities
or phishing.
III-A Impact of Current Attacks on NDN
We now examine some popular types of DoS attacks that work
against current TCP/IP-based Internet and assess their putative effects on NDN.
Reflection Attacks.
A reflection attack involves three parties: the adversary, a victim host, and a
set of secondary victims (reflectors). The goal of the adversary is to use the reflectors to overwhelm the
victim host with traffic.
To do so, a reflection attack uses IP packets with forged addresses: the adversary replaces its own source
address with the address of its
intended victim, and sends these packets to the secondary victims.
Responses to such packets are not routed back to the adversary, and overwhelm the victim instead.
To be effective, such attacks require some form of
amplification, i.e., the amount of data used by the adversary to
perform the attack must be significantly smaller than the amount of
data received by the victim.
NDN is generally resilient to this type of attack due to the
symmetric nature of the path taken by each interest and the
corresponding content. A content packet must follow, in reverse, the
path established by the preceding interest. However, note that an NDN
router is allowed to broadcast an incoming interest on some or
all of its interfaces. (In other words, an interest broadcast can
occur at any hop). Nonetheless, even if each hop (including the
consumer) broadcasts the interest, the maximum number of content
copies a consumer can receive is bounded by the number of its interfaces,
and not by the number of entities that receive an interest.
Consequently, the only effective reflection-style attack requires the
adversary to be on the same physical network (e.g., Ethernet or WLAN)
as the intended victim. Then, the adversary broadcasts one or more
interests on all available interfaces (with the victim’s
layer-2 address as the source) and the victim subsequently receives multiple
copies of requested content. This attack seems somewhat effective
since the adversary’s amplification factor is based on the
comparatively small size of interests with respect to that of
content. However, NDN routers incorporate a useful suppression feature:
Whenever an NDN router “overhears” a content packet on a broadcast
interface $IF_{x}$ for which it has a current PIT entry with the
incoming interface $IF_{x}$, it caches the said content and flushes the
PIT entry. Note that another copy of the same content might be later
delivered via the outgoing interface ($IF_{y}$) for that PIT entry; it
will be discarded since, by that time, the corresponding PIT entry
will have been flushed.
This feature ensures that multiple NDN routers on the same broadcast
domain do not serve the same content more than once, even if the
original interest was broadcast.
Bandwidth Depletion.
In a typical coordinated distributed attack, adversary-controlled
zombies flood their victims with IP traffic in order to saturate their network resources. The usual goal is to make the victims
unreachable by others and/or, more generally, to inhibit victims’
ability to communicate. Normally, such attacks are carried out via
TCP, UDP or ICMP and rely on sending a stream of packets to the
victim at the maximum data rate.222TCP-based attacks also exploit
connection-based nature of the protocol: each packet sent by zombies
tries to open a new connection, which, in turn, requires the victim
to create and store corresponding state, thus saturating its
resources.
A similar kind of attack can be mounted against NDN by directing a
large number of zombies to request existing content from a certain victim.
However, it is easy to see that the effectiveness of this attack
would be very limited. Once the content is initially pulled from its
producer, it is cached at intervening routers and subsequent
interests retrieve it from these routers’ caches. Therefore, the network itself
would limit the number of interests that reach the victim.
Black-Holing and Prefix Hijacking.
In a prefix hijacking [3] attack, a misconfigured,
compromised or malicious autonomous system (AS) advertises invalid
routers so as to motivate other AS-s to forward their traffic to it.
This can result in so-called “black-holing” whereby all traffic sent
to the malicious AS is simply discarded. This attack is effective in
IP networks, since, once routing information is polluted, it is difficult for
routers to detect, and recover from, the problem.
While countermeasures have been proposed (e.g.,
[26]) this remains a serious threat to the current Internet.
NDN is resilient to black-holing implemented via prefix hijacking.
NDN routers have access to strictly more information than their IP
counterparts and can use such information to detect anomalies in the
content distribution process. Since each content follows the
same path as the interest that requested it, the number
of unsatisfied (expired) interests can be used to determine whether a
particular prefix has been hijacked. Also, NDN routers maintain
statistics about performance of each link and interface with respect
to a particular prefix and change their forwarding strategy according
to such statistics.
Loop detection and elimination allows routers to explore topological
redundancy through multipath forwarding. Multipath routing further
reduces impact of prefix hijacking, since it allows routers to try
alternative paths as a reaction to attacks. This increases the
probability of forwarding interests through a path that has not been
affected by the attack. In contrast with IP, advertising fake routes in NDN does
not allow the adversary to mount a loop-holing attack
To sum up, the NDN forwarding plane allows routers to: (1) quickly
(i.e., at RTT scale) detect and react to network failures; (2)
detect and circumvent hijacking attacks and (3) incorporate
congestion into forwarding decisions.
DNS Cache Poisoning.
In the current Internet, DNS servers translates human-readable names
to the corresponding IP address and vice-versa. For performance
reasons, DNS servers usually store the output of previous requests in
their cache. There is a well know attack, called DNS cache
poisoning [11], which allows the adversary to insert
corrupted entries in a DNS server’s cache in order to control the
server responses for a set of DNS names. The best countermeasure
against this attack is the use of the DNS Security Extensions
protocol, i.e., DNSSEC [13]; however, as of today DNSSEC has
not been widely deployed on the Internet.
Packet names in NDN are routed directly, rather than being converted
to addresses. While this implies that there is no need for services
that perform name resolution (and therefore such service can not be
corrupted), it still is possible to conceive an attack analogous to
DNS poisoning on NDN. We believe that the closest counterpart of DNS
cache poisoning in NDN is a combination of route hijacking and
content poisoning: the adversary would force a routing change (if
necessary) that allows it to be on the path for a set of namespaces
that are going to be affected by the attack. Then, it answers
interests with data packets carrying an arbitrary payload.
Intuitively, signatures on data packets allow consumers to
determine whether the content has been poisoned. Sections
V and following are devoted to this kind of attack
and its countermeasures.
III-B Towards New DoS Attacks
Despite their seeming lack of effectiveness or greatly reduced
impact, variations of aforementioned current DoS attacks might be quite
effective against NDN.
Recall that two key features distinguish current Internet
routers from their NDN counterparts: (1) pending interest state (PIT entries) needed to
perform content routing and (2) the use of content caches.
In subsequent sections, we describe two classes of attacks – Interest Flooding
and Content/Cache Poisoning – that capitalize on these NDN features.
In addition, we anticipate other types of DoS-related NDN attacks,
such as those that focus on routing. However, we believe that attacks
targeting these two specific key features of NDN represent most serious
and immediate threats, thus warranting an in-depth investigation.
IV Interest Flooding
Routing of content is performed using state in PIT-s established by
interests, i.e., the name of each incoming content packet is used
to look up the PIT and identify a corresponding entry.
The adversary can take advantage of this state to mount an
effective DoS attack, which we term “interest flooding”. In such an attack,
the adversary (controlling a set of possibly geographically distributed zombies)
generates a large number of closely-spaced interests, aiming to (a)
overwhelm (PIT-s) in routers, in order to
prevent them from handling legitimate interests, and/or (b)
swamp the targeted content producer(s).
Since NDN interests lack source address and are not secured (e.g., not signed)
by design, it is difficult to determine the attack originator(s)
and take targeted countermeasures.
We identify three types of interest flooding attacks, based on the
type of content requested: (1) existing or static, (2)
dynamically-generated, and (3) non-existent (i.e., unsatisfiable interests).
In all cases, the adversary uses zombies to generate a large number
of interests requesting content from targeted producers.
While attacks (1) and (3) are mostly aimed at the network infrastructure,
(2) affects both network and application-layer functionalities.
Similarly to bandwidth depletion attacks discussed in the previous section,
the impact of type (1) attacks is quite limited since
in-network content caching provides a built-in countermeasure.
Suppose that there are several zombies, each with independent path to
the targeted producer. After the initial “wave” of interests from
these zombies, content settles in all intervening routers’ caches.
Subsequent interests for the same content do not propagate to the
producer(s) since they are satisfied via cached copies.
In case of type (2) attacks, benefits of in-network content caching are
lost. Since requested content is dynamic, all interests are routed to
content producer(s), thus consuming bandwidth and router PIT state. Also,
if generating dynamic content is expensive – signing content is
a good example relatively expensive per-packet operation – content producers might waste
significant computational resources. (One concrete example of this
attack class could target a web server that allows site-wide
searches: each zombie issues an interest that requests the victim
server to search its entire site for a random string.)
The direct outcome of this attack type is that the producer wastes
resources to satisfy malicious, rather than legitimate, interests.
The impact on routers varies with their distance from the
targeted content producer: the closer a router is to the producer,
the greater the effect on its PIT due to more concentrated attack traffic.
Type (3) attacks involve zombies issuing unsatisfiable interests for
non-existent content. Such interests can not be collapsed by routers,
and are routed to the targeted content producers.
The latter can quickly ignore such interests
without incurring significant overhead. However, such
interests will linger and take up space in router
PIT-s until they eventually expire. We consider routers to be primary
intended victims of this attack type. Given an existing prefix /ndn/prefix,
there are several easy ways to construct unsatisfiable interests:
1.
Set the name in the interest to: /ndn/prefix/nonce,
where the suffix nonce is a random value. Since NDN performs
longest-prefix matching, interests will be forwarded all the way to the producer
and never satisfied.
2.
Set the PublisherPublicKeyDigest field to a random value.
Since no public key would match this value, the interest will remain unsatisfied.
3.
Set the interest Exclude filter to exclude all existing content starting with
/ndn/prefix, e.g., by using a Bloom Filter with almost all bits set to 1.
Such an interest can not be satisfied since it simultaneously requests and
excludes the same content.
In fact, there is another way of mounting type (3) attacks that does not even
require the adversary to recruit any zombies.
Recall that each data packet has a KeyLocator field (see Section II).
This field can contain the NDN name of the data packet’s verification key. This feature
can be abused to mount a type (3) attack as follows:
The adversary publishes a large amount of content (e.g., a video), that is split into
numerous content packets. Each packet references a distinct non-existent verification key,
such that an interest for this key is not satisfiable.
To increase the effectiveness of this attack, all spurious key names must be routed towards the same
producer. This way, a large number of unsuspecting consumers are “forced” to issue interests
that overwhelm one or more victim’s PIT-s. We point out that NDN automatically
mitigates such attacks through interest collapsing. Since the pool of fake key-names is limited by the
number of packets comprising malicious content, rather than by the combined bandwidth of the
zombies, interests with the same name will be collapsed by routers. Therefore, the total number of
fake key-names represents an upper bound on the amount of space that can be consumed in
the victims’ PIT-s.
IV-A Tentative Countermeasures
We consider several potential countermeasures. The ease of
interest flooding attacks is partly due to lack of
authentication of interests. Anyone can generate a stream of
interests and a given NDN router only knows that a particular
interest entered on a specific interface. There is no other
information about its source. One trivial solution might be to require
signatures on interests. However, this would immediately
raise serious privacy concerns, as discussed in
[12] and would also introduce new DoS vulnerabilities due to
the computational overhead of signature verification.
On the other hand, we believe that potential problems and DoS attacks
due to interest flooding can be addressed without requiring source
authentication. This is because NDN routers are stateful
and can learn much more information about carried traffic than
their current IP counterparts.
Router Statistics.
NDN routers
can easily keep track of unsatisfied (expired) interests and use this
information to limit the following:
•
# of pending interests per outgoing interface: NDN creates flow
balance between interests and content. For each interest sent
upstream, at most one data packet satisfying that interest
can flow downstream. Based on that property, it is trivial
for each router to calculate the maximum number of pending
interest per outgoing interface that the downstream
connection can satisfy before they time out. Thus, a router
should calculate and never send more interests than an
interface can satisfy based on average content package size,
timeout for interests and bandwidth-delay product for the
corresponding link.
•
# of interests per incoming interface: From the same flow
balance principle, a router can easily detect when a
downstream router is sending too many interests that can not
be all satisfied due to the physical limitations of the
downstream link.
•
# of pending interests per namespace: When a certain prefix is
under a DoS attack, routers on the way (especially those
closer to the data producer) can easily detect unusual number
of unsatisfied interests in their PIT-s for that prefix.
In that case, routers can limit the total number of pending
interests for that prefix and throttle down the number of
pending interests for incoming interfaces that have sent too
many unsatisfied interests for that prefix.
Although these countermeasures seem intuitive and possibly effective, we believe
that implementing and testing them will be quite difficult. Most of all,
combining the above three limiting strategies into one algorithm and choosing
appropriate parameters for maximum effectiveness against attacks and
minimum impairment of legitimate traffic is a challenge.
We leave the design and testing of the actual algorithms to future work.
Push-back Mechanisms.
We also consider one router-based countermeasure to interest flooding –
a push-back mechanism that allows routers to isolate attack source(s).
When a router suspects an on-going attack for a particular namespace
(e.g., when it reaches its PIT quota for that namespace on a given
interface), it throttles any new interests for that namespace and reports
this to routers connected on that interface. These routers, in turn, can
propagate such information upstream towards offending interfaces,
while also limiting the rate of forwarded interests for the namespace
under attack. The goal is to push an attack back all the way to
its source(s), or at least to the location where it is detectable.
This countermeasure can be implemented without any modifications to the current
NDN infrastructure.
V Content/Cache Poisoning
We now shift focus to DoS attacks that target content.
In this context, the adversary’s goal is to cause routers to forward and cache
corrupted or fake data packets, consequently preventing consumers from
retrieving legitimate content. We say that a data packet is corrupted if
its signature is invalid. Whereas, a data packet is fake if it has a valid
signature, however, generated with a wrong (private) key.
As mentioned in Section II, all data packets in NDN are signed. This
provides the following security guarantees:
•
Integrity – a valid signature guarantees that the signed data packet is intact;
•
Origin Authentication – since a signature is uniquely bound to
the public key of the signer, anyone can verify whether content originates with the claimed producer;
•
Correctness – a signature binds data packet name with its
payload, thus allowing a consumer to securely determine whether a
data packet is a “correct answer” for the interest that requested it.
Consumers are expected to perform signature verification on every data packet
before accepting it. Also, any NDN router can elect to perform signature verification
for any content it forwards and caches.
Upon receiving and identifying a corrupted or fake data packet, a consumer can
re-request a different (possibly valid) copy of the same data packet using the Exclude
field in NDN interest packet.
In theory, content signatures provide an effective and simple means for detecting
content poisoning attacks, since “bad” content can be easily identified via signature
verification. In other words, NDN should be immune to content poisoning attacks.
However, in practice, this assertion might not hold. While a consumer can afford
to verify all content signatures, NDN routers face two challenges: (1) signature
verification overhead; and (2) trust management, i.e., what key should be
used to verify a given data packet?
While routers can choose to verify signatures on each data packet they
forward and/or store, for performance reasons, they are not required
to do so. Our tests show
that an optimized software implementation of RSA-1024 signature verification running
on Intel Core 2 Duo 2.53 GHz CPU allows us to verify about 150 Mbps of
traffic, assuming $1,500$-byte content packets. (Smaller packets would impose
even higher verification overhead).
Note that we use the smallest possible RSA public exponent – $3$ – thereby incurring
only two modular multiplications per signature verification. Routers with multiple
Gigabit-speed (or faster) interfaces would need an unrealistic amount of
computing power to verify packets at wire rate.
Content signatures also trigger the issue of global trust management architecture.
Without it, routers can not determine the public key needed to verify the data packet
signature. This creates a tension between flexibility (since an application can adopt
an arbitrary trust model for its content) and security
(any NDN router must be able to, if its chooses, verify any data packet’s signature).
Even though each NDN data packet contains a reference to its signature
verification (public) key, such references can not be trusted as they can be easily
abused by the adversary.
V-A Attack Variants
The impracticality of NDN routers verifying all signatures on forwarded or cached
data packets opens the door for content poisoning
attacks. As mentioned before, one can not push poisoned content
unilaterally, i.e., without any prior interest requesting that
content. Consequently, we identify two attacks variants:
1.
Suppose that the adversary is aware of current
(pending) interests for particular content, e.g., because it controls
some NDN routers. Compromised routers that
receive interests for that content simply inject (satisfy
interests with) poisoned content, which may then be cached by
other intervening routers.
2.
Suppose that the adversary anticipates interests in
particular content, e.g., a major news-story is about to break on
CNN or a patch for a popular operating system is about
to being released. We also assume that the name of the corresponding content is
predictable. The adversary, via numerous distributed
zombies, issues many near-simultaneous “legitimate”
interests for that content. Next, a compromised host or router (that receives
one or more such interests) replies with
poisoned content. Then, caches of routers (that processed preceding interests)
become populated with copies of poisoned content.
Subsequent interests for the same content will return a cached
version of the same poisoned content.
In [38] Xie et al. consider
a different technique for introducing poisoned content in caches. An
adversary, who controls a set of zombies, forces them to request
content produced by the adversary. Such content will take space in
caches that could otherwise be used more effectively to store “real”
popular content, i.e., this is a locality-disruption attack.
In this work, we do not consider such attacks, for two reasons: (1) content
injected by the adversary is never delivered to consumers who do not
explicitly request it; and (2) this attack can be considered as
legitimate use of NDN; caching policies should be
designed to deal with this consumer behavior.
While the two aforementioned poisoned content
attack variants require different adversarial capabilities,
their impact on the network is almost identical.
For this reason, we design countermeasures that address the effect of
both.
V-B Tentative Countermeasures
We now discuss tentative countermeasures to content poisoning attacks.
First we focus on the construction of a strong binding between
interests and corresponding data packets. We introduce two
constructions, based on standard NDN features, and analyze their benefits and drawbacks.
Then, we propose further countermeasures based on
heuristics, inter-router communication and user feedback.
V-B1 Self-Certifying and Human-Readable Naming
Self-certifying naming [15] (SCN) allows parties to verify the
association between a name and the corresponding object without relying on
auxiliary information, such as Public Key Certificates and a PKI.
This makes SCN an effective countermeasure against content poisoning
attacks [17].
There are a few well-known approaches in the literature for implementing SCN. The two
most popular ones are geared for static [16] and dynamic
content [15], respectively. In the former, an object name is computed
as the hash of its content. In the latter, an object name is constructed as:
H$(pk)\!:\!L$ where H$(pk)$ is the hash of the producer’s public key
$pk$ and $L$ is a human-readable label.
Users are not expected to handle self-certifying names directly. Instead, SCN
requires a secure indirection mechanism to map from names familiar to users
to the corresponding self-certifying names.
NDN uses hierarchical Human-Readable Naming (HRN)
for content. Human-readable names are designed to be user-friendly, i.e.,
allow consumers to anticipate, guess and remember the name of content they
wish to retrieve. As discussed in [33], HRN’s advantages over SCN
can be summarized as:
1.
More efficient routing: SCN provides a flat, location-free namespace,
which makes it difficult to efficiently retrieve a nearby (cached) copy of
content corresponding to a particular name [5]. Whereas,
SCN-based architectures resolve names using a location-
independent mechanism, such as DHTs [34, 39];
2.
Better usability: consumers can easily
understand the relationship between an object and its human-readable name;
3.
Less complex infrastructure: HRN does not require the use of a trusted
name resolution mechanism to map human-readable to network-intelligible names.
Unfortunately, human readability precludes a strong (i.e., cryptographic)
binding between a name and a corresponding object. In order to determine
whether a human-readable name is appropriate for an object, additional
mechanisms (e.g., a PKI) must be in place.
We consider whether it is possible to integrate the functionalities of SCN
into NDN, without changing its naming structure. To this end, we introduce
“Self-Certifying Interests/Data packets” (SCID), a mechanism that allows
routers to efficiently and securely determine whether a piece of content is the
“correct answer” for particular interest. Two variants of SCID: one for static
(S-SCID) and one for dynamic (D-SCID) content, are described below.
V-B2 Static Content
One of the components automatically appended to the name of each
NDN data packet upon its creation is a cryptographic hash computed over
its data, name (up to the hash itself) and the signature. A consumer requesting a
data packet by name, can elect to use this last hash component in an NDN interest.
(Assuming, of course, that the consumer somehow knows this hash ahead of time.)
NDN routers can easily and efficiently determine whether
a returned data packet corresponds to its requested name with very low overhead.
In fact, routers in the current NDN prototype always verify content hashes.
Our results show that a software implementation of SHA-256 can achieve throughput of
1.5Gbps of traffic (assuming $1,500$-byte packets) on the Intel Core 2 Duo platform,
as in Section V. This is stark contrast with the measly 150Mbps throughput
we observed in verifying RSA-1024 signatures.
This technique, which we refer to as S-SCID, prevents the adversary from serving
corrupted or fake data packets in response to an interest: the hash of the wrong
content can not match the one expressed by the consumer.
Linking multiple data packets is quite simple. For example, let
$CO_{1},\ldots,CO_{m}$ be the collection of data packets corresponding to a
large file. $CO_{i}$ includes (in its payload) the hash of $CO_{i+1}$. If
the hash of $CO_{1}$ can be obtained beforehand, all $CO_{i}$-s can
be retrieved securely with no danger of fetching the wrong or poisoned content.
The problem is thus reduced to discovering the hash of the initial fragment $CO_{1}$.
Also, when fetching a large file, a consumer might wish to have several simultaneously
outstanding interests, in order to maximize bandwidth usage. Therefore, it is insufficient
for data packets to be singly-linked, as described above. Instead, $CO_{i}$
needs to reference $CO_{i+1},...CO_{i+u}$ where $u$ is the highest number of
concurrently pending interests. We expect that $u$ is set by the content
producer based on the nature of specific content. Determining appropriate
values for $u$ is outside the scope of this paper.
While simple and efficient, S-SCID has several limitations.
Clearly, a consumer can not be expected to anticipate, guess, remember or recognize the
hash of content it is about to request. This translates into a
classical chicken-and-egg problem.
The usual SCN solution is to rely on a trusted
infrastructure for mapping human-readable to self-certifying names, akin to
what DNS does today. We discuss how to address this issue (without requiring
such infrastructures) in the next section.
S-SCID also imposes restrictions on inter-packet dependencies.
In order for packet $A$ to link to packet $B$ ($A\rightarrow B$), the latter must be created and
named first. This issue makes it impossible for packets to be linked in a cycle, e.g.,
$A\rightarrow B\rightarrow C\rightarrow A$. Consequently, it is unclear how to support
current Web applications that often involve loops in content linkage.
Also, this technique is unsuitable for dynamic content.
In other words, a consumer has no means of
foretelling the hash of an packet that does not exist at the time of request, e.g.,
the desired packet is the result of a Web search.
V-B3 Dynamic Content
Settings that involve cyclically linked and/or dynamic content require a different flavor of SCID.
NDN interests include the PublisherPublicKeyDigest field, as discussed in
Section II. This field contains the (SHA-256) hash of the public key of
the producer of the matching data packet. Thus, a consumer can (optionally) specify the
public key that it associates with a desired content name. If this field is
present in an interest, each intervening NDN router must make sure that
the corresponding data packet references the same public key.
We call this technique D-SCID.
Unlike C-SCID, data packets that use D-SCID can include arbitrary
references to other data packets, including cyclic links or links to
dynamically generated content. Also, once a consumer learns the hash
of a producer’s public key, it can use it to request all content from that
producer. Therefore, the nature of links between data packets does not limit the
number of concurrent pending interests that consumers can issue to retrieve a
piece of content.
D-SCID prevents adversaries from injecting fake content in response to an interest.
However, corrupted content can still be returned as long as it
references the appropriate producer’s public key. (This is, again, because
NDN routers are not mandated to verify content signatures.)
While S-SCID requires producers to explicitly specify inter-packet links,
D-SCID does not have such requirement.
Both flavors of SCID combine the benefits of self-certifying and human-readable names.
SCID does not mandate any particular trust model.
Also, S-SCID and D-SCID are not mutually exclusive.
Let $CO_{1},\ldots,CO_{m}$ be a collection of data packets
corresponding to a large file, created according to S-SCID – i.e., each $CO_{i}$ contains a reference to the hash of $CO_{i+1}$. A consumer first retrieves the content producer’s public key $pk$ via its preferred public key distribution mechanism. The hash of $pk$ is used to set the PublisherPublicKeyDigest field
of the interest for $CO_{1}$. Once $CO_{1}$ is retrieved,
the consumer extracts the hash of $CO_{2}$ from $CO_{1}$ and issues an
interests for $CO_{2}$ using this hash as last component.
Subsequent interests are issued similarly.
SCN-based architectures generally assume the existence of a trusted
infrastructure that performs mapping between real-world entities and
corresponding self-certifying names. Under the same assumption, SCID is a very effective countermeasure against content poisoning attacks; in particular, in the case of static content the exposure of the
network to such attacks is drastically reduced since
SCID prevents distribution of fake content. As far as corrupted content,
only the first in a collection of packets can be corrupted.
Trust in the first packet of a collection can be bootstrapped using a traditional PKI, as shown in the previous example, or with other mechanisms such as web of trust [2], SPKI/SDSI [14, 1], etc.
We believe that a combination of the two SCID flavors offers a flexible, trust-model independent solution for securing NDN
against content poisoning.
To the best of our knowledge, no current SCN-based system allows naming
content using both “static” and “dynamic” self-certifying names.
V-C Traffic Sampling for Signature Verification
We now discuss some probabilistic and collaborative techniques
for verifying content signatures by NDN routers.
Probabilistic Independent Verification.
Routers verify a random subset of cached content.
Corrupted packets are immediately removed, while
those with valid signatures are marked as such
and never verified again.
Let $r_{1},\ldots,r_{n}$ be a collection of routers. Let $pkt$ be a
data packet stored in all these routers’ caches and $1/v_{i}$ –
the fraction of packets in $r_{i}$’s cache that are verified at any given time.
$pkt$ is checked by at least one router with probability
$\mathbb{P}=1-\prod_{i=1}^{n}(1-1/v_{i})$.
Probabilistic Disjoint Verification.
A more effective strategy involves evenly distributing the
verification load among a set of routers belonging to the same
organization. Let $r_{1},\ldots,r_{n}$ be routers in the same organization,
and let $h_{CO}$ be the least significant 32 bits of the hash of data packet $CO$.
Router $r_{i}$ verifies $CO$ if $h_{CO}\equiv i\bmod n$.
Assuming that $h_{CO}$ is distributed uniformly between $0$ and $2^{32}-1$,
all routers need to verify roughly the same number of packets.
Unfortunately, the adversary can significantly reduce the effectiveness of
this strategy by generating data packets that are only verified by one router.
Specifically, the adversary picks an arbitrary value $x\in[1,n]$,
creates random data packets and injects them into the network only
if $h\bmod n=x$.
To prevent this attack, we replace the hash function used to generate $h$ with a
keyed hash function (HMAC [4]), as follows. All routers belonging to the same
organization share a secret key $k$. Let $h^{k}_{CO}$ be the 32 least significant bits of
HMAC${}_{k}($H$(CO))$. Router $r_{i}$ verifies $CO$ if $h^{k}_{CO}\bmod n=i$.
Since HMAC${}_{k}(\cdot)$ is a pseudorandom function, the adversary can
mount the attack only if it knows the secret key $k$.
Let $1/v_{i}$ (with $v_{i}<n$ for all $i\in[1,n]$) be the fraction of cached
data packets that a router can verify. Given a packet $pkt$, stored in all
caches of routers in the same organization, $pkt$ is verified with probability
$\mathbb{P}=1-\prod_{i=1}^{n}(1-n/v_{i})$.
Neighbor Verification Feedback.
To maximize utility of individual router’s signature verification, we consider a
cooperative approach whereby nodes actively exchange information about
validity of individual data packets. By having a large number of routers
verifying packets and cooperating, cryptographic operations can be applied
less frequently without lowering network’s resistance to content poisoning attacks.
Basically, each router (as above) verifies its cached packets probabilistically and
independently. However, if it determines that a given data packet is corrupted, a
router issues a special warning interest on all its interfaces. A warning references
/ndn/warning/hCO, where /ndn/warning/ corresponds to a
special reserved namespace and hCO represents the hash of the
corrupted data packet. The scope field of a warning interest is set to 2,
i.e., this interest type is not forwarded past one hop.
When a router receives a warning interest, it checks whether its cache contains a
referenced packet with hash hCO. If not, the router discards the warning.
Otherwise it verifies the content it with some probability $p$ that might depend,
on its current router CPU load. If signature verification fails, the router issues its own
warning to its neighbors. Otherwise, further warnings from the same interface are
ignored for a pre-defined period.
To prevent the adversary from injecting fraudulent warnings, every pair of
adjacent routers could share a symmetric key and use it to authenticate warnings,
e.g., using a MAC.
Consumer Feedback.
Recall that consumers verify all signatures on data packets.
We take advantage of this property to design a feedback-based verification
strategy for routers. Consumer feedback can be implemented similarly
to Neighbor Verification Feedback discussed above, i.e., through specially
scoped interests. However, allowing consumers to provide feedback
prompts several new challenges: (1) there is no pairwise trust relationship
between a router and consumers, even if they are one hop away; (2) consumers
are more likely to be compromised than routers; (3) consumers have almost no
accountability: it might not be possible to determine which consumer issued a
false warning.
The intuition behind our strategy is that consumer feedback should not trigger
immediate action by a router. However, a router should monitor collective (aggregated)
consumer feedback and act whenever its volume exceeds some threshold.
Our strategy is based on a probabilistic trust value $T\in~{}[0,1]$, associated with
each content in a router’s cache. $T=1$ indicates that the corresponding content
packet has been verified, while $T\approx 0$ indicates that it
should be selected for verification with probability proportional to $1-T$,
or deleted if the cache becomes full. New data packets are assigned $T=0.5$.
This value increases every time the data packet is retrieved, and decreases
whenever the router receives negative feedback from a consumer.
VI Related Work
NDN is an instantiation of the Content-Centric Networking (CCN) paradigm.
(An alternative term “Information-Centric Networking” is largely synonymous.)
Other related architectures include the Data-Oriented Network Architecture
(DONA) [25] and TRIAD.
DONA is based on “flat” self-certifying names,
computed as the cryptographic hash of the producer’s public key and
a (possibly) human-readable label. Such label, however, is not cryptographically bound
to the content.
New content is published – i.e., registered – with a tree of
trusted resolution handlers to enable retrieval. Resolution handlers
maintain a forwarding table that provides next-hop information for
pieces of content in the network. As such, DONA does not support
dynamically generated content.
Similar to NDN, TRIAD [10] names content using
human-readable, location-independent names.
It maps names to available replicas of data using an integrated
directory. It then forwards requests until a copy of the data is found. The
data location is returned to the client, who retrieves it using standard
HTTP/TCP. TRIAD relies on trusted directories to authenticate content
lookups (but not content itself). For additional security, the authors
of [10] recommend to limit the network to mutually trusting content
routers.
NDN caching performance optimization has been recently investigated
with respect to various metrics including energy
impact [21, 32, 27].
To the best of our knowledge, the work of Xie, et
al. [38] is the first to address cache
robustness in NDN. It introduces CacheShield, a
mechanism that helps routers to prevent caching unpopular content and
therefore maximizing the use of cache for popular content.
There is lots of previous work on DoS attacks on the current Internet
infrastructure. Current literature addresses both attacks
and countermeasures on the routing infrastructure [20],
packet flooding [23], reflection
attacks [30], DNS cache
poisoning [31] and SYN flooding
attacks [37]. Proposed
counter-measures are based on various
strategies and heuristics, including: anomaly
detection [6], ingress/egress
filtering [36], IP trace
back [28, 35],
ISP collaborative defenses [9] and user-collaborative defenses [18].
VII Summary and Future Work
In this paper, we perform initial analysis of NDN’s resilience to DoS
attacks. In doing so, we start by considering attacks on the
current Internet and assess their impact on NDN. .
We then identify two new type of attacks specific to NDN: interest
flooding and cache/content poisoning. For type, we discuss effects and
potential countermeasures.
Clearly, this paper represents only the first step towards mitigation of
DoS in the context of NDN. Much more work is needed to evaluate the
effectiveness of proposed countermeasures. In particular, extensive simulation-
and testbed-based experiments must be conducted in order to determine
optimal parameters for the instantiations of our countermeasures.
Finally, we intend to assess how other content-centric architectures
fare with respect to DoS attacks.
References
[1]
Martin Abadi.
On SDSI’s Linked Local Name Spaces.
Journal of Computer Security, 6(1-2):3–21, October 1998.
[2]
A Abdul-Rahman.
The PGP Trust Model, 1997.
[3]
Hitesh Ballani, Paul Francis, and Xinyang Zhang.
A Study of Prefix Hijacking and Interception in the Internet.
SIGCOMM Comput. Commun. Rev., 37(4):265–276, August 2007.
[4]
M. Bellare, R. Canetti, and H. Krawczyk.
Keying Hash Functions for Message Authentication.
In CRYPTO, 1996.
[5]
Matthew Caesar, Tyson Condie, Jayanthkumar Kannan, Karthik Lakshminarayanan,
Ion Stoica, and Scott Shenker.
Rofl: routing on flat labels.
In In SIGCOMM, pages 363–374, 2006.
[6]
Glenn Carl, George Kesidis, Richard R. Brooks, and Suresh Rai.
Denial-of-service attack-detection techniques.
IEEE Internet Computing, 10(1):82–89, jan 2006.
[7]
CCNx Content Object.
http://www.ccnx.org/releases/latest/doc/technical/ContentObject.html.
[8]
CCNx Interests Message.
http://www.ccnx.org/releases/latest/doc/technical/InterestMessage.html.
[9]
Yu Chen, Kai Hwang, and Wei-Shinn Ku.
Collaborative detection of ddos attacks over multiple network
domains.
IEEE Trans. Parallel Distrib. Syst., 18(12):1649–1662, 2007.
[10]
D.R. Cheriton and M. Gritter.
Triad: A new next-generation internet architecture, 2000.
[11]
David Dagon, Manos Antonakakis, Kevin Day, Xiapu Luo, Christopher P. Lee, and
Wenke Lee.
Recursive dns architectures and vulnerability implications.
In In Network and Distributed System Security Symposium
(NDSS09), 2009.
[12]
S. DiBenedetto, P. Gasti, G. Tsudik, and E. Uzun.
ANDaNA: Anonymous named data networking application.
In NDSS, 2012.
[13]
The DNSSEC Protocol.
http://tools.ietf.org/html/rfc2535.
[14]
Carl M. Ellison, Bill Frantz, Butler Lampson, Ron Rivest, Brian M. Thomas, and
Tatu Ylonen.
SPKI Certificate Theory, September 1999.
RFC2693.
[15]
David Mazi Eres, Michael Kaminsky, M. Frans Kaashoek, and Emmett Witchel.
Separating key management from file system security.
In In Proc. SOSP, pages 124–139, 1999.
[16]
Kevin Fu, M. Frans Kaashoek, and David Mazieres.
Fast and secure distributed read-only file system.
In ACM Transactions on Computer Systems, pages 181–196, 2000.
[17]
Ali Ghodsi, Teemu Koponen, Jarno Rajahalme, Pasi Sarolahti, and Scott Shenker.
Naming in content-oriented architectures.
In Proceedings of the ACM SIGCOMM workshop on
Information-centric networking, ICN ’11, pages 1–6, New York, NY, USA,
2011. ACM.
[18]
C. Gkantsidis and P. Rodriguez.
Cooperative security for network coding file distribution.
In INFOCOM 2006. 25th IEEE International Conference on Computer
Communications, Joint Conference of the IEEE Computer and Communications
Societies, 2006.
[19]
M. Gritter and D.R. Cheriton.
An architecture for content routing support in the internet.
In USENIX Symposium on Internet Technologies and Systems.
USENIX Association, 2001.
[20]
John Ioannidis and Steven M. Bellovin.
Implementing pushback: Router-based defense against ddos attacks.
In In Proceedings of Network and Distributed System Security
Symposium, 2002.
[21]
V. Jacobson, D. Smetters, N. Briggs, M. Plass, J. Thornton, and R. Braynard.
Voccn: Voice-over content centric networks.
2009.
[22]
V. Jacobson, D. Smetters, J. Thornton, M. Plass, N. Briggs, and R. Braynard.
Networking named content.
The 5th international conference on Emerging networking
experiments and technologies, 2009.
[23]
Jae-Hyun Jun, Hyunju Oh, and Sung-Ho Kim.
Ddos flooding attack detection through a step-by-step investigation.
Networked Embedded Systems for Enterprise Applications, IEEE
International Conference on, 0:1–5, 2011.
[24]
T. Koponen, M. Chawla, B.G. Chun, A. Ermolinskiy, K.H. Kim, S. Shenker, and
I. Stoica.
A data-oriented (and beyond) network architecture.
ACM SIGCOMM Computer Communication Review, 37(4):181–192,
2007.
[25]
Teemu Koponen, Mohit Chawla, Byung-Gon Chun, Andrey Ermolinskiy, Kye Hyun Kim,
Scott Shenker, and Ion Stoica.
A data-oriented (and beyond) network architecture.
In SIGCOMM ’07, pages 181–192, New York, NY, USA, 2007. ACM.
[26]
Mohit Lad, Dan Massey, Dan Pei, Yiguo Wu, Beichuan Zhang, and Lixia Zhang.
Phas: a prefix hijack alert system.
USENIX Security, August 2006.
[27]
Uichin Lee, Ivica Rimac, and Volker Hilt.
Greening the internet with content-centric networking.
In e-Energy, pages 179–182, 2010.
[28]
Liming Lu, Mun Choon Chan, and Ee-Chien Chang.
A general model of probabilistic packet marking for ip traceback.
In Proceedings of the 2008 ACM symposium on Information,
computer and communications security, ASIACCS ’08, pages 179–188, New York,
NY, USA, 2008. ACM.
[29]
Named data networking project (NDN).
http://named-data.org.
[30]
Vern Paxson.
An analysis of using reflectors for distributed denial-of-service
attacks.
SIGCOMM Comput. Commun. Rev., 31(3):38–47, July 2001.
[31]
V. Ramasubramanian and E. Sirer.
Perils of transitive trust in the domain name system.
In Proc. International Measurement Conference, 2005.
[32]
Elisha J. Rosensweig, Jim Kurose, and Don Towsley.
Approximate models for general cache networks.
In Proceedings of the 29th conference on Information
communications, INFOCOM’10, pages 1100–1108, Piscataway, NJ, USA, 2010.
IEEE Press.
[33]
Diana K. Smetters and Van Jacobson.
Securing network content.
Palo Alto Research Center, 2009.
[34]
Ion Stoica, Robert Morris, David Karger, M. Frans Kaashoek, and Hari
Balakrishnan.
Chord: A scalable peer-to-peer lookup service for internet
applications.
pages 149–160, 2001.
[35]
Robert Stone.
Centertrack: an ip overlay network for tracking dos floods.
In Proceedings of the 9th conference on USENIX Security
Symposium - Volume 9, SSYM’00, pages 15–15, Berkeley, CA, USA, 2000. USENIX
Association.
[36]
Udaya Kiran Tupakula and Vijay Varadharajan.
A practical method to counteract denial of service attacks.
In Proceedings of the 26th Australasian computer science
conference - Volume 16, ACSC ’03, pages 275–284, Darlinghurst, Australia,
Australia, 2003. Australian Computer Society, Inc.
[37]
Haining Wang, Danlu Zhang, and Kang G. Shin.
Detecting syn flooding attacks.
In In Proceedings of the IEEE Infocom, pages 1530–1539. IEEE,
2002.
[38]
M. Xie, I. Widjaja, and H. Wang.
Enhancing cache robustness for content-centric networks.
In In Proceedings of the IEEE Infocom, 2012.
[39]
Ben Y. Zhao, John D. Kubiatowicz, and Anthony D. Joseph.
Tapestry: An infrastructure for fault-tolerant wide-area location and
routing.
Technical Report UCB/CSD-01-1141, EECS Department, University of
California, Berkeley, Apr 2001. |
Differentiable Scattering Matrix for Optimization of Photonic Structures
Ziwei Zhu
Department of Computer Science, Columbia University, New York, New York 10027, USA
Changxi Zheng
Abstract
The scattering matrix, which quantifies the optical reflection and
transmission of a photonic structure, is pivotal for understanding the
performance of the structure. In many photonic design tasks, it is also
desired to know how the structure’s optical performance changes with
respect to design parameters, that is, the scattering matrix’s derivatives (or gradient).
Here we address this need. We present a new algorithm for computing
scattering matrix derivatives accurately and robustly. In particular, we
focus on the computation in semi-analytical methods (such as rigorous
coupled-wave analysis). To compute the scattering matrix of a structure,
these methods must solve an eigen-decomposition problem. However, when it
comes to computing scattering matrix derivatives, differentiating the
eigen-decomposition poses significant numerical difficulties.
We show that the differentiation of the eigen-decomposition
problem can be completely sidestepped, and thereby propose a robust
algorithm. To demonstrate its efficacy, we use our algorithm to optimize
metasurface structures and reach various optical design goals.
1 introduction
The scattering matrix is a fundamental concept in many fields. It relates the
input state and the output state of a physical system undergoing a scattering
process. Particularly revealing in optics, the scattering matrix has been
widely used for analyzing photonic structures such as
waveguides [19, 9, 18] and
metasurface units [16, 43, 8].
Once the scattering matrix of a photonic structure is known, the structure’s
optical performance (e.g., mode conversion efficiency and phase shift) can be
directly obtained.
Because of its vital importance,
many numerical methods have been developed to compute the scattering matrix
of a photonic structure.
Among them, a popular class is the semi-analytical methods,
such as the method of
lines [28] and rigorous coupled-wave analysis (RCWA) [24].
These methods exploit
the fact that many photonic structures in practice (such as waveguides and metasurface units)
have a piecewise constant cross-sectional shape along the transmission direction (denoted as $z$-direction).
Thus, to solve Maxwell’s equations, they only need to discretize the 2D cross-sectional region,
reducing Maxwell’s equations into a set of
continuous differential equations along $z$-direction, whose solution can be
expressed through an eigenvalue analysis.
Thanks to the semi-discretization, these methods often enable
faster computation in comparison to full discretization methods
(such as finite-element- and finite-volume-based methods).
Indeed, methods like RCWA have been widely used in designing
various photonic structures, such as
metasurfaces [3, 16, 4, 11],
metagratings [13, 1],
holograms [43, 44],
polarimeters [5],
solar cells [40],
radiative cooling structures [30],
color structures [34], photonic crystals[41],
and waveguides [19, 9].
In this work, we extend the semi-analytical methods to obtain the higher-order information of scattering matrices, namely the scattering matrix’s derivatives
(or gradient). Provided a photonic structure specified by certain design
parameters, we aim to compute not only its scattering matrix but its
derivatives with respect to the design parameters.
The scattering matrix derivatives depict the changes of the
structure’s optical behaviors as its design parameters vary.
This higher-order information,
if robustly and efficiently computed,
finds many applications in photonic design. Most notable is the optimization of
photonic structures.
The derivatives provide guidance on how we can adjust the parameters (e.g., through
the gradient descent algorithm [31]) to improve the structure’s optical
performance [20] or to find a design robust to
fabrication error [2, 39, 45].
Unfortunately, the computation of scattering matrix derivatives is nontrivial.
The difficulty is rooted in the fact that the permissible optical modes in
a photonic structure are eigenfunctions of a linear (Hermitian) operator
determined by Maxwell’s equations [15].
Thus, to compute the scattering matrix, semi-analytical methods must solve an
eigen-decomposition problem:
its eigenvalues describe the propagation constants (or effective indices) of the modes
and its eigenvectors indicate propagating modal patterns.
Differentiating the scattering matrix, by chain rule,
requires the derivatives of eigenvalues and eigenvectors.
It is the need of eigenvector derivatives that renders the scattering matrix differentiation ill-posed:
when there exist repeated eigenvalues,
the corresponding eigenvectors are not uniquely defined.
As the parameter changes, the numerical results of the eigenvectors
may change discontinuously, and their derivatives become undefined
(see more discussion in Section 3).
Not merely does this issue exist as a corner case; many photonic structures
in practice have geometric and material symmetries, from which repeated
eigenvalues and thus ill-defined eigenvector derivatives emerge (see Fig. 2).
Consequently,
one must carefully choose eigenvectors such that they
vary smoothly with respect to the design parameters.
This choice, albeit attainable, demands complex and expensive computational effort [38].
In this paper, we question the necessity of eigenvector derivatives for
differentiating scattering matrices.
We show that while eigen-decompositions
are needed for computing a photonic structure’s scattering matrix,
eigenvector derivatives can be fully sidestepped for differentiating the scattering
matrix.
Based on our new derivation, we present a fast and robust algorithm that,
without resorting to eigenvector derivatives, computes the scattering matrix
derivatives with respect to any design parameters.
Our method is designed for scattering matrices in general,
independent from any specific basis representation;
nor is it bound to any particular geometric parameterization.
To demonstrate the use of our method, we apply the scattering matrix
derivatives for optimizing the design of metasurface units.
We can choose different design parameterizations and
use gradient-based optimization to reach various light transmission goals.
We also propose a general parameterization of the meta-unit’s cross-sectional shape
that can be optimized using our method.
2 Background: Scattering Matrix
We start by briefly reviewing the classic notion of scattering matrix in computational photonics,
to pave the way toward its differentiation.
To numerically analyze a photonic structure (such as a waveguide),
the structure is often discretized along the wave propagation direction (i.e., $z$-direction)
into a series of layers each with a uniform cross-sectional material distribution (Fig. 1-b).
Consider optical waves of a specific frequency.
Their propagation in each layer is characterized by a scattering matrix $\mathbf{S}$,
which relates waves incident on the layer from left and right sides (Fig. 1-a) to the waves scattered out in either direction.
Concretely, let $\mathbf{a}_{\text{L}}$ and $\mathbf{a}_{\text{R}}$ denote vectors describing
incident waves on the layer from left and right sides, respectively.
Here $\mathbf{a}_{\text{L}}$ and $\mathbf{a}_{\text{R}}$ stack coefficients that represent the
waves under a chosen basis, whose construction will be outlined shortly.
Under the same basis, we use $\mathbf{b}_{\text{L}}$ and $\mathbf{b}_{\text{R}}$ to denote
the scattered waves in left and right directions.
With these notations, the incident and scattered waves are related through
$$\begin{bmatrix}\mathbf{b}_{\text{L}}\\
\mathbf{b}_{\text{R}}\end{bmatrix}=\mathbf{S}\begin{bmatrix}\mathbf{a}_{\text{%
L}}\\
\mathbf{a}_{\text{R}}\end{bmatrix},\text{ where }\mathbf{S}\coloneqq\begin{%
bmatrix}\mathbf{R}_{\text{L}}&\mathbf{T}_{\text{RL}}\\
\mathbf{T}_{\text{LR}}&\mathbf{R}_{\text{R}}\end{bmatrix}.$$
(1)
Here $\mathbf{S}$ is decomposed into four submatrices: $\mathbf{R}_{\text{L}}$ and
$\mathbf{R}_{\text{R}}$ indicate how the incident wave from left or right direction
is reflected by the layer, while $\mathbf{T}_{\text{RL}}$ and $\mathbf{T}_{\text{LR}}$
describe how the incident wave (from either direction) transmits through the layer.
The computation of scattering matrix starts with a semi-discretization of the
frequency-domain Maxwellâs equations of a photonic layer, namely,
$$-jk_{0}\frac{\partial}{\partial z}\mathbf{e}=\mathbf{P}\mathbf{h}\;\textrm{ %
and }\;-jk_{0}\frac{\partial}{\partial z}\mathbf{h}=\mathbf{Q}\mathbf{e},$$
(2)
where $k_{0}$ is the free-space wave number, and the vectors $\mathbf{e}$ and $\mathbf{h}$ describe the electric and magnetic fields of the photonic structure
under a chosen basis—for example, RCWA uses the 2D Fourier basis on the
cross-section of the wave propagation direction. The matrices $\mathbf{P}$ and $\mathbf{Q}$ encode the cross-sectional distributions of material permeability
and permittivity.
The semi-discretization (2) is a common form in many
numerical analysis methods for photonic structures (such as the method of line [28] and RCWA [24]).
The difference across those methods only lies in the specific ways of constructing $\mathbf{P}$ and $\mathbf{Q}$
(e.g., see Supplement 1 for their construction in RCWA).
Once $\mathbf{P}$ and $\mathbf{Q}$ are determined, the scattering matrix $\mathbf{S}$ can be constructed.
A key step of this construction is to solve an eigenvalue problem,
$(\mathbf{P}\mathbf{Q})\mathbf{W}=\mathbf{W}\bm{\Gamma}$, to obtain eigenvectors $\mathbf{W}$ and the diagonal eigenvalue matrix
$\bm{\Gamma}$. As we will discuss in Section 3, it is this eigenproblem
that renders the differentiation of the scattering matrix ill-posed.
To understand the challenges and how we overcome them,
we first present the recipe of computing $\mathbf{S}$ from $\mathbf{W}$ and $\bm{\Gamma}$,
as follows.
Let $\bm{\Omega}\coloneqq(\mathbf{P}\mathbf{Q})^{\frac{1}{2}}$. Then, its eigenvalue matrix is $\bm{\Lambda}=\bm{\Gamma}^{\frac{1}{2}}$.
As derived in [32], the formulas of computing the scattering matrix $\mathbf{S}$ defined
in (1) are
$$\displaystyle\mathbf{R}_{\textrm{L}}=\mathbf{R}_{\textrm{R}}$$
$$\displaystyle=\left(\mathbf{A}-\mathbf{X}\mathbf{B}\mathbf{A}^{-1}\mathbf{X}%
\mathbf{B}\right)^{-1}\left(\mathbf{X}\mathbf{B}\mathbf{A}^{-1}\mathbf{X}%
\mathbf{A}-\mathbf{B}\right),$$
(3a)
$$\displaystyle\mathbf{T}_{\text{LR}}=\mathbf{T}_{\text{RL}}$$
$$\displaystyle=\left(\mathbf{A}-\mathbf{X}\mathbf{B}\mathbf{A}^{-1}\mathbf{X}%
\mathbf{B}\right)^{-1}\mathbf{X}\left(\mathbf{A}-\mathbf{B}\mathbf{A}^{-1}%
\mathbf{B}\right),$$
(3b)
where the matrices $\mathbf{X}$, $\mathbf{A}$, and $\mathbf{B}$ have the following forms:
$$\displaystyle\mathbf{X}$$
$$\displaystyle=e^{j\bm{\Lambda}\frac{L}{k_{0}}},$$
(4a)
$$\displaystyle\mathbf{A}$$
$$\displaystyle=\mathbf{W}^{-1}\mathbf{W}_{0}+\mathbf{V}^{-1}\mathbf{V}_{0},\;%
\text{ and }\mathbf{B}=\mathbf{W}^{-1}\mathbf{W}_{0}-\mathbf{V}^{-1}\mathbf{V}%
_{0}.$$
(4b)
Here we use $L$ to denote the layer thickness (Fig. 1), and the matrix $\mathbf{V}$
is related to $\mathbf{W}$ through $\mathbf{V}=\mathbf{Q}\mathbf{W}\bm{\Lambda}^{-1}$.
$\mathbf{W}$ and $\mathbf{V}$ together form a basis of electric and magnetic components
for the optical waves in the layer.
Similarly, $\mathbf{W}_{0}$ and $\mathbf{V}_{0}$ form a basis for free space propagation,
independent from the photonic structure. They are constant values
for computing the derivatives of $\mathbf{S}$.
The vectors, $\mathbf{a}_{\text{L}}$, $\mathbf{a}_{\text{R}}$ $\mathbf{b}_{\text{L}}$, and $\mathbf{b}_{\text{R}}$,
in (1) are coefficients under this free-space basis to describe incident and scattered waves.
Once the scattering matrices of individual layers are computed,
they are combined using the Redheffer star product [29]
into the total scattering matrix, one that indicates the optical response of the entire
photonic structure.
Remark. The formulas in Eqs. (3) and (4)
assume that the current photonic layer is sandwiched by two free-space layers.
This assumption is by no means a limitation. In an arbitrary photonic
structure, the layers can be
treated as if they are interleaved with free-space layers—each of which has a zero thickness.
3 Differentiable Scattering Matrix
The geometry or material distribution of photonic structure is specified by its
structural (design) parameters (e.g., see Fig. 2). These
parameters determine the structure’s permittivity and permeability
distributions described by $\mathbf{P}$ and $\mathbf{Q}$ in (2). Thus, one can
compute their derivatives, $\mathbf{P}^{\prime}$ and $\mathbf{Q}^{\prime}$, with respect to an arbitrary
parameter. Given $\mathbf{P}^{\prime}$ and $\mathbf{Q}^{\prime}$, we now address the question of how to
compute the scattering matrix derivative $\mathbf{S}^{\prime}$ with respect to the same
parameter.
3.1 Challenges in Scattering Matrix Differentiation
The construction of scattering matrix $\mathbf{S}$ needs to solve an eigenvalue problem $(\mathbf{P}\mathbf{Q})\mathbf{W}=\mathbf{W}\bm{\Gamma}$,
as the eigenvalues $\bm{\Gamma}$ and eigenvectors $\mathbf{W}$ appear in
Eqs. (3) and (4) for computing $\mathbf{S}$.
Thus, for the differentiation of $\mathbf{S}$, it seems also needed to compute
the derivatives of the eigenvalues $\bm{\Gamma}$ and eigenvectors $\mathbf{W}$.
Unfortunately, the derivatives of eigenvectors in many cases are ill-defined.
Most notable is when there exist repeated eigenvalues.
Repeated eigenvalues are not uncommon: many photonic devices have certain structural
symmetries, from which eigenvalue repetition naturally emerges (see Fig. 2).
For those repeated eigenvalues, their eigenvectors (up to a scale) are not
uniquely determined; any set of linearly independent vectors that span the
same subspace are valid eigenvectors. Because of the ambiguity, as the
structural parameter changes, those eigenvectors may change discontinuously
(see an examples in Supplement 1), and thus their derivatives may not be well-defined.
As a result, one must carefully choose eigenvectors in the subspace of repeated
eigenvalues such that the eigenvectors change continuously with respect to the
structural parameter.
This choice, however, is computationally expensive.
As derived in [38],
to ensure well-defined eigenvector derivatives,
one must compute higher-order derivatives of the eigenvalues and
the matrix $\mathbf{P}\mathbf{Q}$: if the repeated eigenvalues have repeated
derivatives up to the $n$-th order (see Fig. 2), then
derivatives up to the $(n+1)$-th
order of both eigenvalues and the matrix $\mathbf{P}\mathbf{Q}$ must be computed to
determine first-order eigenvector derivatives.
3.2 Differentiation without Resort to Eigenvector Derivatives
We now present
a new algorithm for computing the scattering matrix derivative $\mathbf{S}^{\prime}$.
Even in the presence of repeated eigenvalues and their derivatives,
our method requires only the first-order derivatives of the matrices $\mathbf{P}$ and $\mathbf{Q}$,
completely sidestepping the differentiation of eigenvalues and eigenvectors.
In comparison to the way that takes eigenvalue derivatives (as described above),
our method
is more robust and efficient.
First, we rewrite the commonly used expressions of scattering matrix
components, shown in (3), in new forms,
$$\displaystyle\mathbf{R}_{\textrm{L}}=\mathbf{R}_{\textrm{R}}$$
$$\displaystyle=\left(\mathbf{I}-\mathbf{D}_{1}^{2}\right)^{-1}\left(\mathbf{D}_%
{1}\mathbf{D}_{2}-\mathbf{D}_{3}\right),$$
(5a)
$$\displaystyle\mathbf{T}_{\text{LR}}=\mathbf{T}_{\text{RL}}$$
$$\displaystyle=\left(\mathbf{I}-\mathbf{D}_{1}^{2}\right)^{-1}\left(\mathbf{D}_%
{2}-\mathbf{D}_{1}\mathbf{D}_{3}\right),$$
(5b)
where $\mathbf{D}_{1}$, $\mathbf{D}_{2}$, and $\mathbf{D}_{3}$ denote the following matrix
multiplications, respectively:
$$\displaystyle\mathbf{D}_{1}$$
$$\displaystyle\coloneqq\mathbf{A}^{-1}\mathbf{X}\mathbf{B}=\left(\mathbf{W}_{0}%
+\mathbf{T}\mathbf{V}_{0}\right)^{-1}\mathbf{W}\mathbf{X}\mathbf{W}^{-1}\left(%
\mathbf{W}_{0}-\mathbf{T}\mathbf{V}_{0}\right),$$
(6a)
$$\displaystyle\mathbf{D}_{2}$$
$$\displaystyle\coloneqq\mathbf{A}^{-1}\mathbf{X}\mathbf{A}=\left(\mathbf{W}_{0}%
+\mathbf{T}\mathbf{V}_{0}\right)^{-1}\mathbf{W}\mathbf{X}\mathbf{W}^{-1}\left(%
\mathbf{W}_{0}+\mathbf{T}\mathbf{V}_{0}\right),$$
(6b)
$$\displaystyle\mathbf{D}_{3}$$
$$\displaystyle\coloneqq\mathbf{A}^{-1}\mathbf{B}=\left(\mathbf{W}_{0}+\mathbf{T%
}\mathbf{V}_{0}\right)^{-1}\left(\mathbf{W}_{0}-\mathbf{T}\mathbf{V}_{0}\right).$$
(6c)
The derivation of these new expressions (5) and (6)
are provided in Supplement 1.
In Eqs. (6), the equalities are reached by applying (4)
and using $\mathbf{T}$ that denotes $\mathbf{T}\coloneqq\bm{\Omega}\mathbf{Q}^{-1}$.
The expressions in (5) present a new route for computing
scattering matrix derivative.
They indicate that the scattering matrix $\mathbf{S}$ is determined by
the three matrices, $\mathbf{D}_{1}$, $\mathbf{D}_{2}$, and $\mathbf{D}_{3}$. As a result, to compute its derivative $\mathbf{S}^{\prime}$ using the chain rule,
we need to compute the derivatives of $\mathbf{D}_{1}$, $\mathbf{D}_{2}$, and $\mathbf{D}_{3}$
with respect to the structural parameter.
In Eqs. (6), both $\mathbf{W}_{0}$ and $\mathbf{V}_{0}$ (introduced in (4)) are constant matrices.
Thus, the derivatives of $\mathbf{D}_{1}$, $\mathbf{D}_{2}$, and $\mathbf{D}_{3}$
only depend on the derivatives of two other matrices in Eqs. (6),
namely $\mathbf{T}$ and $\mathbf{W}\mathbf{X}\mathbf{W}^{-1}$.
We now describe how to compute the derivatives of the two matrices, respectively.
Derivative of $\mathbf{T}$.
The matrix $\mathbf{T}\coloneqq\bm{\Omega}\mathbf{Q}^{-1}$ is related to $\mathbf{Q}$ and $\bm{\Omega}\coloneqq\left(\mathbf{P}\mathbf{Q}\right)^{{1}/{2}}$
but not the eigenvalues and eigenvectors.
Its derivative can be expressed as
$$\mathbf{T}^{\prime}=\bm{\Omega}^{\prime}\mathbf{Q}^{-1}+\bm{\Omega}\left(%
\mathbf{Q}^{-1}\right)^{\prime}=\bm{\Omega}^{\prime}\mathbf{Q}^{-1}-\bm{\Omega%
}\mathbf{Q}^{-1}\mathbf{Q}^{\prime}\mathbf{Q}^{-1},$$
(7)
where $\mathbf{Q}$ depends on the material permittivity
distributions of the photonic structure, and therefore its
derivative $\mathbf{Q}^{\prime}$ with respect to a design parameter
can be directly computed (see examples in Section 4).
The way of computing $\bm{\Omega}^{\prime}$ can be derived by taking the derivatives
on both sides of the relation $\bm{\Omega}^{2}=\mathbf{P}\mathbf{Q}$, which yields
$$\bm{\Omega}^{\prime}\bm{\Omega}+\bm{\Omega}\bm{\Omega}^{\prime}=\mathbf{P}^{%
\prime}\mathbf{Q}+\mathbf{P}\mathbf{Q}^{\prime}.$$
(8)
Given $\mathbf{P}^{\prime}$ and $\mathbf{Q}^{\prime}$, the right-hand side of this equation can be directly computed.
To compute $\bm{\Omega}^{\prime}$, we rewrite the left-hand side by denoting
$\bm{\Omega}^{\prime}$ as $\bm{\Omega}^{\prime}=\mathbf{W}\mathbf{Y}\mathbf{W}^{-1}$ for some unknown $\mathbf{Y}$,
where $\mathbf{W}$ is the eigenvector matrix of $\mathbf{P}\mathbf{Q}$. Using the fact that
$\bm{\Omega}=(\mathbf{P}\mathbf{Q})^{1/2}=\mathbf{W}\bm{\Lambda}\mathbf{W}^{-1}$, we obtain a simplified form
of (8):
$$\mathbf{Y}\bm{\Lambda}+\bm{\Lambda}\mathbf{Y}=\mathbf{W}^{-1}(\mathbf{P}^{%
\prime}\mathbf{Q}+\mathbf{P}\mathbf{Q}^{\prime})\mathbf{W}.$$
(9)
From (9), $\mathbf{Y}$ can be easily solved by noticing that $\bm{\Lambda}$ is a diagonal
matrix, and therefore (9) can be written element-wise as $(\lambda_{i}+\lambda_{j})\mathbf{Y}_{ij}=\mathbf{C}_{ij}$, where $\lambda_{i}$ is the $i$-th eigenvalue in $\bm{\Lambda}$,
and $\mathbf{C}$ denote the matrix on the right-hand side of (9).
In other words, the elements of $\mathbf{Y}$ can be obtained by solving $n^{2}$ 1D linear equations
in parallel. Once $\mathbf{Y}$ is obtained, $\bm{\Omega}^{\prime}$ is computed using
$\bm{\Omega}^{\prime}=\mathbf{W}\mathbf{Y}\mathbf{W}^{-1}$.
We note that while this process of computing $\bm{\Omega}^{\prime}$ requires the
eigenvectors $\mathbf{W}$ and eigenvalues $\bm{\Lambda}$, they are also needed for computing
the scattering matrix in the first place.
Our solving process does not require the derivatives of eigenvectors.
Therefore, it introduces no additional effort in terms of eigen-decomposition.
Derivative of $\mathbf{W}\mathbf{X}\mathbf{W}^{-1}$.
In the first glance, the derivative of $\mathbf{W}\mathbf{X}\mathbf{W}^{-1}$
depends on the eigenvectors $\mathbf{W}$.
However, from the definition of $\mathbf{X}$ in (4a), we notice that
$\mathbf{W}\mathbf{X}\mathbf{W}^{-1}=e^{j\bm{\Omega}{L}/{k_{0}}}$,
which suggests an alternative approach: take the derivative
of the matrix exponential $e^{j\bm{\Omega}{L}/{k_{0}}}$ with respect to $\bm{\Omega}$.
A common approach of computing the matrix exponential
$e^{j\bm{\Omega}{L}/{k_{0}}}$ is through the eigen-decomposition
of $\bm{\Omega}$ followed by the exponential of the resulting eigenvalues.
If we take this approach, the derivative computation must involve
the derivatives of eigenvectors, which might not be well-defined.
Another approach, used by Feynman [12]
and others [17, 7, 35], expresses the derivative of a matrix exponential using
an integral that in itself involves matrix exponentials.
Yet, numerically evaluating the matrix exponentials and the integral are expensive.
Instead, our proposed method for computing the derivative is based on the following
proposition originally proved in [25].
Proposition 1.
Consider an $n\times n$ matrix $\bm{\Omega}$ and its derivative $\bm{\Omega}^{\prime}$ with respect to an arbitrary parameter.
If
$$\mathbf{G}=\begin{bmatrix}\bm{\Omega}&\bm{\Omega}^{\prime}\\
\mathbf{0}&\bm{\Omega}\end{bmatrix},\;\textrm{then }e^{j\mathbf{G}{L}/{k_{0}}}%
=\begin{bmatrix}e^{j\bm{\Omega}{L}/{k_{0}}}&\left(e^{j\bm{\Omega}{L}/{k_{0}}}%
\right)^{\prime}\\
\mathbf{0}&e^{j\bm{\Omega}{L}/{k_{0}}}\end{bmatrix},$$
(10)
where the top-right $n\times n$ block matrix in $e^{j\mathbf{G}{L}/{k_{0}}}$ is the derivative
of the matrix exponential $e^{j\bm{\Omega}{L}/{k_{0}}}$.
In our problem, $\bm{\Omega}^{\prime}$ is computed as described above (by solving (9)),
and the common way of computing $e^{j\mathbf{G}{L}/{k_{0}}}$ is by taking the
eigen-decomposition of $\mathbf{G}$, which is again what we wish to avoid.
We therefore take a different approach, the scaling and squaring method [14],
to compute $e^{j\mathbf{G}{L}/{k_{0}}}$—without the need of eigen-decomposition.
The scaling and squaring method exploits the relation
$e^{\mathbf{A}}=\left(e^{\mathbf{A}/\sigma}\right)^{\sigma}$ for any $n\times n$ matrix $\mathbf{A}$.
In practice, $\sigma$ is chosen to be $\sigma=2^{s}$ for some non-negative integer $s$.
The idea is to have the norm of $\mathbf{A}/\sigma$ sufficiently small such that
$e^{\mathbf{A}/\sigma}$ can be well approximated by a Padé approximant near the
origin. The Padé approximant is a rational polynomial of $\mathbf{A}$.
Its evaluation requires only matrix multiplications and inverse, but no eigen-decomposition.
The scaling and squaring method is robust and accurate,
and has been used in many numerical tools (such as MATLAB’s expm function).
When applying this method,
we further exploit the specific structure of $\mathbf{G}$ (i.e., its bottom-left block
matrix vanishes, and its two diagonal block matrices are identical) to tailor
the method for improving computational performance.
Supplement 1 presents our detailed derivations and computational steps.
4 Results
This section presents our numerical results.
First, we validate our algorithm for computing scattering matrix derivatives.
Next, to demonstrate the use of scattering matrix derivatives in photonics,
we optimize the geometry of photonic metasurface units (also called
meta-atoms).
Meta-atoms are the building blocks of a metasurface,
often designed based on physical intuitions and manually crafted
libraries [42, 27, 26].
More recently, inverse design methods of meta-atom structures have also been
explored—e.g., through finite-difference-based gradient descent [6],
adjoint-based level-set method [23], and
topological optimization [33, 22].
Due to fabrication constraints, meta-atoms often have constant cross-sectional
shapes along one direction (i.e., $z$-direction, as shown in Fig. 3-a).
Thus, the semi-analytical methods (such as RCWA) are particularly efficient for
simulating meta-atoms, thanks to their ability of not discretizing along
$z$-direction [24]. Our method, for the first time, enables the semi-analytical methods
to also compute scattering matrix derivatives with respect to design parameters.
Here, in the framework of RCWA, we demonstrate automatic discovery of meta-atom structures that reach various amplitude and phase goals.
4.1 Validation
To validate our algorithm, we consider a dielectric meta-atom used in
metasurface holography [26, 27].
Its structure is shown in Fig. 3-a.
We use Eqs. (5) to compute the
scattering matrix, for which the matrices $\mathbf{P}$ and $\mathbf{Q}$ (introduced in (2))
are constructed using RCWA. The scattering matrix allows us to compute
light propagation properties of the meta-atom, which are in turn compared
to the results from finite difference time domain (FDTD) method implemented
in Lumerical [36].
We first scan the light wavelength from $1.2\mu\text{m}$ to $1.6\mu\text{m}$. For
each wavelength, we compute, using our scattering matrix and FDFD respectively,
the effective index of the fundamental mode propagating in the meta-atom.
The results from our method agree with FDFD results (see Fig. 3-b).
Furthermore, we consider the far-field light transmission through the meta-atom,
and compute the phase shift and amplitude change for each wavelength. Again, the
results from our method and FDTD match closely, as shown in Fig. 3-c.
These experiments confirm that our scattering matrix computation is as accurate as
FDTD in Lumerical. In terms of computational cost, our method takes about 0.15 seconds for each monochromatic simulation, and a few seconds
for the entire $1.2\mu\text{m}$-$1.6\mu\text{m}$ wavelength range, whereas the FDTD simulation takes several minutes.
Next, we validate our derivative computation. We
consider again the meta-atom structure shown in Fig. 3-a, and choose
the parameter $\alpha$ to be the size of the hollow square.
Using our method, we compute the derivative of the structure’s scattering matrix
with respect to $\alpha$. Meanwhile, since there is no analytic expression of the scattering matrix derivative,
we approximate it using finite difference (FD) estimation, that is,
$$\frac{\partial\mathbf{S}}{\partial\alpha}\approx\frac{\mathbf{S}(\alpha+\Delta%
\alpha)-\mathbf{S}(\alpha-\Delta\alpha)}{2\Delta\alpha}.$$
(11)
We estimate $\frac{\partial\mathbf{S}}{\partial\alpha}$ using a sweeping range of $\Delta\alpha$ values, and compare them
to the derivative resulted from our method.
The results are illustrated in Fig. 4. The accuracy of FD approximation
largely depends on the choice of $\Delta\alpha$.
Only when $\Delta\alpha$ is chosen within a certain range,
FD approximation is accurate enough to agree with our derivative results.
This agreement confirms the correctness of our method.
But for different elements in the scattering matrix, the valid $\Delta\alpha$ range
varies (indicated in light green in Fig. 4), suggesting that FD approximation is impractical:
it is hard, if not impossible, to choose a proper $\Delta\alpha$ to produce accurate
derivative estimations for all elements in the scattering matrix.
In contrast, our method is robust for computing the derivatives.
Computational cost.
In addition to the robustness, our method is also faster than the FD method.
In the FD method, computing a matrix derivative requires the computation of two scattering
matrices $\mathbf{S}(\alpha+\Delta\alpha)$ and $\mathbf{S}(\alpha-\Delta\alpha)$. In contrast,
our method, in addition to computing $\mathbf{S}(\alpha)$,
only requires a few matrix multiplications and inverses (recall Section 3.3.2).
On our workstation computer, the overhead of computing a scattering matrix derivative
is about $30\%\sim 40\%$ of the cost for computing the scattering matrix itself.
4.2 Use Case: Optimization of Meta-atom Structure
Controlling phase and amplitude of monochromatic light.
First, we optimize meta-atom structures to reach specific transmitted amplitudes and phases
for a monochromatic light (at 1.55$\mu\text{m}$ wavelength, $x$-polarized).
The cross-sectional shape is shown in Fig. 5-a, determined by two parameters.
The objective function for the inverse design is defined as
$$\mathcal{L}={\left|\mathbf{T}_{\text{LR}}(m,m)-t_{m}\right|}^{2},$$
(12)
where $\mathbf{T}_{\text{LR}}$ is the transmission submatrix in the scattering matrix (recall (1)),
$m$ is the mode index for the incident and outgoing light in free space,
thus $\mathbf{T}_{\text{LR}}(m,m)$ denotes the $m$-th diagonal element of the matrix.
Also, $t_{m}$ is a complex constant specifying the target amplitude and phase of
the transmission.
Here we consider the fundamental mode (the way to choose corresponding $m$ is given in Supplemental 1),
which describes the far-field light transmission along the $z$-direction.
To verify the robustness of our method and the enabled optimization,
we evenly sample different targets $t_{m}$ on a circle on the complex plane
(see Fig. 5-b). For each target, we find meta-atom’s shape parameters by minimizing (12)
through a gradient-descent algorithm [31],
for which the gradients of (12) with respect to the design parameters are
computed using our method.
As shown in Fig. 5-b, we are able to automatically discover structures
that reach these targets closely.
Controlling phases for both $x-$ and $y-$polarized light.
Next, we optimize meta-atom structures to obtain target responses for
$x$- and $y$-polarized light, simultaneously. This type of meta-atoms
has been used to construct metasurface holograms [10].
In our example, the light wavelength is 1.3$\mu\text{m}$; the meta-atoms have
a fixed height of 2.0$\mu\text{m}$ and a period of 2.5$\mu\text{m}$ along $x$- and $y$-direction.
The cross-sectional shape of the meta-atoms are specified by two parameters shown in
Fig. 7-a. We determine the parameters by minimizing the following objective function:
$$\mathcal{L}=-\frac{\mathbf{T}_{\text{LR}}(m_{x},m_{x})}{\left|\mathbf{T}_{%
\text{LR}}(m_{x},m_{x})\right|}t_{x}^{*}-\frac{\mathbf{T}_{\text{LR}}(m_{y},m_%
{y})}{\left|\mathbf{T}_{\text{LR}}(m_{y},m_{y})\right|}t_{y}^{*},$$
(13)
where the subscript $x$ (and $y$) indicates light polarization;
$t_{x}$ (and $t_{y}$) are the target phase changes from $x$-polarized (and $y$-polarized) incident light to
the outgoing light with the same polarization (i.e., $t_{x}=\exp(i\phi_{x})$ for some $\phi_{x}$).
The first term in (13) measures, for the $x$-polarized light, the cosine difference (through dot product on complex plane)
between the $m$-th mode’s phase change and the target phase change,
and similarly for the second term.
The optimized structures for different $x$- and $y$-polarized phase targets are
shown in Fig. 7. In all cases, the residual between the target
and the resulting phase change is within 7% of one period ($2\pi$), and in most cases within 1%.
Controlling amplitudes for multiple wavelengths.
We also demonstrate inverse design of meta-atoms for another type of optical response:
obtain two target amplitude responses at two separate wavelengths, simultaneously.
This type of responses have proven useful for making colored metasurface
holograms [26, 27].
Here we consider two archetypes used in [27], each described
by two parameters (see Fig. 6-a). The two wavelengths under consideration are
1.2$\mu\text{m}$ (labeled as blue) and 1.6$\mu\text{m}$ (red), and the objective function is defined as
$$\mathcal{L}=\left[\left|\mathbf{T}_{\text{LR},1}(m,m)\right|^{2}-A^{2}_{1}%
\right]^{2}+\left[\left|\mathbf{T}_{\text{LR},2}(m,m)\right|^{2}-A^{2}_{2}%
\right]^{2}.$$
(14)
Here the subscript “1” and “2” indicate the blue (1.2$\mu\text{m}$) and red (1.6$\mu\text{m}$) wavelength, respectively.
The first term accounts for the blue wavelength:
$\mathbf{T}_{\text{LR},1}$ is the transmission submatrix of the scattering matrix
and $A_{1}$ is the desired amplitude. Similar is the second term.
More terms can be added in (14) to incorporate more than two wavelengths.
For each archetype, we find its parameter values via a gradient-descent algorithm that minimizes
(14), and choose between the two archetypes one that produces a smaller objective value.
The optimized structures and their performances are shown in Fig. 6.
For almost all the experiments (each with a different amplitude target),
the resulting amplitudes by the inversely designed meta-atoms match closely to
their targets.
General cross-sectional shape design.
Lastly, we introduce a new way to inverse design the meta-atom’s
cross-sectional shape under a general representation.
We use the star-convex polygon [37] to represent the cross-sectional shape.
Such a shape can be discretized by sampling $N$
points on its boundary so that the polar angles of these points are evenly
distributed over $[0,2\pi]$. In other words,
the $(k+1)$-th point has the coordinate $p_{k}\left[\cos{\left(2k\pi/N\right)},-\sin{\left(2k\pi/N\right)}\right]$, where $p_{k}$ is a non-negative value (see
Fig. 8-a), and the shape is specified by $N$ parameters $p_{1},\ldots,p_{N}$.
A large $N$ offers many degrees of freedom to represent a complex shape,
but meanwhile renders exhaustive search through the entire parameter space
too expensive—one must rely on numerical optimization methods to determine the parameter
values.
This shape representation is particularly suitable for RCWA-based analysis, as it allows for a closed-form
2D Fourier transform of the shape (and thus the permittivity distribution) [21].
In RCWA framework, 2D Fourier transform of the cross-sectional permittivity distribution
is needed for computing the matrices, $\mathbf{P}$ and $\mathbf{Q}$, as well as their derivatives with respect
to the $p_{k}$ parameters. Supplemental 1 provides the details of this process.
As examples, we optimize octagons ($N=8$) to obtain desired optical responses
in different scattering directions.
First, we specify the target scattering directions.
Notice that to predict optical behavior of a single meta-atom in simulation,
periodic boundary condition is often used. Under this condition, the meta-atom
is effectively a 2D grating structure, for which
we can use diffraction orders to specify different scattering directions:
the output light with diffraction order $(p,q)$ is along the direction
$$\vec{k}=\left(\frac{2\pi p}{L_{x}},\frac{2\pi q}{L_{y}},1\right),$$
(15)
where $L_{x}$ and $L_{y}$ are periods along $x$- and $y$-axis, respectively ($L_{x}=L_{y}=1\mu\text{m}$ in our examples).
We consider $x$-polarized light with the wavelength of 1.55$\mu\text{m}$.
The goal here is to obtain specified far-field phases and amplitudes
at two scattering directions—ones that correspond to the diffraction orders, $(-1,0)$
and $(1,0)$, as shown in Fig. 8-b.
We further restrict $p_{k}$ to be in the range $[0.15\mu\text{m},0.45\mu\text{m}]$, and determine
$p_{k}$ values by minimizing
$$\mathcal{L}={\left|\mathbf{T}_{\text{LR}}(m,n_{1})-t_{1}\right|}^{2}+{\left|%
\mathbf{T}_{\text{LR}}(m,n_{2})-t_{2}\right|}^{2},$$
(16)
where $n_{1}$ and $n_{2}$ are mode indices for the diffraction orders $(-1,0)$ and $(1,0)$, respectively;
and $t_{1}$ and $t_{2}$ specify the target phases and amplitudes (as complex values) in the two outgoing directions.
We perform two experiments for two sets of $t_{1}$ and $t_{2}$ goals.
The optimization convergence curves and resulting shapes are shown in Fig. 8.
5 Conclusion
We have presented an algorithm for computing the derivatives of the scattering
matrices of a photonic structure with respect to its structural parameters.
Our method is built on the framework of semi-analytical methods for analyzing photonic
structures. A key step in semi-analytical methods
for computing scattering matrices is the eigen-decomposition.
However,
to compute scattering matrix derivatives,
directly differentiating the eigenvalue analysis
poses significant difficulties.
We show a new route to compute scattering matrix derivatives
without the need of differentiating the eigen-decomposition process.
The scattering matrix derivatives present how a photonic structure’s
performance changes as its structural parameters vary.
While we demonstrated their use in the optimization of meta-atom units,
they are useful in many other applications.
Therefore, our method may serve as a useful analysis tool in a wide range of
photonic design tasks.
Funding
National Science Foundation (CAREER-1453101, 1717178, 1816041).
Acknowledgments
We thank Nanfang Yu for valuable suggestions.
Disclosures
The authors declare no conflicts of interest.
See Supplement 1 for supporting content.
References
[1]
Ahmed, A., Liscidini, M., and Gordon, R.
Design and analysis of high-index-contrast gratings using coupled
mode theory.
IEEE Photonics Journal 2, 6 (2010), 884–893.
[2]
Akel, H., and Webb, J.
Design sensitivities for scattering-matrix calculation with
tetrahedral edge elements.
IEEE transactions on magnetics 36, 4 (2000), 1043–1046.
[3]
Arbabi, A., Horie, Y., Bagheri, M., and Faraon, A.
Dielectric metasurfaces for complete control of phase and
polarization with subwavelength spatial resolution and high transmission.
Nature nanotechnology 10, 11 (2015), 937–943.
[4]
Arbabi, E., Arbabi, A., Kamali, S. M., Horie, Y., and Faraon, A.
Multiwavelength polarization-insensitive lenses based on dielectric
metasurfaces with meta-molecules.
Optica 3, 6 (2016), 628–633.
[5]
Arbabi, E., Kamali, S. M., Arbabi, A., and Faraon, A.
Full-stokes imaging polarimetry using dielectric metasurfaces.
ACS Photonics 5, 8 (2018), 3132–3140.
[6]
Backer, A. S.
Computational inverse design for cascaded systems of metasurface
optics.
Optics express 27, 21 (2019), 30308–30331.
[7]
Bellman, R.
Introduction to matrix analysis, vol. 19.
Siam, 1997.
[8]
Blau, Y., Eitan, M., Egorov, V., Boag, A., Hanein, Y., and Scheuer, J.
In situ real-time beam monitoring with dielectric meta-holograms.
Optics express 26, 22 (2018), 28469–28483.
[9]
Cao, Q., Lalanne, P., and Hugonin, J.-P.
Stable and efficient bloch-mode computational method for
one-dimensional grating waveguides.
JOSA A 19, 2 (2002), 335–338.
[10]
Chen, W. T., Yang, K.-Y., Wang, C.-M., Huang, Y.-W., Sun, G., Chiang,
I.-D., Liao, C. Y., Hsu, W.-L., Lin, H. T., Sun, S., Zhou, L., Liu, A. Q.,
and Tsai, D. P.
High-efficiency broadband meta-hologram with polarization-controlled
dual images.
Nano letters 14, 1 (2014), 225–230.
[11]
Colburn, S., Zhan, A., and Majumdar, A.
Varifocal zoom imaging with large area focal length adjustable
metalenses.
Optica 5, 7 (2018), 825–831.
[12]
Feynman, R. P.
An operator calculus having applications in quantum electrodynamics.
Physical Review 84, 1 (1951), 108.
[13]
Fitio, V., Yaremchuk, I., Bendzyak, A., and Bobitski, Y.
Application rcwa for studying plasmon resonance under diffraction by
metal gratings.
In 2017 IEEE First Ukraine Conference on Electrical and Computer
Engineering (UKRCON) (2017), IEEE, pp. 667–670.
[14]
Higham, N. J.
The scaling and squaring method for the matrix exponential revisited.
SIAM Journal on Matrix Analysis and Applications 26, 4 (2005),
1179–1193.
[15]
Joannopoulos, J., Meade, R., and Winn, J.
Photonic crystals.
Molding the flow of light (1995).
[16]
Kamali, S. M., Arbabi, E., Arbabi, A., Horie, Y., Faraji-Dana, M., and
Faraon, A.
Angle-multiplexed metasurfaces: encoding independent wavefronts in a
single metasurface under different illumination angles.
Physical Review X 7, 4 (2017), 041056.
[17]
Karplus, R., and Schwinger, J.
A note on saturation in microwave spectroscopy.
Physical Review 73, 9 (1948), 1020.
[18]
Kwiecien, P., Richter, I., and Čtyrokỳ, J.
Rcwa/arcwa-an efficient and diligent workhorse for
nanophotonic/nanoplasmonic simulations-can it work even better?
In 2015 17th International Conference on Transparent Optical
Networks (ICTON) (2015), IEEE, pp. 1–8.
[19]
Lalanne, P., and Silberstein, E.
Fourier-modal methods applied to waveguide computational problems.
Optics Letters 25, 15 (2000), 1092–1094.
[20]
Lalau-Keraly, C. M., Bhargava, S., Miller, O. D., and Yablonovitch, E.
Adjoint shape optimization applied to electromagnetic design.
Optics express 21, 18 (2013), 21693–21701.
[21]
Lee, S.-W., and Mittra, R.
Fourier transform of a polygonal shape function and its application
in electromagnetics.
IEEE Transactions on Antennas and Propagation 31, 1 (1983),
99–103.
[22]
Lin, Z., Liu, V., Pestourie, R., and Johnson, S. G.
Topology optimization of freeform large-area metasurfaces.
Optics express 27, 11 (2019), 15765–15775.
[23]
Mansouree, M., and Arbabi, A.
Metasurface design using level-set and gradient descent optimization
techniques.
In 2019 International Applied Computational Electromagnetics
Society Symposium (ACES) (2019), IEEE, pp. 1–2.
[24]
Moharam, M., and Gaylord, T.
Rigorous coupled-wave analysis of planar-grating diffraction.
JOSA 71, 7 (1981), 811–818.
[25]
Najfeld, I., and Havel, T. F.
Derivatives of the matrix exponential and their computation.
Advances in applied mathematics 16, 3 (1995), 321–375.
[26]
Overvig, A., Shrestha, S., Xiao, C., Zheng, C., and Yu, N.
Two-color and 3d phase-amplitude modulation holograms.
In CLEO: QELS_Fundamental Science (2018), Optical Society of
America, pp. FF1F–6.
[27]
Overvig, A. C., Shrestha, S., Malek, S. C., Lu, M., Stein, A., Zheng, C.,
and Yu, N.
Dielectric metasurfaces for complete and independent control of the
optical amplitude and phase.
Light: Science & Applications 8, 1 (2019), 1–12.
[28]
Pregla, R., and Pascher, W.
The method of lines.
Numerical techniques for microwave and millimeter wave passive
structures 1 (1989), 381–446.
[29]
Redheffer, R.
Difference equations and functional equations in transmission-line
theory.
Modern mathematics for the engineer 12 (1961), 282–337.
[30]
Rephaeli, E., Raman, A., and Fan, S.
Ultrabroadband photonic structures to achieve high-performance
daytime radiative cooling.
Nano letters 13, 4 (2013), 1457–1461.
[31]
Ruder, S.
An overview of gradient descent optimization algorithms.
arXiv preprint arXiv:1609.04747 (2016).
[32]
Rumpf, R. C.
Improved formulation of scattering matrices for semi-analytical
methods that is consistent with convention.
Progress In Electromagnetics Research 35 (2011), 241–261.
[33]
Sell, D., Yang, J., Doshay, S., Yang, R., and Fan, J. A.
Large-angle, multifunctional metagratings based on freeform multimode
geometries.
Nano letters 17, 6 (2017), 3752–3757.
[34]
Shen, Y., Rinnerbauer, V., Wang, I., Stelmakh, V., Joannopoulos, J. D.,
and Soljacic, M.
Structural colors from fano resonances.
Acs Photonics 2, 1 (2015), 27–32.
[35]
Snider, R.
Perturbation variation methods for a quantum boltzmann equation.
Journal of Mathematical Physics 5, 11 (1964), 1580–1587.
[36]
Solutions, F.
Lumerical solutions inc.
Vancouver, Canada (2003).
[37]
Stanek, J. C.
A characterization of starshaped sets.
Canadian Journal of Mathematics 29, 4 (1977), 673–680.
[38]
van der Aa, N., Ter Morsche, H., and Mattheij, R.
Computation of eigenvalue and eigenvector derivatives for a general
complex-valued eigensystem.
The electronic journal of linear algebra 16 (2007).
[39]
Veronis, G., Dutton, R. W., and Fan, S.
Method for sensitivity analysis of photonic crystal devices.
Optics letters 29, 19 (2004), 2288–2290.
[40]
Wang, K. X., Yu, Z., Liu, V., Cui, Y., and Fan, S.
Absorption enhancement in ultrathin crystalline silicon solar cells
with antireflection and light-trapping nanocone gratings.
Nano letters 12, 3 (2012), 1616–1619.
[41]
Wang, Z., and Yu, Z.
Spectral analysis based on compressive sensing in nanophotonic
structures.
Optics Express 22, 21 (2014), 25608–25614.
[42]
Yu, N., Aieta, F., Genevet, P., Kats, M. A., Gaburro, Z., and Capasso, F.
A broadband, background-free quarter-wave plate based on plasmonic
metasurfaces.
Nano letters 12, 12 (2012), 6328–6333.
[43]
Zhao, R., Sain, B., Wei, Q., Tang, C., Li, X., Weiss, T., Huang, L., Wang,
Y., and Zentgraf, T.
Multichannel vectorial holographic display and encryption.
Light: Science & Applications 7, 1 (2018), 1–9.
[44]
Zhou, Y., Kravchenko, I. I., Wang, H., Zheng, H., Gu, G., and Valentine,
J.
Multifunctional metaoptics based on bilayer metasurfaces.
Light: Science & Applications 8, 1 (2019), 1–9.
[45]
Zhu, Z., Dave, U. D., Lipson, M., and Zheng, C.
Inverse geometric design of fabrication-robust nanophotonic
waveguides.
In CLEO: Science & Innovations (2020), Optical Society of
America.
See pages - of supplement.pdf |
$\alpha_{s}$ and the $\tau$ hadronic width
Matthias Jamin
Institució Catalana de Recerca i Estudis
Avançats (ICREA), IFAE, Theoretical Physics Group, UAB,
E-08193 Bellaterra, Barcelona, Spain
Abstract
Different choices exist for the renormalisation group resummation in the
determination of $\alpha_{s}$ from hadronic $\tau$ decays: namely fixed-order (FOPT)
and contour-improved perturbation theory (CIPT). The two approaches lead to
systematic differences in the resulting $\alpha_{s}$. On the basis of a model for
higher-order terms in the perturbative series, which incorporates well-known
structure from renormalons, it is found that while CIPT is unable to account
for the fully resummed series, FOPT smoothly approaches the Borel sum.
Employing the model to determine $\alpha_{s}$ yields $\alpha_{s}(M_{\tau})=0.316\pm 0.006$,
which after evolution leads to $\alpha_{s}(M_{Z})=0.1180\pm 0.0008$.
1 INTRODUCTION
Due to its particular mass of $M_{\tau}\approx 1.8\,\mbox{\rm GeV}$, the $\tau$ lepton
constitutes an excellent system to study QCD at low energies, as in about 65%
of all cases it decays into hadrons, while the QCD description remains
largely perturbative. In the seminal work [1], the framework for
a precise determination of the QCD coupling $\alpha_{s}$ from the total $\tau$
hadronic width
$$R_{\tau}\,\equiv\,\frac{\Gamma[\tau^{-}\to{\rm hadrons}\,\nu_{\tau}(\gamma)]}{%
\Gamma[\tau^{-}\to e^{-}\overline{\nu}_{e}\nu_{\tau}(\gamma)]}\,=\,3.640(10)\,,$$
(1)
was developed, while afterwards invariant mass distributions were incorporated
into the analysis as well [2, 3]. The most recent study of the
ALEPH spectral function data on the basis of the final full LEP data set yielded
$\alpha_{s}(M_{\tau})=0.344\pm 0.005_{\rm exp}\pm 0.007_{\rm th}$ [4], which
after evolution to the $Z$-boson mass scale results in
$\alpha_{s}(M_{Z})=0.1212(11)$. The dominant quantifiable theory uncertainty resides
in as yet uncalculated higher-order QCD corrections and improvements of the
perturbative series through renormalisation group (RG) methods.
Most suitable for the $\alpha_{s}$ determination is the $\tau$ decay rate into light
$u$ and $d$ quarks $R_{\tau,V/A}$ via a vector or axialvector current, since
in this case power corrections are especially suppressed. Theoretically,
$R_{\tau,V/A}$ takes the form [1]
$$\displaystyle R_{\tau,V/A}$$
$$\displaystyle\!\!=$$
$$\displaystyle\frac{N_{c}}{2}\,S_{\rm EW}\,|V_{ud}|^{2}\,\Big{[}\,1+\delta^{(0)}$$
(2)
$$\displaystyle+\,\delta_{\rm EW}^{\prime}+\sum\limits_{D\geq 2}\delta_{ud,V/A}^%
{(D)}\,\Big{]}\,,$$
where $S_{\rm EW}=1.0198(6)$ [5] and $\delta_{\rm EW}^{\prime}=0.0010(10)$
[6] are electroweak corrections, $\delta^{(0)}$ comprises the
perturbative QCD correction, and the $\delta_{ud,V/A}^{(D)}$ denote quark
mass and higher $D$-dimensional operator corrections which arise in the
framework of the operator product expansion (OPE).
When computing the total $\tau$ hadronic width a phase-space integral over the
physical spectrum has to be calculated, which by analyticity can be related to
a contour integral over QCD correlators in the complex $s$-plane, where $s$ is
the invariant mass of the final state hadronic system. As we also intend to
RG improve the perturbative series, the questions arises if the RG resummation
should be performed before or after evaluating the contour integral? If the
“true” all-order result were available, both treatments should agree, but to
any finite order in perturbation theory, significant differences may arise.
The approach of first RG-improving the correlators and then performing the
contour integral was introduced in [7, 8] and termed contour-improved
perturbation theory (CIPT), as unarguably large running effects of
$\alpha_{s}(\sqrt{s})$ along the contour are resummed. However, it is known that the
QCD series is divergent, being asymptotic at best. Hence, also the explicit
expansion coefficients of the perturbative series are bound to become large.
If cancellations between explicit coefficients and running effects occur, just
resumming the running effects may not lead to a good approximation, but it
could be better to perform a consistent expansion in powers of $\alpha_{s}(M_{\tau})$,
being called fixed-order perturbation theory (FOPT).
While it can be demonstrated that, as expected, CIPT is a good approximation,
and CIPT as well as FOPT are compatible, when the running effects dominate
[9], in the large-$\beta_{0}$ approximation, in which $\delta^{(0)}$ is
exactly calculable, FOPT gives a better approximation to the true result and
CIPT is not seen to be compatible with it [10, 11]. Thus it is not a
priori clear which behaviour prevails in real QCD. It should be obvious that
this is a question about perturbative orders beyond the presently known
${\cal O}(\alpha_{s}^{4})$ term [12].
To further investigate the difference between CIPT and FOPT, the contribution
of higher perturbative orders has to be modelled. This then allows to
investigate the following questions:
i)
Are FO and CI perturbation theory found to be compatible, once terms
beyond the currently known perturbative coefficients of the series for
$\delta^{(0)}$ are included?
ii)
How do FO and CI perturbation theory at a particular order compare
to the true result for $\delta^{(0)}$, and is the closest approach
to the true result related to the minimal terms in the respective series?
iii)
And finally, which of the two methods, FOPT or CIPT provides the
closer approach to the true value at order ${\cal O}(\alpha_{s}^{4})$, and in
general?
The working assumption in all this is that the “true” result is approximated
with reasonable accuracy by the Borel sum of the model series under
consideration, since the power corrections to $R_{\tau}$ are known to be small.
The model employed below will be constructed such as to incorporate the general
structure of the Adler function in the Borel plane, dictated by the OPE and
the RG.
2 PERTURBATIVE CORRECTION $\delta^{(0)}$
In the following, we shall only be concerned with the purely perturbative
correction $\delta^{(0)}$ which gives the dominant contribution to
$R_{\tau,V/A}$. In FOPT it takes the general form
$$\delta^{(0)}_{\rm FO}\,=\,\sum\limits_{n=1}^{\infty}a(M_{\tau}^{2})^{n}\sum%
\limits_{k=1}^{n}k\,c_{n,k}\,J_{k-1}\,,$$
(3)
where $a(\mu^{2})\equiv a_{\mu}\equiv\alpha_{s}(\mu)/\pi$, and $c_{n,k}$ are the
coefficients which appear in the perturbative expansion of the vector
correlation function,
$$\Pi_{V}(s)\,=\,-\,\frac{N_{c}}{12\pi^{2}}\sum\limits_{n=0}^{\infty}a_{\mu}^{n}%
\sum\limits_{k=0}^{n+1}c_{n,k}\ln^{k}\!\left(\frac{-s}{\mu^{2}}\right).$$
(4)
At each perturbative order, the coefficients $c_{n,1}$ can be considered
independent, while all other $c_{n,k}$ with $k\geq 2$ are calculable from the
RG equation. Further details can for example be found in ref. [13].
Finally, the $J_{l}$ are contour integrals which are defined by
$$J_{l}\,\equiv\,\frac{1}{2\pi i}\!\!\oint\limits_{|x|=1}\!\!\frac{dx}{x}\,(1-x)%
^{3}\,(1+x)\ln^{l}(-x)\,.$$
(5)
The first three which are required up to ${\cal O}(\alpha_{s}^{3})$ take the numerical values
$$J_{0}\,=\,1\,,\quad J_{1}\,=\,-\,\mbox{$\frac{19}{12}$}\,,\quad J_{2}\,=\,%
\mbox{$\frac{265}{72}$}-\mbox{$\frac{1}{3}$}\,\pi^{2}\,.$$
(6)
At order $\alpha_{s}^{n}$ FOPT contains unsummed logarithms of order
$\ln^{l}(-x)\sim\pi^{l}$ with $l<n$ related to the contour integrals $J_{l}$.
CIPT sums these logarithms, which yields
$$\delta^{(0)}_{\rm CI}\,=\,\sum\limits_{n=1}^{\infty}c_{n,1}\,J_{n}^{a}(M_{\tau%
}^{2})$$
(7)
in terms of the contour integrals $J_{n}^{a}(M_{\tau}^{2})$ over the running coupling,
defined as:
$$J_{n}^{a}(M_{\tau}^{2})\,\equiv\,\frac{1}{2\pi i}\!\!\oint\limits_{|x|=1}\!\!%
\frac{dx}{x}\,(1-x)^{3}\,(1+x)\,a^{n}(-M_{\tau}^{2}x)\,.$$
(8)
In contrast to FOPT, for CIPT each order $n$ just depends on the corresponding
coefficient $c_{n,1}$. Thus, all contributions proportional to the coefficient
$c_{n,1}$ which in FOPT appear at all perturbative orders equal or greater than
$n$ are resummed into a single term.
Numerically, the two approaches lead to significant differences. Using
$\alpha_{s}(M_{\tau})=0.34$ in eqs. (3) and (7), one finds
$$\displaystyle\delta^{(0)}_{\rm FO}$$
$$\displaystyle\!\!=$$
$$\displaystyle 0.2200\;(0.2288)\,,$$
(9)
$$\displaystyle\vbox{\vskip 11.381102pt}\delta^{(0)}_{\rm CI}$$
$$\displaystyle\!\!=$$
$$\displaystyle 0.1984\;(0.2021)\,,$$
(10)
where the first number in both cases employs the known coefficients up to
${\cal O}(\alpha_{s}^{4})$ [12] and the numbers in brackets include an estimate of
the ${\cal O}(\alpha_{s}^{5})$ term with $c_{5,1}\approx 283$ [13]. Inspecting the
individual contributions from each order, up to ${\cal O}(\alpha_{s}^{5})$ the CIPT series
appears to be better convergent. However, around the seventh order, the
contour integrals $J_{n}^{a}(M_{\tau}^{2})$ change sign and thus at this order the
contributions are bound to become small. Therefore, the faster approach to the
minimal term does not necessarily imply that CIPT gives the closer approach to
the true result for the resummed series.
3 A PHYSICAL MODEL
To clarify whether FOPT or CIPT results in a better approximation to
$\delta^{(0)}$, one needs to construct a physically motivated model for its
series. The corresponding model will be based on the Borel transform of the
Adler function $D_{V}(s)$:
$$D_{V}(s)\,\equiv\,-\,s\,\frac{d}{ds}\,\Pi_{V}(s)\,\equiv\,\frac{N_{c}}{12\pi^{%
2}}\,\big{[}1+\widehat{D}(s)\big{]}\,.$$
(11)
In the following discussion it is slightly more convenient to utilise the
related function $\widehat{D}(s)$. Its Borel transform $B[\widehat{D}](t)$ is defined by
the relation
$$\widehat{D}(\alpha)\,\equiv\,\int\limits_{0}^{\infty}dt\,{\rm e}^{-t/\alpha}\,%
B[\widehat{D}](t)\,.$$
(12)
The integral $\widehat{D}(\alpha)$, if it exists, gives the Borel sum of the original
divergent series. It was found that the Borel-transformed Adler function
$B[\widehat{D}](t)$ obtains infrared (IR) and ultraviolet (UV) renormalon poles at
positive and negative integer values of the variable $u\equiv 9t/(4\pi)$,
respectively [14, 15]. (With the exception of $u=1$.)
Apart from very low orders, where a dominance of renormalon poles close to
$u=0$ has not yet set in, intermediate orders should be dominated by the
leading IR renormalon poles, while the leading UV renormalon, being closest
to $u=0$, dictates the large-order behaviour of the perturbative expansion.
Assuming that only the first two orders are not yet dominated by the lowest
IR renormalons, one is led to the ansatz
$$\displaystyle B[\widehat{D}](u)$$
$$\displaystyle\!\!=$$
$$\displaystyle B[\widehat{D}_{1}^{\rm UV}](u)+B[\widehat{D}_{2}^{\rm IR}](u)+$$
(13)
$$\displaystyle B[\widehat{D}_{3}^{\rm IR}](u)+d_{0}^{\rm PO}+d_{1}^{\rm PO}u\,,$$
which includes one UV renormalon at $u=-1$, the two leading IR renormalons at
$u=2$ and $u=3$, as well as polynomial terms for the two lowest perturbative
orders. Explicit expressions for the UV and IR renormalon pole terms
$B[\widehat{D}_{p}^{\rm UV}](u)$ and $B[\widehat{D}_{p}^{\rm IR}](u)$ can be found in section 5
of ref. [13].
Apart from the residues $d_{p}^{\rm UV}$ and $d_{p}^{\rm IR}$, the full structure
of the renormalon pole terms is dictated by the OPE and the RG. Therefore, the
model (13) depends on five parameters, the three residua $d_{1}^{\rm UV}$,
$d_{2}^{\rm IR}$ and $d_{3}^{\rm IR}$, as well as the two polynomial parameters
$d_{0}^{\rm PO}$ and $d_{1}^{\rm PO}$. These parameters can be fixed by matching
to the perturbative expansion of $\widehat{D}(s)$ up to ${\cal O}(\alpha_{s}^{5})$. Thereby one
also makes use of the estimate for $c_{5,1}$. The parameters of the model
(13) are then found to be:
$$d_{1}^{\rm UV}=\,-\,1.56\cdot 10^{-2}\,,\;d_{2}^{\rm IR}=\,3.16\,,\;d_{3}^{\rm
IR%
}=\,-\,13.5\,,\\
$$
$$d_{0}^{\rm PO}=\,0.781\,,\;d_{1}^{\rm PO}=\,7.66\cdot 10^{-3}\,.$$
(14)
The fact that the parameter $d_{1}^{\rm PO}$ turns out to be small implies that
the coefficient $c_{2,1}$ is already reasonably well described by the
renormalon pole contribution, although it was not used to fix the residua.
Therefore, one could set $d_{1}^{\rm PO}=0$ and actually work with a model which
only has four parameters. The predicted value $c_{5,1}=280$ in this model turns
out very close to the estimate, which can be viewed as one test of the stability
of the model.
A graphical account of the model (13) for the (reduced) Adler function
$\widehat{D}(M_{\tau}^{2})$ is displayed in figure 1. The full circles
denote the partial sums of the perturbative series up to order $n$. The minimal
term of the series at the 5th order is marked by a grey diamond. The
perturbative results are compared with the Borel sum of the model (straight
line).111Also shown as the shaded band is an estimate of the uncertainty
inferred from the complex ambiguity which arises while defining the Borel
integral over the IR renormalon poles. For details see appendix A of
ref. [13]. Generally, figure 1 shows that the model is
well-behaved: the series goes through a number of small terms such that the
truncated series nicely agrees with its Borel sum, before the sign-alternating
asymptotic behaviour takes over around $n=10$.
The implications of the model (13) for $\delta^{(0)}$ in FOPT and CIPT is
graphically represented in figure 2. The full circles denote the
result for $\delta^{(0)}_{\rm FO}$ and the grey circles the one for
$\delta^{(0)}_{\rm CI}$, as a function of the order $n$ up to which the
perturbative series has been summed. The straight line corresponds to the
principal value Borel sum of the series, $\delta^{(0)}_{\rm BS}=0.2371$, and
the shaded band provides an error estimate based on the imaginary part divided
by $\pi$. The order at which the series have their smallest terms is indicated
by the grey diamonds. As is obvious from figure 2, like for the
Adler function itself, FOPT displays the behaviour expected from an asymptotic
series: the terms decrease up to a certain order around which the closest
approach to the resummed result is found, and for even higher orders, the
divergent large-order behaviour of the series sets in. For CIPT, on the other
hand, the asymptotic behaviour sets in earlier, and the series is never able
to come close to the Borel sum.
The finding that in the model (13) CIPT misses the full Borel sum can be
traced back to the fact that in CIPT the running effects along the complex
contour are resummed to all orders, while explicit contributions of the
$c_{n,1}$ at a certain order are dropped. However, being an asymptotic series,
also the Adler function coefficients $c_{n,1}$ become large, and cancellations
between the explicit contributions and the running effects take place. As was
shown in section 4 of [13], in the large-$\beta_{0}$ approximation, for
$R_{\tau}$ the leading IR renormalon cancels completely, and also the large-order
divergence of the series is softened. Even though in real QCD the leading IR
does not anymore cancel completely, for $R_{\tau}$ it is still suppressed by a
factor $1/n^{2}$, and furthermore a sign-alternating UV renormalon component does
not yet show up in the known coefficients. Thus, the cancellations between
running effects and explicit coefficients are also expected to prevail in full
QCD.
The deficiency of CIPT to approach the Borel sum of the series, which leads to
the marked differences of $\delta^{(0)}_{\rm FO}$ and $\delta^{(0)}_{\rm CI}$,
can also be observed when the Adler function is inspected along the complex
circle. This is discussed in detail in appendix B of ref. [13]. While
FOPT converges to the Borel sum on the full circle (though rather badly close
to the Minkowskian axis), in some regions of the circle CIPT largely differs
from the resummed result, again due to the missed cancellations.
As the behaviour of CIPT versus FOPT hinges on the contribution of the leading
IR renormalon at $u=2$, in principal also models can be constructed for which
CIPT provides a good account of the Borel sum. These would generally be models
where $d_{2}^{\rm IR}$ is much smaller than the value quoted in eq. (14).
While such models can at present not be excluded, the pattern of the individual
contributions appears more unnatural than in the model (13): the known
$c_{n,1}$ can only be reproduced when one allows for large cancellations
between the individual terms. Thus, the behaviour generally expected from the
presence of renormalon poles, namely dominance of leading IR poles at
intermediate orders, would be lost.
4 DETERMINATION OF $\alpha_{s}$
The starting point for a determination of $\alpha_{s}$ from hadronic $\tau$ decays
is eq. (2) for the decay rate of the $\tau$ lepton into light $u$
and $d$ quarks. The analysis will be based on FOPT together with the ansatz
(13) for higher-order terms discussed in the last section. Due to the
observation that CIPT is not able to approach the resummed series, it will
not be employed below.
The first step of the $\alpha_{s}$ analysis consists in estimating the values of the
power corrections $\delta_{ud,V+A}^{(D)}$, which arise from higher-dimensional
operators in the framework of the OPE. Given these estimates and experimental
data, a phenomenological value of $\delta^{(0)}$ can be calculated using
eq. (2). This allows to determine the value of $\alpha_{s}(M_{\tau})$ by
requiring that the theoretical value $\delta^{(0)}_{\rm FO}$ matches the
phenomenological value $\delta^{(0)}_{\rm phen}$. Errors are estimated by
varying all parameters within their uncertainties.
In view of the smallness of the light quark masses $m_{u}$ and $m_{d}$, as well as
the suppression of the dimension-4 contribution in $R_{\tau}$, the dominant power
correction arises from the six dimensional 4-quark condensates. As the number
of contributing operators is too large to treat the 4-quark condensates
individually, conventionally the so-called vacuum-saturation approximation
(VSA) [16] is employed. Then the corresponding contribution takes the
form
$$\displaystyle\delta_{\langle\bar{q}q\bar{q}q\rangle,V+A}^{(6)}$$
$$\displaystyle\!\!=$$
$$\displaystyle-\,\frac{512}{27}\,\pi^{3}\alpha_{s}\,\frac{\rho\langle\bar{q}q%
\rangle^{2}}{M_{\tau}^{6}}$$
(15)
$$\displaystyle\!\!=$$
$$\displaystyle(-\,4.8\pm 2.9)\cdot 10^{-3}\,,$$
where for the numerical estimate the required quark condensate is taken from
the GMOR relation [17, 18], and $\rho=2\pm 1$ was assumed to take
into account violations of the VSA. This choice includes most estimates of the
four-quark condensate present in the literature.
To complete the estimate of power corrections to $R_{\tau}$, the longitudinal
contribution which arises from the pseudoscalar correlator still has to be
included.222The scalar correlator, being proportional to $(m_{u}-m_{d})^{2}$,
is completely negligible. Because the perturbative series for this correlator
does not converge very well, the approach of refs. [19, 20] will
be followed. The main idea is to replace the QCD expression for the pseudoscalar
correlator by a phenomenological representation. The dominant contribution to
the pseudoscalar spectral function stems from the well-known pion pole, giving
$$\delta_{ud,S+P}^{\pi}\,=\,-\,16\pi^{2}\,\frac{f_{\pi}^{2}M_{\pi}^{2}}{M_{\tau}%
^{4}}\Biggl{(}1-\frac{M_{\pi}^{2}}{M_{\tau}^{2}}\Biggr{)}^{\!2}\,,$$
(16)
plus small corrections from higher-excited pionic resonances. Repeating the
analysis of ref. [19] and updating the input parameters, one finds
$$\delta_{ud,S+P}\,=\,(-\,2.64\pm 0.05)\cdot 10^{-3}\,.$$
(17)
Collecting all contributions, and adding the errors in quadrature, one arrives
at the total estimate of all power corrections:
$$\delta_{\rm PC}\,=\,(-\,7.1\pm 3.1)\cdot 10^{-3}\,.$$
(18)
The value (18) is consistent with the most recent fit to the $\tau$
spectral functions performed in ref. [4].
As a matter of principle, the OPE of correlation functions in the complex
$s$-plane could be inflicted with so-called “duality violations” [21].
These arise from the contour integral close to the physical region which even
though suppressed in $R_{\tau}$ could be sizeable [22]. Nevertheless,
before a possible additional duality violating contribution can be extracted
consistently from a combined fit to spectral moments, it shall be omitted.
Employing the value $R_{\tau,V+A}=3.479\pm 0.011$, which results from
eq. (1) in conjunction with $R_{\tau,S}=0.1615\pm 0.0040$
[4], as well as $|V_{ud}|=0.97418\pm 0.00026$ [23], from
eqs. (2) and (18) the phenomenological value for $\delta^{(0)}$
can be derived:
$$\delta^{(0)}_{\rm phen}\,=\,0.2042\pm 0.0050\,.$$
(19)
The dominant experimental uncertainty in (19) is due to $R_{\tau,V+A}$
and the theoretical one to the dimension-6 condensate. The final step in the
extraction of $\alpha_{s}(M_{\tau})$ now consists in finding the values of $\alpha_{s}$ for
which $\delta^{(0)}_{\rm phen}$ matches the theoretical prediction, which
yields [13]
$$\alpha_{s}(M_{\tau})\,=\,0.3156\pm 0.0030_{\rm exp}\pm 0.0051_{\rm th}\,.$$
(20)
Evolving this result to the $Z$-boson mass scale finally leads to
$$\alpha_{s}(M_{Z})\,=\,0.11795\pm 0.00076\,,$$
(21)
in perfect agreement with the world average [24].
Acknowledgements
The author would like to thank Martin Beneke for a most enjoyable collaboration.
This work has been supported in parts by EU Contract No. MRTN-CT-2006-035482
(FLAVIAnet), by CICYT-FEDER-FPA2005-02211, and by the Spanish Consolider-Ingenio
2010 Programme CPAN (CSD2007-00042).
References
[1]
E. Braaten, S. Narison, and A. Pich, Nucl. Phys. B373 (1992) 581.
[2]
ALEPH Collaboration, S. Schael et al., Phys. Rept. 421
(2005) 191,
[hep-ex/0506072].
[3]
M. Davier, A. Höcker, and Z. Zhang, Rev. Mod. Phys. 78 (2006)
1043, [hep-ph/0507078].
[4]
M. Davier, S. Descotes-Genon, A. Höcker, B. Malaescu, and Z. Zhang,
Eur. Phys. J. C56 (2008) 305,
arXiv:0803.0979 [hep-ph].
[5]
W. Marciano and A. Sirlin, Phys. Rev. Lett. 61 (1988) 1815.
[6]
E. Braaten and C. S. Li, Phys. Rev. D42 (1990) 3888.
[7]
A. A. Pivovarov, Z. Phys. C53 (1992) 461,
[hep-ph/0302003].
[8]
F. Le Diberder and A. Pich, Phys. Lett. B286 (1992) 147.
[9]
M. Jamin, JHEP 09 (2005) 058,
[hep-ph/0509001].
[10]
P. Ball, M. Beneke, and V. M. Braun, Nucl. Phys. B452 (1995)
563, [hep-ph/9502300].
[11]
M. Beneke, Phys. Rept. 317 (1999) 1,
[hep-ph/9807443].
[12]
P. A. Baikov, K. G. Chetyrkin, and J. H. Kühn, Phys. Rev. Lett.
101 (2008) 012002, arXiv:0801.1821 [hep-ph].
[13]
M. Beneke and M. Jamin, JHEP 09 (2008) 044,
arXiv:0801.1821 [hep-ph].
[14]
M. Beneke, Nucl. Phys. B405 (1993) 424.
[15]
D. J. Broadhurst, Z. Phys. C58 (1993) 339.
[16]
M. A. Shifman, A. I. Vainshtein, and V. I. Zakharov, Nucl. Phys.
B147 (1979) 385, 448.
[17]
M. Gell-Mann, R. J. Oakes, and B. Renner, Phys. Rev. 175 (1968)
2195.
[18]
M. Jamin, Phys. Lett. B538 (2002) 71,
[hep-ph/0201174].
[19]
E. Gámiz, M. Jamin, A. Pich, J. Prades, and F. Schwab,
J. High Energy Phys. 01 (2003) 060,
[hep-ph/0212230].
[20]
E. Gámiz, M. Jamin, A. Pich, J. Prades, and F. Schwab,
Phys. Rev. Lett. 94 (2005) 011803,
[hep-ph/0408044].
[21]
M. A. Shifman,
hep-ph/0009131. Boris
Ioffe Festschrift, At the Frontier of Particle Physics, Handbook of QCD,
M.A. Shifman (ed.), World Scientific, Singapore, 2001.
[22]
O. Cata, M. Golterman, and S. Peris, Phys. Rev. D77 (2008) 093006,
arXiv:0803.0246.
[23]
I. S. Towner and J. C. Hardy,
Phys. Rev. C77 (2008) 025501,
arXiv:0710.3181.
[24]
W.-M. Yao et al., Review of Particle Physics, Journal of
Physics G 33 (2006) 1. |
$I$ in generalized supergravity
T. Araujo
Asia Pacific Center for Theoretical Physics, Postech, Pohang 37673, Korea
E. Ó Colgáin
Asia Pacific Center for Theoretical Physics, Postech, Pohang 37673, Korea
J. Sakamoto
Department of Physics, Kyoto University, Kitashirakawa, Kyoto 606-8502, Japan
M. M. Sheikh-Jabbari
School of Physics, Institute for Research in Fundamental Sciences (IPM), P.O.Box 19395-5531, Tehran, Iran
K. Yoshida
Department of Physics, Kyoto University, Kitashirakawa, Kyoto 606-8502, Japan
Abstract
We showed in previous work that for Yang-Baxter (YB) deformations of AdS${}_{5}\times$S${}^{5}$, the open string metric and coupling, and as a result the closed string density $e^{-2\Phi}\sqrt{g}$, remains undeformed. In this work we identify the Page forms as the open string counterpart for RR fields and demonstrate case by case that the non-zero Page forms remain invariant under YB deformations. We show that these results remain true, not only for homogeneous YB deformations, but also for modified YB deformations associated with $\eta$-deformations. We give a physical meaning to the Killing vector $I$ of generalized supergravity and show for all YB deformations: 1) $I$ appears as a current for center of mass motion on the worldvolume of a D-branes probing the background, 2) $I$ is equal to the divergence of the noncommutativity parameter, 3) $I$ exhibits “holographic” behavior, where the radial component of $I$ vanishes at the AdS boundary, and 4) in pure spinor formalism $I$ is related to a certain state in the BRST cohomology.
††preprint: APCTP Pre2017 - 015††preprint: KUNS-2696††preprint: IPM/P-2017/028
I Introduction
Over recent years we have witnessed an explosion in interest in integrability preserving deformations of AdS${}_{5}\times$S${}^{5}$ superstring, collectively dubbed Yang-Baxter (YB) deformations Klimcik:2002zj ; Klimcik:2008eq ; Delduc:2013fga ; Matsumoto:2015jja , which includes T-duality shift T-duality (TsT) transformations Lunin:2005jy as a special case. As a by-product of this breakthrough, a large family of supergravity solutions, or geometries, of varying complexity can be defined. In Araujo:2017jkb ; Araujo:2017jap , it was observed that within the ensuing geometric madness, there was a simplifying principle: in open string frame the metric and string coupling were undeformed and all information about the deformation resided in a noncommutativity (NC) parameter, or antisymmetric bivector $\Theta$, thus unlocking a key to the AdS/CFT interpretation (see also vanTongeren:2015uha ; vanTongeren:2016eeb ).
Concurrently, significant progress has been made in the Green-Schwarz (GS) formulation of superstring theory. From old work Grisaru:1985fv , the on-shell condition of type II supergravities ensures the invariance of the GS string action under kappa-symmetry (which is a local fermionic symmetry and necessary to get the right number of physical spacetime fermions). On the other hand, the reverse statement had not been clarified even after more than three decades. In groundbreaking work by Tseytlin and Wulff Wulff:2016tju , it has been shown that the kappa-symmetry of the GS string action defined on an arbitrary background leads to generalized type II supergravities including an extra Killing vector field $I$, rather than the usual, well-known supergravities.
The derivation of generalized supergravity is robust and the result is so striking that the fundamental aspects of superstring theory must be carefully reconsidered. Hence, the study of generalized type II supergravities is a fundamental issue concerned with the formulation of superstring theory. Here, we confine our attention to generalized type IIB supergravity. We reproduce the equations of motion (EOM) of the NSNS sector in the appendix.
In generalized type IIB supergravity, there are a lot of open questions, including, what is the physical interpretation of $I$? In order to tackle these issues, it is useful to employ the YB deformation because this can be regarded as a solution generation technique in generalized supergravity. Although we appreciate the profound derivation of the generalized type IIB supergravity from the kappa-invariance nowadays, it is remarkable that its discovery was a by-product of an enriched understanding of integrable deformations of the AdS${}_{5}\times$S${}^{5}$ superstring.
The integrable deformation technique that concerns us was invented by Klimcik for the principal chiral models based on the modified classical Yang-Baxter equation (mCYBE) Klimcik:2002zj ; Klimcik:2008eq . According to this procedure, an integrable deformation is specified by taking a classical $r$-matrix satisfying the mCYBE and hence this procedure is often called the Yang-Baxter deformation. Klimcik’s work was subsequently generalized to the symmetric coset case Delduc:2013fga and the homogeneous CYBE Matsumoto:2015jja . Since the classical action of the AdS${}_{5}\times$S${}^{5}$ superstring Metsaev:1998it has a construction based on the supercoset
$$\displaystyle\frac{PSU(2,2|4)}{SO(1,4)\times SO(5)},$$
(1)
the $\mathbf{Z}_{4}$-grading of the superalgebra leads to classical integrability Bena:2003wd .
Thus, the YB deformation is also applicable to the AdS${}_{5}\times$S${}^{5}$ superstring theory Delduc:2013qra ; Kawaguchi:2014qwa . The pioneering work of Delduc, Magro and Vicedo Delduc:2013qra , based on the mCYBE, is the standard $q$-deformation of $\mathfrak{su}(2,2|4)$ with the classical $r$-matrix of Drinfeld-Jimbo type.
This is often called the $\eta$-deformation. After performing supercoset construction, the full background, including the RR sector, was worked out Arutyunov:2015qva , which led to the surprise that the resulting background was not a solution of the usual type IIB supergravity, but a generalized variant Arutyunov:2015mqj .
To date, a considerable amount of community effort has focused on the largely technical exercise of mapping out the dictionary between $r$-matrices and geometries, or YB deformations, including connections to (non-Abelian) T-duality Matsumoto:2014nra ; Matsumoto:2014gwa ; Matsumoto:2014ubv ; Matsumoto:2015uja ; Kawaguchi:2014fca ; vanTongeren:2015soa ; Osten:2016dvf ; Hoare:2016wca ; Kyono2016jqy ; Hoare:2016ibq ; Hoare:2016hwh ; Orlando:2016qqu ; Borsato:2016ose . Following the latter thread further, the relation between Double Field Theory (DFT) Siegel:1993th ; Siegel:1993xq ; Hull:2009mi was elucidated in Sakatani:2016fvh ; Baguet:2016prz ; Sakamoto:2017wor ; Sakamoto:2017cpu and some implications for the AdS/CFT dual gauge theory, including whether planar integrability is preserved Beisert:2005if ; Guica:2017mtd , have been teased out in Araujo:2017jkb ; Araujo:2017jap ; vanTongeren:2015uha ; vanTongeren:2016eeb ; Roychowdhury:2017oed ; Roychowdhury:2017vdo . For studies of integrability, including classical solutions on the $\eta$-deformed background, see AM-NR ; Kame-coord ; magnon ; Bozhilov:2016owo ; Arutyunov:2016ysi ; Ahn:2016egk ; Hernandez:2017raj ; Klabbers:2017vtw . In this letter, we return to the physical interpretation of the Killing vector $I$.
We recall for homogeneous YB deformations that the Killing vector $I$ is related to the divergence of the noncommutativity (NC) parameter, or bivector $\Theta$ Araujo:2017jkb ; Araujo:2017jap ,
$$I^{N}=\nabla_{M}\Theta^{MN},$$
(2)
where $\nabla$ is the covariant derivative with respect to the open string metric. In this letter, we extend this result to the known $\eta$-deformations Delduc:2013qra ; Arutyunov:2013ega ; Delduc:2014kha ; Arutyunov:2015qva and provide a derivation in terms of D3-branes probing the background, where $I$ may be interpreted as a current for the center of mass motion on the D3-brane worldvolume. The conclusion that (2) holds for all YB deformations, including $\eta$-deformations, hinges on the observation that Page forms Page , special combinations of RR fluxes and NSNS two-form that appear in the D-brane action, facilitate a simple rewriting of the equations of generalized supergravity and that the Page five-form associated to the D3-brane is invariant, which we have confirmed on a case by case basis. More generally, we conjecture that the non-zero Page forms are invariants of YB deformations. In addition, through complementary comments on the role of $I$ as a certain state in the BRST cohomology in pure spinor formalism, our work provides $I$ with a more physical flavor.
In the remainder of the letter, we study $\eta$-deformations. Concretely, we demonstrate that the appropriate inclusion of pure gauge contributions to the NSNS two-form indeed results in an open string metric that is undeformed. For the explicit Arutyunov, Borsato, Frolov (ABF) solution Arutyunov:2013ega ; Arutyunov:2015qva , we check that the well-known T-duality invariant $e^{-2\Phi}\sqrt{g}$ is also an invariant of the deformation. We use this fact and (2) to determine the NSNS sector of $\eta$-deformations of AdS${}_{5}\times$S${}^{5}$ Delduc:2014kha 111In Delduc:2014kha only the metric and NSNS two-form are computed.
The remainder of the solution, including the dilaton and RR sector can be determined by performing the supercoset construction directly, as explained in Arutyunov:2015qva ; Kyono2016jqy . Alternatively, it can also be fixed by analytic continuations Hoare:2016ibq , starting from the original $\eta$-deformed background Arutyunov:2013ega ; Arutyunov:2015qva ., in the process finding that the open string metric is undeformed, in line with our expectations. We check all expressions against the dilaton EOM, which we show is satisfied in all cases.
II Open string frame
In the context of string theory and its related effective field theories, depending on the probes used and the associated degrees of freedom, one may conveniently describe the system in different frames. These frames are related by field redefinitions and carry the name of the associated probe. For example, closed strings probe string frame and particles probe Einstein frame.
Within string theory, especially when we have D-brane probes, or when we have spacetimes with causal boundary like AdS geometry, one may choose to probe spacetime with D-branes or “open strings”. It is known, especially after the seminal work of Seiberg and Witten Seiberg:1999vs , that the open string and closed string frames in the NSNS sector are different if we have a Kalb-Ramond $B$-field in the background. In this section, we first review the open string frame in the NSNS sector and then introduce the open string frame fields for the RR sector.
II.1 Open string frame, NSNS sector
Given a triple ($g_{MN},B_{MN},\Phi$) comprising the closed string metric $g$, closed string coupling $g_{s}e^{\Phi}$ and NSNS two-form $B$, one can define an open string metric $G$, an antisymmetric bivector $\Theta$ and open string coupling $G_{s}$ Seiberg:1999vs 222See Duff:1989tf for an earlier incidence of open string metric and coupling.:
$$\displaystyle G_{MN}$$
$$\displaystyle=$$
$$\displaystyle\left(g-Bg^{-1}B\right)_{MN},$$
(3)
$$\displaystyle\Theta^{MN}$$
$$\displaystyle=$$
$$\displaystyle-\left(({g+B})^{-1}B({g-B})^{-1}\right)^{MN},$$
(4)
$$\displaystyle G_{s}$$
$$\displaystyle=$$
$$\displaystyle g_{s}{\rm e}^{\Phi}\left(\frac{\det(g+B)}{\det g}\right)^{\frac{%
1}{2}}.$$
(5)
The above are called open string fields because they are the combinations naturally appearing in the DBI part of the brane action describing open strings. One can also directly show that these are the fields appearing in the low-energy effective limit of open string scatterings Seiberg:1999vs ; Open-string-scattering .
For $r$-matrix solutions to the homogeneous CYBE it was noted in Araujo:2017jkb ; Araujo:2017jap that following the frame change, the open string metric is simply undeformed AdS${}_{5}$ with constant open string coupling $G_{s}=g_{s}$ and all information about the YB deformation resides in the bivector $\Theta$. Invariance of the open string metric and coupling under $O(d,d)$ transformations had been observed earlier Berman:2000jw . In addition, the bivector exhibits holographic commutativity, namely $\Theta$ depends on the holographic direction $z$, but all dependence drops out at the boundary $z=0$. At $z=0$ one is expected to make contact with a NC deformation of $\mathcal{N}=4$ super Yang-Mills: in support of this correspondence, it has been shown case by case that the bivector, evaluated at $z=0$, agrees with the Moyal bracket arising from the Drinfeld twist Drinfeld of the conformal algebra. At the level of algebra, this establishes an equivalence between homogeneous YB deformations and conformal twists, thereby substantiating an earlier conjecture of van Tongeren vanTongeren:2015uha .
A corollary of these results is the relation Araujo:2017jap
$$\sqrt{\det G}=e^{-2\Phi}\sqrt{\det g}.$$
(6)
Since $G$ is undeformed AdS${}_{5}\times$S${}^{5}$ metric, this implies that the density $e^{-2\Phi}\sqrt{\det g}$, in addition to being a well-known invariant of Buscher’s T-duality, is more generally an invariant of YB deformations based on $r$-matrix solutions to the homogeneous CYBE. We will show later that this extends to $\eta$-deformations Delduc:2013qra ; Arutyunov:2013ega ; Delduc:2014kha .
It was shown in Araujo:2017jkb ; Araujo:2017jap that for homogeneous YB deformations we always have equation (2), thus relating $I$ to the open string bivector $\Theta$. This relation was explicitly checked for an exhaustive set of backgrounds associated with conformal twists and moreover, we gave a symmetry based argument for (2). We will demonstrate below that (2), as well as the symmetry based argument leading to it, remains valid for the mCYBE case associated with $\eta$-deformations. In addition, we demonstrate that an expression for the dilaton can be read off from (6), once one uses the fact that $G$ is undeformed.
II.2 Open string field, RR sector
In the closed string theory we have RR form fields too and one may wonder what are their corresponding fields in the open string frame. Similarly to the NSNS sector, one may look again into the D-brane action, but now the Chern-Simons or Wess-Zumino part of it which contains RR-forms. Here we will conveniently denote this part of a $p$-brane action by $S_{WZ}$, which, when field strength of the gauge field on the brane is set to zero, takes the form:
$$S_{WZ}=\int_{\Sigma_{p}}\sum_{n\leq p+1}C_{n}\wedge e^{B},$$
(7)
where $C_{n}$ are $n$-form RR fields pulled back to the Dp-brane worldvolume $\Sigma_{p}$.
The above suggests that combinations of RR-forms and $B$-field are appropriate for open string frame RR forms. When $B$ is a constant exterior derivative (“field strength”), the above sum is related to a $p+2$-form Page form Page :
$$\displaystyle Q_{1}$$
$$\displaystyle=$$
$$\displaystyle F_{1},\quad Q_{3}=F_{3}+BF_{1},\quad Q_{5}=F_{5}+BF_{3}+\frac{1}%
{2}B^{2}F_{1},$$
$$\displaystyle Q_{7}$$
$$\displaystyle=$$
$$\displaystyle*F_{3}-BF_{5}-\frac{1}{2}B^{2}F_{3}-\frac{1}{3!}B^{3}F_{1},$$
(8)
$$\displaystyle Q_{9}$$
$$\displaystyle=$$
$$\displaystyle*F_{1}-B*F_{3}+\frac{1}{2}B^{2}F_{5}+\frac{1}{3!}B^{3}F_{3}+\frac%
{1}{4!}B^{4}F_{1},$$
where we have omitted wedge products and employed $B^{2}=B\wedge B$, etc. Closure of $Q_{n}$ is guaranteed by the Bianchis/EOMs of usual type IIB supergravity. We take Page forms (8) as the open string frame fields corresponding to RR forms.
One may view the pull back of the above Page forms on the brane as “Page charge densities” and define Page charges Page as integrals of them over compact cycles (where the branes wrap). Page charges have been shown to be
localized, conserved and most importantly quantized Marolf:2000cb , so that they may be identified with the rank of gauge groups in the dual theory.
III Page forms and generalized supergravity
It is known that homogeneous YB deformations, which are equivalent to (generalized) T-dualities Hoare:2016wsk ; Borsato:2016pas ; Borsato:2017qsx , result in supergravity solutions provided a unimodular condition holds Borsato:2016ose . For non-unimodular YB deformations one finds a solution to generalized supergravity Arutyunov:2015mqj , where, as explained in the appendix, the EOMs involve an extra vector field $I$, which is a Killing vector of the original and the deformed background. The full set of field equations of generalized supergravity may be found in Wulff:2016tju ; Arutyunov:2015mqj .
Here, we make the observation that the Bianchis and EOMs of generalized supergravity simplify considerably once expressed in terms of Page forms (8).
When $I\neq 0$, the equations of generalized supergravity read Arutyunov:2015mqj ,
$$\displaystyle\mbox{d}F_{1}$$
$$\displaystyle=$$
$$\displaystyle i_{I}F_{3}+i_{I}B\wedge F_{1},$$
(9)
$$\displaystyle\mbox{d}F_{3}+H\wedge F_{1}$$
$$\displaystyle=$$
$$\displaystyle i_{I}F_{5}+i_{I}B\wedge F_{3},$$
(10)
$$\displaystyle\mbox{d}F_{5}+H\wedge F_{3}$$
$$\displaystyle=$$
$$\displaystyle-i_{I}*F_{3}+i_{I}B\wedge F_{5},$$
(11)
$$\displaystyle\mbox{d}*F_{3}-H\wedge F_{5}$$
$$\displaystyle=$$
$$\displaystyle-i_{I}*F_{1}+i_{I}B\wedge*F_{3},$$
(12)
$$\displaystyle\mbox{d}*F_{1}-H\wedge*F_{3}$$
$$\displaystyle=$$
$$\displaystyle i_{I}B\wedge*F_{1}.$$
(13)
Recast in terms of the Page forms (8), the modified equations take a remarkably simple form
$$\displaystyle\mbox{d}Q_{1}$$
$$\displaystyle=$$
$$\displaystyle i_{I}Q_{3},\quad\mbox{d}Q_{3}=i_{I}Q_{5},\quad\mbox{d}Q_{5}=-i_{%
I}Q_{7},$$
$$\displaystyle\quad\mbox{d}Q_{7}$$
$$\displaystyle=$$
$$\displaystyle-i_{I}Q_{9}.$$
(14)
We omit $\mbox{d}Q_{9}$ due to its length. It is now clear that the Page forms are no longer closed. Nevertheless, the above are a consistent set of equations if $I$ is a Killing vector field.
To get a better feel for the rewriting, let us explicitly demonstrate how the first two equations in (III) come about. Similar logic applies to the later equations. From $i_{I}F_{1}=0$, which is a property of all generalized supergravity solutions Arutyunov:2015mqj , we see that (9) is simply the first equation in (III). Furthermore, combining (9) and (10), we have:
$$\displaystyle\mbox{d}(F_{3}+BF_{1})$$
$$\displaystyle=$$
$$\displaystyle i_{I}F_{5}+i_{I}BF_{3}+B(i_{I}F_{3}+i_{I}BF_{1}),$$
(15)
$$\displaystyle=$$
$$\displaystyle i_{I}F_{5}+i_{I}(BF_{3})+\frac{1}{2}i_{I}(B^{2}F_{1})$$
$$\displaystyle=$$
$$\displaystyle i_{I}Q_{5},$$
which explains the second equation in (III).
It is easy to convince oneself through explicit calculation, either case by case or employing a general ansatz, that the non-zero Page forms and associated Page charges are invariants of TsT transformations. For more general YB deformations, this can be checked for explicit solutions case by case, which leads us to the conjecture:
Non-zero Page forms and associated Page charges are invariants of all Yang-Baxter deformations.
The above extends our earlier result for invariance of open string metric and coupling to the RR-sector.
At an intuitive level, this statement is somewhat obvious and expected: in the current setting, we start with AdS${}_{5}\times$S${}^{5}$, a geometry sourced exclusively by D3-branes, we perform a special transformation that introduces a continuous constant deformation parameter, which is not quantized. In this light, it is reasonable to expect that the non-zero Page forms and charges are invariant.
It is instructive to illustrate our point in the simplest example of the Hashimoto-Itzhaki, Maldacena-Russo geometry Hashimoto:1999ut ; Maldacena:1999mh (see also Alishahiha:1999ci ),
$$\displaystyle\mbox{d}s^{2}$$
$$\displaystyle=$$
$$\displaystyle\frac{(-\mbox{d}t^{2}+\mbox{d}x_{3}^{2}+\mbox{d}z^{2})}{z^{2}}+%
\frac{z^{2}(\mbox{d}x_{1}^{2}+\mbox{d}x_{2}^{2})}{(z^{4}+\eta^{2})}+\mbox{d}s^%
{2}(S^{5}),$$
$$\displaystyle B$$
$$\displaystyle=$$
$$\displaystyle\frac{\eta}{z^{4}+\eta^{2}}\mbox{d}x_{1}\wedge\mbox{d}x_{2},\quad
e%
^{2\Phi}=\frac{z^{4}}{(z^{4}+\eta^{2})},$$
$$\displaystyle F_{3}$$
$$\displaystyle=$$
$$\displaystyle\frac{4\eta}{z^{5}}\mbox{d}t\wedge\mbox{d}x_{3}\wedge\mbox{d}z,$$
(16)
$$\displaystyle F_{5}$$
$$\displaystyle=$$
$$\displaystyle(1+*)\frac{4}{z(z^{4}+\eta^{2})}\mbox{d}t\wedge\mbox{d}x_{1}%
\wedge\mbox{d}x_{2}\wedge\mbox{d}x_{3}\wedge\mbox{d}z.$$
A short calculation reveals that the only non-zero Page forms associated with this geometry are
$$Q_{3}=\frac{4\eta}{z^{5}}\mbox{d}t\wedge\mbox{d}x_{3}\wedge\mbox{d}z,\quad Q_{%
5}=(1+*)4\mbox{vol}(AdS_{5}).$$
(17)
The important point to take away from this example is that $Q_{5}$ has not changed in the deformation and is simply the original undeformed five-form flux. Although we provide no general proof, one may check by direct computation to see that $Q_{5}$ is the undeformed five-form flux for all YB deformations, homogeneous or modified, of AdS${}_{5}\times$S${}^{5}$ (for $\eta$-deformations associated with mCYBE this can be checked using the ABF solution Arutyunov:2013ega ). Our conjecture applies equally to YB deformations of different geometries, where different Page forms will be invariant.
IV Physical meaning of $I$
In this section, we arrive at the main point of this letter, namely to address the physical meaning of the Killing vector $I$ of generalized supergravity. We will now give two complementary perspectives: we will identify $I$ as a current on the D-brane worldvolume and elucidate its meaning in the pure spinor formalism of the string $\sigma$-model.
IV.1 $\Lambda$-symmetry and $I$ on the brane
In Araujo:2017jkb ; Araujo:2017jap we argued that (2) may be derived from a symmetry principle, the $\Lambda$-symmetry: (generalized) supergravity equations are invariant under the $\Lambda$-symmetry,
$$B\rightarrow B+\mbox{d}\Lambda,$$
(18)
where $\Lambda$ is an arbitrary one-form. At the level of generalized gravity this is manifest because the EOMs depend only on the field strength $H=\mbox{d}B$ (see appendix B of Araujo:2017jap ). In the presence of open strings or D-branes, however, we have a $U(1)$ gauge field on the brane, $A$, with field strength $F=\mbox{d}A$.
If we have a stack of $N$ coincident D-branes this $U(1)$ symmetry gets enhanced to $U(N)$ Witten:1995im and one may decompose it into $SU(N)$ fields and the “center of mass” $U(1)$ part. In what follows we set the $SU(N)$ part of the gauge fields to zero, since it does not mix with the closed string fields, and only focus on this center of mass $U(1)$ part.
In the presence of D-branes the $\Lambda$-symmetry (18) is replaced with Witten:1995im ; SheikhJabbari:1999ba
$$B\rightarrow B+\mbox{d}\Lambda,\qquad A\rightarrow A-\Lambda.$$
(19)
Then, besides $H$ there is another $\Lambda$-gauge invariant combination
$${\cal F}=B+F=B+\mbox{d}A,$$
(20)
and one can check that the DBI part of the brane action only involves ${\cal F}$ Leigh:1989jq :
$$S_{DBI}=\int_{\Sigma_{p}}\frac{e^{-\Phi}}{g_{s}}\sqrt{\det(g+{\cal F})}.$$
(21)
We comment in passing that ${\cal F}$ and hence the DBI action, is invariant under the $U(1)$ center of mass gauge symmetry, the $\lambda$-symmetry, under which $B$ remains invariant and $A\rightarrow A+\mbox{d}\lambda$, where $\lambda$ is an arbitrary scalar function.
The NC description for the DBI action with the open string field $\Theta$ as NC parameter, as considered in Seiberg:1999vs , is only a valid description in the specific $\Lambda$-gauge $\Lambda=A$. In this gauge ${\cal F}=B$. Hereafter, we will be working in this $\Lambda$-gauge. In fact, the closed string-open string frame map (3) - (5), expressed in this gauge Seiberg:1999vs , yields
$$\frac{1}{G_{s}}\sqrt{\det G}=\frac{e^{-\Phi}}{g_{s}}\sqrt{\det(g+B)}.$$
(22)
We will prove below that (2) is a consequence of $\Lambda$-symmetry, of course once we fix the ${\cal F}=B$ gauge. To this end, we first note that $M,N$ indices are bulk AdS${}_{5}$ indices, while the fields appearing in the brane action are pullbacks on the brane worldvolume. So, we may consider a brane that is AdS-filling, for example a D7-brane wrapping three dimensions on the
S${}^{5}$ piece. Alternatively, we may consider a D3-brane on the AdS${}_{5}$ part, and recall that the brane worldvolume is codimension-one on this 5D spacetime.
We shall choose the latter, where it is easy to show that if the D3-brane has no expansion along the radial direction in Poincaré patch of AdS${}_{5}$, parameterized by $z$ coordinate, we have
$$S_{DBI}=4\int_{\text{AdS}_{5}}\frac{e^{-\Phi}}{g_{s}}\sqrt{\det(g+B)}=4\int_{%
\text{AdS}_{5}}\frac{1}{G_{s}}\sqrt{\det G},$$
(23)
where we used (22). In addition, we used the facts that open string metric is simply the undeformed AdS${}_{5}$ metric, $G_{s}$ is a constant, and the factor of 4 has appeared as a result of exterior differentiation of the D3-brane DBI, which produces a 5D action from a 4D action. Our treatment here mirrors similar analysis in Schwarz:2013wra .
Once again we point out that the action (23) is written in a specific $\Lambda$-gauge, and that, recalling (19), we have
$$\delta_{\Lambda}S_{DBI}=-\delta_{A}S_{DBI},$$
(24)
and hence the EOM for the gauge field $A$ is related to the $\Lambda$ invariance of the $\Lambda$-gauge fixed action. We shall return to this point momentarily.
The full D3-brane action besides the DBI part, also involves a Wess-Zumino part, which for a D3-brane in ${\cal F}=B$ $\Lambda$-gauge may be written as
$$S_{WZ}=\int_{\text{AdS}_{5}}(F_{5}+B\wedge F_{3}+\frac{1}{2}B^{2}\wedge F_{1}).$$
(25)
As a side comment, we note that with the factor 4 in the DBI part (23), we get the “no force” condition cancelation of DBI and the WZ parts for flat, undeformed D3-branes.
Now, we vary the overall action $S_{DBI}+S_{WZ}$ with respect to $\Lambda$. From the variation of the DBI term, we get Araujo:2017jap
$$\delta_{\Lambda}S_{DBI}=-\frac{\sqrt{\det G}}{G_{s}}\nabla_{M}\Theta^{MN}%
\delta\Lambda_{N},$$
(26)
where we have used (22) and $\Theta^{MN}=(g+B)^{[MN]}$, whereas from the variation of $S_{WZ}$ we get
$$\displaystyle\delta_{\Lambda}S_{WZ}$$
$$\displaystyle=$$
$$\displaystyle-\frac{1}{4}(F_{3}+B\wedge F_{1})\wedge\mbox{d}\delta\Lambda$$
$$\displaystyle=$$
$$\displaystyle-\frac{1}{4}i_{I}Q_{5}\wedge\delta\Lambda,$$
$$\displaystyle=$$
$$\displaystyle-\frac{1}{g_{s}}i_{I}\mbox{vol}(AdS_{5})\wedge\delta\Lambda=\frac%
{1}{g_{s}}\sqrt{\det G}I^{N}\delta\Lambda_{N}.$$
In the second line, we have used (III) and in the third line, we have employed
$$Q_{5}=\frac{4}{g_{s}}[\mbox{vol}(AdS_{5})+\mbox{vol}(S^{5})],$$
(28)
where we have reinstated $g_{s}$. Recalling that $G_{s}=g_{s}$, combining the two contributions, we arrive at (2). In other words, (2) is nothing but the EOM for the gauge field $A$ in the $\Lambda=A$ gauge, or in other words, (2) is the condition for consistency of $\Lambda=A$ gauge fixing. We stress that the $\lambda$-symmetry is still unfixed and that our arguments above do not depend on $\lambda$-symmetry gauge fixing.
Given the above result, one may ask if it can be written in a different $\Lambda$-gauge. The answer is simply yes: we may work in a $B=0$ gauge and using (24), one can readily see that the EOM for the gauge field $A$ of the action
$$S=\int_{\Sigma_{4}}\frac{1}{G_{s}}\sqrt{G+F}-A_{\mu}I^{\mu}$$
(29)
yields the pullback of (2) on the brane worldvolume. The above action is interesting because it provides a new perspective on the vector field $I$: $I$ is a current coupled to the center of mass $U(1)$.
Our derivation above is true for all YB deformations of AdS${}_{5}\times$S${}^{5}$ as long as (28) holds and one may check that this is the case by the exhaustive set of examples associated with CYBE discussed in Araujo:2017jap . For the mCYBE case, we show this explicitly in the next section.
IV.2 $I$ and pure spinor formulation
In the previous section we gave two meanings to the $I$ vector in the open string frame: we derived (2) and argued that $I$ may be viewed as a current for the center of mass $U(1)$ gauge symmetry that we are left with after fixing the $\Lambda$-gauge. In Wulff:2016tju it was suggested that the vector field $I$ may be related to some non-physical states that appear in the pure spinor analysis of supergravity Bedoya:2010qz ; Mikhailov:2012id ; Mikhailov:2014qka . In this section, we clarify this relation.
Type IIB string worldsheet theory can be characterized by the following data Berkovits:2001ue :
i) A conformally invariant action $S_{IIB}$,
ii) a pair of nilpotent BRST operators $Q_{L}$ and $Q_{R}$ satisfying
$$Q^{2}_{L}=Q^{2}_{R}=\{Q_{L},Q_{R}\}=0$$
(30)
and a grading, the ghost number, for which $\mathtt{gh}(Q_{L})=\mathtt{gh}(Q_{R})=+1$,
iii) and a pair of $b$-ghosts $b$ and $\bar{b}$ satisfying $\{Q,b\}=T$ and $\{Q,\bar{b}\}=\bar{T}$,
where $Q=Q_{L}+Q_{R}$. Let us now assume the existence of a theory satisfying these axioms.
Infinitesimal deformations parametrized by the integrated vertex operators $V_{i}^{(2)}$ are defined by
$$\begin{split}\displaystyle S&\displaystyle=S_{IIB}+\eta\int V_{1}^{(2)}+\eta^{%
2}\int V_{2}^{(2)}+\cdots\\
\displaystyle Q&\displaystyle=Q+\eta Q_{1}+\eta^{2}Q_{2}+\cdots\end{split}$$
(31)
For sigma models on $\mathfrak{g}=\mathfrak{psu}(2,2|4)$, the $\beta$-deformations Lunin:2005jy are described by vertex operators,
$$V_{1}^{(2)}=\frac{1}{2}r^{ab}j_{a}\wedge j_{b}\;,$$
(32)
where $j_{a}$ are conserved currents of global symmetries, $r^{ab}$ are components of the tensor $r\in(\mathfrak{g}\wedge\mathfrak{g})_{0}/\mathfrak{g}$, and $(\mathfrak{g}\wedge\mathfrak{g})_{0}$ stands for the subspace of $(\mathfrak{g}\wedge\mathfrak{g})$ generated by objects of the form $a\wedge b$ with $[a,b]=0$. An important feature in the present analysis is that the second order expansion in $\eta$ implies that $r$ satisfies the homgeneous CYBE. The simplest example is given by the case when $r\in su(4)$. As we know, this deformation gives the Lunin-Maldacena background Lunin:2005jy that can be obtained by a TsT transformation. See Bedoya:2010qz for further details.
It is easy to write a perturbative expansion similar to the deformations by integrated vertex (31) for the YB deformed $\sigma$-model action. Using the projector $P_{-}^{\alpha\beta}=\frac{1}{2}(\gamma^{\alpha\beta}-\epsilon^{\alpha\beta})$ where $\gamma_{\alpha\beta}$ is the worldsheet metric, the YB deformations Delduc:2013qra ; Kawaguchi:2014qwa of AdS${}_{5}\times S^{5}$ are defined by
$${\cal L}=\frac{(1+c\eta^{2})^{2}}{2(1-c\eta^{2})}P^{\alpha\beta}_{-}\textrm{%
Str}\left(A_{\alpha}\circ\mbox{d}\circ\frac{1}{1-\eta R\circ\mbox{d}}\circ A_{%
\beta}\right)\;,$$
(33)
where
$$\mbox{d}\equiv P_{1}+\frac{2}{1-c\eta^{2}}P_{2}-P_{3}\;,$$
(34)
and $P_{i}$ are projectors onto the $\mathbb{Z}_{4}$ grading of the algebra $\mathfrak{psu}(2,2|4)$, with $c=0$ for deformations based on the homogeneous CYBE Kawaguchi:2014qwa and
$c=1$ for the mCYBE considered in Delduc:2013qra .
The perturbative expansion for the case $c=0$ is obvious: since the perturbation by integrated vertex in the pure spinor formalism (31) is related to the homogeneous CYBE at ${\cal O}(\eta^{2})$, it is clear that both pictures must be physically equivalent.
In what follows, the analysis of the $I$ field refers to deformations corresponding to homogeneous CYBE solutions, but we briefly discuss what we can expect from the analysis of deformations built from mCYBE solutions. In this case, given an infinitesimal parameter $\eta$ we have the expansion
$$\begin{split}\displaystyle{\cal L}&\displaystyle=-\frac{1}{2}P_{-}^{\alpha%
\beta}\textrm{Str}\left(A_{\alpha}\circ\mbox{d}_{0}\circ A_{\beta}\right)-%
\frac{1}{2}\eta P_{-}^{\alpha\beta}\textrm{Str}\left[A_{\alpha}\circ(\mbox{d}_%
{0}\circ R\circ\mbox{d}_{0})\circ A_{\beta}\right]\\
&\displaystyle\frac{1}{2}\eta^{2}P_{-}^{\alpha\beta}\textrm{Str}\left[A_{%
\alpha}\circ(\mbox{d}_{0}\circ R\circ\mbox{d}_{0}\circ R\circ\mbox{d}_{0}+2P_{%
2}+3\mbox{d}_{0})\circ A_{\beta}\right]+{\cal O}(\eta^{3})\end{split}$$
(35)
where $\mbox{d}_{0}\equiv\mbox{d}\ |_{\eta=0}$ and the first term is the Metsaev-Tseytlin action Metsaev:1998it . Therefore, based on our previous experience, we may conjecture that there exists a perturbation by a vertex in the pure spinor formalism of the form
$$\widetilde{V}_{1}^{(2)}=\frac{1}{2}\tilde{r}^{ab}j_{a}\wedge j_{b}\;,$$
(36)
where $\tilde{r}$ satisfies the mCYBE. All in all, we see that these two deformations pictures are consistent with each other, provided this mild conjecture is satisfied.
Now, let us return to the homogeneous CYBE picture. Consistency with supergravity imposes the additional condition on the vertex operator (32):
$$r^{ab}f_{ab}^{\phantom{ab}c}=0\;,$$
(37)
where $f_{ab}^{\phantom{ab}c}$ are the structure constants of the Lie algebra $\mathfrak{g}$. The authors of Bedoya:2010qz have shown that when this condition is not satisfied, there are vertex operators associated with states, we call $\eta$-states 333Non-physical states in their terminology., which are not present in the type IIB spectrum.
In Mikhailov:2012id , Mikhailov considers the flat space limit of the deformation (31) and performs an analysis of the $\eta$-states. But since his analysis is in flat space and at first order in the $\eta$-expansion, there are many problems to be solved for finite transformations and curved backgrounds. There is one particular open question that can be solved using the connection between vertex operators and YB deformations: Do the $\eta$-states survive at all perturbative orders in $\eta$ in curved backgrounds?
From the YB deformation viewpoint, the condition (37), called unimodularity condition, appears in Borsato:2016ose and implies that deformed backgrounds satisfy the type II supergravity equations of motion, if and only if (37) is satisfied. Failure of the unimodularity condition (37) means that these YB deformed backgrounds satisfy the so called generalized supergravity equations of motion, that take into account a covector $X$ with components
$$X_{M}\equiv\partial_{M}\Phi+(g_{NM}+B_{NM})I^{N},$$
(38)
where $I^{M}$ is a Killing vector in the generalized type II background. Furthermore, in the special case $I=0$, we get $X=\partial\Phi$, which can be interpreted as the gradient of the dilaton field $\Phi$. In this case, we recover the usual type II supergravity equations of motion.
Therefore, just one of the $\eta$-states in Bedoya:2010qz survives at all perturbative orders in $\eta$, and it should correspond to the vector field $I$ in the YB deformation context. In order to make this correspondence more explicit, we follow a simple strategy: we know that type IIB supergravity does not have any vector field in the physical spectrum, then we should keep track of states that give origin to vectors in the BRST cohomology of Mikhailov:2014qka . Just one of these vector-like states may be realized as a descendant of a physical type IIB state, namely, the gradient of the dilaton $V=\partial\Phi$.
As usual, kappa-symmetry plays the crucial role restricting the extra degrees of freedom in a supergravity theory, including the generalized supergravity in the absence of Weyl invariance of the worldsheet theory. In pure spinor formalism, the correct number of degrees of freedom is carried and imposed through BRST charges and one would hence expect to find a counterpart of $I$ through BRST cohomology analysis.
The BRST cohomology in the flat space limit 444This corresponds to a Taylor expansion of the AdS${}_{5}\times$S${}_{5}$ solution around a point $x_{\ast}$ Mikhailov:2012id and at order $\eta$ has been performed in Mikhailov:2014qka . In our analysis, the essential states in this cohomology are the covectors $A^{+}$ and $A^{-}$, which satisfy the conditions
$$\begin{split}\displaystyle\partial_{[M}A_{N]}^{+}&\displaystyle=0\\
\displaystyle\partial_{(M}A_{N)}^{-}&\displaystyle=0\;.\end{split}$$
(39)
The vector field $A^{+}$ has been identified as the gradient of the dilaton, that is
$$A_{M}^{+}:=\partial_{M}\Phi\;.$$
(40)
The state $A^{-}$ is clearly an $\eta$-state, as there is no additional vector field in the type IIB string spectrum. Furthermore, through some mild assumptions, it is reasonable to conjecture that this state is linearly related to the vector $I$, that is $I\sim A^{-}$.
In the undeformed limit $\eta=0$, we recover a type IIB supergravity solution, where $I=0$. Therefore, the perturbative expansion of this field must be of the form $I=\eta I^{(1)}+\eta^{2}I^{(2)}+{\cal O}(\eta^{3})$. In addition, we already know that the vector field $I$ is a Killing vector in the generalized supergravity solution, and that the flat space limit of the Killing vector equation is simply
$$\nabla_{M}I_{N}+\nabla_{N}I_{M}=0\ \ \xrightarrow[\texttt{flat space}]{\ }\ %
\partial_{M}I_{N}+\partial_{N}I_{M}=0\;,$$
(41)
which is, obviously, the second equation in (39). Therefore, it is easy to see that the $\eta$-state $A^{-}$ corresponds to the first order component of $I$ in its perturbative expansion, that is
$$A_{M}^{-}=I_{M}^{(1)}\;.$$
(42)
We can assume that in a complete BRST analysis, all perturbative states of $\eta$-states combines to give the Killing vector $I$ in the Yang-Baxter deformation context. Additionally, all other $\eta$-states in the BRST cohomology are obstructed by higher orders corrections and are, using the terminology of Bedoya:2010qz , truly non-physical states. Finally, equations (41) and (42) realize explicitly the observation of Mikhailov:2014qka that $\eta$-states are in correspondence with global symmetries of the background.
V $\eta$-deformations and mCYBE
In previous work Araujo:2017jkb ; Araujo:2017jap , we have largely discussed YB deformations based on $r$-matrix solutions to the homogeneous CYBE. For the analysis of the bulk D3-brane action, we were ambivalent to whether the $r$-matrix solved the homogeneous CYBE or mCYBE. In principle, we could check $Q_{5}$ for all the known $\eta$-deformations Delduc:2014kha , but apart from the ABF solution Arutyunov:2013ega ; Arutyunov:2015qva , where $Q_{5}$ coincides with the undeformed five-form flux, the explicit form of the RR fluxes are not known. So, here we will take a different track and demonstrate that equation (2) is valid for all known $\eta$-deformations.
To be clear, our approach is to take $g$ and $B$, which can be read off from the string $\sigma$-model Arutyunov:2013ega ; Delduc:2014kha , determine $\Theta$ and then use equation (2) to work out $I$, which we will directly compare with the Killing vector from the generalized supergravity solution. For the YB deformations identified by Delduc et al. Delduc:2014kha , we will determine both $I$ and $\Phi$ and check that the dilaton EOM Arutyunov:2015mqj ,
$$R-\frac{1}{12}H_{MNP}H^{MNP}+4\nabla_{M}X^{M}-4X_{M}X^{M}=0,$$
(43)
with $X$ defined in (38), is satisfied. This provides a powerful consistency check on our results.
We recall the $\eta$-deformed geometry for AdS${}_{5}\times$S${}^{5}$ Arutyunov:2013ega ,
$$\displaystyle\mbox{d}s^{2}$$
$$\displaystyle=$$
$$\displaystyle-\frac{(1+\rho^{2})\mbox{d}t^{2}}{1-\kappa^{2}\rho^{2}}+\frac{%
\mbox{d}\rho^{2}}{(1+\rho^{2})(1-\kappa^{2}\rho^{2})}+\frac{\rho^{2}\mbox{d}%
\zeta^{2}}{1+\kappa^{2}\rho^{4}\sin^{2}\zeta}+\frac{\rho^{2}\cos^{2}\zeta\mbox%
{d}\psi_{1}^{2}}{1+\kappa^{2}\rho^{4}\sin^{2}\zeta}+\rho^{2}\sin^{2}\zeta\mbox%
{d}\psi_{2}^{2}$$
(44)
$$\displaystyle+$$
$$\displaystyle\frac{(1-r^{2})\mbox{d}\phi^{2}}{1+\kappa^{2}r^{2}}+\frac{\mbox{d%
}r^{2}}{(1-r^{2})(1+\kappa^{2}r^{2})}+\frac{r^{2}\mbox{d}\xi^{2}}{1+\kappa^{2}%
r^{4}\sin^{2}\xi}+\frac{r^{2}\cos^{2}\xi\mbox{d}\phi_{1}^{2}}{1+\kappa^{2}r^{4%
}\sin^{2}\xi}+r^{2}\sin^{2}\xi\mbox{d}\phi_{2}^{2},$$
$$\displaystyle B$$
$$\displaystyle=$$
$$\displaystyle-\frac{\kappa\rho^{4}\sin 2\zeta}{2(1+\kappa^{2}\rho^{4}\sin^{2}%
\zeta)}\mbox{d}\zeta\wedge\mbox{d}\psi_{1}+\frac{\kappa\rho}{1-\kappa^{2}\rho^%
{2}}\mbox{d}t\wedge\mbox{d}\rho-\frac{\kappa r^{4}\sin 2\xi}{2(1+\kappa^{2}r^{%
4}\sin^{2}\xi)}\mbox{d}\phi_{1}\wedge\mbox{d}\xi+\frac{\kappa r}{1+\kappa^{2}r%
^{2}}\mbox{d}\phi\wedge\mbox{d}r.$$
Relative to its original incarnation, we have retained the pure gauge terms in the $B$-field (see also Delduc:2014kha ). It turns out that these terms are important for the open string metric to be undeformed. It is worth noting that there is a simple analytic continuation that takes the AdS${}_{5}$ deformation to the S${}^{5}$ deformation. Interestingly, just as for $\lambda$-deformations Sfetsos:2014cea ; Demulder:2015lva , it can be shown that the dilaton equation forces the same deformation parameter in AdS${}_{5}$ and S${}^{5}$. Since $\eta$-deformations correspond to scaling limits of $\lambda$-deformations Hoare:2015gda , this is not too surprising.
From (4), it is an easy exercise to determine $\Theta$:
$$\displaystyle\Theta^{\zeta\psi_{1}}$$
$$\displaystyle=$$
$$\displaystyle\kappa\tan\zeta,\quad\Theta^{t\rho}=\kappa\rho,$$
$$\displaystyle\Theta^{\xi\phi_{1}}$$
$$\displaystyle=$$
$$\displaystyle-\kappa\tan\xi,\quad\Theta^{\phi r}=-\kappa r,$$
(45)
and $I$ using (2),
$$I=\kappa(-4\partial_{t}+2\partial_{\psi_{1}}+4\partial_{\phi}-2\partial_{\phi_%
{1}}),$$
(46)
which precisely agrees with the ABF result Arutyunov:2015qva . In contrast to ABF, the pure gauge terms in the $B$-field mean the dilaton is different, though $X$ of course is the same.
To determine the dilaton, we will now introduce a simple trick that has been largely overlooked. It is well-known that $e^{-2\Phi}\sqrt{\det g}$ is an invariant of the Buscher rules of T-duality. Existing results extend this to homogeneous YB deformations Araujo:2017jap . Taking a small leap, we can apply this to the ABF solution (44), with the result,
$$e^{-2\Phi}=(1-\kappa^{2}\rho^{2})(1+\kappa^{2}\rho^{4}\sin^{2}\zeta)(1+\kappa^%
{2}r^{2})(1+\kappa^{2}r^{4}\sin^{2}\xi).$$
For consistency, we have checked the dilaton equation (43), which suggests the density is a universal YB invariant.
Emboldened by this success, we proceed to analyze the $\eta$-deformations identified in Delduc:2014kha . Note, for these geometries the corresponding $I$, $\Phi$ and RR sector are not known. We proceed as before: we check that open string metric is undeformed in the presence of pure gauge contributions 555This allows us to identify and correct typos in Delduc:2014kha ., determine the dilaton from the invariant and deduce the Killing vector from (2). In each case, we will show that the picture is complete by checking the dilaton EOM. One special feature of the $\eta$-deformations Delduc:2014kha is that the $r$-matrix is a permutation of the ABF $r$-matrix and only the AdS${}_{5}$ part of the deformation differs from the ABF solution.
Focusing on the AdS${}_{5}$ deformation, which is new relative to ABF, the first $\eta$-deformation may be written as Delduc:2014kha
$$\displaystyle\mbox{d}s^{2}$$
$$\displaystyle=$$
$$\displaystyle-\frac{(1+\rho^{2})}{s}\mbox{d}t^{2}+\frac{1+\kappa^{2}\sin^{2}%
\zeta-\kappa^{2}\rho^{2}(1+\rho^{2})\cos^{2}\zeta\sin^{2}\zeta}{(1+\rho^{2})fs%
}\mbox{d}\rho^{2}+\frac{\rho^{2}(1+\kappa^{2}(1+\rho^{2})\cos^{2}\zeta-\kappa^%
{2}\rho^{2}\sin^{4}\zeta)}{fs}\mbox{d}\zeta^{2}$$
$$\displaystyle+$$
$$\displaystyle\frac{2\kappa^{2}\rho(1+\rho^{2}\sin^{2}\zeta)\cos\zeta\sin\zeta}%
{fs}\mbox{d}\rho\mbox{d}\zeta+\frac{\rho^{2}\cos^{2}\zeta}{f}\mbox{d}\psi_{1}^%
{2}+\rho^{2}\sin^{2}\zeta\mbox{d}\psi_{2}^{2},$$
$$\displaystyle B$$
$$\displaystyle=$$
$$\displaystyle-\frac{1}{2\kappa}\mbox{d}\log f\wedge\mbox{d}\psi_{1}-\frac{%
\kappa}{s}\left(\rho\sin^{2}\zeta\mbox{d}\rho+\rho^{2}(1+\rho^{2})\sin\zeta%
\cos\zeta\mbox{d}\zeta\right)\wedge\mbox{d}t.$$
(47)
where we have introduced the functions:
$$\displaystyle f(\rho,\zeta)$$
$$\displaystyle=$$
$$\displaystyle 1+\kappa^{2}+\kappa^{2}\rho^{2}\cos^{2}\zeta,$$
$$\displaystyle s(\rho,\zeta)$$
$$\displaystyle=$$
$$\displaystyle 1-\kappa^{2}\rho^{2}(1+\rho^{2}\cos^{2}\zeta)\sin^{2}\zeta,$$
$$\displaystyle h(\rho,\zeta)$$
$$\displaystyle=$$
$$\displaystyle 1+\kappa^{2}(1+\rho^{2})+\kappa^{2}\rho^{2}(1+\rho^{2})\cos^{2}\zeta.$$
(48)
The open string metric is undeformed, as expected, and $\Theta$ takes the form:
$$\displaystyle\Theta^{t\rho}$$
$$\displaystyle=$$
$$\displaystyle\kappa\rho\sin^{2}\zeta,\quad\Theta^{t\zeta}=\kappa\cos\zeta\sin\zeta,$$
$$\displaystyle\Theta^{\rho\psi_{1}}$$
$$\displaystyle=$$
$$\displaystyle\kappa(\rho^{-1}+\rho),\quad\Theta^{\zeta\psi_{1}}=-\kappa\rho^{-%
2}\tan\zeta.$$
(49)
Including the contribution from the internal space, the Killing vector may be determined from (2),
$$I=\kappa(-2\partial_{t}+4\partial_{\psi_{1}}+4\partial_{\phi}-2\partial_{\phi_%
{1}}).$$
(50)
Finally, the dilaton can be calculated from the invariant $e^{-2\Phi}\sqrt{\det g}$:
$$e^{-2\Phi}=fs(1+\kappa^{2}r^{2})(1+\kappa^{2}r^{4}\sin^{2}\xi).$$
(51)
To confirm that our treatment is correct, we have checked that the dilaton EOM (43) is satisfied.
Finally, we consider the remaining $\eta$-deformation Delduc:2014kha :
$$\displaystyle\mbox{d}s^{2}$$
$$\displaystyle=$$
$$\displaystyle-(1+\rho^{2})\mbox{d}t^{2}+\frac{1+\kappa^{2}+\kappa^{2}\rho^{2}(%
2+\rho^{2})\cos^{2}\zeta}{(1+\rho^{2})fh}\mbox{d}\rho^{2}+\frac{\rho^{2}(1+%
\kappa^{2}(1+\rho^{2}))}{fh}\mbox{d}\zeta^{2}-\frac{\kappa^{2}\rho^{3}\sin(2%
\zeta)}{fh}\mbox{d}\rho\mbox{d}\zeta$$
$$\displaystyle+$$
$$\displaystyle\frac{\rho^{2}\cos^{2}\zeta}{f}\mbox{d}\psi_{1}^{2}+\frac{\rho^{2%
}\sin^{2}\zeta}{h}\mbox{d}\psi_{2}^{2},$$
$$\displaystyle B$$
$$\displaystyle=$$
$$\displaystyle-\frac{1}{2\kappa}\mbox{d}\log f\wedge\mbox{d}\psi_{1}-\frac{%
\kappa}{h}\left(\rho\sin^{2}\zeta\mbox{d}\rho+\rho^{2}(1+\rho^{2})\sin\zeta%
\cos\zeta\mbox{d}\zeta\right)\wedge\mbox{d}\psi_{2}.$$
(52)
One can check that the open string metric is again undeformed and the external bivector takes the form,
$$\displaystyle\Theta^{\rho\psi_{i}}$$
$$\displaystyle=$$
$$\displaystyle\kappa(\rho^{-1}+\rho),\quad\Theta^{\zeta\psi_{1}}=-\kappa\rho^{-%
2}\tan\zeta,$$
$$\displaystyle\Theta^{\zeta\psi_{2}}$$
$$\displaystyle=$$
$$\displaystyle\kappa\rho^{-2}(1+\rho^{2})\cot\zeta.$$
(53)
Including the internal directions, the Killing vector is
$$I=\kappa(4\partial_{\psi_{1}}+2\partial_{\psi_{2}}+4\partial_{\phi}-2\partial_%
{\phi_{1}}),$$
(54)
and the dilaton is
$$e^{-2\Phi}=fh(1+\kappa^{2}r^{2})(1+\kappa^{2}r^{4}\sin^{2}\xi).$$
(55)
Again, we have checked the dilaton equation.
We end this section with some comments. It is easy to check that the Jacobi identify for the bivector, or more concretely,
$$\Theta^{[LP}\partial_{P}\Theta^{MN]}=0,$$
(56)
is satisfied for the three examples of $\eta$-deformations we have considered. We observe that this is a necessary condition for the bivector $\Theta$ to correspond to an NC parameter.
Furthermore, in Araujo:2017jap it was noted that the bivector $\Theta$ exhibits holographic noncommutativity: although it may depend on the holographic direction $z$, working in Poincaré patch with the boundary at $z=0$, the dependence on the holographic direction drops out consistently at the boundary. To see that the same picture applies equally well to the ABF solution (44), one can employ the following map from (unit radius) global AdS${}_{5}$ to the Poincaré metric,
$$\displaystyle\rho$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{2z}\sqrt{4\vec{x}^{2}+(z^{2}-1+\vec{x}^{2}-t^{2})^{2}},$$
$$\displaystyle\cos\tau$$
$$\displaystyle=$$
$$\displaystyle\frac{z^{2}+1+\vec{x}^{2}-t^{2}}{\sqrt{4t^{2}+(z^{2}+1+\vec{x}^{2%
}-t^{2})^{2}}},$$
$$\displaystyle\cos\zeta$$
$$\displaystyle=$$
$$\displaystyle\frac{2\sqrt{(x_{1}^{2}+x_{2}^{2})}}{\sqrt{4(x_{1}^{2}+x_{2}^{2}+%
x_{3}^{2})+(z^{2}-1+\vec{x}^{2}-t^{2})^{2}}},$$
$$\displaystyle\cos\psi_{1}$$
$$\displaystyle=$$
$$\displaystyle\frac{x_{1}}{\sqrt{x_{1}^{2}+x_{2}^{2}}},$$
$$\displaystyle\cos\psi_{2}$$
$$\displaystyle=$$
$$\displaystyle\frac{2x_{3}}{\sqrt{4x_{3}^{2}+(z^{2}-1+\vec{x}^{2}-t^{2})^{2}}},$$
(57)
where $\vec{x}\equiv(x_{1},x_{2},x_{3})$. Evaluating the ABF bivector (V) in Poincaré coordinates at the boundary, $\Theta^{\mu\nu}=\Theta^{MN}|_{z=0}$, we have confirmed that all $z$-dependence disappears leaving the following non-zero contributions to the bivector in Poincaré patch:
$$\displaystyle\Theta^{tx_{1}}$$
$$\displaystyle=$$
$$\displaystyle-\kappa\frac{tx_{2}}{2}(x_{\mu}x^{\mu}-1),\quad\Theta^{tx_{2}}=%
\kappa\frac{tx_{1}}{2}(x_{\mu}x^{\mu}-1),$$
$$\displaystyle\Theta^{x_{1}x_{2}}$$
$$\displaystyle=$$
$$\displaystyle-\kappa\left(x_{3}^{2}+\frac{1}{4}(x_{\mu}x^{\mu}-1)(x_{\mu}x^{%
\mu}-x_{1}^{2}-x_{2}^{2}-1)\right),$$
$$\displaystyle\Theta^{x_{1}x_{3}}$$
$$\displaystyle=$$
$$\displaystyle\kappa\frac{x_{2}x_{3}}{2}(x_{\mu}x^{\mu}+1),$$
$$\displaystyle\Theta^{x_{2}x_{3}}$$
$$\displaystyle=$$
$$\displaystyle-\kappa\frac{x_{1}x_{3}}{2}(x_{\mu}x^{\mu}+1).$$
(58)
We expect a similar outcome for other $\eta$-deformations. The corresponding expression for the Killing vector $I$ can be written in terms of the Killing vectors for the Poincaré metric of AdS${}_{5}$ as
$$I=2\kappa(P_{0}+K_{0}+M_{12}),$$
(59)
where $P_{0}$ and $K_{0}$ denote translations and special conformal transformations in the temporal direction, respectively, and $M_{12}$ is a rotation in the $(x_{1},x_{2})$-plane. We refer the reader to (A.2) of Araujo:2017jap for details of the notation. Note, the holographic behavior of $I$ is dictated by (2).
VI Conclusions
One of the main results of this letter is a derivation of equation (2) from the requirement that the action of D3-branes is invariant under $\Lambda$-symmetry. An interesting feature of the calculation is that the equations of generalized supergravity are incorporated through a bulk AdS${}_{5}$ Wess-Zumino term, which couples the D3-brane to the background. This provides an explanation for the relation, which was observed to hold for all homogeneous Yang-Baxter deformations on a case by case basis. In this letter we show that $\eta$-deformations have an open string metric that is undeformed and satisfy the same equation. This allows us to interpret $I$ as a current for the center of mass motion on the worldvolume of the D3-brane.
As a secondary result, we noted that the Bianchi identities and EOMs of generalized supergravity take a strikingly simple form when expressed in terms of Page forms. We conjectured that the non-zero Page forms, as well as associated Page charges, and the well-known T-duality invariant $e^{-2\Phi}\sqrt{g}$, are invariants of YB deformations. We rigorously test the latter claim by using it to determine the dilaton for the $\eta$-deformations and confirm that this leads to a solution to the dilaton EOM of generalized supergravity.
Lastly, using the deformation of the pure spinor action through integrated vertex operators, we have identified the states in the BRST cohomology that give origin to the vector field $I$ in the $\eta$-deformation context. As a by-product, we have shown that other non-physical states in the BRST cohomology of Mikhailov:2014qka are suppressed by higher order corrections in the curvature and/or deformation parameter $\eta$.
It is a straightforward exercise to extend the analysis presented here to $\eta$-deformations of AdS${}_{3}\times$S${}^{3}$ and AdS${}_{2}\times$S${}^{2}$ geometries Hoare:2014pna ; Lunin:2014tsa . Once pure gauge contributions to the NSNS two-form are introduced, we expect the open string metric to be undeformed and for equation (2) to hold with the slight story twist that it will arise from variations of the actions of D1-branes, D5-branes, intersecting D3-branes, etc.
One problem that has yet to be unraveled is the interpretation of the $\eta$-deformed geometries Delduc:2013qra ; Arutyunov:2013ega in the context of AdS/CFT. This problem is thorny, since in contrast to simpler homogeneous YB deformations, one has to consider both AdS${}_{5}$ and S${}^{5}$ deformations. As a result, there is less residual global symmetry, but the consensus is that the dual gauge theory, if it exists, is expected to be some NC deformation of $\mathcal{N}=4$ super Yang-Mills. In this letter, we have extracted a candidate NC parameter for the ABF solution Arutyunov:2013ega , shown that it is associative in the sense that the Jacobi (56) is satisfied, which means that it is consistent with a non-commuting open string description. Moreover, we have checked in Poincaré patch that the holographic direction decouples at the boundary, where the putative dual gauge theory is expected to reside.
Acknowledgements
We thank I. Bakhmatov, D. Berman, M. Duff, B. Hoare and D.C. Thompson for discussion on related projects. E. Ó C acknowledges the kind hospitality of the workshop “Recent Advances in T/U-dualities and Generalized Geometries”, June 6-9 2017, Zagreb, where this work was initiated.
The work of J.S. is supported by the Japan Society for the Promotion of Science (JSPS).
M.M. Sh-J. is supported in part by the Saramadan Iran Federation, the junior research chair on black hole physics by the Iranian NSF
and the ICTP Simons Associate fellowship and ICTP network project NT-04 K.Y. acknowledges the Supporting Program for Interaction-based Initiative Team Studies (SPIRITS)
from Kyoto University and a JSPS Grant-in-Aid for Scientific Research (C) No. 15K05051.
This work is also supported in part by the JSPS Japan-Russia Research Cooperative Program.
Appendix A Review of Generalized Supergravity
The equations of motion for the metric, NSNS two-form and dilaton in the generalized type IIB supergravity are given by
(60)
where the spacetime indices $M,N,\ldots$ run from 0 to 9, $H$ is the field strength of the NSNS two-form. The appearance of a vector field $X$ indicates the generalization. Note here that the dilaton $\Phi$ is implicitly included in $X$ with a spacetime derivative and an extra vector field $I$ as explained later.
The flux contribution $T_{MN}$ is defined as
$$\displaystyle T_{MN}$$
$$\displaystyle\equiv$$
$$\displaystyle\frac{1}{2}\mathcal{F}_{M}\mathcal{F}_{N}+\frac{1}{4}\mathcal{F}_%
{MKL}\mathcal{F}_{N}{}^{KL}+\frac{1}{4\times 4!}\mathcal{F}_{MPQRS}\mathcal{F}%
_{N}{}^{PQRS}$$
(61)
$$\displaystyle-\frac{1}{4}G_{MN}(\mathcal{F}_{K}\mathcal{F}^{K}+\frac{1}{6}%
\mathcal{F}_{PQR}\mathcal{F}^{PQR})\,.$$
Here $\mathcal{F}_{M}\,,\mathcal{F}_{MNK}\,,\mathcal{F}_{MNKPQ}$ are
the rescaled RR field strengths, i. e.
$$\mathcal{F}_{n_{1}n_{2}\ldots}={\rm e}^{\Phi}F_{n_{1}n_{2}\ldots}\,.$$
(62)
The vector $X$ is decomposed as
$$X_{M}\equiv I_{M}+Z_{M}\,,$$
(63)
where $I$ and $Z$ satisfy relations
$$\displaystyle D_{M}I_{N}+D_{N}I_{M}=0\,,$$
(64)
$$\displaystyle I^{K}\,H_{KMN}+D_{M}Z_{N}-D_{N}Z_{M}=0\,,$$
(65)
$$\displaystyle I^{M}\,Z_{M}=0\,,$$
(66)
where $D$ is the closed string covariant derivative.
Assuming that the Lie derivative of the NSNS two-form along the $I$-direction vanishes, one obtains the following expression:
$$\displaystyle Z_{M}=\partial_{M}\Phi-B_{MN}I^{N}\,.$$
(67)
Thus, in total, $X$ can be regarded as a composite field comprising the dilaton derivative, the NSNS two-form and
the extra vector field $I$.
References
(1)
C. Klimcik,
JHEP 0212, 051 (2002)
[hep-th/0210095]
(2)
C. Klimcik,
J. Math. Phys. 50, 043508 (2009)
[arXiv:0802.3518 [hep-th]].
(3)
F. Delduc, M. Magro and B. Vicedo,
JHEP 1311 (2013) 192
[arXiv:1308.3581 [hep-th]].
(4)
T. Matsumoto and K. Yoshida,
Nucl. Phys. B 893 (2015) 287
[arXiv:1501.03665 [hep-th]].
(5)
O. Lunin and J. M. Maldacena,
JHEP 0505, 033 (2005)
[hep-th/0502086].
(6)
T. Araujo, I. Bakhmatov, E. Ó Colgáin, J. Sakamoto, M. M. Sheikh-Jabbari and K. Yoshida,
Phys. Rev. D 95, no. 10, 105006 (2017)
[arXiv:1702.02861 [hep-th]].
(7)
T. Araujo, I. Bakhmatov, E. Ó Colgáin, J. Sakamoto, M. M. Sheikh-Jabbari and K. Yoshida,
arXiv:1705.02063 [hep-th].
(8)
S. J. van Tongeren,
Nucl. Phys. B 904, 148 (2016)
[arXiv:1506.01023 [hep-th]].
(9)
S. J. van Tongeren,
Phys. Lett. B 765, 344 (2017)
[arXiv:1610.05677 [hep-th]].
(10)
M. T. Grisaru, P. S. Howe, L. Mezincescu, B. Nilsson and P. K. Townsend,
Phys. Lett. 162B (1985) 116.
(11)
L. Wulff and A. A. Tseytlin,
JHEP 1606, 174 (2016)
[arXiv:1605.04884 [hep-th]].
(12)
R. R. Metsaev and A. A. Tseytlin,
Nucl. Phys. B 533, 109 (1998)
[hep-th/9805028].
(13)
I. Bena, J. Polchinski and R. Roiban,
Phys. Rev. D 69, 046002 (2004)
[hep-th/0305116].
(14)
F. Delduc, M. Magro and B. Vicedo,
Phys. Rev. Lett. 112, no. 5, 051601 (2014)
[arXiv:1309.5850 [hep-th]].
(15)
I. Kawaguchi, T. Matsumoto and K. Yoshida,
JHEP 1404, 153 (2014)
[arXiv:1401.4855 [hep-th]].
(16)
G. Arutyunov, R. Borsato and S. Frolov,
JHEP 1512, 049 (2015)
[arXiv:1507.04239 [hep-th]].
(17)
G. Arutyunov, S. Frolov, B. Hoare, R. Roiban and A. A. Tseytlin,
Nucl. Phys. B 903, 262 (2016)
[arXiv:1511.05795 [hep-th]].
(18)
T. Matsumoto and K. Yoshida,
JHEP 1406, 135 (2014)
[arXiv:1404.1838 [hep-th]].
(19)
T. Matsumoto and K. Yoshida,
JHEP 1406, 163 (2014)
[arXiv:1404.3657 [hep-th]].
(20)
T. Matsumoto and K. Yoshida,
JHEP 1503, 137 (2015)
[arXiv:1412.3658 [hep-th]].
(21)
T. Matsumoto and K. Yoshida,
JHEP 1504, 180 (2015)
[arXiv:1502.00740 [hep-th]].
(22)
I. Kawaguchi, T. Matsumoto and K. Yoshida,
JHEP 1406, 146 (2014)
[arXiv:1402.6147 [hep-th]].
(23)
S. J. van Tongeren,
JHEP 1506, 048 (2015)
[arXiv:1504.05516 [hep-th]].
(24)
D. Osten and S. J. van Tongeren,
Nucl. Phys. B 915, 184 (2017)
[arXiv:1608.08504 [hep-th]].
(25)
B. Hoare and D. C. Thompson,
JHEP 1702, 059 (2017)
[arXiv:1611.08020 [hep-th]].
(26)
H. Kyono and K. Yoshida,
PTEP 2016, no. 8, 083B03 (2016)
[arXiv:1605.02519 [hep-th]].
(27)
B. Hoare and S. J. van Tongeren,
J. Phys. A 49, no. 48, 484003 (2016)
[arXiv:1605.03552 [hep-th]].
(28)
B. Hoare and S. J. van Tongeren,
J. Phys. A 49, no. 43, 434006 (2016)
[arXiv:1605.03554 [hep-th]].
(29)
D. Orlando, S. Reffert, J. Sakamoto and K. Yoshida,
J. Phys. A 49 (2016) no.44, 445403
[arXiv:1607.00795 [hep-th]].
(30)
R. Borsato and L. Wulff,
JHEP 1610, 045 (2016)
[arXiv:1608.03570 [hep-th]].
(31)
W. Siegel,
Phys. Rev. D 48, 2826 (1993)
[hep-th/9305073].
(32)
W. Siegel,
Phys. Rev. D 47, 5453 (1993)
[hep-th/9302036].
(33)
C. Hull and B. Zwiebach,
JHEP 0909, 099 (2009)
[arXiv:0904.4664 [hep-th]].
(34)
Y. Sakatani, S. Uehara and K. Yoshida,
arXiv:1611.05856 [hep-th].
(35)
A. Baguet, M. Magro and H. Samtleben,
arXiv:1612.07210 [hep-th].
(36)
J. Sakamoto, Y. Sakatani and K. Yoshida,
PTEP 2017 (2017) no.5, 053B07
[arXiv:1703.09213 [hep-th]].
(37)
J. Sakamoto, Y. Sakatani and K. Yoshida,
arXiv:1705.07116 [hep-th].
(38)
N. Beisert and R. Roiban,
JHEP 0508, 039 (2005)
[hep-th/0505187].
(39)
M. Guica, F. Levkovich-Maslyuk and K. Zarembo,
arXiv:1706.07957 [hep-th].
(40)
D. Roychowdhury,
arXiv:1706.02625 [hep-th].
(41)
D. Roychowdhury,
arXiv:1707.07172 [hep-th].
(42)
G. Arutyunov and D. Medina-Rincon,
JHEP 1410 (2014) 50
[arXiv:1406.2536 [hep-th]].
(43)
T. Kameyama and K. Yoshida,
J. Phys. A 48 (2015) 7, 075401
[arXiv:1408.2189 [hep-th]];
J. Phys. A 48 (2015) 24, 245401
[arXiv:1410.5544 [hep-th]];
PTEP 2016 (2016) no.6, 063B01
[arXiv:1602.06786 [hep-th]].
(44)
M. Khouchen and J. Kluson,
Phys. Rev. D 90 (2014) 066001
[arXiv:1405.5017 [hep-th]];
C. Ahn and P. Bozhilov,
$\eta$-deformed AdS${}_{5}\times$S${}^{5}$,”
Phys. Lett. B 737 (2014) 293
[arXiv:1406.0628 [hep-th]];
A. Banerjee and K. L. Panigrahi,
JHEP 1409 (2014) 048
[arXiv:1406.3642 [hep-th]].
(45)
P. Bozhilov,
arXiv:1601.00127 [hep-th].
(46)
G. Arutyunov, M. Heinze and D. Medina-Rincon,
J. Phys. A 50, no. 3, 035401 (2017)
[arXiv:1607.05190 [hep-th]].
(47)
C. Ahn,
Phys. Lett. B 767, 121 (2017)
[arXiv:1611.09992 [hep-th]].
(48)
R. Hernandez and J. M. Nieto,
arXiv:1707.08032 [hep-th].
(49)
R. Klabbers and S. J. van Tongeren,
arXiv:1708.02894 [hep-th].
(50)
G. Arutyunov, R. Borsato and S. Frolov,
JHEP 1404, 002 (2014)
[arXiv:1312.3542 [hep-th]].
(51)
F. Delduc, M. Magro and B. Vicedo,
JHEP 1410, 132 (2014)
[arXiv:1406.6286 [hep-th]].
(52)
D. N. Page,
Phys. Rev. D 28, 2976 (1983).
(53)
N. Seiberg and E. Witten,
JHEP 9909, 032 (1999)
[hep-th/9908142].
(54)
M. J. Duff,
Nucl. Phys. B 335, 610 (1990).
doi:10.1016/0550-3213(90)90520-N
(55)
M. M. Sheikh-Jabbari,
Phys. Lett. B 450, 119 (1999)
[hep-th/9810179].
(56)
D. S. Berman, V. L. Campos, M. Cederwall, U. Gran, H. Larsson, M. Nielsen, B. E. W. Nilsson and P. Sundell,
JHEP 0105, 002 (2001)
[hep-th/0011282].
(57)
V. G. Drinfeld, Leningrad Math. J. 1 (1990) 321.
(58)
D. Marolf,
hep-th/0006117.
(59)
B. Hoare and A. A. Tseytlin,
J. Phys. A 49, no. 49, 494001 (2016)
[arXiv:1609.02550 [hep-th]].
(60)
R. Borsato and L. Wulff,
Phys. Rev. Lett. 117 (2016) no.25, 251602
[arXiv:1609.09834 [hep-th]].
(61)
R. Borsato and L. Wulff,
arXiv:1706.10169 [hep-th].
(62)
A. Hashimoto and N. Itzhaki,
Phys. Lett. B 465, 142 (1999)
[hep-th/9907166].
(63)
J. M. Maldacena and J. G. Russo,
JHEP 9909, 025 (1999)
[hep-th/9908134].
(64)
M. Alishahiha, Y. Oz and M. M. Sheikh-Jabbari,
JHEP 9911, 007 (1999)
[hep-th/9909215].
(65)
E. Witten,
Nucl. Phys. B 460, 335 (1996)
[hep-th/9510135].
(66)
M. M. Sheikh-Jabbari,
Phys. Lett. B 477, 325 (2000)
[hep-th/9910258].
(67)
R. G. Leigh,
Mod. Phys. Lett. A 4, 2767 (1989).
(68)
J. H. Schwarz,
JHEP 1401, 088 (2014)
[arXiv:1311.0305 [hep-th]].
(69)
O. A. Bedoya, L. I. Bevilaqua, A. Mikhailov and V. O. Rivelles,
Nucl. Phys. B 848, 155 (2011)
[arXiv:1005.0049 [hep-th]].
(70)
A. Mikhailov,
JHEP 1211, 082 (2012)
[arXiv:1203.0677 [hep-th]].
(71)
A. Mikhailov,
Nucl. Phys. B 907, 509 (2016)
[arXiv:1401.3783 [hep-th]].
(72)
N. Berkovits and P. S. Howe,
Nucl. Phys. B 635, 75 (2002)
[hep-th/0112160].
(73)
K. Sfetsos and D. C. Thompson,
JHEP 1412, 164 (2014)
[arXiv:1410.1886 [hep-th]].
(74)
S. Demulder, K. Sfetsos and D. C. Thompson,
JHEP 1507, 019 (2015)
[arXiv:1504.02781 [hep-th]].
(75)
B. Hoare and A. A. Tseytlin,
Nucl. Phys. B 897, 448 (2015)
[arXiv:1504.07213 [hep-th]].
(76)
B. Hoare, R. Roiban and A. A. Tseytlin,
JHEP 1406, 002 (2014)
[arXiv:1403.5517 [hep-th]].
(77)
O. Lunin, R. Roiban and A. A. Tseytlin,
Nucl. Phys. B 891 (2015) 106
[arXiv:1411.1066 [hep-th]]. |
hep-th/0111150
The quantum Hilbert space of a chiral two-form
in $d=5+1$ dimensions
Måns Henningson
Institute of Theoretical Physics, Chalmers University of Technology
S-412 96 Göteborg, Sweden
mans@fy.chalmers.se
Abstract:
We consider the quantum theory of a two-form gauge field on a space-time which is a direct product of time and a spatial manifold, taken to be a compact five-manifold with no torsion in its cohomology. We show that the Hilbert space of this non-chiral theory is a certain subspace of a tensor product of two spaces, that are naturally interpreted as the Hilbert spaces of a chiral and anti-chiral two-form theory respectively. We also study the observable operators in the non-chiral theory that correspond to the electric and magnetic field strengths, the Hamiltonian, and the exponentiated holonomy of the gauge-field around a spatial two-cycle. All these operators can be decomposed into contributions pertaining to the chiral and anti-chiral sectors of the theory.
1 Introduction
Giving a proper definition of the still rather mysterious interacting $(2,0)$ superconformal quantum theories in $d=5+1$ dimensions [1][2] appears as a major challenge for the future. These theories are quite interesting in their own right. (See e.g. [3] or [4] for a review.) They also seem to provide the right framework for gaining a better understanding of some aspects of $N=4$ super Yang-Mills theory in $d=3+1$ dimensions, to which they reduce upon compactification on a two-torus. Finally, the study of these theories could be viewed as an viable approach to the broader problem of understanding string and $M$-theory by studying them in a context in which the subtle conceptual difficulties of quantum gravity are absent.
At a generic point in its moduli space, an interacting $(2,0)$ theory can be described as a set of $(2,0)$ massless tensor multiplets strongly coupled to self-dual tensile strings that have acquired their tension by a mechanism somehow related to the Higgs mechanism for a spontaneously broken gauge symmetry. A possible approach to studying the interacting theory would thus be to first obtain a thorough understanding of the theory of free second-quantized tensor multiplets. One would then couple these, first to classical currents, then to currents from first-quantized tensile strings, and finally to currents from some second-quantized theory, possibly describing self-dual strongly interacting tensionless strings.
Already the first step in this programme, namely the quantization of a free tensor multiplet, is a rather subtle and interesting problem, in particular because of the chiral two-form gauge field (i.e. with self-dual three-form field strength) that is part of it. Such a field cannot be described by a covariant Lagrangian in the ordinary sense, but it is by now generally agreed on that the quantum theory is still well-defined as a subsector of the quantum theory of an ordinary non-chiral two-form (which of course has a Lagrangian description) [5]. Sofar, this procedure has mostly been carried through with a functional integral formalism in six-dimensional spaces with Euclidean signature. This means that the duality operator acting on three-forms squares to minus the identity and thus acts as a complex structure. Objects pertaining to the chiral and the anti-chiral subsector of the theory are then characterized by being holomorphic and anti-holomorphic respectively with respect to this complex structure [6].
This ‘holomorphic factorization’ approach in Euclidean space is however not quite satisfactory. Indeed, chiral two-forms (and the symplectic Majorana spinors that are also part of the tensor multiplets) really only exist in space-times of Minkowski signature. It would therefore be desirable to directly construct the quantum theory of a chiral two-form on such a space-time. Formally, one could also here work with a functional integral formalism, but in practice this would have to be viewed as a Wick rotation of a Euclidean functional integral, and could thus only be carried through on a space-time $Y$ that is a direct product of time ${\bf R}$ and a spatial five-manifold $M$ endowed with some smooth Riemannian metric $g$, i.e.
$$Y={\bf R}\times M.$$
(1)
A more serious objection is maybe that it is unclear to what extent the functional integral formalism can be applied to the case of the interacting $(2,0)$ theories, where we do not even know whether a description in terms of a set of fields is possible or not. In this paper, we will therefore instead use the canonical formalism (which of course also limits us to space-times $Y$ of the same form as in (1)) to describe the quantum theory of a chiral two-form. This means that we should construct the Hilbert space of the theory, describe what the observables are, and define how the corresponding linear operators act on the Hilbert space. This way of describing a quantum theory should be sufficiently general to allow generalization to the interacting $(2,0)$ theories. The present paper possibly appears as rather pedantic, and some of the results have already appeared in a different form in the literature, but we still hope that our methods will prove useful for future developments, in particular for understanding the coupling of a chiral theory to external currents.
Our approach to the problem of defining the quantum theory of a chiral two-form is again to view this theory as a subsector of the theory of a non-chiral two-form gauge field $B$. For the latter theory, it is straightforward to construct its Hilbert space ${\cal V}$ and describe the action of the observable operators on it. The observables that we will be considering in this paper correspond to the ‘electric’ field-strength two-form $G$, the ‘magnetic’ field-strength three-form $H$, the Hamiltonian ${\cal H}$, and finally the exponentiated holonomy or Wilson surface observable $W(\Sigma)$ of the two-form gauge-field around a spatial two-cycle $\Sigma$. These quantities are given by
$$\displaystyle G$$
$$\displaystyle=$$
$$\displaystyle\dot{B}$$
(2)
$$\displaystyle H$$
$$\displaystyle=$$
$$\displaystyle dB,$$
(3)
$${\cal H}=\frac{1}{8\pi}\int_{M}\left(G\wedge*G+H\wedge*H\right),$$
(4)
and
$$W(\Sigma)=\exp i\int_{\Sigma}B.$$
(5)
(Here a dot denotes the derivative with respect to time, $d$ is the exterior derivative on $M$, and * denotes the Hodge duality operator that derives from the metric $g$ on $M$ and maps two-forms and three-forms on $M$ into each other. We work in the gauge $e\cdot B=0$, where $e$ is the vector field that generates time translations.) The quantities $G$, $H$, and ${\cal H}$ are real and correspond to Hermitian operators, whereas $W(\Sigma)$ is $U(1)$-valued and corresponds to a unitary operator.
Since the non-chiral two-form theory is free, one would expect its Hilbert space ${\cal V}$ to be the tensor product of the Hilbert spaces ${\cal V}_{+}$ and ${\cal V}_{-}$ of the chiral and anti-chiral theories. It will turn out, however, that because of a subtle correlation between these sectors, ${\cal V}$ is rather a subspace of ${\cal V}_{+}\otimes{\cal V}_{-}$, i.e.
$${\cal V}\subset{\cal V}_{+}\otimes{\cal V}_{-}.$$
(6)
One would also expect that the observable operators $G$, $H$, ${\cal H}$ and $W(\Sigma)$ can be decomposed into operators acting within ${\cal V}_{+}$ or ${\cal V}_{-}$ as
$$\displaystyle G$$
$$\displaystyle=$$
$$\displaystyle G_{+}+G_{-}$$
(7)
$$\displaystyle H$$
$$\displaystyle=$$
$$\displaystyle H_{+}+H_{-},$$
(8)
$${\cal H}={\cal H}_{+}+{\cal H}_{-},$$
(9)
and
$$W(\Sigma)=W_{+}(\Sigma)W_{-}(\Sigma).$$
(10)
As we will see, this is indeed true, provided that we regard $W_{+}(\Sigma)$ and $W_{-}(\Sigma)$ as elements of the sets of unitary operators on ${\cal V}_{+}$ and ${\cal V}_{-}$ modulo $\pm 1$. (Their product $W(\Sigma)$ is well-defined as a unitary operator on ${\cal V}$, though.)
We end this introduction by discussing the choice of spatial five-manifold $M$ that we will work on. As usual in quantum mechanics, it is convenient to work on a compact manifold $M$, so that various operators have discrete rather than continuous spectra with normalizable eigenvectors that form a complete set. The simplest possibility is then of course a topologically trivial compact $M$. However, if one later wishes to couple the theory to prescribed magnetic currents supported on some one-dimensional strings, one will effectively remove the locus of these strings from $M$ and thus create a (non-compact) space with homologically non-trivial three-cycles. It is therefore natural to include the effects of non-trivial spatial topology already now. In this paper, we will consider a compact manifold $M$ with non-trivial second and third homology groups, which for simplicity we take to be torsion-free, i.e.
$$H_{2}(M,{\bf Z})\simeq H_{3}(M,{\bf Z})\simeq{\bf Z}^{b}$$
(11)
for some integer $b$. We endow $M$ with a smooth Riemannian metric $g$.
2 The phase space
To quantize the theory of a non-chiral two-form gauge field $B$, it is important to note that $B$ is only locally (i.e. over a single coordinate patch) a two-form. Globally, it is rather a connection on a 1-gerbe over the space-time manifold (1), and may thus undergo a gauge transformation $B\rightarrow B+\Delta B$ from one coordinate patch to another. The gauge parameter $\Delta B$ is here a closed two-form such that the de Rham cohomology class $\left[\frac{\Delta B}{2\pi}\right]$ is the image of an integer class on the overlap of the two coordinate patches.
We may now proceed with a standard canonical analysis starting from a covariant Lagrangian of Maxwell type. We choose the gauge $e\cdot B=0$, where $e$ is the vector field that generates time translations. The remaining dynamical variable is then the restriction of $B$ to $M$ at some time $t$, which we henceforth denote just $B$. The canonical momentum conjugate to $B$ is some multiple of the two-form
$$G=\dot{B},$$
(12)
i.e. the electric field strength. This means that the equal time Poisson bracket $\left\{.,.\right\}$ between functionals of $B$ only or functionals of $G$ only vanish, whereas
$$\left\{\int_{\Sigma}B,\int_{M}G\wedge*\alpha\right\}=4\pi\int_{\Sigma}\alpha$$
(13)
for an arbitrary two-cycle $\Sigma$ and an arbitrary two-form $\alpha$ on $M$. In terms of
$$H=dB,$$
(14)
i.e. the magnetic field strength, we have that the Poisson bracket between functionals of $B$ only or functionals of $H$ only vanish, whereas
$$\left\{\int_{M}H\wedge*\beta,\int_{M}G\wedge*\alpha\right\}=4\pi\int_{M}\beta%
\wedge*d\alpha$$
(15)
for an arbitrary two-form $\alpha$ and an arbitrary three-form $\beta$ on $M$. It is important for the sequel that we choose the coupling constant in the Lagrangian so that the numerical constant in the right hand side of these expressions takes precisely the value $4\pi$. The Hamiltonian ${\cal H}$ is then given by (4).
The phase-space of this system is the space of connections $B$, modulo gauge transformations, that satisfy the generalized Maxwell equations
$$\displaystyle\dot{G}$$
$$\displaystyle=$$
$$\displaystyle-d^{*}H$$
(16)
$$\displaystyle\dot{H}$$
$$\displaystyle=$$
$$\displaystyle dG$$
(17)
$$\displaystyle d^{*}G$$
$$\displaystyle=$$
$$\displaystyle 0$$
(18)
$$\displaystyle dH$$
$$\displaystyle=$$
$$\displaystyle 0.$$
(19)
One must remember, though, that two gauge inequivalent connections may differ by a ‘flat’ connection, in which case they have the same electric and magnetic fields strengths. To distinguish them, we will, in addition to the field strengths $G$ and $H$, specify also the holonomy
$$\int_{[\Sigma]^{0}}B$$
(20)
of $B$ around a particular chosen representive $[\Sigma]^{0}$ of each homology class $[\Sigma]\in H_{2}(M,{\bf Z})$. Because of the gauge ambiguity of $B$, this holonomy is well-defined only as an element of ${\bf R}/2\pi{\bf Z}$, i.e. $\exp i\int_{\Sigma_{0}}B$ is a well-defined element of $U(1)$. The Wilson surface observable $W(\Sigma)$ of an arbitrary two-cycle
$$\Sigma=[\Sigma]^{0}+\partial D$$
(21)
in the homology class $[\Sigma]\in H_{2}(M,{\bf Z})$ can now be written in terms of these phase space variables as
$$W(\Sigma)=\exp i\int_{\Sigma}B=\exp i\int_{[\Sigma]^{0}}B\exp i\int_{D}H,$$
(22)
where we have used Stokes’ theorem. Note that three-chain $D$ is only defined modulo a three-cycle by (21), but the periods of the magnetic field strength $H$ over such a three-cycle is a multiple of $2\pi$, so $W(\Sigma)$ is still well-defined as a $U(1)$-valued function of $\Sigma$.
By the Hodge theorem, the last two equations in (19) imply that we may write $G$ and $H$ as
$$\displaystyle G$$
$$\displaystyle=$$
$$\displaystyle G^{0}+G^{\prime}$$
(23)
$$\displaystyle H$$
$$\displaystyle=$$
$$\displaystyle H^{0}+H^{\prime},$$
(24)
where $G_{0}$ and $H_{0}$ are harmonic two- and three-forms respectively, $G^{\prime}$ is a coexact two-form, and $H^{\prime}$ is an exact three-form. The Hamiltonian decomposes accordingly as
$${\cal H}={\cal H}^{0}+{\cal H}^{\prime}$$
(25)
with
$$\displaystyle{\cal H}^{0}$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{8\pi}\int_{M}\left(G^{0}\wedge*G^{0}+H^{0}\wedge*H^{0}\right)$$
(26)
$$\displaystyle{\cal H}^{\prime}$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{8\pi}\int_{M}\left(G^{\prime}\wedge*G^{\prime}+H^{\prime%
}\wedge*H^{\prime}\right).$$
(27)
Similarly, the Wilson surface observable decomposes as
$$W(\Sigma)=W^{0}(\Sigma)W^{\prime}(\Sigma)$$
(28)
with
$$\displaystyle W^{0}(\Sigma)$$
$$\displaystyle=$$
$$\displaystyle\exp i\int_{[\Sigma]^{0}}B\exp i\int_{D}H^{0}$$
(29)
$$\displaystyle W^{\prime}(\Sigma)$$
$$\displaystyle=$$
$$\displaystyle\exp i\int_{D}H^{\prime}.$$
(30)
In fact, the Hilbert space ${\cal V}$ is a tensor product of Hilbert spaces ${\cal V}^{0}$ and ${\cal V}^{\prime}$ corresponding to the harmonic and non-harmonic modes respectively:
$${\cal V}={\cal V}^{0}\otimes{\cal V}^{\prime}.$$
(31)
The decomposition into chiral and anti-chiral sectors described in the introduction can be carried through separately for the harmonic and non-harmonic modes.
3 The harmonic modes
In this section, we will construct the quantum Hilbert space ${\cal V}^{0}$ corresponding to the classical phase-space variables $H^{0}$, $G^{0}$ and $\exp i\int_{[\Sigma]^{0}}B$,where again $H^{0}$ and $G^{0}$ are the harmonic parts of the magnetic and electric field strengths, and $\exp i\int_{[\Sigma]^{0}}B$ is the exponentiated holonomy around a chosen representative $[\Sigma]^{0}$ of each homology class $[\Sigma]\in H_{2}(M,{\bf Z})$.
We have already mentioned the fact that the periods of $H$ are multiples of $2\pi$, already at the classical level. Since the periods of the exact three-form $H^{\prime}$ are trivial, this means that
$$H^{0}=2\pi m^{0},$$
(32)
where $m^{0}$ denotes the unique harmonic representative of the de Rham cohomology class that is the image of a class $m\in H^{3}(M,{\bf Z})$. In the quantum theory, this equation is interpreted as a statement about the spectrum of the corresponding operator-valued three-form.
Classically, $G^{0}$ can be an arbitrary harmonic two-form, and $*G^{0}$ is thus an arbitrary harmonic three-form. In the quantum theory, the eigenvalues of the corresponding operator-valued three-form are of the form
$$*G^{0}=4\pi n^{0},$$
(33)
where $n^{0}$ denotes the unique harmonic representative of the de Rham cohomology class that is the image of a class $n\in H^{3}(M,{\bf Z})$. To see this, we take the two-cycle $\Sigma$ in (13) to equal the chosen representative $[\Sigma]^{0}$ for some $[\Sigma]\in H_{2}(M,{\bf Z})$, and take $\alpha$ to equal $\frac{1}{4\pi}$ times the harmonic two-form that is dual to $[\Sigma]^{0}$. We then get that
$$\left\{\int_{[\Sigma]^{0}}B,\int_{M}*G^{0}\wedge\alpha\right\}=1,$$
(34)
i.e. $\int_{M}*G^{0}\wedge\alpha$ is the canonically conjugate momentum of $\int_{[\Sigma]^{0}}B$. But the latter phase-space variable is periodic with period $2\pi$, so as usual in quantum theory the spectrum of $\int_{M}*G_{0}\wedge\alpha$ is quantized in integer units, from which statement (33) follows.
The operators $H^{0}$ and $*G^{0}$ commute, so as an orthonormal basis in the Hilbert space ${\cal V}^{0}$ we may choose their simultaneous eigenvectors
$$\left|m,n\right>,$$
(35)
where $m,n\in H^{3}(M,{\bf Z})$. The corresponding eigenvalues are given by (32) and (33) respectively. These states are also eigenstates of the Hamiltonian with eigenvalues
$${\cal H}^{0}=2\pi\int_{M}\left(\frac{1}{4}m^{0}\wedge*m^{0}+n^{0}\wedge*n^{0}%
\right).$$
(36)
Finally we should describe the action of the Wilson surface observable operator $W^{0}(\Sigma)$ on these states. From the above, it follows that
$$W^{0}(\Sigma)\left|m,n\right>=\left|m,n+1\right>\exp 2\pi i\int_{D}m^{0}.$$
(37)
Again, although the three-chain $D$ is only well-defined modulo a three-cycle by (21), the integral of $m^{0}$ over $D$ is well-defined modulo an integer, so $W^{0}(\Sigma)$ is a well-defined unitary operator that only depends on $\Sigma$.
We will now describe how the Hilbert space ${\cal V}^{0}$ can be viewed as a subspace of ${\cal V}^{0}_{+}\otimes{\cal V}^{0}_{-}$, where ${\cal V}^{0}_{+}$ and ${\cal V}^{0}_{-}$ are Hilbert spaces pertaining to the chiral and anti-chiral theories respectively. We first note that, by our assumption about $M$, $H^{3}(M,{\bf Z}_{2})$ is simply the mod 2 reduction of $H^{3}(M,{\bf Z})$. We let $[k]\in H^{3}(M,{\bf Z}_{2})$ denote the mod 2 reduction of an element $k\in H^{3}(M,{\bf Z})$. We also choose some fixed lifting $\overline{a}\in H^{3}(M,{\bf Z})$ of elements $a\in H^{3}(M,{\bf Z}_{2})$. (This means that $[\overline{a}]=a$ for all $a\in H^{3}(M,{\bf Z}_{2})$.) The spaces ${\cal V}^{0}_{+}$ and ${\cal V}^{0}_{-}$ are spanned by the orthonormal states
$$\left|k_{+},a_{+}\right>$$
(38)
and
$$\left|k_{-},a_{-}\right>$$
(39)
respectively, where $k_{+},k_{-}\in H^{3}(M,{\bf Z})$ and $a_{+},a_{-}\in H^{3}(M,{\bf Z}_{2})$. The space ${\cal V}^{0}$ is now isomorphic to the subspace $\hat{\cal V}\subset{\cal V}^{0}_{+}\otimes{\cal V}^{0}_{-}$ given by states of the form
$$\left|k_{+},a_{+}\right>\otimes\left|k_{-},a_{-}\right>$$
(40)
subject to the conditions
$$\displaystyle[k_{+}]$$
$$\displaystyle=$$
$$\displaystyle[k_{-}]$$
(41)
$$\displaystyle a_{+}$$
$$\displaystyle=$$
$$\displaystyle a_{-}.$$
(42)
The isomorphism between ${\cal V}$ and $\hat{\cal V}$ is given by
$$\left|m,n\right>=\left|k_{+},a_{+}\right>\otimes\left|k_{-},a_{-}\right>$$
(43)
where
$$\displaystyle m$$
$$\displaystyle=$$
$$\displaystyle k_{+}+k_{-}+\overline{a_{+}}$$
(44)
$$\displaystyle n$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{2}(k_{+}-k_{-})$$
(45)
or equivalently
$$\displaystyle k_{+}$$
$$\displaystyle=$$
$$\displaystyle n+\frac{1}{2}(m-\overline{[m]})$$
(46)
$$\displaystyle k_{-}$$
$$\displaystyle=$$
$$\displaystyle-n+\frac{1}{2}(m-\overline{[m]})$$
(47)
$$\displaystyle a_{+}$$
$$\displaystyle=$$
$$\displaystyle[m]$$
(48)
$$\displaystyle a_{-}$$
$$\displaystyle=$$
$$\displaystyle[m].$$
(49)
This construction is justified by showing how the various observable operators can be decomposed into contributions acting within ${\cal V}_{+}$ or ${\cal V}_{-}$. Beginning with the electric and magnetic field strengths, we have $G^{0}=G^{0}_{+}+G^{0}_{-}$ and $H^{0}=H^{0}_{+}+H^{0}_{-}$. The states $\left|k_{+},a_{+}\right>$ are eigenstates of $*G^{0}_{+}$ and $H^{0}_{+}$ with eigenvalues
$$\displaystyle*G^{0}_{+}$$
$$\displaystyle=$$
$$\displaystyle\pi(2k_{+}+\overline{a_{+}})^{0}$$
(50)
$$\displaystyle H^{0}_{+}$$
$$\displaystyle=$$
$$\displaystyle\pi(2k_{+}+\overline{a_{+}})^{0},$$
(51)
so $*G^{0}_{+}=H^{0}_{+}$ on the Hilbert space ${\cal V}^{0}_{+}$ of the chiral theory. (Superscript ${}^{0}$ on a class in $H^{3}(M,{\bf Z})$ denotes the unique harmonic representative of the corresponding de Rham cohomology class.) Similarly, the states $\left|k_{-},a_{-}\right>$ are eigenstates of $*G^{0}_{-}$ and $H^{0}_{-}$ with eigenvalues
$$\displaystyle*G^{0}_{-}=-\pi(2k_{-}+\overline{a_{-}})^{0}$$
(52)
$$\displaystyle H^{0}_{-}=\pi(2k_{-}+\overline{a_{-}})^{0},$$
(53)
so that $*G^{0}_{-}=-H^{0}_{-}$ on ${\cal V}^{0}_{-}$ as is appropriate for the anti-chiral theory. One should note that the periods of $H^{0}_{+}$ and $H^{0}_{-}$ may be half-integer multiples of $2\pi$ as in [7]. The states $\left|k_{+},a_{+}\right>$ and $\left|k_{-},a_{-}\right>$ are also eigenstates of the Hamiltonians of the chiral and anti-chiral theories respectively with eigenvalues
$$\displaystyle{\cal H}^{0}_{+}$$
$$\displaystyle=$$
$$\displaystyle\frac{\pi}{4}\int_{M}(2k_{+}+\overline{a_{+}})^{0}\wedge*(2k_{+}+%
\overline{a_{+}})^{0}$$
(54)
$$\displaystyle{\cal H}^{0}_{-}$$
$$\displaystyle=$$
$$\displaystyle\frac{\pi}{4}\int_{M}(2k_{-}+\overline{a_{-}})^{0}\wedge*(2k_{-}+%
\overline{a_{-}})^{0}$$
(55)
so that indeed ${\cal H}^{0}={\cal H}^{0}_{+}+{\cal H}^{0}_{-}$. Finally we decompose the Wilson surface observable $W^{0}(\Sigma)$ as $W^{0}(\Sigma)=W^{0}_{+}(\Sigma)W^{0}_{-}(\Sigma)$ with
$$\displaystyle W^{0}_{+}(\Sigma)\left|k_{+},a_{+}\right>$$
$$\displaystyle=$$
$$\displaystyle\left|k_{+}+1,a_{+}\right>\exp i\pi\int_{D}(2k_{+}+\overline{a_{+%
}})^{0}$$
(56)
$$\displaystyle W^{0}_{-}(\Sigma)\left|k_{-},a_{-}\right>$$
$$\displaystyle=$$
$$\displaystyle\left|k_{-}-1,a_{-}\right>\exp i\pi\int_{D}(2k_{-}+\overline{a_{-%
}})^{0}.$$
(57)
But since $D$ is only well-defined modulo a three-cycle by (21), $W^{0}_{+}(\Sigma)$ and $W^{0}_{-}(\Sigma)$ are only well-defined as unitary operators modulo $\pm 1$. However, their product $W^{0}(\Sigma)$ is of course well-defined as a unitary operator on $\hat{\cal V}\simeq{\cal V}$, since $a_{+}=a_{-}$ on this space.
Finally, we note that ${\cal V}^{0}_{+}$ may be decomposed as
$${\cal V}^{0}_{+}=\bigoplus_{a_{+}\in H^{3}(M,{\bf Z})}{\cal V}^{0}_{a_{+}},$$
(58)
where ${\cal V}^{0}_{a_{+}}$ is spanned by states of the form (38) for $k_{+}\in H^{3}(M,{\bf Z})$. The observables that we have been considering do not mix these subspaces, each of which could thus be interpreted as the Hilbert space of a different version of the chiral theory. The fact that there are inequivalent chiral theories was first seen on a general six-dimensional space-time in [5], where it was pointed out that there is no canonical way of picking one of the theories. In our case, where the six-dimensional general covariance is explicitly broken by the form of the space-time (1), there is however a distingished theory, namely the $a_{+}=0$ theory.
4 The non-harmonic modes
We now turn our attention to the phase-space variables $G^{\prime}$ and $H^{\prime}$, i.e. the coexact part of the electric field strength and the exact part of the magnetic field strength respectively. Our aim is to construct the Hilbert space ${\cal V}^{\prime}_{+}$ of the chiral sector of the theory and describe the action of the various observable operators on it.
To begin with, we will discuss the spectra of the Laplacians
$$\displaystyle\Delta^{(2)}_{\rm coexact}$$
$$\displaystyle=$$
$$\displaystyle dd^{*}+d^{*}d=d^{*}d$$
(59)
$$\displaystyle\Delta^{(3)}_{\rm exact}$$
$$\displaystyle=$$
$$\displaystyle dd^{*}+d^{*}d=dd^{*}$$
(60)
acting on coexact two-forms and exact three-forms on $M$ respectively. They can be described as follows: The eigenvalues of both Laplacians are of the form $\omega^{2}$ for $\omega\in\Omega$, where $\Omega$ is some discrete set of positive numbers. For each $\omega\in\Omega$ there are two linearly independent coexact two-forms $G_{\omega}$ and $\hat{G}_{\omega}$ that are eigenvectors of $\Delta^{(2)}_{\rm coexact}$ and two linearly independent exact three-forms $H_{\omega}$ and $\hat{H}_{\omega}$ that are eigenvectors of $\Delta^{(3)}_{\rm exact}$. We can choose them to fulfill the orthonormality conditions
$$\displaystyle\int_{M}G_{\omega}\wedge*G_{\omega^{\prime}}$$
$$\displaystyle=$$
$$\displaystyle\delta_{\omega\omega^{\prime}}$$
(61)
$$\displaystyle\int_{M}\hat{G}_{\omega}\wedge*\hat{G}_{\omega^{\prime}}$$
$$\displaystyle=$$
$$\displaystyle\delta_{\omega\omega^{\prime}}$$
(62)
$$\displaystyle\int_{M}G_{\omega}\wedge*\hat{G}_{\omega^{\prime}}$$
$$\displaystyle=$$
$$\displaystyle 0$$
(63)
and
$$\displaystyle\int_{M}H_{\omega}\wedge*H_{\omega^{\prime}}$$
$$\displaystyle=$$
$$\displaystyle\delta_{\omega\omega^{\prime}}$$
(64)
$$\displaystyle\int_{M}\hat{H}_{\omega}\wedge*\hat{H}_{\omega^{\prime}}$$
$$\displaystyle=$$
$$\displaystyle\delta_{\omega\omega^{\prime}}$$
(65)
$$\displaystyle\int_{M}H_{\omega}\wedge*\hat{H}_{\omega^{\prime}}$$
$$\displaystyle=$$
$$\displaystyle 0.$$
(66)
There are also the corresponding completeness relations of $G_{\omega}$ and $\hat{G}_{\omega}$ for $\omega\in\Omega$ in the space of coexact two-forms, and $H_{\omega}$ and $\hat{H}_{\omega}$ for $\omega\in\Omega$ in the space of exact three-forms. The forms $G_{\omega}$, $\hat{G}_{\omega}$, $H_{\omega}$ and $\hat{H}_{\omega}$ are related to each other in the following way:
$$\begin{array}[]{ccc}&{\scriptstyle\frac{1}{\omega}*d}&\cr G_{\omega}&\stackrel%
{{\scriptstyle\longleftarrow}}{{\longrightarrow}}&\hat{G}_{\omega}\cr&{%
\scriptstyle-\frac{1}{\omega}*d}&\cr{\scriptstyle\frac{1}{\omega}d}\downarrow%
\uparrow{\scriptstyle\frac{1}{\omega}d^{*}}&&{\scriptstyle\frac{1}{\omega}d}%
\downarrow\uparrow{\scriptstyle\frac{1}{\omega}d^{*}}\cr&{\scriptstyle\frac{1}%
{\omega}d*}&\cr H_{\omega}&\stackrel{{\scriptstyle\longleftarrow}}{{%
\longrightarrow}}&\hat{H}_{\omega}\cr&{\scriptstyle-\frac{1}{\omega}d*}&\end{%
array}.$$
(67)
Thus $H_{\omega}=*\hat{G}_{\omega}$ and $\hat{H}_{\omega}=-*G_{\omega}$.
We can now write the general solution to the equations (19) in the form (24) with
$$\displaystyle G^{\prime}$$
$$\displaystyle=$$
$$\displaystyle\sum_{\omega\in\Omega}\left(\left(a_{\omega}\cos\omega t-b_{%
\omega}\sin\omega t\right)G_{\omega}+\left(\hat{a}_{\omega}\cos\omega t-\hat{b%
}_{\omega}\sin\omega t\right)\hat{G}_{\omega}\right)$$
(68)
$$\displaystyle H^{\prime}$$
$$\displaystyle=$$
$$\displaystyle\sum_{\omega\in\Omega}\left(\left(a_{\omega}\sin\omega t+b_{%
\omega}\cos\omega t\right)H_{\omega}+\left(\hat{a}_{\omega}\sin\omega t+\hat{b%
}_{\omega}\cos\omega t\right)\hat{H}_{\omega}\right),$$
(69)
where $a_{\omega}$, $b_{\omega}$, $\hat{a}_{\omega}$ and $\hat{b}_{\omega}$ are arbitrary constants. In particular, at time $t=0$, we have
$$\displaystyle G^{\prime}|_{t=0}$$
$$\displaystyle=$$
$$\displaystyle\sum_{\omega\in\Omega}\left(a_{\omega}G_{\omega}+\hat{a}_{\omega}%
\hat{G}_{\omega}\right)$$
(70)
$$\displaystyle H^{\prime}|_{t=0}$$
$$\displaystyle=$$
$$\displaystyle\sum_{\omega\in\Omega}\left(b_{\omega}H_{\omega}+\hat{b}_{\omega}%
\hat{H}_{\omega}\right),$$
(71)
so that
$$\displaystyle a_{\omega}$$
$$\displaystyle=$$
$$\displaystyle\int_{M}G\wedge*G_{\omega}$$
(72)
$$\displaystyle b_{\omega}$$
$$\displaystyle=$$
$$\displaystyle\int_{M}H\wedge*H_{\omega}$$
(73)
$$\displaystyle\hat{a}_{\omega}$$
$$\displaystyle=$$
$$\displaystyle\int_{M}G\wedge*\hat{G}_{\omega}$$
(74)
$$\displaystyle\hat{b}_{\omega}$$
$$\displaystyle=$$
$$\displaystyle\int_{M}H\wedge*\hat{H}_{\omega}$$
(75)
with the integrals evaluated at $t=0$. It then follows from (13) that the only non-vanishing Poisson brackets between the variables $a_{\omega}$, $b_{\omega}$, $\hat{a}_{\omega}$ and $\hat{b}_{\omega}$ are
$$\displaystyle\left\{a_{\omega},b_{\omega^{\prime}}\right\}$$
$$\displaystyle=$$
$$\displaystyle-4\pi\omega\delta_{\omega,\omega^{\prime}}$$
(76)
$$\displaystyle\left\{\hat{a}_{\omega},\hat{b}_{\omega^{\prime}}\right\}$$
$$\displaystyle=$$
$$\displaystyle-4\pi\omega\delta_{\omega,\omega^{\prime}}.$$
(77)
In order to disentangle the chiral and anti-chiral sectors of the theory, we introduce the linear combinations $c^{+}_{\omega}$, $c^{-}_{\omega}$, $\hat{c}^{+}_{\omega}$, and $\hat{c}^{-}_{\omega}$ defined by
$$\displaystyle c^{\pm}_{\omega}$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{\sqrt{2}}(a_{\omega}\mp\hat{b}_{\omega})$$
(78)
$$\displaystyle\hat{c}^{\pm}_{\omega}$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{\sqrt{2}}(\hat{a}_{\omega}\pm b_{\omega}).$$
(79)
The only non-vanishing Poisson brackets between these variables are
$$\displaystyle\left\{c^{+}_{\omega},\hat{c}^{+}_{\omega^{\prime}}\right\}$$
$$\displaystyle=$$
$$\displaystyle-4\pi\omega\delta_{\omega,\omega^{\prime}}$$
(80)
$$\displaystyle\left\{c^{-}_{\omega},\hat{c}^{-}_{\omega^{\prime}}\right\}$$
$$\displaystyle=$$
$$\displaystyle 4\pi\omega\delta_{\omega,\omega^{\prime}}.$$
(81)
We can now decompose the electric and magnetic field strengths into contributions from the chiral and anti-chiral sectors of the theory, i.e. $G^{\prime}=G^{\prime}_{+}+G^{\prime}_{-}$ and $H^{\prime}=H^{\prime}_{+}+H^{\prime}_{-}$, where
$$\displaystyle H^{\prime}_{+}$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{\sqrt{2}}\sum_{\omega\in\Omega}\left(\left(\hat{c}^{+}_{%
\omega}\cos\omega t+c^{+}_{\omega}\sin\omega t\right)H_{\omega}+\left(-c^{+}_{%
\omega}\cos\omega t+\hat{c}^{+}_{\omega}\sin\omega t\right)\hat{H}_{\omega}\right)$$
(82)
$$\displaystyle G^{\prime}_{+}$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{\sqrt{2}}\sum_{\omega\in\Omega}\left(\left(c^{+}_{\omega%
}\cos\omega t-\hat{}c^{+}_{\omega}\sin\omega t\right)G_{\omega}+\left(\hat{c}^%
{+}_{\omega}\cos\omega t+c^{+}_{\omega}\sin\omega t\right)\hat{G}_{\omega}\right)$$
(83)
and
$$\displaystyle H^{\prime}_{-}$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{\sqrt{2}}\sum_{\omega\in\Omega}\left(\left(-\hat{c}^{+}_%
{\omega}\cos\omega t+c^{+}_{\omega}\sin\omega t\right)H_{\omega}+\left(c^{+}_{%
\omega}\cos\omega t+\hat{c}^{+}_{\omega}\sin\omega t\right)\hat{H}_{\omega}\right)$$
(84)
$$\displaystyle G^{\prime}_{-}$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{\sqrt{2}}\sum_{\omega\in\Omega}\left(\left(c^{+}_{\omega%
}\cos\omega t+\hat{c}^{+}_{\omega}\sin\omega t\right)G_{\omega}+\left(\hat{c}^%
{+}_{\omega}\cos\omega t-c^{+}_{\omega}\sin\omega t\right)\hat{G}_{\omega}%
\right).$$
(85)
We see that indeed $*G^{\prime}_{+}=H^{\prime}_{+}$ and $*G^{\prime}_{-}=-H^{\prime}_{-}$ as desired. The Hamiltonian also decomposes as ${\cal H}^{\prime}={\cal H}^{\prime}_{+}+{\cal H}^{\prime}_{-}$, where
$$\displaystyle{\cal H}^{\prime}_{+}$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{8\pi}\sum_{\omega\in\Omega}\left((c^{+}_{\omega})^{2}+(%
\hat{c}^{+}_{\omega})^{2}\right)$$
(86)
$$\displaystyle{\cal H}^{\prime}_{-}$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{8\pi}\sum_{\omega\in\Omega}\left((c^{-}_{\omega})^{2}+(%
\hat{c}^{-}_{\omega})^{2}\right).$$
(87)
Finally, we decompose the Wilson surface observable as $W^{\prime}(\Sigma)=W^{\prime}_{+}(\Sigma)W^{\prime}_{-}(\Sigma)$, where
$$\displaystyle W^{\prime}_{+}(\Sigma)$$
$$\displaystyle=$$
$$\displaystyle\exp i\int_{D}H^{\prime}_{+}$$
(88)
$$\displaystyle=$$
$$\displaystyle\exp\frac{i}{\sqrt{2}}\sum_{\omega\in\Omega}\frac{1}{\omega}\int_%
{\Sigma-[\Sigma]^{0}}\left(\left(\hat{c}^{+}_{\omega}\cos\omega t+c^{+}_{%
\omega}\sin\omega t\right)G_{\omega}+\left(-c^{+}_{\omega}\cos\omega t+\hat{c}%
^{+}_{\omega}\sin\omega t\right)\hat{G}_{\omega}\right)$$
(89)
$$\displaystyle W^{\prime}_{-}(\Sigma)$$
$$\displaystyle=$$
$$\displaystyle\exp i\int_{D}H^{\prime}_{-}$$
(90)
$$\displaystyle=$$
$$\displaystyle\exp\frac{i}{\sqrt{2}}\sum_{\omega\in\Omega}\frac{1}{\omega}\int_%
{\Sigma-[\Sigma]^{0}}\left(\left(-\hat{c}^{-}_{\omega}\cos\omega t+c^{-}_{%
\omega}\sin\omega t\right)G_{\omega}+\left(c^{-}_{\omega}\cos\omega t+\hat{c}^%
{-}_{\omega}\sin\omega t\right)\hat{G}_{\omega}\right).$$
(91)
Here we have used the exactness of $H_{\omega}$ and $\hat{H}_{\omega}$ to write $W^{\prime}_{+}(\Sigma)$ and $W^{\prime}_{-}(\Sigma)$ in a form where it is manifest that the choice of $D$ fulfilling (21) is immaterial.
By the usual procedure of canonical quantization, real functions on the classical phase space correspond to Hermitian operators on the quantum Hilbert space, with the operator commutator $\left[.,.\right]$ related to the Poisson bracket as $i\left\{.,.\right\}$. We will now carry out this procedure for the chiral sector of the theory. We introduce the annihilitation operators $\alpha_{\omega}$ and their Hermitian conjugates the creation operators $\alpha_{\omega}^{\dagger}$ for $\omega\in\Omega$ as
$$\displaystyle\alpha_{\omega}$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{\sqrt{8\pi\omega}}(c^{+}_{\omega}+i\hat{c}^{+}_{\omega})$$
(92)
$$\displaystyle\alpha_{\omega}^{\dagger}$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{\sqrt{8\pi\omega}}(c^{+}_{\omega}-i\hat{c}^{+}_{\omega}).$$
(93)
They obey the harmonic oscillator commutation relations
$$\displaystyle[\alpha_{\omega},\alpha_{\omega^{\prime}}]$$
$$\displaystyle=$$
$$\displaystyle 0$$
(94)
$$\displaystyle[\alpha_{\omega}^{\dagger},\alpha_{\omega^{\prime}}^{\dagger}]$$
$$\displaystyle=$$
$$\displaystyle 0$$
(95)
$$\displaystyle[\alpha_{\omega},\alpha_{\omega^{\prime}}^{\dagger}]$$
$$\displaystyle=$$
$$\displaystyle\delta_{\omega,\omega^{\prime}}.$$
(96)
This algebra can be represented on a Hilbert space with basis vectors of the form
$$\left|\left\{n_{\omega}\right\}\right>=\bigotimes_{\omega\in\Omega}\left|n_{%
\omega}\right>=\bigotimes_{\omega\in\Omega}\left(\alpha^{\dagger}_{\omega}%
\right)^{n_{\omega}}\left|0\right>,$$
(97)
where the $n_{\omega}$ are non-negative integers. These states are eigenvectors of the normal-ordered occupation-number operators $N_{\omega}=\alpha^{\dagger}_{\omega}\alpha_{\omega}$ with eigenvalues $n_{\omega}$.
We now turn to the construction of observable operators that act on this Hilbert space. The magnetic and electric field strength operators are straightforward:
$$\displaystyle H^{\prime}_{+}$$
$$\displaystyle=$$
$$\displaystyle\sum_{\omega\in\Omega}\sqrt{\pi\omega}\left(\alpha^{\dagger}_{%
\omega}e^{-i\omega t}(-\hat{H}_{\omega}+iH_{\omega})+\alpha_{\omega}e^{i\omega
t%
}(-\hat{H}_{\omega}-iH_{\omega})\right)$$
(98)
$$\displaystyle G^{\prime}_{+}$$
$$\displaystyle=$$
$$\displaystyle\sum_{\omega\in\Omega}\sqrt{\pi\omega}\left(\alpha^{\dagger}_{%
\omega}e^{-i\omega t}(G_{\omega}+i\hat{G}_{\omega})+\alpha_{\omega}e^{i\omega t%
}(G_{\omega}-i\hat{G}_{\omega})\right).$$
(99)
Being bilinear in the field strengths, the Hamilton operator suffers from an ordering ambiguity. We fix this by the usual choice of zero-point energy so that
$${\cal H}^{\prime}_{+}=\sum_{\omega\in\Omega}\omega\left(N_{\omega}+\frac{1}{2}%
\right).$$
(100)
The exponentiated holonomy operator of course suffers from a more serious ordering problem. In this paper, we will define it as
$$W^{\prime}_{+}(\Sigma)=\exp i\int_{D}H^{\prime}_{+}$$
(101)
with $H^{\prime}_{+}$ given by (99). Again, we may of course use the exactness of $H^{\prime}_{+}$ to write $W^{\prime}_{+}(\Sigma)$ in form where the apparent dependence on a choice of $D$ disappears.
By construction, $W^{\prime}_{+}(\Sigma)$ is formally unitary, i.e.
$$W^{\prime}_{+}(\Sigma)(W^{\prime}_{+}(\Sigma))^{\dagger}=(W^{\prime}_{+}(%
\Sigma))^{\dagger}W^{\prime}_{+}(\Sigma)=1.$$
(102)
If we introduce a second two-cycle $\Sigma^{\prime}=[\Sigma^{\prime}]^{0}+\partial D^{\prime}$, the operators $W^{\prime}_{+}(\Sigma)$, $W^{\prime}_{+}(\Sigma^{\prime})$, and $W^{\prime}_{+}(\Sigma+\Sigma^{\prime})$ fulfill an interesting algebra. To investigate this, we start by computing the commutator of the operators $\int_{D}H^{\prime}_{+}$ and $\int_{D^{\prime}}H^{\prime}_{+}$:
$$\displaystyle\left[\int_{D}H^{\prime}_{+}\right.,$$
$$\displaystyle\left.\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\int_{D^{\prime}}H^{\prime}_{%
+}\right]$$
(103)
$$\displaystyle=$$
$$\displaystyle 2\pi i\sum_{\omega\in\Omega}\omega\left(\int_{D}H_{\omega}\int_{%
D^{\prime}}\hat{H}_{\omega}-\int_{D}\hat{H}_{\omega}\int_{D^{\prime}}H_{\omega%
}\right)$$
(104)
$$\displaystyle=$$
$$\displaystyle 2\pi i\sum_{\omega\in\Omega}\left(\int_{D}H_{\omega}\int_{D^{%
\prime}}d*H_{\omega}+\int_{D}\hat{H}_{\omega}\int_{D^{\prime}}d*\hat{H}_{%
\omega}\right)$$
(105)
$$\displaystyle=$$
$$\displaystyle 2\pi i\sum_{\omega\in\Omega}\left(\int_{D}H_{\omega}\int_{\Sigma%
^{\prime}-[\Sigma^{\prime}]^{0}}*H_{\omega}+\int_{D}\hat{H}_{\omega}\int_{%
\Sigma^{\prime}-[\Sigma^{\prime}]^{0}}*\hat{H}_{\omega}\right)$$
(106)
$$\displaystyle=$$
$$\displaystyle 2\pi i\sum_{\omega\in\Omega}\left(\int_{M}*P_{D}\wedge*H_{\omega%
}\int_{M}P_{\Sigma^{\prime}-[\Sigma^{\prime}]^{0}}\wedge*H_{\omega}+\int_{M}*P%
_{D}\wedge*\hat{H}_{\omega}\int_{M}P_{\Sigma^{\prime}-[\Sigma^{\prime}]^{0}}%
\wedge*\hat{H}_{\omega}\right)$$
(107)
$$\displaystyle=$$
$$\displaystyle 2\pi i\int_{M}P_{D}\wedge P_{\Sigma^{\prime}-[\Sigma^{\prime}]^{%
0}}$$
(108)
$$\displaystyle=$$
$$\displaystyle 2\pi iD\cdot(\Sigma^{\prime}-[\Sigma^{\prime}]^{0})$$
(109)
$$\displaystyle=$$
$$\displaystyle 2\pi iL(\Sigma-[\Sigma]^{0},\Sigma^{\prime}-[\Sigma^{\prime}]^{0%
}).$$
(110)
In the fourth line, we have introduced the Poincaré duals $P_{D}$ and $P_{\Sigma^{\prime}-[\Sigma^{\prime}]^{0}}$ of $D$ and $\Sigma^{\prime}-[\Sigma^{\prime}]^{0}$ respectively, and in the next line we have used the completeness relation of $H_{\omega}$ and $\hat{H}_{\omega}$ for $\omega\in\Omega$ in the space of exact three-forms on $M$. In the sixth line, $D\cdot(\Sigma^{\prime}-[\Sigma^{\prime}]^{0})$ denotes the intersection number of $D$ and $\Sigma^{\prime}-[\Sigma^{\prime}]^{0}$, which by definition equals the linking number $L(\Sigma-\Sigma^{0},\Sigma^{\prime}-[\Sigma^{\prime}]^{0})$ of $\Sigma-[\Sigma]^{0}$ and $\Sigma^{\prime}-[\Sigma^{\prime}]^{0}$. This is an integer, so the commutator we have calculated is a multiple of $2\pi i$. (Essentially the same calculation was performed in [8] for the case when $M$ is flat Euclidean five-space.) It now follows from the Baker-Hausdorff formula that
$$W^{\prime}_{+}(\Sigma)W^{\prime}_{+}(\Sigma^{\prime})=W^{\prime}_{+}(\Sigma^{%
\prime})W^{\prime}_{+}(\Sigma)=(-1)^{L(\Sigma-[\Sigma]^{0},\Sigma^{\prime}-[%
\Sigma^{\prime}]^{0})}W^{\prime}_{+}(\Sigma+\Sigma^{\prime}),$$
(111)
i.e. $W^{\prime}_{+}(\Sigma)$ and $W^{\prime}_{+}(\Sigma^{\prime})$ commute, and their product is up to a sign equal to $W^{\prime}_{+}(\Sigma+\Sigma^{\prime})$. One should note that this sign in the multiplication law appears also on a manifold $M$ of trivial topology, in contrast to the sign ambiguity in the definition of $W^{0}_{+}(\Sigma)$ on a topologically non-trivial $M$ that we discussed in the previous section. Altogether, on a topologically non-trivial manifold, we should think of the complete operator $W_{+}(\Sigma)=W^{0}_{+}(\Sigma)W^{\prime}_{+}(\Sigma)$ as an element of the set of unitary operators modulo $\pm 1$, in which case the the sign factor in the multiplication law (111) of course is irrelevant.
Finally, we will briefly discuss the divergences of the operator $W^{\prime}_{+}(\Sigma)$. This operator is trivially a product over $\omega\in\Omega$ of commuting factors $W^{\prime}_{\omega}(\Sigma)$. Each of these may by the Baker-Hausdorff formula be rewritten as a product of an Hermitian $c$-number pre-factor, a factor involving the creation operator $\alpha^{\dagger}_{\omega}$, and a factor involving the annihilation operator $\alpha_{\omega}$ (in that order). It is then clear that $W^{\prime}_{\omega}(\Sigma)$ has finite matrix elements $\left<n_{\omega}\right|W_{\omega}^{\prime}(\Sigma)\left|n^{\prime}_{\omega}\right>$ (which for the case of $n_{\omega}=n^{\prime}_{\omega}=0$ equals the $c$-number prefactor). However, the infinite product over $\omega\in\Omega$ will in general diverge. To get a finite result, one would have to regulate the product (e.g. by introducing some kind of cutoff on $\omega$) and then modify the definition of $W^{\prime}(\Sigma)$ by including some Hermitian $c$-number factor so that the regulator may be removed. Some of the formal properties of $W^{\prime}(\Sigma)$, such as its unitarity and also its scale invariance, will be lost in this process, though. (See [9] for a related discussion on the Weyl anomaly of Wilson surface observables in the non-chiral theory.)
This research was supported by the Swedish Research Council.
References
[1]
E. Witten, ‘Some comments on string dynamics’, Los Angeles 1995, Future perspectives in string theory 501, hep-th/9507121.
[2]
A. Strominger, ‘Open $p$-branes’, ‘it Phys. Lett. B383 (1996) 44, hep-th/9512059.
[3]
O. Aharony, ‘A brief review of little string theories’, Class. Quant. Grav. 17 (2000) 929, hep-th/9911147.
[4]
N. Seiberg, ‘Notes on theories with 16 supercharges’, Nucl. Phys. Proc. Suppl. 67 (1998) 158, hep-th/9705117.
[5]
E. Witten, ‘Five-brane effective action in $M$-theory’, J. Geom. Phys. 22 (1997) 103, hep-th/9610234.
[6]
M. Henningson, B.E.W. Nilsson, and P. Salomonson, ‘Holomorphic factorization of correlation functions in $(4k+2)$-dimensional $(2k)$-form gauge theory’, JHEP 9909 (1999) 008, hep-th/9908107.
[7]
E. Witten, ‘Duality relations among topological effects in string theory’, JHEP 0005 (2000) 031, hep-th/9912086.
[8]
M. Henningson, ‘Commutation relations for surface operators in six-dimensional $(2,0)$ theory’, JHEP 0103 (2001) 011, hep-th/0012070.
[9]
M. Henningson and K. Skenderis, ‘Weyl anomaly for Wilson surfaces’, JHEP 9906 (1999) 012, hep-th/9905163. |
Abstract
In this paper we use methods developed in Part 1 of The Dipion Cocktail, to fit
the $p_{t}$ dependence of dipions for mid-central Au-Au collisions at
$\sqrt{s_{NN}}$=200 GeV. For the minijet fragmentation part we use PYTHIA
fragmentation as described in Part 1. For the thermal resonance production we
use an exponential growth behavior. The interference between the direct
production of dipion pairs from non-resonance minijet fragmentation and
re-scattering through resonance states gives a measure of the size of the
re-scattering region. This size is contain in the $\alpha$ parameter of Part1.
We assumed a relationship between the $\alpha$ parameter and the mass shift of
the $\rho$ resonance. The data used for the fits comes from the RHIC collider
as measured in the STAR experiment.
A Heavy Ion Fireball freeze-out Dipion Cocktail for Au-Au Collisions at $\sqrt{s_{NN}}$=200 GeV $p_{t}$ dependence(Part 2).
R.S. Longacre${}^{a}$
${}^{a}$Brookhaven National Laboratory, Upton, NY 11973, USA
1 Introduction
The ultra-relativistic heavy ion collision starts out as a state of high
density nuclear matter called the Quark Gluon Plasma(QGP) and expands rapidly
to freeze-out. During the freeze-out phase quarks and gluons form a system of
strongly interacting hadrons. These hadrons continue to expand in a thermal
manner until no further scattering is possible because the system becomes to
dilute. However this transition from quarks and gluons(partons) into hadrons is
not a smooth affair. The expansion is very rapid and some faster or hard
scattered partons fragment directly into hadron through a minijet[1]
process. Thus we have thermal and minijet hadrons present in the last
scattering of the hadrons. The Dipion Cocktail Part 1 considered this mixture
of sources and applied it to the dipion mass spectrum of the heavy ion fireball
formed in Au-Au collisions at $\sqrt{s_{NN}}$=200. Part 1 showed that both
thermal or soft production of hadrons and the minijet fragmented hadrons can
be described through a set of unified formal equations. Part 2(this paper)
applies this formalism to the $p_{t}$ dependence of dipions for Au-Au collisions
at $\sqrt{s_{NN}}=$ 200 GeV and 40% to 80% centrality.
The paper is organized in the following manner:
Sec. 1 Introduction. Sec. 2 Review of the two component model which we use to
fit the dipion data within a set of $p_{t}$ ranges. Sec. 3 Discussion of the
relationship between the $\alpha$ parameter and the mass and widths of
resonances. Sec. 4 we present a fit to 19 $p_{t}$ ranges for Au-Au collisions at
$\sqrt{s_{NN}}=$ 200 GeV and 40% to 80% centrality. Sec. 5 Summary and
Discussion.
2 Two component model with Breit-Wigner parameters
In this section we will alter equation 6 of Part 1 so it can use Breit-Wigner
parameters (mass, width) instead of phase shifts. We will also need to
modify the re-scattering part of the equation in order to have the correct
threshold behavior we have introduced in Part 1 for the minijet partial waves.
The phase shift can be written for the $\ell^{th}$ wave as
$$cot\delta_{\ell}=\frac{(M_{\ell}^{2}-M_{\pi\pi}^{2})}{M_{\ell}\Gamma_{\ell}},$$
(1)
where $M_{\ell}$ is the mass of the resonance in the $\ell$wave and
$\Gamma_{\ell}$ is its total width.
$$\Gamma_{\ell}=\Gamma_{0\ell}{\frac{qB_{\ell}(q/q_{s})}{M_{\pi\pi}}\over{\frac{%
q_{\ell}B_{\ell}(q_{\ell}/q_{s})}{M_{\ell}}}}$$
(2)
with $\Gamma_{0\ell}$ the total width at resonance, $B_{\ell}$ is the
Blatt-Weisskopf-barrier factor[2] for the $\ell$ of the resonance,
$q$ is the $\pi\pi$ center mass momentum, $q_{\ell}$ is $q$ at resonance,
$M_{\ell}$ is the mass of the resonance, and $q_{s}$ is center mass momentum
related to the size(1.0 fm is used $q_{s}$ = .200 GeV/c).
Using equation 1 we rewrite equation 6 of Part 1 as
$$|T_{\ell}|^{2}=|D_{\ell}|^{2}\frac{sin^{2}\delta_{\ell}}{PS_{\ell}}+\frac{|A_{%
\ell}|^{2}sin^{2}\delta_{\ell}}{PS_{\ell}}\left|\alpha+PS_{\ell}cot\delta_{%
\ell}\right|^{2}$$
(3)
The $D_{\ell}$ is the thermal production term and is constant except for the
Boltzmann weight(see equation 13 in Part 1). The expected threshold behavior
$q^{2\ell+1}$ comes from the $sin\delta_{\ell}$ term. Since there is
$sin^{2}\delta_{\ell}$ one of the $q^{2\ell+1}$ is killed off by dividing by
$PS_{\ell}$. In Figure 6 of Part 1 we see we have put into our minijet
$A_{\ell}$ the correct threshold $q^{2\ell+1}$ so we need to kill off the
$q^{2\ell+1}$ of the other $sin\delta_{\ell}$ term. Therefore the above
equation for our minijet $A_{\ell}$ we will use
$$|T_{\ell}|^{2}=|D_{\ell}|^{2}\frac{sin^{2}\delta_{\ell}}{PS_{\ell}}+\frac{|A_{%
\ell}|^{2}sin^{2}\delta_{\ell}}{PS_{\ell}^{2}}\left|\alpha+PS_{\ell}cot\delta_%
{\ell}\right|^{2}$$
(4)
Rewriting equation 6 of Part 1 for each partial wave with Breit-Wigner
parameters the first term becomes
$$|T_{\ell}|_{1}^{2}=|D_{\ell}|^{2}\frac{M_{\pi\pi}^{2}}{\sqrt{M_{\pi\pi}^{2}+p^%
{2}_{t}}}exp\frac{-\sqrt{M_{\pi\pi}^{2}+p^{2}_{t}}}{T}\frac{M_{\ell}\Gamma_{%
\ell}}{(M_{\ell}^{2}-M_{\pi\pi}^{2})^{2}+M_{\ell}^{2}\Gamma_{\ell}^{2}},$$
(5)
while the second term
$$|T_{\ell}|_{2}^{2}=|A_{\ell}|^{2}\frac{M_{\ell}^{2}\Gamma_{\ell}^{2}}{(M_{\ell%
}^{2}-M_{\pi\pi}^{2})^{2}+M_{\ell}^{2}\Gamma_{\ell}^{2}}\left|\alpha+\frac{2qB%
_{\ell}(\frac{q}{q_{s}})(M_{\ell}^{2}-M_{\pi\pi}^{2})}{M_{\pi\pi}M_{\ell}%
\Gamma_{\ell}}\right|^{2}\left(\frac{M_{\pi\pi}^{2}}{4q^{2}B_{\ell}^{2}(\frac{%
q}{q_{s}})}\right).$$
(6)
$$|T|^{2}=\sum_{\ell}|T_{\ell}|^{2}$$
(7)
where
$$|T_{\ell}|^{2}=|T_{\ell}|_{1}^{2}+|T_{\ell}|_{2}^{2}$$
(8)
and $|A_{0}|^{2}=S(M_{\pi^{+}\pi^{-}})$,$|A_{1}|^{2}=P(M_{\pi^{+}\pi^{-}})$,$|A_{2}|^{2}=D(M_{\pi^{+}\pi^{-}})$, and $|A_{3}|^{2}=F(M_{\pi^{+}\pi^{-}})$. S, P, D and F comes from
subsection 5.2 of Part 1.
3 A Relationship between the $\alpha$ parameter and the mass and widths of resonances.
Equation 6 of Part 1 has an important factor the coefficient $\alpha$.
This coefficient is related to the real part of the $\pi\pi$ re-scattering
loop and is given by equation 9. When the pions re-scatter or interact at
a close distance or a point the real part of the loop $\alpha$ has its maximum
value of $\alpha_{0}$. While if the pions re-scatter or interact at a distance
determined by the diffractive limit the value of $\alpha$ is zero. The $\alpha$
which is the real part of the re-scattering factor has a simple form given by
$$\alpha=(1.0-\frac{r^{2}}{r_{0}^{2}})\alpha_{0}$$
(9)
where $r$ is the radius of re-scattering in fm’s and $r_{0}$ is 1.0
fm or the limiting range of the strong interaction ranging to $r$ = 0.0 for
point like interactions.
When $\pi\pi$ pairs interact at the diffractive limit their phase shift should
be the same as the phase shift of the vacuum. The same statement is true for
$\pi\pi$ interacting at a point since asymptotic freedom demands that the
strong interaction should have no effect. However values of $\alpha$ in between
zero and $\alpha_{0}$ represent a confined volume where strongly interacting
gluons, quarks and virtual mesons may influence the phase shift of the
$\pi\pi$ system. Phase shifts under a Breit-Wigner parameters assumption
depend on the mass and width of the resonance parameter. In the next
section we use fits to data to determine the relationship between Breit-Wigner
parameters and the value of $\alpha$.
4 STAR data dipion $p_{t}$ range (0.2 GeV/c $<$ $p_{t}$ $<$ 4.0 GeV/c)
We have fitted 19 dipion $p_{t}$ ranges(see Table I) using equation 7 above for
the STAR Au-Au collisions at $\sqrt{s_{NN}}=$ 200 GeV 40% to 80%
centrality data. We included minijets up to $\ell$ = 3 and resonances $\sigma$
$\ell$ = 0, $\rho(770)$ $\ell$ = 1, and $f_{2}(1270)$ $\ell$ = 2. Using the
arguments of Sec. 3 of Part 1, we added the $f_{0}$ as a direct thermal term
($|T_{0}|_{1}^{2}$) and only the $\sigma$ interfered with $\ell$ = 0 minijet
background. Two other thermal terms are present in the cocktail, the $k^{0}_{s}$
and the $\omega_{0}$. All the thermal terms have an exponential behavior with
dipion $p_{t}$. The spectrum of the minijet partial waves is obtained from
PYTHIA[3](see Sec. 5.2 of Part 1). We let the data determine which
minijet partial wave to add. We find only Swave minijet background is important
until $p_{t}$ equal to 1.1 GeV/c. Above 1.5 GeV/c all four minijet partial waves
are used up to Fwave. It should be noted Dwave and Fwave are small effects.
We used PDG[4] for the $f_{2}(1270)$ mass = 1.275 GeV and width =
.185 GeV. The $f_{0}$ was fitted obtaining mass = 0.9727 $\pm$ .0039 GeV and
width = 0.04512 $\pm$ 0.01128 GeV. The $\sigma$ mass and width used was fixed
because it was ill determined. The mass used was mass = 1.011 GeV and width =
1.015 GeV. The $\rho$ mass and width is explained below.
Finally the threshold effective mass region .280 GeV to .430 GeV is dominated
by the Swave and receives contributions from minijet fragmentation, $\pi\pi$
Swave phase shift, $\eta$ decay, HBT adding to the like sign $\pi\pi$
distribution that has been subtracted away from the unlike sign $\pi\pi$ and
the coulomb correction between the charged pions. The minijet fragmentation is
the least known of the effects since we relied on PYTHIA, however there are
large uncertainty in all the other effects. So for these fits we let the
minijet fragmentation be free to fit the data and let the Breit-Wigner
parameters for the $\sigma$ determine the Swave phase shifts plus leaving
out all other effects.
For the $\alpha$ parameter in $p_{t}$ bins up to 1.1 GeV/c the minijet Swave
interference is the determining factor. Above 1.1 GeV/c the Pwave interference
becomes most important. The values of $\alpha$ which gives a reasonable fit are
shown in Table II.
We have determined that the $\sigma$ pole or Breit-Wigner parameters is so far
away from the real axis thus it is too short lived to be influenced by hadronic
interactions. The $\rho$ phase shift being of a life time comparable to
hadronic interaction taking place becomes most sensitive. We have found as a
function of $\alpha$ the best of $\rho$ width is always 0.147 GeV with an error
of $\pm$ .007 GeV. The mass however decreases as $\alpha$ grows, reaching a
minimum of 0.738 GeV at an $\alpha$ of 0.907. This is a mass shift of 37 MeV.
An $\alpha$ of 0.504 is the smallest $\alpha$ we find in our fits. A mass of
0.775 GeV is the best fit when the value of $\alpha$ is at 0.504. Using
equation 9 in Table II we determine the radius of $\pi\pi$ re-scattering for
each $p_{t}$ bin. The value of $\alpha_{0}$ used in Table II is equal to 2.0 as
determined in Appendix B of Part 1. Table II shows an interesting density
effect around dipion $p_{t}$ of 0.6 to 1.0 GeV/c. If one consider that $p_{t}$
maybe related to fireball size through the idea of hubble flow, then pions with
a $p_{t}$ of around 0.4 GeV/c maybe coming from a less dense region in the
central part of the fireball. This could be a density wave effect.
Table I. The $p_{t}$ ranges bins we fit.
Table I
$$p_{t}$$ bin number
lower edge(GeV/c)
upper edge(GeV/c)
1
0.2
0.4
2
0.4
0.6
3
0.6
0.8
4
0.8
1.0
5
1.0
1.2
6
1.2
1.4
7
1.4
1.6
8
1.6
1.8
9
1.8
2.0
10
2.0
2.2
11
2.2
2.4
12
2.4
2.6
13
2.6
2.8
14
2.8
3.0
15
3.0
3.2
16
3.2
3.4
17
3.4
3.6
18
3.6
3.8
19
3.8
4.0
Table II. The $\alpha$ value, $\rho$ mass value and radius in each $p_{t}$ range.
Table II
$$p_{t}$$(GeV/c)
$$\alpha$$
$$\rho$$ mass(GeV)
radius(f)
0.3
0.907 $$\pm$$ .028
0.738 $$\pm$$ .004
.739 $$\pm$$ .010
0.5
0.806 $$\pm$$ .025
0.748 $$\pm$$ .003
.773 $$\pm$$ .008
0.7
0.706 $$\pm$$ .022
0.755 $$\pm$$ .002
.804 $$\pm$$ .007
0.9
0.706 $$\pm$$ .022
0.755 $$\pm$$ .002
.804 $$\pm$$ .007
1.1
0.806 $$\pm$$ .025
0.748 $$\pm$$ .003
.773 $$\pm$$ .008
1.3
0.806 $$\pm$$ .025
0.748 $$\pm$$ .003
.773 $$\pm$$ .008
1.5
0.806 $$\pm$$ .025
0.748 $$\pm$$ .003
.773 $$\pm$$ .008
1.7
0.806 $$\pm$$ .025
0.748 $$\pm$$ .003
.773 $$\pm$$ .008
1.9
0.806 $$\pm$$ .025
0.748 $$\pm$$ .003
.773 $$\pm$$ .008
2.1
0.605 $$\pm$$ .019
0.759 $$\pm$$ .002
.835 $$\pm$$ .006
2.3
0.504 $$\pm$$ .016
0.775 $$\pm$$ .001
.865 $$\pm$$ .004
2.5
0.504 $$\pm$$ .016
0.775 $$\pm$$ .001
.865 $$\pm$$ .004
2.7
0.504 $$\pm$$ .016
0.775 $$\pm$$ .001
.865 $$\pm$$ .004
2.9
0.504 $$\pm$$ .016
0.775 $$\pm$$ .001
.865 $$\pm$$ .004
3.1
0.504 $$\pm$$ .016
0.775 $$\pm$$ .001
.865 $$\pm$$ .004
3.3
0.504 $$\pm$$ .016
0.775 $$\pm$$ .001
.865 $$\pm$$ .004
3.5
0.504 $$\pm$$ .016
0.775 $$\pm$$ .001
.865 $$\pm$$ .004
3.7
0.504 $$\pm$$ .016
0.775 $$\pm$$ .001
.865 $$\pm$$ .004
3.9
0.504 $$\pm$$ .016
0.775 $$\pm$$ .001
.865 $$\pm$$ .004
The 19 dipion spectrum for the $p_{t}$ bins are shown in Figure 1 through Figure
19.
The direct cross sectional yield of the thermally produced states into $\pi^{+}$
$\pi^{-}$ is shown in Figure 20. The states are $k_{s}$, $\rho$, $f_{0}$ and $f_{2}$
and the yields come from the exponential fits to the direct thermal component
of equation 7.
5 Summary and Discussion
The Dipion Cocktail Part 2 applies a formalism derived in The Dipion Cocktail
Part 1 which considered a mixture of sources present in the dipion
mass spectrum of the heavy ion fireball. Part 1 showed that both thermal or
soft production of hadrons and the minijet fragmented hadrons can
be described through a set of unified formal equations. Part 2(this paper)
applies this Part 1 formalism to the $p_{t}$ dependence of dipions for Au-Au
collisions at $\sqrt{s_{NN}}=$ 200 GeV and 40% to 80% centrality.
Part 1 started with the basic definition of elastic $\pi\pi$
scattering. Next showed how re-scattering of pions depends on the unitary
condition that interactions present in the phase shift of an orbital state
must interact all the time. The process of parton fragmentation into dipion
states through unitarity leads to a equation of production and re-scattering
in a given orbital quantum number. This equation(equation 7) has two
components in each orbital state: one being the thermal production of
resonances in a dipion orbital state, the other is the re-scattering of
dipions coming from parton or minijet fragmentation into the dipion orbital
state which do not come directly from the resonance. Unitarity requires that
there most be re-scatter through resonance phase shifts which we defined
through Breit-Wigner parameters (mass, width).
We have fitted 19 dipion $p_{t}$ ranges(see Table I) using equation 7. We
included minijets up to $\ell$ = 3 and resonances $\sigma$ $\ell$ = 0,
$\rho(770)$ $\ell$ = 1, and $f_{2}(1270)$ $\ell$ = 2. Using the arguments of
Sec. 3 of Part 1, we added the $f_{0}$ as a direct thermal term ($|T_{0}|_{1}^{2}$) and
only the $\sigma$ interfered with $\ell$ = 0 minijet background. Two other
thermal terms are present in the cocktail, the $k^{0}_{s}$ and the $\omega_{0}$.
All the thermal terms have an exponential behavior with dipion $p_{t}$ and are
shown Figure 20. The spectrum of the minijet partial waves is obtained from
PYTHIA[3](see Sec. 5.2 of Part 1). We let the data determine which
minijet partial wave to add. We find only Swave minijet background is important
until $p_{t}$ equal to 1.1 GeV/c. Above 1.5 GeV/c all four minijet partial waves
are used up to Fwave. It should be noted Dwave and Fwave are small effects.
We used PDG[4] for the $f_{2}(1270)$ mass = 1.275 GeV and width =
.185 GeV. The $f_{0}$ was fitted obtaining mass = 0.9727 $\pm$ .0039 GeV and
width = 0.04512 $\pm$ 0.01128 GeV. The $\sigma$ mass and width used was fixed
because it was ill determined. The mass used was mass = 1.011 GeV and width =
1.015 GeV.
For the $\alpha$ parameter in $p_{t}$ bins up to 1.1 GeV/c the minijet Swave
interference is the determining factor. Above 1.1 GeV/c the Pwave interference
becomes most important. The values of $\alpha$ which gives a reasonable fit are
shown in Table II.
We have determined that the $\sigma$ pole or Breit-Wigner parameters is so far
away from the real axis thus it is too short lived to be influenced by hadronic
interactions. The $\rho$ phase shift being of a life time comparable to
hadronic interaction taking place becomes most sensitive. We have found as a
function of $\alpha$ the best of $\rho$ width is always 0.147 GeV with an error
of $\pm$ .007 GeV. The mass however decreases as $\alpha$ grows, reaching a
minimum of 0.738 GeV at an $\alpha$ of 0.907. This is a mass shift of 37 MeV.
An $\alpha$ of 0.504 is the smallest $\alpha$ we find in our fits. A mass of
0.775 GeV is the best fit when the value of $\alpha$ is at 0.504. Using
equation 9 in Table II we determine the radius of $\pi\pi$ re-scattering for
each $p_{t}$ range. Table II shows an interesting density effect around dipion
$p_{t}$ of 0.6 to 1.0 GeV/c. If one consider that $p_{t}$ maybe related to fireball
size through the idea of hubble flow, then pions with a $p_{t}$ of around
0.4 GeV/c maybe coming from a less dense region in the central part of the
fireball. This could be a density wave effect.
6 Acknowledgments
This research was supported by the U.S. Department of Energy under Contract No.
DE-AC02-98CH10886. The author thanks William Love for the STAR analysis of the
angular correlation data from Run 4. Also for his assistance in the production
of figures. It is sad that he is gone.
References
[1]
T. Trainor, Phys. Rev. C 80 (2009) 044901.
[2]
F. von Hippel and C. Quigg, Phys. Rev. 5 (1972) 624.
[3]
T. Sjostrand, M. van Zijil, Phys. Rev. D 36 (1987) 2019.
[4]
J. Beringer et al. (Particle Data Group), J. Phys. D86 (2012)
010001. |
CCTest: Testing and Repairing Code Completion Systems
Zongjie Li${}^{a}$, Chaozheng Wang${}^{b}$, Zhibo Liu${}^{a}$, Haoxuan Wang${}^{c}$, Shuai Wang${}^{a}$, Cuiyun Gao${}^{b}$
${}^{a}$ The Hong Kong University of Science and Technology, Hong Kong SAR
${}^{b}$ Harbin Institute of Technology, Shenzhen, China
${}^{c}$ Swiss Federal Institute of Technology Lausanne, Switzerland
{zligo,zliudc,shuaiw}@cse.ust.hk, {wangchaozheng}@stu.hit.edu.cn
{gaocuiyun}@hit.edu.cn, {haoxuan.wang}@epfl.ch
Abstract
Code completion, a highly valuable topic in the software development domain, has
been increasingly promoted for use by recent advances in large language models
(LLMs). To date, visible LLM-based code completion frameworks like GitHub
Copilot and GPT are trained using deep learning over vast quantities of
unstructured text and open source codes. As the paramount component and the
cornerstone in daily programming tasks, code completion has largely boosted
professionals’ efficiency in building real-world software systems.
In contrast to this flourishing market, we find that code completion models
often output suspicious results, and to date, an automated testing and
enhancement framework for code completion models is not available.
This research proposes CCTest, a framework to test and repair code completion
systems in blackbox settings. CCTest features a novel mutation strategy, namely
program structure-consistency (PSC) mutations, to generate mutated code
completion inputs. Then, it detects inconsistent outputs, representing likely
erroneous cases, from all the completed code cases. Moreover, CCTest repairs the
code completion outputs by selecting the output that mostly reflects the
“average” appearance of all output cases, as the final output of the code
completion systems.
We detected a total of 33,540 inputs that can trigger likely erroneous cases
from eight popular LLM-based code completion systems. With repairing, we show
that the performance of code completion models notably increased by 53.51%
on average.
I Introduction
Large language models (LLMs) such as GitHub Copilot [23], OpenAI’s
Codex [1] and Tabnine [5] are increasingly being promoted
for use within the software development domain. Such models are built using
machine learning (ML) over vast quantities of unstructured text, including
websites, books, and open source codes. This enables them to produce
“completions” given some input prompt made up of code and comments
(documentation). To date, de facto LLM-based code completion frameworks are
advocated with the aim to provide an “AI pair programmer” capable of
automatically generating programs from natural language specifications or code
snippets.
Despite being the highly powerful and promising component in augmenting modern
software development, our observation is that code completion systems are not
perfect, and they frequently generate confusing and likely erroneous results. We
argue that the “sub-optimal” and even buggy behavior of code completion
systems are undesirable, undermining the performance and usability of code
completion. Nevertheless, it is yet neglected by today’s research community,
whereas preliminary studies on LLM-based code completion frameworks are
primarily on its security implication, cost reduction, or potential extension in
different
domains [68, 46, 44, 29, 50].
Considering Fig. 1(a), where Codegen generates a code
snippet as the completion of the prompt input. Moreover, by slightly tweaking the
prompt, we find that Codegen outputs a dramatically different code snippet, as
illustrated in Fig. 1(b).
While it is generally obscure to directly decide the “correctness” of these
two completed code snippets, given that the two inputs are (semantically)
similar to a human observer, the high distinction of two completed code snippets
indicates that Codegen’s outputs are of low consistency, which is a sign of
erroneous outputs.
Overall, this obvious inconsistency in the generated code completions
motivates us to test and repair code completion systems. Thus, our intuition is
to form a testing oracle, with the aim of checking the structure-level
consistency in completed code snippets.
This paper introduces CCTest, an automated testing and repairing framework for
code completion systems.
Our research thrusts are two-fold. First, we design a set of program mutation
strategies, namely program structure-consistency (PSC) mutations, to generate
mutated code snippets with similar or identical program structures. Given the
corresponding set $O$ of code completion outputs for each (mutated) code
snippet, we identify erroneous cases (i.e., outliers) in $O$ by defining and
comparing program distances of generated code completions. The mutated program
whose derived code completion output has long distances compared to
other mutated completion outputs will be deemed as spurious.
We further design a code repairing scheme, forming a post-processing process to
repair the code completion outputs by selecting the code completion output
$\hat{o}$ that mostly reflects the “average” appearance of $O$. We show that
$\hat{o}$ generally manifests high consistency and quality with the reference
inputs, extensively improving the accuracy of code completion systems.
Furthermore, our testing and repairing scheme treats the code completion models
as a “black box”, and we do not assume any specific implementation details of
the code completion systems or their underlying LLMs.
CCTest offers an up-to-date assessment of de facto LLM-based code completion
systems and the correctness of their outputs, which may impede reaching the full
potential of modern “AI pair programmer” in software development.
From a total of 182,464 test inputs used for this study, we found 33,540
programs exposing code completion errors from eight widely-used LLM-based code
completion systems, one of which (Copilot) is a popular commercial tool,
and the other seven are either actively developed and maintained by the
non-profit organization (CodeParrot [18]) or hosted by the
industrial companies (EleutherAI’s GPT-Neo [8] and Salesforce’s
Codegen [42]). With repairing, we show that the
average performance of code completion models notably increased by 40.1%
and 67 % on two metrics. In sum, this research makes the following
contributions:
•
To study LLM-based code completion systems in a realistic setting and to
delineate their up-to-date capabilities, we introduce and advocate a new
focus, conducting comprehensive and large-scale testing and repairing on their
outputs. The findings obtained in this study will guide future research that aims
to use and improve code completion tools.
•
We propose CCTest, an automated testing and repairing framework for code
completion systems. CCTest features PSC, a novel testing scheme that is
particularly designed to test code completion systems. We further propose a
black-box repairing scheme to augment the accuracy of code completion outputs.
CCTest incorporates a set of design principles and optimizations to deliver an
efficient testing and repairing workflow.
•
Empirically, we evaluate one commercial with seven popular free code
completion systems, and we successfully found 33,540 programs causing
code completion errors. Our repairing scheme further enhances the accuracy of
the code completion models with much higher accuracy. We made various
observations and obtained inspiring findings regarding modern LLM-based code
completion systems.
II Preliminary
This section introduces the background of code completion systems to deliver a
self-contained paper. Note that the automated testing and repairing pipeline
shipped by CCTest treats each code completion system as a black box,
and we do not assume a specific internal. Nevertheless, given the
dominating usage of LLMs in forming today’s code completion systems, we
primarily introduce LLM-based code completion models. Our evaluations are
accordingly designed to test LLM-based code completion models.
Fig. 2 presents a holistic view of the LLM-based code completion
system, and explains how CCTest fits the workflow to test and repair a code
completion system. Benefit from the prosperous development and major success of
transformer-based natural language models such as OpenAI’s GPT2 and
GPT3 [31, 13, 9], the code
completion task, as a typical conditioned open-ended language generation task,
are extensively improved with much higher accuracy.111A full introduction
of transformer models and how it is trained to boost open-ended language
generation is beyond the scope of this paper. Interested audiences can refer
to [55].
Though details of training datasets are often obscure, modern LLMs-based code
completion systems are advertised as being trained with over millions or even
billions of lines of code [23]. Typically, the input training data are
dissected into sentences and further into tokens, where each token comprises a
sequence of characters before being fed to LLMs. For instance, the tokenizer of
Codex and GPT-3 is configured to produce tokens with an average of four
characters. Tokens are encoded with unique numeric identifiers, up to
user-defined vocabulary size. This process is often referred to as byte pair
encoding (BPE) [20], allowing the models to ingest more text into
their (fixed-size) input window. Various techniques have been proposed to
configure and improve the performance of LLMs, such as learning from rare tokens
and deciding a proper stop word.
During the query phase, the code completion system’s input is often referred to
as a prompt, denoting an incomplete code snippet. Similarly, the prompt
code snippet is first abstracted into sentences and further into tokens, for
which the code completion system can predict a list of code completion
suggestions (ranked by the confidence scores) that mostly likely
continue/complete the input prompt. For instance, the code completion system is
frequently assessed by completing a function body, given the function prototypes
and some statements in the function prologue.
Note that modern code completion systems can process prompts of different types,
including code snippets and natural language descriptions (comments) of the
expected program. For natural language sentences, they are usually divided into
background, input and output to describe a competitive programming problem [33].
CCTest is designed to test the code completion system by mutating the
prompts into a variant set $O$ and identifying bug-triggering prompt
variants using an implicitly-formed oracle. Moreover, CCTest can repair the code
completion output, by analyzing outputs $O$ of prompt variants to select cases
that are most close to the “average” appearance of $O$. We now introduce
the workflow of CCTest in Sec. III.
III Approach Overview
Fig. 3 presents an overview of CCTest in terms of testing and
repairing code completion systems. In particular,
① Prompt Variants Generation. CCTest launches PSC and generates
a set of structural-consistent variants $P$ of an input prompt $p_{0}$. Here, we
propose a novel testing scheme, PSC, that mutates a prompt with code
structural-invariant transformations. The mutated prompts manifest closely
related structural representations from a human observer’s perspective.
② Testing Oracle Generation. The target code completion model
outputs a set of code completion outputs in accordance with the prompt variants.
Here, we form a testing oracle by comparing the structural-level consistency
(similarity) among completion outputs. The key observation is that, for prompt
variants with closely similar structures, the code completion outputs should
consistently manifest high (visual) consistency. Thus, “outlier” code
completion outputs are deemed erroneous, whose corresponding prompt variants are
defect-inducing inputs.
③ Inconsistent Completion Repairing. We note that the
aforementioned testing pipeline can be extended to automatically repair the
inconsistent code completion outputs. To this end, CCTest identifies output
$\hat{o}$ that is mostly close to the “average” appearance of $O$ (after
excluding outliers in ②). Note that this step is also applicable to
black-box models.
Study Scope. We primarily target the erroneous code completion
outputs, denoting “stealthy logic bugs” of code completion systems. As
detailed in Sec. III-B, our testing approach can automatically
expose inconsistency defects in code completion systems, meaning that when given
a set of structurally-consistent prompts, the completion outputs are deemed to
share closely similar appearances as well.
During preliminary study, we also find that when feeding a set of
structurally-consistent prompts to code completion systems, certain prompt
variants may somehow impede the code completion systems from generating any
completion outputs. In some sense, this is comparable to identifying a “crash” of
the code completion system. Considering that fuzzing or random testing may
likely trigger such salience states during in-house development, they are not
the primary focus of CCTest. Nevertheless, we still record and report all such
defects we encountered during the evaluation.
Note that we are not using extreme (broken) prompts to stress code
completion systems. Modern code completion models are on the basis of LLMs, and
our preliminary study shows providing some trivial, broken prompts may make
the code completion system to generate meaningless outputs. The current
implementation of CCTest supports to generate syntactic valid Python code snippets as
prompts for testing, given the popularity and representativeness of the Python language.
Nevertheless, it is easy to see that our method is not limited to Python
code; we leave extending CCTest to support other programming languages as one
future work.
III-A Program Structure-Consistency (PSC) Mutations
We first introduce the PSC mutation scheme, which performs structural-consistent
mutations over an input prompt. Overall, our key observation is
that programs with similar/identical control structures will retain such
consistency in the corresponding completion code. Hence, by observing certain
code completion outputs manifesting inconsistent control structures, potential
code completion errors can be flagged.
Fig. 3 depicts the workflow of PSC testing. Starting from an
input prompt $p_{0}$, we conduct a two-step approach to 1) performing
structural-consistent mutation to generate a set of mutated prompts with
similar, if not identical, control structures and collect the code completion
output $o_{k}$ from each mutated prompt $p_{k}$, and 2) employing a testing
oracle to cross compare the similarity of the code completion outputs among
$o_{k}$ and identify outliers. We identify outlier $o_{i}$ such that its control
structure has an abnormally long distance compared to the others; these outliers
are deemed as wrong code completion errors. In this section, we detail the
design of PSC code mutations.
To generate structural-consistent mutation, one can replace one code fragment
$s_{0}$ (e.g., a statement) in the seed prompt $p_{0}$ with similar fragments
using one PSC transformation and generate one mutated prompt $p_{k}$. The
mutated prompt is expected to retain program structures compared to $p_{0}$.
III-A1 Mutation Schemes
At this step, we implement a set of PSC mutations, which perform transformations
on different levels of the program hierarchical structure, ranging from identifiers,
instructions, to basic blocks. Table I lists all the
PSC mutations adopted in this work. We now introduce each mutation below. Note
that due to the limited space, here we only discuss the design and
implementation essential.
Design Goal. Overall, the mutation scheme is designed as incremental and
retains the structural-level representation of the mutated prompts. However, we
make an encouraging observation that such straightforward mutation imposes notable
challenges for code completion systems to comprehend the inputs and accordingly
generate consistent completion outputs over prompts and the mutated variants;
see sample flaws in Sec. V-B.
REP_R & REP_C. Our first mutation scheme performs
renaming toward function parameters. Overall, this pass will replace one
parameter in the function’s parameters list with a new identifier, and every
usage of this parameter will be replaced accordingly. The way of naming can
either be “regulate” or “context”. Considering the following case:
where for the “regular” scheme (REP_R), we replace the
function parameter b with Param1. As REP_C which
takes the specific context into account, we extend the function parameter
b with a new identifier that subsumes both function name add
and b.
REL_R & REL_C. In addition to mutating the function
parameters, we also propose another mutation scheme at the identifier level to
rename local variables. Similar to replacing parameters, this pass will randomly
select a local variable whose scope is in the function. Then, we replace
all of its references with a new identifier. Considering the following case:
where for REL_R, we replace the local variable res
with LocalVar1. As for REL_C which takes the context
information into account, we rewrite the local variable res with a new
identifier that subsumes both function name compare and res.
IRR. This scheme implements a set of mapping rules, such that it
will search and replace certain common arithmetic operations into its semantics
equivalent expressions, but with different syntactic forms. Considering the
following case,
where the += statement is replaced with a standard addition
expression. Overall, we implement 4 replacement rules over different
common arithmetic expressions at this step.
RTF. In addition to mutating arithmetic expressions, we also
implement RTF schemes to mutate boolean expressions, particularly
expressions used in forming branch conditions, with its semantics equivalent
versions. Considering the following case:
where the boolean expression is extended with an always-true condition.
This would not alter the functionality, nor does it largely change the program
structures. However, we find that such schemes can effectively impede code
completion systems from generating structural-consistent outputs, as shown in
Sec. V.
GRA_R & GRA_C. This mutation inserts a small chunk of
“garbage code” into the program, which does not alter the program semantics,
but slightly increases the program’s control flow complexity. In particular, we use
always-false conditions to form an if condition, where we insert a small set of
statements into the newly formed branch, which will never be executed. Similar
to mutations mentioned above, creating garbage code may take context information
into account. Considering the following case:
where GRA_R inserts a branch whose if condition
expression is “False” with a new variable named TempVar. As for
GRA_C that takes context into consideration, we create an always-false
condition on the basis of the syntactic form of the function parameters, and use
it to form the if condition expression. Similarly, variable names in
the enclosed branch also subsume both function name and TempVar.
INI. The last mutation pass inserts a “print” statement into
the prompt. Overall, this scheme is designed based on the observation that
programmers often insert such “print” statements during the debugging process,
such that they print the status of some variables to help debugging. See the
following example:
where we insert print into the prompt. Though this scheme
does not change the program structures, we find that it effectively improves the
performance of popular code completion systems; see details in
Sec.V-C.
Alternative: Mutating Natural Language Comments. It is worth noting that
besides processing prompts in the source code form, modern code completion
systems can also generate code completions given prompts in natural language
sentences. In such cases, the input sentences are usually code comments or
descriptions of the intended code functionality.
Careful audiences may wonder about the feasibility of mutating natural language
comments to test code completion models, whose expected workflow may be similar
to recent advances in testing machine translation
systems [52, 54].
In fact, we tentatively explored this direction. We clarify that unlike machine
translation system testing, whose inputs are arbitrary natural language
sentences, code comments are often in limited forms, and a vast
majority of code comments have no
adjectives [47, 45].
This imposes a major difficulty in comprehensively mutating “synonyms” in code
comments for testing purposes.
III-B PSC Testing: Forming Testing Oracles
Given a list of code completion outputs $\mathcal{O}$ in accordance with the
mutated prompts $\mathcal{P}$, we compare each output $o_{i}\in\mathcal{O}$ with the remaining cases in $\mathcal{O}$ and decide if it is an
“outlier.” The full algorithm is shown in Algorithm 1.
We first iterate each case in $\mathcal{O}$, decide its similarity with the
remaining cases (lines 2–3), and normalize the scores (line 4). Here, we compute
the similarity score using the Levenshtein distance implemented by fuzzywuzzy, a
common algorithm to decide the edit distance among two programs.
Then, we employ a threshold $T$ to decide whether a case exhibits anomalously low
similarity with other cases for more than $T$ times (lines 5–12). and if so, the
case will be deemed as an outlier. The corresponding $p_{i}$ deems as an
error-inducing input.
Overall, given that we have implemented $N$ ($N=9$ in the current implementation
of CCTest) passes to mutate a prompt, $T$ is a configurable hyper-parameter ($T\leq N$), such that a code completion output deems an outlier, if its distance
with $T$ code completion outputs are longer than the average distance of code
completion output pairs.
$T$ will be decided with empirical evidences, as will be discussed in
Sec. V-B. After performing Algorithm 1, we keep the
remaining code completion outputs $\mathcal{O}^{*}=\mathcal{O}\setminus\mathcal{L}$ for usage in the repairing phase, as will be explained in
Sec. III-C.
Alterantive Testing Oracles. This section primarily focuses on the
establishment of an implicit testing oracle, by comparing the
consistency among code completion outputs $O$; outliers will be deemed
as errors, given their corresponding input prompts are visually similar under
our carefully designed PSC schemes. Given that said, careful readers may wonder
about the feasibility of forming alternative testing oracles, which may be more
obvious. Below, we discuss two alternative testing oracles and analyze their
(in-)capability in our research context.
Comparing Syntactic Form According to Reference. One straightforward way
of forming a test oracle is to cut a reference program $p$ into two chunks $p_{1}$
and $p_{2}$, whereas $p_{1}$ serves the prompt, and $p_{2}$ serves the code completion
reference; we decide the error of code completion model by comparing its
completion output $o_{1}$ with $p_{2}$.
We clarify that this oracle is often too strict.
Fig. 4 compares the code completion output of Copilot and the
reference code snippet (this code snippet is from LeetCode). The reference code
snippet leverages a helper function inorder to deliver a classic
inorder traversal of the input binary tree. Copilot’s completion output appears
to present a code completion output on the basis of recursive calls. Overall,
though the code completion output appears (visually) distinct with the reference
code snippet, the code completion output is functional valid.
We believe that strictly comparing the syntactic equality of code completion
outputs with the reference code, though straightforward, may not necessarily and
faithfully reflect the true defects of code completion systems. In other words, such
a strict comparison may induce an extensive amount of false positive cases.
Checking Functionality According to Reference. To form the testing
oracle for code completion systems, recent works have also explored strictly
checking the functionality correctness of code completion outputs in accordance
with the reference code snippet. In particular, [41]
forms the prompt using function prototypes and some code snippets from leetcode
to Copilot, and counts the number of passed test cases over the code completion
outputs (when being used together with the prompts). More passed test cases
indicate that the code completion outputs are of higher quality. We admit that
this approach is more flexible than the aforementioned alternative, given that
it allows some (syntactic) changes in the output, as long as it does not alter
the functionality. Nevertheless, we respectively argue that this may be less
desirable than our formed oracle. In principle, modern code completion systems
are not championed to replace human programmers; rather, it aids human
programmers with suggested code completion outputs that are useful. In other
words, we believe strict functionality accuracy may not be the first design
choice for a code completion system (e.g., some sloppy arithmetic errors in
completion outputs may be easily fixed by the users in a post-process phase).
Thus, we believe it is more desirable to cross compare the structural-level
consistency (as in CCTest), rather than strictly comparing the functionality
correctness.
III-C Repairing
In this section, we present the technical pipeline of repairing the code
completion outputs. With the testing launched in Sec. III-B, we have
collected a set of code completion outputs $\mathcal{O}^{*}$ with outliers being
excluded.
At this step, we aim to identify a code completion output that appears mostly
close to the “average appearance” of $O^{*}$. To this end, we measure the
average pair-wise edit distance $\hat{d}$ of every $o_{i},o_{j}\in O^{*}$. Then,
we search for an $\hat{o}\in O^{*}$, whose average pair-wise distance with all
other elements in $O^{*}$ is the closest to $\hat{d}$. This $\hat{o}$ will be
the repaired code completion output returned to the users.
IV Implementation and Evaluation Setup
CCTest is implemented in Python, with about 5k LOC. CCTest is currently
implemented to mutate Python code, given the popularity of this language in
software development and the matureness of corresponding code completion
systems. We now discuss the implementation and evaluation setup.
Parsing and Mutating Programs. We parse Python code with
tree-sitter [10], a mature and dependency-free parser generator
tool with an open-source license. It is widely used in code-related projects such
as Codebert [19], and it does not require the input code to be executable, meaning
that incomplete code fragments without a building script can also be parsed. We
first parse the prompt code into a concrete syntax tree, and conduct several
sanity checks (see in Sec. V-A) on the target code to see which transformation
pass could be performed. Then, after applying feasible transformations, CCTest will output the corresponding transformed code with IDs of the applied
transformation passes.
Seed Programs. Consistent with most research in this field, we form our
evaluation dataset from a hosted Leetcode answers repo [3] and CodesearchNet
[28]. Leetcode is an online platform for practicing algorithmic coding
challenges designed to prepare software engineers for technical interviews.
As shown in III, since the LLM-based code completion systems have the limitation of the max token size,
we only select the programs whose token length is between 64 and 2048.
Overall, each seed program contains a medium size function with
presumably complex control structures. As shown in Table II, we
picked up 613 code snippets from the Github repo as the seed programs.
The total number of generated variants for Leetcode programs is 4,296.
We also take another commonly-used dataset,
CodeSearchNet [28], in the evaluation. This dataset is
particularly designed and widely-used in the research of code representation
learning. The Python component of this dataset contains train, validation, and
test splits; we use the test split for the evaluation.
Besides the similar data filtering process for Leetcode, we only keep one code snippet
for the codes in the same “path” in the test split to keep the result balanced.
Each program in this dataset contains a Python function with medium
size and non-trivial control structures. We use a total of 2,297 code samples
from this dataset, as shown in Table II.
We clarify that both LeetCode and CodeSearchNet are deemed proper
for neural code learning tasks like code
completion [28]. Thus, errors and repairing revealed
in our evaluation should mostly reflect common obstacles and enhancements users
can expect during daily usage of code completion systems. In contrast, too
complex prompts (e.g., real-world complex software) may inherently impede the
understanding and assessment of LLM-based code completion systems.
Statistics of Test Cases. Table II reports statistics of the Python
programs used in the study. We use Leetcode and CodeSearchNet to generate 2,910 programs.
These programs are the seed inputs of our mutation scheme for each LLM-based code completion system.
The total number of generated variants is 19,898 (see Table V
for the breakdown).
Each seed program contains a medium size funtion with presumably complex
control structures and many global variables. For each seed program with its variants,
we equally split the function into two parts, the first part is used as “prompt” and the
remaining is used as “oracle” to measure the quality of the result.
Code Completion Systems. Table III reports the code completion
systems used in this study.
First, we use Github Copilot, one highly visible commercial code completion
system that generates quite a buzz in the community [63]. We
purchase the standard commercial license (for single user) to unleash its full
potential. As “black-box” commercial products, it is unclear about
their implementation details (marked $?$ in Table III).
We also evaluate several well-known LLM models for code completion, including
CodeParrot [18], GPT-Neo [8], GPT-J [17], and
CodeGen [42]. All of these models are deemed
employing large-scale language models, given that up to billions of parameters
are involved in their underlying models, and tens of thousands of vocabularies
are considered. CodeParrot is a GPT-2 model that is trained specifically for Python
code generation. We use two variants, CodeParrot-small and CodeParrot. The
smaller variant contains less amount of parameters and is trained over less
data. Nevertheless, we find that both variants manifest a high level of code
generation capability. As for GPT-Neo, we use two variants, GPT-Neo-125M and
GPT-Neo-13B, which are both GPT3-like models. Our observation shows that
comparing to its larger variant, GPT-Neo-125M is prone to generate code snippets
with less diversity but more straightforward.
Given that said, we believe that both models are well-suited for code
generation, and they indeed manifest a reasonably high robustness under our
testing campaign. GPT-J is also a transformer model; the pre-trained model used
in our research contains 6B parameters, and is often referred to as GPT-J-6B.
Two versions of CodeGen are evaluated, and it is
specially designed for program synthesis. We use two variants,
CodeGen-2B-mono and CodeGen-6B-mono, which are trained on a large corpus with Python code,
to generate the code snippets.
All these systems are stated to be trained on a large corpus of open-source
source code. For instance, Copilot is noted to include the vast majority of
GitHub’s open-source code. GPT-J and GPT-Neo are trained on the Pile
dataset [22], a diverse language modeling dataset.
For all LLMs except Copilot, we download their pre-trained models from
huggingface [2] and run the code completion locally.
In all, LLM-based code completion systems presented in Table III, to the
best of our knowledge and experience, represent the best systems available to
the public. We indeed tentatively explored other code completion solutions based
on conventional machine learning or rule-based approaches. We clarify that these
conventional methods are seen to produce much worse and shallow code completion
outputs compared with these LLM-based systems.
V Findings
In evaluation, we mainly explore the following research questions.
RQ1: Can CCTest generate high-quality and structural-consistent prompt variants?
RQ2: How effective is CCTest to detect code completion defects?
RQ3: To what extent can CCTest enhance the quality of code completion outputs?
We answer each research question in one subsection below.
V-A RQ1: Effectiveness on Input Generation
Answering RQ1 requires assessing the quality of mutated prompts. At
this step, for each seed prompt, we generate mutants and first check whether
they pass the compilation. In particular, for each sentence, we generate up to 9
mutants, leading to a total of 19,898 mutant prompts generated on top of 2,910
seed prompts. As a quick sanity check on the validity of transformed code, we
use the standard ast module in Python to compile all transformed
Python prompts. We report that all these generated mutant prompts can pass the
compilation, indicating that they are valid prompts.
To illustrate the structural-level consistency of mutated prompts, we compute
and cross compare the distances of the mutated prompts from the same seed
prompt. Ideally, the smaller distance indicates that the mutants manifest
closely related structures, thus justifying the consistency of derived code
completion outputs. We use pycode-similar [4], a
well-performing tool to decide Python code similarity based on AST tree
matching. Let the prompt AST have $n$ nodes, where $m$ nodes are matched toward
nodes on the mutated prompt’s AST. pycode-similar returns ratio
$\frac{m}{n}$, denoting how similar two programs are. We report the distribution
of the “distance score” (distance is computed as $1-\frac{m}{n}$) in
Table IV. Overall, it evident that for the vast majority of
mutated prompts (over 98%), structural-level distance is less than 10%.
Recall that several transformation passes in CCTest (Sec. III-A1) introduce only identifier-level changes. As
expected, the distance scores between their mutated prompts and the seed prompts
are zero, implying that program structures are retained.
A few cases manifest a relatively high distance. With manual inspection, we find
that it is primarily due to the fact that the size of the original prompt is
short, leading to an increased distance $1-\frac{m}{n}$ where $n$ is small.
Answer to RQ1: CCTest can generate highly quality prompt mutants that
are grammatically valid and structurally consistent.
V-B RQ2: Bug Detection
To answer RQ2, we launch testing to detect code completion defects.
Table V provides an overview of our findings. Note that we use two
datasets for the testing. For instance, 4909+17899 in the Copilot “#Prompts”
cell denotes that we generate 4,909 mutated prompts over the LeetCode dataset,
and 17,899 mutated prompts are generated from the CodeSearchNet dataset.
Similarly, “3003 + 12101” in the first “#Defect $T=1$” cell means that
3,003 outliers are found using the LeetCode mutated inputs, whereas 12,101
outliers are detected by using the other test input set. For the outlier
detection evaluation, we assess the performance under different values of the
hyper-parameter $T$ (see details below). Recall CCTest offers two metrics,
bleu4 score and edit distance score, to assess the effectiveness of
repairing. For each model (under different outlier thresholds $T$), we quantify
the repairing effectiveness under both metrics, and also with respect to the two
leveraged datasets.
First, As shown in the second column of Table V, while nearly all
mutated prompts can be processed, we still find 58 (0.03%) mutated
prompts that trigger “no response” for the tested code completion systems. As
expected, such cases are rare, given the high capability and comprehensiveness
of LLM-based production code completion systems.
As clarified in Sec. III-B, CCTest leverages a threshold $T$
to form the testing oracle: a code completion output is deemed as an
outlier, if it has a long distance with $T$ code completion
outputs mutated from the same seed prompt. Therefore, $T$ is an integer ranging
from 1 to the total number of passes ($9$ in our implementation).
Table V reports the number of uncovered outliers in accordance with
different models and thresholds. Overall, out of in total 182,464 test cases, a
different number of defects are found under five thresholds $T$. In particular,
when setting the threshold $T$ as 9, we find 5,912 and 27,628 defects for 8 LLMs
on the LeetCode and CodeSearchNet datasets, respectively.
As expected, we observe that with the increment number of $T$, the number of
detected defects (outliers) decreases. As we illustrated in
Algorithm 1, the stringency of selection for outliers depends on the
threshold $T$, where a larger $T$ represents a more strict standard. For
example, compared to $T=5$, the number of the detected outlier for codeparrot on
LeetCode is only 904, less than a quarter of the former (3,631). And if we set
$T=1$, it is evident that nearly all the prompts are deemed as “outliers”,
which implies a high false positive rate.
Copilot outperforms the other seven LLMs, given less number of inconsistency
defects (293+2184 for $T=9$). However, Copilot has the most “no response”
failures (41 out of 58). We believe the reason behind such observation is that
its output is related to the network quality and it would be possible that
Copilot refuse to return anything even query for ten times. Despite the small number
of “no response” failures, all the other cases can be analyzed and completed
with some non-trivial outputs.
Overall, Table V illustrates that the inconsistency bugs in code
completion systems are pervasive, and a large number of defects can be found
even if bugs remain even for highly permissive consistency thresholds. We
present two representative cases in Fig. 5 and
Fig. 6, respectively.
In particular, Fig. 5 presents a case, such that the code
completion output of the seed prompt is largely deviated from the ground truth.
Note that when only given the code completion output in
Fig. 5(b), it is inherently challenging to decide if the
deviation between code completion outputs in Fig. 5(a) and
Fig. 5(b) are due to model capacity or bugs. Nevertheless, when
referring to Fig. 5(c), it becomes evident that the tested code
completion system, Copilot, is capable of generating high-quality code
completion outputs that manifest closer structural representation to the ground
truth. Note that comparing to the seed prompt, we only tweak a little on the
parameter name for the prompt used in Fig. 5(c). Thus, it becomes
accurate to assume that Fig. 5(b) denotes a bug of code
completion, which may be likely fixable.
Moreover, Fig. 6(a) presents another case, such that the code
completion output of the seed prompt appears to be highly similar to the
ground truth (Fig. 6(b)). In contrast, when applying the REL_R
scheme to mutate a local variable’s name, the code completion output of a
mutated prompt (Fig. 6(c)) becomes largely distinct from the
ground truth. This clearly denotes a bug of the code completion system.
Processing Time & Cost. We employ a GPU server for
running the involved models locally. The server has an Intel Xeon Patinum 8276
CPU, 256 GB memory and 4 NVIDIA A100 GPU. Although processing time is in general
not a concern for this study, we record and report that it takes on average 0.3
CPU seconds to generate one mutated prompt and about 35 CPU seconds to infer and
obtain the results. Nevertheless, for commercial models, Table V
has revealed that roughly 10.86% percent of its code completion outputs
are inconsistent and spurious when we set the highest threshold T.
In other words, we interpret that about
10.86% percent of the code completion outputs are highly confusing to the
users and are thus “wasted.” This may indicate an undesirable situation and
financial loss, given that modern cloud-based code completion systems
may feature a “pay-as-you-go” mode, where users are charged based on how many
queries they send to the services.
Answer to RQ2: CCTest identifies a large number of defects when being
used to test different (commercial) code completion systems, despite the varying
thresholds used in deciding the distances. In production usage, we recommend
configuring $T=9$ as a presumably proper threshold to detect outliers.
V-C RQ3: Repairing Effectiveness
To answer RQ3, we first present the accuracy improvement in terms of
different evaluated code completion systems. Then, we assess the potency of each
PSC scheme implemented in CCTest across different models in terms of their
contribution to repairing.
Accuracy Enhancement After Repairing. As noted in RQ2, we report
the accuracy improvement of each model with respect to both edit distance and
bleu4 score in Table V. Note that both edit distance and bleu4
metrics are commonly used in relevant research to access the performance of
LLMs; a higher edit distance score or bleu4 score indicates better performance.
To clarify, Table V reports the enhancement ratio. For instance,
when assessing Copilot against the LeetCode dataset, let the edit distance or
bleu4 score be $s$. We compute the enhancement ratio as $r=\frac{s^{\prime}-s}{s}$,
where $s^{\prime}$ is the edit distance/bleu4 score after repairing.
Moreover, for all evaluated models under both metrics, the improvement ratios
are generally improved with $T$ growing. We mark the best improvement of each
evaluation setting in blue.
Not all the cases are the same, for Copilot on the LeetCode dataset, the bleu4
score reaches the peak when $T=5$. Similar observations are made on
Codegen-6B-mono where corresponding $T=7$. We attribute this observation to the
fact that Copilot and Codegen have relatively better performance than the
remaining code completion systems. Therefore, deciding “outliers” under a
strict threshold ($T=9$) may likely overlook outliers and undermine the
opportunities of enhancing accuracy.
Potency of Transformation Passes. At this step, we measure the potency of
all nine transformation passes implemented in CCTest.
Let code completion output $o_{i}$ be the repaired output, such that $o_{i}$ is
generated by using the prompt mutated from transformation pass $t_{i}$. We deem
$t_{i}$ under such circumstance as the “optimal” transformation that
successfully contributes to the code completion repairing.
For both LeetCode and CodeSearchNet, we report the distribution of different
transformations selected as “optimal” under different models in
Fig.7 and Fig. 8, respectively. Overall, all
transformations introduced in Sec.III are extensively served as the
“optimal” in both datasets. In particular, nearly all passes, except “IRR”
and “RTF,” are chosen for roughly equal amount of times. IRR has a relatively
smaller application scope, given that IRR looks for specific arithmetic
operations like “+=” that may not be pervasively used. Similarly, RTF
requires the existence of an if condition with a relatively simple condition,
which may not be available in our test cases.
Overall, we interpret the evaluation results are highly encouraging, showing
that all the designed passes manifest high applicability and effectiveness in
enhancing different models. When using the LeetCode dataset, “IRR” is obviously
more useful in contributing “optimal” cases. With manual inspection, we find
that the codes in LeetCode, an online judge (OJ) platform, are more likely to
contain arithmetic operations such as “+=” in comparison to the code in the
CodeSearchNet dataset.
In addition to reporting the distribution of the “optimal” transformations, we
also report to what extent each transformation, when contributing to the code
completion repairing, can enhance the bleu4 score of code completion
systems.
Fig. 9 and Fig. 10 report the enhancement
ratios across all code completion systems under two datasets. Aligned with the
observation in Fig. 7 and Fig. 8, we find that
all transformation passes manifest a promising enhancement ratio under different
settings and datasets. That is, instead of identifying one or few dominating
passes that significantly contribute to enhancing the output of code completion,
each pass offers a reasonable degree of contribution. “INI” delivers
particularly outstanding improvement for the Copilot/LeetCode evaluation.
Again, we interpret the evaluation results are highly encouraging, justifying
the necessity of every transformation pass designed in CCTest.
Answer to RQ3: CCTest can successfully improve the quality of different
code completion systems. We also find that instead of one or few prompt mutation
schemes that significantly contribute to enhancing code completion, all schemes
are shown as effective in improving the performance of code completion systems.
VI Discussion
Limitations and Threats to Validity. We now give a discussion
of the validity and shortcomings of this paper’s approach. In this research,
construct validity denotes the degree to which our metrics actually
reflect the correctness of code completion systems. Overall, we conduct dynamic
testing and manual inspection to study the outputs of de facto code completion
systems. Hence, while this practical approach detects their defects and reveals
chances of repairing their outputs, the most possible threat is that our testing
approach cannot guarantee the correctness of code completion systems. We clarify
that our work roots the same assumption as previous testing works that aim to
detect flaws of deep learning-based applications with dynamic testing rather
than verification.
We check code completion outputs by comparing a set of outputs that are supposed
to exhibit consistent structures. The evaluation shows that the focus of
structural-consistency effectively unveils a large number of defects. However, a
possible threat is that defects can be neglected in the completion output, in
case all outputs share aligned yet erroneous code patterns. We deem this as a
general and well-known hurdle for invariant property-based testing techniques. We leave
exploring solutions to this challenge for future work.
Besides, there exists the potential threat that the proposed testing and
repairing framework, CCTest, may not adapt to other types of code completion
systems. Nevertheless, we mitigate this threat to external validity by
designing an approach that is system and algorithm independent. As a result, our
approach is anticipated to be applicable to other settings outside the current
scope. We believe the proposed technique is general, and we give further
discussions regarding other settings in this section.
Cross Comparison of Code Completion Outputs. Overall, CCTest individually tests each code completion system. Holistically speaking,
the proposed approach constitutes program property-based testing (or
metamorphic testing).
With this regard, careful readers may wonder about the feasibility of conducting a
differential testing, by processing the same prompt with different code
completion systems and differentiating their outputs. However, we note that the
code completion outputs can have drastically different representations since
different code completion systems have different model training data and LLM
model capacity. For instance, Copilot is seen to produce a large chunk of code
snippets (with multiple statements), whereas some other well-known systems are
prone to giving more succinct outputs for the same input prompts. Our
preliminary exploration also shows that they manifest different tactics and
translation templates in code generation. Thus, the similarity among the code
completion outputs are deemed as low across different code completion systems.
Overall, we leave it as one future work to explore practical methods to perform
cross comparison, for instance, by extracting certain “semantics-level”
signatures or regulating their output code patterns first.
Other Settings. The main focus of this study is Python code completion,
one challenging and popular task commonly encountered in real-world software
engineering missions. Overall, we envision the feasibility of smoothly migrating
the prototype of CCTest to test code completion models of other languages, such
as C and Java. While extending CCTest to handle other programming languages
demands new parsers and re-implementation of current PSC schemes, we see that the
key technical pipeline, including mutation, outlier detection, and repairing,
are mostly model and language independent. Note that these are
engineering endeavors rather than open-ended research problems. We leave it as
one future work to support other languages.
VII Related Work
Testing & Repairing Neural Models. In recent several years, many works
have been applying software testing methods to neural
models [67, 57, 48, 69, 43, 60, 16, 40]. To date, the tested neural
models include typical computer vision tasks like image classification, object
detection [61, 51, 56],
auto-driving [67, 69], as well as natural
language processing tasks like sentiment
analysis [49, 38, 21, 58],
question answering [14], and machine
translation [26, 27, 25, 53, 12],
and specific properties like fairness [15]. CCTest is
inspired by recent advances in machine translation testing. It addresses domain
specific challenges in mutating program inputs, and for the first time, provides
a systematic framework for testing and repairing code completion systems in
blackbox settings.
Neural Code Comprehension. In addition to using LLMs for code completion,
deep learning-based methods have been extensively used in related important code
comprehension tasks and have achieved promising results. For instance, function
naming decides the function name by summarizing the function
body [6, 7, 71]. Often, code
paths on the function ASTs are extracted for embedding and name prediction, and
given the large volume of available paths, optimization schemes like attention
are often used to speed up the processing. Code classification and
code search are two popular tasks in this field.
De facto methods explore learning from ASTs and CFGs to
leverage structural information for code
classification [39, 37, 62] and for code search
[11, 24].
Tree-based convolutional neural networks and graphics neural networks are
leveraged in these tasks [59, 36].
Furthermore, many security downstream applications have been built on the basis
of neural code comprehension models, including code cloning and plagiarism
detection [32, 35], malware clustering [30],
software component
analysis [66, 64, 65], vulnerability
detection [34, 70].
VIII Conclusion
We conducted a systematic investigation on the output consistency of modern code
completion systems. We offer CCTest, a testing framework to perform
structural-consistency mutations on input prompts, and we further design a
black-box repairing scheme to enhance the accuracy of code completion systems.
Our evaluation of eight prominent (commercial) code completion systems identified
thousands of inconsistency defects, whereas our repairing scheme effectively
enhances their accuracy. This work may serve as a roadmap for researchers and
users interested in utilizing and improving code completion systems.
References
[1]
Codex.
https://openai.com/blog/openai-codex/.
[2]
Hugging Face.
https://huggingface.co/.
[3]
Leetcode Python.
https://github.com/JiayangWu/LeetCode-Python.
[4]
pycode-similar.
https://pypi.org/project/pycode-similar/.
[5]
Tabnine.
https://www.tabnine.com/.
[6]
Uri Alon, Shaked Brody, Omer Levy, and Eran Yahav.
code2seq: Generating sequences from structured representations of
code.
arXiv preprint arXiv:1808.01400, 2018.
[7]
Uri Alon, Meital Zilberstein, Omer Levy, and Eran Yahav.
code2vec: Learning distributed representations of code.
Proceedings of the ACM on Programming Languages, 3(POPL):1–29,
2019.
[8]
Sid Black, Leo Gao, Phil Wang, Connor Leahy, and Stella Biderman.
GPT-Neo: Large Scale Autoregressive Language Modeling with
Mesh-Tensorflow, March 2021.
If you use this software, please cite it using these metadata.
[9]
Tom Brown, Benjamin Mann, Nick Ryder, Melanie Subbiah, Jared D Kaplan, Prafulla
Dhariwal, Arvind Neelakantan, Pranav Shyam, Girish Sastry, Amanda Askell,
et al.
Language models are few-shot learners.
Advances in neural information processing systems,
33:1877–1901, 2020.
[10]
Max Brunsfeld.
Tree-sitter.
https://github.com/tree-sitter/tree-sitter.
[11]
José Cambronero, Hongyu Li, Seohyun Kim, Koushik Sen, and Satish Chandra.
When deep learning met code search.
In Marlon Dumas, Dietmar Pfahl, Sven Apel, and Alessandra Russo,
editors, Proceedings of the ACM Joint Meeting on European Software
Engineering Conference and Symposium on the Foundations of Software
Engineering, ESEC/SIGSOFT FSE 2019, Tallinn, Estonia, August 26-30,
2019, pages 964–974. ACM, 2019.
[12]
Jialun Cao, Meiziniu Li, Yeting Li, Ming Wen, and Shing-Chi Cheung.
SemMT: A semantic-based testing approach for machine translation
systems.
arXiv preprint arXiv:2012.01815, 2020.
[13]
Mark Chen, Jerry Tworek, Heewoo Jun, Qiming Yuan, Henrique Ponde de Oliveira
Pinto, Jared Kaplan, Harri Edwards, Yuri Burda, Nicholas Joseph, Greg
Brockman, et al.
Evaluating large language models trained on code.
arXiv preprint arXiv:2107.03374, 2021.
[14]
Songqiang Chen, Shuo Jin, and Xiaoyuan Xie.
Testing your question answering software via asking recursively.
In Proceedings of the 36th IEEE/ACM International Conference on
Automated Software Engineering, ASE ’21, page 104–116. IEEE Press, 2021.
[15]
Zhenpeng Chen, Jie M Zhang, Max Hort, Federica Sarro, and Mark Harman.
Fairness testing: A comprehensive survey and analysis of trends.
arXiv e-prints, pages arXiv–2207, 2022.
[16]
Anurag Dwarakanath, Manish Ahuja, Samarth Sikand, Raghotham M. Rao, R. P.
Jagadeesh Chandra Bose, Neville Dubash, and Sanjay Podder.
Identifying implementation bugs in machine learning based image
classifiers using metamorphic testing.
In ISSTA, 2018.
[17]
EleutherAI.
Gpt-j.
https://huggingface.co/EleutherAI/gpt-j-6B.
[18]
Hugging Face.
Codeparrot.
https://huggingface.co/codeparrot/codeparrot.
[19]
Zhangyin Feng, Daya Guo, Duyu Tang, Nan Duan, Xiaocheng Feng, Ming Gong, Linjun
Shou, Bing Qin, Ting Liu, Daxin Jiang, and Ming Zhou.
Codebert: A pre-trained model for programming and natural
languages.
In Trevor Cohn, Yulan He, and Yang Liu, editors, Findings of the
Association for Computational Linguistics: EMNLP 2020, Online Event, 16-20
November 2020, volume EMNLP 2020 of Findings of ACL, pages
1536–1547. Association for Computational Linguistics, 2020.
[20]
Philip Gage.
A new algorithm for data compression.
C Users Journal, 12(2):23–38, 1994.
[21]
Sainyam Galhotra, Yuriy Brun, and Alexandra Meliou.
Fairness testing: testing software for discrimination.
In ACM ESEC/FSE, pages 498–510. ACM, 2017.
[22]
Leo Gao, Stella Biderman, Sid Black, Laurence Golding, Travis Hoppe, Charles
Foster, Jason Phang, Horace He, Anish Thite, Noa Nabeshima, Shawn Presser,
and Connor Leahy.
The Pile: An 800gb dataset of diverse text for language modeling.
arXiv preprint arXiv:2101.00027, 2020.
[23]
GitHub.
Copilot.
[24]
Wenchao Gu, Zongjie Li, Cuiyun Gao, Chaozheng Wang, Hongyu Zhang, Zenglin Xu,
and Michael R. Lyu.
Cradle: Deep code retrieval based on semantic dependency learning.
Neural Networks, 141:385–394, 2021.
[25]
Shashij Gupta, Pinjia He, Clara Meister, and Zhendong Su.
Machine translation testing via pathological invariance.
In Proceedings of the 28th ACM Joint Meeting on European
Software Engineering Conference and Symposium on the Foundations of Software
Engineering, pages 863–875, 2020.
[26]
Pinjia He, Clara Meister, and Zhendong Su.
Structure-invariant testing for machine translation.
In ICSE, 2020.
[27]
Pinjia He, Clara Meister, and Zhendong Su.
Testing machine translation via referential transparency.
In 2021 IEEE/ACM 43rd International Conference on Software
Engineering (ICSE), pages 410–422. IEEE, 2021.
[28]
Hamel Husain, Ho-Hsiang Wu, Tiferet Gazit, Miltiadis Allamanis, and Marc
Brockschmidt.
Codesearchnet challenge: Evaluating the state of semantic code
search.
arXiv preprint arXiv:1909.09436, 2019.
[29]
Saki Imai.
Is github copilot a substitute for human pair-programming? an
empirical study.
In 2022 IEEE/ACM 44th International Conference on Software
Engineering: Companion Proceedings (ICSE-Companion), pages 319–321. IEEE,
2022.
[30]
Mahmoud Kalash, Mrigank Rochan, Noman Mohammed, Neil DB Bruce, Yang Wang, and
Farkhund Iqbal.
Malware classification with deep convolutional neural networks.
In 2018 9th IFIP international conference on new technologies,
mobility and security (NTMS), pages 1–5. IEEE, 2018.
[31]
Klemens Lagler, Michael Schindelegger, Johannes Böhm, Hana Krásná,
and Tobias Nilsson.
GPT2: Empirical slant delay model for radio space geodetic
techniques.
Geophysical research letters, 40(6):1069–1073, 2013.
[32]
Maggie Lei, Hao Li, Ji Li, Namrata Aundhkar, and Dae-Kyoo Kim.
Deep learning application on code clone detection: A review of
current knowledge.
Journal of Systems and Software, 184:111141, 2022.
[33]
Yujia Li, David H. Choi, Junyoung Chung, Nate Kushman, Julian Schrittwieser,
Rémi Leblond, Tom Eccles, James Keeling, Felix Gimeno, Agustin Dal
Lago, Thomas Hubert, Peter Choy, Cyprien de Masson d’Autume, Igor Babuschkin,
Xinyun Chen, Po-Sen Huang, Johannes Welbl, Sven Gowal, Alexey Cherepanov,
James Molloy, Daniel J. Mankowitz, Esme Sutherland Robson, Pushmeet Kohli,
Nando de Freitas, Koray Kavukcuoglu, and Oriol Vinyals.
Competition-level code generation with alphacode.
CoRR, abs/2203.07814, 2022.
[34]
Zhen Li, Deqing Zou, Shouhuai Xu, Xinyu Ou, Hai Jin, Sujuan Wang, Zhijun Deng,
and Yuyi Zhong.
Vuldeepecker: A deep learning-based system for vulnerability
detection.
arXiv preprint arXiv:1801.01681, 2018.
[35]
Zongjie Li, Pingchuan Ma, Huaijin Wang, Shuai Wang, Qiyi Tang, Sen Nie, and Shi
Wu.
Unleashing the power of compiler intermediate representation to
enhance neural program embeddings.
In 44th IEEE/ACM 44th International Conference on Software
Engineering, ICSE 2022, Pittsburgh, PA, USA, May 25-27, 2022, pages
2253–2265. ACM, 2022.
[36]
Shangqing Liu, Xiaofei Xie, Lei Ma, Jing Kai Siow, and Yang Liu.
Graphsearchnet: Enhancing gnns via capturing global dependency for
semantic code search.
CoRR, abs/2111.02671, 2021.
[37]
Shuai Lu, Daya Guo, Shuo Ren, Junjie Huang, Alexey Svyatkovskiy, Ambrosio
Blanco, Colin Clement, Dawn Drain, Daxin Jiang, Duyu Tang, et al.
Codexglue: A machine learning benchmark dataset for code
understanding and generation.
arXiv preprint arXiv:2102.04664, 2021.
[38]
Pingchuan Ma, Shuai Wang, and Jin Liu.
Metamorphic testing and certified mitigation of fairness violations
in nlp models.
In IJCAI, pages 458–465, 2020.
[39]
Lili Mou, Ge Li, Lu Zhang, Tao Wang, and Zhi Jin.
Convolutional neural networks over tree structures for programming
language processing.
In Thirtieth AAAI conference on artificial intelligence, 2016.
[40]
Shin Nakajima and Tsong Yueh Chen.
Generating biased dataset for metamorphic testing of machine learning
programs.
In IFIP-ICTSS, 2019.
[41]
Nhan Nguyen and Sarah Nadi.
An empirical evaluation of github copilot’s code suggestions.
In IEEE/ACM 19th International Conference on Mining Software
Repositories, MSR 2022, Pittsburgh, PA, USA, May 23-24, 2022, pages 1–5.
IEEE, 2022.
[42]
Erik Nijkamp, Bo Pang, Hiroaki Hayashi, Lifu Tu, Huan Wang, Yingbo Zhou, Silvio
Savarese, and Caiming Xiong.
A conversational paradigm for program synthesis.
arXiv preprint arXiv:2203.13474, 2022.
[43]
Augustus Odena and Ian Goodfellow.
Tensorfuzz: Debugging neural networks with coverage-guided fuzzing.
arXiv preprint arXiv:1807.10875, 2018.
[44]
Hammond Pearce, Benjamin Tan, Baleegh Ahmad, Ramesh Karri, and Brendan
Dolan-Gavitt.
Can openai codex and other large language models help us fix security
bugs?
arXiv preprint arXiv:2112.02125, 2021.
[45]
Hammond Pearce, Benjamin Tan, Baleegh Ahmad, Ramesh Karri, and Brendan
Dolan-Gavitt.
Can openai codex and other large language models help us fix security
bugs?
CoRR, abs/2112.02125, 2021.
[46]
Hammond Pearce, Benjamin Tan, Prashanth Krishnamurthy, Farshad Khorrami, Ramesh
Karri, and Brendan Dolan-Gavitt.
Pop quiz! can a large language model help with reverse engineering?
arXiv preprint arXiv:2202.01142, 2022.
[47]
Hammond Pearce, Benjamin Tan, Prashanth Krishnamurthy, Farshad Khorrami, Ramesh
Karri, and Brendan Dolan-Gavitt.
Pop quiz! can a large language model help with reverse engineering?
CoRR, abs/2202.01142, 2022.
[48]
Kexin Pei, Yinzhi Cao, Junfeng Yang, and Suman Jana.
DeepXplore: Automated whitebox testing of deep learning systems.
In Proceedings of the 26th Symposium on Operating Systems
Principles, SOSP ’17, pages 1–18, New York, NY, USA, 2017. ACM.
[49]
Marco Tulio Ribeiro, Tongshuang Wu, Carlos Guestrin, and Sameer Singh.
Beyond accuracy: Behavioral testing of nlp models with checklist.
arXiv preprint arXiv:2005.04118, 2020.
[50]
Sami Sarsa, Paul Denny, Arto Hellas, and Juho Leinonen.
Automatic generation of programming exercises and code explanations
with large language models.
arXiv preprint arXiv:2206.11861, 2022.
[51]
Jinyang Shao.
Testing object detection for autonomous driving systems via 3d
reconstruction.
In 2021 IEEE/ACM 43rd International Conference on Software
Engineering: Companion Proceedings (ICSE-Companion), pages 117–119. IEEE,
2021.
[52]
Zeyu Sun, Jie M. Zhang, Mark Harman, Mike Papadakis, and Lu Zhang.
Automatic testing and improvement of machine translation.
In Gregg Rothermel and Doo-Hwan Bae, editors, ICSE ’20: 42nd
International Conference on Software Engineering, Seoul, South Korea, 27 June
- 19 July, 2020, pages 974–985. ACM.
[53]
Zeyu Sun, Jie M Zhang, Mark Harman, Mike Papadakis, and Lu Zhang.
Automatic testing and improvement of machine translation.
In Proceedings of the ACM/IEEE 42nd International Conference on
Software Engineering, pages 974–985, 2020.
[54]
Zeyu Sun, Jie M. Zhang, Yingfei Xiong, Mark Harman, Mike Papadakis, and
Lu Zhang.
Improving machine translation systems via isotopic replacement.
In 44th IEEE/ACM 44th International Conference on Software
Engineering, ICSE 2022, Pittsburgh, PA, USA, May 25-27, 2022, pages
1181–1192. ACM, 2022.
[55]
Yi Tay, Mostafa Dehghani, Dara Bahri, and Donald Metzler.
Efficient transformers: A survey.
ACM Computing Surveys (CSUR), 2020.
[56]
Yongqiang Tian, Shiqing Ma, Ming Wen, Yepang Liu, Shing-Chi Cheung, and Xiangyu
Zhang.
To what extent do dnn-based image classification models make
unreliable inferences?
Empirical Software Engineering, 26(5):1–40, 2021.
[57]
Yuchi Tian, Kexin Pei, Suman Jana, and Baishakhi Ray.
DeepTest: Automated testing of deep-neural-network-driven
autonomous cars.
ICSE ’18, 2018.
[58]
Sakshi Udeshi, Pryanshu Arora, and Sudipta Chattopadhyay.
Automated directed fairness testing.
ASE, 2018.
[59]
Huanting Wang, Guixin Ye, Zhanyong Tang, Shin Hwei Tan, Songfang Huang, Dingyi
Fang, Yansong Feng, Lizhong Bian, and Zheng Wang.
Combining graph-based learning with automated data collection for
code vulnerability detection.
IEEE Transactions on Information Forensics and Security,
16:1943–1958, 2020.
[60]
Jingyi Wang, Guoliang Dong, Jun Sun, Xinyu Wang, and Peixin Zhang.
Adversarial sample detection for deep neural network through model
mutation testing.
ICSE, 2019.
[61]
Shuai Wang and Zhendong Su.
Metamorphic object insertion for testing object detection systems.
In ASE, 2020.
[62]
Yue Wang, Weishi Wang, Shafiq Joty, and Steven CH Hoi.
Codet5: Identifier-aware unified pre-trained encoder-decoder models
for code understanding and generation.
arXiv preprint arXiv:2109.00859, 2021.
[63]
Dror Weiss.
Comparison between copilot and tabnine.
https://twitter.com/drorwe/status/1539329682851733505.
[64]
Seunghoon Woo, Sunghan Park, Seulbae Kim, Heejo Lee, and Hakjoo Oh.
Centris: A precise and scalable approach for identifying modified
open-source software reuse.
In 2021 IEEE/ACM 43rd International Conference on Software
Engineering (ICSE), pages 860–872. IEEE, 2021.
[65]
Zeping Yu, Wenxin Zheng, Jiaqi Wang, Qiyi Tang, Sen Nie, and Shi Wu.
Codecmr: Cross-modal retrieval for function-level binary source code
matching.
Advances in Neural Information Processing Systems,
33:3872–3883, 2020.
[66]
Xian Zhan, Lingling Fan, Sen Chen, Feng We, Tianming Liu, Xiapu Luo, and Yang
Liu.
Atvhunter: Reliable version detection of third-party libraries for
vulnerability identification in android applications.
In 2021 IEEE/ACM 43rd International Conference on Software
Engineering (ICSE), pages 1695–1707. IEEE, 2021.
[67]
Mengshi Zhang, Yuqun Zhang, Lingming Zhang, Cong Liu, and Sarfraz Khurshid.
DeepRoad: GAN-based Metamorphic Testing and Input Validation
Framework for Autonomous Driving Systems.
In ASE, 2018.
[68]
Zhaowei Zhang, Hongyu Zhang, Beijun Shen, and Xiaodong Gu.
Diet code is healthy: Simplifying programs for pre-trained models of
code.
arXiv preprint arXiv:2206.14390, 2022.
[69]
Husheng Zhou, Wei Li, Yuankun Zhu, Yuqun Zhang, Bei Yu, Lingming Zhang, and
Cong Liu.
Deepbillboard: Systematic physical-world testing of autonomous
driving systems.
ICSE, 2020.
[70]
Yaqin Zhou, Shangqing Liu, Jingkai Siow, Xiaoning Du, and Yang Liu.
Devign: Effective vulnerability identification by learning
comprehensive program semantics via graph neural networks.
Advances in neural information processing systems, 32, 2019.
[71]
Daniel Zügner, Tobias Kirschstein, Michele Catasta, Jure Leskovec, and
Stephan Günnemann.
Language-agnostic representation learning of source code from
structure and context.
arXiv preprint arXiv:2103.11318, 2021. |
Preprint RIMS 965
January 1994
hep-th/9402051
Extended reflection equation algebras, the braid
group on
a handlebody and associated link polynomials
Christian Schwiebert††⋆ Supported by the Science and Technology Fellowship Programme for
Japan under the auspices of the Commission of the European
Communities††♢ e-mail:
cbs@kurims.kyoto-u.ac.jp
Research Institute for Mathematical Sciences
Kyoto University, Sakyo-ku, 606 Kyoto, Japan
ABSTRACT
The correspondence of the braid group on a handlebody of arbitrary
genus to the algebra of Yang-Baxter and extended reflection equation
operators is shown. Representations of the infinite dimensional
extended reflection equation algebra in terms of direct products of
quantum algebra generators are derived, they lead to a representation
of this braid group in terms of $R$-matrices. Restriction to the
reflection equation operators only gives the coloured braid group.
The reflection equation operators, describing the effect of handles
attached to a 3-ball, satisfy characteristic equations which give rise
to additional skein relations and thereby invariants of links on
handlebodies. The origin of the skein relations is explained and they
are derived from an adequately adapted handlebody version of the Jones
polynomial. Relevance of these results to the construction of link
polynomials on closed 3-manifolds via Heegard splitting and surgery is
indicated.
\chapter
INTRODUCTION
In this paper††† A completely rewritten and largely
extended version of ref. [10] we should like to explain
the relation between quantum groups (QG) and the braid group on a
three dimensional manifold of arbitrary genus with boundary. As a
consequence we will be able to define invariants of links on such
manifolds. A three-manifold having a genus $g$ Riemann surface as
boundary is conventionally named a genus $g$ handlebody. The braid
group on such a handlebody can be formulated in terms of the usual
braid group generators $\sigma_{i}$ for genus zero 3-manifolds
\Ref\rArtE. Artin, Ann. Math. 48 (1947) 101 plus additional generators $\tau_{\alpha}$ implementing
windings around handles. We shall make use of the results in
\Ref\rSosA. B. Sossinsky, in:
“Euler international mathematical institute on quantum
groups”, ed. P. P. Kulish, Lect. Notes Math. 1510,
Springer, Berlin 1992 where such a description was given, explicitly, the
braid group $B^{g}_{n}$ on a handlebody comprises the following relations:
$$\eqalign{\sigma_{i}\sigma_{i+1}\sigma_{i}&=\sigma_{i+1}\sigma_{i}\sigma_{i+1},%
\quad i=1,\ldots,n-1\cr\sigma_{i}\sigma_{j}&=\sigma_{j}\sigma_{i},\quad\mid i-%
j\mid\geq 2\cr\sigma_{i}\tau_{\alpha}&=\tau_{\alpha}\sigma_{i},\quad i\geq 2,%
\ \alpha=1,\ldots,g\cr\sigma_{1}\tau_{\alpha}\sigma_{1}\tau_{\alpha}&=\tau_{%
\alpha}\sigma_{1}\tau_{\alpha}\sigma_{1},\cr\sigma_{1}\tau_{\alpha}\sigma_{1}^%
{-1}\tau_{\beta}&=\tau_{\beta}\sigma_{1}\tau_{\alpha}\sigma_{1}^{-1},\quad%
\alpha<\beta.}\eqn\bralgf$$
The first two equations define the well known Artin braid group $B_{n}$
acting on $n$ strands in a topologically trivial 3-manifold
\refmark\rArt. For each handle there is a new generator $\tau_{\alpha}$
having nontrivial commutation relations only with $\sigma_{1}$ and the $\tau$
generators. The last equation is absent in the case of a solid torus,
i.e. for genus one. The first equation of \bralgf corresponds to the
Yang-Baxter equation written in braid form
$$\widehat{R}_{12}\widehat{R}_{23}\widehat{R}_{12}=\widehat{R}_{23}\widehat{R}_{%
12}\widehat{R}_{23},\eqn\ybef$$
providing a link to the theory of quantum groups as $\sigma_{i}$ can be
represented in terms of the $\widehat{R}$-matrix. The fourth equation of
\bralgf is also related to quantum groups, it can be considered
both as a comodule invariant w.r.t. QG coaction and as a way of
describing the quantum algebra, we will use it in the form
$$R_{12}K_{1}R_{21}K_{2}=K_{2}R_{12}K_{1}R_{21}.\eqn\ref$$
There exist also spectral parameter dependent versions of it which play
prominent roles in quantum inverse scattering \Ref\rSklaE. K. Sklyanin, J. Phys. A21 (1988) 2375,
describing the commutation relations of monodromy matrices. Actually,
they appeared first in the study of two particle scattering on a
half-line, with matrix $K(\theta)$ describing reflection of a particle
at the endpoint and $R(\theta-{\theta}^{\prime})$ describing two particle
scattering \Ref\rCheI. V. Cherednik, Teor. Mat. Fiz. 61 (1984) 55. Hence the name reflection equation (RE),
suggested in \Ref\rKulaP. P. Kulish, in: “Proceedings of the second
international Wigner symposium”, eds. H. D. Doebner et. al.,
World Scientific, Singapore 1993, where also the connection of \ref with $B^{1}_{n}$ was mentioned. We should mention that there exists still
another spectral parameter independent reflection equation
\REF\rKSSP. P. Kulish, R. Sasaki and C. Schwiebert, J. Math. Phys.
34 (1993) 286 \REF\rSklbP. P. Kulish and E. K. Sklyanin, J. Phys. A25 (1992)
5963 \refmark\rKSS,\rSklb, which is
invariant w.r.t. different QG comodule transformations compared to
\ref. The last equation of \bralgf is close to the RE, it is a
compatibility condition for solutions of the RE s.t. these can be
combined into new solutions of the RE. In QG language it looks like
$$R_{12}K_{1}R^{-1}_{12}{K^{\prime}}_{2}={K^{\prime}}_{2}R_{12}K_{1}R^{-1}_{12},\eqn\retwif$$
and its properties and connection to $B^{g}_{n}$ were
discussed in \REF\rKulbP. P. Kulish, Kyoto Univ. preprint YITP/K-92-984 \REF\rKuSaP. P. Kulish and R. Sasaki, Durham Univ. preprint DTP-92-53 \REF\rSchC. Schwiebert, Kyoto Univ. preprint YITP/U-92-38, unpublished
\refmark\rKulb,\rKuSa,\rSch under different aspects.
When discussing representations of the braid group \bralgf we will
naturally be led to representations of $\tau_{\alpha}$ in terms of
$R$-matrices s.t. $B^{g}_{n}$ can be viewed as a subgroup of $B_{n+g}$.
Equivalently, $\tau_{\alpha}$ can be expressed in terms of quantum algebra
generators. Indeed we will derive whole series of new solutions of both
\ref and \retwif in terms of quantum algebra generators, and they
precisely correspond to the description of $\tau_{\alpha}$ in terms of
$R$-matrices. Furthermore, we derive quadratic characteristic
equations for the matrix $K$, and hence the additional generators
$\tau_{\alpha}$, similar to the Hecke algebra relation for $\sigma_{i}$. They can be
interpreted as an additional skein relation when considering closed
braids on the handlebody and, in principle, they recursively define
link invariants for closed braids on arbitrary genus 3-manifolds with
boundary \refmark\rSch. This in turn, via Heegaard splitting, might
be a way of constructing invariant polynomials of links on arbitrary
3-manifolds without boundary. Since we know the representation of
$\tau_{\alpha}$ in terms of $R$-matrices we can also write down a trace
formula for the link invariants, this is equivalent to using the
quantum trace that is defined for the matrix $K$ \refmark\rKSS.
Further we show that the characteristic equation for $\tau_{\alpha}$ is
actually a consequence of the one for $\sigma_{i}$.
The plan of the paper is as follows. We introduce the RE in section
two and discuss its properties as an associative quadratic algebra,
then we extend it by \retwif and derive new solutions of the combined
system in terms of quantum algebra generators. In section three we
review some results of \refmark\rSos concerning the braid group on a
handlebody and obtain the representation of $\tau_{\alpha}$ in terms of
$R$-matrices and quantum algebra generators. We also discuss there the
connection between the Hecke algebra relation and the quadratic
equation for $\tau_{\alpha}$. In the fourth section we look at closed braids
on handlebodies and their invariants, notably by means of new skein
relations and quantum traces for the additional generators. Finally,
in the fifth section we mention some implications and possible
applications of our results.
\chapter
ALGEBRAS OF REFLECTION EQUATION OPERATORS
We will study the properties of the following reflection
equation††† We assume familiarity of the reader with
basic quantum group terminology as introduced in \REF\rFRTL. D. Faddeev, N. Yu. Reshetikhin and L. A. Takhtajan,
Alg. i Anal. 1 (1989) 178 (in Russian, English transl.:
Leningrad Math. J. 1 (1990) 193)
\REF\rTakL. A. Takhtajan, in: “Introduction to quantum
groups and integrable massive models of quantum field theory”,
eds. M. L. Ge and B. H. Zhao, World Scientific, Singapore 1991 \refmark\rFRT,\rTak, for example. Throughout this
paper when giving explicit examples we only use $sl_{q}(2)$ for simplicity,
generalizations should be obvious.
$$RK_{1}\widetilde{R}K_{2}=K_{2}RK_{1}\widetilde{R},\eqn\re$$
where $\widetilde{R}=PRP$ and $P$ is the permutation operator. Its basic
property and a guideline for its construction is invariance w.r.t. the
QG coaction, i.e. $K_{T}=TKT^{-1}$ is also a solution of this RE if
all elements of $K$ and $T$ commute, $[K_{ij},T_{mn}]=0$, and $T$ obeys the QG relations
$$RT_{1}T_{2}=T_{2}T_{1}R.\eqn\rtt$$
Just as in the case of the defining relations \rtt of the QG we can
view \re as an associative quadratic algebra. If we use the $sl_{q}(2)$
$R$-matrix
$${R^{ij}}_{kl}=\pmatrix{q&0&0&0\cr 0&1&0&0\cr 0&\omega&1&0\cr 0&0&0&q\cr},\quad%
\omega=q-q^{-1}\eqn\rmat$$
being a solution of the Yang-Baxter equation
$$R_{12}R_{13}R_{23}=R_{23}R_{13}R_{12},\eqn\ybe$$
then we find that the commutation relations for the
entries of the matrix $K=$ $a\,b\choose c\,d$
are given by
$$\eqalign{ab&=q^{-2}ba,\cr ac&=q^{2}ca,\cr}\qquad\eqalign{ad&=da,\cr bc-cb&=q^{%
-1}\omega(ad-a^{2}),\cr}\qquad\eqalign{bd-db&=-q^{-1}\omega ab,\cr cd-dc&=q^{-%
1}\omega ca.\cr}\eqn\alg$$
This algebra has two central elements, the quantum trace and the
quantum determinant which we set equal to one
$$c_{1}=q^{-1}a+qd,\qquad c_{2}=ad-q^{2}cb\equiv 1.\eqn\cent$$
The normalization of $c_{1}$ is chosen such that $K$ and $K^{-1}$ have
equal quantum trace. Using these relations the ‘antipode’ $S(K)\equiv K^{-1}$ can be found to be
$$K^{-1}=\pmatrix{q^{2}d-q\omega a&-q^{2}b\cr-q^{2}c&a\cr}.\eqn\inv$$
Then we easily establish a relation (characteristic equation) between
$K$ and $K^{-1}$
$$qK+q^{-1}K^{-1}-c_{1}I=0,\eqn\cheq$$
which will give rise to a skein relation later on.
A few remarks about the properties of the above algebra follow.
\nextline(i)vii The RE algebra \alg depends only on $q^{2}$.
\nextline(ii)vi If $q$ is a root of unity, $q^{2p}=1$, then
$a^{p}$ is a further central element. \nextline(iii)v The $sl_{q}(2)$ RE algebra has two constant or
one-dimensional representations, one of them clearly is the identity
matrix and the other one a lower-right triangular matrix with
arbitrary constants $b,c,d$. Constant solutions of \re were studied
in \refmark\rKSS. \nextline(iv)ii The $K$-matrix can be considered as a
product of two suitable quantum planes \REF\rManYu. I. Manin, Montreal Univ. preprint CRM-1561 (1988)
\REF\rWeZuJ. Wess and B. Zumino, Nucl. Phys. (Proc. Suppl.)
B18 (1990) 302 \refmark\rMan,\rWeZu
$x^{i}x^{j}=q^{-1}{R^{ji}}_{kl}x^{k}x^{l}$ and
$y_{i}y_{j}=q^{-1}y_{k}y_{l}{R^{kl}}_{ji}$
invariant w.r.t. the QG coaction
${x^{\prime}}^{i}={T^{i}}_{j}x^{j}$ and ${y^{\prime}}_{i}=y_{j}{(T^{-1})^{j}}_{i}$
respectively. Commutation relations between $x^{i}$ and $y_{j}$ can be
determined using \rtt as $x^{i}y_{j}=({\it const.})\ y_{k}x^{l}{R^{ik}}_{lj}$ and hence those of their product
${K^{i}}_{j}=x^{i}y_{j}$ which coincide with \alg. This also gives a
better understanding of the comodule property $K_{T}=TKT^{-1}$
of the RE. \nextline(v)iii If we impose suitable reality conditions on
$x^{i},y_{j}$ and hence ${K^{i}}_{j}$ then a linear combination of the
elements of \alg is just the $q$-deformed Minkowski space
\REF\rCWSSWU. Carow-Watamura et. al., Z. Phys. C48 (1990) 159 \REF\rSmWZW. B. Schmidke, J. Wess and B. Zumino, Z. Phys. C52
(1991) 471 \refmark\rCWSSW,\rSmWZ, where
$c_{1}$ is the time coordinate and $c_{2}$ the invariant length. Various
reality conditions are discussed in \refmark\rSklb, they parallel
those of $sl_{q}(2)$. \nextline(vi)ii Truncation of algebra \alg by $c_{1}=0$ can
be shown to lead to the quantum 2-sphere, a quantum analogue
of homogeneous spaces \Ref\rPodP. Podles, Lett. Math. Phys. 14 (1987) 193. \nextline(vii)i It is possible to introduce an index free
notation for quantum planes and extend it to the $K$-matrix, such that
the RE can be rewritten in exchange algebra form with four $R$-matrices
on one side \REF\rMadaS. Majid, J. Math. Phys. 32 (1991) 3246 \refmark\rSklb,\rMada. \nextline(viii) The monodromy $M=Pexp\bigl{(}{2\pi i\over k}\int^{2\pi}_{0}J(x)\,dx\bigr{)}$ of the $sl_{q}(2)$ Kac-Moody current
satisfies the RE when regularized on a one-dimensional lattice with
periodic boundary conditions \Ref\rAFSA. Yu. Alekseev, L. D. Faddeev and M. A.
Semenov-Tian-Shansky, Commun. Math. Phys. 149 (1992) 335. As this commutation
relation of the regularized monodromy holds for arbitrary numbers of
sites it might be expected to survive the continuum limit.
\nextline(ix)ii Neither $K^{-1}$ nor $K^{2}$ is a solution of the
RE.
The last remark leads us to a very important property of the RE,
namely, given two different solutions of the RE satisfying a
certain compatibility condition then one can use them to construct new
solutions \refmark\rKulb,\rKuSa,\rSch. Explicitly, let $K$ and $K^{\prime}$ be
solutions of \re then
$$(i)\quad{\widetilde{K}}=KK^{\prime}\qquad{\rm and}\qquad(ii)\quad{\widetilde{%
\widetilde{K}}}=KK^{\prime}K^{-1}\eqn\prod$$
are also solutions of \re provided $K$ and $K^{\prime}$ commute as follows
$$RK_{1}R^{-1}K^{\prime}_{2}=K^{\prime}_{2}RK_{1}R^{-1}.\eqn\retwi$$
This equation is invariant under the coaction $K_{T}=TKT^{-1}$
and $K^{\prime}_{S}=SK^{\prime}S^{-1}$ if $S$ also obeys QG relations \rtt and in
addition $RT_{1}S_{2}=S_{2}T_{1}R$, especially we can put $S$
equal to $T$.
Note that the second composite solution in ∏ is not a trivial
consequence of the first since $R^{-1}$ does not solve the RE. This
process of building up new solutions can obviously be continued using
newly constructed solutions if they satisfy \retwi, but some care has
to be taken to keep track of the ordering as \retwi is not symmetric
under exchange of $K$ and $K^{\prime}$. This will become clearer when we
discuss systems of solutions of both \re and \retwi. We will
sometimes refer to both equations as extended RE algebra.
Equation \retwi gives 16 commutation relations between the elements
of $K$ and $K^{\prime}$
$$\eqalign{a^{\prime}a&=aa^{\prime}-q\omega bc^{\prime},\cr a^{\prime}b&=ba^{%
\prime},\cr a^{\prime}c&=ca^{\prime}+q\omega(a-d)c^{\prime},\cr a^{\prime}d&=%
da^{\prime}+q^{-1}\omega bc^{\prime},\cr b^{\prime}a&=ab^{\prime}+q\omega b(a^%
{\prime}-d^{\prime}),\cr b^{\prime}b&=q^{2}bb^{\prime},\cr b^{\prime}c&=q^{-2}%
cb^{\prime}+(1+q^{-2})\omega^{2}bc^{\prime}\cr&{\phantom{=}}\ -q^{-1}\omega(a-%
d)(a^{\prime}-d^{\prime}),\cr b^{\prime}d&=db^{\prime}-q^{-1}\omega b(a^{%
\prime}-d^{\prime}),\cr}\qquad\eqalign{c^{\prime}a&=ac^{\prime},\cr c^{\prime}%
b&=q^{-2}bc^{\prime},\cr c^{\prime}c&=q^{2}cc^{\prime},\cr c^{\prime}d&=dc^{%
\prime},\cr d^{\prime}a&=ad^{\prime}+q^{-1}\omega bc^{\prime},\cr d^{\prime}b&%
=bd^{\prime},\cr d^{\prime}c&=cd^{\prime}-q^{-1}\omega(a-d)c^{\prime},\cr d^{%
\prime}d&=dd^{\prime}-q^{-3}\omega bc^{\prime},\cr}\eqn\recom$$
and they only depend on $q^{2}$. Note that $K$ and $K^{\prime}$ are commuting
for $q=1$ even if one linearizes them, whereas the RE in this
case produces the undeformed $sl(2)$ Lie algebra relations
\REF\rJurB. Jurčo, Lett. Math. Phys. 27 (1991) 177 \REF\rZumB. Zumino, in: “Proceedings of the ${\rm X^{th}}$
congress on mathematical physics”, ed. K. Schmüdgen, Springer,
Berlin 1992 \refmark\rJur,\rZum.
The extended RE algebra was implicitly contained also in
the construction of complex quantum groups recently
\Ref\rCDSWZC. Chryssomalakos et. al., Commun. Math. Phys. 147
(1992) 634, where relations \recom describe commutation
relations among the generators of the quantum algebra and their
complex conjugates, while both sets individually satisfy \alg.
Algebra \alg was also constructed in \refmark\rMada in the
framework of braided tensor categories.††† This approach to
the RE algebra takes the point of view that one has an extra braiding
between elements of the two copies of the algebra in the coproduct
$\Delta(K)=K{\dot{\otimes}}K$ s.t. $(1\otimes a)\cdot(a\otimes 1)=a\otimes a-q\omega b\otimes c$, for example, note the
new term on the RHS. Denoting $1\otimes a$ as $a^{\prime}$ and $a\otimes 1$
as $a$, etc., this relation is identical with the first of \recom. Then
this ‘braided coproduct’ for the RE algebra is compatible with the algebra
relations and is of the same form as for the QG and quantum algebra.
This so-called ‘braided group’ which seemingly can be associated to
any QG was cast into RE form in \Ref\rMadbS. Majid, J. Math. Phys. 34 (1993) 1176. \nextlineAn important point is further that the central elements of $K$ and $K^{\prime}$
are mutually central in both algebras, i.e.
$$[{K^{i}}_{j},c^{\prime}_{m}]=[{{K^{\prime}}^{i}}_{j},c_{m}]=0,\quad m=1,2.\eqn\mutcen$$
It is obvious that we have central elements for the combined solutions
and also characteristic equations, for example
$$qKK^{\prime}+q^{-1}(KK^{\prime})^{-1}-C_{1}I=0,\eqn\comcen$$
where $C_{1}=q^{-1}(aa^{\prime}+bc^{\prime})+q(cb^{\prime}+dd^{\prime})$.
It is known that the RE algebra has a representation in terms of the
quantum algebra generators \Ref\rMoReG. Moore and N. Yu. Reshetikhin, Nucl. Phys. B328
(1989) 557. The $sl_{q}(2)$ algebra
dual to the QG \rtt can similarly be written in matrix form
\refmark\rFRT
$$\widetilde{R}L^{\varepsilon_{1}}_{1}L^{\varepsilon_{2}}_{2}=L^{\varepsilon_{2}%
}_{2}L^{\varepsilon_{1}}_{1}\widetilde{R},\qquad(\varepsilon_{1},\varepsilon_{%
2})\in\{(+,+),(+,-),(-,-)\}\eqn\rll$$
where
$$L^{+}=\pmatrix{q^{H/2}&q^{-1/2}\omega X^{-}\cr 0&q^{-H/2}\cr},\qquad L^{-}=%
\pmatrix{q^{-H/2}&0\cr-q^{1/2}\omega X^{+}&q^{H/2}\cr}\eqn\lplm$$
and this gives the $sl_{q}(2)$ algebra
$$[H,X^{\pm}]=\pm 2X^{\pm},\qquad[X^{+},X^{-}]=\omega^{-1}(q^{H}-q^{-H})\eqn\sltwo$$
with antipode $S(H)=-H$, $S(X^{\pm})=-q^{\mp 1}X^{\pm}$
and coproduct $\Delta(L^{\pm})=L^{\pm}{\dot{\otimes}}L^{\pm}$.
It is easy to show using \rll that $K=S(L^{-})L^{+}$ represents a
solution of the RE, explicitly $K$ is given by
$$K=\pmatrix{q^{H}&q^{-1/2}\omega q^{H/2}X^{-}\cr q^{-1/2}\omega X^{+}q^{H/2}&q^%
{-H}+q^{-1}\omega^{2}X^{+}X^{-}\cr}.\eqn\lsol$$
This representation of the RE algebra has quantum determinant $c_{2}=1$
and the quantum trace $c_{1}$ is just the quadratic Casimir operator of
the quantum algebra $sl_{q}(2)$.
We now find whole towers of new representations of the RE algebra in
terms of quantum algebra generators and generalize them to the
extended RE algebra. They will be useful for representing the braid
group \bralgf. To avoid clumsy notation we introduce the abbreviation
$S^{\pm}\equiv S(L^{\pm})$.
There is a simple way to produce further representations of the RE
algebra, namely by means of the coproduct $\Delta$ which gives
representations on tensor products of spaces, in the simplest case of
two spaces we obtain
$$\eqalign{\Delta({K^{i}}_{j})&=\Delta({{S^{-}}^{i}}_{k})\Delta({{L^{+}}^{k}}_{j%
})\cr&={K^{m}}_{n}\otimes{{S^{-}}^{i}}_{m}{{L^{+}}^{n}}_{j}\cr&={(1{\dot{%
\otimes}}S^{-})^{i}}_{m}{(K{\dot{\otimes}}1)^{m}}_{n}{(1{\dot{\otimes}}L^{+})^%
{n}}_{j}\cr&\equiv{{S^{-}_{2}}^{i}}_{m}{{K_{1}}^{m}}_{n}{{L^{+}_{2}}^{n}}_{j}}\eqn\copr$$
or in matrix notation simply $\Delta(K)=S^{-}_{2}K_{1}L^{+}_{2}$.
Note that this coproduct for $K$ is a consequence of the one for
$L^{\pm}$ and cannot be expressed as $\Delta(K)=K_{1}K_{2}$ (cf. footnote
after eqn. \recom). We stress that whenever we write down tensor
products in the following then entries of different spaces are
strictly commuting. \nextlineWe can get a whole string of solutions of the RE algebra generalizing \copr by repeatedly applying the coproduct,
and in addition embed it into the $g$-fold tensorproduct of the
universal enveloping algebra of $sl_{q}(2)$ leading to the definition
$$K^{0}_{(m)}=S^{-}_{g}\cdots S^{-}_{g-(m-2)}K_{g-(m-1)}L^{+}_{g-(m-2)}\cdots L^%
{+}_{g},\qquad 1\leq m\leq g\eqn\kzerm$$
where $K_{i}=S^{-}_{i}L^{+}_{i}$ and
$L^{+}_{i}=1{\dot{\otimes}}\cdots 1{\dot{\otimes}}L^{+}{\dot{\otimes}}1\cdots{%
\dot{\otimes}}1$ with $L^{+}$ inserted
into the $i$-th position, etc. In the simplest case of $sl_{q}(2)$ all
objects on the RHS of \kzerm are $2\times 2$ matrices, matrix
multiplication being understood, and their entries take values in
the $g$-fold tensor product $U_{q}^{\otimes g}(sl(2))$. So \kzerm defines $g$ operators $K^{0}_{(1)}=K_{g},\ K^{0}_{(2)}=S^{-}_{g}K_{g-1}L^{+}_{g},\ \ldots,\ K^{0}_{(g%
)}=S^{-}_{g}\cdots S^{-}_{2}K_{1}L^{+}_{2}\cdots L^{+}_{g}$. We keep
$g$ arbitrary but fixed, it will correspond to the genus of the
handlebody later on. Formally one may envisage the limit $g\rightarrow\infty$ as one has a natural sequence of embeddings of
tensor products into higher ones. Incidentally, we found that a
variant of the operators $K^{0}_{(m)}$ has been employed in the formulation
of quantum differential geometry \Ref\rSWZP. Schupp, P. Watts and B. Zumino, Commun. Math. Phys.
157 (1993) 305.
\nextlineHowever, there is a drawback because $K^{0}_{(m)}$ and $K^{0}_{(n)}$ do
not satisfy \retwi for $m\not=n$ but instead again the RE
$$R(K^{0}_{(m)})_{1}\widetilde{R}(K^{0}_{(n)})_{2}=(K^{0}_{(n)})_{2}R(K^{0}_{(m)%
})_{1}\widetilde{R},\qquad m\geq n\eqn\regen$$
and hence $K^{0}_{(m)}$ cannot be used to represent the generators $\tau_{\alpha}$.
Fortunately, this construction gives us a hint how to solve the
problem. We define two more sets of operators $K^{+}_{(m)}$ and $K^{-}_{(m)}$ by
$$K^{\pm}_{(m)}=S^{\pm}_{g}\cdots S^{\pm}_{g-(m-2)}K_{g-(m-1)}L^{\pm}_{g-(m-2)}%
\cdots L^{\pm}_{g},\eqn\kpmm$$
they are also solutions of the RE and have the following commutation
relations
$$\eqalign{R(K^{\pm}_{(m)})_{1}\widetilde{R}(K^{\pm}_{(m)})_{2}&=(K^{\pm}_{(m)})%
_{2}R(K^{\pm}_{(m)})_{1}\widetilde{R},\cr R(K^{+}_{(m)})_{1}R^{-1}(K^{+}_{(n)}%
)_{2}&=(K^{+}_{(n)})_{2}R(K^{+}_{(m)})_{1}R^{-1},\qquad m<n\cr R(K^{-}_{(m)})_%
{1}R^{-1}(K^{-}_{(n)})_{2}&=(K^{-}_{(n)})_{2}R(K^{-}_{(m)})_{1}R^{-1},\qquad m%
>n\cr R(K^{-}_{(m)})_{1}R^{-1}(K^{+}_{(n)})_{2}&=(K^{+}_{(n)})_{2}R(K^{-}_{(m)%
})_{1}R^{-1},\qquad m\not=n}\eqn\recomgen$$
corresponding to \re and \retwi while the last equation gives the
commutation relations between the two sets of operators. They also
have definite commutation relations with $K^{0}_{(m)}$ given by
$$\eqalign{R(K^{0}_{(m)})_{1}\widetilde{R}(K^{\pm}_{(n)})_{2}&=(K^{\pm}_{(n)})_{%
2}R(K^{0}_{(m)})_{1}\widetilde{R},\cr R(K^{0}_{(m)})_{1}R^{-1}(K^{+}_{(n)})_{2%
}&=(K^{+}_{(n)})_{2}R(K^{0}_{(m)})_{1}R^{-1},\cr R(K^{-}_{(m)})_{1}R^{-1}(K^{0%
}_{(n)})_{2}&=(K^{0}_{(n)})_{2}R(K^{-}_{(m)})_{1}R^{-1}.\cr}\qquad\eqalign{&m%
\geq n\cr&m<n\cr&m>n\cr}\eqn\regentwi$$
These relations can be verified because we obviously have
$$\eqalign{\widetilde{R}(L^{\varepsilon_{1}}_{m})_{1}(L^{\varepsilon_{2}}_{m})_{%
2}&=(L^{\varepsilon_{2}}_{m})_{2}(L^{\varepsilon_{1}}_{m})_{1}\widetilde{R},%
\cr(L^{\varepsilon_{1}}_{m})_{1}(L^{\varepsilon_{2}}_{n})_{2}&=(L^{\varepsilon%
_{2}}_{n})_{2}(L^{\varepsilon_{1}}_{m})_{1},\qquad\quad m\not=n\cr}\eqn\rllgen$$
as a consequence of \rll. Thus there are two sets of operators
expressed in terms of quantum algebra generators that can be used to
represent the braid group generators $\tau_{\alpha}$. Their meaning will be
clarified in the next section. In principle \recomgen constitutes
an infinite dimensional algebra and it might have representations
other than by quantum algebra generators.
Some of the relations in \recomgen and \regentwi have the form of
\retwi and so the new solutions of the RE in ∏ can be built.
For example, if one considers only the $K^{+}_{(m)}$ series then it can be
seen that the product $K^{+}_{(m)}K^{+}_{(n)}$, $n>m$, has also commutation
relation \retwi with $K^{+}_{(p)}$ if $p>n$, and this behaviour persists for
operators with appropriate ordering. This is most easily seen in terms
of diagrams to be introduced in the next section. \nextlineFor these new solutions (but not for $K^{0}_{(m)}$) the characteristic
equation \cheq holds as well
$$qK^{\pm}_{(m)}+q^{-1}(K^{\pm}_{(m)})^{-1}-c_{1}I=0,\eqn\chargen$$
with $(K^{\pm}_{(m)})^{-1}=S^{\pm}_{g}\cdots S^{\pm}_{g-(m-2)}K^{-1}_{g-(m-1)}L^{\pm%
}_{g-(m-2)}\cdots L^{\pm}_{g}$ and $K^{-1}_{i}=S^{+}_{i}L^{-}_{i}$. The central element $c_{1}$ as the quadratic casimir
operator of $sl_{q}(2)$ remains unchanged. Finally, as representations of the
extended RE algebra their commutation relations are automatically invariant
w.r.t. the comodule transformation $({K^{\pm}_{(m)}})_{T}=TK^{\pm}_{(m)}T^{-1}$.
We remark that instead of \copr we could have used the permuted
coproduct $\Delta^{\prime}(K)=S^{-}_{1}K_{2}L^{+}_{1}$ having a
generalization as $K^{0}_{(m)}=S^{-}_{1}\cdots S^{-}_{m-1}K_{m}L^{+}_{m-1}\cdots L^{+}_{1}$, and hence $K^{\pm}_{(m)}=S^{\pm}_{1}\cdots S^{\pm}_{m-1}K_{m}L^{\pm}_{m-1}\cdots L^{\pm}_%
{1}$. They correspond to a reverse ordering of spaces and
differ from \kpmm, but do also satisfy \regen, \recomgen and
\regentwi. However, $K^{\pm}_{(m)}$ in this simpler form is not consistent
with our conventions in the next section, so we do not consider this
further.
The RE algebra therefore plays different roles, it is a comodule w.r.t.
the QG and on the other hand it acts via \lsol on representations of
the quantum algebra dual to the QG. Further applicatons of the RE algebra were mentioned in \refmark\rKSS.
\chapter
REPRESENTATIONS OF THE BRAID GROUP ON HANDLEBODIES
The braid group $B^{g}_{n}$ on a solid handlebody $H_{g}$ of genus $g$ was
described in \refmark\rSos. In addition to the generators $\sigma_{i},i=1,\ldots,n-1$ of the braid group $B_{n}$ defined on a 3-dimensional
manifold of genus zero there are generators $\tau_{\alpha},\alpha=1,\ldots,g$
implementing windings around the $g$ handles. The algebra
is given by
$$\eqalign{\sigma_{i}\sigma_{i+1}\sigma_{i}&=\sigma_{i+1}\sigma_{i}\sigma_{i+1},%
\quad i=1,\ldots,n-1\cr\sigma_{i}\sigma_{j}&=\sigma_{j}\sigma_{i},\quad\mid i-%
j\mid\geq 2\cr\sigma_{i}\tau_{\alpha}&=\tau_{\alpha}\sigma_{i},\quad i\geq 2,%
\ \alpha=1,\ldots,g\cr\sigma_{1}\tau_{\alpha}\sigma_{1}\tau_{\alpha}&=\tau_{%
\alpha}\sigma_{1}\tau_{\alpha}\sigma_{1},\cr\sigma_{1}\tau_{\alpha}\sigma_{1}^%
{-1}\tau_{\beta}&=\tau_{\beta}\sigma_{1}\tau_{\alpha}\sigma_{1}^{-1},\quad%
\alpha<\beta\cr}\eqn\bralg$$
and the first two relations define the well known Artin braid group
\refmark\rArt. We refer to \refmark\rSos for details and
references, here we only explain conventions briefly which should
make \bralg fairly transparent.
On the handlebody (Fig.1) it is possible, without loss of generality,
to prescribe a fixed ordering of the points where the strands begin
(resp. end) having coordinates $P_{i}^{(1)}=({i\over{n+1}},{1\over 2},1)$ (resp. $P_{j}^{(0)}=({j\over{n+1}},{1\over 2},0)$,
$i,j=1,\ldots,n$ in a lefthanded $(x,y,z)$-coordinate system. So the
unit cube in the positive octant is contained in $H_{g}$ and the usual
braids are obtained by connecting points $P_{i}^{(1)}$ and $P_{j}^{(0)}$
by strands confined to the unit cube. The braid diagram is obtained by
projecting on the $x$-$z$-plane. The handles are positioned, say, to
the left of the unit cube around coordinates $h_{\alpha}=({-\alpha\over{g+1}},y,1-{\alpha\over{g+1}}),\ \alpha=1,\ldots,g$. For the braid group
on $H_{g}$ the strands are allowed to leave the unit cube at height $z$
and go around the handle $h_{\alpha}$ counterclockwise for $\tau_{\alpha}$
(clockwise for $\tau_{\alpha}^{-1}$) and then come back to the unit cube at
height $z-\delta$, $\delta$ small. The convention††† We have chosen
conventions slightly different from \refmark\rSos, especially in the
last equation of \bralg the condition in \refmark\rSos is $\alpha>\beta$,
and also ‘strands should be over if $z_{2}>z_{1}$’. is that strands
leaving or entering the unit cube at height $z_{1}$ should be over those
doing so at $z_{2}$ in the projection onto the $x$-$z$-plane if $z_{1}>z_{2}$. Within the unit cube strands can only go downward in the
negative $z$-direction. This definition can be further formalized, but
everything is rather intuitive.
Fig.1: The 2-braid $\tau_{2}^{-1}\sigma_{1}^{2}\tau_{1}$ and its
closure (dotted lines)
For our arguments it is more appropriate to think of piercing long
bars through the handles and after that forget about them. Then, if
we rotate the bars by $\pi/4$ around the $x$-axis
counterclockwise to $h^{\prime}_{\alpha}=({-\alpha\over{g+1}},{\alpha\over{g+1}}-1,z)$
we can depict the braiding in a more systematic way by projecting on
the $x$-$z$-plane. For example, the fourth equation of \bralg can be
represented graphically as in Fig.2 where, as usual, $\sigma_{1}$ has been
represented by a crossing of two strands.
Fig.2: Graphical representation of the
reflection equation (for later
convenience numbering of spaces is indicated)
xxxxx
Similarly, the last equation of \bralg can be represented as in
Fig.3 and is proven easily this way by pulling lines appropriately.
Fig.3: Graphical representation of compatibility
condition \retwi
The strands leaving the unit cube to wind around the bars always
belong to the first space $V_{1}$ of the tensor product $V(n)=V_{1}\otimes\cdots\otimes V_{n}$ on which the $\sigma_{i}$ act, and this explains
why only $\sigma_{1}$ is non-commuting with $\tau_{\alpha}$. It also means that
$\tau_{\alpha}$ is acting non-trivially only in $V_{1}$ by some operator
$K_{(\alpha)}$, so we put
$$\eqalign{\sigma_{i}&=q\ 1\otimes\ldots 1\otimes\widehat{R}_{i,i+1}\otimes 1%
\ldots\otimes 1,\quad i=1,\ldots,n-1\cr\tau_{\alpha}&=q^{3}\ K_{(\alpha)}%
\otimes 1\ldots\otimes 1,\quad\alpha=1,\ldots,g\cr}\eqn\ident$$
where as usual $\sigma_{i}$ is acting non-trivially only in $V_{i}\otimes V_{i+1}$ as $\sigma_{i}=PR_{i,i+1}\equiv\widehat{R}_{i,i+1}$. The factor $q^{3}$
is inserted for later convenience to match the factor $q$ in the
definition of $\sigma_{i}$. Using this we can show explicitly that \bralg is equivalent to the Yang-Baxter equation and to the extented RE algebra,
by identifying $\sigma_{1}=\widehat{R}_{12}$ and $\tau_{\alpha}=(K_{(\alpha)})_{1}$, plus
two other rather obvious consistency conditions as given in \bralg.
This equivalence is also explained in \refmark\rKulb,\rKuSa.
Thus $\sigma_{i}$ has an explicit matrix representation, but what about
$\tau_{\alpha}$? Because $\sigma_{i}$ is represented by a $R$-matrix one should
expect that the same holds true for $\tau_{\alpha}$, and this is supported
by Fig.2 which suggests to represent the effect of a handle on a
strand going around it by the square of a $R$-matrix. Let us consider
first the genus one case, i.e. only the RE has to be taken into
account. As shown in the previous section $K=S^{-}L^{+}$ is a
solution of the RE and this tells us how to represent $\tau_{\alpha}$ because
$S^{-}$ and $L^{+}$ are related to the $R$-matrix. In fact, looking at
the universal $R$-matrix of $sl_{q}(2)$ acting on $V_{1}\otimes V_{2}$
$$R_{U}=q^{{1\over 2}H\otimes H}\sum^{\infty}_{n=0}{(1-q^{-2})^{n}\over[n;q^{-2}%
]!}\bigl{(}q^{-{1\over 2}H}X^{-}\bigr{)}^{n}\otimes\bigl{(}q^{{1\over 2}H}X^{+%
}\bigr{)}^{n},\qquad[n;q]={(1-q^{n})\over(1-q)}\eqn\ru$$
we can represent the $sl_{q}(2)$ generators either on $V_{1}$ or $V_{2}$
(the fundamental representation of $sl_{q}(2)$ is
$\rho_{f}(H)=$ $1\,\ 0\ \choose 0\,-1$,
$\rho_{f}(X^{+})=$ $0\,1\choose 0\,0$,
$\rho_{f}(X^{-})=$ $0\,0\choose 1\,0$) giving
$$\eqalign{\rho_{f}(R_{U})\big{|}_{V_{1}}&=\pmatrix{q^{H/2}&0\cr q^{-1/2}\omega X%
^{+}&q^{-H/2}\cr}=S^{-},\cr\rho_{f}(R_{U})\big{|}_{V_{2}}&=\pmatrix{q^{H/2}&q^%
{-1/2}\omega X^{-}\cr 0&q^{-H/2}\cr}=L^{+}.}\eqn\alrep$$
And further representing in a second step the ‘semiuniversal’
operators $S^{-}$ and $L^{+}$ we get
$$\rho_{f}(S^{-})=q^{-1/2}R,\qquad\rho_{f}(L^{+})=q^{-1/2}\widetilde{R},\qquad%
\rho_{f}(S^{-}L^{+})=q^{-1}{\widehat{\widetilde{R}}}{}^{2}\eqn\oprep$$
where $\widehat{\widetilde{R}}=P\widehat{R}P=RP$. This means that we have to represent a
strand ‘interacting’ with the handle like in Fig.2 by
${K_{(1)}{}^{i}}_{j}=q^{-1}({\widehat{\widetilde{R}}}{}^{2}{}^{i}{}_{j})^{m}{}_%
{n}\equiv q^{-1}({\widehat{R}}^{2}{}^{m}{}_{n})^{i}{}_{j}$, where the $(ij)$ indices are in the first
space of $V(n)$ and therefore ${K_{(1)}{}^{i}}_{j}=q^{-1}({\widehat{R}}^{2}{}^{m}{}_{n})^{i}{}_{j}$ suits the graphical representation in Fig.2.
In effect we have translated a topological property of the solid torus
into a quantum algebra operator acting on $V_{1}$ and an additional
‘internal’ space $V_{1^{\prime}}$ embedded into the space $V(1;n)=V_{1^{\prime}}\otimes V_{1}\otimes\cdots V_{n}$. As a consequence we have a
two-dimensional representation of the RE algebra given by matrices
$$\eqalign{{a^{m}}_{n}&=\pmatrix{q&0\cr 0&q^{-1}\cr},\cr{c^{m}}_{n}&=\pmatrix{0&%
q^{-1}\omega\cr 0&0\cr},}\qquad\quad\eqalign{{b^{m}}_{n}&=\pmatrix{0&0\cr q^{-%
1}\omega&0\cr},\cr{d^{m}}_{n}&=\pmatrix{q^{-1}(1+\omega^{2})&0\cr 0&q\cr},}\eqn\twdrep$$
which indeed satisfy \alg. The operator $K_{(1)}=S^{-}L^{+}|_{\rho}$ appeared also in the context of conformal field theory
\refmark\rMoRe and was used there, for example, in connection with
topology changing amplitudes in Chern-Simons field theory.
It is easy to check from \rmat that both $\widehat{R}$ and $\widehat{\widetilde{R}}$
obey a quadratic equation of the form
$$\widehat{R}^{2}-\omega\widehat{R}-I=0.\eqn\rchar$$
This, in turn, leads to a quadratic equation for $\widehat{R}^{2}$
$$\widehat{R}^{2}+\widehat{R}^{-2}-(q^{2}+q^{-2})I=0,\eqn\rsqch$$
and this is nothing but the characteristic equation \cheq for
$K_{(1)}=q^{-1}\widehat{R}^{2}_{1^{\prime}1}$. Comparing with \cheq we can identify
$c_{1}(K)=q^{2}+q^{-2}$ (discarding the $2\times 2$ identity matrix),
and this can be verified by calculating the quantum trace of the
explicit representation \twdrep. Hence, for this representation of
$K_{(1)}$ its characteristic equation follows from the one for the
$R$-matrix. Therefore the somewhat ad hoc assumption of considering
the characteristic equation of the $K$-matrix as a skein relation for
lines going around handles in \refmark\rSch is justified because
\rchar has an interpretation as a skein relation. Even more so as we
have seen that handles themselves can be considered as kind of lines
in topologically trivial regions, we will focus on this in the next
section.
In order to explain the case of arbitrary genus it is sufficient to
look at $g=2$. Now we also need to take into account the last equation
of \bralg. From Fig.3 we can guess that $K_{(1)}$ is as before, but
$K_{(2)}$ should be represented by a product of four $R$-matrices
acting on $V(2;n)=V_{2^{\prime}}\otimes V_{1^{\prime}}\otimes V_{1}\otimes\cdots V_{n}$ as
$$K_{(1)}=q^{-1}I_{2^{\prime}}\otimes\widehat{R}^{2}_{1^{\prime}1},\qquad K_{(2)%
}=q^{-1}\widehat{R}^{-1}_{1^{\prime}1}\widehat{R}^{2}_{2^{\prime}1^{\prime}}%
\widehat{R}_{1^{\prime}1}.\eqn\kontw$$
The indices characterizing operators $a,b,c,d$ belong to $V_{1}$, all
primed indices refer to ‘internal’ spaces related to the handles of
the manifold. Thus we can read off from \kontw the explicit
four-dimensional representation using \rmat
$$\eqalign{a_{(1)}&=\pmatrix{q&0&0&0\cr 0&q^{-1}&0&0\cr 0&0&q&0\cr 0&0&0&q^{-1}%
\cr},\cr c_{(1)}&=\pmatrix{0&q^{-1}\omega&0&0\cr 0&0&0&0\cr 0&0&0&q^{-1}\omega%
\cr 0&0&0&0\cr},\cr&\cr a_{(2)}&=\pmatrix{q&0&0&0\cr 0&q&-\omega^{2}&0\cr 0&0&%
q^{-1}&0\cr 0&0&0&q^{-1}\cr},\cr c_{(2)}&=\pmatrix{0&0&\omega&0\cr 0&0&0&q^{-2%
}\omega\cr 0&0&0&0\cr 0&0&0&0\cr},\cr}\qquad\quad\eqalign{b_{(1)}&=\pmatrix{0&%
0&0&0\cr q^{-1}\omega&0&0&0\cr 0&0&0&0\cr 0&0&q^{-1}\omega&0\cr},\cr d_{(1)}&=%
\pmatrix{q^{-1}(1+\omega^{2})&0&0&0\cr 0&q&0&0\cr 0&0&q^{-1}(1+\omega^{2})&0%
\cr 0&0&0&q\cr},\cr&\cr b_{(2)}&=\pmatrix{0&0&0&0\cr q^{-2}\omega^{2}&0&0&0\cr
q%
^{-2}\omega&0&0&0\cr 0&\omega&-\omega^{2}&0\cr},\cr d_{(2)}&=\pmatrix{q^{-1}(1%
+\omega^{2})&0&0&0\cr 0&q^{-1}(1+\omega^{2})&q^{-2}\omega^{2}&0\cr 0&0&q&0\cr 0%
&0&0&q\cr}.\cr}\eqn\fodrep$$
These matrices do not only satisfy \alg but really
provide a non-trivial explicit representation of \recom (with
$K_{(1)}=K,\ K_{(2)}=K^{\prime}$), proving that \kontw is indeed a
representation of \bralg. Furthermore, using the representation
$\rho_{f}$ of $H$ and $X^{\pm}$ it can be verified that \fodrep may
equally well be obtained from quantum algebra solution \kpmm of the
extended RE algebra
$$K_{(1)}=\rho_{f}(K_{2})\equiv\rho_{f}(K^{+}_{(1)}),\qquad K_{(2)}=\rho_{f}(S^{%
+}_{2}K_{1}L^{+}_{2})\equiv\rho_{f}(K^{+}_{(2)}).\eqn\kfotw$$
As shown in section 2 all quantum algebra solutions satisfy the same
characteristic equation, so $K_{(1)}$ and $K_{(2)}$ do satisfy
\rsqch which is obvious from \kontw anyway. Then one might wonder
what the meaning is of $K^{-}_{(m)}$, it can be seen that it plays the
same role as $K^{+}_{(m)}$ but corresponds to a different incompatible
set of conventions compared to those given in the beginning (i.e. $\alpha>\beta$ in \bralg, $z_{2}>z_{1}$ for strands leaving or entering the unit
cube, clockwise rotation of bars corresponding to handles, interchange
of figures for $K$ and $K^{-1}$). This choice gives nothing new and
needs not to be considered, for example, $K_{(2)}=\rho_{f}(K^{-}_{(2)})$
is the same as in \fodrep but with all four matrices transposed and
$b_{(2)}$ interchanged with $c_{(2)}$. The last relation of \bralg in this case can be depicted as in Fig.4, the major difference being
some lines now going under the bars.
Fig.4: Compatibility condition \retwi with $\tau_{\alpha}$
represented by the $K^{-}_{(m)}$ series
We still have to explain the meaning of ∏ in terms of the braid
group generators. Property $(i)$ relates to successive application of
$K_{(1)}$ and $K_{(2)}$ leading to a new move encircling both bars as
displayed in Fig.5.
Fig.5: The product $\tau_{1}\tau_{2}$ (equivalent to
representing $\tau_{2}$ by $K^{0}_{(2)}$)
It is immediately clear that the move $K_{(1)}K_{(2)}$ again
satisfies the RE as can be seen just by inserting an additional bar
appropriately into Fig.2. Property $(ii)$ can be understood similarly
by looking at $K_{(1)}K_{(2)}K^{-1}_{(1)}$ shown in Fig.6. If the results of
multiplying (products of) braid group generators obey the braid group
relations again it then graphically comes down to being able to ‘pull
lines’ in a simple way.
Fig.6: The product $\tau_{1}\tau_{2}\tau^{-1}_{1}$ (equivalent to representing $\tau_{2}$ by $K^{-}_{(2)}$)
The results of figs.5,6 show that although we have chosen the $K^{+}_{(m)}$
series to represent the braid group generators $\tau_{\alpha}$, we now see the
$K^{-}_{(m)}$ and $K^{0}_{(m)}$ series appearing because the combined solutions are
precisely given by $K^{0}_{(2)}$ and $K^{-}_{(2)}$ as can be checked by formulas.
Even though we start from $K^{+}_{(m)}$ solely its properties as an extended
RE algebra force us to consider the whole system \regen, \recomgen and
\regentwi. As a side remark we mention here that if we may use graphs
for both $K^{+}_{(m)}$ and $K^{-}_{(n)}$ it is easy to see why they have no
commutation relation for $m=n$, it is just not possible to disentangle
the lines.
Now it should be clear how this generalizes to the case of arbitrary
genus. We will have $g$ bars corresponding to the handles and $\tau_{\alpha}$
is represented by a strand going from first space over the first $(\alpha-1)$ bars to wind around the one corresponding to the handle $h_{\alpha}$
and going back again to $V_{1}$ over the first $(\alpha-1)$ bars. An
example is shown in Fig.7. It is obvious that the $\tau_{\alpha}$ satisfy
the defining relations \bralg and this can be proven analogously to
figs.3,4.
Fig.7: The generator $\tau_{4}$ for the $g=5$ case
It is clear that $\tau_{\alpha}$ acting on $V(g;n)=V_{g^{\prime}}\otimes\cdots V_{1^{\prime}}\otimes V_{1}\otimes\cdots V_%
{n}$ is represented as in \ident with the non-trivial part of $K_{(\alpha)}$ given by (identify indices
$0^{\prime}\equiv 1$ where neccessary)
$$\eqalign{K_{(\alpha)}&=q^{-1}\widehat{R}^{-1}_{1^{\prime}1}\widehat{R}^{-1}_{2%
^{\prime}1^{\prime}}\cdots\widehat{R}^{-1}_{(\alpha-1)^{\prime}(\alpha-2)^{%
\prime}}\widehat{R}^{2}_{{\alpha}^{\prime}(\alpha-1)^{\prime}}\widehat{R}_{(%
\alpha-1)^{\prime}(\alpha-2)^{\prime}}\cdots\widehat{R}_{2^{\prime}1^{\prime}}%
\widehat{R}_{1^{\prime}1}\cr&=\rho_{f}(K^{+}_{(\alpha)})\ ,}\eqn\krepgen$$
and $K^{+}_{(\alpha)}$ is expressed in terms of quantum algebra generators as
in \kpmm. Note that numbering the primed spaces is by convention, and
we have chosen a more natural one with opposite ordering compared to
\kpmm where it is fixed (cf. remark at end of section 2). All that
was said about the $g=2$ case above can be generalized to arbitrary
genus, there are no new features emerging.
We finish this section by noting that the generators $\tau_{\alpha}$ in the
representation \krepgen can be considered as a subgroup of the
coloured braid group $C_{g+1}$. By definition, the coloured braid group
$C_{g+1}$ is the kernel of the mapping from the Artin braid group
$B_{g+1}$ to the permutation group $P_{g+1}$ having
${1\over 2}g(g+1)$ elements $\kappa_{\alpha\beta}$ that can
be taken in our notation as
$$\kappa_{\alpha\beta}=\sigma^{-1}_{(\beta+1)^{\prime}}\cdots\sigma^{-1}_{(%
\alpha-1)^{\prime}}\ \sigma^{2\phantom{-}}_{\alpha^{\prime}}\sigma_{(\alpha-1)%
^{\prime}}\cdots\sigma_{(\beta+1)^{\prime}}\ ,\qquad 0\leq\beta<\alpha\leq g\eqn\colbr$$
acting on $V(g;1)=V_{g^{\prime}}\otimes\cdots V_{1^{\prime}}\otimes V_{1}$. This
corresponds to $g$ bars plus one strand. Of course, in our case the
bars cannot wind around each other (but see the interpretation of
bars as lines in the next section) so we have to fix $\beta=0$ and let
$1\leq\alpha\leq g$, s.t. $\kappa_{\alpha 0}$ gives
precisely the $g$ generators in \krepgen as a subset of the
generators of $C_{g+1}$. Therefore we can also think of $B^{g}_{n}$ as a
subgroup of the braid group $B_{g+n}$ having generators
$\sigma_{g^{\prime}},\ldots,\sigma_{1^{\prime}},\sigma_{1},\ldots,\sigma_{n-1}$. From our
experiences in section 2 it is obvious how to represent the full
coloured braid group in terms of quantum algebra generators, namely
the equivalent of \colbr is given by
$$\eqalign{K^{+}_{(j;m)}=S^{+}_{g-j}\cdots S^{+}_{g-j-(m-2)}K_{g-j-(m-1)}L^{+}_{%
g-j-(m-2)}\cdots L^{+}_{g-j},\qquad j&=0,\ldots,g-1\cr m&=1,\ldots,g-j}\eqn\fcolbr$$
and similarly for $K^{-}_{(j;m)}$ which gives \colbr but with all braid
group generators except $\sigma^{2}_{\alpha^{\prime}}$ replaced by their inverses. For
each fixed value of $j$ they have commutation relations \recomgen.
Of course, we can write down a similar formula for $K^{0}_{(j;m)}$ but
they do not represent the coloured braid group. It was already noted
in \refmark\rSWZ that the coloured braid group has a representation
within tensor products of the universal enveloping algebra of $sl_{q}(N)$.
\chapter
INVARIANTS OF LINKS ON HANDLEBODIES
We will now define links on the handlebody $H_{g}$ and then try to find
invariant polynomials. A $g$-link $L_{g}$ on $H_{g}$ is obtained as the
closure of a $g$-braid by connecting $P^{(0)}_{i}$ with $P^{(1)}_{i}$
outside the unit cube in the $x>0$ region (Fig.1). Citing a
theorem \refmark\rSos, every $g$-link can be obtained as the
closure of a $g$-braid. Markov moves for $B_{n}^{g}$ are defined
in the same manner as the usual ones for $B_{n}$, i.e. $B\rightarrow B^{\prime}B(B^{\prime})^{-1}$ for arbitrary $B^{\prime}\in B^{g}_{n}$ (Markov I) and $B\rightarrow B\sigma_{n}^{\pm 1}$ with $\sigma_{n}\in B^{g}_{n+1}$ (Markov II).
Then the Markov theorem would state that two $g$-braids have
equivalent closures iff there is a finite sequence of Markov moves of
type I and II taking one $g$-braid to the other. However, the Markov
theorem for $g>1$ was only stated as a conjecture in \refmark\rSos,
it holds for $g=1$ (we shall not need it in what follows).
There are several approaches to the construction of link polynomials,
one may roughly distinguish them in the following way (a convenient
access to original literature is \Ref\rKoha“New developments in the theory of knots”,
ed. T. Kohno, World Scientific, Singapore 1989, basic accounts of
knot theory and the relation to quantum groups are e.g.
\REF\rGuaaE. Guadagnini, in: “Proceedings of the
${\rm 14^{th}}$ John Hopkins workshop on current problems in particle
theory”, eds. G. Domokos et. al., World Scientific, Singapore 1991
\REF\rAGSL. Alvarez-Gaume, C. Gomez and G. Sierra, Nucl. Phys. B330 (1990) 347 \refmark\rGuaa,\rAGS). It is well known that the
expression of $\sigma_{i}$ in terms of $\widehat{R}$ gives rise to a Hecke algebra
representation of the braid group $B_{n}$ and the characteristic
equation of the $\widehat{R}$-matrix together with the first two equations of
\bralg comprise just the relations of the Hecke algebra $H(q^{2},n)$
with generators $\sigma_{i}$ (we would have to rescale $q\rightarrow q^{1/2}$ to make contact with the usual convention). One defines a
linear functional on $H(q^{2},n)$, the Ocneanu trace, which is the main
ingredient in the definition of the invariant link polynomial
\Ref\rJonV. F. R. Jones, Bull. Am. Math. Soc. 12 (1985) 103;
Ann. Math. 126 (1987) 335. This is just the quantum trace of the braid
group generators represented by $\widehat{R}$ and the last step is then
proving invariance of it w.r.t Markov moves \REF\rTurV. G. Turaev, Invent. Math. 92 (1988) 527
\REF\rReTuN. Yu. Reshetikhin and V. G. Turaev, Commun. Math. Phys.
127 (1991) 1; Invent. Math. 103 (1991) 547 \REF\rADWY. Akutsu, T. Deguchi and M. Wadati,
J. Phys. Soc. Jpn. 56 (1987) 839, 3039, 3464;
57 (1988) 757, 1173, 1905 \refmark\rTur,\rReTu,\rADW.
Further it is possible to define link polynomials recursively using
skein relations \REF\rAlexJ. W. Alexander, Trans. Am. Math. Soc. 20 (1923) 275;
Proc. Natl. Acad. Sci. 9 (1928) 93 \REF\rConJ. H. Conway, in “Computational problems in abstract
algebra ”, Pergamon, New York 1970 \REF\rKaufL. H. Kauffman, Topology 26 (1987) 395
\refmark\rAlex,\rCon,\rKauf. Finally, there is the Chern-Simons
field theory approach \REF\rWitaE. Witten, Commun. Math. Phys. 121 (1989) 351 \REF\rGMME. Guadagnini, M. Martellini and M. Mintchev, in: “Proceedings of the ${\rm 13^{th}}$ John Hopkins workshop on current
problems in particle theory”, ed. L. Lusanna, World Scientific,
Singapore 1990
\refmark\rWita,\rGMM. In view of this we might expect that the
explicit representation \ident can be used to define an invariant
link polynomial on $H_{g}$ by means of quantum traces of generators
$\sigma_{i}$ and $\tau_{\alpha}$. Also their characteristic equations should give
rise to skein relations.
We recall the definition of the Jones polynomial \refmark\rJon
$$V(B)=q^{-3w({\hat{B}})}Tr\big{|}_{V(n)}(B\mu^{\otimes n}),\eqn\jopol$$
which is a class function on the braid group $B_{n}$ that is invariant
w.r.t Markov moves of type I and II. Because of inclusion of type II
moves it is an ambient isotopic invariant, i.e. locks in a line are
irrelvant. In \jopol we denoted the closure of the braid $B\in B_{n}$
by ${\hat{B}}$, and $w({\hat{B}})$, the writhe or Tait number, is
the number of overcrossings minus the number of undercrossings in
${\hat{B}}$. Finally, the matrix $\mu$ is defined via an element of the
quantum algebra
$$\mu=\pmatrix{q^{-1}&0\cr 0&q}\equiv\rho_{f}(q^{-H}),\eqn\mdef$$
so the trace in \jopol could equally well be called the quantum trace
of $B$. For instance, the quantum trace of the RE algebra can be expressed
as $c_{1}=Tr(K\mu)$. Invariance of \jopol under Markov I rests on the
property $[\sigma_{i},\mu^{\otimes n}]=0$, and for Markov II
relies on the key property of the quantum trace $Tr\big{|}_{V_{n+1}}(\sigma^{\pm 1}_{n}(I^{\otimes n}\otimes\mu))=q^{\pm 3}I^{%
\otimes n}$.
The Jones polynomial satisfies a skein relation which is obtained from
the characteristic equation of $\sigma_{i}$
$$q^{-1}\sigma_{i}-q\sigma_{i}^{-1}-\omega I=0\eqn\schar$$
by using linearity of the trace, the result is for arbitrary $B,B^{\prime}\in B_{n}$
$$q^{2}V(B\sigma_{i}B^{\prime})-q^{-2}V(B\sigma_{i}^{-1}B^{\prime})-(q-q^{-1})V(%
BB^{\prime})=0.\eqn\jposk$$
This can be depicted as
Fig.8: Skein relation of the Jones polynomial
where each pictogram means the polynomial of the closed braid
differing only at the crossing between $B$ and $B^{\prime}$ as indicated.
By simply closing the lines the normalization of the unknot is
obtained as $N=q+q^{-1}$, this is the same as in \refmark\rWita up
to rescaling of $q$ and corresponds to standard framing (see the
discussion in \Ref\rWitbE. Witten, Nucl. Phys. B322 (1989) 629 on the effects of framing).
There is a simple possibility to make use of this formalism in our
context as we have a representation of $\tau_{\alpha}$ in terms of
$\widehat{R}$-matrices. We just extend the definition of the Jones polynomial
from $V(n)$ to $V(g;n)$ and get an ambient isotopy invariant
polynomial for $g$-links on $H_{g}$ in terms of the Jones polynomial of
links in ${\bf{\rm R}}^{3}$ (or a 3-ball, for that matter). Proof of
invariance w.r.t. Markov I,II works as before. We are forced to
interprete bars corresponding to the handles as lines and, because of
the trace in $V(B)$, close them to obtain the link in ${\bf{\rm R}}^{3}$ which is associated to the $g$-link on $H_{g}$. Because we have
an description of the crossings by $\widehat{R}$-matrices the direction of the
bars is fixed downwards. This procedure is obviously well defined as
the equivalence class of a $g$-link is mapped uniquely onto the class
of the associated ordinary link \refmark\rSos. The polynomial for
$g$-links is then given by
$$V_{g}(B)=q^{-3w({\hat{B}})}Tr\big{|}_{V(g;n)}(B\mu^{\otimes(g+n)}),\eqn\gjopol$$
where $B\in B_{n}^{g}$ is a word in the generators $\sigma_{i}$ and
$\tau_{\alpha}$, its closure is obtained via the representation \ident and
\krepgen of $\tau_{\alpha}$. It is easy to find examples of $g$-links which
belong to different classes but share the same value of the
polynomial.
It might appear as if one were back to the usual situation where one
deals with the generators $\sigma_{i}$ only. However, keeping generators
$\tau_{\alpha}$ is a great advantage, the reason is that they are very special
expressions in $\sigma_{i}$. They all obey the characteristic equation
$$q^{-2}\tau_{\alpha}+q^{2}\tau_{a}^{-1}-c_{1}I=0,\eqn\tchar$$
which, as before, leads to a skein relation that follows from \gjopol
$$q^{4}V_{g}(B\tau_{\alpha}B^{\prime})+q^{-4}V_{g}(B\tau_{\alpha}^{-1}B^{\prime}%
)-(q^{2}+q^{-2})V_{g}(BB^{\prime})=0\eqn\gjposk$$
in addition to the $\sigma_{i}$ skein relation for $V_{g}(B)$. Here we
inserted the value $c_{1}=q^{2}+q^{-2}$ for the
fundamental representation of $sl_{q}(2)$. Above relation can be depicted as
Fig.9: The additional skein relation for generators
$\tau_{\alpha}$
where we displayed only the $g=1$ case. It is known
\refmark\rAlex,\rCon that by recursively using skein relations the
invariant polynomial can be calculated uniquely. So it is reasonable
to suggest that both skein relations \jposk and \gjposk suffice to
calculate a well defined polynomial for any $g$-link \refmark\rSch,
knowing the origin of the additional skein relation it is obvious
that this statement is correct.
If we use skein relations to calculate the polynomial we can fix the
procedure as follows. First untie the knot in the topological trivial
region using \jposk, this is clearly possible and it eventually gives
unknots going around the bars. If an unknot winds around a bar $n$
times, $(\tau_{\alpha})^{n}$, then it always can be reduced to $n=1$ with the
help of \gjposk, regardless whether $n$ is positive or negative.
Similarly if it winds around several bars in the right order by using
the analogue \comcen of \gjposk for a product of several generators,
otherwise the correct order must be established first by using \gjposk.
This way the reduction process can be much simplified, but
nevertheless in the end \jposk has to be used again to untie the
simple loops around the $S^{1}$-factors (closed bars). If there is only
one loop simple rules can be established as indicated in Fig.10a.
Fig.10: (a) Evaluation of a simple loop encircling $n$
(closed) bars ($N=q+q^{-1}$ is the
normalization of the unknot), (b) Relation between
$\tau_{\alpha}^{-1}$ and $\tau_{\alpha}$ xxxx
The virtue of Fig.10a is that it connects loops going around handles
to loops in topologically trivial regions, we can just reinsert
instead of the factor $N$ an unknot in ${\bf{\rm R}}^{3}$ (if there are
bars not being encircled by the loop they contribute factors of $N$ on
the RHS). In a way, it looks as if this were related to the surgery
method described in \refmark\rWita, see also \REF\rGuabE. Guadagnini, Nucl. Phys. B375 (1992) 381
\REF\rGuacE. Guadagnini and S. Panicucci, Nucl. Phys. B388
(1992) 159 \refmark\rGuab,\rGuac.
One can also express a loop originating from $\tau_{\alpha}^{-1}$ directly in
terms of one originating from $\tau_{\alpha}$ (only the $g=1$ case is shown in
Fig.10b), if the loop encircles $n$ closed bars corresponding to a
ordered product of $n$ generators $\tau_{\alpha}$ the factor will be $q^{6n}$.
This can be taken a little further but as we do not have concrete
calculations in mind we do not elaborate on it.
Of course, even though it is possible to define the above invariant
polynomial, we would have prefered to define it intrinsically on the
handlebody keeping the information about the topology strictly, i.e.
without transforming holes into lines carrying some representations.
But this is not so easy, because if we use \jopol defined on $V(n)$
only, but with $g$-braids $B$ containing generators $\tau_{\alpha}$, then the
trace is no longer invariant with respect to Markov I. This follows
from the fact that $[K,\mu]\not=0$, the only
difference occurs in first space and Markov II is still valid. We are
presently looking for a modification of the polynomial that would
employ only the algebraic properties of the RE algebra and make no use of
the representation discussed above, whether this is possible and
whether it would lead to an inequivalent invariant is an open
question.
\chapter
DISCUSSION
There are a few topics that can be mentioned in connection with the
present work. The motivation in \refmark\rSos was to define
invariant polynomials of links intrinsically on any closed 3-manifold
$M$. The prerequisite for this is a polynomial defined on
handlebodies, it would then be neccessary to investigate how it
transforms w.r.t. the Heegard homeomorphism $\psi:\partial H_{g}\rightarrow\partial H_{g}$ since any closed compact 3-manifold $M$ can
be obtained by the Heegard decomposition $M=H_{g}\cup_{\psi}H^{\prime}_{g},\ H_{g}\cap H^{\prime}_{g}=\partial H_{g}=%
\partial H^{\prime}_{g}$. Every link in $M$ is
isotopic to a closed braid in $H_{g}$, but the braid depends on the
Heegard splitting. One would then need to use (a subgroup of) the
mapping class group of a genus $g$ Riemann surface in order to study
the behaviour of the polynomial w.r.t. the Heegard homeomorphism,
maybe the approach in \Ref\rKohbT. Kohno, Topology 31 (1992) 203 could be useful where the
homeomorphisms of a handlebody were expressed essentially in terms of
$R$-matrices. Related to this subject is the surgery method because
the Heegard splitting could be used also to transform invariants that
are defined on a certain closed 3-manifold to a different one. Whether
such an approach would be more tractable compared to \refmark\rWita
is not clear a priori. The presence of the generators $\tau_{\alpha}$ suggests
the idea of an ‘algebraization’ of the surgery method.
A further problem is whether there exist representations of $B^{g}_{n}$
other than in terms of $R$-matrices. The RE algebra as written in \alg,
especially existence of central elements and characteristic equations,
are a consequence of the Hecke algebra representation of the
generators $\sigma_{i}$. It is not clear whether it is possible to represent
$\tau_{\alpha}$ differently from $\sigma_{i}$. The problem to find representations
of $B^{g}_{n}$ is of course related to the difficulty in definig invariant
link polynomials, because the Markov trace is dependent on classes of
(irreducible) representations. In this context it is worthwhile noting
that at least $B^{1}_{n}$ is a Coxeter group \refmark\rSos.
Also, we would like to draw attention again to the (infinite
dimensional) extended RE algebra and the new representations of it that we
constructed in terms of quantum algebra generators which satisfy the
system of commutation relations \recomgen. This might have some
applications other than discussed here, e.g. in the description of
differential geometry on quantum groups or quantum spin chains.
During typesetting this manuscript we came across two preprints which
bear some similarity to our work. In \Ref\rAleA. Yu. Alekseev, Uppsala Univ. preprint, hep-th/9311074 the monodromies
of flat connections around the cycles of a Riemann surface with marked
points are considered, they obey some variant of the extended RE algebra.
In \Ref\rKaZaM. Karowski and A. Zapletal, FU Berlin preprint,
hep-th/9312008 a quantum group invariant $n$-state vertex model
on a torus is constructed which has a topological interaction of the
vertices with the interior of the torus, the grahical notation is also
reminiscent of ours.
\refout |
CMB Delensing Beyond the B Modes
Daniel Green,${}^{\bigstar,\clubsuit}$ Joel Meyers${}^{\clubsuit}$, and Alexander van Engelen ${}^{\clubsuit}$
${}^{\bigstar}$ University of California, Berkeley, California 94720, USA
${}^{\clubsuit}$ Canadian Institute for Theoretical Astrophysics, Toronto, ON M5S 3H8, Canada
Abstract
Gravitational lensing by large-scale structure significantly impacts observations of the cosmic microwave background (CMB): it smooths the acoustic peaks in temperature and $E$-mode polarization power spectra, correlating previously uncorrelated modes; and it converts $E$-mode polarization into $B$-mode polarization. The act of measuring and removing the effect of lensing from CMB maps, or delensing, has been well studied in the context of $B$ modes, but little attention has been given to the delensing of the temperature and $E$ modes. In this paper, we model the expected delensed $T$ and $E$ power spectra to all orders in the lensing potential, demonstrating the sharpening of the acoustic peaks and a significant reduction in lens-induced power spectrum covariances. We then perform cosmological forecasts, demonstrating that delensing will yield improved sensitivity to parameters with upcoming surveys. We highlight the breaking of the degeneracy between the effective number of neutrino species and primordial helium fraction as a concrete application. We also show that delensing increases cosmological information as long as the measured lensing reconstruction is included in the analysis. We conclude that with future data, delensing will be crucial not only for primordial $B$-mode science but for a range of other observables as well.
Contents
1 Introduction
2 All-Orders Delensing
2.1 Non-Perturbative Delensing for Real Data
2.2 Approximate Delensed Power Spectra
2.3 Polarization
3 Numeric Spectra and Covariances
3.1 Spectra
3.2 Power Spectrum Covariance
4 Delensing and Cosmological Parameters
4.1 Forecasting Methodology
4.2 Implications
5 Conclusion
A Gradient Expansion
B Filters
B.1 Noise and Filtering
B.2 Conservation of Total Power
B.3 Optimal Filters
B.4 Polarization
B.5 Filtering in $Q/U$ versus $E/B$
C Numeric Computation of Polarization Spectra
D Calculating the Covariance
E Fisher Information and Delensing
F Exact Delensing and Real Data
1 Introduction
Weak gravitational lensing of the cosmic microwave background (CMB) [1, 2, 3, 4] is now a highly significant feature, seen in both the power spectra [5, 6, 7, 8, 9, 10, 11] and the higher-order statistics [12, 13, 14, 15, 16, 17, 18]. Depending on the question of interest, CMB lensing can be either a nuisance or a tool. For instance, the sum of the neutrino masses can be measured from the reconstruction of the lensing potential [19]. For some other parameters, error bars improve when the effect of lensing is removed from the power spectra (delensing). Delensing the $B$-mode polarization to search for primordial gravitational waves is one example that has been studied in great detail [20, 21, 22, 23, 24], but delensing is a more broadly useful tool that has been explored to a much smaller extent in temperature ($T$) and $E$-mode polarization ($E$). Recently the delensing of the small-scale temperature field was demonstrated on Planck CMB data using Planck maps of the cosmic infrared background [25]. In this paper we present a computation of the delensed small-scale CMB power spectra to all orders in the lensing potential, calculate the associated covariance matrices of delensed power spectra, and forecast parameter constraints from upcoming CMB surveys when analyzing the delensed spectra.
One of the motivations for studying delensing in this regime is for future measurements of the effective number of neutrino species, $N_{\rm eff}$. Free streaming radiation, such as neutrinos, is known to induce a phase shift in the acoustic peaks of the primary CMB [26, 27, 28]. Lensing is known to smooth the acoustic peaks [29, 30] which, in turn, reduces the accuracy of the measurements of the peak locations. In fact, the benefit of delensing is quite analogous to BAO reconstruction [31], as illustrated in Figure 1. For this reason, forecasts for future CMB experiments show that unlensed spectra lead to better measurements of $N_{\rm eff}$ [28]. In reality, delensing is an imperfect procedure and therefore any proper treatment of forecasting or analysis should predict the delensed, rather than unlensed, spectra.
One of the great technical simplifications of CMB lensing is that the process is local in the observed direction. In the flat sky limit,
$$\tilde{T}(\vec{x}\mskip 2.0mu \vphantom{x})=T(\vec{x}\mskip 2.0mu \vphantom{x}%
+\vec{\alpha}(\vec{x}\mskip 2.0mu \vphantom{x}))\simeq T(\vec{x}\mskip 2.0mu %
\vphantom{x})+\vec{\alpha}(\vec{x}\mskip 2.0mu \vphantom{x})\cdot\vec{\nabla}T%
(\vec{x}\mskip 2.0mu \vphantom{x})+\ldots\ .$$
(1.1)
where $\tilde{T}$ ($T$) is the lensed (unlensed) temperature map, $\vec{\alpha}=\vec{\nabla}\phi$ is the deflection angle, and $\phi$ is the lensing potential. Given an observed temperature map ($T^{\rm obs}$) and lensing map ($\vec{\alpha}\,{}^{\rm obs}$), we can certainly imagine a perturbative approach to delensing where
$$T^{\rm d}(\vec{x}\mskip 2.0mu \vphantom{x})\approx T^{\rm obs}(\vec{x}\mskip 2%
.0mu \vphantom{x})-\vec{\alpha}\,{}^{\rm obs}\cdot\vec{\nabla}T^{\rm obs}(\vec%
{x}\mskip 2.0mu \vphantom{x})\ ,$$
(1.2)
where $T^{\rm d}$ is the delensed temperature map.
In practice, modeling lensing of the CMB power spectra requires more accuracy than the simple perturbative description. Fortunately, $\phi$ is Gaussian to good approximation, which makes an all-orders description of the lensed spectra calculable [29, 30]. One would therefore expect that delensing could be treated by a similar all-orders procedure to predict the delensed power spectra (see also [25] for a related discussion), given a non-perturbative description of the method for delensing.
In this paper, we will provide an all-orders description of delensing for both temperature and polarization. We will first describe a non-perturbative approach to delensing that reproduces the unlensed CMB in the limit of no noise. This procedure is naturally generalized to account for the noise in the temperature, polarization, and lensing maps. We are careful to use filtered maps as part of the delensing procedure, which we show is necessary for improving parameters constraints. In principle, one can then produce the all-orders delensed spectra. In practice, the exact expressions are difficult to calculate due to the non-local relationship between the observed data and the true location of the underlying lenses. Fortunately, on the scales of interest, the lensing potential varies slowly compared to the CMB maps and these non-local effects can be neglected or included in a perturbative expansion. This will allow us to provide simple expressions for the delensed power spectra that we also implement numerically.
The most immediate application of these all-orders results is for forecasting future CMB experiments. We include forecasts covering a range of possible experimental configurations to illustrate the impact of delensing on $N_{\rm eff}$ and other cosmological parameters. Our goal is to understand to what degree forecasts using unlensed spectra are achievable given realistic noise levels in the lensing map. This is especially important for forecasts of $N_{\rm eff}$ for CMB Stage IV, which are tantalizingly close to the theoretical threshold of $\Delta N_{\rm eff}=0.027$ (see e.g. [32, 33, 34, 35, 36, 37] for discussion). We will also show that delensing reduces the covariance between the lensing power spectrum and the observed temperature and polarization spectra. Proper forecasting must thus account for both the delensed spectra and covariance matrix [38, 39].
This paper is organized as follows. In Section 2, we present the theoretical framework for computing the delensed CMB spectra. We apply these results in Section 3 to show the numerically computed spectra and covariance matrices. In Section 4, we use these results in forecasts for future CMB experiments. We highlight the impact of delensing by comparing forecasts with lensed, unlensed, and delensed spectra. We conclude in Section 5.
The main text is supplemented by six appendices. Appendix A explores the validity of our expansion in gradients of $\phi$. We compute the optimal filters in Appendix B for both the temperature and polarization spectra in various limits and explain the choice of filters used in the main text. Appendix C gives the expressions for efficient numerical computation of the delensed polarization spectra. In Appendix D, we show how to calculate the delensed covariance matrix. In Appendix E, we explore the effect of delensing on the Fisher information. We argue that, as long as the lensing potential is included in the likelihood, one should gain information by delensing. Appendix F explores an alternate all-orders approach to delensing that is exact in the limit of no noise.
We will use the following conventions throughout: we define Stage II, III, and IV to be 1 arcmin resolution experiments with 10, 5, and 1 $\mu$K-arcmin temperature noise respectively. The lensing noise for these experiments is determined assuming the minimum variance quadratic estimator [40], which combines information in the lensed temperature and polarization fields, including the improvement from iterative delensing with the $EB$ reconstruction [23]. We typically show power spectra in terms of ${\mathcal{D}}_{\ell}\equiv\ell(\ell+1)C_{\ell}/(2\pi)$. We will use ${\vec{\ell}\mskip 2.0mu \vphantom{\ell}}$ to label harmonics of the CMB temperature and polarization but we use ${\vec{L}\mskip 2.0mu \vphantom{L}}$ for the harmonics of the lensing potential.
2 All-Orders Delensing
The goal of this section is to present an all-orders theoretical framework for delensing, both in principle and with real data. Our discussion will focus on temperature at first, but we will also include the generalization to polarization. We will assume the flat sky limit for simplicity. Nevertheless, our primary interest is in delensing temperature and $E$-mode polarization where the modes of most interest are well approximated by the flat sky limit. Previous work aimed at reconstructing the primary (unlensed) CMB simultaneously with the lensing potential include global maximum-likelihood approaches [41, 42], local maximum-likelihood approaches [43], and Bayesian techniques [44].
Our approach to delensing removes the effect of the lensing directly from the CMB maps, rather than simply deconvolving the estimated lensing deflection power spectrum from the CMB power spectrum. The realization of the lensing deflection field (including its scatter about the mean) affects the smoothing of the acoustic peaks and is responsible for moving Fisher information from the power spectrum to higher-order statistics. It is therefore important that delensing is a map-level procedure rather than a deconvolution of the power spectra.
2.1 Non-Perturbative Delensing for Real Data
Lensing is a local process in real space. Given an unlensed temperature map $T(\vec{x}\mskip 2.0mu \vphantom{x})$ and a lensing field $\vec{\alpha}(\vec{x}\mskip 2.0mu \vphantom{x})$, the lensed temperature map is given by
$$\tilde{T}(\vec{x}\mskip 2.0mu \vphantom{x})=T(\vec{x}\mskip 2.0mu \vphantom{x}%
+\vec{\alpha}(\vec{x}\mskip 2.0mu \vphantom{x}))=\int\frac{d^{2}\ell}{(2\pi)^{%
2}}e^{i{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}\cdot\big{(}\vec{x}\mskip 2.0mu%
\vphantom{x}+\vec{\alpha}(\vec{x}\mskip 2.0mu \vphantom{x})\big{)}}T_{\vec{%
\ell}\mskip 2.0mu \vphantom{\ell}}\ .$$
(2.1)
Suppose now that we have a perfect measurement of the lensing map (i.e. $\vec{\alpha}\,{}^{\rm obs}=\vec{\alpha}$) and would like to reconstruct $T(\vec{x}\mskip 2.0mu \vphantom{x})$. We can define an exact delensing procedure as
$$\displaystyle T^{\rm d}(\vec{x}\mskip 2.0mu \vphantom{x})=\int d^{2}x^{\prime}%
J(\vec{x}\mskip 2.0mu \vphantom{x}{}^{\prime})\delta^{2}(\vec{x}\mskip 2.0mu %
\vphantom{x}-\vec{\alpha}(\vec{x}\mskip 2.0mu \vphantom{x}^{\prime})-\vec{x}%
\mskip 2.0mu \vphantom{x}^{\prime})\tilde{T}(\vec{x}\mskip 2.0mu \vphantom{x}^%
{\prime})\ ,$$
(2.2)
where $J(\vec{x}\mskip 2.0mu \vphantom{x})=\det\partial_{i}(x_{j}+\alpha_{j}(\vec{x}%
\mskip 2.0mu \vphantom{x}))$ is the Jacobian for the change of variables $\vec{y}\mskip 2.0mu \vphantom{y}=\vec{x}\mskip 2.0mu \vphantom{x}+\vec{\alpha}%
(\vec{x}\mskip 2.0mu \vphantom{x})$ and $T^{\rm d}$ is our delensed map. By a change of variables, it is clear that this procedure perfectly inverts the lensing
$$T^{\rm d}(\vec{x}\mskip 2.0mu \vphantom{x})=\int d^{2}x^{\prime}J(\vec{x}%
\mskip 2.0mu \vphantom{x}^{\prime})\delta(\vec{x}\mskip 2.0mu \vphantom{x}-%
\vec{\alpha}(\vec{x}\mskip 2.0mu \vphantom{x}^{\prime})-\vec{x}\mskip 2.0mu %
\vphantom{x}^{\prime})T(\vec{x}\mskip 2.0mu \vphantom{x}^{\prime}+\vec{\alpha}%
(\vec{x}\mskip 2.0mu \vphantom{x}^{\prime}))=\int d^{2}y^{\prime}\delta(\vec{x%
}\mskip 2.0mu \vphantom{x}-\vec{y}\mskip 2.0mu \vphantom{y}^{\prime})T(\vec{y}%
\mskip 2.0mu \vphantom{y}^{\prime})=T(\vec{x}\mskip 2.0mu \vphantom{x})\ .$$
(2.3)
We see that, in the absence of noise, lensing can be perfectly inverted.
At a practical level, this exact procedure is challenging to implement. In order to make use of the Jacobian, we need a map of the lensing deflection field, and we also require a map of its gradients at small scales which is typically more contaminated with noise (see Appendix A for more details). While a simple solution is to neglect the Jacobian in (2.3), this approximation introduces an error of ${\cal O}(\vec{\nabla}\cdot\vec{\alpha})$.
Perhaps the more obvious and convenient approach is to invert lensing locally in the map as follows,
$$T^{\rm d}(\vec{x}\mskip 2.0mu \vphantom{x})=\tilde{T}\big{(}\vec{x}\mskip 2.0%
mu \vphantom{x}-\vec{\alpha}(\vec{x}\mskip 2.0mu \vphantom{x})\big{)}=T\Big{(}%
\vec{x}\mskip 2.0mu \vphantom{x}+\vec{\alpha}\big{(}\vec{x}\mskip 2.0mu %
\vphantom{x}-\vec{\alpha}(\vec{x}\mskip 2.0mu \vphantom{x})\big{)}-\vec{\alpha%
}(\vec{x}\mskip 2.0mu \vphantom{x})\Big{)}=T\Big{(}\vec{x}\mskip 2.0mu %
\vphantom{x}+{\cal O}\big{(}(\vec{\alpha}\cdot\vec{\nabla})\vec{\alpha}\big{)}%
\Big{)}\ .$$
(2.4)
One beneficial feature of this approach is that it is very easy to implement given the maps $\tilde{T}(\vec{x}\mskip 2.0mu \vphantom{x})$ and $\phi(\vec{x}\mskip 2.0mu \vphantom{x})$. While this procedure does not delens the temperature exactly, it does so up to an error of order ${\cal O}\left((\vec{\alpha}\cdot\vec{\nabla})\vec{\alpha}\right)$. This error is acceptably small for our applications as we show explicitly in Appendix A. We could even iteratively correct this procedure by hand, $T^{\rm d}(\vec{x}\mskip 2.0mu \vphantom{x})=\tilde{T}\left(\vec{x}\mskip 2.0mu%
\vphantom{x}-\vec{\alpha}(\vec{x}\mskip 2.0mu \vphantom{x})+(\vec{\alpha}%
\cdot\vec{\nabla})\vec{\alpha}\right)$, to push the error to progressively higher orders in $\vec{\alpha}\cdot\vec{\nabla}$.
In reality, we observe neither the temperature (polarization) nor the lensing potential perfectly. Given our limited knowledge, we must provide a procedure for delensing an observed map that is well suited for the strengths and limitations of our observations. We would like any delensing procedure to have the following properties:
•
In the limit where the noise vanishes, delensing should be accurate: $T^{\rm d}(\vec{x}\mskip 2.0mu \vphantom{x})\approx T(\vec{x}\mskip 2.0mu %
\vphantom{x})$.
•
We do not add or remove power from the map. Lensing itself conserves total power and we wish maintain this property of the delensing as well.
•
We should be allowed to filter our maps to minimize the impact111One often filters maps simply to project out the noisy modes altogether. Such a procedure does not conserve total power, and we show explicitly in Section 4 that such a procedure weakens parameter constraints.
The role of filtering here is to avoid introducing additional noise in the maps from delensing itself, due to noise in $\vec{\alpha}\,{}^{\rm obs}$ and/or $T^{\rm obs}$. See Appendix B for more discussion. of noisy modes.
With real data, we typically want to filter the maps before delensing to avoid using noisy modes, both in $T$ and $\phi$. Therefore,
given an observed temperature map $T^{\rm obs}(\vec{x}\mskip 2.0mu \vphantom{x})$, and an observed lensing map $\vec{\alpha}\,{}^{\rm obs}$, our delensed map will be given by
$$T^{\rm d}(\vec{x}\mskip 2.0mu \vphantom{x})=\bar{h}\star T^{\rm obs}(\vec{x}%
\mskip 2.0mu \vphantom{x})+h\star T^{\rm obs}(\vec{x}\mskip 2.0mu \vphantom{x}%
-g\star\vec{\alpha}\,{}^{\rm obs}(\vec{x}\mskip 2.0mu \vphantom{x}))$$
(2.5)
where
$$a\star b(\vec{x}\mskip 2.0mu \vphantom{x})=\int d^{2}x\,a(\vec{x}\mskip 2.0mu %
\vphantom{x}-{\vec{x}\mskip 2.0mu \vphantom{x}}^{\prime})b(\vec{x}\mskip 2.0mu%
\vphantom{x}^{\prime})=\int\frac{d^{2}\ell}{(2\pi)^{2}}e^{i{\vec{\ell}\mskip 2%
.0mu \vphantom{\ell}}\cdot\vec{x}\mskip 2.0mu \vphantom{x}}a_{\vec{\ell}\mskip
2%
.0mu \vphantom{\ell}}\ b_{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}\ .$$
(2.6)
The functions $\bar{h}$, $h$, and $g$ are filters that we will discuss shortly. If we want to satisfy the above requirements, we can see that $\bar{h}$ must be determined by $g$ and $h$. In particular, there is a constraint imposed on the filters to ensure that we not add or remove power $\langle T^{\rm d}(0)^{2}\rangle=\langle T^{\rm obs}(0)^{2}\rangle$. We will determine the explicit expression for of $\bar{h}$ once we discuss the power spectrum of $T^{\rm d}$ for general filters222We discuss optimal and near-optimal filtering schemes in Appendix B. However, the delensing procedure should make sense independent of the precise choice of filters.. In the limit of no noise and $g,h\to 1$, we must impose a constraint $\bar{h}\to 0$ in order to reproduce the procedure in Equation (2.4). An alternative procedure that uses Equation (2.3) as the starting point is discussed in Appendix F.
A similar delensing procedure was recently discussed in [25], applied to Planck data. The approach taken there is equivalent to Equation 2.5 with $\bar{h}=0$ and $h=1$. This choice conserves total power and matches our procedure in the limit of no noise. Including $h\neq 1$ and $\bar{h}\neq 0$ is important for minimizing the noise induced in the delensed maps, as shown in Appendix B.1. Furthermore, we will show in Section 4 that filtering with $\bar{h}=0$ produces significantly weaker constraints on cosmological parameters than when we allow $\bar{h}\neq 0$ for noisy modes.
The expressions for the delensed maps are easier to work with in terms of harmonics,
$$T^{\rm d}_{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}=\bar{h}_{\vec{\ell}\mskip 2%
.0mu \vphantom{\ell}}T^{\rm obs}_{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}+\int
d%
^{2}x\frac{d^{2}\ell_{1}}{(2\pi)^{2}}h_{{\vec{\ell}\mskip 2.0mu \vphantom{\ell%
}}_{1}}T^{\rm obs}_{{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}_{1}}e^{-i{\vec{%
\ell}\mskip 2.0mu \vphantom{\ell}}\cdot\vec{x}\mskip 2.0mu \vphantom{x}}e^{i{%
\vec{\ell}\mskip 2.0mu \vphantom{\ell}}_{1}\cdot(\vec{x}\mskip 2.0mu \vphantom%
{x}-g\star\vec{\alpha}(\vec{x}\mskip 2.0mu \vphantom{x}))}\ .$$
(2.7)
From here on, we will assume isotropic noise, and therefore isotropic filters, so we will take $h_{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}=h_{\ell}$ and $\bar{h}_{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}=\bar{h}_{\ell}$.
We can formally write the delensed $C_{\ell}$ as
$$\displaystyle\langle T^{\rm d}_{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}T^{\rm d%
}_{\vec{\ell}\mskip 2.0mu \vphantom{\ell}^{\prime}}\rangle$$
$$\displaystyle=$$
$$\displaystyle(2\pi)^{2}\delta({\vec{\ell}\mskip 2.0mu \vphantom{\ell}}+\vec{%
\ell}\mskip 2.0mu \vphantom{\ell}^{\prime})|\bar{h}_{\ell}|^{2}C_{\ell}^{\rm
obs%
}+\Bigg{[}\int d^{2}x\frac{d^{2}\ell_{1}}{(2\pi)^{2}}e^{-i({\vec{\ell}\mskip 2%
.0mu \vphantom{\ell}}-{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}_{1})\cdot\vec{x%
}\mskip 2.0mu \vphantom{x}}\times$$
(2.8)
$$\displaystyle\qquad\qquad\bigg{(}\bar{h}_{\ell^{\prime}}h_{\ell_{1}}\langle e^%
{-i{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}_{1}\cdot g\star\vec{\alpha}\,{}^{%
\rm obs}(\vec{x}\mskip 2.0mu \vphantom{x})}T^{\rm obs}_{{\vec{\ell}\mskip 2.0%
mu \vphantom{\ell}}_{1}}T^{\rm obs}_{\vec{\ell}\mskip 2.0mu \vphantom{\ell}^{%
\prime}}\rangle\bigg{)}+\{{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}%
\leftrightarrow\vec{\ell}\mskip 2.0mu \vphantom{\ell}^{\prime}\}\Bigg{]}$$
$$\displaystyle+\int d^{2}x^{\prime}d^{2}x^{\prime\prime}\frac{d^{2}\ell_{1}d^{2%
}\ell_{2}}{(2\pi)^{4}}e^{-i({\vec{\ell}\mskip 2.0mu \vphantom{\ell}}-{\vec{%
\ell}\mskip 2.0mu \vphantom{\ell}}_{1})\cdot\vec{x}\mskip 2.0mu \vphantom{x}}e%
^{-i(\vec{\ell}\mskip 2.0mu \vphantom{\ell}^{\prime}-{\vec{\ell}\mskip 2.0mu %
\vphantom{\ell}}_{2})\cdot\vec{x}\mskip 2.0mu \vphantom{x}^{\prime}}h_{\ell_{1%
}}h_{\ell_{2}}\times$$
$$\displaystyle\qquad\qquad\langle e^{-i{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}%
_{1}\cdot g\star\vec{\alpha}\,{}^{\rm obs}(\vec{x}\mskip 2.0mu \vphantom{x})}e%
^{-i{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}_{2}\cdot g\star\vec{\alpha}\,{}^{%
\rm obs}(\vec{x}\mskip 2.0mu \vphantom{x}^{\prime})}T^{\rm obs}_{{\vec{\ell}%
\mskip 2.0mu \vphantom{\ell}}_{1}}T^{\rm obs}_{{\vec{\ell}\mskip 2.0mu %
\vphantom{\ell}}_{2}}\rangle\ .$$
In principle, this defines the all-orders result, if one can evaluate all of the correlation functions exactly and perform the integrals. In the next section, we will discuss several approximations that will simplify the calculations and allow for simpler analytic results that can be efficiently computed numerically. We can also systematically improve these (or any other) approximations by treating them as a perturbative expansion of this equation.
While these results hold for any choice of the $g_{L}$, $h_{\ell}$, and $\bar{h}_{\ell}$ filters, one should optimize these choices to minimize the impact of noise. We will discuss this optimization in Appendix B but we will ultimately use signal-to-noise filtering of the form
$$\displaystyle g_{L}=\frac{C_{L}^{\phi\phi}}{C_{L}^{\phi\phi,{\rm obs}}}\qquad%
\qquad h_{\ell}=\frac{\tilde{C}_{\ell}^{TT}}{C_{\ell}^{TT,{\rm obs}}}\ ,$$
(2.9)
where $\tilde{C}^{TT}_{\ell}$ is the lensed spectrum. We determine $\bar{h}_{\ell}$ in terms of $h_{\ell}$ and $g_{L}$ from the conservation of total power in Appendix B.2. We will also need to define a filter for polarization, $h^{P}_{\ell}$, which is similarly filtered in terms of $C_{\ell}^{EE}$. These choices are optimal in certain limits, but are also easy to implement and have simple interpretations.
2.2 Approximate Delensed Power Spectra
Two well-motivated approximations that will greatly simplify calculations are (1) small lensing gradients, $(\vec{\alpha}\cdot\vec{\nabla})\,\vec{\alpha}\ll\vec{\alpha}$ and (2) Gaussianity of the lensing potential. These two approximations alone allow for simple all-orders expressions for the delensed power spectra.
Let us start by dropping gradients of $\vec{\alpha}$. As described above, in the limit of no noise we can make the approximation $\tilde{T}(\vec{x}\mskip 2.0mu \vphantom{x}-g\star\vec{\alpha}\,{}^{\rm obs}(%
\vec{x}\mskip 2.0mu \vphantom{x}))\approx T(\vec{x}\mskip 2.0mu \vphantom{x}+%
\vec{\alpha}-g\star\vec{\alpha}\,{}^{\rm obs}(\vec{x}\mskip 2.0mu \vphantom{x}))$ up to errors that we can safely ignore (see Appendix A for details). However, we are still filtering the maps before delensing, which makes it more challenging to implement this approximation. Specifically, we have
$$\displaystyle h\star T^{\rm obs}(\vec{x}\mskip 2.0mu \vphantom{x}-g\star\vec{%
\alpha}\,{}^{\rm obs}(\vec{x}\mskip 2.0mu \vphantom{x}))$$
$$\displaystyle=$$
$$\displaystyle\int d^{2}x^{\prime}\frac{d^{2}\ell_{1}}{(2\pi)^{2}}h_{\ell_{1}}e%
^{i{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}_{1}\cdot(\vec{x}\mskip 2.0mu %
\vphantom{x}-\vec{x}\mskip 2.0mu \vphantom{x}^{\prime}-g\star\vec{\alpha}\,{}^%
{\rm obs}(\vec{x}\mskip 2.0mu \vphantom{x}))}T^{\rm obs}(\vec{x}\mskip 2.0mu %
\vphantom{x}^{\prime})$$
$$\displaystyle=$$
$$\displaystyle\int\frac{d^{2}\ell_{1}}{(2\pi)^{2}}h_{\ell_{1}}e^{i{\vec{\ell}%
\mskip 2.0mu \vphantom{\ell}}_{1}\cdot(\vec{x}\mskip 2.0mu \vphantom{x}-g\star%
\vec{\alpha}\,{}^{\rm obs}(\vec{x}\mskip 2.0mu \vphantom{x}))}\Big{[}T^{\rm N}%
_{{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}_{1}}+e^{-i{\vec{\ell}\mskip 2.0mu %
\vphantom{\ell}}_{1}\cdot\vec{x}\mskip 2.0mu \vphantom{x}}\int d^{2}x^{\prime}%
\frac{d^{2}\ell_{2}}{(2\pi)^{2}}e^{i{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}_{%
2}\cdot(\vec{x}\mskip 2.0mu \vphantom{x}^{\prime}+\vec{\alpha}(\vec{x}\mskip 2%
.0mu \vphantom{x}^{\prime}))}T_{{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}_{2}}%
\Big{]}\ ,$$
where we used $T^{\rm obs}=\tilde{T}+T^{\rm N}$. For a completely general filter, $h_{\ell}$, this expression can be challenging to work with. In practice, for the range of interest, the filters change very slowly. For $\ell_{1},\ell_{2}$ that dominate these integrals, we expect that $h_{\ell_{1}}\simeq h_{\ell_{2}}$, so we can perform the integral over $\ell_{1}$ (and relabel $\ell_{2}\to\ell_{1}$) to find
$$\displaystyle h\star T^{\rm obs}(\vec{x}\mskip 2.0mu \vphantom{x}-g\star\vec{%
\alpha}\,{}^{\rm obs}(\vec{x}\mskip 2.0mu \vphantom{x}))$$
$$\displaystyle\approx$$
$$\displaystyle\int\frac{d^{2}\ell_{1}}{(2\pi)^{2}}h_{\ell_{1}}\Big{[}e^{i{\vec{%
\ell}\mskip 2.0mu \vphantom{\ell}}_{1}\cdot(\vec{x}\mskip 2.0mu \vphantom{x}-g%
\star\vec{\alpha}\,{}^{\rm obs}(\vec{x}\mskip 2.0mu \vphantom{x}))}T^{\rm N}_{%
{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}_{1}}+e^{i{\vec{\ell}\mskip 2.0mu %
\vphantom{\ell}}_{1}\cdot(\vec{x}\mskip 2.0mu \vphantom{x}-g\star\vec{\alpha}%
\,{}^{\rm obs}(\vec{x}\mskip 2.0mu \vphantom{x})+\vec{\alpha}(\vec{x}\mskip 2.%
0mu \vphantom{x}))}T_{{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}_{1}}\Big{]}\ ,$$
$$\displaystyle\bar{h}\star T^{\rm obs}(\vec{x}\mskip 2.0mu \vphantom{x})$$
$$\displaystyle\approx$$
$$\displaystyle\int\frac{d^{2}\ell_{1}}{(2\pi)^{2}}\bar{h}_{\ell_{1}}\Big{[}e^{i%
{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}_{1}\cdot\vec{x}\mskip 2.0mu \vphantom%
{x}}T^{\rm N}_{{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}_{1}}+e^{i{\vec{\ell}%
\mskip 2.0mu \vphantom{\ell}}_{1}\cdot(\vec{x}\mskip 2.0mu \vphantom{x}+\vec{%
\alpha}(\vec{x}\mskip 2.0mu \vphantom{x}))}T_{{\vec{\ell}\mskip 2.0mu %
\vphantom{\ell}}_{1}}\Big{]}\ .$$
(2.11)
In position space, we are assuming that $h(\vec{x}\mskip 2.0mu \vphantom{x})$ is highly localized (i.e. approximately a delta function) compared to the scales over which $\vec{\alpha}(\vec{x}\mskip 2.0mu \vphantom{x})$ varies. In $\ell$-space, this means that $h_{\ell}\simeq h_{\ell\pm 100}$ since the lensing power spectrum peaks at large angular scales $L\lesssim 100$.
Putting this all together and transforming to harmonics gives
$$\displaystyle T^{\rm d}_{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}$$
$$\displaystyle=$$
$$\displaystyle\bar{h}_{\ell}T^{\rm N}_{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}+%
\int d^{2}x\frac{d^{2}\ell_{1}}{(2\pi)^{2}}\Bigg{[}h_{\ell_{1}}e^{-i{\vec{\ell%
}\mskip 2.0mu \vphantom{\ell}}\cdot\vec{x}\mskip 2.0mu \vphantom{x}}\Big{(}e^{%
i{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}_{1}\cdot(\vec{x}\mskip 2.0mu %
\vphantom{x}-g\star\vec{\alpha}\,{}^{\rm obs}(\vec{x}\mskip 2.0mu \vphantom{x}%
))}T^{\rm N}_{{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}_{1}}+e^{i{\vec{\ell}%
\mskip 2.0mu \vphantom{\ell}}_{1}\cdot(\vec{x}\mskip 2.0mu \vphantom{x}-g\star%
\vec{\alpha}\,{}^{\rm obs}(\vec{x}\mskip 2.0mu \vphantom{x})+\vec{\alpha}(\vec%
{x}\mskip 2.0mu \vphantom{x}))}T_{{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}_{1}%
}\Big{)}$$
(2.12)
$$\displaystyle +\bar{h}_{\ell_{1}}e^{-i{\vec%
{\ell}\mskip 2.0mu \vphantom{\ell}}\cdot\vec{x}\mskip 2.0mu \vphantom{x}}e^{i{%
\vec{\ell}\mskip 2.0mu \vphantom{\ell}}_{1}\cdot(\vec{x}\mskip 2.0mu \vphantom%
{x}+\vec{\alpha}(\vec{x}\mskip 2.0mu \vphantom{x}))}T_{{\vec{\ell}\mskip 2.0mu%
\vphantom{\ell}}_{1}}\Bigg{]}\ .$$
The noise and unlensed temperature are independent of the lensing potential, and therefore the power spectrum becomes
$$\displaystyle\left\langle T^{\rm d}_{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}T^%
{\rm d}_{\vec{\ell}\mskip 2.0mu \vphantom{\ell}^{\prime}}\right\rangle$$
$$\displaystyle=$$
$$\displaystyle(2\pi)^{2}\delta({\vec{\ell}\mskip 2.0mu \vphantom{\ell}}+\vec{%
\ell}\mskip 2.0mu \vphantom{\ell}^{\prime})|\bar{h}_{\ell}|^{2}C_{\ell}^{\rm
obs%
}+\Bigg{[}\int d^{2}x\,\bar{h}_{\ell^{\prime}}h_{-\ell^{\prime}}C^{\rm N}_{%
\ell^{\prime}}e^{-i({\vec{\ell}\mskip 2.0mu \vphantom{\ell}}+\vec{\ell}\mskip 2%
.0mu \vphantom{\ell}^{\prime})\cdot\vec{x}\mskip 2.0mu \vphantom{x}}\left%
\langle e^{-i\vec{\ell}\mskip 2.0mu \vphantom{\ell}^{\prime}\cdot g\star\vec{%
\alpha}\,{}^{\rm obs}(\vec{x}\mskip 2.0mu \vphantom{x})}\right\rangle+\{{\vec{%
\ell}\mskip 2.0mu \vphantom{\ell}}\leftrightarrow\vec{\ell}\mskip 2.0mu %
\vphantom{\ell}^{\prime}\}\Bigg{]}$$
(2.13)
$$\displaystyle+\int d^{2}xd^{2}x^{\prime}\frac{d^{2}\ell_{1}}{(2\pi)^{2}}\,\bar%
{h}_{\ell_{1}}h_{-\ell_{1}}C_{\ell_{1}}\times$$
$$\displaystyle\qquad\qquad\Bigg{[}e^{-i{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}%
\cdot\vec{x}\mskip 2.0mu \vphantom{x}-i\vec{\ell}\mskip 2.0mu \vphantom{\ell}^%
{\prime}\cdot\vec{x}\mskip 2.0mu \vphantom{x}^{\prime}+i{\vec{\ell}\mskip 2.0%
mu \vphantom{\ell}}_{1}\cdot(\vec{x}\mskip 2.0mu \vphantom{x}-\vec{x}\mskip 2.%
0mu \vphantom{x}^{\prime})}\left\langle e^{i{\vec{\ell}\mskip 2.0mu \vphantom{%
\ell}}_{1}\cdot(\vec{\alpha}(\vec{x}\mskip 2.0mu \vphantom{x})-g\star\vec{%
\alpha}\,{}^{\rm obs}(\vec{x}\mskip 2.0mu \vphantom{x}))}e^{-i{\vec{\ell}%
\mskip 2.0mu \vphantom{\ell}}_{1}\cdot\vec{\alpha}(\vec{x}\mskip 2.0mu %
\vphantom{x}^{\prime})}\right\rangle+\{{\vec{\ell}\mskip 2.0mu \vphantom{\ell}%
}\leftrightarrow\vec{\ell}\mskip 2.0mu \vphantom{\ell}^{\prime}\}\Bigg{]}$$
$$\displaystyle+\int d^{2}xd^{2}x^{\prime}\frac{d^{2}\ell_{1}}{(2\pi)^{2}}\,|h_{%
\ell_{1}}|^{2}C_{\ell_{1}}\times$$
$$\displaystyle\qquad\qquad\Bigg{[}e^{-i{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}%
\cdot\vec{x}\mskip 2.0mu \vphantom{x}-i\vec{\ell}\mskip 2.0mu \vphantom{\ell}^%
{\prime}\cdot\vec{x}\mskip 2.0mu \vphantom{x}^{\prime}+i{\vec{\ell}\mskip 2.0%
mu \vphantom{\ell}}_{1}\cdot(\vec{x}\mskip 2.0mu \vphantom{x}-\vec{x}\mskip 2.%
0mu \vphantom{x}^{\prime})}\left\langle e^{i{\vec{\ell}\mskip 2.0mu \vphantom{%
\ell}}_{1}\cdot(\vec{\alpha}(\vec{x}\mskip 2.0mu \vphantom{x})-g\star\vec{%
\alpha}\,{}^{\rm obs}(\vec{x}\mskip 2.0mu \vphantom{x}))}e^{-i{\vec{\ell}%
\mskip 2.0mu \vphantom{\ell}}_{1}\cdot(\vec{\alpha}(\vec{x}\mskip 2.0mu %
\vphantom{x}^{\prime})-g\star\vec{\alpha}\,{}^{\rm obs}(\vec{x}\mskip 2.0mu %
\vphantom{x}^{\prime}))}\right\rangle\Bigg{]}\ .$$
In principle, this reduces the problem to the correlation functions of $\vec{\alpha}$ and $\vec{\alpha}\,{}^{\rm obs}$, but because of the exponentials, it will involve an infinite series of terms.
Now we will assume333The CMB lensing field is expected to be mostly Gaussian because the matter fluctuations that lens the CMB are mainly at high redshift $z\sim 2$ and on large scales $k\sim 10^{2}$Mpc${}^{-1}$, where the perturbations are well-described by linear theory . This is supported by simulations [45, 46] and analytical calculations [47, 48, 49]. Non-Gaussianity from post-Born corrections should also be small [50]. Non-Gaussian contributions may also be included as perturbative corrections to our Gaussian approximation. that $\vec{\alpha}$ is Gaussian. We can then evaluate the correlation functions using
$$\langle e^{iy}\rangle=\int_{-\infty}^{\infty}dye^{iy}\frac{1}{\sqrt{2\pi\sigma%
^{2}}}e^{-y^{2}/2\sigma^{2}}=e^{-\langle y^{2}\rangle/2}\ ,$$
(2.14)
which holds for any Gaussian random variable $y$.
We can now reduce the problem to computing the two-point correlation functions of $\vec{\alpha}$ and $\vec{\alpha}\,{}^{\rm obs}$ in terms of $C^{\phi\phi}_{\ell}$ and $C^{\phi\phi,{\rm obs}}_{\ell}$ where $\vec{\alpha}=\vec{\nabla}\phi$. This boils down to evaluating
$$\left\langle\alpha^{X}_{i}(\vec{x}\mskip 2.0mu \vphantom{x})\alpha^{Y}_{j}(%
\vec{x}\mskip 2.0mu \vphantom{x}^{\prime})\right\rangle=\frac{1}{2}\delta_{ij}%
C^{Z}_{0}(r)-(\hat{r}^{i}\hat{r}^{j}-\frac{1}{2}\delta^{ij})C^{Z}_{2}(r)$$
(2.15)
where $X,Y=\{\,,{\rm obs}\}$ and $Z=\{\,,{\rm cross},{\rm obs}\}$ for $\{X=Y\neq{\rm obs}\,,X\neq Y,X=Y={\rm obs}\}$ respectively. With these conventions, we will define the filter to be ${\bf g}_{L}^{Z}=\{1,g_{L},|g_{L}|^{2}\}$ and lensing power $C_{L}^{\phi\phi,Z}=\{C^{\phi\phi}_{L},C_{L}^{\phi\phi},C_{L}^{\phi\phi,{\rm obs%
}}\}$. We have assumed isotropic reconstruction noise so that $g_{{\vec{L}\mskip 2.0mu \vphantom{L}}}=g_{L}$. In terms of these quantities, our correlation functions are given by
$$\displaystyle C^{Z}_{0}(r)\equiv\left\langle\vec{\alpha}(\vec{x}\mskip 2.0mu %
\vphantom{x})\cdot\vec{\alpha}(\vec{x}\mskip 2.0mu \vphantom{x}^{\prime})\right\rangle$$
$$\displaystyle=$$
$$\displaystyle\int\frac{d^{2}L}{(2\pi)^{2}}L^{2}{\bf g}^{Z}_{L}C^{\phi\phi,Z}_{%
L}e^{i{\vec{L}\mskip 2.0mu \vphantom{L}}\cdot\vec{r}\mskip 2.0mu \vphantom{r}}$$
(2.16)
$$\displaystyle=$$
$$\displaystyle\int\frac{dL}{2\pi}L^{3}{\bf g}^{Z}_{L}C^{\phi\phi,Z}_{L}J_{0}(Lr%
)\ ,$$
and
$$\displaystyle C^{Z}_{2}(r)\equiv-2(\hat{r}^{i}\hat{r}^{j}-\frac{1}{2}\delta^{%
ij})\left\langle\alpha_{i}(\vec{x}\mskip 2.0mu \vphantom{x})\alpha_{j}(\vec{x}%
\mskip 2.0mu \vphantom{x}^{\prime})\right\rangle$$
$$\displaystyle=$$
$$\displaystyle-\int\frac{d^{2}L}{(2\pi)^{2}}L^{2}\cos 2\varphi\,{\bf g}^{Z}_{L}%
C^{\phi\phi,Z}_{L}e^{i{\vec{L}\mskip 2.0mu \vphantom{L}}\cdot\vec{r}\mskip 2.0%
mu \vphantom{r}}$$
(2.17)
$$\displaystyle=$$
$$\displaystyle\int\frac{dL}{2\pi}L^{3}J_{2}(Lr){\bf g}^{Z}_{L}C^{\phi\phi,Z}_{L%
}\ ,$$
where ${\vec{L}\mskip 2.0mu \vphantom{L}}\cdot\hat{r}=L\cos\varphi$. In practice, these quantities are all very similar, and they differ only in the sense that we are filtering the observed map (but not the underlying lensing field $\vec{\alpha}(\vec{x}\mskip 2.0mu \vphantom{x})$) and that the noise in our measurement only affects the power spectrum of $\vec{\alpha}\,{}^{\rm obs}$. The notation in terms of $C_{0,2}^{Z}$ captures that there are two power spectra and one cross-correlation.
Putting this all together, we get
$$\displaystyle\left\langle T^{\rm d}_{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}T^%
{\rm d}_{\vec{\ell}\mskip 2.0mu \vphantom{\ell}^{\prime}}\right\rangle$$
$$\displaystyle=$$
$$\displaystyle(2\pi)^{2}\delta({\vec{\ell}\mskip 2.0mu \vphantom{\ell}}+\vec{%
\ell}\mskip 2.0mu \vphantom{\ell}^{\prime})\Bigg{[}|\bar{h}_{\ell}|^{2}C_{\ell%
}^{\rm obs}+(h_{\ell}\bar{h}_{-\ell}+h_{-\ell}\bar{h}_{\ell})e^{-\tfrac{1}{4}%
\ell^{2}C^{\rm obs}_{0}(0)}C_{\ell}^{\rm N}$$
$$\displaystyle+\int d^{2}r\frac{d^{2}\ell_{1}}{(2\pi)^{2}}|h_{\ell_{1}}|^{2}C_{%
\ell_{1}}^{\rm N}e^{-i({\vec{\ell}\mskip 2.0mu \vphantom{\ell}}-{\vec{\ell}%
\mskip 2.0mu \vphantom{\ell}}_{1})\cdot\vec{r}\mskip 2.0mu \vphantom{r}}e^{-%
\frac{\ell_{1}^{2}}{2}(C^{\rm obs}_{0}(0)-C^{\rm obs}_{0}(r)+\cos 2\varphi_{1}%
C^{\rm obs}_{2}(r))}$$
$$\displaystyle+\int d^{2}r\frac{d^{2}\ell_{1}}{(2\pi)^{2}}C_{\ell_{1}}e^{-i({%
\vec{\ell}\mskip 2.0mu \vphantom{\ell}}-{\vec{\ell}\mskip 2.0mu \vphantom{\ell%
}}_{1})\cdot\vec{r}\mskip 2.0mu \vphantom{r}}e^{-\frac{\ell^{2}_{1}}{2}(C_{0}(%
0)-C_{0}(r)+\cos 2\varphi_{1}C_{2}(r))}$$
$$\displaystyle\times\Bigg{(}(h_{\ell_{1}}\bar{h}_{-\ell_{1}}+h_{-\ell_{1}}\bar{%
h}_{\ell_{1}})e^{-\frac{\ell_{1}^{2}}{4}C^{\rm obs}_{0}(0)+\frac{\ell_{1}^{2}}%
{2}(C^{\rm cross}_{0}(0)-C^{\rm cross}_{0}(r))+\tfrac{\ell_{1}^{2}}{2}\cos 2%
\varphi_{1}C^{\rm cross}_{2}(r))}$$
$$\displaystyle+|h_{\ell_{1}}|^{2}e^{-\frac{\ell_{1}^{2}}{2}(C^{\rm obs}_{0}(0)-%
C^{\rm obs}_{0}(r)+\cos 2\varphi_{1}C^{\rm obs}_{2}(r))+\ell_{1}^{2}(C^{\rm
cross%
}_{0}(0)-C^{\rm cross}_{0}(r))+\ell_{1}^{2}\cos 2\varphi_{1}C^{\rm cross}_{2}(%
r)}\Bigg{)}\Bigg{]}\ .$$
where $\hat{\ell}_{1}\cdot\hat{r}=\cos\varphi_{1}$. When computing spectra in later sections, we will often consider the case where cross-correlations are used to effectively set $C^{N}_{\ell}=0$ in this expression. However, when computing the covariance matrices the noise terms cannot be removed.
From these expressions, we can also determine the $\bar{h}_{\ell}$ that conserves total power. This is derived in Appendix B.2 and give the result
$$\bar{h}_{\ell}=\sqrt{1-h_{\ell}^{2}\left(1-e^{-\frac{\ell^{2}}{2}C^{\rm obs}_{%
0}(0)}\right)}-h_{\ell}e^{-\frac{\ell^{2}}{4}C^{\rm obs}_{0}(0)}\ .$$
(2.19)
To zeroth order in $C^{\rm obs}_{0}(0)$, this expression is $\bar{h}_{\ell}\approx 1-h_{\ell}$.
2.3 Polarization
Lensing of the polarization field is quite familiar when written in terms of $Q$ and $U$,
$$\big{[}Q\pm iU\big{]}(\vec{x}\mskip 2.0mu \vphantom{x})=-\int\frac{d^{2}\ell}{%
(2\pi)^{2}}\big{[}E\pm iB\big{]}_{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}\,e^{%
\pm 2i\varphi_{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}}e^{i{\vec{\ell}\mskip 2%
.0mu \vphantom{\ell}}\cdot\vec{x}\mskip 2.0mu \vphantom{x}}\ ,$$
(2.20)
where the factor $e^{\pm 2i\varphi_{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}}$ converts between the fixed reference basis in real space and the natural basis in harmonic space. Like temperature, lensing moves the points on this map via
$$\big{[}\tilde{Q}\pm i\tilde{U}\big{]}(\vec{x}\mskip 2.0mu \vphantom{x})=\big{[%
}Q\pm iU\big{]}(\vec{x}\mskip 2.0mu \vphantom{x}+\vec{\alpha}(\vec{x}\mskip 2.%
0mu \vphantom{x}))=-\int\frac{d^{2}\ell}{(2\pi)^{2}}\big{[}E\pm iB\big{]}_{%
\vec{\ell}\mskip 2.0mu \vphantom{\ell}}\,e^{\pm 2i\varphi_{\vec{\ell}\mskip 2.%
0mu \vphantom{\ell}}}e^{i{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}\cdot(\vec{x}%
\mskip 2.0mu \vphantom{x}+\vec{\alpha}(\vec{x}\mskip 2.0mu \vphantom{x}))}\ .$$
(2.21)
The all-orders delensing procedure at this level is therefore to take the $Q$ and $U$ maps and apply the same procedure as we did for temperature:
$$\displaystyle\big{[}Q^{\rm d}\pm iU^{\rm d}\big{]}(\vec{x}\mskip 2.0mu %
\vphantom{x})=\bar{h}^{(P)}\star\big{[}Q^{\rm obs}\pm iU^{\rm obs}\big{]}(\vec%
{x}\mskip 2.0mu \vphantom{x})+h^{(P)}\star\big{[}Q^{\rm obs}\pm iU^{\rm obs}%
\big{]}(\vec{x}\mskip 2.0mu \vphantom{x}-g\star\vec{\alpha}\,{}^{\rm obs}(\vec%
{x}\mskip 2.0mu \vphantom{x}))\ .$$
(2.22)
We note that, a priori, $\bar{h}^{(P)}$ and $h^{(P)}$ are not required to be real functions. Yet, in practice, delensing is a procedure we apply to the $Q$ and $U$ maps, and if the noise is isotropic then there should be no distinction between the filtering of $Q$ and $U$ nor should they be mixed under filtering. As result, we will choose our filters to be real (see Appendix B.5 for further discussion).
Making the same approximations as we did for temperature, the delensed $E$ and $B$ are given by
$$\displaystyle(E^{\rm d}\pm iB^{\rm d})_{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}$$
$$\displaystyle=$$
$$\displaystyle\bar{h}^{(P)}_{\ell}(E^{\rm obs}\pm iB^{\rm obs})_{\vec{\ell}%
\mskip 2.0mu \vphantom{\ell}}+\int d^{2}x\frac{d^{2}\ell_{1}}{(2\pi)^{2}}h^{(P%
)}_{\ell_{1}}e^{\pm 2i(\varphi_{{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}_{1}}-%
\varphi_{\vec{\ell}\mskip 2.0mu \vphantom{\ell}})}e^{-i({\vec{\ell}\mskip 2.0%
mu \vphantom{\ell}}-{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}_{1})\cdot\vec{x}%
\mskip 2.0mu \vphantom{x}}$$
(2.23)
$$\displaystyle\times\Bigg{[}(E^{\rm N}\pm iB^{\rm N})_{{\vec{\ell}\mskip 2.0mu %
\vphantom{\ell}}_{1}}e^{-i{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}_{1}\cdot g%
\star\vec{\alpha}\,{}^{\rm obs}(\vec{x}\mskip 2.0mu \vphantom{x})}+(E\pm iB)_{%
{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}_{1}}e^{i{\vec{\ell}\mskip 2.0mu %
\vphantom{\ell}}_{1}\cdot(\vec{\alpha}(\vec{x}\mskip 2.0mu \vphantom{x})-g%
\star\vec{\alpha}\,{}^{\rm obs}(\vec{x}\mskip 2.0mu \vphantom{x}))}\Bigg{]}\ .$$
Now we can isolate $E^{\rm d}$ (or equivalently $B^{\rm d}$) as
$$\displaystyle E^{\rm d}_{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}$$
$$\displaystyle=$$
$$\displaystyle\bar{h}^{(P)}_{\ell}E^{\rm obs}_{\vec{\ell}\mskip 2.0mu \vphantom%
{\ell}}+\int d^{2}x\frac{d^{2}\ell_{1}}{(2\pi)^{2}}h^{(P)}_{\ell_{1}}e^{i({%
\vec{\ell}\mskip 2.0mu \vphantom{\ell}}_{1}-{\vec{\ell}\mskip 2.0mu \vphantom{%
\ell}})\cdot\vec{x}\mskip 2.0mu \vphantom{x}}$$
(2.24)
$$\displaystyle\times\Bigg{[}\cos(2(\varphi_{{\vec{\ell}\mskip 2.0mu \vphantom{%
\ell}}_{1}}-\varphi_{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}))\bigg{(}E^{\rm N%
}_{{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}_{1}}e^{-i{\vec{\ell}\mskip 2.0mu %
\vphantom{\ell}}_{1}\cdot g\star\vec{\alpha}\,{}^{\rm obs}(\vec{x}\mskip 2.0mu%
\vphantom{x})}+E_{{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}_{1}}e^{i{\vec{\ell%
}\mskip 2.0mu \vphantom{\ell}}_{1}\cdot(\vec{\alpha}(\vec{x}\mskip 2.0mu %
\vphantom{x})-g\star\vec{\alpha}\,{}^{\rm obs}(\vec{x}\mskip 2.0mu \vphantom{x%
}))}\bigg{)}$$
$$\displaystyle+\sin(2(\varphi_{{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}_{1}}-%
\varphi_{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}))B^{\rm N}_{{\vec{\ell}\mskip
2%
.0mu \vphantom{\ell}}_{1}}e^{-i{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}_{1}%
\cdot g\star\vec{\alpha}\,{}^{\rm obs}(\vec{x}\mskip 2.0mu \vphantom{x})}\Bigg%
{]}\ .$$
where we have assumed there are no primordial B-modes. Repeating the same steps as for temperature, we find
$$\displaystyle\langle E^{\rm d}_{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}E^{\rm d%
}_{\vec{\ell}\mskip 2.0mu \vphantom{\ell}^{\prime}}\rangle$$
$$\displaystyle=$$
$$\displaystyle(2\pi)^{2}\delta({\vec{\ell}\mskip 2.0mu \vphantom{\ell}}+\vec{%
\ell}\mskip 2.0mu \vphantom{\ell}^{\prime})\Bigg{[}\left|\bar{h}^{(P)}_{\ell}%
\right|^{2}C^{EE,{\rm obs}}_{\ell}+(h^{(P)}_{\ell}\bar{h}^{(P)}_{-\ell}+h^{(P)%
}_{-\ell}\bar{h}^{(P)}_{\ell})e^{-\tfrac{1}{4}\ell^{2}C^{\rm obs}_{0}(0)}C_{%
\ell}^{EE,\rm{N}}$$
$$\displaystyle+\int d^{2}r\frac{d^{2}\ell_{1}}{(2\pi)^{2}}e^{-i({\vec{\ell}%
\mskip 2.0mu \vphantom{\ell}}-{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}_{1})%
\cdot\vec{r}\mskip 2.0mu \vphantom{r}}e^{-\frac{\ell_{1}^{2}}{2}(C^{\rm obs}_{%
0}(0)-C^{\rm obs}_{0}(r)+\cos 2\varphi_{1}C^{\rm obs}_{2}(r))}$$
$$\displaystyle\times\left|h^{(P)}_{\ell_{1}}\right|^{2}\Big{[}\cos^{2}(2(%
\varphi_{{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}_{1}}-\varphi_{\vec{\ell}%
\mskip 2.0mu \vphantom{\ell}}))C_{\ell_{1}}^{EE,\rm{N}}+\sin^{2}(2(\varphi_{{%
\vec{\ell}\mskip 2.0mu \vphantom{\ell}}_{1}}-\varphi_{\vec{\ell}\mskip 2.0mu %
\vphantom{\ell}}))C_{\ell_{1}}^{BB,\rm{N}}\Big{]}$$
$$\displaystyle+\int d^{2}r\frac{d^{2}\ell_{1}}{(2\pi)^{2}}\cos^{2}(2(\varphi_{{%
\vec{\ell}\mskip 2.0mu \vphantom{\ell}}_{1}}-\varphi_{\vec{\ell}\mskip 2.0mu %
\vphantom{\ell}}))C_{\ell_{1}}^{EE}e^{-i({\vec{\ell}\mskip 2.0mu \vphantom{%
\ell}}-{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}_{1})\cdot\vec{r}\mskip 2.0mu %
\vphantom{r}}e^{-\frac{\ell_{1}^{2}}{2}(C_{0}(0)-C_{0}(r)+\cos 2\varphi_{1}C_{%
2}(r))}$$
$$\displaystyle\times\Bigg{(}(h^{(P)}_{\ell_{1}}\bar{h}^{(P)}_{-\ell_{1}}+h^{(P)%
}_{-\ell_{1}}\bar{h}^{(P)}_{\ell_{1}})e^{-\frac{\ell_{1}^{2}}{4}C^{\rm obs}_{0%
}(0)+\frac{\ell_{1}^{2}}{2}\left(C^{\rm cross}_{0}(0)-C^{\rm cross}_{0}(r)+%
\cos 2\varphi_{1}C^{\rm cross}_{2}(r)\right)}$$
$$\displaystyle+\left|h^{(P)}_{\ell_{1}}\right|^{2}e^{-\frac{\ell_{1}^{2}}{2}%
\left(C^{\rm obs}_{0}(0)-C^{\rm obs}_{0}(r)+\cos 2\varphi_{1}C^{\rm obs}_{2}(r%
)\right)+\ell_{1}^{2}\left(C^{\rm cross}_{0}(0)-C^{\rm cross}_{0}(r)+\cos 2%
\varphi_{1}C^{\rm cross}_{2}(r)\right)}\Bigg{)}\Bigg{]}\ ,$$
where we used $\varphi_{-{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}}=\varphi_{\vec{\ell}\mskip 2%
.0mu \vphantom{\ell}}+\pi$. Similarly we can compute the $TE$ correlation, which simplifies significantly
$$\displaystyle\left\langle T^{\rm d}_{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}E^%
{\rm d}_{\vec{\ell}\mskip 2.0mu \vphantom{\ell}^{\prime}}\right\rangle$$
$$\displaystyle=$$
$$\displaystyle(2\pi)^{2}\delta({\vec{\ell}\mskip 2.0mu \vphantom{\ell}}+\vec{%
\ell}\mskip 2.0mu \vphantom{\ell}^{\prime})\Bigg{[}\bar{h}_{\ell}\bar{h}^{(P)}%
_{-\ell}\tilde{C}^{TE}_{\ell}$$
$$\displaystyle+\int d^{2}r\frac{d^{2}\ell_{1}}{(2\pi)^{2}}\cos(2(\varphi_{{\vec%
{\ell}\mskip 2.0mu \vphantom{\ell}}_{1}}-\varphi_{\vec{\ell}\mskip 2.0mu %
\vphantom{\ell}}))C_{\ell_{1}}^{TE}e^{-i({\vec{\ell}\mskip 2.0mu \vphantom{%
\ell}}-{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}_{1})\cdot\vec{r}\mskip 2.0mu %
\vphantom{r}}e^{-\frac{\ell_{1}^{2}}{2}(C_{0}(0)-C_{0}(r)+\cos 2\varphi_{1}C_{%
2}(r))}$$
$$\displaystyle\times\Bigg{(}(h_{\ell_{1}}\bar{h}^{(P)}_{-\ell_{1}}+h^{(P)}_{-%
\ell_{1}}\bar{h}_{\ell_{1}})e^{-\frac{\ell_{1}^{2}}{4}C^{\rm obs}_{0}(0)+\frac%
{\ell_{1}^{2}}{2}\left(C^{\rm cross}_{0}(0)-C^{\rm cross}_{0}(r)+\cos 2\varphi%
_{1}C^{\rm cross}_{2}(r)\right)}$$
$$\displaystyle+h_{\ell_{1}}h^{(P)}_{-\ell_{1}}e^{-\frac{\ell_{1}^{2}}{2}\left(C%
^{\rm obs}_{0}(0)-C^{\rm obs}_{0}(r)+\cos 2\varphi_{1}C^{\rm obs}_{2}(r)\right%
)+\ell_{1}^{2}\left(C^{\rm cross}_{0}(0)-C^{\rm cross}_{0}(r)+\cos 2\varphi_{1%
}C^{\rm cross}_{2}(r)\right)}\Bigg{)}\Bigg{]}\ .$$
The cross-correlation in this case will have some unusual features because we have separate filters acting on $T$ and $E$. At the map level, this means there are modes where we delens the temperature but not the polarization ($h_{\ell}\approx 1$ and $\bar{h}^{(P)}_{\ell}\approx 1$). The temperature and polarization information is carried by the same photons yet this delensing procedure can break this physical relationship between the lensing in the two maps. Nevertheless, our filtering scheme is chosen to minimize the error-bars in our cosmological parameters and some counter-intuitive features arise as a consequence of minimizing the noise.
One can also apply the same technique to delensing the $B$ modes. Delensing of the $B$ modes is substantially simplified compared to $T$ and $E$ because the unlensed signal is often assumed to vanish. Alternate delensing schemes, such as iterative delensing [23], are likely to be effective in this context and can also be used to generate the $\phi$-map.
Summary: Together with $C^{\phi\phi}_{L}$, the delensed spectra given by Equations (2.2), (2.3) and (2.3) will form the basis for forecasting presented in Section 4. Forecasts involving the tensor-to-scalar ratio, $r$, should include $C^{{\rm d},BB}_{\ell}$ but is beyond the scope of this work. In principle, these same techniques can also be generalized to any $N$-point function or corrected perturbatively to improve accuracy.
3 Numeric Spectra and Covariances
In the previous section, we gave analytic formulas for the $TT$, $TE$, and $EE$ spectra given the unlensed spectra and the noise power spectra for the CMB and lensing maps (assuming Gaussian, isotropic noise). Evaluating these expressions must be done numerically but this is straightforward given the existing techniques used to compute the lensed spectra [4]. Furthermore, from these expressions, we can also compute the covariance matrix for the delensed spectra. In this section we will show the numeric calculation of the spectra and covariance matrix.
3.1 Spectra
The method we use to numerically compute the delensed power spectra is very similar to the procedure for computation of the flat-sky lensed spectra in CAMB [51]. Numerical results are more stable if one computes first the change to the unlensed spectrum due to the combination of lensing and delensing, and then adds the result to the unlensed spectrum. Starting from Eq. (2.2), we expand to first order in $C_{2}(r)$ and find for the change to the delensed temperature correlation function
$$\displaystyle\Delta\xi_{T}^{\rm d}(r)$$
$$\displaystyle=$$
$$\displaystyle\int\frac{d\ell}{2\pi}\ell C_{\ell}\Bigg{[}-J_{0}(\ell r)$$
$$\displaystyle+\left|\bar{h}_{\ell}\right|^{2}\exp\left[-\frac{\ell^{2}}{2}(C_{%
0}(0)-C_{0}(r))\right]\left(J_{0}(\ell r)+\frac{\ell^{2}}{2}C_{2}(r)J_{2}(\ell
r%
)\right)$$
$$\displaystyle+\left(h_{\ell}\bar{h}_{-\ell}+h_{-\ell}\bar{h}_{\ell}\right)\exp%
\left[-\frac{\ell^{2}}{2}\left((C_{0}(0)-C_{0}(r))-(C^{\rm cross}_{0}(0)-C^{%
\rm cross}_{0}(r))+\frac{1}{2}C^{\rm obs}_{0}(0)\right)\right]$$
$$\displaystyle\quad\times\left(J_{0}(\ell r)+\frac{\ell^{2}}{2}\left(C_{2}(r)-C%
^{\rm cross}_{2}(r)\right)J_{2}(\ell r)\right)$$
$$\displaystyle+\left|h_{\ell}\right|^{2}\exp\left[-\frac{\ell^{2}}{2}\left((C_{%
0}(0)-C_{0}(r))-2(C^{\rm cross}_{0}(0)-C^{\rm cross}_{0}(r))+(C^{\rm obs}_{0}(%
0)-C^{\rm obs}_{0}(r))\right)\right]$$
$$\displaystyle\quad\times\left(J_{0}(\ell r)+\frac{\ell^{2}}{2}\left(C_{2}(r)-2%
C^{\rm cross}_{2}(r)+C^{\rm obs}_{2}(r)\right)J_{2}(\ell r)\right)\Bigg{]}\ .$$
One could include higher orders in $C_{2}(r)$ for improved accuracy, however most scales in the CMB are well approximated with the first-order approach used here [52, 4]. These expressions are evaluated using $h_{\ell}=\tilde{C}_{\ell}/(\tilde{C}_{\ell}+C^{N}_{\ell})$, as described in Appendix B. These spectra also require the noise for both $\phi$ and $T$ which is shown in Figure 2.
The delensed temperature power spectrum is then given by
$$C_{\ell}^{\rm d}=C_{\ell}+2\pi\int rdrJ_{0}(\ell r)\Delta\xi_{T}^{\rm d}(r)\,.$$
(3.2)
Similar remarks apply for the polarization spectra, which are discussed in Appendix C.
The delensed spectra that result from this calculation are shown in Figure 3. Qualitatively, the delensed spectra follow our basic expectations. At low $\ell$, where the temperature and polarization are measured with high signal-to-noise, our ability to delens is limited only by the lensing reconstruction noise. At high $\ell$, when the temperature and polarization noise in the left panel of Figure 2 begins to dominate over the signal, delensing becomes difficult and delensed spectra follow the lensed spectra. Both statements are consistent with expectations from the form of the filters.
A more detailed understanding of the spectra can be gained from isolating the various contributions. At the power spectrum level, if we had a perfect lensing measurement, we would qualitatively expect a transition from mostly unlensed at large angular scales to mostly lensed spectra at small angular scales. These limits are represented by the regions of the spectra where the filters take the values $h_{\ell}=1$ and $\bar{h}_{\ell}=1$ respectively. Figure 4 shows the contributions from each filter and indeed low $\ell$ is dominated by $h_{\ell}^{2}$ and high $\ell$ is dominated by $\bar{h}_{\ell}^{2}$. However, at intermediate values the $\bar{h}_{\ell}h_{\ell}$ cross-term dominates, yet it has no simple interpretation as the lensed or unlensed spectrum. This is a direct consequence of working at the map level, but this is crucial for removing the effect of the realization of $\phi$, rather than just the lensing power spectrum. This will lead to some unexpected features in the covariance, but we will find it is necessary for improving constraints through delensing.
Our ability to delens is also affected by the noise in our lensing reconstruction. The smearing of the acoustic peaks of the lensed spectra is due mostly to the peak of the lensing power spectrum around $L\sim 40$. At high $\ell$, there is an excess of power in the lensed spectrum as compared to the unlensed spectrum due to the presence of small scale lenses. One can see, for example, in the $TT$ panel of Figure 3 that around $\ell\sim 2500$ for the Stage IV experiment, the peak smearing has mostly been removed from the delensed spectrum, but there remains an excess of power compared to the unlensed spectrum since the small scale lenses are not resolved with high significance (see the right panel of Figure 2).
3.2 Power Spectrum Covariance
In order to obtain forecasts on cosmological parameter constraints, we also account for the covariances between observed CMB and lensing power spectra, calculating the reduction in these covariances when delensing is performed.
Ignoring the non-trivial off-diagonal lens-induced covariance in a cosmological analysis will double count information encoded in the lensing field, and can overestimate the constraints on cosmological parameters that are sensitive to $C^{\phi\phi}_{L}$. We will see that delensing removes these off-diagonal covariances when the noise is small; however, since our lensing map is not perfect the off-diagonal covariances are not removed perfectly, or at all for the noisiest modes.
To compute the power spectrum covariance matrices, we will use the analytic approximation proposed by [38],
$$\displaystyle\mathrm{Cov}(C_{\ell}^{{\rm d},XY},\,C_{\ell^{\prime}}^{{\rm d},%
WZ})$$
$$\displaystyle=$$
$$\displaystyle\frac{f_{\rm sky}}{2\ell+1}\left[C_{\ell}^{{\rm d},XW}C_{\ell}^{{%
\rm d},YZ}+C_{\ell}^{{\rm d},XZ}C_{\ell}^{{\rm d},YW}\right]\delta_{\ell\,\ell%
^{\prime}}$$
$$\displaystyle+f_{\rm sky}\sum_{L}\Bigg{[}\frac{\partial C_{\ell}^{{\rm d},XY}}%
{\partial C_{L}^{\phi\phi}}{\rm Cov}_{{L}\,{L}^{\prime}}^{\phi\phi,\phi\phi}%
\frac{\partial C_{\ell^{\prime}}^{{\rm d},WZ}}{\partial C_{{L}^{\prime}}^{\phi%
\phi}}\Bigg{]}\ ,$$
where $XY,WZ\in\{TT,TE,EE\}$, $C^{{\rm d},XY}_{\ell}$ are the delensed spectra, and $f_{\rm sky}$ is the observed sky fraction. In principle, our all-orders approach can also be applied directly to the covariance matrix but is beyond the scope of this work. The advantage of this approximate form is that it can be computed from the delensed spectra and derivatives thereof.
In practice, we will typically consider the case where we compute cross correlations of subsets of data that experience different realizations of the noise. This removes the noise when computing these cross-spectra. For the covariance matrix, this amounts to considering $X=W=T$ and $Y=Z=T^{\prime}$ where $C^{TT^{\prime}}_{\ell}$ is the delensed spectrum with $C^{{\rm N},TT^{\prime}}_{\ell}=0$. In Equation (3.2), we see that the noise still enters in the diagonal term via $C^{XW=TT}_{\ell}C^{YZ=T^{\prime}T^{\prime}}_{\ell}$ but does not enter in the off-diagonal terms.
Calculating the derivative of the CMB spectra with respect to the lensing spectrum is tedious but straightforward. Essentially, it is a simple application of the result for the lensed $TT$ spectrum
$$\displaystyle\frac{\partial\tilde{C}_{\ell}^{TT}}{\partial C^{\phi\phi}_{L}}$$
$$\displaystyle=$$
$$\displaystyle\int d^{2}r\frac{d^{2}\ell_{1}}{(2\pi)^{2}}C_{\ell_{1}}e^{-i({%
\vec{\ell}\mskip 2.0mu \vphantom{\ell}}-{\vec{\ell}\mskip 2.0mu \vphantom{\ell%
}}_{1})\cdot\vec{r}\mskip 2.0mu \vphantom{r}}\frac{\partial}{\partial C^{\phi%
\phi}_{L}}\left[e^{-\frac{\ell^{2}_{1}}{2}(C_{0}(0)-C_{0}(r)+\cos 2\varphi_{1}%
C_{2}(r))}\right]$$
$$\displaystyle=$$
$$\displaystyle\int d^{2}r\frac{d^{2}\ell_{1}}{(2\pi)^{2}}C_{\ell_{1}}\frac{\ell%
^{2}_{1}{L}^{3}}{2}\bigg{(}J_{0}({L}r)-1-\cos 2\varphi_{1}J_{2}({L}r)\bigg{)}e%
^{-i({\vec{\ell}\mskip 2.0mu \vphantom{\ell}}-{\vec{\ell}\mskip 2.0mu %
\vphantom{\ell}}_{1})\cdot\vec{r}\mskip 2.0mu \vphantom{r}}e^{-\frac{\ell^{2}_%
{1}}{2}(C_{0}(0)-C_{0}(r)+\cos 2\varphi_{1}C_{2}(r))}\ .$$
The full result for the delensed $TT$ spectrum is shown in Appendix D and is trivially generalized to the polarization spectra. As with the power spectra, we expand to first order in $C_{2}(r)$ for the numerical computation of the covariance matrices.
The covariance is most easily visualized through the dimensionless correlation matrix shown in Figure 5, defined as
$$\mathrm{Corr}\left(C_{\ell}^{XY},\,C_{\ell^{\prime}}^{WZ}\right)\equiv\frac{%
\mathrm{Cov}(C_{\ell}^{XY},\,C_{\ell^{\prime}}^{WZ})}{\sqrt{\mathrm{Cov}(C_{%
\ell}^{XY},\,C_{\ell}^{XY})\,\mathrm{Cov}(C_{\ell^{\prime}}^{WZ},\,C_{\ell^{%
\prime}}^{WZ})}}\,,$$
(3.5)
where the contribution from the CMB noise is included in the covariance. For the lensed spectra, we see the characteristic checkerboard pattern in the $TT$, $TE$, and $EE$ spectra from the lens-induced mode coupling [38]. We also see that the correlation between lensing power spectra and the CMB power spectra is dominated by the low-$L$ lensing modes over a wide range of scales in the CMB [39]. In contrast, the delensed spectra show essentially none of these features at low $\ell$ where we expect the delensing to be effective. This is precisely what we would expect by removing the effect of lensing. However, at moderate values of $\ell$ we see that delensing has produced large off-diagonal correlations that have no analog in the lensed spectra.
The large off-diagonal terms are a result of the $h_{\ell}\bar{h}_{\ell}$ term of Eq. 2.12 shown in Figure 4. Effects of lensing that remain in the $\bar{h}$-filtered maps lead to a broadband lensing contribution in the spectra when cross correlated with the $h$-filtered maps. This in turn causes off-diagonal covariance between modes which contain a significant contribution from the $h_{\ell}\bar{h}_{\ell}$ term. This explains why these off-diagonal correlations are confined to intermediate value of $\ell$, as it is due to the transition from $h_{\ell}$ to $\bar{h}_{\ell}$ where both are important. Furthermore, these regions would not appear if we were simply to set $\bar{h}_{\ell}=0$ throughout. However, our goal is to get the best possible measurement of cosmological parameters, not minimize the off-diagonal correlations. We will see in Section 4 that these regions do not negatively impact the forecasts and, in fact, attempts to remove the off-diagonal correlations by taking $\bar{h}_{\ell}=0$ significantly weaken the constraints. As a result, while these off-diagonal features may seem undesirable, they are an inevitable consequence of choices that optimize delensing both for map making and for cosmological parameters.
4 Delensing and Cosmological Parameters
The primary applications of our all-orders delensed spectra are to forecasting and data analysis. A real experiment will produce temperature, polarization, and lensing maps with the goal of measuring cosmological parameters. We have seen that delensing does sharpen acoustic peaks and removes some of the lensing-induced non-Gaussian covariance. Of course, the practical value of delensing is seen in the error bars of cosmological parameters. In this section, we will explore forecasts using delensed spectra for a variety of cosmological parameters. We will see that forecasts always improve with delensed spectra, providing a more unambiguous reason to delens the CMB maps (see also Appendix E for discussion).
4.1 Forecasting Methodology
All unlensed, lensed, and deflection power spectra used in forecasts are computed using CAMB [51]. Delensed spectra are computed from the CAMB output by using Eqs. (3.1), (3.2), and (C-C.4).
For the noise in the CMB survey, we assume Gaussian noise spectra of the form
$$C_{\ell}^{TT,\rm{N}}=\Delta_{T}^{2}\exp\left(\ell(\ell+1)\frac{\theta_{\mathrm%
{FWHM}}^{2}}{8\log 2}\right)\,,$$
(4.1)
where $\Delta_{T}$ is the instrumental noise in $\mu$K-radians and $\theta_{\mathrm{FWHM}}$ is the beamsize in radians. We assume fully polarized detectors, such that the polarization noise spectra are $C_{\ell}^{EE,\rm{N}}=C_{\ell}^{BB,\rm{N}}=2C_{\ell}^{TT,\rm{N}}$.
For the noise in the lensing reconstruction, we assume that the given CMB survey is used to obtain a lensing map with standard quadratic estimator techniques [40]. We note, however, that maps obtained from tracers of large-scale structure [23], such as the emission from the cosmic far-infrared background [24], can yield higher-fidelity maps of lensing than those obtained internally from the CMB for some upcoming experiments. We use the minimum variance quadratic estimator, which combines information in the lensed temperature and polarization fields. For the $EB$ estimator, which dominates the lensing information in a high-sensitivity experiment, our calculation of the lensing reconstruction noise includes the improvement from iterative delensing [23]. We use this iterative technique only for the purpose of minimizing lensing reconstruction noise, and we use our all-orders method when computing delensed spectra.
Although we model the lens reconstruction noise as coming from the given CMB survey, we are ignoring terms that would arise in both the spectra and covariances if the lensing field were obtained from quadratic combinations of the same $T$ and $E$ modes that are being delensed. Including these terms will generically lead to biases on the delensed power spectra. These biases can be avoided using independent maps of the lensing field such as those from large scale structure [23, 24] or by using CMB modes in the lens reconstruction that are disjoint from those being delensed. This latter technique is analogous to a method for avoiding the bias on measured lensing power auto-spectra originating from the disconnected CMB four-point function [53, 15]. Avoiding these biases also has the benefit of avoiding off-diagonal lens-lens covariance [54] as well as obviating the need for additional terms in the temperature- and polarization-lens power cross-covariance [39] in Eq. (3.2).
In order to forecast constraints on a set of cosmological parameters ${\lambda_{i}}$, we compute Fisher matrices using
$$F_{ij}=\sum_{\ell,\ell^{\prime}}\sum_{WX,YZ}{\partial C_{\ell}^{XY}\over%
\partial\lambda_{i}}\mathrm{Cov}^{-1}(C_{\ell}^{XY},C_{\ell^{\prime}}^{WZ}){%
\partial C_{\ell^{\prime}}^{WZ}\over\partial\lambda_{j}}.$$
(4.2)
In this sum over power spectra, as well as when computing reconstructed lensing maps, we take $\ell_{\mathrm{min}}=30$ and $\ell_{\mathrm{max}}=5000$, except for $TT$ spectra, for which we use $\ell_{\mathrm{max}}^{TT}=3000$, due to the presence of foregrounds in the CMB temperature, such as radio and emission high-redshift galaxies and the thermal and kinetic Sunyaev-Zel’dovich effects. We take the sky fraction to be $f_{\mathrm{sky}}=0.7$, and we assume a 1 arcminute beam.
We have simplified the calculation of the Fisher matrix in Equation (4.2) by ignoring derivatives of the inverse covariance. This approximation is typically very reliable, but turns out to create complications with $C^{{\rm d},TE}_{\ell}$. Specifically, when $h_{\ell}\neq h^{(P)}_{\ell}$ delensing changes the RMS power in $C^{TE}_{\ell}$ by an amount that depends on $C_{0}(0)$, unlike $C^{TT}_{\ell}$ and $C^{EE}_{\ell}$ which conserve total power. This can be seen as excess power at high-$\ell$ in Figure 3. In forecasts, this excess power should cancel between signal and the noise but this cancellation is missed in the Fisher matrix when dropping derivatives of the covariance. To avoid this technical challenge we will use $h^{T}_{\ell}=h^{(P)}_{\ell}$ in $C^{{\rm d},TE}_{\ell}$. This choice has the additional advantage that the $T$ and $E$ fluctuations are shifted by the same amount in the delensing process, preserving the $TE$ correlation. In principle, one might imagine improvements in parameter constraints by using $h^{T}_{\ell}$ instead but we will see that at Stage IV noise levels there is little room for improvement in most parameters.
Fiducial cosmological parameters and step sizes for numerical derivatives are listed in Table 1. We use $TT$, $TE$, $EE$, and $dd$ power spectra in all forecasts ($d\equiv\ell\phi$). For the covariance matrix $\mathrm{Cov}(C_{\ell}^{XY},C_{\ell^{\prime}}^{WZ})$ for lensed and delensed CMB spectra, we include non-Gaussian covariance term given in Eq. (3.2).
Note that we have not included external data such as BAO information or a prior on the optical depth $\tau$ in order to highlight the various aspects of delensing. On the other hand, this choice results in forecasts which are weaker than other published results, especially for $\sum m_{\nu}$.
4.2 Implications
The behavior of cosmological parameters in our forecasts ultimately splits into two categories: parameters that are constrained by the primary CMB (e.g. $\theta_{s}$, $N_{\rm eff}$, and $Y_{p}$) and those that benefit from lensing information (e.g. $\tau$, $A_{s}$, and $\sum m_{\nu}$). These extreme cases will both be important in highlighting how delensing affects information and the role played by the non-Gaussian covariances. We find that delensing always increases the Fisher information. We expand on some of these points in Appendix E.
The qualitative effects of delensing are seen most easily in Figure 6 which shows forecasted errors on an 8-parameter $\Lambda$CDM+$\sum m_{\nu}$+$N_{\rm eff}$ model, where $Y_{p}$ is fixed to be consistent with the predictions of standard big bang nucleosynthesis (BBN) for the given values of $\Omega_{b}h^{2}$ and $N_{\rm eff}$. Constraints on all the parameters are seen to improve at least marginally with delensing, in many cases coming close to the unlensed constraints in the limit of no instrumental noise. This confirms for this model that delensing always increases Fisher information. However, we also see that the improvements for a given parameter depend sensitively on the detailed effect it has on the power spectrum. One can typically tell from the unlensed and lensed constraints which parameters ought to improve with delensing, although $\Omega_{c}h^{2}$ shows that this is by no means guaranteed.
Delensing most clearly improves the measurement of parameters that affect the acoustic oscillations, as was anticipated in Sec. 1. This is seen explicitly in Figure 7, which shows the forecasts for $\theta_{s}$, $N_{\rm eff}$, and $Y_{p}$ for the 9-parameter model where $Y_{p}$ is free. The angular scale of the acoustic horizon, $\theta_{s}$, directly determines the peak locations and benefits most from delensing, due to peak sharpening. This is clearly seen in the forecasts, as the measurement of $\theta_{s}$ smoothly interpolates between the lensed and unlensed forecasts as we lower the noise of the experiment (and therefore the lensing reconstruction noise as seen in Figure 2). In this case, delensing is literally playing the same role as BAO-reconstruction in sharpening the BAO peak of the correlation function and reducing the error in $\theta_{s}$, as shown in Figure 1.
Similar behavior is seen for other cosmological parameters that affect the acoustic peaks, but the effect is most significant when we isolate the effect on the acoustic oscillations. A parameter of particular interest is $N_{\rm eff}$ which affects the peak locations [26, 27, 28] but also alters the damping scale [26, 55]. The effect on the damping tail is degenerate with $Y_{p}$ and by marginalizing over $Y_{p}$ we can isolate the phase shift. In particular we see that the error on $N_{\rm eff}$ in the 9-parameter $\Lambda$CDM+$\sum m_{\nu}$+$N_{\rm eff}$+$Y_{p}$ model for a Stage IV experiment improves from $\sigma(N_{\rm eff})=0.085$ with lensed spectra to $\sigma(N_{\rm eff})=0.067$ with delensed spectra, an improvement of 21%. Figure 7 also shows the constraints on $N_{\rm eff}$ and $Y_{p}$, showing the same improvement for these parameters that we observed with $\theta_{s}$. We also see that the residual non-Gaussian covariances induced by our filtering scheme shown in Figure 5 have no meaningful impact on our constraints on these parameters. Specifically, the forecasts with Gaussian covariances give essentially the same results indicating a negligible effect form the off-diagonal terms. Non-Gaussian covariances will be important for parameters that are sensitive to the lensing power spectrum, as we discuss below and in Appendix E.
We can further isolate the effect of delensing on the phase shift by examining the contours in the $N_{\rm eff}$-$Y_{p}$ plane. The phase shift induced by $N_{\rm eff}$ breaks the degeneracy between $N_{\rm eff}$ and $Y_{p}$ and therefore we expect delensing to have a larger effect along the line of degeneracy. Figure 8 shows that this is precisely what happens in our forecasts. We also see that prediction of BBN consistency, in which $Y_{p}$ is determined in terms of $N_{\rm eff}$, assuming otherwise standard BBN, is not aligned with to the degenerate direction. As a consequence, forecasts for $N_{\rm eff}$ that assume BBN consistency show only a marginal improvement in going from lensed to delensed spectra (see Figure 6). A larger difference between lensed and unlensed forecasts for $N_{\rm eff}$ had been noticed previously [28] although in that case it was likely due to information in the damping tail of the unlensed spectra whereas the delensed spectra discussed here are much closer to the lensed spectra at small angular scales444It is also worth noting that unlike in Ref. [28] we have set $\ell_{\rm max}^{TT}=3000$ which removes all of the high-$\ell$ information in temperature..
For parameters that are directly influenced by $C^{\phi\phi}_{L}$, the benefit of delensing is less clear. Delensing removes the information about the lensing potential and one could worry that delensing could weaken constraints. The intuitive reason that delensing does not weaken constraints is that $\phi$ is a Gaussian field and therefore all cosmological information is encoded in $C^{\phi\phi}_{\ell}$, which we also include in our likelihood. Any information that we are removing from the CMB spectra is then being included through the lensing power spectrum. We see this explicitly in the second row of Figure 6, where the constraints on $\tau$, $A_{s}$, and $\sum m_{\nu}$ are all seen to improve with delensing and even saturate the unlensed forecasts. However, in this case we see that it is always important to include non-Gaussian covariances. Failing to include lensing-induced non-Gaussian covariances gives overly optimistic constraints and it would appear that delensing actually weakens parameter constraints. The unlensed spectra do not contain any non-Gaussian covariances and it is therefore noteworthy that the delensed constraints reproduce the unlensed result at low noise. This shows that the residual off-diagonal covariances in Figure 5 have no meaningful impact on these measurements.
Our forecasts clearly demonstrate the benefits of delensing. However, one might question whether the full filtering scheme introduced in Equation (2.5) is necessary to achieve similar results. Of particular concern is the need for both the $\bar{h}$ and $h$ terms in our map. In Figure 5, we saw that the interplay between these two terms added significant off-diagonal correlations that are not present in the unlensed or lensed covariances. One could eliminate much of this by setting $\bar{h}_{\ell}=0$ but leaving $h_{\ell}$ unchanged (this does not conserve total power because $h_{\ell}\neq 1$). The downside of this procedure is that we are suppressing both the signal and the noise and one may ultimately be losing information. Forecasts shown in Figure 9 demostrate that this is indeed what occurs and setting $\bar{h}=0$ significantly reduces the sensitivity of the experiment. In this sense, we see that our filtering scheme is important for achieving the benefits of delensing.
5 Conclusion
In this paper, we have shown that future CMB experiments will be sufficiently sensitive to CMB lensing that the delensing of all of the spectra (and not just $B$ modes) can meaningfully improve the constraints on cosmological parameters. Delensing sharpens in the acoustic peaks, improving the measurement of peak locations and any cosmological parameters that affect the acoustic structure. Delensing also removes the lens-induced covariances for the modes measured with high significance.
We have shown how to compute the predictions for the delensed spectra and covariances to all orders in the lensing potential. We used these results to model the impact of delensing on cosmological parameters of interest. The most notable improvements occurred for parameters sensitive to peak locations and associated parameters that would otherwise be degenerate. In $\Lambda$CDM, the most dramatic improvements occurred for $\theta_{s}$ which is directly a measurement of the peak locations. When $Y_{p}$ and $N_{\rm eff}$ are both free, the phase shift due to $N_{\rm eff}$ breaks the degeneracy between these two parameters and delensing is seen to substantially improve error bars, showing an improvement of roughly 20% for both parameters with a Stage IV experiment when compared to forecasts with lensed spectra. More generally, we show that when the residual lens-induced covariances are included, Fisher information always increases when using delensed, rather than lensed, spectra.
Looking forward, delensed spectra will ultimately be necessary not just for forecasting but also for any likelihood analysis with delensed data. However, unlike lensed or unlensed spectra, the theoretical predictions depend also on the experimental noise. The analysis presented here computes these spectra for more idealized experiments. In principle, the approach taken here will generalize to any experiment, but real data may violate some of the technical assumptions needed to simplify our analytic predictions. More optimistically, we did not fully solve the problem of how to optimize our filters to maximize the Fisher information and one might imagine even more information may yet be available. As delensing of the CMB becomes more commonplace, these and other extensions of this work will deserve further exploration.
Acknowledgements
We thank Daniel Baumann, Raphael Flauger, Marcel Schmittfull, Neelima Sehgal, Blake Sherwin, and Kendrick Smith for helpful discussions. D.G. and J.M. also thank Daniel Baumann and Benjamin Wallisch for collaboration on related work that inspired this project. D.G. was supported by an NSERC Discovery Grant and the Canadian Institute for Advanced Research. J.M. was supported by the Vincent and Beatrice Tremaine Fellowship.
Appendix A Gradient Expansion
In Section 2, we presented two procedures for delensing. In the limit of a noiseless measurement, these two procedures are
$$T^{\rm d}_{1}{}^{,(J)}(\vec{x}\mskip 2.0mu \vphantom{x})=\int d^{2}x^{\prime}J%
(\vec{x}\mskip 2.0mu \vphantom{x}^{\prime})\delta(\vec{x}\mskip 2.0mu %
\vphantom{x}-\vec{x}\mskip 2.0mu \vphantom{x}^{\prime}-\vec{\alpha}(\vec{x}%
\mskip 2.0mu \vphantom{x}^{\prime}))\tilde{T}(\vec{x}\mskip 2.0mu \vphantom{x}%
^{\prime})\qquad{\rm and}\qquad T^{\rm d}_{2}(\vec{x}\mskip 2.0mu \vphantom{x}%
)=\tilde{T}(\vec{x}\mskip 2.0mu \vphantom{x}-\vec{\alpha}(\vec{x}\mskip 2.0mu %
\vphantom{x}))\ ,$$
(A.1)
where $J(\vec{x}\mskip 2.0mu \vphantom{x})=\det\partial_{i}(x_{j}+\alpha_{j}(\vec{x}%
\mskip 2.0mu \vphantom{x}))$. The advantage of working with $T^{\rm d}_{1}{}^{,(J)}$ is that it reproduces the unlensed map, while $T^{\rm d}_{2}$ disagrees with the unlensed map due to gradients. In practice, $T^{\rm d}_{1}{}^{,(J)}$ is a difficult procedure to implement, in part because we must compute the determinant $J(\vec{x}\mskip 2.0mu \vphantom{x})$ for the observed map $\vec{\alpha}\,{}^{\rm obs}(\vec{x}\mskip 2.0mu \vphantom{x})$ (see Appendix F for details). Yet, we are dropping “gradients” so one might imagine that we can approximate $T^{\rm d}_{1}{}^{(J)}$ by taking $J(\vec{x}\mskip 2.0mu \vphantom{x})\to 1$. In this appendix, we will show that error made by dropping gradients in $T^{\rm d}_{2}$ is acceptably small while $T^{\rm d}_{1}{}^{,(J=1)}$ is not sufficiently accurate.
First let us estimate the error made by setting $J(\vec{x}\mskip 2.0mu \vphantom{x})=1$ by defining
$$T^{\rm d}_{1}(\vec{x}\mskip 2.0mu \vphantom{x})=\int d^{2}x^{\prime}\delta(%
\vec{x}\mskip 2.0mu \vphantom{x}-\vec{x}\mskip 2.0mu \vphantom{x}^{\prime}-%
\vec{\alpha}(\vec{x}\mskip 2.0mu \vphantom{x}^{\prime}))\tilde{T}(\vec{x}%
\mskip 2.0mu \vphantom{x}^{\prime})\ ,$$
(A.2)
which is given in terms of harmonics by
$$T^{\rm d}_{{\vec{\ell}\mskip 2.0mu \vphantom{\ell}},1}=\int d^{2}x^{\prime}%
\frac{d^{2}\ell_{1}}{(2\pi)^{2}}e^{-i({\vec{\ell}\mskip 2.0mu \vphantom{\ell}}%
-{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}_{1})\cdot(\vec{x}\mskip 2.0mu %
\vphantom{x}^{\prime}+\vec{\alpha}(\vec{x}\mskip 2.0mu \vphantom{x}^{\prime}))%
}T_{{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}_{1}}\ .$$
(A.3)
The residual lensing in the power spectrum due to the gradients (i.e the error in setting $J\to 1$) is given by
$$\Delta C_{1,\ell}^{\rm d}=\int d^{2}r\frac{d^{2}\ell^{\prime}}{(2\pi)^{2}}C_{|%
{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}-\vec{\ell}\mskip 2.0mu \vphantom{\ell%
}^{\prime}|}e^{-i\vec{\ell}\mskip 2.0mu \vphantom{\ell}^{\prime}\cdot\vec{r}}%
\Big{[}e^{-\frac{\ell^{\prime}{}^{2}}{2}(C_{0}(0)-C_{0}(r)+\cos 2\varphi^{%
\prime}C_{2}(r))}-1\Big{]}\ ,$$
(A.4)
where we made the change of variables $\vec{\ell}\mskip 2.0mu \vphantom{\ell}^{\prime}={\vec{\ell}\mskip 2.0mu %
\vphantom{\ell}}-{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}_{1}$ and defined $\hat{\ell}\mskip 2.0mu \vphantom{\ell}^{\prime}\cdot\hat{r}\equiv\cos\varphi^{\prime}$ and $\Delta C_{\ell}^{\rm d}\equiv C_{\ell}^{\rm d}-C_{\ell}$. To estimate the size of the error, we will Taylor expand in $\sigma(r)=C_{0}(0)-C_{0}(r)$ and drop $C_{2}(r)\ll C_{0}(r)$ to get
$$\displaystyle\Delta C_{1,\ell}^{\rm d}$$
$$\displaystyle\approx$$
$$\displaystyle\int d^{2}r\frac{d^{2}\ell^{\prime}}{(2\pi)^{2}}C_{|{\vec{\ell}%
\mskip 2.0mu \vphantom{\ell}}-\vec{\ell}\mskip 2.0mu \vphantom{\ell}^{\prime}|%
}e^{-i\vec{\ell}\mskip 2.0mu \vphantom{\ell}^{\prime}\cdot\vec{r}}\frac{\ell^{%
\prime}{}^{2}}{2}\big{[}C_{0}(r)-C_{0}(0)\big{]}$$
(A.5)
$$\displaystyle\approx$$
$$\displaystyle\int\frac{d^{2}L}{(2\pi)^{2}}C_{|{\vec{\ell}\mskip 2.0mu %
\vphantom{\ell}}-{\vec{L}\mskip 2.0mu \vphantom{L}}|}\frac{L{}^{4}}{2}C_{L}^{%
\phi\phi}$$
We should compare this to the perturbative correction from lensing
$$\displaystyle\Delta\tilde{C}_{\ell}$$
$$\displaystyle\approx$$
$$\displaystyle\int\frac{d^{2}L}{(2\pi)^{2}}\,\frac{L{}^{2}}{2}C_{L}^{\phi\phi}%
\Big{[}\left|{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}-{\vec{L}\mskip 2.0mu %
\vphantom{L}}\right|^{2}C_{|{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}-{\vec{L}%
\mskip 2.0mu \vphantom{L}}|}-\ell^{2}C_{\ell}\Big{]}\ .$$
(A.6)
If the integrals in Eqs. (A.5) and (A.6) are both dominated in by $L\ll\ell$, then the gradient term, Eq. (A.5), would be suppressed by $(L/\ell)^{2}$. While this is true for the effect of lensing in Eq. (A.6), $L^{6}C^{\phi\phi}_{L}$ grows with increasing $L$ and the gradient term is actually dominated by $L\sim\ell$ and therefore is unsuppressed. We see that $J(\vec{x}\mskip 2.0mu \vphantom{x})$ is not a small correction but is crucial555In practice, we also filter $\vec{\alpha}\,{}^{\rm obs}$ such that we are often not integrating up to $L\sim\ell$ for $\ell>1000$. Nevertheless, the integral is still dominated by the largest $L$ allowed by the filter. In this sense, the intuition that the gradients are controlled by the peak of $C^{\phi\phi}_{L}$ around $L\sim 40$ is not correct and the suppression only from the filter is not sufficiently accurate. for avoiding this large error.
Now let us compare to the error made by dropping gradients in $T^{\rm d}_{2}$. To first order in gradients we have
$$T^{\rm d}_{{\vec{\ell}\mskip 2.0mu \vphantom{\ell}},2}=\int d^{2}x^{\prime}%
\frac{d^{2}\ell_{1}}{(2\pi)^{2}}e^{-i({\vec{\ell}\mskip 2.0mu \vphantom{\ell}}%
-{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}_{1})\cdot\vec{x}\mskip 2.0mu %
\vphantom{x}^{\prime}}e^{i\ell_{1}^{j}\nabla^{i}\alpha_{j}(\vec{x}\mskip 2.0mu%
\vphantom{x}^{\prime})\alpha_{i}(\vec{x}\mskip 2.0mu \vphantom{x}^{\prime})}T%
_{{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}_{1}}\ .$$
(A.7)
It is easy to check that translation and rotation invariance requires $\langle\nabla^{i}\alpha_{j}(\vec{x}\mskip 2.0mu \vphantom{x}^{\prime})\alpha_{%
i}(\vec{x}\mskip 2.0mu \vphantom{x}^{\prime})\rangle=0$. Therefore the leading correction is ${\cal O}(\alpha^{4})$ and is given by
$$\displaystyle\Delta C^{\rm d}_{2,\ell}$$
$$\displaystyle\approx$$
$$\displaystyle\int\frac{d^{2}\ell^{\prime}d^{2}r}{(2\pi)^{2}}\left|{\vec{\ell}%
\mskip 2.0mu \vphantom{\ell}}-\vec{\ell}\mskip 2.0mu \vphantom{\ell}^{\prime}%
\right|^{2}C_{|{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}-\vec{\ell}\mskip 2.0mu%
\vphantom{\ell}^{\prime}|}e^{i\vec{\ell}\mskip 2.0mu \vphantom{\ell}^{\prime}%
\cdot\vec{r}}\frac{1}{4}\Big{[}C^{\nabla^{2}}_{0}(r)C_{0}(r)-C^{\nabla^{2}}_{0%
}(0)C_{0}(0)\Big{]}\ ,$$
(A.8)
where we have dropped $C_{2}(r)$ and similar terms and defined
$$C^{\nabla^{2}}_{0}(r)=\int\frac{d^{2}L}{(2\pi)^{2}}L^{4}C_{L}^{\phi\phi}e^{i{%
\vec{L}\mskip 2.0mu \vphantom{L}}\cdot\vec{r}}\ .$$
(A.9)
We can integrate over $r$ and one of the momenta to find
$$\displaystyle\Delta C^{\rm d}_{2,\ell}$$
$$\displaystyle\approx$$
$$\displaystyle\int\frac{d^{2}L_{1}d^{2}L_{2}}{(2\pi)^{4}}\frac{L_{2}^{2}}{2}C_{%
L_{2}}^{\phi\phi}\,\frac{L_{1}^{4}}{2}C_{L_{1}}^{\phi\phi}\,\Bigg{[}\left|{%
\vec{\ell}\mskip 2.0mu \vphantom{\ell}}-{\vec{L}\mskip 2.0mu \vphantom{L}}_{1}%
-{\vec{L}\mskip 2.0mu \vphantom{L}}_{2}\right|^{2}C_{|{\vec{\ell}\mskip 2.0mu %
\vphantom{\ell}}-{\vec{L}\mskip 2.0mu \vphantom{L}}_{1}-{\vec{L}\mskip 2.0mu %
\vphantom{L}}_{2}|}-\ell^{2}C_{\ell}\Big{]}\ ,$$
(A.10)
Like $T^{\rm d}_{1}$, we see that the contribution from the derivative causes the $L_{1}$ integral to be peaked at $L_{1}\sim\ell$ and is therefore unsuppressed. However, $T^{\rm d}_{2}$ is additionally suppressed by $C_{0}(0)\simeq 10^{-7}$ which keeps the effect from the gradients small compared to the effect of lensing. Equation (A.10) can be integrated numerically, as shown in Figure 10. We see that gradients are suppressed by two orders of magnitude relative to the perturbative effect of lensing.
Appendix B Filters
In this appendix, we will explain how we choose the filters $h$ , $\bar{h}$ and $g$, which select for, respectively, the CMB modes we delens, the CMB modes we do not delens, and the $\phi$ modes we include when delensing. We will first motivate the need for filtering in our procedure for delensing. We will then determine $\bar{h}$ in terms of $h$ and $g$ by demanding that the total power is conserved by delensing. Finally, we discuss how to optimize the choice of $h$ and $g$ to minimize the variance induced in the maps from lensing / delensing and the generalization to the polarization maps.
B.1 Noise and Filtering
When delensing a temperature or polarization map, it is intuitively clear that we should filter the lensing map used for delensing. After all, we are trying to remove the effect of the physical lens, not introduce more noise into the maps. On the other hand, filtering the temperature and polarization maps may not be as obvious. We often take cross-correlations between different subsets of the data in order to cancel the noise and avoid noise-bias in the resulting spectra. In this sense, we can remove the noise with filtering.
When it comes to constraining cosmological parameters, the noise will always enter through the covariance matrix. For example, given a perfect measurement of $\phi$, delensing an unfiltered temperature map ($h=1$, $\bar{h}=0$) gives us the map
$$T^{\rm d}(\vec{x}\mskip 2.0mu \vphantom{x})=T(\vec{x}\mskip 2.0mu \vphantom{x}%
)+T^{\rm N}(\vec{x}\mskip 2.0mu \vphantom{x}-\vec{\alpha}(\vec{x}\mskip 2.0mu %
\vphantom{x}))\ .$$
(B.1)
While we have removed the lensing from the signal, we have lensed the noise in the process. The delensed noise power spectrum becomes
$$C^{d,N}_{\ell}=\int d^{2}r\frac{d^{2}\ell_{1}}{(2\pi)^{2}}C_{\ell_{1}}^{\rm N}%
e^{-i({\vec{\ell}\mskip 2.0mu \vphantom{\ell}}-{\vec{\ell}\mskip 2.0mu %
\vphantom{\ell}}_{1})\cdot\vec{r}\mskip 2.0mu \vphantom{r}}e^{-\frac{\ell_{1}^%
{2}}{2}(C^{\rm obs}_{0}(0)-C^{\rm obs}_{0}(r)+\cos 2\varphi_{1}C^{\rm obs}_{2}%
(r))}\ .$$
(B.2)
Lensing therefore moves the power in $C_{\ell}^{\rm N}$ to from one scale to another. Most importantly, it will move the noise power from regions with high noise to those with low noise. As a result, if we do not filter before delensing, we will be allowing the noisiest modes to corrupt the cleaner modes by moving noise power around. This effect is illustrated in Figure 11, where we have shown the noise curves before and after delensing with and without filtering. We have added a region with large non-white noise to show how this noise corrupts modes that were measured with high signal-to-noise before delensing.
The goal of filtering is therefore to isolate the noisiest modes from the cleanest modes. There are many ways one could imagine doing this, depending on the goals. Regardless of motivations, one is always left to find an optimal filtering procedure. Defining a (near)-optimal choice of filters will become the focus of the rest of this section. Our goal is not to throw away information but to combine the available maps to produce the best possible measurements of cosmological parameters.
B.2 Conservation of Total Power
We want our delensing procedure to conserve total power to mimic the properties of lensing. Furthermore, conserving power ensures we are not throwing any information away but simply moving it around. This constraint determines $\bar{h}$ in terms of $h$ and $g$ from the requirement that
$$\langle T^{\rm d}(0)^{2}\rangle=\langle T^{\rm obs}(0)^{2}\rangle=\int\frac{d^%
{2}\ell}{(2\pi)^{2}}(C_{\ell}+C_{\ell}^{N})\ .$$
(B.3)
To simplify the calculations, we work with the simplified (dropping gradients) form of delensing
$$T^{\rm d}(\vec{x}\mskip 2.0mu \vphantom{x})\simeq\int\frac{d^{2}\ell}{(2\pi)^{%
2}}\Big{(}e^{i{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}\cdot(\vec{x}\mskip 2.0%
mu \vphantom{x}+\vec{\alpha}(\vec{x}\mskip 2.0mu \vphantom{x}))}\left(\bar{h}_%
{\ell}+e^{-i{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}\cdot g\star\vec{\alpha}\,%
{}^{\rm obs}(\vec{x}\mskip 2.0mu \vphantom{x})}h_{\ell}\right)T_{{\vec{\ell}%
\mskip 2.0mu \vphantom{\ell}}}+e^{i{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}%
\cdot\vec{x}\mskip 2.0mu \vphantom{x}}\left(\bar{h}_{\ell}+e^{-i{\vec{\ell}%
\mskip 2.0mu \vphantom{\ell}}\cdot g\star\vec{\alpha}\,{}^{\rm obs}(\vec{x}%
\mskip 2.0mu \vphantom{x})}h_{\ell}\right)T^{N}_{{\vec{\ell}\mskip 2.0mu %
\vphantom{\ell}}}\Big{)}$$
(B.4)
and therefore
$$\langle T^{\rm d}(0)^{2}\rangle=\int\frac{d^{2}\ell}{(2\pi)^{2}}\Big{(}\left(|%
\bar{h}_{\ell}|^{2}+|h_{\ell}|^{2}\right)C_{\ell}^{\rm obs}+\left(\bar{h}_{%
\ell}h_{-\ell}+h_{\ell}\bar{h}_{-\ell}\right)e^{-\frac{\ell^{2}}{4}C^{\rm obs}%
_{0}(0)}C_{\ell}^{\rm obs}\Big{)}\ .$$
(B.5)
We solve Equation (B.3) for $\bar{h}_{\ell}$ and find
$$\bar{h}_{\ell}=\sqrt{1-h_{\ell}^{2}\left(1-e^{-\frac{\ell^{2}}{2}C^{\rm obs}_{%
0}(0)}\right)}-h_{\ell}e^{-\frac{\ell^{2}}{4}C^{\rm obs}_{0}(0)}\ .$$
(B.6)
This formula is not quite correct, due to neglecting gradients of $\vec{\alpha}$ but this is a small effect (especially at the level of the filters).
B.3 Optimal Filters
In order to find the optimal result, we need to first define what we are trying to minimize. In the interest of providing all-orders expressions, we will define our procedure such that we choose $h$ and $g$ to minimize
$$\left\langle\big{(}T^{\rm d}_{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}-\langle T%
^{\rm d}_{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}\rangle_{\phi,\phi^{\rm N}}%
\big{)}\big{(}T^{\rm d}_{-{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}}-\langle T^%
{\rm d}_{-{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}}\rangle_{\phi,\phi^{\rm N}}%
\big{)}\right\rangle_{T,\phi,\phi^{\rm N}}$$
(B.7)
where $\langle\ldots\rangle_{X}$ means the statistical average with respect to $X$, holding everything else fixed. The advantage of this choice is that we are minimizing something that is manifestly positive and that it will reproduce the filters used in the perturbative limit.
The intuition for this choice is as follows: we are demanding that we minimize how much $T^{\rm d}$ varies with each realization of the lensing field and the reconstruction noise. In the limit where there is no noise, this means that we have removed the lensing from $T^{\rm d}$, as the map does not change under different realizations of the lensing potential. In the limit of a very noisy reconstruction, our procedure minimizes how much noise is introduced into the delensed temperature maps. As we eventually want to use the delensed maps for cosmological constraints, minimizing the noise in the maps is also a desirable feature. Finally, we will see in Appendix E that this minimization procedure determines an approximate local extremum of the Fisher information, and should be close to providing the best possible limits on cosmological parameters of interest.
In principle, we can minimize Equation B.7 to determine $g_{L}$ and $h_{\ell}$ given any noise levels for $T^{\rm obs}$ and $\phi^{\rm obs}$. Unfortunately, solving these equations in complete generality is challenging, even numerically. However, the solutions simplify in a number of limits that will allow us to gain intuition for the behavior of the optimal solution. In practice, we want filters that are easy to implement on real data and therefore we want a simple filtering scheme that approximates the various limits of the optimal filters.
Perturbative Limit: Let us start by expanding $T^{\rm d}$ to linear order in $\phi$ (ignoring gradients) to get
$$\displaystyle T^{\rm d}_{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}$$
$$\displaystyle\simeq$$
$$\displaystyle(1-h_{\ell})\Big{[}T^{\rm N}_{\vec{\ell}\mskip 2.0mu \vphantom{%
\ell}}+T_{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}+\int\frac{d^{2}L}{(2\pi)^{2}%
}\big{(}-{\vec{L}\mskip 2.0mu \vphantom{L}}\cdot({\vec{\ell}\mskip 2.0mu %
\vphantom{\ell}}-{\vec{L}\mskip 2.0mu \vphantom{L}})\big{)}\phi_{{\vec{L}%
\mskip 2.0mu \vphantom{L}}}T_{{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}-{\vec{L%
}\mskip 2.0mu \vphantom{L}}}\Big{]}$$
(B.8)
$$\displaystyle+h_{\ell}\Big{[}T^{\rm N}_{\vec{\ell}\mskip 2.0mu \vphantom{\ell}%
}+T_{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}+\int\frac{d^{2}L}{(2\pi)^{2}}\big%
{(}-{\vec{L}\mskip 2.0mu \vphantom{L}}\cdot({\vec{\ell}\mskip 2.0mu \vphantom{%
\ell}}-{\vec{L}\mskip 2.0mu \vphantom{L}})\big{)}\big{(}\phi_{{\vec{L}\mskip 2%
.0mu \vphantom{L}}}T_{{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}-{\vec{L}\mskip 2%
.0mu \vphantom{L}}}-g_{|{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}-{\vec{\ell}%
\mskip 2.0mu \vphantom{\ell}}_{1}|}\phi^{\rm obs}_{{\vec{L}\mskip 2.0mu %
\vphantom{L}}}(T_{{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}-{\vec{L}\mskip 2.0%
mu \vphantom{L}}}+T^{\rm N}_{{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}-{\vec{L}%
\mskip 2.0mu \vphantom{L}}})\big{)}\Big{]}$$
$$\displaystyle\simeq$$
$$\displaystyle T^{\rm N}_{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}+T_{\vec{\ell}%
\mskip 2.0mu \vphantom{\ell}}+\int\frac{d^{2}L}{(2\pi)^{2}}\big{(}-{\vec{L}%
\mskip 2.0mu \vphantom{L}}\cdot({\vec{\ell}\mskip 2.0mu \vphantom{\ell}}-{\vec%
{L}\mskip 2.0mu \vphantom{L}})\big{)}\big{(}\phi_{{\vec{L}\mskip 2.0mu %
\vphantom{L}}}T_{{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}-{\vec{L}\mskip 2.0mu%
\vphantom{L}}}-h_{|{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}-{\vec{L}\mskip 2.%
0mu \vphantom{L}}|}g_{L}\phi^{\rm obs}_{L}(T_{{\vec{\ell}\mskip 2.0mu %
\vphantom{\ell}}-{\vec{L}\mskip 2.0mu \vphantom{L}}}+T^{\rm N}_{{\vec{\ell}%
\mskip 2.0mu \vphantom{\ell}}-{\vec{L}\mskip 2.0mu \vphantom{L}}})\big{)}$$
where we used $\bar{h}_{\ell}=1-h_{\ell}+{\cal O}(C^{\phi\phi}_{L})$, $\vec{\alpha}(\vec{x}\mskip 2.0mu \vphantom{x})=\int\frac{d^{2}L}{(2\pi)^{2}}e^%
{i{\vec{L}\mskip 2.0mu \vphantom{L}}\cdot\vec{x}\mskip 2.0mu \vphantom{x}}\,i{%
\vec{L}\mskip 2.0mu \vphantom{L}}\phi_{{\vec{L}\mskip 2.0mu \vphantom{L}}}$ and $h_{\ell}\to h_{|{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}-{\vec{L}\mskip 2.0mu %
\vphantom{L}}|}$ (which is the small gradient expansion). We will simplify this expression by assuming $h_{\ell}$ and $g_{L}$ are real (i.e. real in both position- and $\ell$-space). Now we minimize with respect to $h_{q}$ (noticing that $\langle T^{\rm d}\rangle_{\phi,\phi^{\rm N}}\simeq T^{\rm N}_{\ell}+T_{\ell}$ is independent of $h$) to find
$$\partial_{h}\langle T^{\rm d}_{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}T^{\rm d%
}_{-{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}}\rangle^{\prime}=-2g_{|{\vec{\ell%
}\mskip 2.0mu \vphantom{\ell}}-{\vec{q}\mskip 2.0mu \vphantom{q}}|}\,C_{q}C^{%
\phi\phi}_{|{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}-{\vec{q}\mskip 2.0mu %
\vphantom{q}}|}+2h_{q}\,g_{|{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}-{\vec{q}%
\mskip 2.0mu \vphantom{q}}|}^{2}\,(C_{q}+C_{q}^{N})C^{\phi\phi,{\rm obs}}_{|{%
\vec{\ell}\mskip 2.0mu \vphantom{\ell}}-{\vec{q}\mskip 2.0mu \vphantom{q}}|}=0\ ,$$
(B.9)
where $\langle\ldots\rangle^{\prime}$ means we have removed the $(2\pi)^{2}\delta(0)$. To solve this equation, we note that if we write
$$\frac{g_{|{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}-{\vec{q}\mskip 2.0mu %
\vphantom{q}}|}C^{\phi\phi,{\rm obs}}_{|{\vec{\ell}\mskip 2.0mu \vphantom{\ell%
}}-{\vec{q}\mskip 2.0mu \vphantom{q}}|}}{C^{\phi\phi}_{|{\vec{\ell}\mskip 2.0%
mu \vphantom{\ell}}-{\vec{q}\mskip 2.0mu \vphantom{q}}|}}=\frac{C_{q}}{h_{q}(C%
_{q}+C_{q}^{N})}\ ,$$
(B.10)
then the left and right hand sides must be constants (independent of ${\vec{\ell}\mskip 2.0mu \vphantom{\ell}}$ and ${\vec{q}\mskip 2.0mu \vphantom{q}}$), otherwise it would be impossible to find a solution. This means that
$$\displaystyle g_{\ell}=a\frac{C^{\phi\phi}_{L}}{C^{\phi\phi,{\rm obs}}_{L}}%
\qquad\qquad h_{\ell}=a^{-1}\frac{C_{\ell}}{C_{\bar{\ell}}+C_{\ell}^{N}}\ ,$$
(B.11)
where $a$ is a constant. In the limit of a perfect measurement, we would like $g=h=1$, so we should choose $a=1$.
Technically speaking, we should have included the order $\phi^{2}$ term in the expansion of $T^{\rm d}$ because it will contribute to the power spectrum at this order. This would contribute a term of the from $\kappa h_{\ell}T_{\ell}$ where $\kappa\propto\int d^{2}L\,L^{2}C^{\phi\phi}_{L}$ is a constant independent of $\ell$. In what follows, an important feature of this minimization procedure is that if we want to solve $a(q)b(\ell-q)+c(q)d(\ell)=0$ (where $a,b,c,d$ are functions of one variable) for all $q,\ell$, then both terms must be independent of $q$ and $\ell$ which means we can look at each term independently.
The take-away from this calculation is that our definition of optimal filtering for delensing matches the perturbative result [23] in the appropriate limit.
Noisy Lensing Reconstruction: Now suppose we have a noisy measurement of the lensing potential such that the perturbative result suggests we should take $g\ll 1$. Now we can expand in $g$ as the small number while keeping all orders in $\phi$. Working to linear order in $g$, we have
$$\displaystyle T^{\rm d}_{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}$$
$$\displaystyle\simeq$$
$$\displaystyle(1-h_{\ell})\Big{[}T^{\rm N}_{\vec{\ell}\mskip 2.0mu \vphantom{%
\ell}}+\tilde{T}_{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}\Big{]}+h_{\ell}T^{%
\rm N}_{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}+h_{\ell}\tilde{T}_{\vec{\ell}%
\mskip 2.0mu \vphantom{\ell}}+\int\frac{d^{2}L}{(2\pi)^{2}}{\vec{L}\mskip 2.0%
mu \vphantom{L}}\cdot({\vec{\ell}\mskip 2.0mu \vphantom{\ell}}-{\vec{L}\mskip 2%
.0mu \vphantom{L}})g_{L}\phi_{{\vec{L}\mskip 2.0mu \vphantom{L}}}^{\rm obs}h_{%
|{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}-{\vec{L}\mskip 2.0mu \vphantom{L}}|}%
(\tilde{T}_{{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}-{\vec{L}\mskip 2.0mu %
\vphantom{L}}}+T^{\rm N}_{{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}-{\vec{L}%
\mskip 2.0mu \vphantom{L}}})$$
(B.12)
$$\displaystyle=$$
$$\displaystyle T^{\rm N}_{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}+\tilde{T}_{%
\vec{\ell}\mskip 2.0mu \vphantom{\ell}}+\int\frac{d^{2}L}{(2\pi)^{2}}{\vec{L}%
\mskip 2.0mu \vphantom{L}}\cdot({\vec{\ell}\mskip 2.0mu \vphantom{\ell}}-{\vec%
{L}\mskip 2.0mu \vphantom{L}})g_{L}\phi_{{\vec{L}\mskip 2.0mu \vphantom{L}}}^{%
\rm obs}h_{|{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}-{\vec{L}\mskip 2.0mu %
\vphantom{L}}|}(\tilde{T}_{{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}-{\vec{L}%
\mskip 2.0mu \vphantom{L}}}+T^{\rm N}_{{\vec{\ell}\mskip 2.0mu \vphantom{\ell}%
}-{\vec{L}\mskip 2.0mu \vphantom{L}}})$$
Now expanding the power spectrum and taking a derivative with respect to $g_{q}$ we get
$$2({\vec{\ell}\mskip 2.0mu \vphantom{\ell}}-{\vec{q}\mskip 2.0mu \vphantom{q}})%
\cdot{\vec{q}\mskip 2.0mu \vphantom{q}}\,h_{|{\vec{\ell}\mskip 2.0mu \vphantom%
{\ell}}-{\vec{q}\mskip 2.0mu \vphantom{q}}|}\langle\phi^{\rm obs}_{\vec{q}%
\mskip 2.0mu \vphantom{q}}\tilde{T}_{{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}-%
{\vec{q}\mskip 2.0mu \vphantom{q}}}\tilde{T}_{-{\vec{\ell}\mskip 2.0mu %
\vphantom{\ell}}}\rangle^{\prime}-2g_{q}C_{q}^{\rm\phi\phi,{\rm obs}}(({\vec{%
\ell}\mskip 2.0mu \vphantom{\ell}}-{\vec{q}\mskip 2.0mu \vphantom{q}})\cdot{%
\vec{q}\mskip 2.0mu \vphantom{q}})^{2}|h_{|{\vec{\ell}\mskip 2.0mu \vphantom{%
\ell}}-{\vec{q}\mskip 2.0mu \vphantom{q}}|}|^{2}C^{\rm obs}_{|{\vec{\ell}%
\mskip 2.0mu \vphantom{\ell}}-{\vec{q}\mskip 2.0mu \vphantom{q}}|}=0\ .$$
(B.13)
The statement $g_{L}\ll 1$ implies that $C^{\phi\phi,N}_{L}\gg C^{\phi\phi}_{L}$ and therefore it is consistent to drop terms proportional to $g_{L}^{2}C^{\phi\phi,N}_{L}$ while keeping terms of order $g_{L}C^{\phi\phi,N}_{L}\sim C^{\phi\phi}_{L}$.
The last thing we need to evaluate is $\langle\phi^{\rm obs}_{\vec{q}\mskip 2.0mu \vphantom{q}}\tilde{T}_{{\vec{\ell}%
\mskip 2.0mu \vphantom{\ell}}-{\vec{q}\mskip 2.0mu \vphantom{q}}}\tilde{T}_{-{%
\vec{\ell}\mskip 2.0mu \vphantom{\ell}}}\rangle$. Using Gaussian statistics, we see that for a Gaussian random field $y$,
$$\displaystyle\langle y\,f(y)\rangle$$
$$\displaystyle=$$
$$\displaystyle\int dy\frac{1}{\sqrt{2\pi}\sigma}(-\sigma^{2})\left(\partial_{y}%
e^{-\frac{y^{2}}{2\sigma^{2}}}\right)f(y)$$
(B.14)
$$\displaystyle=$$
$$\displaystyle\sigma^{2}\langle f^{\prime}(y)\rangle\ .$$
Now we can use (being careful to note that when $y$ is complex we take a derivative with respect to $y^{*}$),
$$\displaystyle\partial_{\phi_{-{\vec{q}\mskip 2.0mu \vphantom{q}}}}\tilde{T}_{{%
\vec{\ell}\mskip 2.0mu \vphantom{\ell}}}$$
$$\displaystyle=$$
$$\displaystyle\int d^{2}x_{1}\frac{d^{2}\ell_{1}}{(2\pi)^{2}}e^{-i{\vec{\ell}%
\mskip 2.0mu \vphantom{\ell}}\cdot\vec{x}\mskip 2.0mu \vphantom{x}_{1}}({\vec{%
\ell}\mskip 2.0mu \vphantom{\ell}}_{1}\cdot{\vec{q}\mskip 2.0mu \vphantom{q}})%
e^{i{\vec{q}\mskip 2.0mu \vphantom{q}}\cdot\vec{x}\mskip 2.0mu \vphantom{x}_{1%
}}e^{i{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}_{1}\cdot\vec{x}\mskip 2.0mu %
\vphantom{x}_{1}+\int\frac{d^{2}L}{(2\pi)^{2}}e^{i{\vec{L}\mskip 2.0mu %
\vphantom{L}}\cdot\vec{x}\mskip 2.0mu \vphantom{x}_{1}}({\vec{\ell}\mskip 2.0%
mu \vphantom{\ell}}_{1}\cdot{\vec{L}\mskip 2.0mu \vphantom{L}})\phi_{\vec{L}%
\mskip 2.0mu \vphantom{L}}}$$
(B.15)
$$\displaystyle=$$
$$\displaystyle(i{\vec{q}\mskip 2.0mu \vphantom{q}})\cdot(\widetilde{\nabla T})_%
{{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}+{\vec{q}\mskip 2.0mu \vphantom{q}}}\ ,$$
where $\widetilde{\nabla T}=\nabla T(\vec{x}\mskip 2.0mu \vphantom{x}+\vec{\alpha}(%
\vec{x}\mskip 2.0mu \vphantom{x}))$. Now if we define
$$\left\langle(\widetilde{\nabla T})_{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}\,%
\tilde{T}_{\vec{\ell}\mskip 2.0mu \vphantom{\ell}^{\prime}}\right\rangle=i{%
\vec{\ell}\mskip 2.0mu \vphantom{\ell}}\tilde{C}_{\ell}^{T\nabla T}(2\pi)^{2}%
\delta({\vec{\ell}\mskip 2.0mu \vphantom{\ell}}+\vec{\ell}\mskip 2.0mu %
\vphantom{\ell}^{\prime})\ ,$$
(B.16)
then we find
$$\langle\phi^{\rm obs}_{{\vec{q}\mskip 2.0mu \vphantom{q}}}\tilde{T}_{{\vec{%
\ell}\mskip 2.0mu \vphantom{\ell}}-{\vec{q}\mskip 2.0mu \vphantom{q}}}\tilde{T%
}_{-{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}}\rangle^{\prime}=({\vec{\ell}%
\mskip 2.0mu \vphantom{\ell}}-{\vec{q}\mskip 2.0mu \vphantom{q}})\cdot{\vec{q}%
\mskip 2.0mu \vphantom{q}}\,C_{q}^{\phi\phi}\tilde{C}^{T\nabla T}_{\ell-q}-{%
\vec{\ell}\mskip 2.0mu \vphantom{\ell}}\cdot{\vec{q}\mskip 2.0mu \vphantom{q}}%
C_{q}^{\phi\phi}\tilde{C}^{T\nabla T}_{\ell}\ .$$
(B.17)
Putting this together we find
$$g_{\ell}=\frac{C^{\phi\phi}_{L}}{C^{\phi\phi,{\rm obs}}_{L}}\,,\qquad\qquad h_%
{\ell}=\frac{\tilde{C}_{\ell}^{T\nabla T}}{C_{\ell}^{\rm obs}}\ .$$
(B.18)
It turns out that $\tilde{C}_{\ell}^{T\nabla T}\simeq\tilde{C}_{\ell}$ [56] and therefore this explains why taking the perturbative result and making the replacement $C_{\ell}\to\tilde{C}_{\ell}$ is useful approximation to the non-perturbative result, at least in the limit of a noisy lensing map.
Ideal Lensing Map: In order to get intuition for the limits of an ideal filter, we will finally consider the case where we have a (nearly) perfect lensing map in the presence of noisy temperature (or polarization) data.
The complication presented by noisy data is that we can perfectly delens the underlying CMB modes, but we will also shift around the noise in the process. In harmonic space, we will have
$$T^{\rm d}_{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}=\bar{h}_{\ell}(\tilde{T}_{%
\vec{\ell}\mskip 2.0mu \vphantom{\ell}}+T^{\rm N}_{\vec{\ell}\mskip 2.0mu %
\vphantom{\ell}})+h_{\ell}(T_{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}+\tilde{T%
}^{\rm N}_{{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}})$$
(B.19)
where $\tilde{T}^{\rm N}(\vec{x}\mskip 2.0mu \vphantom{x})=T^{\rm N}(\vec{x}\mskip 2.%
0mu \vphantom{x}-\vec{\alpha}(\vec{x}\mskip 2.0mu \vphantom{x}))$. We notice that
$$\Delta T^{\rm d}_{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}\equiv T^{\rm d}_{%
\vec{\ell}\mskip 2.0mu \vphantom{\ell}}-\langle T^{\rm d}\rangle_{\alpha}=\bar%
{h}_{\ell}(\tilde{T}_{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}-e^{-\frac{\ell^{%
2}}{4}C_{0}(0)}T_{\vec{\ell}\mskip 2.0mu \vphantom{\ell}})+h_{\ell}(\tilde{T}^%
{\rm N}_{{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}}-e^{-\frac{\ell^{2}}{4}C_{0}%
(0)}T_{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}^{\rm N})$$
(B.20)
so that
$$\displaystyle\left\langle\Delta T^{\rm d}_{\vec{\ell}\mskip 2.0mu \vphantom{%
\ell}}\Delta T^{\rm d}_{\vec{\ell}\mskip 2.0mu \vphantom{\ell}^{\prime}}\right\rangle$$
$$\displaystyle=$$
$$\displaystyle(2\pi)^{2}\delta({\vec{\ell}\mskip 2.0mu \vphantom{\ell}}+\vec{%
\ell}\mskip 2.0mu \vphantom{\ell}^{\prime})\Big{[}|\bar{h}_{\ell}|^{2}(\tilde{%
C}_{\ell}-e^{-\frac{\ell^{2}}{2}C_{0}(0)}C_{\ell})+|h_{\ell}|^{2}(\tilde{C}_{%
\ell}^{\rm N}-e^{-\frac{\ell^{2}}{2}C_{0}(0)}C_{\ell}^{\rm N})$$
(B.21)
Because of the relatively complicated form of $\bar{h}_{\ell}$ in terms of $h_{\ell}$ this is not especially easy to minimize with respect to $h_{\ell}$. However, we are mostly interested in the behavior that defined $h_{\ell}\ll 1$ so we can expand $\bar{h}_{\ell}\simeq 1-h_{\ell}e^{-\frac{\ell^{2}}{4}C_{0}(0)}$. Minimizing with respect to $h_{\ell}$ we find
$$h_{\ell}\sim\frac{e^{-\frac{\ell^{2}}{4}C_{0}(0)}\left(\tilde{C}_{\ell}-e^{-%
\frac{\ell^{2}}{2}C_{0}(0)}C_{\ell}\right)}{(\tilde{C}^{\rm N}_{\ell}-e^{-%
\frac{\ell^{2}}{2}C_{0}(0)}C^{\rm N}_{\ell})}\ .$$
(B.22)
In writing this expression, we assumed that the temperature map is noisy, in order to be consistent with the assumption $h_{\ell}\ll 1$. We can evaluate this expression using
$$\displaystyle\Delta\tilde{C}^{({\rm N})}_{\ell}$$
$$\displaystyle\equiv$$
$$\displaystyle\left(\tilde{C}^{({\rm N})}_{\ell}-e^{-\frac{\ell^{2}}{2}C_{0}(0)%
}C^{({\rm N})}_{\ell}\right)$$
(B.23)
$$\displaystyle=$$
$$\displaystyle e^{-\frac{\ell^{2}}{2}C_{0}(0)}\int d^{2}r\frac{d^{2}\ell_{1}}{(%
2\pi)^{2}}e^{-i({\vec{\ell}\mskip 2.0mu \vphantom{\ell}}-{\vec{\ell}\mskip 2.0%
mu \vphantom{\ell}}_{1})\cdot\vec{r}\mskip 2.0mu \vphantom{r}}\left(e^{\frac{%
\ell_{1}^{2}}{2}(C_{0}(r)-\cos 2\varphi C_{2}(r))}-1\right)C_{\ell_{1}}^{({\rm
N%
})}\ .$$
The main takeaway is that in the limit of a perfect lensing map, the optimal filter as the temperature map becomes noisy is controlled by the ratio of the lensed power to the delensed noise.
Summary: While our all-orders spectra suggest an all-orders method for choosing filters, in practice the the optimal choice is difficult to determine analytically and therefore of limited practical utility. However, the all-orders approach simplifies in a number of limits of interest and therefore we can choose our filters by matching the appropriate limit. For the lensing potential, the optimal fiter in all cases is
$$g_{L}=\frac{C_{L}^{\phi\phi}}{C_{L}^{\phi\phi}+C_{L}^{\phi\phi,{\rm N}}}\ .$$
(B.24)
The case is more complicated for temperature and polarization, as there are three different types of spectra to consider: lensed, unlensed, and delensed spectra. Intuitively, when we are signal dominated for all of the spectra, the choice $g_{L},h_{\ell},h^{(P)}_{\ell}\simeq 1$ is free of any subtlety. Furthermore, for $T$ and $E$, as we go to larger $L$ and $\ell$, the noise in $C_{L}^{\phi\phi,{\rm obs}}$ usually dominates before we are limited by the noise in $T$ or $E$. Therefore, we will typically be in the situation where $g\ll 1$ when the choice when there is a noticeable difference in the filters for $T$ and $E$. We calculated the optimal filters perturbatively in $g$ and found
$$h_{\ell}=\frac{\tilde{C}_{\ell}^{T\nabla T}}{C_{\ell}^{\rm obs}}\simeq\frac{%
\tilde{C}_{\ell}^{TT}}{C_{\ell}^{\rm obs}}$$
(B.25)
This choice seems to behave appropriately in the limits applicable to a typical CMB experiment and would seem to be the appropriate choice for our forecasting purposes. The corresponding $\bar{h}$ filter is
$$\bar{h}_{\ell}=\sqrt{1-h_{\ell}^{2}\left(1-e^{-\frac{\ell^{2}}{2}C^{\rm obs}_{%
0}(0)}\right)}-h_{\ell}e^{-\frac{\ell^{2}}{4}C^{\rm obs}_{0}(0)}\ .$$
(B.26)
B.4 Polarization
Having fully explored the choice of filters for temperature, we will now repeat the process for polarization (in an abbreviated form). Our delensed polarization field is defined to be
$$\displaystyle\big{[}Q^{\rm d}\pm iU^{\rm d}\big{]}(\vec{x}\mskip 2.0mu %
\vphantom{x})=\bar{h}^{(P)}\star\big{[}Q^{\rm obs}\pm iU^{\rm obs}\big{]}(\vec%
{x}\mskip 2.0mu \vphantom{x})+h^{(P)}\star\big{[}Q^{\rm obs}\pm iU^{\rm obs}%
\big{]}(\vec{x}\mskip 2.0mu \vphantom{x}-g\star\vec{\alpha}\,{}^{\rm obs}(\vec%
{x}\mskip 2.0mu \vphantom{x}))\ .$$
(B.27)
As discussed in the main text, we are choosing a common filter for both $Q$ and $U$ and we will take $\bar{h}^{(P)}$ and $h^{(P)}$ to be real. We choose a common filter because isotropic noise implies the noise is the same for both $Q$ and $U$. Furthermore, since lensing acts locally on $Q$ and $U$, we want to avoid filtering that mixes $Q$ into $U$ or vise versa. A direct consequence of these choices is that there is a common filter for $E$ and $B$. We will expand on this choice in the next subsection.
The first constraint is that we require that the total power is unchanged, which for polarization means we want to keep $Q^{2}(0)+U^{2}(0)$ fixed. This implies that
$$\displaystyle\left\langle\left|Q^{\rm d}\pm iU^{\rm d}\right|^{2}\right\rangle%
(0)$$
$$\displaystyle=$$
$$\displaystyle\int\frac{d^{2}\ell}{(2\pi)^{2}}\Big{(}C_{\ell}^{EE,{\rm obs}}+C_%
{\ell}^{BB,{\rm obs}}\Big{)}$$
$$\displaystyle\qquad\qquad\times\left[\left(\left|\bar{h}^{(P)}_{\ell}\right|^{%
2}+\left|h^{(P)}_{\ell}\right|^{2}\right)+\left(\bar{h}^{(P)}_{\ell}h^{(P)}_{-%
\ell}+h^{(P)}_{\ell}\bar{h}^{(P)}_{-\ell}\right)e^{-\frac{\ell^{2}}{4}C^{\rm
obs%
}_{0}(0)}\right]$$
Requiring that this is the same as the observed power implies that
$$\bar{h}^{(P)}_{\ell}=\sqrt{1-{h^{(P)}_{\ell}}^{2}\left(1-e^{-\frac{\ell^{2}}{2%
}C^{\rm obs}_{0}(0)}\right)}-h^{(P)}_{\ell}e^{-\frac{\ell^{2}}{4}C^{\rm obs}_{%
0}(0)}\ .$$
(B.29)
Perhaps unsurprisingly, this is identical to the constraint we found for the temperature filters.
Now we will compute the filter in the noisy lens limit by minimizing
$$\left\langle\big{(}E^{\rm d}_{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}-\langle E%
^{\rm d}_{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}\rangle_{\phi,\phi^{\rm N}}%
\big{)}\big{(}E^{\rm d}_{-{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}}-\langle E^%
{\rm d}_{-{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}}\rangle_{\phi,\phi^{\rm N}}%
\big{)}\right\rangle_{E,\phi,\phi^{\rm N}}\ .$$
(B.30)
Expanding in small $g_{\ell}$ we find
$$\displaystyle E^{\rm d}_{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}$$
$$\displaystyle=$$
$$\displaystyle E^{\rm obs}_{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}+\int\frac{d%
^{2}L}{(2\pi)^{2}}\Big{[}h^{(P)}_{|{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}-{%
\vec{L}\mskip 2.0mu \vphantom{L}}|}{\vec{L}\mskip 2.0mu \vphantom{L}}\cdot({%
\vec{\ell}\mskip 2.0mu \vphantom{\ell}}-{\vec{L}\mskip 2.0mu \vphantom{L}})g_{%
L}\phi^{\rm obs}_{{\vec{L}\mskip 2.0mu \vphantom{L}}}$$
(B.31)
$$\displaystyle\times\Bigg{[}\cos(2(\varphi_{{\vec{\ell}\mskip 2.0mu \vphantom{%
\ell}}-{\vec{L}\mskip 2.0mu \vphantom{L}}}-\varphi_{\vec{\ell}\mskip 2.0mu %
\vphantom{\ell}}))E^{\rm obs}_{{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}-{\vec{%
L}\mskip 2.0mu \vphantom{L}}}+\sin(2(\varphi_{{\vec{\ell}\mskip 2.0mu %
\vphantom{\ell}}-{\vec{L}\mskip 2.0mu \vphantom{L}}}-\varphi_{\vec{\ell}\mskip
2%
.0mu \vphantom{\ell}}))B^{\rm N}_{{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}-{%
\vec{L}\mskip 2.0mu \vphantom{L}}}\Bigg{]}$$
The angles introduce a small complication compared to temperature. However, in practice the lensing is peaked at low multipoles such that $|{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}-{\vec{\ell}\mskip 2.0mu \vphantom{%
\ell}}_{1}|\ll\ell$ or $\cos(\varphi_{{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}_{1}}-\varphi_{{\vec{%
\ell}\mskip 2.0mu \vphantom{\ell}}})\simeq 1$. Therefore, to simplify the discussion we can set $(\varphi_{{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}_{1}}-\varphi_{\vec{\ell}%
\mskip 2.0mu \vphantom{\ell}})=0$ such that
$$\displaystyle\langle E^{\rm d}_{{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}}E^{%
\rm d}_{\vec{\ell}\mskip 2.0mu \vphantom{\ell}^{\prime}}\rangle$$
$$\displaystyle\sim$$
$$\displaystyle(2\pi)^{2}\delta({\vec{\ell}\mskip 2.0mu \vphantom{\ell}}+\vec{%
\ell}\mskip 2.0mu \vphantom{\ell}^{\prime})C_{\ell}^{EE,{\rm obs}}+\int\frac{d%
^{2}L}{(2\pi)^{2}}\Big{[}h^{(P)}_{|{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}-{%
\vec{L}\mskip 2.0mu \vphantom{L}}|}{\vec{L}\mskip 2.0mu \vphantom{L}}\cdot({%
\vec{\ell}\mskip 2.0mu \vphantom{\ell}}-{\vec{L}\mskip 2.0mu \vphantom{L}})g_{%
L}\left\langle\phi^{\rm obs}_{{\vec{L}\mskip 2.0mu \vphantom{L}}}\tilde{E}_{{%
\vec{\ell}\mskip 2.0mu \vphantom{\ell}}-{\vec{L}\mskip 2.0mu \vphantom{L}}}%
\tilde{E}_{\vec{\ell}\mskip 2.0mu \vphantom{\ell}^{\prime}}\right\rangle+\{{%
\vec{\ell}\mskip 2.0mu \vphantom{\ell}}\leftrightarrow\vec{\ell}\mskip 2.0mu %
\vphantom{\ell}^{\prime}\}\Big{]}$$
(B.32)
$$\displaystyle+\int\frac{d^{2}L}{(2\pi)^{2}}\left|h^{(P)}_{|{\vec{\ell}\mskip 2%
.0mu \vphantom{\ell}}-{\vec{L}\mskip 2.0mu \vphantom{L}}|}\right|^{2}\left|g_{%
L}\right|^{2}\left[{\vec{L}\mskip 2.0mu \vphantom{L}}\cdot({\vec{\ell}\mskip 2%
.0mu \vphantom{\ell}}-{\vec{L}\mskip 2.0mu \vphantom{L}})\right]^{2}\,C^{\phi%
\phi,{\rm obs}}_{L}C^{EE,{\rm obs}}_{|{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}%
-{\vec{L}\mskip 2.0mu \vphantom{L}}|}\ .$$
Now taking a derivative with respect to $g_{q}$ we get
$$\left(h^{(P)}_{|{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}-{\vec{q}\mskip 2.0mu %
\vphantom{q}}|}{\vec{q}\mskip 2.0mu \vphantom{q}}\cdot({\vec{\ell}\mskip 2.0mu%
\vphantom{\ell}}-{\vec{q}\mskip 2.0mu \vphantom{q}})\left\langle\phi^{\rm obs%
}_{{\vec{q}\mskip 2.0mu \vphantom{q}}}\tilde{E}_{{\vec{\ell}\mskip 2.0mu %
\vphantom{\ell}}-{\vec{q}\mskip 2.0mu \vphantom{q}}}\tilde{E}_{\vec{\ell}%
\mskip 2.0mu \vphantom{\ell}^{\prime}}\right\rangle+\{{\vec{\ell}\mskip 2.0mu %
\vphantom{\ell}}\leftrightarrow\vec{\ell}\mskip 2.0mu \vphantom{\ell}^{\prime}%
\}\right)=2\left|h^{(P)}_{|{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}-{\vec{q}%
\mskip 2.0mu \vphantom{q}}|}\right|^{2}g_{q}\left[{\vec{q}\mskip 2.0mu %
\vphantom{q}}\cdot({\vec{\ell}\mskip 2.0mu \vphantom{\ell}}-{\vec{q}\mskip 2.0%
mu \vphantom{q}})\right]^{2}\,C^{\phi\phi,{\rm obs}}_{1}C^{EE,{\rm obs}}_{|{%
\vec{\ell}\mskip 2.0mu \vphantom{\ell}}-{\vec{q}\mskip 2.0mu \vphantom{q}}|}$$
(B.33)
We can evaluate the left hand side using $\langle yf(y)\rangle=\sigma^{2}\langle f^{\prime}(y)\rangle$ for Gaussian $y$ as we did for temperature. We will define
$$\widetilde{\nabla E}_{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}\equiv i\left[{%
\rm Re}\int d^{2}x\frac{d^{2}\ell_{1}}{(2\pi)^{2}}\,{\vec{\ell}\mskip 2.0mu %
\vphantom{\ell}}_{1}E_{{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}_{1}}e^{-i({%
\vec{\ell}\mskip 2.0mu \vphantom{\ell}}-{\vec{\ell}\mskip 2.0mu \vphantom{\ell%
}}_{1})\cdot\vec{x}\mskip 2.0mu \vphantom{x}}e^{2i(\varphi_{{\vec{\ell}\mskip 2%
.0mu \vphantom{\ell}}_{1}}-\varphi_{{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}})%
}e^{i{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}_{1}\cdot\vec{\alpha}(\vec{x}%
\mskip 2.0mu \vphantom{x})}\right]$$
(B.34)
and
$$\left\langle\widetilde{\nabla E}_{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}%
\tilde{E}_{\vec{\ell}\mskip 2.0mu \vphantom{\ell}^{\prime}}\right\rangle=i{%
\vec{\ell}\mskip 2.0mu \vphantom{\ell}}\tilde{C}^{E\nabla E}_{\ell}(2\pi)^{2}%
\delta({\vec{\ell}\mskip 2.0mu \vphantom{\ell}}+\vec{\ell}\mskip 2.0mu %
\vphantom{\ell}^{\prime})$$
(B.35)
Following the same procedure as we did for temperature, we have
$$\displaystyle\partial_{\phi_{-{\vec{q}\mskip 2.0mu \vphantom{q}}}}\tilde{E}_{{%
\vec{\ell}\mskip 2.0mu \vphantom{\ell}}}$$
$$\displaystyle=$$
$$\displaystyle(i{\vec{q}\mskip 2.0mu \vphantom{q}})\cdot(\widetilde{\nabla E})_%
{{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}+\vec{q}}\ .$$
(B.36)
and therefore
$$\left\langle\phi^{\rm obs}_{{\vec{q}\mskip 2.0mu \vphantom{q}}}\tilde{E}_{{%
\vec{\ell}\mskip 2.0mu \vphantom{\ell}}-{\vec{q}\mskip 2.0mu \vphantom{q}}}%
\tilde{T}_{\vec{\ell}\mskip 2.0mu \vphantom{\ell}^{\prime}}\right\rangle=(2\pi%
)^{2}\delta({\vec{\ell}\mskip 2.0mu \vphantom{\ell}}+\vec{\ell}\mskip 2.0mu %
\vphantom{\ell}^{\prime})\Big{[}({\vec{\ell}\mskip 2.0mu \vphantom{\ell}}-{%
\vec{q}\mskip 2.0mu \vphantom{q}})\cdot{\vec{q}\mskip 2.0mu \vphantom{q}}\,C_{%
q}^{\phi\phi}\tilde{C}^{E\nabla E}_{\ell-q}-{\vec{\ell}\mskip 2.0mu \vphantom{%
\ell}}\cdot{\vec{q}\mskip 2.0mu \vphantom{q}}\,C_{q}^{\phi\phi}\tilde{C}^{E%
\nabla E}_{\ell}\Big{]}\ .$$
(B.37)
The optimal filters are then
$$g_{L}=\frac{C^{\phi\phi}_{L}}{C^{\phi\phi,{\rm obs}}_{L}}\,,\qquad\qquad h^{(P%
)}_{\ell}=\frac{\tilde{C}_{\ell}^{E\nabla E}}{C_{\ell}^{EE,{\rm obs}}}\approx%
\frac{\tilde{C}_{\ell}^{EE}}{C_{\ell}^{EE,{\rm obs}}}\ .$$
(B.38)
It is reassuring that $g$ in unchanged from the temperature filters. One can check that the approximation $\tilde{C}_{\ell}^{E\nabla E}\approx\tilde{C}_{\ell}^{EE}$ is the same one that allowed us to neglect the angular terms in Equation B.31, namely that ${\vec{\ell}\mskip 2.0mu \vphantom{\ell}}-{\vec{L}\mskip 2.0mu \vphantom{L}}%
\simeq{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}$.
B.5 Filtering in $Q/U$ versus $E/B$
We have chosen to filter locally in terms of the $Q$ and $U$ maps. Locally, the polarization defines a vector field with independent components and isotropic noise, so this was a natural choice. However, one could instead imagine converting the map into $E$ and $B$ modes and filtering each separating. In this subsection, we will explain the relationship between these two approaches in order to further explain the meaning of our choice in filtering $Q$ and $U$.
To simplify our discussion, let us ignore delensing and simply discuss the meaning of the filters directly. After converting $P(\vec{x}\mskip 2.0mu \vphantom{x})$ into $E_{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}$ and $B_{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}$, it is natural to consider the filtered fields
$$E_{\ell}^{f}=h^{E}_{\ell}E_{\ell}\qquad\qquad B_{\ell}^{f}=h^{B}_{\ell}B_{\ell%
}\ .$$
(B.39)
To make the connection to the original $Q$ and $U$, let us define
$$(Q\pm iU)_{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}=\int d^{2}xe^{-i\vec{x}%
\mskip 2.0mu \vphantom{x}\cdot{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}}(Q\pm iU%
)(\vec{x}\mskip 2.0mu \vphantom{x})$$
(B.40)
such that
$$\displaystyle e^{\pm i2\varphi_{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}}(E\pm
iB%
)_{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}=Q_{\vec{\ell}\mskip 2.0mu \vphantom%
{\ell}}\pm iU_{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}\ .$$
(B.41)
Now if we define the filtered maps as $Q^{f}_{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}\pm iU^{f}_{\vec{\ell}\mskip 2.0%
mu \vphantom{\ell}}\equiv e^{\pm i2\varphi_{\vec{\ell}\mskip 2.0mu \vphantom{%
\ell}}}(E^{f}\pm iB^{f})_{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}$ we have
$$\displaystyle Q^{f}_{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}+iU^{f}_{\vec{\ell%
}\mskip 2.0mu \vphantom{\ell}}=\frac{1}{2}(h^{E}_{\vec{\ell}\mskip 2.0mu %
\vphantom{\ell}}+h^{B}_{\vec{\ell}\mskip 2.0mu \vphantom{\ell}})(Q+iU)_{\vec{%
\ell}\mskip 2.0mu \vphantom{\ell}}+\frac{1}{2}(h^{E}-h^{B})_{\vec{\ell}\mskip 2%
.0mu \vphantom{\ell}}e^{4i\varphi_{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}}(Q-%
iU)_{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}$$
(B.42)
One can immediately see that the first term does not mix $Q$ and $U$, while the second term rotates the polarization vector.
We can now map this back position space in terms of $P(\vec{x}\mskip 2.0mu \vphantom{x})=Q(\vec{x}\mskip 2.0mu \vphantom{x})+iU(%
\vec{x}\mskip 2.0mu \vphantom{x})$ and $P^{f}(\vec{x}\mskip 2.0mu \vphantom{x})=Q^{f}(\vec{x}\mskip 2.0mu \vphantom{x}%
)+iU^{f}(\vec{x}\mskip 2.0mu \vphantom{x})$ as
$$P^{f}(\vec{x}\mskip 2.0mu \vphantom{x})=\int d^{2}x^{\prime}\left[h^{E+B}(x-x^%
{\prime})P(x^{\prime})+h^{E-B}(x-x^{\prime})P^{*}(x^{\prime})\right]$$
(B.43)
where
$$\displaystyle h^{E+B}(\vec{x}\mskip 2.0mu \vphantom{x})$$
$$\displaystyle\equiv$$
$$\displaystyle\int d^{2}\ell e^{i{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}\cdot%
\vec{x}\mskip 2.0mu \vphantom{x}}\frac{1}{2}(h^{E}+h^{B})_{\ell}$$
(B.44)
$$\displaystyle h^{E-B}(\vec{x}\mskip 2.0mu \vphantom{x})$$
$$\displaystyle\equiv$$
$$\displaystyle\int d^{2}\ell e^{i{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}\cdot%
\vec{x}\mskip 2.0mu \vphantom{x}}e^{i4\varphi_{\vec{\ell}\mskip 2.0mu %
\vphantom{\ell}}}\frac{1}{2}(h^{E}-h^{B})_{\ell}\ .$$
(B.45)
We see that the $h^{E+B}$ filter is an ordinary scalar under rotations while the $h^{E-B}$ transforms like a spin-4 object.
The properties of these two filters are relatively easy to understand from isotropy. Since we want to isotropy to be preserved by filtering, our filters should also decompose into representations of the rotations. Furthermore, in order for the filtered field to transform in the same way as the unfiltered field, the filters in Equation (B.43) are limited to spin-0 and spin-4.
Our filtering scheme is equivalent to only the first term in Equation (B.43) and therefore we have set $h^{E-B}=0$ or $h^{E}=h^{B}$. This choice also matches the optimal perturbative result [23]. We made this choice to preserve the local nature of the filter in real space. Specifically, since $\delta(\vec{x}\mskip 2.0mu \vphantom{x})$ is a scalar under rotations, in the limit where the filter is trivial we can only have the scalar piece. This is desirable for delensing because delensing itself is local in $\vec{x}\mskip 2.0mu \vphantom{x}$.
Although we have chosen to drop the $h^{E-B}$ filter, one could imagine situations where it provides useful information. Because the primordial $E$-mode signal is much larger than the $B$-mode signal, there are correlations in the signal between $U$ and $Q$ at separated points. One could imagine this non-local information being useful in weighting the signal-to-noise in the filtered maps, especially for noisy maps. Whether this improves constraints is a question of whether our filtering scheme is truly optimal. It is possible that this more elaborate scheme that includes the spin-4 filter could improve constraints; although, as we saw in Section 4, there is often very little room for improvement given how close the delensed and unlensed forecasts are when using the simpler filters.
Appendix C Numeric Computation of Polarization Spectra
Here we give the expressions used for the numeric computation of delensed polarization spectra. As for the temperature, we expand to first order in $C_{2}(r)$ and compute the change to the correlation functions due to lensing and delensing
$$\displaystyle\Delta\xi_{+}^{\rm d}(r)$$
$$\displaystyle=$$
$$\displaystyle\int\frac{d\ell}{2\pi}\ell(C_{\ell}^{EE}+C_{\ell}^{BB})\Bigg{[}-J%
_{0}(\ell r)$$
$$\displaystyle+\left|\bar{h}^{(P)}_{\ell}\right|^{2}\exp\left[-\frac{\ell^{2}}{%
2}(C_{0}(0)-C_{0}(r))\right]\left(J_{0}(\ell r)+\frac{\ell^{2}}{2}C_{2}(r)J_{2%
}(\ell r)\right)$$
$$\displaystyle+\left(h^{(P)}_{\ell}\bar{h}^{(P)}_{-\ell}+h^{(P)}_{-\ell}\bar{h}%
^{(P)}_{\ell}\right)\exp\left[-\frac{\ell^{2}}{2}\left((C_{0}(0)-C_{0}(r))-(C^%
{\rm cross}_{0}(0)-C^{\rm cross}_{0}(r))+\frac{1}{2}C^{\rm obs}_{0}(0)\right)\right]$$
$$\displaystyle\quad\times\left(J_{0}(\ell r)+\frac{\ell^{2}}{2}\left(C_{2}(r)-C%
^{\rm cross}_{2}(r)\right)J_{2}(\ell r)\right)$$
$$\displaystyle+\left|h^{(P)}_{\ell}\right|^{2}\exp\left[-\frac{\ell^{2}}{2}%
\left((C_{0}(0)-C_{0}(r))-2(C^{\rm cross}_{0}(0)-C^{\rm cross}_{0}(r))+(C^{\rm
obs%
}_{0}(0)-C^{\rm obs}_{0}(r))\right)\right]$$
$$\displaystyle\quad\times\left(J_{0}(\ell r)+\frac{\ell^{2}}{2}\left(C_{2}(r)-2%
C^{\rm cross}_{2}(r)+C^{\rm obs}_{2}(r)\right)J_{2}(\ell r)\right)\Bigg{]}\ ,$$
$$\displaystyle\Delta\xi_{-}^{\rm d}(r)$$
$$\displaystyle=$$
$$\displaystyle\int\frac{d\ell}{2\pi}\ell(C_{\ell}^{EE}-C_{\ell}^{BB})\Bigg{[}-J%
_{4}(\ell r)$$
$$\displaystyle+\left|\bar{h}^{(P)}_{\ell}\right|^{2}\exp\left[-\frac{\ell^{2}}{%
2}(C_{0}(0)-C_{0}(r))\right]\left(J_{4}(\ell r)+\frac{\ell^{2}}{4}C_{2}(r)(J_{%
2}(\ell r)+J_{6}(\ell r))\right)$$
$$\displaystyle+\left(h^{(P)}_{\ell}\bar{h}^{(P)}_{-\ell}+h^{(P)}_{-\ell}\bar{h}%
^{(P)}_{\ell}\right)\exp\left[-\frac{\ell^{2}}{2}\left((C_{0}(0)-C_{0}(r))-(C^%
{\rm cross}_{0}(0)-C^{\rm cross}_{0}(r))+\frac{1}{2}C^{\rm obs}_{0}(0)\right)\right]$$
$$\displaystyle\quad\times\left(J_{4}(\ell r)+\frac{\ell^{2}}{4}\left(C_{2}(r)-C%
^{\rm cross}_{2}(r)\right)(J_{2}(\ell r)+J_{6}(\ell r))\right)$$
$$\displaystyle+\left|h^{(P)}_{\ell}\right|^{2}\exp\left[-\frac{\ell^{2}}{2}%
\left((C_{0}(0)-C_{0}(r))-2(C^{\rm cross}_{0}(0)-C^{\rm cross}_{0}(r))+(C^{\rm
obs%
}_{0}(0)-C^{\rm obs}_{0}(r))\right)\right]$$
$$\displaystyle\quad\times\left(J_{4}(\ell r)+\frac{\ell^{2}}{4}\left(C_{2}(r)-2%
C^{\rm cross}_{2}(r)+C^{\rm obs}_{2}(r)\right)(J_{2}(\ell r)+J_{6}(\ell r))%
\right)\Bigg{]}\ ,$$
$$\displaystyle\Delta\xi_{X}^{\rm d}(r)$$
$$\displaystyle=$$
$$\displaystyle\int\frac{d\ell}{2\pi}\ell C_{\ell}^{TE}\Bigg{[}-J_{2}(\ell r)$$
$$\displaystyle+\bar{h}_{\ell}\bar{h}^{(P)}_{-\ell}\exp\left[-\frac{\ell^{2}}{2}%
(C_{0}(0)-C_{0}(r))\right]\left(J_{2}(\ell r)+\frac{\ell^{2}}{4}C_{2}(r)(J_{0}%
(\ell r)+J_{4}(\ell r))\right)$$
$$\displaystyle+\left(h_{\ell}\bar{h}^{(P)}_{-\ell}+h^{(P)}_{-\ell}\bar{h}_{\ell%
}\right)\exp\left[-\frac{\ell^{2}}{2}\left((C_{0}(0)-C_{0}(r))-(C^{\rm cross}_%
{0}(0)-C^{\rm cross}_{0}(r))+\frac{1}{2}C^{\rm obs}_{0}(0)\right)\right]$$
$$\displaystyle\quad\times\left(J_{2}(\ell r)+\frac{\ell^{2}}{4}\left(C_{2}(r)-C%
^{\rm cross}_{2}(r)\right)(J_{0}(\ell r)+J_{4}(\ell r))\right)$$
$$\displaystyle+h_{\ell}h^{(P)}_{-\ell}\exp\left[-\frac{\ell^{2}}{2}\left((C_{0}%
(0)-C_{0}(r))-2(C^{\rm cross}_{0}(0)-C^{\rm cross}_{0}(r))+(C^{\rm obs}_{0}(0)%
-C^{\rm obs}_{0}(r))\right)\right]$$
$$\displaystyle\quad\times\left(J_{2}(\ell r)+\frac{\ell^{2}}{4}\left(C_{2}(r)-2%
C^{\rm cross}_{2}(r)+C^{\rm obs}_{2}(r)\right)(J_{0}(\ell r)+J_{4}(\ell r))%
\right)\Bigg{]}\ .$$
The delensed polarization spectra are then
$$\displaystyle C_{\ell}^{{\rm d},EE}$$
$$\displaystyle=$$
$$\displaystyle C_{\ell}^{EE}+2\pi\int rdr\frac{1}{2}\left(J_{0}(\ell r)\Delta%
\xi_{+}^{\rm d}(r)+J_{4}(\ell r)\Delta\xi_{-}^{\rm d}(r)\right)\,,$$
(C.4)
$$\displaystyle C_{\ell}^{{\rm d},BB}$$
$$\displaystyle=$$
$$\displaystyle C_{\ell}^{BB}+2\pi\int rdr\frac{1}{2}\left(J_{0}(\ell r)\Delta%
\xi_{+}^{\rm d}(r)-J_{4}(\ell r)\Delta\xi_{-}^{\rm d}(r)\right)\,,$$
$$\displaystyle C_{\ell}^{{\rm d},TE}$$
$$\displaystyle=$$
$$\displaystyle C_{\ell}^{TE}+2\pi\int rdrJ_{2}(\ell r)\Delta\xi_{X}^{\rm d}(r)\,.$$
Appendix D Calculating the Covariance
In our forecasts, we use the approximate form of the covariance matrix
$$\displaystyle{\rm Cov}_{\ell\,\ell^{\prime}}^{XY,WZ}$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{2\ell+1}\left[C_{\ell}^{XW}C_{\ell}^{YZ}+C_{\ell}^{XZ}C_%
{\ell}^{YW}\right]\delta_{\ell\,\ell^{\prime}}+\sum_{L}\Bigg{[}\frac{\partial C%
_{\ell}^{XY}}{\partial C_{L}^{\phi\phi}}{\rm Cov}_{{L}\,{L}}^{\phi\phi,\phi%
\phi}\frac{\partial C_{\ell^{\prime}}^{WZ}}{\partial C_{L}^{\phi\phi}}\Bigg{]}\ ,$$
(D.1)
where $XY,WZ=TT,TE,EE$ and $C^{XY,WZ}_{\ell}$ are the delensed spectra. In order to evaluate the covariance, we therefore need to compute the derivatives $\frac{\partial C_{\ell}^{XY}}{\partial C_{L}^{\phi\phi}}$. As a warm up, consider this derivative acting on the lensed $TT$ power spectrum,
$$\tilde{C}_{\ell}^{TT}=\int d^{2}r\frac{d^{2}\ell_{1}}{(2\pi)^{2}}C_{\ell_{1}}e%
^{-i({\vec{\ell}\mskip 2.0mu \vphantom{\ell}}-{\vec{\ell}\mskip 2.0mu %
\vphantom{\ell}}_{1})\cdot\vec{r}\mskip 2.0mu \vphantom{r}}e^{-\frac{\ell^{2}_%
{1}}{2}(C_{0}(0)-C_{0}(r)+\cos 2\varphi_{1}C_{2}(r))}$$
(D.2)
such that
$$\frac{\partial\tilde{C}_{\ell}^{TT}}{\partial C^{\phi\phi}_{L}}=\int d^{2}r%
\frac{d^{2}\ell_{1}}{(2\pi)^{2}}C_{\ell_{1}}\frac{\ell^{2}_{1}{L}^{3}}{2}\bigg%
{(}J_{0}({L}r)-1-\cos 2\varphi_{1}J_{2}({L}r)\bigg{)}e^{-i({\vec{\ell}\mskip 2%
.0mu \vphantom{\ell}}-{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}_{1})\cdot\vec{r%
}\mskip 2.0mu \vphantom{r}}e^{-\frac{\ell^{2}_{1}}{2}(C_{0}(0)-C_{0}(r)+\cos 2%
\varphi_{1}C_{2}(r))}\ .$$
(D.3)
Now we can the same procedure to the delensed spectra, holding the filters fixed. We find
$$\displaystyle\frac{\partial C_{\ell}^{{\rm d},TT}}{\partial C^{\phi\phi}_{L}}$$
$$\displaystyle=$$
$$\displaystyle|\bar{h}_{\ell}|^{2}\frac{\partial\tilde{C}_{\ell}^{TT}}{\partial
C%
^{\phi\phi}_{L}}+(h_{\ell}\bar{h}_{-\ell}+h_{-\ell}\bar{h}_{\ell})(-\tfrac{1}{%
4}g_{L}^{2}\ell^{2}{L}^{3})e^{-\tfrac{1}{4}\ell^{2}C^{\rm obs}_{0}(0)}C_{\ell}%
^{\rm N}$$
$$\displaystyle+\int d^{2}r\frac{d^{2}\ell_{1}}{(2\pi)^{2}}|h_{\ell_{1}}|^{2}C_{%
\ell_{1}}^{\rm N}e^{-i({\vec{\ell}\mskip 2.0mu \vphantom{\ell}}-{\vec{\ell}%
\mskip 2.0mu \vphantom{\ell}}_{1})\cdot\vec{r}\mskip 2.0mu \vphantom{r}}e^{-%
\frac{\ell_{1}^{2}}{2}(C^{\rm obs}_{0}(0)-C^{\rm obs}_{0}(r)+\cos 2\varphi_{1}%
C^{\rm obs}_{2}(r))}$$
$$\displaystyle\qquad\qquad\times g_{L}^{2}\frac{\ell^{2}_{1}{L}^{3}}{2}\bigg{(}%
J_{0}({L}r)-1-\cos 2\varphi_{1}J_{2}({L}r)\bigg{)}$$
$$\displaystyle+\int d^{2}r\frac{d^{2}\ell_{1}}{(2\pi)^{2}}C_{\ell_{1}}e^{-i({%
\vec{\ell}\mskip 2.0mu \vphantom{\ell}}-{\vec{\ell}\mskip 2.0mu \vphantom{\ell%
}}_{1})\cdot\vec{r}\mskip 2.0mu \vphantom{r}}e^{-\frac{\ell^{2}_{1}}{2}(C_{0}(%
0)-C_{0}(r)+\cos 2\varphi_{1}C_{2}(r))}$$
$$\displaystyle\qquad\times\frac{\ell^{2}_{1}{L}^{3}}{2}\bigg{[}\Big{(}J_{0}({L}%
r)-1-\cos 2\varphi_{1}J_{2}({L}r)\Big{)}(1-g_{L})-\frac{1}{2}\bigg{]}$$
$$\displaystyle\times\Bigg{(}(h_{\ell_{1}}\bar{h}_{-\ell_{1}}+h_{-\ell_{1}}\bar{%
h}_{\ell_{1}})e^{-\frac{\ell_{1}^{2}}{4}C^{\rm obs}_{0}(0)+\frac{\ell_{1}^{2}}%
{2}(C^{\rm cross}_{0}(0)-C^{\rm cross}_{0}(r))+\tfrac{\ell_{1}^{2}}{2}\cos 2%
\varphi_{1}C^{\rm cross}_{2}(r))}\Bigg{)}$$
$$\displaystyle+\int d^{2}r\frac{d^{2}\ell_{1}}{(2\pi)^{2}}C_{\ell_{1}}e^{-i({%
\vec{\ell}\mskip 2.0mu \vphantom{\ell}}-{\vec{\ell}\mskip 2.0mu \vphantom{\ell%
}}_{1})\cdot\vec{r}\mskip 2.0mu \vphantom{r}}e^{-\frac{\ell^{2}_{1}}{2}(C_{0}(%
0)-C_{0}(r)+\cos 2\varphi_{1}C_{2}(r))}$$
$$\displaystyle\qquad\times\frac{\ell^{2}_{1}{L}^{3}}{2}\bigg{[}\Big{(}J_{0}({L}%
r)-1-\cos 2\varphi_{1}J_{2}({L}r)\Big{)}(1-g_{L})^{2}\bigg{]}$$
$$\displaystyle\times\Bigg{(}|h_{\ell_{1}}|^{2}e^{-\frac{\ell_{1}^{2}}{2}(C^{\rm
obs%
}_{0}(0)-C^{\rm obs}_{0}(r)+\cos 2\varphi_{1}C^{\rm obs}_{2}(r))+\ell_{1}^{2}(%
C^{\rm cross}_{0}(0)-C^{\rm cross}_{0}(r))+\ell_{1}^{2}\cos 2\varphi_{1}C^{\rm
cross%
}_{2}(r)}\Bigg{)}$$
In practice, we will always consider the case where the spectra involved are cross-correlations of the from $X=W=T$ and $Y=Z=T^{\prime}$ such that $C^{{\rm N},TT^{\prime}}_{\ell}=0$. Since the off-diagonal terms in Equation (D.1) will involve only these cross-correlations, we may drop the noise terms in Equation (D) and similarly for covariances including the E-modes.
Appendix E Fisher Information and Delensing
From some perspectives, it is not clear that errors bars should improve by delensing the temperature and/or polarization. Cosmological parameters have an impact on lensing and one might worry that delensing could remove this information. Alternatively, one might argue that all of the cosmological information is in the lensing power spectrum and this information is available whether or not we delens, as long as we include the lensing likelihood, and there should be no improvement by delensing. Neither of these arguments would suggest that delensing is adding information. The purpose of this appendix is to address these concerns and show that delensing should always add Fisher Information and reduce error bars (or at least leave them unchanged).
We will assume that $\phi$ is a Gaussian random field, as we did in the main text. As a result, all information about cosmological parameters encoded in $\phi$ is determined by $C^{\phi\phi}_{L}$. Although it does not carry cosmological information itself, this does not imply that the the specific realization of $\phi$ cannot impact the measurement of cosmological parameters. The realization of $\phi$ lenses the CMB in a way that changes the sensitivity of the lensed spectra to cosmological parameters666This is most obvious in the case of $r$, where lensing $B$ modes act as a foreground. For sufficiently low $r$, without delensing with the realization of $\phi$, we could not distinguish $r$ from cosmic variance of the lensing potential. . Measuring $\phi$ and removing its effects from the CMB maps can increase the Fisher information by restoring information that was originally in the unlensed spectra.
We will first demonstrate that our delensing procedure produces a local extremum of the Fisher information. The easiest way to see this is to note that $\bar{h}^{(P)}=\bar{h}=1$ and $h^{(P)}=h=0$ is equivalent to not delensing. Therefore, any statement about Fisher information with or without delensing is equivalent to a statement about the choice of filters. The Fisher matrix for a set of cosmological parameters $\lambda_{i}$ is given by
$$\displaystyle F_{ij}=\sum_{X,Y,W,Z}\frac{\partial C_{\ell}^{{\rm d},XY}}{%
\partial\lambda_{i}}{\rm Cov}^{-1}_{XY,WZ;\ell,\ell^{\prime}}\frac{\partial C_%
{\ell^{\prime}}^{{\rm d},WZ}}{\partial\lambda_{j}}$$
(E.1)
where $X,Y,W,Z\supset\{T,E,B,\phi\}$ and $C^{{\rm d},\phi\phi}_{\ell}\equiv C^{\phi\phi}_{L}$. We will assume the covariance matrix is given in terms of $C^{{\rm d},XY}_{\ell}$ as explained in Appendix D. We can determine if delensing will improve the constraint on a given a cosmological parameter by computing
$$\frac{\partial}{\partial{\bf h}}F_{ii}|_{{\bf h}=0}\approx\sum_{X,Y,W,Z}2\left%
(\left.\frac{\partial^{2}C_{\ell}^{{\rm d},XY}}{\partial\lambda_{i}\partial{%
\bf h}}\right|_{{\bf h}=0}\right)\,{\rm Cov}^{-1}_{XY,WZ;\ell,\ell^{\prime}}%
\frac{\partial C_{\ell^{\prime}}^{{\rm d},WZ}}{\partial\lambda_{i}}$$
(E.2)
where ${\bf h}=\{h_{\ell},h^{(P)}_{\ell}\}$. We have assumed that the change to the Fisher information in $\lambda_{i}$ is dominated by the change to $C_{\ell}^{XY}$ rather than to the covariance matrix. Now we compute
$$\left.\frac{\partial^{2}C_{\ell}^{{\rm d},XY}}{\partial\lambda_{i}\partial{\bf
h%
}}\right|_{{\bf h}=0}=\frac{\partial}{\partial\lambda_{i}}\Big{(}\left\langle X%
_{{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}}^{\rm obs}Y^{\rm obs}(\vec{x}\mskip
2%
.0mu \vphantom{x}-g\star\vec{\alpha}\,{}^{\rm obs}(\vec{x}\mskip 2.0mu %
\vphantom{x}))_{-{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}}\right\rangle^{%
\prime}-e^{-\frac{\ell^{2}}{4}C^{\rm obs}_{0}(0)}C_{\ell}^{XY,{\rm obs}}\Big{)%
}+\{X\leftrightarrow Y\}$$
(E.3)
where $X,Y\supset\{T,E,B\}$. Note that $e^{-\frac{\ell^{2}}{4}C^{\rm obs}_{0}(0)}C_{\ell}^{{\rm d},XY}$ is the change to $C^{XY}_{\ell}$ by the average effect of lensing on $X$- and $Y$-maps. The term in brackets is therefore the effect of removing the realization of $\phi^{\rm obs}$ from the maps (i.e. the total minus the average). It is easy to see that this does not vanish and will depend on the unlensed spectra. Therefore, any cosmological parameter that affects the unlensed spectra should have $\frac{\partial}{\partial{\bf h}}F_{ii}|_{{\bf h}=0}\neq 0$. As a result, we are guaranteed that delensing changes the Fisher information for some of the cosmological parameters (since $h$ can take either sign, we can always increase the Fisher information and therefore decrease the error bars).
We also have good reason to think that our filters are nearly maximizing the Fisher information. We can see that Equation E.1 is a function only of the delensed spectra. As a result $\partial_{\bf h}F_{ij}\propto\partial_{\bf h}C^{\rm{d},XY}$. One can check that our optimal filters, defined by minimizing Equation B.7, satisfy
$$\partial_{\bf h}C^{\rm{d},XX}=0\to\partial_{\bf h}F_{ij}\approx 0\ .$$
(E.4)
In the last step we have assumed that the diagonal terms in the delensed covariance play the dominant role in the Fisher information. Under these circumstances, optimizing our filters is roughly the same as extremizing the Fisher information. This is confirmed through our forecasts in Section 4 where we see that for parameters like $N_{\rm eff}$, $Y_{p}$, and $\theta_{s}$ that affect the primary CMB, the off-diagonal covariances have essentially no effect on the forecasts.
The question that remains is whether the local extremum is a minimum or a maximum. For parameters that affect only the unlensed spectra, we should maximize information with perfect delensing and therefore this extremum should be a maximum even for imperfect delensing. On the other hand, it is less obvious that this is a maximum for cosmological parameters that affect $C_{L}^{\phi\phi}$ without introducing large effects in the unlensed CMB. In such cases, it would seem surprising that delensing increases the Fisher information as we are removing information about the lensing potential from the spectra. The intuitive reason that delensing does not remove information is that the information that allows us to delens also gives a direct measurement of $C^{\phi\phi}_{L}$. For a Gaussian random field, the power spectrum should contain all of the cosmological information encoded in $\phi$. Therefore, as long as $C^{\phi\phi,{\rm obs}}_{L}$ is included in the likelihood, we should not gain or lose information encoded in $C_{L}^{\phi\phi}$ by delensing the other spectra.
To see this this another way, there is also nothing that forbids us form increasing the amount of lensing by changing the sign of $g_{L}$. Therefore, if we lost information by delensing, then we should increase information by increasing the amount of lensing in the spectra. Any such procedure is just some operation performed on a Gaussian random field, $f(\alpha^{\rm obs})$. As long as we are only interested information that is contained directly in $C^{\phi\phi,{\rm obs}}_{L}$, then adding $f(\alpha^{\rm obs})$ is just repeating the same information and should not be double counted. Of course, it is the covariance matrix that should correct for this.
To see the role of the covariance for delensing, we will assume that we can measure $\vec{\alpha}$ without noise. The first case we consider is where $C^{\phi\phi}_{L}$ is not included in our likelihood and we include only the delensed $T,E$, and $B$. When delensing can be performed perfectly, we have
$$\left.\frac{\partial^{2}C_{\ell}^{{\rm d},XY}}{\partial\lambda_{i}\partial{\bf
h%
}}\right|_{{\bf h}=0}=\frac{\partial}{\partial\lambda_{i}}\Big{(}e^{-\frac{%
\ell^{2}}{4}C_{0}(0)}C_{\ell}^{XY}-e^{-\frac{\ell^{2}}{4}C_{0}(0)}\tilde{C}_{%
\ell}^{XY}\Big{)}+\{X\leftrightarrow Y\}\ .$$
(E.5)
Since $C_{\ell}-\tilde{C}_{\ell}\sim{\cal O}(C^{\phi\phi}_{L})$, we can ignore the derivatives that act on the exponents (to first approximation) and therefore if $\partial_{\lambda_{i}}C^{XY}_{\ell}=0$ we have
$$\displaystyle\frac{\partial}{\partial{\bf h}}F_{ii}|_{{\bf h}=0}$$
$$\displaystyle\approx$$
$$\displaystyle\sum_{X,Y,W,Z}-2e^{-\frac{\ell^{2}}{4}C_{0}(0)}\left(\left.\frac{%
\partial\tilde{C}_{\ell}^{XY}}{\partial\lambda_{i}\partial{\bf h}}\right|_{{%
\bf h}=0}\right)\,{\rm Cov}^{-1}_{XY,WZ;\ell,\ell^{\prime}}\frac{\partial%
\tilde{C}_{\ell^{\prime}}^{WZ}}{\partial\lambda_{i}}$$
(E.6)
$$\displaystyle\approx$$
$$\displaystyle-2e^{-\frac{\ell^{2}}{4}C_{0}(0)}F_{ii}<0$$
This is intuitively clear, we are removing the information about $C^{\phi\phi}_{L}$ and therefore the information decreases.
To see that no information is lost when we include the measurement of $C^{\phi\phi}_{L}$ we must account for the off-diagonal contributions to the covariance matrix. Inverting the covariance matrix is of course very numerically challenging in practice, but we can schematically understand the effect as
$${\rm Cov}^{-1}_{XX,\phi\phi}\approx-{\rm Cov}_{XX,XX}^{-1}\frac{\partial C^{{%
\rm d},XX}}{\partial C^{\phi\phi}}$$
(E.7)
which is inspired by Equation 3.2 and inverted as a $2\times 2$ matrix assuming we are looking at a single additional spectrum $C^{{\rm d},XX}$. We also assumed the off-diagonal covariances are small compared to the diagonal terms. The final step is to notice that if the only dependence on $\lambda_{i}$ is through lensing, then
$$\frac{\partial C^{{\rm d},XY}}{\partial\lambda_{i}}=\frac{\partial C^{{\rm d},%
XY}}{\partial C^{\phi\phi}}\frac{\partial C^{\phi\phi}}{\partial\lambda_{i}}$$
(E.8)
Now combining the off-diagonal and diagonal contributions,
$$\displaystyle F_{ii}$$
$$\displaystyle\approx$$
$$\displaystyle\frac{\partial C^{\phi\phi}}{\partial\lambda_{i}}{\rm Cov}_{\phi%
\phi,\phi\phi}^{-1}\frac{\partial C^{\phi\phi}}{\partial\lambda_{i}}+\sum_{X}%
\frac{\partial C^{\phi\phi}}{\partial\lambda_{i}}\frac{\partial C^{XX}}{%
\partial C^{\phi\phi}}{\rm Cov}_{XX,XX}^{-1}\frac{\partial C^{XX}}{\partial C^%
{\phi\phi}}\frac{\partial C^{\phi\phi}}{\partial\lambda_{i}}$$
(E.9)
$$\displaystyle\qquad\qquad-\frac{\partial C^{\phi\phi}}{\partial\lambda_{i}}%
\frac{\partial C^{XX}}{\partial C^{\phi\phi}}{\rm Cov}_{XX,XX}^{-1}\frac{%
\partial C^{XX}}{\partial C^{\phi\phi}}\frac{\partial C^{\phi\phi}}{\partial%
\lambda_{i}}$$
$$\displaystyle\approx$$
$$\displaystyle\frac{\partial C^{\phi\phi}}{\partial\lambda_{i}}{\rm Cov}_{\phi%
\phi,\phi\phi}^{-1}\frac{\partial C^{\phi\phi}}{\partial\lambda_{i}}$$
This was a vast oversimplification of the problem, but we see that schematically, the purpose of the off-diagonal terms in the covariance is to remove the information that is already included in $C^{\phi\phi}_{\ell}$, which is all of the non-trivial information in this case. Therefore, we only see a decrease in the information when we delens if we neglect to include the observed lensing power spectrum.
The additional complication of real data (beyond inverting large matrices) is that we do not measure $\phi$ perfectly, and there can be residual information about $C^{\phi\phi}_{L}$ left in the delensed spectrum. For example, suppose we use a sub-optimal measurement of $\phi$ and therefore both the delensing procedure and $C^{\phi\phi,{\rm obs}}_{L}$ miss important information that is in the lensed spectra. This information is still encoded in the temperature and polarization spectra, but delensing using $\phi^{\rm obs}$ may decrease the Fisher information by making it more difficult to extract this information. Furthermore, if we did not filter the noisy modes in $\phi$, then we could imagine the induced error in the delensed spectra could dominate over the error in the observed spectra.
In practice, most information about $C^{\phi\phi}_{L}$ can be determined through known procedures for extracting $\vec{\alpha}\,{}^{\rm obs}$. We expect that any information about lensing that is encoded in the map should allow for a reconstruction of $\vec{\alpha}$ with low noise. Therefore, while it is possible for the Fisher information to decrease with delensing, we expect this to occur only when sub-optimal methods are used for reconstruction and/or filtering.
Appendix F Exact Delensing and Real Data
Given an observed temperature map, $T^{\rm obs}(\vec{x}\mskip 2.0mu \vphantom{x})$, and an observed lensing map $\vec{\alpha}\,{}^{\rm obs}$, an alternate approach to delensing is to define [44]
$$T^{\rm d}(\vec{x}\mskip 2.0mu \vphantom{x})=\bar{h}\star T^{\rm obs}(\vec{x}%
\mskip 2.0mu \vphantom{x})+\int d^{2}x^{\prime}J^{\rm obs}(\vec{x}\mskip 2.0mu%
\vphantom{x}^{\prime})\delta(\vec{x}\mskip 2.0mu \vphantom{x}-g\star\vec{%
\alpha}\,{}^{\rm obs}(\vec{x}\mskip 2.0mu \vphantom{x}^{\prime})-\vec{x}\mskip
2%
.0mu \vphantom{x}^{\prime})h\star T^{\rm obs}(\vec{x}\mskip 2.0mu \vphantom{x}%
^{\prime})\ ,$$
(F.1)
where $J^{\rm obs}=\det\partial_{i}(x_{j}+g\star\alpha^{\rm obs}_{j})$. As discussed in Section 2, in the absence of noise with $\bar{h}=0$ and $h_{\ell}=g_{\ell}=1$, this choice of delensing has $T^{\rm d}(\vec{x}\mskip 2.0mu \vphantom{x})=T(\vec{x}\mskip 2.0mu \vphantom{x})$ even including gradients.
Assuming the noise is uncorrelated with the lensing noise and the signal, we can formally write the delensed $C_{\ell}$ as
$$\displaystyle\langle T^{\rm d}_{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}T^{\rm d%
}_{\vec{\ell}\mskip 2.0mu \vphantom{\ell}^{\prime}}\rangle$$
$$\displaystyle=$$
$$\displaystyle(2\pi)^{2}\delta({\vec{\ell}\mskip 2.0mu \vphantom{\ell}}+\vec{%
\ell}\mskip 2.0mu \vphantom{\ell}^{\prime})|\bar{h}_{\ell}|^{2}C_{\ell}^{\rm
obs%
}+\Bigg{[}\int d^{2}x^{\prime}\frac{d^{2}x_{1}d^{2}\ell_{1}d^{2}x_{2}}{(2\pi)^%
{2}}e^{i{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}_{1}\cdot(\vec{x}\mskip 2.0mu %
\vphantom{x}^{\prime}-\vec{x}\mskip 2.0mu \vphantom{x}_{1})}e^{-i{\vec{\ell}%
\mskip 2.0mu \vphantom{\ell}}\cdot\vec{x}\mskip 2.0mu \vphantom{x}^{\prime}}e^%
{-i\vec{\ell}\mskip 2.0mu \vphantom{\ell}^{\prime}\cdot\vec{x}\mskip 2.0mu %
\vphantom{x}_{2}}\times$$
(F.2)
$$\displaystyle\qquad\qquad\bigg{(}\bar{h}_{\ell^{\prime}}h_{\ell_{1}}\left%
\langle J^{\rm obs}(\vec{x}\mskip 2.0mu \vphantom{x}^{\prime})e^{-i{\vec{\ell}%
\mskip 2.0mu \vphantom{\ell}}\cdot g\star\vec{\alpha}\,{}^{\rm obs}(\vec{x}%
\mskip 2.0mu \vphantom{x}^{\prime})}T^{\rm obs}(\vec{x}\mskip 2.0mu \vphantom{%
x}_{1})T^{\rm obs}(\vec{x}\mskip 2.0mu \vphantom{x}_{2})\right\rangle\bigg{)}+%
\{{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}\leftrightarrow\vec{\ell}\mskip 2.0%
mu \vphantom{\ell}^{\prime}\}\Bigg{]}$$
$$\displaystyle+\int d^{2}x^{\prime}d^{2}x^{\prime\prime}\frac{d^{2}x_{1}d^{2}%
\ell_{1}d^{2}x_{2}d^{2}\ell_{2}}{(2\pi)^{4}}e^{-i{\vec{\ell}\mskip 2.0mu %
\vphantom{\ell}}\cdot\vec{x}\mskip 2.0mu \vphantom{x}^{\prime}-i\vec{\ell}%
\mskip 2.0mu \vphantom{\ell}^{\prime}\cdot\vec{x}\mskip 2.0mu \vphantom{x}^{%
\prime\prime}}e^{i{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}_{1}\cdot(\vec{x}%
\mskip 2.0mu \vphantom{x}^{\prime}-\vec{x}\mskip 2.0mu \vphantom{x}_{1})}h_{%
\ell_{1}}h_{\ell_{2}}\times$$
$$\displaystyle\qquad\qquad\left\langle J^{\rm obs}(\vec{x}\mskip 2.0mu %
\vphantom{x}^{\prime})J^{\rm obs}(\vec{x}\mskip 2.0mu \vphantom{x}^{\prime%
\prime})e^{-i{\vec{\ell}\mskip 2.0mu \vphantom{\ell}}\cdot g\star\vec{\alpha}%
\,{}^{\rm obs}(\vec{x}\mskip 2.0mu \vphantom{x}^{\prime})-i\vec{\ell}\mskip 2.%
0mu \vphantom{\ell}^{\prime}\cdot g\star\vec{\alpha}\,{}^{\rm obs}(\vec{x}%
\mskip 2.0mu \vphantom{x}^{\prime\prime})}T^{\rm obs}(\vec{x}\mskip 2.0mu %
\vphantom{x}_{1})T^{\rm obs}(\vec{x}\mskip 2.0mu \vphantom{x}_{2})\right%
\rangle\ .$$
This result can be used as the starting point for a variety of calculations. In principle, this defines the all-orders result, if one can evaluate all of the correlation functions exactly and perform the integrals. One may also use this as a starting point where one can systematically include small effects as a perturbative expansion.
There are two main technical challenges of this approach: (1) computing $J^{\rm obs}(\vec{x}\mskip 2.0mu \vphantom{x})$ with real data and (2) evaluating the integral in Equation F.1. As we saw in Appendix A, including $J^{\rm obs}(\vec{x}\mskip 2.0mu \vphantom{x})$ is necessary for controlling the effects of small scale gradients. The advantage of our approximate approach in Equation 2.5 is that we avoid both complications by simply moving the points in real space.
References
[1]
A. Blanchard and J. Schneider, “Gravitational lensing effect on the
fluctuations of the cosmic background radiation,” A&A 184 (Oct., 1987) 1–6.
[2]
F. Bernardeau, “Weak lensing detection in CMB maps,” Astron.
Astrophys. 324 (1997) 15–26,
arXiv:astro-ph/9611012
[astro-ph].
[3]
M. Zaldarriaga and U. Seljak, “Reconstructing projected matter density from
cosmic microwave background,”
Phys. Rev.
D59 (1999) 123507,
arXiv:astro-ph/9810257
[astro-ph].
[4]
A. Lewis and A. Challinor, “Weak gravitational lensing of the cmb,”
Phys. Rept.
429 (2006) 1–65,
arXiv:astro-ph/0601594
[astro-ph].
[5]
E. Calabrese, A. Slosar, A. Melchiorri, G. F. Smoot, and O. Zahn, “Cosmic
Microwave Weak lensing data as a test for the dark universe,”
Phys. Rev.
D77 (2008) 123531,
arXiv:0803.2309 [astro-ph].
[6]
C. L. Reichardt et al., “High resolution CMB power spectrum from the
complete ACBAR data set,”
Astrophys. J.
694 (2009) 1200–1219,
arXiv:0801.1491 [astro-ph].
[7]
R. Keisler et al., “A Measurement of the Damping Tail of the Cosmic
Microwave Background Power Spectrum with the South Pole Telescope,”
Astrophys. J.
743 (2011) 28,
arXiv:1105.3182
[astro-ph.CO].
[8]
Atacama Cosmology Telescope Collaboration, J. L. Sievers et al., “The Atacama Cosmology Telescope: Cosmological parameters from
three seasons of data,”
JCAP 1310 (2013) 060,
arXiv:1301.0824
[astro-ph.CO].
[9]
K. T. Story et al., “A Measurement of the Cosmic Microwave Background
Damping Tail from the 2500-square-degree SPT-SZ survey,”
Astrophys. J.
779 (2013) 86,
arXiv:1210.7231
[astro-ph.CO].
[10]
Planck Collaboration, P. A. R. Ade et al., “Planck 2013
results. XVI. Cosmological parameters,”
Astron. Astrophys.
571 (2014) A16,
arXiv:1303.5076
[astro-ph.CO].
[11]
Planck Collaboration, P. A. R. Ade et al., “Planck 2015
results. XIII. Cosmological parameters,”
Astron. Astrophys.
594 (2016) A13,
arXiv:1502.01589
[astro-ph.CO].
[12]
K. M. Smith, O. Zahn, and O. Dore, “Detection of Gravitational Lensing in the
Cosmic Microwave Background,”
Phys. Rev.
D76 (2007) 043510,
arXiv:0705.3980 [astro-ph].
[13]
C. M. Hirata, S. Ho, N. Padmanabhan, U. Seljak, and N. A. Bahcall,
“Correlation of CMB with large-scale structure: II. Weak lensing,”
Phys. Rev.
D78 (2008) 043520,
arXiv:0801.0644 [astro-ph].
[14]
S. Das et al., “Detection of the Power Spectrum of Cosmic Microwave
Background Lensing by the Atacama Cosmology Telescope,”
Phys. Rev.
Lett. 107 (2011) 021301,
arXiv:1103.2124
[astro-ph.CO].
[15]
A. van Engelen et al., “A measurement of gravitational lensing of the
microwave background using South Pole Telescope data,”
Astrophys. J.
756 (2012) 142,
arXiv:1202.0546
[astro-ph.CO].
[16]
POLARBEAR Collaboration, P. A. R. Ade et al., “Measurement
of the Cosmic Microwave Background Polarization Lensing Power Spectrum with
the POLARBEAR experiment,”
Phys. Rev.
Lett. 113 (2014) 021301,
arXiv:1312.6646
[astro-ph.CO].
[17]
SPT Collaboration, K. T. Story et al., “A Measurement of
the Cosmic Microwave Background Gravitational Lensing Potential from 100
Square Degrees of SPTpol Data,”
Astrophys. J.
810 no. 1, (2015) 50,
arXiv:1412.4760
[astro-ph.CO].
[18]
Planck Collaboration, P. A. R. Ade et al., “Planck 2015
results. XV. Gravitational lensing,”
Astron. Astrophys.
594 (2016) A15,
arXiv:1502.01591
[astro-ph.CO].
[19]
M. Kaplinghat, L. Knox, and Y.-S. Song, “Determining neutrino mass from the
CMB alone,” Phys. Rev. Lett. 91 (2003) 241301,
arXiv:astro-ph/0303344
[astro-ph].
[20]
L. Knox and Y.-S. Song, “A Limit on the detectability of the energy scale of
inflation,” Phys. Rev. Lett. 89 (2002) 011303,
arXiv:astro-ph/0202286
[astro-ph].
[21]
M. Kesden, A. Cooray, and M. Kamionkowski, “Separation of gravitational wave
and cosmic shear contributions to cosmic microwave background
polarization,” Phys. Rev. Lett. 89 (2002) 011304,
arXiv:astro-ph/0202434
[astro-ph].
[22]
U. Seljak and C. M. Hirata, “Gravitational lensing as a contaminant of the
gravity wave signal in CMB,”
Phys. Rev.
D69 (2004) 043005,
arXiv:astro-ph/0310163
[astro-ph].
[23]
K. M. Smith, D. Hanson, M. LoVerde, C. M. Hirata, and O. Zahn,
“Delensing CMB polarization with external datasets,”
J. Cosmology Astropart. Phys 6 (June, 2012) 014, arXiv:1010.0048.
[24]
B. D. Sherwin and M. Schmittfull, “Delensing the CMB with the Cosmic Infrared
Background,” Phys.
Rev. D92 no. 4, (2015) 043005,
arXiv:1502.05356
[astro-ph.CO].
[25]
P. Larsen, A. Challinor, B. D. Sherwin, and D. Mak, “A first demonstration of
CIB delensing,”
arXiv:1607.05733
[astro-ph.CO].
[26]
S. Bashinsky and U. Seljak, “Neutrino perturbations in CMB anisotropy and
matter clustering,”
Phys. Rev.
D69 (2004) 083002,
arXiv:astro-ph/0310198
[astro-ph].
[27]
B. Follin, L. Knox, M. Millea, and Z. Pan, “First Detection of the Acoustic
Oscillation Phase Shift Expected from the Cosmic Neutrino Background,”
Phys. Rev.
Lett. 115 no. 9, (2015) 091301,
arXiv:1503.07863
[astro-ph.CO].
[28]
D. Baumann, D. Green, J. Meyers, and B. Wallisch, “Phases of New Physics in
the CMB,” JCAP
1601 (2016) 007,
arXiv:1508.06342
[astro-ph.CO].
[29]
U. Seljak, “Gravitational lensing effect on cosmic microwave background
anisotropies: A Power spectrum approach,”
Astrophys. J. 463
(1996) 1,
arXiv:astro-ph/9505109
[astro-ph].
[30]
M. Zaldarriaga and U. Seljak, “Gravitational lensing effect on cosmic
microwave background polarization,”
Phys. Rev.
D58 (1998) 023003,
arXiv:astro-ph/9803150
[astro-ph].
[31]
D. J. Eisenstein, H.-j. Seo, E. Sirko, and D. Spergel, “Improving
Cosmological Distance Measurements by Reconstruction of the Baryon Acoustic
Peak,” Astrophys. J.
664 (2007) 675–679,
arXiv:astro-ph/0604362
[astro-ph].
[32]
C. Brust, D. E. Kaplan, and M. T. Walters, “New Light Species and the CMB,”
JHEP 12
(2013) 058,
arXiv:1303.5379 [hep-ph].
[33]
A. Salvio, A. Strumia, and W. Xue, “Thermal axion production,”
JCAP 1401 (2014) 011,
arXiv:1310.6982 [hep-ph].
[34]
M. Kawasaki, M. Yamada, and T. T. Yanagida, “Observable dark radiation from a
cosmologically safe QCD axion,”
Phys. Rev.
D91 no. 12, (2015) 125018,
arXiv:1504.04126 [hep-ph].
[35]
Z. Chacko, Y. Cui, S. Hong, and T. Okui, “Hidden dark matter sector, dark
radiation, and the CMB,”
Phys. Rev.
D92 (2015) 055033,
arXiv:1505.04192 [hep-ph].
[36]
P. Adshead, Y. Cui, and J. Shelton, “Chilly Dark Sectors and Asymmetric
Reheating,” JHEP
06 (2016) 016,
arXiv:1604.02458 [hep-ph].
[37]
D. Baumann, D. Green, and B. Wallisch, “A New Target for Cosmic Axion
Searches,”
arXiv:1604.08614
[astro-ph.CO].
[38]
A. Benoit-Levy, K. M. Smith, and W. Hu, “Non-Gaussian structure of the lensed
CMB power spectra covariance matrix,”
Phys. Rev.
D86 (2012) 123008,
arXiv:1205.0474
[astro-ph.CO].
[39]
M. M. Schmittfull, A. Challinor, D. Hanson, and A. Lewis, “Joint analysis of
CMB temperature and lensing-reconstruction power spectra,”
Phys. Rev.
D88 no. 6, (2013) 063012,
arXiv:1308.0286
[astro-ph.CO].
[40]
W. Hu and T. Okamoto, “Mass reconstruction with cmb polarization,”
Astrophys. J. 574
(2002) 566–574,
arXiv:astro-ph/0111606
[astro-ph].
[41]
C. M. Hirata and U. Seljak, “Analyzing weak lensing of the cosmic microwave
background using the likelihood function,”
Phys. Rev.
D67 (2003) 043001,
arXiv:astro-ph/0209489
[astro-ph].
[42]
C. M. Hirata and U. Seljak, “Reconstruction of lensing from the cosmic
microwave background polarization,”
Phys. Rev.
D68 (2003) 083002,
arXiv:astro-ph/0306354
[astro-ph].
[43]
E. Anderes, L. Knox, and A. van Engelen, “Mapping gravitational lensing of
the CMB using local likelihoods,”
Phys. Rev.
D83 (2011) 043523,
arXiv:1012.1833
[astro-ph.CO].
[44]
E. Anderes, B. Wandelt, and G. Lavaux, “Bayesian inference of CMB
gravitational lensing,”
Astrophys. J.
808 no. 2, (2015) 152,
arXiv:1412.4079
[astro-ph.CO].
[45]
C. Antolini, Y. Fantaye, M. Martinelli, C. Carbone, and C. Baccigalupi,
“N-body lensed CMB maps: lensing extraction and characterization,”
JCAP 1402 (2014) 039,
arXiv:1311.7112
[astro-ph.CO].
[46]
J. Liu, J. C. Hill, B. D. Sherwin, A. Petri, V. Böhm, and Z. Haiman, “CMB
Lensing Beyond the Power Spectrum: Cosmological Constraints from the
One-Point PDF and Peak Counts,”
arXiv:1608.03169
[astro-ph.CO].
[47]
V. Böhm, M. Schmittfull, and B. D. Sherwin, “A bias to CMB lensing
measurements from the bispectrum of large-scale structure,”
arXiv:1605.01392
[astro-ph.CO].
[48]
T. Namikawa, “CMB Lensing Bispectrum from Nonlinear Growth of the Large Scale
Structure,” Phys.
Rev. D93 no. 12, (2016) 121301,
arXiv:1604.08578
[astro-ph.CO].
[49]
A. Lewis and G. Pratten, “Effect of lensing non-Gaussianity on the CMB power
spectra,”
arXiv:1608.01263
[astro-ph.CO].
[50]
G. Pratten and A. Lewis, “Impact of post-Born lensing on the CMB,”
JCAP 1608 no. 08, (2016) 047,
arXiv:1605.05662
[astro-ph.CO].
[51]
A. Lewis, A. Challinor, and A. Lasenby, “Efficient Computation of
Cosmic Microwave Background Anisotropies in Closed Friedmann-Robertson-Walker
Models,” ApJ 538
(Aug., 2000) 473–476,
astro-ph/9911177.
[52]
A. Challinor and A. Lewis, “Lensed CMB power spectra from all-sky correlation
functions,” Phys.
Rev. D71 (2005) 103010,
arXiv:astro-ph/0502425
[astro-ph].
[53]
B. D. Sherwin and S. Das, “CMB Lensing - Power Without Bias,”
arXiv:1011.4510
[astro-ph.CO].
[54]
D. Hanson, A. Challinor, G. Efstathiou, and P. Bielewicz, “CMB temperature
lensing power reconstruction,”
Phys. Rev.
D83 (2011) 043005,
arXiv:1008.4403
[astro-ph.CO].
[55]
Z. Hou, R. Keisler, L. Knox, M. Millea, and C. Reichardt, “How Massless
Neutrinos Affect the Cosmic Microwave Background Damping Tail,”
Phys. Rev.
D87 (2013) 083008,
arXiv:1104.2333
[astro-ph.CO].
[56]
A. Lewis, A. Challinor, and D. Hanson, “The shape of the CMB lensing
bispectrum,” JCAP 1103 (2011) 018,
arXiv:1101.2234
[astro-ph.CO]. |
Self-biased current, magnetic interference response and superconducting vortices in tilted Weyl semimetals with disorder
Mohammad Alidoust
Department of Physics, K.N. Toosi University of Technology, Tehran 15875-4416, Iran
(December 2, 2020)
Abstract
We have generalized a quasiclassical model for Weyl semimetals with a tilted band in the presence of an externally applied magnetic field. This model is applicable to ballistic, moderately disordered, and samples containing a high density of nonmagnetic impurities. We employ this formalism and show that a self-biased supercurrent, creating a $\varphi_{0}$-junction, can flow through a triplet channel in Weyl semimetals. Furthermore, our results demonstrate that multiple supercurrent reversals are accessible through varying the junction thickness and parameters characterizing the Weyl semimetals. We discuss the influence of different parameters on the Fraunhofer response of charge supercurrent, and how these parameters are capable of shifting the locations of proximity-induced vortices in the triplet channel.
I introduction
The topological state of matter has been a striking topic during the past decade and attracted extensive attention
both theoretically and experimentally rev1 . The topological phases can host topologically protected intriguing phenomena and exotic particles, which offer promising prospects to practical arena such as topological
quantum computation rev1 . The research efforts in this context have so far been fruitful and led to the exploration of topological insulators ti1 ; ti2 and Weyl semimetals wyl1 ; wyl2 ; wyl3 ; wyl5 ; wyl6 ; wyl7 ; Xu1 , for instance. A topological insulator possesses insulating characteristics in its bulk material and shows perfect conducting features in its surface channels. Also, the band touching points of Weyl semimetals are the so-called Weyl nodes where the Fermi surface, encompassing the nodes, has a nonzero Chern number, thus topologically is nontrivial. The interplay of topological phase with superconductivity is expected to result in topological superconductivity, hosting Majorana fermions governed by non-Abeian statisticsramon ; shapiro ; Banerjee .
The conventional BCS superconductivity in Weyl semimetals can occur due to
the inter-valley couplings while the unconventional triplet
correlations may arise by the intra-valley pairings sc1_theor ; sc2_theor ; sc3_theor ; sc4_theor ; sc5_theor ; sc6_theor ; sc7_theor ; sc8_theor ; sc9_theor ; sc10_theor ; sc12_theor ; sc13_theor ; alidoustBP1 ; alidoustBP2 ; Gorbar ; Zhou11 . The
latter case, if energetically favorable, might create superconducting
correlations with finite momentum that places these correlations in the
Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase sc1_theor ; sc5_theor . The
FFLO phenomenon was first predicted for conventional BCS superconductors near their
critical magnetic field where the BCS superconductivity is suppressed
and the amplitude of singlet Cooper pairs is highly
oscillatoryfflo . Considering the topological phase transition, which
is inherent to the bulk material of ballistic Weyl semimetals, the existence of
superconductivity in such materials can provide a unique platform to
reveal the interplay of superconductivity and topology sc1_exp ; sc2_exp ; sc3_exp ; vol1 ; vol2 that has both theory and experiment attentionSirohi ; Soluyanov_app ; Ruan_app ; Zhang ; Zhang2 ; Chen ; Xu ; Fang ; alidoustBP1 ; alidoustBP2 ; Bovenzi ; Kononov ; Kononov2 ; Bachmann_app ; Xiao_app ; shapiro2 ; Hou_app ; Li ; Das ; wylsuperc_exp1 ; Teknowijoyo ; Lu ; Guguchia ; Takahashi ; QiaoLi . In particular, quite recent experimental progresses have observed enhancements
in the critical temperature of superconducting $\rm MoTe_{2}$, from
$0.1$ K to $8.2$ K, under pressures of order of $11.0$ GPa or from $0.1$ K to
$1.3$ K through partially substituting the tellurium ions by sulfur
sc1_exp ; sc2_exp . This enhancement is attributed to the
interplay of topology and superconductivity sc1_exp ; sc2_exp . Nonetheless, this enhancement might be due to the emergence of type-II Weyl semimetal phase in which the transition from type-I to type-II phase increases the available density of states near the Weyl nodes as recently explored in theory MA_type2 ; shapiro ; volovik . This transition can be achieved by tensile stress or dopping MA_type2 ; shapiro ; volovik .
Experimentally, the presence of disorder and nonmagnetic impurities in majority of samples is inevitable and may highly influence data analyses of physical quantities sensitive to them. A prominent example is the surface of topological insulators that are expected to be ballistic, showing conductance values equal to those of theory predicts. However, experimental measurement of the conductance of these surface channels was inconsistent with theory predictions. This seemingly discrepancy was resolved through magnetic scanning methods and further conductance spectroscopy analyses. It was demonstrated that disorder and impurities in these surface channels are practically unavoidable and highly alter its conductanceexp_ti4c . To properly model realistic surface channels of topological insulators with different densities of nonmagnetic impurities, a quasiclassical approach was recently generalized in the presence of superconductivity and arbitrary magnetization patterns, addressing both equilibrium and non-equilibrium states zu1 ; zu2 ; zu3 . Likewise, the focus of literature has so far been ideal systems and less attention paid to disordered Weyl semimetals in the presence of superconductivity. In this paper, we develop a quasiclassical model for Weyl semimetals with the inclusion of a tilting parameter MA_type2 and derive Eilenberger and Usadel equations for superconducting Weyl semimetals subject to an externally exerted magnetic field. The Eilenberger equation supports ballistic systems and samples with a moderate density of impurities while the Usadel model covers samples with a high density of impurities and disorder that make the motion of quasiparticles diffusive. As a practical application of the developed model, we apply the Usadel equation to Weyl semimetal mediated Josephson junctions. We study crossovers in charge supercurrent and demonstrate that a spontaneous supercurrent can flow through a triplet channel, creating a $\varphi_{0}$-junction, which is well controlled via junction length and the material parameters pertaining to the Weyl semimetals including spin-orbit coupling that provide experimentally efficient control knobs. We also consider a two-dimensional junction subject to an external magnetic field and show that the charge current has a decaying oscillatory behavior by increasing the external magnetic field, constituting Fraunhofer-modulated diffraction patterns. In all cases, we evaluate the influence of the tilting parameter on our findings.
The paper is organized as follows. In Sec. II, starting from low energy Hamiltonian, we present main steps of formulating a quasiclassical model for Weyl semimetals. Considering a standard model for nonmagnetic impurities, we derive the Eilenberger equation which is applicable to ballistic and moderately disordered samples. Next, we consider samples with a high density of impurities, average the Eilenberger equation over particles’ momentum, and derive the Usadel equation. We then discuss the tunneling boundary conditions and derive a charge current relationship for the model Hamiltonian we start with. In Sec. III, we consider a Josephson configuration, find solutions to the Green’s function, and derive an analytical expression to the supercurrent phase relation. We study the spontaneous supercurrent, current reversals, and the effects of tilting parameter on them. In Sec. IV, we consider a two-dimensional junction subject to an external magnetic field and study the behavior of supercurrent flow and superconducting vortices. Finally, we give concluding remarks in Sec. V.
II Eilenberger and Usadel equations
We first discuss the Hamiltonian of normal state Weyl semimetal and next incorporate superconductivity. The model Hamiltonian that governs the dynamics of low energy particles inside a ballistic Weyl semimetal subject to an external magnetic field reads:
$$\begin{split}&\displaystyle H=\sum_{\sigma\sigma^{\prime}}\int\frac{d\textbf{k%
}}{(2\pi)^{3}}\psi_{\sigma}^{\dagger}(\textbf{k})\Big{\{}\gamma\Big{[}(\text{k%
}_{k}+e\text{A}_{k})^{2}-Q^{2}\Big{]}\sigma_{z}\\
&\displaystyle+\beta\Big{[}(\text{k}_{k}+e\text{A}_{k})^{2}-Q^{2}\Big{]}+%
\alpha_{k}{(\text{k}_{k}+e\text{A}_{k})}{\sigma_{k}}-\mu\Big{\}}_{\sigma\sigma%
^{\prime}}\psi_{\sigma^{\prime}}(\textbf{k}),\end{split}$$
(1)
in which indices $\sigma,\sigma^{\prime}\equiv\uparrow,\downarrow$, $k\equiv x,y,z$, $\gamma$ characterizes the Weyl semimetal and breaks the time reversal symmetry, $Q$ is the splitting of Weyl nodes, $\beta$ describes the tilt of the Weyl cones, $\alpha_{k}$ is the strength of the inversion symmetry breaking parameter, and $\mu$ stands for the chemical potential. The particles’ momentum is denoted by $\textbf{k}=(\text{k}_{x},\text{k}_{y},\text{k}_{z})$ and the external magnetic field is given through its associated vector potential $\textbf{A}=(\text{A}_{x},\text{A}_{y},\text{A}_{z})$. Here, ${\bm{\sigma}}=(\sigma_{x},\sigma_{y},\sigma_{z})$ are the Pauli matrices and $e$ is the elementary charge.
To describe a system made of Weyl semimetal, we define propagators;
$$\displaystyle G_{\sigma\sigma^{\prime}}(t,t^{\prime};\mathbf{r},\mathbf{r}^{%
\prime})$$
$$\displaystyle=$$
$$\displaystyle-\langle{\cal T}\Psi_{\sigma}(t,\mathbf{r}^{\prime})\Psi_{\sigma^%
{\prime}}^{{\dagger}}(t^{\prime},\mathbf{r}^{\prime})\rangle,$$
(2a)
$$\displaystyle\bar{G}_{\sigma\sigma^{\prime}}(t,t^{\prime};\mathbf{r},\mathbf{r%
}^{\prime})$$
$$\displaystyle=$$
$$\displaystyle-\langle{\cal T}\Psi^{{\dagger}}_{\sigma}(t,\mathbf{r})\Psi_{%
\sigma^{\prime}}(t^{\prime},\mathbf{r}^{\prime})\rangle,$$
(2b)
$$\displaystyle F_{\sigma\sigma^{\prime}}(t,t^{\prime};\mathbf{r},\mathbf{r}^{%
\prime})$$
$$\displaystyle=$$
$$\displaystyle+\langle{\cal T}\Psi_{\sigma}(t,\mathbf{r})\Psi_{\sigma^{\prime}}%
(t^{\prime},\mathbf{r}^{\prime})\rangle,$$
(2c)
$$\displaystyle F^{{\dagger}}_{\sigma\sigma^{\prime}}(t,t^{\prime};\mathbf{r},%
\mathbf{r}^{\prime})$$
$$\displaystyle=$$
$$\displaystyle+\langle{\cal T}\Psi^{{\dagger}}_{\sigma}(t,\mathbf{r})\Psi^{{%
\dagger}}_{\sigma^{\prime}}(t^{\prime},\mathbf{r}^{\prime})\rangle,$$
(2d)
where ${\cal T}$ is the time ordering operator, $t,t^{\prime}$ are the imaginary times at $\mathbf{r},\mathbf{r}^{\prime}$, respectively.
We consider elastic impurity scattering potentials $V(\mathbf{r})$ inside the Weyl semimetal by a self-energy term
$$\Sigma_{\text{imp}}(\mathbf{r},\mathbf{r}^{\prime})=\langle V(\mathbf{r})G(%
\mathbf{r},\mathbf{r}^{\prime})V(\mathbf{r}^{\prime})\rangle,$$
(3)
where we average over the positions of impurities. To obtain the above self-energy term, we treat the impurity potentials as perturbation and expand the Green’s function in terms of the unperturbed Green’s function up to the second order. We find the mean free time of particles in the disordered Weyl semimetal as $\tau^{-1}=2\pi n_{i}N_{0}\int d\Omega_{{\bm{n}}_{\text{F}}}(4\pi)^{-1}|v(%
\Omega)|^{2}$ in which $v(\Omega)$ is the Fourier transform of the scattering potential that depends on the relative angle $\Omega$ between the incident and scattered direction of particles, $N_{0}$ is the density of states per spin at the Fermi level of the system and $n_{i}$ is the concentration of impurities. Note that, for the sake of simplicity in the subsequent calculations, we have neglected the intervalley scatterings and anisotropic terms and their effects on the mean free time. In the particle-hole space we find the following equation for the Green’s function:
$$\displaystyle\left(\begin{array}[]{cc}-i\omega_{n}+\hat{H}(\mathbf{r})&-\hat{%
\Delta}(\textbf{r})\\
\hat{\Delta}^{*}(\textbf{r})&i\omega_{n}+\sigma_{y}\hat{H}^{*}(\mathbf{r})%
\sigma_{y}\end{array}\right)\check{G}(\omega_{n};\mathbf{r},\mathbf{r}^{\prime})$$
$$\displaystyle=\delta(\mathbf{r}-\mathbf{r}^{\prime})+\frac{1}{2\pi N_{0}\tau}%
\check{G}(\omega_{n};\mathbf{r},\mathbf{r})\check{G}(\omega_{n};\mathbf{r},%
\mathbf{r}^{\prime}),$$
(4)
in which $\omega_{n}=\pi(2n+1)k_{B}T$ is the Matsubara frequency, $n\in{Z}$, $k_{B}$ is the Boltzman constant, $T$ is temperature, and $\hat{\Delta}(\mathbf{r})$ is the superconducting gap inside the Weyl semimetal. The matrix form of the Green’s function is given by;
$$\check{G}(\omega_{n};\mathbf{r},\mathbf{r}^{\prime})=\left(\begin{array}[]{cc}%
-\hat{G}(\omega_{n};\mathbf{r},\mathbf{r}^{\prime})&-i\hat{F}(\omega_{n};%
\mathbf{r},\mathbf{r}^{\prime})\sigma_{y}\\
-i\sigma_{y}\hat{F}^{\dagger}(\omega_{n};\mathbf{r},\mathbf{r}^{\prime})&%
\sigma_{y}\hat{\bar{G}}(\omega_{n};\mathbf{r},\mathbf{r}^{\prime})\sigma_{y}%
\end{array}\right).$$
We have denoted $2\times 2$ matrices by ‘hat’ symbol, $\hat{\square}$, and $4\times 4$ matrices by ‘check’ symbol, $\check{\square}$.
Next, we subtract from Eq. (II) its conjugate and perform a Fourier transformation with respect to the relative coordinates: $\mathbf{R}=(\mathbf{r}+\mathbf{r}^{\prime})/2$ and $\delta\mathbf{r}=\mathbf{r}-\mathbf{r}^{\prime}$. In order to simplify our calculations, we assume that the Fermi energy is the largest energy scale in the system, and thus the Green’s function is localized at the Fermi level (with Fermi velocity $v_{F}$). Hence, define the quasiclassical Green’s function
$$\displaystyle\check{g}(\omega_{n};\mathbf{R},\mathbf{n}_{\text{F}})=\frac{i}{%
\pi}\int d\xi_{\text{p}}\check{G}(\omega_{n};\mathbf{R},\mathbf{p}),$$
(5)
in which $d\xi_{\text{p}}=v_{F}d\text{p}$. Incorporating these assumptions we finally arrive at the Eilenberger equation eiln :
$$\displaystyle\text{p}_{\text{F}}^{k}\Big{\{}{\cal L},{\check{\tilde{\nabla}}_{%
k}\check{g}}\Big{\}}+\Big{[}\omega_{n}\tau_{z}+\check{\Gamma}_{k}+\frac{1}{2%
\tau}\langle\check{g}\rangle,\check{g}\Big{]}=0,$$
$$\displaystyle{\cal L}=\gamma\sigma_{z}+\beta,$$
(6)
$$\displaystyle\check{\tilde{\nabla}}_{k}{\check{\text{X}}}\equiv\check{\nabla}_%
{k}{\check{\text{X}}}-\Big{[}ie\text{A}_{k}\tau_{z},{\check{\text{X}}}\Big{]},$$
$$\displaystyle\check{\Gamma}_{k}=-i\check{\Delta}-i\text{p}_{\text{F}}^{k}{%
\alpha_{k}}\tau_{z}{\sigma}_{k}+i\gamma Q^{2}\sigma_{z}+i\beta Q^{2},$$
where ${\bm{\tau}}=(\tau_{x},\tau_{y},\tau_{z})$ are Pauli matrices in particle-hole space, $\mathbf{p}_{\text{F}}=(\text{p}_{\text{F}}^{x},\text{p}_{\text{F}}^{y},\text{p%
}_{\text{F}}^{z})$, and $\check{\nabla}_{k}\equiv\check{\partial}_{x,y,z}$. The average over disorder is shown by $\langle...\rangle$. In the above calculations, we have neglected contributions of the order of $L^{-2}$ and $\omega_{c}/\mu\ll 1$, where $\omega_{c}$ is the cyclotron frequency and $L$ is a length scale, large enough compared to the Fermi wave length; $L\gg\lambda_{\text{F}}$. Also, it is assumed that ${\cal L}$ is the leading term in the Hamiltonian.
The Eilenberger equation can be more simplified in systems with numerous impurities so that $1/\tau>|\omega_{n}|,|\Delta|$. In this case, the quasiparticles move diffusively with random directions and trajectories, which is the so-called diffusive regime usadel .
In the diffusive regime, we integrate the quasiclassical Green’s function, Eq. (5), over all possible directions of quasiparticles’ momentum:
$$\displaystyle\langle\check{g}(\omega_{n};\mathbf{R},\mathbf{n}_{\text{F}})%
\rangle\equiv\int\frac{d\Omega_{\mathbf{n}_{\text{F}}}}{4\pi}\check{g}(\omega_%
{n};\mathbf{R},\mathbf{n}_{\text{F}}),\;\mathbf{n}_{\text{F}}=\frac{\mathbf{p}%
_{\text{F}}}{|\mathbf{p}_{\text{F}}|}.$$
(7)
In this regime, the Green’s function can be expanded through the first two harmonics: $s$-wave and $p$-wave
$$\check{g}(\omega_{n};\mathbf{R},\mathbf{n}_{\text{F}})=\check{\text{g}}_{s}(%
\omega_{n};\mathbf{R})+{n}_{\text{F}}^{k}\check{\text{g}}_{p}^{k}(\omega_{n};%
\mathbf{R}),$$
(8)
where the $s$-wave harmonic in the expansion above (8) is isotropic and its magnitude is much larger than the $p$-wave harmonic: $\check{\text{g}}_{s}\gg\check{\text{g}}_{p}^{k}$. By substituting this expanded Green’s function into Eq. (6) and performing an integration over momentum directions we find
$$\displaystyle\check{\text{g}}_{p}^{k}=-\tau\text{p}_{\text{F}}^{k}\check{\text%
{g}}_{s}\Big{\{}{\cal L},\check{\tilde{\nabla}}_{k}\check{\text{g}}_{s}\Big{\}%
}+\tau\text{p}_{\text{F}}^{k}\check{\text{g}}_{s}\Big{[}i\alpha_{k}\tau_{z}{%
\sigma}_{k},\check{\text{g}}_{s}\Big{]}.$$
(9)
Next, by substituting Eq. (9) into Eq. (6) and assuming that ${\nabla}_{k}\gamma={\nabla}_{k}\beta={\nabla}_{k}\alpha_{k}={\nabla}_{k}\text{%
A}_{k}=0$, we find a generalized Usadel model for tilted Weyl semimetals usadel :
$$\frac{\text{p}_{\text{F}}^{k}}{3}\Big{\{}{\cal L},\check{\tilde{\nabla}}_{k}%
\check{\text{g}}_{p}^{k}\Big{\}}-\frac{\text{p}_{\text{F}}^{k}}{3}\Big{[}i%
\alpha_{k}\tau_{z}{\sigma}_{k},\check{\text{g}}_{p}^{k}\Big{]}+\Big{[}\omega_{%
n}\tau_{z}+\check{\Gamma}_{k},\check{\text{g}}_{s}\Big{]}=0.~{}~{}~{}$$
(10)
Solving Eqs. (6) and (10), one finds appropriate Green’s function that contains all information describing observable physical properties of various systems. Hence, we now proceed to apply our formulated quasiclassical model to hybrid structures made of disordered Weyl semimetals and superconductors, which are practical platforms that the Eilenberger and Usadel equations are able to describe them properly. The quasiclassical Eilenbeger and Usadel approaches were also generalized for spin-orbit coupled systems Konschelle ; Bergeret ; Huang ; ali_so1 ; ali_so2 ; gupta , surface channels of topological insulators zu1 ; zu2 ; zu3 , and black phosphorus monolayer (phosphorene) alidoustBP2 in the presence of superconductivity and a Zeeman field. A specific configuration is depicted in Fig. 1. As seen, two superconductors are coupled through a disordered Weyl semimetal of thickness $d$ and width $W$. The interfaces are located at $z=\pm d/2$ in the $xy$ plane. The macroscopic phases of the left and right superconductor terminals are denoted by $\theta_{l}$ and $\theta_{r}$, respectively.
We consider low transparent interfaces (the so-called tunneling limit) between the superconductors and Weyl semimetal and neglect the inverse proximity effect at the interfaces. We therefore find the following expression to the boundary conditions zu1 ; zu2 ; zu3 ; boundary_c1 ; ma_jap
$$\zeta{n}_{k}\check{\text{g}}_{p}^{k}=\Big{[}\check{\text{g}}_{s},\check{\text{%
g}}_{\text{SC}}\Big{]},$$
(11)
in which ${n}_{k}$ is a unit vector perpendicular to a boundary, $\zeta$ controls the opacity of interfaces, and $\check{\text{g}}_{\text{SC}}$ is the Green’s function of the bulk superconductors. To study the quantum transport, we derive an expression to the charge current flow (due to the superconducting phase gradient across the device, in our case). The quantum definition of current density is expressed through the Hamiltonian Eq. (1) as follows
$$\begin{split}\displaystyle\frac{\partial\rho}{\partial t}=\lim\limits_{{\bm{r}%
}\rightarrow{\bm{r}}^{\prime}}\sum\limits_{\sigma\sigma^{\prime}}\frac{1}{i%
\hbar}\Big{[}\psi^{\dagger}_{\sigma}({\bm{r}}^{\prime})H_{\sigma\sigma^{\prime%
}}({\bm{r}})\psi_{\sigma^{\prime}}({\bm{r}})\\
\displaystyle-\psi^{\dagger}_{\sigma}({\bm{r}}^{\prime})H_{\sigma\sigma^{%
\prime}}^{\dagger}({\bm{r}}^{\prime})\psi_{\sigma^{\prime}}({\bm{r}})\Big{]},%
\end{split}$$
(12)
where the left hand side is the time variation of charge density $\rho$. Throughout our calculations, we consider a steady state regime and, therefore, set $\partial\rho/\partial t=0$. We use the Fourier representation of the Keldysh Greenâs function in equilibrium:
$$\sum_{n}\int\frac{d\textbf{p}}{(2\pi)^{3}}e^{i\textbf{p}\cdot\textbf{r}}\check%
{G}^{K}(\omega_{n};\textbf{R},\textbf{p}),$$
(13)
and we finally arrive at an expression for the current density in the ballistic regime. By applying the quasiclassical approximations described above and making use of the harmonics expassion to the Green’s function, Eq. (8), we find the following expression for the current density in the diffusive regime:
$${J}_{k}=\frac{ie\pi}{3}N_{0}\text{p}_{\text{F}}^{k}T\sum_{n}\mathrm{Tr}\Big{[}%
\tau_{z}\big{(}\gamma\sigma_{z}+\beta\big{)}\check{\text{g}}_{p}^{k}\Big{]}.$$
(14)
To derive Eq. (14) we have assumed sufficiently small $\alpha_{k}$ and neglected terms of the order of $\alpha_{k}(\text{p}_{\text{F}}^{k})^{-1}$.
III self-biased supercurrent and supercurrent reversals
In order to find the charge current density, one has to solve either Eq.(6) (in the ballistic regimezu3 ) or Eq. (10) (in the diffusive regimezu1 ; zu2 ) together with proper boundary conditions (11) and substitute the resultant Green’s function into Eq. (12). In the diffusive regime, the Usadel equation (10) results in nonliner boundary value differential equations that has to be solved numerically ma_odfr . To obtain decopled linear differential equations that are simpler to solve and provide analytical solutions, we expand and linearize the Green’s function aournd the bulk solution $\check{\text{g}}_{0}(\omega_{n};\mathbf{R})$, i.e., $\check{\text{g}}(\omega_{n};\mathbf{R})\approx\check{\text{g}}_{0}(\omega_{n};%
\mathbf{R})+\check{f}(\omega_{n};\mathbf{R})$ma_jap . This approximation is experimentally relevant and accessible in a low proximity limit either close to the superconducting critical temperature or devices with low transparent interfaces ma_jap . The external magnetic field is given by its associated vector potential that satisfies the Lorentz gauge ${\bm{\nabla}}\cdot\textbf{A}=0$ and ${H}_{x}={\bm{\nabla}}\times\textbf{A}$. As depicted in Fig. 1, we consider a situation where the external magnetic field is directed towards $x$ direction, perpendicular to the junction plane, and, therefore, can be described by $\textbf{A}=(0,0,y{H}_{x})$. The Usadel equation (10) for the triplet channel in the presence of the external magnetic field within a tilted Weyl semimetal results in the following decoupled linear differential equations:
$$\displaystyle(\beta+\gamma)^{2}{\nabla}^{2}_{k}f_{\uparrow\uparrow}(\omega_{n}%
)-[\alpha_{z}+2e\text{A}_{z}(\beta+\gamma)]^{2}f_{\uparrow\uparrow}(\omega_{n}%
)-2i(\beta+\gamma)[\alpha_{z}+2e\text{A}_{z}(\beta+\gamma)]{\nabla}_{z}f_{%
\uparrow\uparrow}(\omega_{n})+\omega_{n}D^{-2}f_{\uparrow\uparrow}(\omega_{n})%
=0,$$
(15a)
$$\displaystyle(\beta-\gamma)^{2}{\nabla}^{2}_{k}f_{\downarrow\downarrow}(\omega%
_{n})-[\alpha_{z}-2e\text{A}_{z}(\beta-\gamma)]^{2}f_{\downarrow\downarrow}(%
\omega_{n})+2i(\beta-\gamma)[\alpha_{z}-2e\text{A}_{z}(\beta-\gamma)]{\nabla}_%
{z}f_{\downarrow\downarrow}(\omega_{n})+\omega_{n}D^{-2}f_{\downarrow%
\downarrow}(\omega_{n})=0,$$
(15b)
$$\displaystyle(\beta+\gamma)^{2}{\nabla}^{2}_{k}\tilde{f}_{\uparrow\uparrow}(%
\omega_{n})-[\alpha_{z}+2e\text{A}_{z}(\beta+\gamma)]^{2}\tilde{f}_{\uparrow%
\uparrow}(\omega_{n})+2i(\beta+\gamma)[\alpha_{z}+2e\text{A}_{z}(\beta+\gamma)%
]{\nabla}_{z}\tilde{f}_{\uparrow\uparrow}(\omega_{n})+\omega_{n}D^{-2}\tilde{f%
}_{\uparrow\uparrow}(\omega_{n})=0,$$
(15c)
$$\displaystyle(\beta-\gamma)^{2}{\nabla}^{2}_{k}\tilde{f}_{\downarrow\downarrow%
}(\omega_{n})-[\alpha_{z}-2e\text{A}_{z}(\beta-\gamma)]^{2}\tilde{f}_{%
\downarrow\downarrow}(\omega_{n})-2i(\beta-\gamma)[\alpha_{z}+2e\text{A}_{z}(%
\beta-\gamma)]{\nabla}_{z}\tilde{f}_{\downarrow\downarrow}(\omega_{n})+\omega_%
{n}D^{-2}\tilde{f}_{\downarrow\downarrow}(\omega_{n})=0.$$
(15d)
Here, index $k$ runs over $y,z$ in a two-dimensional system and $x,y,z$ in a three-dimensional junction.
To begin, we first set the external magnetic field zero, i.e., ${H}_{x}=0$ in Eqs. (15). By considering the low proximity limit, described above, we find appropriate expressions to the components of Green’s function. For example, the components $f_{\uparrow\uparrow}(\omega_{n};z)$ and $\tilde{f}_{\uparrow\uparrow}(\omega_{n};z)$ become:
$$\displaystyle\begin{split}&\displaystyle f_{\uparrow\uparrow}(\omega_{n};z)={%
\cal F}(\omega_{n})\exp\left({-\frac{2i\alpha_{z}dz}{\beta+\gamma}}\right)%
\left\{\exp\left(\frac{i(\alpha_{z}d(z+1)+\theta_{r}(\beta+\gamma)+\lambda_{n}%
D^{-1}(1-z))}{\beta+\gamma}\right)+\right.\\
&\displaystyle\left.\exp\left(\frac{i(\alpha_{z}d(z+1)+\theta_{r}(\beta+\gamma%
)+\lambda_{n}D^{-1}(z+1))}{\beta+\gamma}\right)+\exp\left({\frac{i(\alpha_{z}%
dz+\beta\theta_{l}+\gamma\theta_{l}-\lambda_{n}D^{-1}(z-2))}{\beta+\gamma}}%
\right)+\exp\left(i\frac{z(\alpha_{z}d+\lambda_{n}D^{-1})}{\beta+\gamma}+i%
\theta_{l}\right)\right\},\end{split}$$
(16a)
$$\displaystyle\begin{split}&\displaystyle\tilde{f}_{\uparrow\uparrow}(\omega_{n%
};z)={\cal F}(\omega_{n})\exp\left(-i\frac{\alpha_{z}d+(\beta+\gamma)(\theta_{%
l}+\theta_{r})}{\beta+\gamma}\right)\left\{\exp\left(\frac{i(\alpha_{z}d(z+1)+%
\theta_{r}(\beta+\gamma)-\lambda_{n}D^{-1}(z-2))}{\beta+\gamma}\right)+\right.%
\\
&\displaystyle\left.\exp\left({\frac{i(\alpha_{z}dz+(\beta+\gamma)\theta_{l}+%
\lambda_{n}D^{-1}(1-z))}{\beta+\gamma}}\right)+\exp\left({\frac{i(\alpha_{z}dz%
+(\beta+\gamma)\theta_{l}+\lambda_{n}D^{-1}(1+z))}{\beta+\gamma}}\right)+\exp%
\left({\frac{i(\alpha_{z}d(z+1)+(\beta+\gamma)\theta_{r}+\lambda_{n}D^{-1}z)}{%
\beta+\gamma}}\right)\right\}.\end{split}$$
(16b)
Here, we have defined ${\cal F}(\omega_{n})=iD\Big{\{}\lambda_{n}\zeta[\exp\left({\frac{2i\lambda_{n}%
D^{-1}}{\beta+\gamma}}\right)-1]\Big{\}}^{-1}f_{t}$, $D^{2}=2\text{p}_{\text{F}}^{2}\tau/3$, and $\lambda_{n}^{2}=-\omega_{n}/d^{2}$. Our analyses of the boundary conditions and the Usadel equation demonstrate that the supercurrent in this system can flow through a triplet channel. Therefore, we assume a triplet component $f_{t}$ to $\hat{\text{g}}_{\text{SC}}$ in Eq. (11). This finding may explain a recent experiment where a long-ranged supercurrent was observed through a Josephson junction made of $\rm WTe_{2}$ Weyl semimetal Kononov2 . In the singlet channel we recover the results of a conventional SNS junction (up to the zero order of $\alpha_{z}\text{p}_{\text{F}}^{-1}$). We only presented two representative components of $\tilde{\hat{f}}(\omega_{n};z)$ and $\hat{f}(\omega_{n};z)$. To obtain the total charge current passing through the junction, $I_{c}$, we substitute these solutions into Eq. (14) and, after performing calculations, find the following charge supercurrent phase relation
$$I_{c}=e\pi N_{0}{\cal A}D^{3}T\sum_{n}\frac{f_{t}^{2}}{\zeta^{2}\lambda_{n}}%
\csc\Big{(}\frac{\lambda_{n}D^{-1}}{\beta-\gamma}\Big{)}\csc\Big{(}\frac{%
\lambda_{n}D^{-1}}{\beta+\gamma}\Big{)}\left\{(\beta-\gamma)\sin\Big{(}\frac{%
\lambda_{n}D^{-1}}{\beta+\gamma}\Big{)}\sin\Big{(}\frac{\alpha_{z}d}{\beta-%
\gamma}-\varphi\Big{)}-(\beta+\gamma)\sin\Big{(}\frac{\lambda_{n}D^{-1}}{\beta%
-\gamma}\Big{)}\sin\Big{(}\frac{\alpha_{z}d}{\beta+\gamma}+\varphi\Big{)}%
\right\},$$
(17)
in which ${\cal A}$ is the cross section of the Weyl semimetal/superconductor interface, $\varphi=\theta_{l}-\theta_{r}$, and we define $I_{0}=e\pi N_{0}{\cal A}$. As seen in Eq. (17), the supercurrent experiences a total phase shift $\Theta_{0}(\beta,\gamma,\alpha_{z})$ made of $\varphi_{0}^{\pm}=d\alpha_{z}/(\beta\pm\gamma)$ that renders the junction grand state into values other than the standard $0$ or $\pi$ states in conventional Josephson junctions. This phase shift causes a self-biased supercurrent at zero phase difference $\varphi=0$. Note that $\varphi_{0}^{\pm}$ are independent of $D$. This finding illustrates that the addressed phase shift is robust against the density of impurities considered in the system, i.e., $\varphi_{0}^{\pm}$ are independent of $\tau$. Hence, this phenomenon can obviously occur in moderately disordered and ballistic regimes as quite recently explored in experimentherve inline with theory predictions for topological insulator surface channelszu1 ; zu2 ; zu3 and black phosphorus monolayer alidoustBP1 ; alidoustBP2 . Also, the explored $\varphi_{0}$ state here relies on inherent parameters of Weyl semimetal without involving Zeeman field zu1 ; zu2 ; zu3 ; alidoustBP1 ; alidoustBP2 ; phi0 ; herve . It is worth mentioning that the appearance of $4\pi$-periodic current phase relation in topological insulator Josephson junctions might be due to the presence of Majorana Fermions although such a $4\pi$-periodic supercurrent phase relation can be theoretically obtained in the trivial regime of a topological topological insulator as well zu3 ; 4pi . Figure 2 illustrates the normalized charge supercurrent as a function of the thickness of Weyl semimetal $d$ normalized by the superconducting coherence length $\xi_{S}$ for differing values of the tilting parameter $\beta$, Fig. 2(a), and strength of inversion symmetry breaking parameter $\alpha_{z}$, Fig. 2(b), at zero phase difference, i.e., $\varphi=0$. In Fig. 2(a), considering representative values, we set $\alpha_{z}=0.5,\gamma=0.1$ fixed and vary $\beta$ whereas in Fig. 2(b), $\beta=0.6,\gamma=0.1$ are set fixed and $\alpha_{z}$ varies. We see that the supercurrent decays and experiences multiple sign changes as a function of $d$ in both cases. The sign change occurs faster by decreasing $\beta$ and increasing $\alpha_{z}$. This can be understood by Eq. (17). Increasing $\alpha_{z}$ or decreasing $\beta$, the phase shift increases and, therefore, by varying $d$, faster oscillations occur. Equation (17) demonstrates that $\beta$, in the absence of $\alpha_{z}$, is unable to induce phase shift in this channel while in the presence of a finite $\alpha_{z}$, the tilting parameter changes the magnitude and sign of the phase shifts, making a total phase shift $\Theta_{0}(\beta,\gamma,\alpha_{z})$.
The inversion symmetry breaking parameter, $\alpha_{z}$, may respond to an external mechanical deformation efficientlyalidoustBP1 ; alidoustBP2 and, thus, the spontaneous supercurrent and current reversals might be effectively controllable through external knobs regardless of the density of impurities and disorder present in the system.
The tilting parameter competes with the inversion symmetry breaking parameter in inducing supercurrent reversals. The increase of $\beta$ results in shifting the locations of current reversal points and, in general, reduces the number of crossovers that can appear in a specific interval of junction thickness, Fig. 2(a). This is opposite to the effect of $\alpha_{z}$ shown in Fig. 2(b). Furthermore, by increasing $\beta$, the supercurrent vs the junction thickness enhances and decays slower whereas $\alpha_{z}$ is unable to influence the degree of supercurrent decay. From Eq. (17), the supercurrent is proportional to $(\beta\pm\gamma)\text{csch}(1/d(\beta-\gamma))\text{csch}(1/d(\beta+\gamma))%
\text{sinh}(1/d(\beta\mp\gamma))\sin(\varphi_{0}^{\pm})$. It is apparent that $\alpha_{z}$ is unable to effectively enhance or suppress the magnitude of total supercurrent while $\beta$ and $\gamma$ can highly alter the total charge current. If we set $\alpha_{z}=0$, the supercurrent as a function of junction thickness $d$ decays with no current reversal, showing long-ranged characteristicsKononov2 . We now proceed to study Weyl semimetal Josephson junctions subject to an external magnetic field.
IV Faunhofer response and proximity vortices
In order to study the response of charge current to an external magnetic field in a Weyl semimetal mediated Josephson junction, we consider a two-dimensional configuration with ${\bm{H}}=({H}_{x},0,0)$ depicted in Fig. 1. We assume that $W\gg d$, define ${\Phi}=\pi Wd{H}_{x}$, ${\Phi}_{0}=h/2e$ a quantum flux, and ${\bm{\Phi}}={\Phi}/{\Phi_{0}}$ ma_odfr ; ma_jap ; zu1 . The resultant Green’s function components; $f_{\uparrow\uparrow}(\omega_{n};z,y),\tilde{f}_{\uparrow\uparrow}(\omega_{n};z%
,y)$ and calculated current density flowing in the $z$ direction are given by;
$$\displaystyle\begin{split}&\displaystyle f_{\uparrow\uparrow}(\omega_{n};z)={%
\cal F}(\omega_{n})\exp\left(-\frac{i(z(\alpha_{z}d+\lambda_{n}D^{-1})-2\beta{%
\bm{\Phi}}y(z-1)-2\gamma{\bm{\Phi}}y(z-1))}{\beta+\gamma}\right)\left\{\exp%
\left({\frac{i(\alpha_{z}d+\theta_{r}(\beta+\gamma)+\lambda_{n}D^{-1}(2z+1))}{%
\beta+\gamma}}\right)+\right.\\
&\displaystyle\left.\exp\left(\frac{i(\alpha_{z}d+\theta_{r}(\beta+\gamma)+%
\lambda_{n}D^{-1})}{\beta+\gamma}\right)+\exp\left(\frac{i(\beta(2{\bm{\Phi}}y%
+\theta_{l})+\gamma(2{\bm{\Phi}}y+\theta_{l})+2\lambda_{n}D^{-1}z)}{\beta+%
\gamma}\right)+\exp{\left(\frac{2i\lambda_{n}D^{-1}}{\beta+\gamma}+2i{\bm{\Phi%
}}y+i\theta_{l}\right)}\right\},\end{split}$$
(18a)
$$\displaystyle\begin{split}&\displaystyle\tilde{f}_{\uparrow\uparrow}(\omega_{n%
};z)={\cal F}(\omega_{n})\exp\left(-\frac{i(\alpha_{z}(d-dz)+\beta(2{\bm{\Phi}%
}yz+\theta_{l}+\theta_{r})+2\gamma{\bm{\Phi}}yz+\gamma(\theta_{l}+\theta_{r})+%
\lambda_{n}D^{-1}z)}{\beta+\gamma}\right)\left\{\exp\left({\frac{i(\alpha_{z}d%
+(\beta+\gamma)\theta_{r}+2\lambda_{n}D^{-1}z)}{\beta+\gamma}}\right)+\right.%
\\
&\displaystyle\left.\exp\left({\frac{i(\alpha_{z}d+\theta_{r}(\beta+\gamma)+2%
\lambda_{n}D^{-1})}{\beta+\gamma}}\right)+\exp\left(\frac{i((\beta+\gamma)(2{%
\bm{\Phi}}y+\theta_{l})+\lambda_{n}D^{-1}(2z+1))}{\beta+\gamma}\right)+\exp%
\left({\frac{i((\beta+\gamma)(2{\bm{\Phi}}y+\theta_{l})+\lambda_{n}D^{-1})}{%
\beta+\gamma}}\right)\right\},\end{split}$$
(18b)
$$\begin{split}\displaystyle J_{z}(y)=e\pi N_{0}D^{3}T\sum_{n}&\displaystyle%
\frac{f_{t}^{2}}{\zeta^{2}\lambda_{n}}\csc\Big{(}\frac{\lambda_{n}D^{-1}}{%
\beta-\gamma}\Big{)}\csc\Big{(}\frac{\lambda_{n}D^{-1}}{\beta+\gamma}\Big{)}%
\times\\
&\displaystyle\left\{(\beta-\gamma)\sin\Big{(}\frac{\lambda_{n}D^{-1}}{\beta+%
\gamma}\Big{)}\sin\Big{(}\frac{\alpha_{z}d}{\beta-\gamma}+2{\bm{\Phi}}y-%
\varphi\Big{)}-(\beta+\gamma)\sin\Big{(}\frac{\lambda_{n}D^{-1}}{\beta-\gamma}%
\Big{)}\sin\Big{(}\frac{\alpha_{z}d}{\beta+\gamma}-2{\bm{\Phi}}y+\varphi\Big{)%
}\right\}.\end{split}$$
(19)
To obtain the total charge current flowing through the junction, we integrate the charge current density flowing in the $z$ direction, Eq. (19), over the $y$ direction (perpendicular to the current direction), i.e., $I_{c}=\int_{-W/2}^{+W/2}J_{z}(y)dy$, assuming that the junction width is equal to $W$. This integration leads to the standard Fraunhofer diffraction pattern vs the external magnetic flux with a multiplication of Eq. (17). Therefore, the charge current in the presence of an external magnetic field perpendicular to the junction plane has a form of
$$I_{c}\propto\Big{(}\frac{{\Phi}}{{\Phi}_{0}}\Big{)}^{-1}\sin\Big{(}\frac{{\Phi%
}}{{\Phi}_{0}}\Big{)}[I_{1}\sin(\varphi_{0}^{+}+\varphi)+I_{2}\sin(\varphi_{0}%
^{-}+\varphi)].$$
(20)
When the inversion symmetry breaking parameter vanishes, the tilting parameter only controls the magnitude of total supercurrent passing through the triplet channel. The presence of inversion symmetry breaking parameter induces more oscillations in the charge current (and thus the Fraunhofer pattern) in an external magnetic field when increasing the tilting parameter.
To gain more insight, we plot the spatial map of absolute value of the Cooper pair wavefunction (i.e. the anomalous Green’s function at equal times) within the junction area in Fig. 3. We have set the external flux fixed at ${\Phi}=2\Phi_{0}$, $\gamma=0.1$, and the inversion symmetry breaking parameter $\alpha_{z}=0.5$. The rest of parameters are equal to those of Fig. 2. From left to right, we increase the tilting parameter $\beta=0.15,0.35,0.45,0.60,0.75,0.90$. The external magnetic field induces two vortices ($n$ vortices for $\Phi=n\Phi_{0}$) along the junction interface in the $y$ direction at the middle of junction $x=0$ ma_jap ; zu1 ; ma_odfr . Increasing $\beta$, the vortices move along the junction width in the $y$ direction. This increase also reforms the vortices so that the point-like cores at $\beta=0.15$ turn to lines with finite sizes expanded in the $z$ direction and the destruction of the pair wavefunction is no longer limited to the junction area and extends into the superconductors. Our further investigates show that varying the strength of inversion symmetry breaking parameter $\alpha_{z}$ only drives the proximity vortices and shifts the locations of vortex cores along the $y$ direction without changing the shape of the vortices’ profile whereas varying the tilting parameter when $\alpha_{z}=0$ only changes the shape of vortices the same as what is shown in Fig. 3 from left to right, ignoring the location shifts. This is fully consistent with the influences of $\alpha_{z}$ and $\beta$ on the charge current discussed in passing and can be directly inferred from the dependence of charge current density, Eq. (19), on $\beta,\gamma$, and $\alpha_{z}$.
V Conclusions
In summary, we have generalized a quasiclassical model, including the Eilenberger and Usadel equations, for Weyl semimetals with impurities subject to an external magnetic field. As a preliminary step in the application of these generalized techniques to practical systems, we have studied supercurrent flow through a diffusive Weyl semimetal Josephson junction. We have found that the supercurrent can be carried through a triplet channel with a nonzero threshold made of $d\alpha_{z}/(\beta\pm\gamma)$ in which $d$ is the thickness of Weyl semimetal, $\alpha_{z}$ is the strength of spin-orbit interaction, and $\beta,\gamma$ characterize the Weyl semimetal. Our results demonstrate that the tilting parameter, $\beta$, can control the induction of current crossovers and the self-biased supercurrent independent of the density of impurities present in the samples. We also consider a two-dimensional Josephson junction and study the effect of $\alpha_{z}$ and $\beta$ on the Cooper pair wavefunction, superconducting vortices, and the response of charge current to an externally applied magnetic field.
Acknowledgements.M.A. is supported by Iran’s National Elites Foundation (INEF). M.A. would like to thank A. Zyuzin for useful discussions and careful reading of the paper. M.A. also thanks G. Sewell for valuable discussions.
References
(1)
M. Z. Hasan and C. L. Kane, Colloquium: topological insulators, Rev. Mod. Phys. 82, 3045 (2010); X.-L. Qi and S.-C. Zhang, Topological insulators and superconductors, Rev. Mod. Phys. 83, 1057 (2011).
(2)
B. A. Bernevig, T. L. Hughes, and S-C. Zhang, Quantum spin Hall effect and topological phase transition in HgTe quantum wells, Science
314, 1757 (2006).
(3)
M. Knig, S. Wiedmann, C. Brne, A. Roth, H. Buhmann,
L. W. Molenkamp, X-L. Qi, and S-C. Zhang, Quantum spin Hall insulator state in HgTe quantum wells, Science 318,
766 (2007).
(4)
S.-Y. Xu, etal., Discovery of a Weyl fermion semimetal and topological Fermi arcs, Science 349, 613 (2015).
(5)
A. A. Burkov and L. Balents, Weyl semimetal in a topological insulator multilayer, Phys. Rev. Lett. 107, 127205 (2011).
(6)
A. A. Zyuzin and A. A. Burkov, Topological response in Weyl semimetals and the chiral anomaly, Phys. Rev. B 86, 115133 (2012).
(7)
S.-M. Huang, etal., A Weyl Fermion semimetal with surface Fermi arcs in the transition metal monopnictide $\rm TaAs$ class, Nat. Comm. 6, 7373 (2015).
(8)
B. Q. Lv, H. M. Weng, B. B. Fu, X. P. Wang, H. Miao, J. Ma, P. Richard, X. C. Huang, L. X. Zhao, G. F. Chen, Z. Fang, X. Dai, T. Qian, and H. Ding, Experimental discovery of Weyl semimetal TaAs, Phys. Rev. X 5, 031013 (2015).
(9)
A. A. Soluyanov, D. Gresch, Z. Wang, Q. Wu, M. Troyer, X.
Dai, and B. A. Bernevig, Type-II Weyl semimetals, Nature (London) 527, 495 (2015).
(10)
C. Xu, B. Li, W. Jiao, W. Zhou, B. Qian, R. Sankar, N. D. Zhigadlo, Y. Qi, D. Qian, F.-C. Chou, and X. Xu, Topological Type-II Dirac Fermions Approaching the Fermi Level in a Transition Metal Dichalcogenide $\rm NiTe_{2}$, Chem. Mater. 30, 4823 (2018).
(11)
R. Aguado, Majorana quasiparticles in condensed matter, La Rivista del Nuovo Cimento 40, 523 (2017).
(12)
D.S. Shapiro, D.E. Feldman, A.D. Mirlin, and
A. Shnirman, Thermoelectric transport in junctions of Majorana and Dirac channels,
Phys. Rev. B
95, 195425 (2017).
(13)
M. Banerjee, M. Heiblum, V. Umansky, D. E. Feldman, Y. Oreg, A. Stern, Observation of half-integer thermal Hall conductance, arXiv:1710.00492.
(14)
G. Y. Cho, J. H. Bardarson, Y.-M. Lu, and J. E. Moore, Superconductivity of doped Weyl semimetals: Finite-momentum pairing and electronic analog of the ${}^{3}$He-A phase,
Phys. Rev. B 86, 214514 (2012).
(15)
H. Wei, S. Chao, and V. Aji, Odd-parity superconductivity in Weyl semimetals, Phys. Rev.
B 89, 014506 (2014).
(16)
G. Bednik, A. A. Zyuzin, and A. A. Burkov, Superconductivity in Weyl metals, Phys. Rev.
B 92, 035153 (2015).
(17)
T. Zhou, Y. Gao, and Z. D. Wang, Superconductivity in Weyl metals, Phys. Rev. B 93,
094517 (2016).
(18)
P. Hosur, X. Dai, Z. Fang, and X.-L. Qi, Time-reversal-invariant topological superconductivity in doped Weyl semimetals, Phys. Rev. B 90,
045130 (2014).
(19)
B. Lu, K. Yada, M. Sato, and Y. Tanaka, Crossed surface flat bands of Weyl semimetal superconductors, Phys. Rev. Lett.
114, 096804 (2015).
(20)
R. Wang, L. Hao, B. Wang, C. S. Ting, Quantum anomalies in superconducting Weyl metals, Phys. Rev. B 93, 184511 (2016).
(21)
Y. Li, F. D. M. Haldane, Topological nodal Cooper pairing in doped Weyl metals, Phys. Rev. Lett. 120, 067003 (2018).
(22)
T. Meng and L. Balents, Weyl superconductors, Phys. Rev. B 86, 054504
(2012).
(23)
T. Das, Weyl semimetal and superconductor designed in an orbital-selective superlattice, Phys. Rev. B 88, 035444 (2013).
(24)
U. Khanna, A. Kundu, S. Pradhan, and
S. Rao, Proximity-induced superconductivity in Weyl semimetals, Phys. Rev. B 90, 195430 (2014).
(25)
S. A. Yang, H. Pan, and F. Zhang, Dirac and Weyl superconductors in three dimensions, Phys. Rev.
Lett. 113, 046401 (2014).
(26)
M. Alidoust, M. Willatzen, A.-P. Jauho, Strain-engineered Majorana zero energy modes and $\varphi_{0}$ Josephson state in black phosphorus, Phys. Rev. B 98, 085414 (2018).
(27)
M. Alidoust, M. Willatzen, A.-P. Jauho, Fraunhofer response and supercurrent spin switching in black phosphorus with strain and disorder, Phys. Rev. B 98, 184505 (2018).
(28)
E. V. Gorbar, V. A. Miransky, I. A. Shovkovy, P. O. Sukhachov, Inter-node superconductivity in strained Weyl semimetals, arXiv:1809.00019.
(29)
T. Zhou, Y. Gao, and Z. D. Wang, Resolving different pairing states in Weyl superconductors through the single-particle spectrum
, Phys. Rev. B 98, 024515 (2018).
(30)
P. Fulde and R. A. Ferrell
Phys. Rev. 135, A550 (1964); A.I. Larkin, Yu.N. Ovchinnikov,
Zh. Eksp. Teor. Fiz. 47, 1136 (1964);
Sov. Phys. JETP 20: 762 (1965).
(31)
Y. Qi, etal., Superconductivity in Weyl semimetal candidate $\rm MoTe_{2}$, Nature Comm. 7, 11038 (2016).
(32)
F. C. Chen, etal., Superconductivity enhancement in the S-doped Weyl semimetal candidate $\rm MoTe_{2}$, Appl. Phys. Lett. 108, 162601 (2016).
(33)
X. Luo, etal., $\rm T_{d}-MoTe_{2}$: A possible topological superconductor, Appl. Phys. Lett. 109, 102601 (2016)
(34)
O. A. Pankratov, S. V. Pakhomov, and B. A. Volkov, Supersymmetry in heterojunctions: Band-inverting contact on the basis of $\rm Pb_{1-x}Sn_{x}Te$ and $\rm Hg_{1-x}Cd_{x}Te$, Solid State
Comm. 61, 93 (1987).
(35)
G. E. Volovik, The Universe in a Helium Droplet (Oxford University Press, Oxford, 2003); G.E. Volovik, Lifshitz transitions, type-II Dirac and Weyl fermions, event horizon and all that, J. Low Temp. Phys. 189, 276 (2017).
(36)
A. A. Soluyanov, D. Gresch, Z. Wang, Q. Wu, M. Troyer, X.
Dai, and B. A. Bernevig, Type-II Weyl semimetals, Nature (London) 527, 495 (2015).
(37)
J. Ruan, S.-K. Jian, H. Yao, H. Zhang, S.-C. Zhang, and D. Xing, Symmetry-protected ideal Weyl semimetal in HgTe-class materials
Nat. Commun. 7, 11136 (2016).
(38)
M. D. Bachmann, N. Nair, F. Flicker, R. Ilan, T. Meng, Nirmal J. Ghimire, E. D. Bauer, F. Ronning, J. G. Analytis, P. Moll, Inducing superconductivity in Weyl semimetal microstructures by selective ion sputtering, Science Advance 3, no. 5 (2017).
(39)
R. C. Xiao, P. L. Gong, Q. S. Wu, W. J. Lu, M. J. Wei, J. Y. Li, H. Y. Lv, X. Luo, P. Tong, X. B. Zhu, and Y. P. Sun, Manipulation of type-I and type-II Dirac points in
${\rm PdTe_{2}}$
superconductor by external pressure, Phys. Rev. B 96, 075101 (2017).
(40)
B. Rosenstein, B.Y. Shapiro, D. Li, I. Shapiro, Magnetic properties of type-I and type-II Weyl semimetals in the superconducting state, Phys. Rev. B 97, 144510 (2018).
(41)
Z. Hou and Q. Sun, Double Andreev Reflections in Type-II Weyl Semimetal-Superconductor Junctions, Phys. Rev. B 96, 155305 (2017).
(42)
X.-S. Li, S.-F. Zhang, X.-R. Sun, W.-J. Gong, Double Andreev reflections and double electron transmissions in a normal-superconductor-normal junction based on type-II Weyl semimetal, arXiv:1805.07928.
(43)
S.-B. Zhang, F. Dolcini, D. Breunig, B. Trauzettel, Appearance of the universal value $e^{2}/h$ of the zero-bias conductance in a Weyl semimetal-superconductor junction, Phys. Rev. B 97, 041116 (2018).
(44)
S.-B. Zhang, J. Erdmenger, B. Trauzettel, Chirality Josephson current due to a novel quantum anomaly in inversion-asymmetric Weyl semimetals, arXiv:1806.08111.
(45)
A. Chen, D. I. Pikulin, and M. Franz, Josephson current signatures of Majorana flat bands on the surface of time-reversal-invariant Weyl and Dirac semimetals, Phys. Rev. B 95, 174505 (2017).
(46)
Y. Xu, S. Uddin, J. Wang, Z. Ma, and J.-F. Liu, Electrically modulated SQUID with a single Josephson junction coupled by a time reversal breaking Weyl semimetal thin film, Phys. Rev. B 97, 035427 (2017).
(47)
J. Fang, W. Duan, J. Liu, C. Zhang, and Z. Ma, Pairing-dependent superconductivity gap and nonholonomic Andreev reflection in Weyl semimetal/Weyl superconductor heterojunctions, Phys. Rev. B 97, 165301 (2018).
(48)
N. Bovenzi, M. Breitkreiz, P. Baireuther, T. E. OâBrien, J. Tworzydlo, I. Adagideli, and C. W. J. Beenakker, Chirality blockade of Andreev reflection in a magnetic Weyl semimetal, Phys. Rev. B 96, 035437 (2017).
(49)
S. Das, Amit, A. Sirohi, L. Yadav, S. Gayen, Y. Singh, and G. Sheet, Conventional Superconductivity in Type-II Dirac Semimetal PdTe2, Phys. Rev. B 97, 014523 (2017).
(50)
A. Kononov, O.O. Shvetsov, S.V. Egorov, A.V. Timonina, N.N. Kolesnikov, E.V. Deviatov, Signature of Fermi arc surface states in Andreev reflection at the $\rm WTe_{2}$ Weyl semimetal surface, EuroPhys. Lett. 122, 27004 (2018).
(51)
A. Kononov, O.O. Shvetsov, S.V. Egorov, A.V. Timonina, N.N. Kolesnikov, E.V. Deviatov, $\pi$ periodicity in Josephson effect for a three-dimensional Weyl semimetal $\rm WTe_{2}$ , arXiv:1809.03902.
(52)
C. Heikes, I.-L. Liu, T. Metz, C. Eckberg, P. Neves, Y. Wu, L. Hung, P. Piccoli, H. Cao, J. Leao, J. Paglione, T. Yildirim, N. P. Butch, W. Ratcliff, Mechanical control of crystal symmetry and superconductivity in Weyl semimetal $\rm MoTe_{2}$, arXiv:1804.09093.
(53)
S. Teknowijoyo, N. H. Jo, M. S. Scheurer, M. A. Tanatar, K. Cho, S. L. Bud’ko, P. P. Orth, P. C. Canfield, R. Prozorov, Nodeless superconductivity in type-II Dirac semimetal $\rm PdTe_{2}$: low-temperature London penetration depth and symmetry analysis, arXiv:1804.00723.
(54)
A. Sirohi, S. Das, R. R. Chowdhuri, A., Y. Singh, S. Gayen, G. Sheet, Mixed type I and type-II superconductivity due to intrinsic electronic inhomogeneities in the type-II Dirac semimetal $\rm PdTe_{2}$, arXiv:1804.10837.
(55)
P. Lu, J.-S. Kim, J. Yang, H. Gao, J. Wu, D. Shao, B. Li, D. Zhou, J. Sun, D. Akinwande, D. Xing, and J.-F. Lin, Origin of superconductivity in the Weyl semimetal $\rm WTe_{2}$ under pressure, Phys. Rev. B 94, 224512 (2016).
(56)
H. Takahashi, T. Akiba, K. Imura, T. Shiino, K. Deguchi, N. K. Sato, H. Sakai, M. S. Bahramy, and S. Ishiwata, Anticorrelation between polar lattice instability and superconductivity in the Weyl semimetal candidate $\rm MoTe_{2}$, Phys. Rev. B 95, 100501(R) (2017).
(57)
Z. Guguchia, F. von Rohr, Z. Shermadini, A. T. Lee, S. Banerjee, A. R. Wieteska, C. A. Marianetti, B. A. Frandsen, H. Luetkens, Z. Gong, S. C. Cheung, C. Baines, A. Shengelaya, G. Taniashvili, A. N. Pasupathy, E. Morenzoni, S. J. L. Billinge, A. Amato, R. J. Cava, R. Khasanov, and Y. J. Uemura, Signatures of the topological $s^{\pm}$ superconducting order parameter in the type-II Weyl semimetal $\rm T_{d}MoTe_{2}$, Nat. Comm. 8, 1082 (2017).
(58)
Q. Li, C. He, Y. Wang, E. Liu, M. Wang, Y. Wang, J. Zeng, Z. Ma, T. Cao, C. Yi, N. Wang, K. Watanabe, T. Taniguchi, L. Shao, Y. Shi, X. Chen, S.-J. Liang, Q.-H. Wang, F. Miao, Proximity-Induced Superconductivity with Subgap Anomaly in Type-II Weyl Semi-Metal $\rm WTe_{2}$, Nano Lett. (2018).
(59)
G. E. Volovik, Exotic Lifshitz transitions in topological materials, arXiv:1701.06435.
(60)
M. Alidoust, K. Halterman, and A. A. Zyuzin, Superconductivity in type-II Weyl semimetals, Phys. Rev. B 95, 155124 (2017).
(61)
D. Li, B. Rosenstein, B. Ya. Shapiro, and I. Shapiro, Effect of the type I to type II Weyl semimetal topological transition on superconductivity, Phys. Rev. B 95, 094513 (2017).
(62)
E. Y. Ma, M. R. Calvo, J. Wang, B. Lian,
M. Muhlbauer, C. Brune, Y.-T. Cui, K. Lai, W. Kundhikanjana,
Y. Yang, M. Baenninger, M. Konig, C. Ames, H. Buhmann, P. Leubner,
L. W. Molenkamp, S.-C. Zhang, D. Goldhaber-Gordon, M. A. Kelly, and
Z.-X. Shen, Unexpected edge conduction in mercury telluride
quantum wells under broken time-reversal symmetry,
Nat. Commun. 6 7252 (2015).
(63)
A. Zyuzin, M. Alidoust, D. Loss, Josephson
junction through a disordered topological insulator with helical
magnetization, Phys. Rev. B 93, 214502 (2016).
(64)
I. V. Bobkova, A. M. Bobkov, A. A. Zyuzin,
M. Alidoust, Magnetoelectrics in disordered topological
insulator Josephson junctions,
Phys. Rev. B
94, 134506 (2016).
(65)
M. Alidoust and H. Hamzehpour, Spontaneous supercurrent and
$\varphi_{0}$ phase shift parallel to magnetized topological insulator interfaces, Phys. Rev. B 96, 165422 (2017).
(66)
V. Braude and Yu. V. Nazarov, Fully developed triplet proximity effect, Phys. Rev. Lett. 98, 077003 (2007); R. Grein, M. Eschrig, G. Metalidis, and G. Schon, Spin-dependent Cooper Pair Phase and Pure Spin Supercurrents
in Strongly Polarized Ferromagnets, Phys. Rev. Lett. 102, 227005 (2009); M. Eschrig, T. Löfwander, Triplet supercurrents in clean and disordered half-metallic ferromagnets, Nature Phys. 4, 138 (2008); F. Konschelle, A. Buzdin, Magnetic moment manipulation by a Josephson current, Phys. Rev. Lett. 102, 017001 (2009); Y. Tanaka, T. Yokoyama, N. Nagaosa, Manipulation of the Majorana fermion, Andreev reflection, and Josephson current on topological insulators, Phys. Rev. Lett. 103, 107002 (2009).
(67)
A. Assouline, C. Feuillet-Palma, N. Bergeal, T. Zhang, A. Mottaghizadeh, A. Zimmers, E. Lhuillier, M. Marangolo, M. Eddrief, P. Atkinson, M. Aprili, H. Aubin, Spin-Orbit induced phase-shift in $\rm Bi_{2}Se_{3}$ Josephson junctions, arXiv:1806.01406.
(68)
G. Eilenberger, Transformation of Gorkov’s
equation for type II superconductors into transport-like
equations, Z. Phys. 214, 195 (1968).
(69)
K. D. Usadel, Generalized diffusion equation for superconducting alloys, Phys. Rev. Lett. 25, 507 (1977).
(70)
F. Konschelle, Transport equations for superconductors in the presence of spin interaction, Euro. Phys. J. B 87, 119
(2014).
(71)
F. S. Bergeret and I. V. Tokatly, Spin-orbit coupling as a source of long-range triplet proximity effect in superconductor-ferromagnet hybrid structures, Phys. Rev. B 89, 134517 (2014); I. V. Tokatly, Usadel equation in the presence of intrinsic spin-orbit coupling: A unified theory
of magnetoelectric effects in normal and superconducting systems, Phys. Rev. B 96, 060502(R) (2017).
(72)
C. Huang, I. V. Tokatly, and F. S. Bergeret, Extrinsic spin-charge coupling in diffusive superconducting systems, Phys. Rev. B 98, 144515 (2018).
(73)
M. Alidoust, K. Halterman, Spontaneous edge
accumulation of spin currents in finite-size two-dimensional
diffusive spin-orbit coupled SFS heterostructures, New J. Phys. 17,
033001 (2015).
(74)
M. Alidoust, K. Halterman, Long-range spin-triplet correlations and edge spin currents in diffusive spin-orbit coupled SNS hybrids with a single spin-active interface, J. Phys: Cond. Matt. 27, 235301 (2015).
(75)
H. Chakraborti, S. Deb, R. Schott, V. Thakur, A. Chatterjee, S. Yadav, R. K. Saroj, A. D. D Wieck, S. M. Shivaprasad, K. Das Gupta, and S. Dhar, Coherent transmission of superconducting carriers through a $\sim 2$ micron polar semiconductor, Superconductor Science and Tech. (2018).
(76)
M. Alidoust and K. Halterman, Proximity induced vortices and
long-range triplet supercurrents in ferromagnetic Josephson
junctions and spin valves,
J. Appl. Phys. 117, 123906 (2015).
(77)
A. V. Zaitsev, Sov. Phys. JETP 59, 1015 (1984); M. Y. Kuprianov and V. F. Lukichev, Sov. Phys. JETP 67, 1163 (1988).
(78)
J. Wiedenmann, E. Bocquillon, R. S. Deacon, S. Hartinger, O. Herrmann, T. M. Klapwijk, L. Maier, C. Ames, C. Brüne, C. Gould, A. Oiwa, K. Ishibashi, S. Tarucha, H. Buhmann and L. W. Molenkamp, $4\pi$-periodic Josephson supercurrent in $\rm HgTe$-based topological Josephson junctions, Nat. Comm. 7, 10303 (2016); C. Li, J. C. de Boer, B. de Ronde, S. V. Ramankutty, E. van Heumen, Y. Huang, A. de Visser, A. A. Golubov, Mark S. Golden and A. Brinkman, $4\pi$-periodic Andreev bound states in a Dirac semimetal, Nat. Mat. 17, 875 (2018).
(79)
M. Alidoust, A. Zyuzin, K.
Halterman, Pure Odd Frequency Superconductivity at the
Cores of Proximity Vortices, Phys. Rev. B 95, 045115 (2017). |
Two-parameter scaling theory of the longitudinal magnetoconductivity in a Weyl metal phase: Chiral anomaly, weak disorder, and finite temperature
Kyoung-Min Kim${}^{1}$, Dongwoo Shin${}^{1}$, M. Sasaki${}^{2}$, Heon-Jung Kim${}^{3}$, Jeehoon Kim${}^{1,4}$, and Ki-Seok Kim${}^{1}$
${}^{1}$Department of Physics, POSTECH, Pohang, Gyeongbuk 790-784, Korea
${}^{2}$Department of Physics, Faculty of Science, Yamagata University, Kojirakawa, Yamagata 990-8560, Japan
${}^{3}$Department of Physics, College of Natural Science, Daegu University, Gyeongbuk 712-714, Korea
${}^{4}$Center for Artificial Low Dimensional Electronic Systems, Institute for Basic Science, 77 Cheongam-Ro, Nam-Gu, Pohang 790-784, Korea
(November 27, 2020)
Abstract
It is at the heart of modern condensed matter physics to investigate the role of a topological structure in anomalous transport phenomena. In particular, chiral anomaly turns out to be the underlying mechanism for the negative longitudinal magnetoresistivity in a Weyl metal phase. Existence of a dissipationless current channel causes enhancement of electric currents along the direction of a pair of Weyl points or applied magnetic fields ($B$). However, temperature ($T$) dependence of the negative longitudinal magnetoresistivity has not been understood yet in the presence of disorder scattering since it is not clear at all how to introduce effects of disorder scattering into the “topological-in-origin” transport coefficient at finite temperatures. The calculation based on the Kubo formula of the current-current correlation function is simply not known for this anomalous transport coefficient. Combining the renormalization group analysis with the Boltzmann transport theory to encode the chiral anomaly, we reveal how disorder scattering renormalizes the distance between a pair of Weyl points and such a renormalization effect modifies the topological-in-origin transport coefficient at finite temperatures. As a result, we find breakdown of $B/T$ scaling, given by $B/T^{1+\eta}$ with $0<\eta<1$. This breakdown may be regarded to be a fingerprint of the interplay between disorder scattering and topological structure in a Weyl metal phase.
I Introduction
Researches on the role of topological-in-origin terms in quantum phases and their transitions have been a driving force for modern condensed matter physics, which cover quantum spin chains 1D_QFT_Textbook and deconfined quantum criticality DQCP_Senthil ; DQCP_Tanaka , quantum Hall effects and topological phases of matter Fradkin_Textbook , Anderson localization for the classification of topological phases and their phase transitions Mirlin_Review , and so on. In particular, renormalization effects of such topological terms are responsible for novel universality classes beyond the Landau-Ginzburg-Wilson paradigm of phase transitions with symmetry breaking. However, it is quite a nontrivial task to perform the renormalization group analysis in the presence of the topological-in-origin term, even if it can be taken into account perturbatively for the contribution of a bulk sometimes. Frequently, non-perturbative effects should be introduced into the renormalization group analysis PP_Transition_IQHE ; Fu_Kane_NLsM , uncontrolled in this situation and thus, being under debates as an open question.
In this study we investigate disorder-driven renormalization of a topological-in-origin term referred to as an inhomogeneous $\theta-$term in three spatial dimensions WM_Review_Kim , defined by
$$\displaystyle\mathcal{F}=-\frac{1}{\beta}\int_{-\infty}^{\infty}dv(\bm{r})P[v(%
\bm{r})]\ln\int D\bar{\psi}(\bm{r},\tau)D\psi(\bm{r},\tau)$$
$$\displaystyle\exp\Big{[}-\int_{0}^{\beta}d\tau\int d^{3}\bm{r}\Big{\{}\bar{%
\psi}(\bm{r},\tau)i\gamma_{\mu}[\partial_{\mu}-ieA_{\mu}(\bm{r},\tau)]\psi(\bm%
{r},\tau)$$
$$\displaystyle+v(\bm{r})\bar{\psi}(\bm{r},\tau)\gamma_{\tau}\psi(\bm{r},\tau)-%
\frac{1}{4}F_{\mu\nu}(\bm{r},\tau)F_{\mu\nu}(\bm{r},\tau)$$
$$\displaystyle+\theta(\bm{r})\frac{e^{2}}{16\pi^{2}}\varepsilon_{\mu\nu\gamma%
\delta}F_{\mu\nu}(\bm{r},\tau)F_{\gamma\delta}(\bm{r},\tau)\Big{\}}\Big{]}.$$
(1)
$\psi(\bm{r},\tau)$ is a four-component Dirac spinor to describe an electron field of spin $1/2$ in two orbitals. Its dynamics is given by a Dirac theory, where $\gamma_{\mu}$ with $\mu=(\tau,x,y,z)$ is a Dirac matrix to satisfy the Clifford algebra. $A_{\mu}(\bm{r},\tau)$ and $F_{\mu\nu}(\bm{r},\tau)=\partial_{\mu}A_{\nu}(\bm{r},\tau)-\partial_{\nu}A_{%
\mu}(\bm{r},\tau)$ are an externally applied electromagnetic field and its field strength tensor, respectively. $v(\bm{r})$ is a potential configuration, given randomly and described by the Gaussian probability distribution $P[v(\bm{r})]=\mathcal{N}\exp\Big{(}-\int d^{3}\bm{r}\frac{[v(\bm{r})]^{2}}{2%
\Gamma}\Big{)}$. $\Gamma$ is the variance of the disorder distribution and $\mathcal{N}$ is a normalization constant, determined by $\int_{-\infty}^{\infty}dv(\bm{r})P[v(\bm{r})]=1$. The last term is an inhomogeneous $\theta-$term, topological in its origin and keeping chiral anomaly that the chiral current is not conserved in the quantum level QFT_Textbook , given by
$$\displaystyle\partial_{\mu}[\bar{\psi}(\bm{r},\tau)\gamma_{\mu}\gamma_{5}\psi(%
\bm{r},\tau)]=-\frac{e^{2}}{16\pi^{2}}\varepsilon_{\mu\nu\gamma\delta}F_{\mu%
\nu}(\bm{r},\tau)F_{\gamma\delta}(\bm{r},\tau).$$
$\gamma_{5}$ is chiral Dirac matrix to anticommute with $\gamma_{\mu}$. Here, the problem is how the inhomogeneous $\theta-$term becomes renormalized via the disorder scattering.
This problem can be cast into more physical terms. Introducing the chiral-anomaly equation into the effective field theory and performing the integration-by-parts for the chiral-current term with the $\theta(\bm{r})$ coefficient Kyoung , we obtain
$$\displaystyle\mathcal{F}=-\frac{1}{\beta}\int_{-\infty}^{\infty}dv(\bm{r})P[v(%
\bm{r})]\ln\int D\bar{\psi}(\bm{r},\tau)D\psi(\bm{r},\tau)$$
$$\displaystyle\exp\Big{[}-\int_{0}^{\beta}d\tau\int d^{3}\bm{r}\Big{\{}\bar{%
\psi}(\bm{r},\tau)i\gamma_{\mu}[\partial_{\mu}-ieA_{\mu}(\bm{r},\tau)]\psi(\bm%
{r},\tau)$$
$$\displaystyle+c_{\mu}(\bm{r},\tau)\bar{\psi}(\bm{r},\tau)\gamma_{\mu}\gamma_{5%
}\psi(\bm{r},\tau)+v(\bm{r})\bar{\psi}(\bm{r},\tau)\gamma_{\tau}\psi(\bm{r},\tau)$$
$$\displaystyle-\frac{1}{4}F_{\mu\nu}(\bm{r},\tau)F_{\mu\nu}(\bm{r},\tau)\Big{\}%
}\Big{]}.$$
(3)
$c_{\mu}(\bm{r},\tau)=\partial_{\mu}\theta(\bm{r})$ is referred to as chiral gauge field, regarded to be a background potential given by the inhomogeneous $\theta$ coefficient. When the background chiral gauge field serves a homogeneous potential, the resulting spectrum turns out to describe dynamics of Weyl electrons. The right-handed helicity part shifts into the right hand side and the left-handed helicity part does into the left Weyl_Metal_I ; Weyl_Metal_II ; Weyl_Metal_III . Physically, this homogeneous chiral-gauge-field potential is realized as $\bm{c}=\bm{\nabla}\theta(\bm{r})=g\bm{B}$, applying a homogeneous magnetic field $\bm{B}$ into a gapless semiconductor described above. The Dirac point separates into a pair of Weyl points along the direction of the applied magnetic field and the distance of the pair of Weyl points is proportional to the strength of the applied magnetic field with a Lande$-g$ factor. See the supplementary material. As a result, the previous mathematically defined problem is actually how the background chiral gauge field, more physically, the distance between a pair of Weyl points becomes renormalized by random elastic scattering.
The renormalization effect of the distance between a pair of Weyl points is measurable experimentally since the information is encoded into the negative longitudinal magnetoresistivity. This anomalous transport phenomena in a Weyl metal phase has been well known for more than thirty years NLMR_Theory_Original and experimentally confirmed firstly in 2013 NLMR_Experiment_Original . The electrical resistivity measured along the direction of the applied magnetic field becomes smaller than that measured in other directions. More quantitatively, the magnetoconductivity is enhanced in the longitudinal setup, i.e., $\bm{E}~{}\|~{}\bm{B}$, as follows
$$\displaystyle\sigma_{L}(B)=\sigma_{D}(1+\mathcal{C}_{W}B^{2}),$$
(4)
where $\bm{E}$ is an applied electric field Boltzmann_Spivak_Tong . $\sigma_{D}$ is the Drude conductivity determined purely by disorder scattering. In real experiments, quantum corrections by weak anti-localization are introduced into the Drude conductivity Boltzmann_Kim_I . $\mathcal{C}_{W}$ is a positive coefficient, discussed later in more detail. An essential point is that the enhancement of the longitudinal magnetoconductivity is given by the square of the distance between the pair of Weyl points. This longitudinal enhancement can be figured out in the following way: There exists a dissipationless current channel as a vacuum state, which connects the pair of Weyl points, responsible for the chiral anomaly. As a result, electrical currents are allowed to flow better along this direction through this vacuum channel although the measured longitudinal magnetoconductivity does not result from such dissipationless electrical currents NLMR_Theory_Original . When the distance between the pair of Weyl points is renormalized by random elastic scattering, the positive coefficient $\mathcal{C}_{W}$ would evolve as a function of an energy scale, here, temperature. It is natural to expect finding a scaling theory for the chiral-anomaly-driven enhanced longitudinal magnetoconductivity.
The above discussion reminds us of a two-parameter scaling theory for the Anderson localization in topological phases of matter Altland_Two_Parameter_Scaling , including the plateau-plateau transition in the integer quantum Hall effect PP_Transition_IQHE . There, the transport phenomenon of the Anderson localization transition is determined by the “transverse” conductivity $\sigma_{xx}$ and the Hall conductivity $\sigma_{xy}$, where the latter encodes the topological information of the integer quantum Hall effect. The present situation is quite analogous to that of the integer quantum Hall effect. $\sigma_{xx}$ in the quantum Hall effect is identified with the Drude conductivity $\sigma_{D}$, determined by disorder scattering directly. On the other hand, $\sigma_{xy}$ in the quantum Hall effect is analogous to the distance between the pair of Weyl points, where the renormalization effect is introduced into the temperature dependence of $\mathcal{C}_{W}$.
In this study we investigate the longitudinal magnetoconductivity at finite temperatures and find a two-parameter scaling theory, where renormalization effects result from random elastic scattering. There is one difficult point in the calculation of the longitudinal magnetoconductivity in a Weyl metal phase. It turns out that a naive Kubo-formula calculation does not incorporate the role of the chiral anomaly in the longitudinal magnetoconductivity TFLT_Kim . As a result, we fail to find the $B^{2}$ enhancement of the longitudinal magnetoconductivity within the Kubo-formula calculation. In this respect our strategy consists of a two-fold way: First, we perform the renormalization group analysis and find how the distance between a pair of Weyl points evolves as a function of an energy scale or temperature. Second, introducing this information into the Boltzmann transport theory with chiral anomaly, we reveal the longitudinal negative magnetoconductivity as a function of both the applied magnetic field and temperature, given by
$$\displaystyle\sigma_{L}(B,T)\approx\sigma_{D}(T)[1+\mathcal{C}_{W}(T)B^{2}].$$
(5)
In particular, we find breakdown of $B/T$ scaling
$$\displaystyle\Delta\sigma_{L}(B,T)\equiv\frac{\sigma_{L}(B,T)-\sigma_{D}(T)}{%
\sigma_{D}(T)}=\mathcal{C}_{W}T_{0}^{2(1+\eta)}\Big{(}\frac{B}{T^{1+\eta}}\Big%
{)}^{2},$$
where $\eta$ is a scaling exponent with $0<\eta<1$ and $T_{0}$ is an energy scale. We claim that this breakdown may be regarded to be a fingerprint of the interplay between disorder scattering and topological structure in a Weyl metal phase.
II Renormalization for the distance between a pair of Weyl points via disorder-driven inter-valley scattering
II.1 Effective field theory for a Weyl metal phase with disorder: Replica theory
We start from an effective Hamiltonian density for a Weyl metal phase with time reversal symmetry breaking
$$\displaystyle\mathcal{H}_{B}=\psi^{\dagger}_{B}(\bm{x})\Big{(}v_{B}\bm{\alpha}%
\cdot(-\imath\nabla)+g_{B}\bm{B}\cdot\bm{\sigma}\otimes I_{2\times 2}\Big{)}%
\psi_{B}(\bm{x}).$$
(7)
$\psi_{B}(\bm{x})=(\psi_{BR}(\bm{x}),\psi_{BL}(\bm{x}))^{T}$ is a four-component Dirac-spinor field in a two-component Weyl-spinor field with right(R)-left(L) chirality, and $v_{B}$ is the velocity of such fermions. $\bm{B}$ is an externally applied magnetic field with a Lande$-g$ factor $g_{B}$, splitting the Dirac band into a pair of Weyl bands (Fig. 1). $\bm{\alpha}$ is a four-by-four matrix, given by $\bm{\alpha}=\bm{\sigma}\otimes\sigma_{z}$, where $\bm{\sigma}$ is a Pauli matrix. The subscript $B$ denotes “bare”, meaning that this effective Hamiltonian density is defined at an ultraviolet (UV) scale.
We consider two types of random potentials, introducing “intra-valley scattering” $\psi_{B}^{\dagger}(\bm{x})V_{B}(\bm{x})\psi_{B}(\bm{x})$ and “inter-valley scattering” $\psi_{B}^{\dagger}(\bm{x})U_{B}(\bm{x})\big{(}I_{2\times 2}\otimes\sigma_{x}%
\big{)}\psi_{B}(\bm{x})$ into the effective Hamiltonian. Then, we obtain the following effective action
$$\displaystyle S_{B}[\bar{\psi}_{B}(x),\psi_{B}(x);V_{B}(\bm{x}),U_{B}(\bm{x})]$$
$$\displaystyle=\int d^{4}x\big{\{}\bar{\psi}_{B}(x)\big{(}\gamma^{0}\partial_{0%
}+\imath v_{B}\gamma^{k}\partial_{k}+c_{{B}\mu}\gamma^{\mu}\gamma^{5}\big{)}%
\psi_{B}(x)$$
$$\displaystyle+\bar{\psi}_{B}(x)\gamma^{0}V_{B}(\bm{x})\psi_{B}(x)+\bar{\psi}_{%
B}(x)U_{B}(\bm{x})\psi_{B}(x)\big{\}}$$
(8)
with $\bar{\psi}_{B}(x)\equiv\psi_{B}^{\dagger}(x)\gamma^{0}$. Here, gamma matrices are given in the Weyl representation, for example, $\gamma^{0}=I_{2\times 2}\otimes\sigma_{x}$. A magnetic field is generalized to be a chiral gauge field $c_{{B}\mu}=(c_{B0},c_{Bk}\equiv g_{B}B_{k})$. $x$ means “space-time”, given by $x^{\mu}=(\tau,\bm{x})$. See the supplementary material.
A physical observable in this system is measured as follows
$$\displaystyle\left\langle\mathcal{O}[\bar{\psi}_{B}(x),\psi_{B}(x)]\right%
\rangle=\int\mathcal{D}V_{B}(\bm{x})\mathcal{D}U_{B}(\bm{x})P_{B}[V_{B}(\bm{x}%
),U_{B}(\bm{x})]$$
$$\displaystyle\frac{\int\mathcal{D}\bar{\psi}_{B}(x)\mathcal{D}\psi_{B}(x)%
\mathcal{O}[\bar{\psi}_{B}(x),\psi_{B}(x)]e^{-S_{B0}[\bar{\psi}_{B}(x),\psi_{B%
}(x)]}e^{-\int d^{4}x\bar{\psi}_{B}(x)[\gamma^{0}V_{B}(\bm{x})+U_{B}(\bm{x})]%
\psi_{B}(x)}}{\int\mathcal{D}\bar{\psi}_{B}(x)\mathcal{D}\psi_{B}(x)e^{-S_{B0}%
[\bar{\psi}_{B}(x),\psi_{B}(x)]}e^{-\int d^{4}x\bar{\psi}_{B}(x)[\gamma^{0}V_{%
B}(\bm{x})+U_{B}(\bm{x})]\psi_{B}(x)}},$$
(9)
where the free part of the effective action is $S_{B0}[\bar{\psi}_{B}(x),\psi_{B}(x)]=\int d^{4}x\bar{\psi}_{B}(x)\big{(}%
\gamma^{0}\partial_{0}+\imath v_{B}\gamma^{k}\partial_{k}+c_{B\mu}\gamma^{\mu}%
\gamma^{5}\big{)}\psi_{B}(x)$. Resorting to the replica trick and performing the average for disorder with the Gaussian distribution function of $P_{B}[V_{B}(\bm{x}),U_{B}(\bm{x})]=N_{B}\exp{\Big{[}-\frac{\int d^{3}\bm{x}V_{%
B}^{2}(\bm{x})}{2\Gamma_{BV}}-\frac{\int d^{3}\bm{x}U_{B}^{2}(\bm{x})}{2\Gamma%
_{BU}}\Big{]}}$, the above expression is reformulated as follows
$$\displaystyle\left\langle\mathcal{O}[\bar{\psi}_{B}(x),\psi_{B}(x)]\right%
\rangle=\lim_{R\to 0}\frac{1}{R}\sum_{a=1}^{R}\int\mathcal{D}\bar{\psi}_{B}^{a%
}(x)\mathcal{D}\psi_{B}^{a}(x)$$
$$\displaystyle\mathcal{O}[\bar{\psi}_{B}^{a}(x),\psi_{B}^{a}(x)]\exp\Big{\{}-%
\sum_{a=1}^{R}S_{B0}[\bar{\psi}_{B}^{a}(x),\psi_{B}^{a}(x)]$$
$$\displaystyle-\sum_{b,c=1}^{R}S_{B\textrm{dis}}[\bar{\psi}_{B}^{b}(x),\psi_{B}%
^{b}(x),\bar{\psi}_{B}^{c}(x),\psi_{B}^{c}(x)]\Big{\}}.$$
(10)
Here, $N_{B}$ is a normalization constant and $\Gamma_{BV(U)}$ is a variance for the disorder distribution. As a result, the effective interaction term induced by disorder scattering is
$$\displaystyle S_{B\textrm{dis}}[\bar{\psi}_{B}^{b}(x),\psi_{B}^{b}(x),\bar{%
\psi}_{B}^{c}(x),\psi_{B}^{c}(x)]$$
$$\displaystyle=$$
$$\displaystyle-\int^{\beta}_{0}d\tau\int^{\beta}_{0}d\tau^{\prime}\int d^{3}\bm%
{x}\frac{\Gamma_{BV}}{2}\bar{\psi}_{B}^{b}(\tau,\bm{x})\gamma^{0}\psi_{B}^{b}(%
\tau,\bm{x})\bar{\psi}_{B}^{c}(\tau^{\prime},\bm{x})\gamma^{0}\psi_{B}^{c}(%
\tau^{\prime},\bm{x})$$
(11)
$$\displaystyle-\int^{\beta}_{0}d\tau\int^{\beta}_{0}d\tau^{\prime}\int d^{3}\bm%
{x}\frac{\Gamma_{BU}}{2}\bar{\psi}_{B}^{b}(\tau,\bm{x})\psi_{B}^{b}(\tau,\bm{x%
})\bar{\psi}_{B}^{c}(\tau^{\prime},\bm{x})\psi_{B}^{c}(\tau^{\prime},\bm{x}).$$
The effective field theory is given by $S_{B}[\bar{\psi}_{B}^{a}(x),\psi_{B}^{a}(x)]=S_{B0}[\bar{\psi}_{B}^{a}(x),\psi%
_{B}^{a}(x)]+S_{B\textrm{dis}}[\bar{\psi}_{B}^{b}(x),\psi_{B}^{b}(x),\bar{\psi%
}_{B}^{c}(x),\psi_{B}^{c}(x)]$.
II.2 Renormalization group analysis: Role of inter-valley scattering in the distance between a pair of Weyl point
In order to perform the renormalization group analysis within the dimensional regularization QFT_Textbook , we rewrite $S_{B}[\bar{\psi}_{B}^{a}(x),\psi_{B}^{a}(x)]$, the effective bare action of bare field variables in terms of $S_{R}[\bar{\psi}_{R}^{a},\psi_{R}^{a}]$, the effective renormalized action of renormalized field variables with $S_{CT}[\bar{\psi}_{R}^{a},\psi_{R}^{a}]$, counter terms of renormalized field variables
$$\displaystyle S_{R}[\bar{\psi}_{R}^{a},\psi_{R}^{a}]$$
$$\displaystyle=$$
$$\displaystyle\int d^{d+1}x\bar{\psi}_{R}^{a}\big{(}\gamma^{0}\partial_{0}+v_{R%
}\imath\gamma^{k}\partial_{k}+c_{R0}\gamma^{0}\gamma^{5}+c_{Rk}\gamma^{k}%
\gamma^{5}\big{)}\psi_{R}^{a}$$
$$\displaystyle-$$
$$\displaystyle\int d\tau\int d\tau^{\prime}\int d^{d}\bm{x}\frac{\Gamma_{RV}}{2%
}(\bar{\psi}_{R}^{b}\gamma^{0}\psi_{R}^{b})_{\tau}(\bar{\psi}_{R}^{c}\gamma^{0%
}\psi_{R}^{c})_{\tau^{\prime}}-\int d\tau\int d\tau^{\prime}\int d^{d}\bm{x}%
\frac{\Gamma_{RU}}{2}(\bar{\psi}_{R}^{b}\psi_{R}^{b})_{\tau}(\bar{\psi}_{R}^{c%
}\psi_{R}^{c})_{\tau^{\prime}},$$
$$\displaystyle S_{CT}[\bar{\psi}_{R}^{a},\psi_{R}^{a}]$$
$$\displaystyle=$$
$$\displaystyle\int d^{d+1}x\bar{\psi}_{R}^{a}\big{(}\delta_{\psi}^{\omega}%
\gamma^{0}\partial_{0}+\delta_{\psi}^{\bm{k}}v_{R}\imath\gamma^{k}\partial_{k}%
+\delta_{c0}c_{R0}\gamma^{0}\gamma^{5}+\delta_{\bm{c}}c_{Rk}\gamma^{k}\gamma^{%
5}\big{)}\psi_{R}^{a}$$
$$\displaystyle-$$
$$\displaystyle\int d\tau\int d\tau^{\prime}\int d^{d}\bm{x}\frac{\delta_{\Gamma
V%
}\Gamma_{RV}}{2}(\bar{\psi}_{R}^{b}\gamma^{0}\psi_{R}^{b})_{\tau}(\bar{\psi}_{%
R}^{c}\gamma^{0}\psi_{R}^{c})_{\tau^{\prime}}-\int d\tau\int d\tau^{\prime}%
\int d^{d}\bm{x}\frac{\delta_{\Gamma U}\Gamma_{RU}}{2}(\bar{\psi}_{R}^{b}\psi_%
{R}^{b})_{\tau}(\bar{\psi}_{R}^{c}\psi_{R}^{c})_{\tau^{\prime}},$$
where $S_{B}[\bar{\psi}_{B}^{a},\psi_{B}^{a}]=S_{R}[\bar{\psi}_{R}^{a},\psi_{R}^{a}]+%
S_{CT}[\bar{\psi}_{R}^{a},\psi_{R}^{a}]$. It is straightforward to see how bare quantities are related with renormalized ones, given by
$$\displaystyle\psi_{B}^{a}=(Z_{\psi}^{\omega})^{\frac{1}{2}}\psi_{R}^{a},~{}~{}%
~{}v_{B}=Z_{\psi}^{\bm{k}}(Z_{\psi}^{\omega})^{-1}v_{R},$$
$$\displaystyle c_{B0}=Z_{c0}(Z_{\psi}^{\omega})^{-1}c_{R0},~{}~{}~{}c_{Bk}=Z_{%
\bm{c}}(Z_{\psi}^{\omega})^{-1}c_{Rk},$$
$$\displaystyle\Gamma_{BV}=Z_{\Gamma V}(Z_{\psi}^{\omega})^{-2}\Gamma_{RV},~{}~{%
}~{}\Gamma_{BU}=Z_{\Gamma U}(Z_{\psi}^{\omega})^{-2}\Gamma_{RU},$$
where $Z_{\psi}^{\omega}=1+\delta_{\psi}^{\omega}$, $Z_{\psi}^{\bm{k}}=1+\delta_{\psi}^{\bm{k}}$, $Z_{c0}=1+\delta_{c0}$, $Z_{\bm{c}}=1+\delta_{\bm{c}}$, $Z_{\Gamma V}=1+\delta_{\Gamma V}$, and $Z_{\Gamma U}=1+\delta_{\Gamma U}$.
Dimensional analysis gives $\textrm{dim}[\Gamma]=2-d$. In this respect we perform the dimensional regularization in $d=2+\varepsilon$ where $\varepsilon$, a “small” parameter to control the present renormalization group analysis, will be analytically continued to $\varepsilon=1$ in the end. Performing the standard procedure for the renormalization group analysis, we find renormalization group equations, where both vertex and self-energy corrections are introduced self-consistently. See Fig. 2, where all quantum corrections are shown as Feynman’s diagrams up to the two-loop order for self-energy corrections and the one-loop order for vertex corrections. All details are shown in the supplementary material. As a result, we find counter terms with
$$\displaystyle\delta_{\psi}^{\omega}=\frac{\Gamma_{V}+\Gamma_{U}}{2\pi%
\varepsilon}-\frac{5\Gamma_{V}^{2}+12\Gamma_{V}\Gamma_{U}+7\Gamma_{U}^{2}}{48%
\pi^{2}\varepsilon},$$
$$\displaystyle\delta_{\bm{c}}=\frac{\Gamma_{V}^{2}-\Gamma_{U}^{2}}{16\pi^{2}%
\varepsilon},~{}~{}\delta_{\Gamma V}=\frac{\Gamma_{V}+\Gamma_{U}}{2\pi%
\varepsilon},~{}~{}\delta_{\Gamma U}=-\frac{\Gamma_{V}+\Gamma_{U}}{2\pi%
\varepsilon},$$
Inserting these divergent coefficients into equations (LABEL:r.renormalization_equations) and performing derivatives with respect to an energy scale for renormalization given by $\ln M$, we find renormalization group equations
$$\displaystyle\frac{d\Gamma_{V}}{d\ln{M}}$$
$$\displaystyle=$$
$$\displaystyle\Gamma_{V}-\frac{a_{\Gamma}}{3}\Gamma_{V}(\Gamma_{V}+\Gamma_{U})$$
$$\displaystyle+$$
$$\displaystyle b_{\Gamma}\Gamma_{V}(\Gamma_{V}+\Gamma_{U})(c_{\Gamma}\Gamma_{V}%
+\Gamma_{U}),$$
$$\displaystyle\frac{d\Gamma_{U}}{d\ln{M}}$$
$$\displaystyle=$$
$$\displaystyle\Gamma_{U}-a_{\Gamma}\Gamma_{U}(\Gamma_{V}+\Gamma_{U})$$
$$\displaystyle+$$
$$\displaystyle b_{\Gamma}\Gamma_{U}(\Gamma_{V}+\Gamma_{U})(c_{\Gamma}\Gamma_{V}%
+\Gamma_{U}),$$
$$\displaystyle\frac{dc_{k}}{d\ln{M}}$$
$$\displaystyle=$$
$$\displaystyle c_{k}\Big{[}-1-a_{\bm{c}}(\Gamma_{V}+\Gamma_{U})$$
(15)
$$\displaystyle+$$
$$\displaystyle b_{\bm{c}}(\Gamma_{V}+\Gamma_{U})(2\Gamma_{V}+\Gamma_{U})\Big{]},$$
where positive numerical constants are given by
$$\displaystyle a_{\Gamma}=\frac{3}{2\pi},~{}~{}b_{\Gamma}=\frac{7}{24\pi^{2}},~%
{}~{}c_{\Gamma}=\frac{5}{7},~{}~{}a_{\bm{c}}=\frac{1}{2\pi},~{}~{}b_{\bm{c}}=%
\frac{1}{12\pi^{2}}.$$
Fig. 3 shows renormalization group flows for physical parameters according to Eq. (15). In the plane of $(\Gamma_{V},\Gamma_{U})$, we find two stable fixed points corresponding to two phases of a disordered Weyl metal state, and one unstable fixed point corresponding to the phase transition point between two phases: (1) The stable fixed point of $(0,\Gamma_{0})$ with $\Gamma_{0}=0$ represents a clean Weyl metal phase, protected for the case of weak disorder by the pseudogap density of states of the Weyl metal state. (2) The stable fixed point of $(0,\Gamma_{2})$ with $\Gamma_{2}=\frac{a_{\Gamma}+\sqrt{a_{\Gamma}^{2}-4b_{\Gamma}}}{2b_{\Gamma}}%
\simeq 13.68$ is identified with a diffusive Weyl metal phase, analogous to the diffusive Fermi-liquid fixed point of a conventional metallic phase CMP_QFT_Textbook . (3) The unstable fixed point of $(0,\Gamma_{1})$ with $\Gamma_{1}=\frac{a_{\Gamma}-\sqrt{a_{\Gamma}^{2}-4b_{\Gamma}}}{2b_{\Gamma}}%
\simeq 2.09$ denotes a critical point to separate the diffusive Weyl metal phase from the clean Weyl metal state, the existence of which originates from the pseudogap density of states. Interestingly, all these fixed points lie at the line of $\Gamma_{V}=0$, which means that inter-valley scattering shows dominant effects over intra-valley scattering for the low-energy physics in the disordered Weyl metallic state. Naively, one may suspect that their roles are similar because of the similarity of their renormalization group equations. However, the magnitude of the one-loop correction for $\Gamma_{U}$ turns out to be three times larger than that for $\Gamma_{V}$, and thus, the renormalization group flow of $(\Gamma_{V},\Gamma_{U})$ is overwhelmed by $\Gamma_{U}$. As a result, there is no chance by which $\Gamma_{V}$ has a non-trivial fixed point value. Detailed analysis on this issue is given in the supplementary material (Fig. 16).
In order to figure out how the distance between the pair of Weyl points renormalizes as a function of an energy scale, we focus on renormalization group equations for $\Gamma_{U}$ and $c_{k}$ at $\Gamma_{V}=0$
$$\displaystyle\frac{d\Gamma_{U}}{d\ln{M}}$$
$$\displaystyle=$$
$$\displaystyle\Gamma_{U}-a_{\Gamma}\Gamma_{U}^{2}+b_{\Gamma}\Gamma_{U}^{3}$$
(16)
$$\displaystyle\frac{dc_{k}}{d\ln{M}}$$
$$\displaystyle=$$
$$\displaystyle c_{k}\big{[}-1-a_{\bm{c}}\Gamma_{U}+b_{\bm{c}}\Gamma_{U}^{2}\big%
{]}.$$
(17)
It is straightforward to solve the first equation and find an approximate solution for $\Gamma_{U}$ near each fixed point at $\Gamma_{V}=0$. Inserting such fixed-point solutions into the second equation, we uncover how the distance between the pair of Weyl points evolves as a function of temperature
$$c_{k}(T)=c_{k}(T_{0})\bigg{(}\frac{T_{0}}{T}\bigg{)}^{\lambda_{\bm{c},fn}},$$
(18)
where the energy scale $M$ has been replaced with temperature $T$. Critical exponents of $\lambda_{\bm{c},fn}$ are found to be
$$\displaystyle\lambda_{\bm{c},f0}$$
$$\displaystyle=$$
$$\displaystyle 1+a_{\bm{c}}\Gamma_{0}-b_{\bm{c}}\Gamma_{0}^{2}=1$$
(19)
$$\displaystyle\lambda_{\bm{c},f1}$$
$$\displaystyle=$$
$$\displaystyle 1+a_{\bm{c}}\Gamma_{1}-b_{\bm{c}}\Gamma_{1}^{2}\simeq 1.34$$
(20)
$$\displaystyle\lambda_{\bm{c},f2}$$
$$\displaystyle=$$
$$\displaystyle 1+a_{\bm{c}}\Gamma_{2}-b_{\bm{c}}\Gamma_{2}^{2}\simeq 1.60.$$
(21)
It turns out that the distance between a pair of Weyl points increases to reach infinity, regarded to be beyond the perturbative renormalization group analysis. However, the infinity should be considered as an artifact of the continuum approximation. If the Brillouin zone is taken into account in the effective field theory, there must be a maximum of the distance within the Brillouin zone. In this respect it is natural to modify the above scaling solution as follows
$$c_{k}(T)=c_{k}(T_{0})\bigg{(}\frac{T_{0}}{T+T_{M}}\bigg{)}^{\lambda_{\bm{c},fn%
}},$$
(22)
where $T_{M}$ is a cutoff scale in the low-energy limit. It is interesting to notice that disorder scattering changes the temperature-dependent exponent of $c_{k}$. Inter-valley scattering gives rise to fast enhancement of the distance between a pair of Weyl points at low temperatures. This looks counter-intuitive, where anti-screening instead of screening arises from inter-valley scattering.
III Two-parameter scaling theory for the longitudinal magnetoconductivity of a disordered Weyl metal phase within Boltzmann transport theory
The question to address in this study is to find a scaling theory for the longitudinal magnetoconductivity. As discussed in the introduction, not only the Drude conductivity but also the distance between a pair of Weyl points or the spatial gradient of the inhomogeneous $\theta(\bm{r})$ coefficient in the topological-in-origin $\bm{E}\cdot\bm{B}$ term should be taken into account for the longitudinal magnetoconductivity in the Weyl metal phase. This situation is analogous to that of a plateau-plateau transition in the integer quantum Hall effect: Not only the Drude conductivity but also the Hall conductivity, a topological $\theta-$term, should be considered on equal footing in order to describe such a quantum phase transition involved with Anderson localization. In this respect we call the scaling theory for the longitudinal magnetoconductivity of a disordered Weyl metal phase two-parameter scaling theory as the Anderson localization transition in the case of the quantum Hall effect.
Previously, we found $\Gamma_{U}(T)$ and $c(T)$, based on the perturbative renormalization group analysis, where $\Gamma_{U}(T)$ gives the Drude conductivity and $c(T)$ describe the enhancement of the longitudinal magnetoconductivity. More precisely, we can address renormalization effects of the longitudinal magnetoconductivity based on the Boltzmann transport theory for a Weyl metal phase Boltzmann_Kim_I ; Boltzmann_Kim_II
$$\displaystyle\frac{\partial n_{\chi}(\bm{p};\bm{r},t)}{\partial t}+\bm{\dot{r}%
}_{\chi}\cdot\bm{\nabla}_{\bm{r}}n_{\chi}(\bm{p};\bm{r},t)+\bm{\dot{p}}_{\chi}%
\cdot\bm{\nabla}_{\bm{p}}n_{\chi}(\bm{p};\bm{r},t)$$
$$\displaystyle=I_{coll}[n_{\chi}(\bm{p};\bm{r},t)].$$
(23)
Here, $n_{\chi}(\bm{p};\bm{r},t)$ is the distribution function at a chiral Fermi surface denoted by $\chi=\pm$, where $\bm{p}$ is the relative momentum of a particle-hole pair near the chiral Fermi surface, and $\bm{r}$ and $t$ are the center of mass position and time of the particle-hole pair.
$\bm{\dot{r}}_{\chi}$ and $\bm{\dot{p}}_{\chi}$ represent the change of position and momentum with respect to time, classically described and given by the so called modified Drude model AHE_Review_I ; AHE_Review_II
$$\displaystyle\dot{\bm{x}}_{F}^{\chi}={\bm{v}}_{F}^{\chi}+\dot{\bm{p}}_{F}^{%
\chi}\times\bm{\mathcal{B}}_{F}^{\chi},$$
$$\displaystyle\dot{\bm{p}}_{F}^{\chi}={\bm{E}}+\dot{\bm{x}}_{F}^{\chi}\times{%
\bm{B}}.$$
(24)
$\bm{\mathcal{B}}_{F}^{\chi}$ represents a momentum-space magnetic field on the chiral Fermi surface, resulting from a momentum-space magnetic charge $\chi$ enclosed by the chiral Fermi surface. We would like to recall that the Berry curvature does not appear on the “normal” Fermi surface that does not enclose a band-touching point. As a result, we reproduce the Drude model with $\bm{\mathcal{B}}_{F}^{\chi}=0$. It is essential to realize the following relation between the applied magnetic field and the distance between the pair of Weyl points
$$\displaystyle\bm{B}\rightarrow g^{-1}\bm{c}(T).$$
(25)
It is straightforward to solve these coupled equations, the solution of which is
$$\displaystyle\dot{\bm{x}}_{F}^{\chi}\approx G^{\chi}_{3}(T)\Big{[}{\bm{v}}_{F}%
^{\chi}+{\bm{E}}\times\bm{\mathcal{B}}_{F}^{\chi}+g^{-1}\big{(}\bm{\mathcal{B}%
}_{F}^{\chi}\cdot{\bm{v}}_{F}^{\chi}\big{)}{\bm{c}}(T)\Big{]},$$
$$\displaystyle\dot{\bm{p}}_{F}^{\chi}\approx G^{\chi}_{3}(T)\Big{[}{\bm{E}}+g^{%
-1}{\bm{v}}_{F}^{\chi}\times{\bm{c}}(T)+g^{-1}\big{(}{\bm{E}}\cdot{\bm{c}}(T)%
\big{)}\bm{\mathcal{B}}_{F}^{\chi}\Big{]},$$
where $G_{3}^{\chi}=\big{(}1+g^{-1}\bm{\mathcal{B}}_{F}^{\chi}\cdot{\bm{c}}(T)\big{)}%
^{-1}$ is a volume factor of the modified phase space with a pair of momentum-space magnetic charges $\chi=\pm$. They are well known the role of anomalous electromagnetic-field-dependent terms in anomalous transport phenomena: (1) The second term of ${\bm{E}}\times\bm{\mathcal{B}}_{F}^{\chi}$ in the first equation is responsible for the anomalous Hall effect, the Hall effect without an applied magnetic field due to an emergent magnetic field referred to as Berry curvature in the momentum space AHE_I ; AHE_II ; AHE_III . (2) The third term of $g^{-1}\big{(}\bm{\mathcal{B}}_{F}^{\chi}\cdot{\bm{v}}_{F}^{\chi}\big{)}{\bm{c}%
}(T)$ in the first equation gives rise to the so called chiral magnetic effect that dissipationless electric currents are driven by applied magnetic fields in the limit of vanishing applied electric fields, proportional to the distance between the pair of Weyl points or applied magnetic fields CME_I ; CME_II ; CME_III ; CME_IV ; CME_V ; CME_VI . (3) The third term of $g^{-1}\big{(}{\bm{E}}\cdot{\bm{c}}(T)\big{)}\bm{\mathcal{B}}_{F}^{\chi}$ in the second equation causes the gauge anomaly for electrons on each chiral Fermi surface that gauge or electric currents on each chiral Fermi surface are not conserved NLMR_Theory_Original ; NLMR_Experiment_Original ; Boltzmann_Spivak_Tong ; Boltzmann_Kim_I ; Boltzmann_Kim_II ; CME_II ; CME_III ; ABJ_Anomaly_I ; ABJ_Anomaly_II ; ABJ_Anomaly_III ; ABJ_Anomaly_IV ; ABJ_Anomaly_V ; ABJ_Anomaly_VI ; ABJ_Anomaly_VII ; ABJ_Anomaly_VIII . Of course, the breakdown of the gauge symmetry should be cured when total electric currents are considered, but chiral “electric” currents are still not conserved, referred to as chiral anomaly.
The collision part is given by
$$\displaystyle I_{coll}[\delta n_{\chi}(\bm{p};\bm{r},t)]$$
$$\displaystyle=$$
$$\displaystyle-\frac{n_{\chi}(\bm{p};\bm{r},t)-n_{\chi}^{eq}(\bm{p})}{\tau_{%
intra}(T)}$$
(27)
$$\displaystyle-$$
$$\displaystyle\frac{n_{\chi}(\bm{p};\bm{r},t)-n_{-\chi}(\bm{p};\bm{r},t)}{\tau_%
{inter}(T)}.$$
The first term describes the intra-valley scattering, and the second represents the inter-valley scattering. In this respect both scattering rates of $1/\tau_{intra}(T)$ and $1/\tau_{intra}(T)$ correspond to $\Gamma_{V}(T)$ and $\Gamma_{U}(T)$, respectively.
Considering homogeneity of the Weyl metal phase under constant electric fields in the dc-limit, we are allowed to solve $\bm{\dot{p}}_{\chi}\cdot\bm{\nabla}_{\bm{p}}n_{\chi}(\bm{p})=I_{coll}[n_{\chi}%
(\bm{p})]$. As a result, we find a two-parameter scaling theory for the longitudinal magnetoconductivity in a disordered Weyl metal phase
$$\displaystyle\sigma_{L}(B,T)=\sigma_{D}(B,T)\big{(}1+const.[c(B,T)]^{2}\big{)},$$
(28)
where $\sigma_{D}(B,T)$ is the Drude conductivity inversely proportional to $\Gamma_{U}(T)$ and $c(B,T)$ is the distance between a pair of Weyl points.
Rewriting the distance between the pair of Weyl points as $c(B,T)\equiv C_{W}^{1/2}(T)B$, we consider
$$\displaystyle\Delta\sigma_{L}(B,T)$$
$$\displaystyle\equiv$$
$$\displaystyle\frac{\sigma_{L}(B,T)-\sigma_{D}(B,T)}{\sigma_{D}(B,T)}$$
(29)
$$\displaystyle=$$
$$\displaystyle C_{W}(T)B^{2}$$
for the universal scaling relation. More explicitly, inserting $C_{W}(T)=C_{W}(T_{0})\big{(}T_{0}/[T+T_{M}]\big{)}^{-2\lambda_{\bm{c},fn}}$ into the above, we find
$$\frac{T_{0}^{-2(1+\eta_{n})}\Delta\sigma_{L}(B,T)}{C_{W}(T_{0})}=\bigg{(}\frac%
{B}{[T+T_{M}]^{1+\eta_{n}}}\bigg{)}^{2},$$
(30)
where “anomalous dimensions” are given by $\eta_{0}=0,\eta_{1}=0.34$ and $\eta_{2}=0.60$, respectively, for each fixed point.
Fig. 4 shows the longitudinal magnetoconductivity, enhanced to be proportional to $B^{2}$, the square of the distance between the pair of Weyl points, at each temperature. Our renormalization group analysis confirms that the distance between the pair of Weyl points is renormalized to increase, lowering temperature, i.e., $C_{W}(T_{H})<C_{W}(T_{L})$ with $T_{H}>T_{L}$. As a result, the degree of enhancement becomes larger as temperature is reduced (Left). Interestingly, these longitudinal transport coefficients turn out to be collapsed into a single universal curve, described by Eq. (30) (Right).
Fig. 5 shows the comparison between $C_{W}(T)$ from an experimental data of $Bi_{1-x}Sb_{x}$ with $x=3\sim 4\%$ and that from our renormalization group analysis WM_Review_Kim ; Exp_in_preparation . Experimentally, the enhancement coefficient $C_{W}(T)$ can be found from fitting the experimental data with Eq. (5) at a given temperature, where the Drude part is replaced with a transport coefficient of weak anti-localization corrections and additional contributions, which have nothing to do with Weyl points, are also introduced NLMR_Experiment_Original . Repeating this fitting procedure for various temperatures, we obtain the temperature dependence of $C_{W}(T)$. The comparison between the experimental $C_{W}(T)$ and the renormalization group analysis Eq. (30) looks appealing.
IV Discussion and conclusion
The original motivation of the present study is to reveal the existence of a topological phase transition from a Weyl metal phase to a normal metal state as a function of the strength of disorder and temperature. Our physical picture for this phase transition is as follows. Disorder scattering, in particular, inter-valley scattering is expected to kill the nature of the Weyl metallic phase since it induces mixing of chirality. We recall that the inter-valley scattering appears as an effective random-mass term. If $\bar{\psi}(x)\psi(x)$ has a nontrivial vacuum expectation value, i.e., $\langle\bar{\psi}(x)\psi(x)\rangle\not=0$, expected to realize in the case of sufficiently strong disorder, the chiral symmetry breaks down even at the classical level and the chiral anomaly loses the physical meaning. As a result, we speculate that the distance between the pair of Weyl points renormalizes to vanish. A diffusive normal metallic state would be realized in the case of sufficiently strong disorder. Since this phase transition is not involved with symmetry breaking, it is identified with a topological phase transition.
This topological phase transition may be translated into Peccei-Quinn symmetry breaking in the context of dynamical generation of axions PQSB ; PQSB_Review . In order to realize the Peccei-Quinn symmetry breaking, there must be a scalar field. When the scalar field does not have its vacuum expectation value, any value of the $\theta-$angle can be canceled by the Peccei-Quinn transformation. On the other hand, the Peccei-Quinn symmetry breaking occurs when the scalar field has its vacuum expectation value. As a result of the continuous symmetry breaking, there exists a Goldstone boson field, referred to as an axion field. When the Peccei-Quinn symmetry is exact and thus, the axion field is massless, any value of the $\theta-$angle can be still canceled by the Peccei-Quinn transformation. However, there are instanton excitations, which do not allow the Peccei-Quinn symmetry not to be exact, giving rise to a mass term in the axion dynamics. Then, the vacuum angle is fixed to be $\theta=0$, minimizing the energy of the system. In the present situation the corresponding scalar field results from the Hubbard-Stratonovich transformation of the random-mass term in the replica effective field theory, conventionally referred to as $Q_{ab}$, where $a$ and $b$ denote the replica index. However, there are two different aspects between the possible topological phase transition and the Peccei-Quinn symmetry breaking in high energy physics: (1) The vacuum angle is given by an inhomogeneous function of position while its gradient identified with a chiral gauge field is a constant. (2) There are no instanton-type excitations in the Weyl metal phase. This direction of research would be an interesting future task.
Unfortunately, the perturbative renormalization group analysis fails to access such an unstable fixed point, identified with the quantum critical point of the topological phase transition. In this respect the naming of the two-parameter scaling theory is not satisfactory in our opinion, basically motivated from the analogy with the plateau-plateau transition in the integer quantum Hall effect. However, it turns out that the longitudinal magnetoconductivity is governed by both parameters of the Drude conductivity and the distance between the pair of Weyl points, renormalized by inter-valley scattering, essentially analogous to $\sigma_{xx}$ and $\sigma_{xy}$ in the quantum Hall effect, respectively. In this respect we may call what we performed two-parameter scaling theory for the longitudinal magnetoconductivity in a disordered Weyl metal phase.
An unexpected result is breakdown of the $B/T$ scaling behavior near the diffusive fixed point although it is fulfilled near the clean fixed point. Actually, we could verify this prediction, comparing the proposed formula Eq. (30) of the two-parameter scaling theory with $C_{W}(T)$ in the experimental data of $Bi_{1-x}Sb_{x}$ with $x=3\sim 4\%$. Here, we took into account modifying the original renormalization group analysis, introducing a cutoff scale into the equation for the distance between the pair of Weyl points as Eq. (22), in order to prohibit the divergence of the length scale within the Brillouin zone. This breakdown may be regarded to be a fingerprint of the interplay between disorder scattering and topological structure in a Weyl metal phase.
Acknowledgement
This study was supported by the Ministry of Education, Science, and Technology (No. NRF-2015R1C1A1A01051629 and No. 2011-0030046) of the National Research Foundation of Korea (NRF) and by TJ Park Science Fellowship of the POSCO TJ Park Foundation. This work was also supported by the POSTECH Basic Science Research Institute Grant (2015). We would like to appreciate fruitful discussions in the APCTP workshop on Delocalisation Transitions in Disordered Systems in 2015.
Appendix A model hamiltonian
A minimal model for a Weyl metal state is given by
$$\displaystyle\mathcal{H}=\psi^{\dagger}(\bm{x})\big{(}v\bm{\alpha}\cdot(-%
\imath\nabla)-\mu+g\bm{B}\cdot\bm{\sigma}\otimes I_{2\times 2}\big{)}\psi(\bm{%
x}).$$
$\psi=(\psi_{R},\psi_{L})^{T}$ is a four-component Dirac-spinor field in a two-component Weyl-spinor field with right(R)-left(L) chirality, and $v$ is the velocity of such fermions. $\mu$ is an electron chemical potential. $\bm{B}$ is an externally applied magnetic field with a Lande$-g$ factor $g$. $\bm{\alpha}$ is a four-by-four matrix, given by $\bm{\alpha}=\bm{\sigma}\otimes\sigma_{z}$, where $\bm{\sigma}$ is a Pauli matrix.
First, we look into a band structure. This block-diagonal matrix can be diagonalized as
$$\displaystyle\mathcal{H}_{\bm{k}}=\phi^{\dagger}_{\bm{k}}\big{(}\left|v\bm{k}+%
g\bm{B}\right|\sigma_{z}\otimes P_{+}-\left|v\bm{k}-g\bm{B}\right|\sigma_{z}%
\otimes P_{-}-\mu\big{)}\phi_{\bm{k}}$$
where $P_{+}=\begin{pmatrix}1&0\\
0&0\end{pmatrix}$ and$P_{-}=\begin{pmatrix}0&0\\
0&1\end{pmatrix}$ are projection matrices, and $\phi_{\bm{k}}=U_{\bm{k}}\psi_{\bm{k}}$ is an eigenstate. The unitary matrix varying with $\bm{k}$ is given by
$$\displaystyle U_{\bm{k}}=\begin{pmatrix}\cos{\frac{\zeta_{+}}{2}}&\sin{\frac{%
\zeta_{+}}{2}}e^{-\imath\eta_{+}}\\
-\sin{\frac{\zeta_{+}}{2}}e^{\imath\eta_{+}}&\cos{\frac{\zeta_{+}}{2}}\end{%
pmatrix}\otimes P_{+}+\begin{pmatrix}\cos{\frac{\zeta_{-}}{2}}&\sin{\frac{%
\zeta_{-}}{2}}e^{-\imath\eta_{-}}\\
-\sin{\frac{\zeta_{-}}{2}}e^{\imath\eta_{-}}&\cos{\frac{\zeta_{-}}{2}}\end{%
pmatrix}\otimes P_{-},$$
where $\zeta_{\pm}$ $(\eta_{\pm})$ is the polar (azimuthal) angle of $\bm{k}\pm\frac{g}{v}\bm{B}$, respectively. If we draw a band structure along some momentum-line, for example, $\bm{k}=(0,0,k_{z})$, then we obtain a pair of Weyl cones as shown in Fig. 1.
Second, we consider two types of random potentials, say, “intra-valley scattering” and “inter-valley scattering”, given by
$$\displaystyle\psi^{\dagger}_{R}(\bm{x})V(\bm{x})\psi_{R}(\bm{x})+\psi^{\dagger%
}_{L}(\bm{x})V(\bm{x})\psi_{L}(\bm{x})=\psi^{\dagger}(\bm{x})V(\bm{x})\psi(\bm%
{x}),$$
$$\displaystyle\psi^{\dagger}_{R}(\bm{x})U(\bm{x})\psi_{L}(\bm{x})+\psi^{\dagger%
}_{R}(\bm{x})U(\bm{x})\psi_{L}(\bm{x})=\psi^{\dagger}(\bm{x})U(\bm{x})\big{(}I%
_{2\times 2}\otimes\sigma_{x}\big{)}\psi(\bm{x}),$$
where $V(\bm{x})$ and $U(\bm{x})$ are disorder potentials for intra-valley scattering and inter-valley scattering, respectively.
Now, the effective action is
$$\displaystyle S[\psi^{\dagger},\psi;V,U]=\int^{\beta}_{0}d\tau\int d^{3}\bm{x}%
\psi^{\dagger}(\tau,\bm{x})\big{\{}\partial_{\tau}+v\bm{\alpha}\cdot(-\imath%
\nabla)-\mu+g\bm{B}\cdot\bm{\sigma}\otimes I_{2\times 2}+V(\bm{x})+U(\bm{x})I_%
{2\times 2}\otimes\sigma_{x}\big{\}}\psi(\tau,\bm{x}),$$
where the corresponding free energy is given by $F[V,U]=-T\ln{\int\mathcal{D}\psi^{\dagger}\mathcal{D}\psi e^{-S[\psi^{\dagger}%
,\psi;V,U]}}$ in a given configuration of random potentials. We represent this effective action in terms of gamma matrices in the Weyl representation
$$\displaystyle\gamma^{0}=I_{2\times 2}\otimes\sigma_{x},~{}\gamma^{k}=\gamma^{0%
}(-\alpha_{k})=\sigma_{k}\otimes\imath\sigma_{y}~{}(k=1,2,3),~{}\gamma^{5}=%
\imath\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}=-I_{2\times 2}\otimes\sigma_{z}.$$
Then, we reach the following expression
$$\displaystyle S[\bar{\psi},\psi;V,U]=\int d^{4}x\big{\{}\bar{\psi}(x)\big{(}%
\gamma^{0}\partial_{0}+\imath v\gamma^{k}\partial_{k}-\mu\gamma^{0}+c_{\mu}%
\gamma^{\mu}\gamma^{5}\big{)}\psi(x)+\bar{\psi}(x)\gamma^{0}V(\bm{x})\psi(x)+%
\bar{\psi}(x)U(\bm{x})\psi(x)\big{\}}$$
with an adjoint spinor-field $\bar{\psi}\equiv\psi^{\dagger}\gamma^{0}$, where we introduced $c_{k}\equiv gB_{k}$ $(k=1,2,3)$ with “time-component” $c_{0}$. “Space-time” of $x$ is $x^{\mu}=(\tau,\bm{x})$ and other four-vectors are defined, similarly. For example, four-momentum is $p^{\mu}=(p^{0},\bm{p})$ with $p^{0}=-\imath\omega_{n}$. Since the action has been formulated in the imaginary time, it is defined on the Euclidean geometry as shown by $p^{\mu}p_{\mu}=-{\omega_{n}}^{2}-\bm{p}^{2}$.
Appendix B effective field theory for renormalization group analysis
B.1 Disorder Average
We define the free part of the effective action as
$$\displaystyle S_{0}[\bar{\psi},\psi]=\int d^{4}x\bar{\psi}(x)\big{(}\gamma^{0}%
\partial_{0}+\imath v\gamma^{k}\partial_{k}-\mu\gamma^{0}+c_{\mu}\gamma^{\mu}%
\gamma^{5}\big{)}\psi(x).$$
Then, a physical observable is measured as follows
$$\displaystyle\left\langle\mathcal{O}(\bar{\psi},\psi)\right\rangle=\int%
\mathcal{D}V\mathcal{D}UP[V,U]\frac{\int\mathcal{D}\bar{\psi}\mathcal{D}\psi%
\mathcal{O}(\bar{\psi},\psi)e^{-S_{0}[\bar{\psi},\psi]}e^{-\int d^{4}x\bar{%
\psi}(x)(\gamma^{0}V(\bm{x})+U(\bm{x}))\psi(x)}}{\int\mathcal{D}\bar{\psi}%
\mathcal{D}\psi e^{-S_{0}[\bar{\psi},\psi]}e^{-\int d^{4}x\bar{\psi}(x)(\gamma%
^{0}V(\bm{x})+U(\bm{x}))\psi(x)}}.$$
This can be reformulated as
$$\displaystyle\left\langle\mathcal{O}(\bar{\psi},\psi)\right\rangle=\int%
\mathcal{D}V\mathcal{D}UP[V,U]\frac{\delta}{\delta J}\biggr{|}_{J=0}\ln{Z[V,U,%
J]},$$
$$\displaystyle Z[V,U,J]=\int\mathcal{D}\bar{\psi}\mathcal{D}\psi e^{-S_{0}[\bar%
{\psi},\psi]}e^{-\int d^{4}x\bar{\psi}(x)(\gamma^{0}V(\bm{x})+U(\bm{x}))\psi(x%
)+\int d^{4}xJ(x)\mathcal{O}(\bar{\psi}(x),\psi(x))},$$
where $J(x)$ is a source field coupled to an operator $O(\bar{\psi},\psi)$, locally.
In order to perform the averaging procedure for disorders, we resort to the replica trick of $\ln{Z}=\lim_{R\to 0}\frac{Z^{R}-1}{R}$
$$\displaystyle\left\langle\mathcal{O}(\bar{\psi},\psi)\right\rangle=\lim_{R\to 0%
}\int\mathcal{D}V\mathcal{D}UP[V,U]\frac{\delta}{\delta J}\biggr{|}_{J=0}\frac%
{Z^{R}[V,U,J]-1}{R},$$
where the replicated partition function is
$$\displaystyle Z^{R}[V,U,J]=\int\mathcal{D}\bar{\psi}^{a}\mathcal{D}\psi^{a}%
\exp{\bigg{[}-\sum_{a=1}^{R}S_{0}[\bar{\psi}^{a},\psi^{a}]-\sum_{a=1}^{R}\int d%
^{4}x\bar{\psi}^{a}(x)\big{(}\gamma^{0}V(\bm{x})+U(\bm{x})\big{)}\psi^{a}(x)+%
\int d^{4}xJ(x)\sum_{a=1}^{R}\mathcal{O}(\bar{\psi}^{a},\psi^{a})\bigg{]}}$$
with a replica index “a”. In this technique a physical observable is given by
$$\displaystyle\left\langle\mathcal{O}(\bar{\psi},\psi)\right\rangle$$
$$\displaystyle=$$
$$\displaystyle\lim_{R\to 0}\frac{1}{R}\int\mathcal{D}V\mathcal{D}UP[V,U]\int%
\mathcal{D}\bar{\psi}^{a}\mathcal{D}\psi^{a}\mathcal{O}(\bar{\psi}^{a},\psi^{a})$$
$$\displaystyle\times\exp{\bigg{[}-\sum_{a=1}^{R}S_{0}[\bar{\psi}^{a},\psi^{a}]-%
\sum_{a=1}^{R}\int d^{4}x\bar{\psi}^{a}(x)\big{(}\gamma^{0}V(\bm{x})+U(\bm{x})%
\big{)}\psi^{a}(x)\bigg{]}}.$$
In this study we take into account static-and Gaussian-distributed disorders, given by
$$\displaystyle P[V,U]=N\exp{\bigg{[}-\frac{\int d^{3}\bm{x}V^{2}(\bm{x})}{2%
\Gamma_{V}}-\frac{\int d^{3}\bm{x}U^{2}(\bm{x})}{2\Gamma_{U}}\bigg{]}},$$
where $N$ is a normalization factor. It is straightforward to perform the Gaussian integral for disorders, resulting in
$$\displaystyle\left\langle\mathcal{O}(\bar{\psi},\psi)\right\rangle=\lim_{R\to 0%
}\frac{1}{R}\sum_{a=1}^{R}\int\mathcal{D}\bar{\psi}\mathcal{D}\psi\mathcal{O}(%
\bar{\psi}^{a},\psi^{a})\exp{\bigg{[}-\sum_{a=1}^{R}S_{0}[\bar{\psi}^{a},\psi^%
{a}]-\sum_{b,c=1}^{R}S_{\textrm{dis}}[\bar{\psi}^{b},\psi^{b},\bar{\psi}^{c},%
\psi^{c}]\bigg{]}},$$
where disorder-driven effective interactions are [Eq. (11)]
$$\displaystyle S_{\textrm{dis}}[\bar{\psi}^{b},\psi^{b},\bar{\psi}^{c},\psi^{c}]$$
$$\displaystyle=$$
$$\displaystyle-\int^{\beta}_{0}d\tau\int^{\beta}_{0}d\tau^{\prime}\int d^{3}\bm%
{x}\frac{\Gamma_{V}}{2}\bar{\psi}^{b}(\tau,\bm{x})\gamma^{0}\psi^{b}(\tau,\bm{%
x})\bar{\psi}^{c}(\tau^{\prime},\bm{x})\gamma^{0}\psi^{c}(\tau^{\prime},\bm{x})$$
$$\displaystyle-\int^{\beta}_{0}d\tau\int^{\beta}_{0}d\tau^{\prime}\int d^{3}\bm%
{x}\frac{\Gamma_{U}}{2}\bar{\psi}^{b}(\tau,\bm{x})\psi^{b}(\tau,\bm{x})\bar{%
\psi}^{c}(\tau^{\prime},\bm{x})\psi^{c}(\tau^{\prime},\bm{x}).$$
B.2 Renormalized perturbation theory
From now on, we focus on the case of a zero-chemical potential. We start from the following effective action
$$\displaystyle S_{B}$$
$$\displaystyle=$$
$$\displaystyle\int d^{d+1}x\bar{\psi}_{B}^{a}(x)(\gamma^{0}\partial_{0}+v_{B}%
\imath\gamma^{k}\partial_{k}+c_{B\mu}\gamma^{\mu}\gamma^{5})\psi_{B}^{a}(x)$$
$$\displaystyle-$$
$$\displaystyle\int^{\beta}_{0}d\tau\int^{\beta}_{0}d\tau^{\prime}\int d^{d}\bm{%
x}\frac{\Gamma_{BV}}{2}\bar{\psi}_{B}^{b}(\tau,\bm{x})\gamma^{0}\psi_{B}^{b}(%
\tau,\bm{x})\bar{\psi}_{B}^{c}(\tau^{\prime},\bm{x})\gamma^{0}\psi_{B}^{c}(%
\tau^{\prime},\bm{x})$$
$$\displaystyle-$$
$$\displaystyle\int^{\beta}_{0}d\tau\int^{\beta}_{0}d\tau^{\prime}\int d^{d}\bm{%
x}\frac{\Gamma_{BU}}{2}\bar{\psi}_{B}^{b}(\tau,\bm{x})\psi_{B}^{b}(\tau,\bm{x}%
)\bar{\psi}_{B}^{c}(\tau^{\prime},\bm{x})\psi_{B}^{c}(\tau^{\prime},\bm{x}),$$
where summations on the replica indices are implied. The subscript $B$ denotes “bare”, meaning that this effective action is defined at an ultraviolet (UV) scale. Note that we have generalized dimensions to “d(space)+1(time)” for dimensional regularization.
Performing the dimensional analysis, where space and time coordinates have $-1$ in mass dimension, we observe
$$\displaystyle\textrm{dim}[\psi]=\frac{d}{2},~{}~{}\textrm{dim}[v]=0,~{}~{}%
\textrm{dim}[c_{\mu}]=1,~{}~{}\textrm{dim}[\Gamma_{V}]=\textrm{dim}[\Gamma_{U}%
]=2-d.$$
In this respect we perform the renormalization group analysis in $d+1=3+\varepsilon$ dimensions, where $\varepsilon$ is a “small” parameter. In the end of the calculation the dimensions are analytically continued to the physical dimensions ($d+1=4$) by setting $\varepsilon=1$.
Taking into account quantum corrections, divergences would be generated. They can be absorbed into renormalization constants by redefining fields and parameters. Rewriting the bare action in terms of renormalized fields and couplings, we obtain
$$\displaystyle S_{B}$$
$$\displaystyle=$$
$$\displaystyle\int d^{d+1}x\bar{\psi}_{R}^{a}(x)\big{(}Z_{\psi}^{\omega}\gamma^%
{0}\partial_{0}+Z_{\psi}^{\bm{k}}v_{R}\imath\gamma^{k}\partial_{k}+Z_{c0}c_{R0%
}\gamma^{0}\gamma^{5}+Z_{\bm{c}}c_{Rk}\gamma^{k}\gamma^{5}\big{)}\psi_{R}^{a}(x)$$
$$\displaystyle-$$
$$\displaystyle\int^{\beta}_{0}d\tau\int^{\beta}_{0}d\tau^{\prime}\int d^{d}\bm{%
x}\frac{Z_{\Gamma V}\Gamma_{RV}}{2}\bar{\psi}_{R}^{b}(\tau,\bm{x})\gamma^{0}%
\psi_{R}^{b}(\tau,\bm{x})\bar{\psi}_{R}^{c}(\tau^{\prime},\bm{x})\gamma^{0}%
\psi_{R}^{c}(\tau^{\prime},\bm{x}),$$
$$\displaystyle-$$
$$\displaystyle\int^{\beta}_{0}d\tau\int^{\beta}_{0}d\tau^{\prime}\int d^{d}\bm{%
x}\frac{Z_{\Gamma U}\Gamma_{RU}}{2}\bar{\psi}_{R}^{b}(\tau,\bm{x})\psi_{R}^{b}%
(\tau,\bm{x})\bar{\psi}_{R}^{c}(\tau^{\prime},\bm{x})\psi_{R}^{c}(\tau^{\prime%
},\bm{x}),$$
where such renormalized fields and parameters are given by
$$\displaystyle\psi_{B}^{a}=(Z_{\psi}^{\omega})^{\frac{1}{2}}\psi_{R}^{a},~{}~{}%
~{}v_{B}=Z_{\psi}^{\bm{k}}(Z_{\psi}^{\omega})^{-1}v_{R},~{}~{}~{}c_{B0}=Z_{c0}%
(Z_{\psi}^{\omega})^{-1}c_{R0},$$
$$\displaystyle c_{Bk}=Z_{\bm{c}}(Z_{\psi}^{\omega})^{-1}c_{Rk},~{}~{}~{}\Gamma_%
{BV}=Z_{\Gamma V}(Z_{\psi}^{\omega})^{-2}\Gamma_{RV},~{}~{}~{}\Gamma_{BU}=Z_{%
\Gamma U}(Z_{\psi}^{\omega})^{-2}\Gamma_{RU}.$$
It is more elaborate to represent this theory by separating the renormalized part from counter terms that are to absorb divergences in the following way [Eq. (II.2)],
$$\displaystyle S_{B}$$
$$\displaystyle=$$
$$\displaystyle S_{R}+S_{CT},$$
$$\displaystyle S_{R}$$
$$\displaystyle=$$
$$\displaystyle\int d^{d+1}x\bar{\psi}_{R}^{a}\big{(}\gamma^{0}\partial_{0}+v_{R%
}\imath\gamma^{k}\partial_{k}+c_{R0}\gamma^{0}\gamma^{5}+c_{Rk}\gamma^{k}%
\gamma^{5}\big{)}\psi_{R}^{a}$$
$$\displaystyle-$$
$$\displaystyle\int d\tau\int d\tau^{\prime}\int d^{d}\bm{x}\frac{\Gamma_{RV}}{2%
}(\bar{\psi}_{R}^{b}\gamma^{0}\psi_{R}^{b})_{\tau}(\bar{\psi}_{R}^{c}\gamma^{0%
}\psi_{R}^{c})_{\tau^{\prime}}-\int d\tau\int d\tau^{\prime}\int d^{d}\bm{x}%
\frac{\Gamma_{RU}}{2}(\bar{\psi}_{R}^{b}\psi_{R}^{b})_{\tau}(\bar{\psi}_{R}^{c%
}\psi_{R}^{c})_{\tau^{\prime}},$$
$$\displaystyle S_{CT}$$
$$\displaystyle=$$
$$\displaystyle\int d^{d+1}x\bar{\psi}_{R}^{a}\big{(}\delta_{\psi}^{\omega}%
\gamma^{0}\partial_{0}+\delta_{\psi}^{\bm{k}}v_{R}\imath\gamma^{k}\partial_{k}%
+\delta_{c0}c_{R0}\gamma^{0}\gamma^{5}+\delta_{\bm{c}}c_{Rk}\gamma^{k}\gamma^{%
5}\big{)}\psi_{R}^{a}$$
$$\displaystyle-$$
$$\displaystyle\int d\tau\int d\tau^{\prime}\int d^{d}\bm{x}\frac{\delta_{\Gamma
V%
}\Gamma_{RV}}{2}(\bar{\psi}_{R}^{b}\gamma^{0}\psi_{R}^{b})_{\tau}(\bar{\psi}_{%
R}^{c}\gamma^{0}\psi_{R}^{c})_{\tau^{\prime}}-\int d\tau\int d\tau^{\prime}%
\int d^{d}\bm{x}\frac{\delta_{\Gamma U}\Gamma_{RU}}{2}(\bar{\psi}_{R}^{b}\psi_%
{R}^{b})_{\tau}(\bar{\psi}_{R}^{c}\psi_{R}^{c})_{\tau^{\prime}},$$
where $Z_{\psi}^{\omega}=1+\delta_{\psi}^{\omega}$, $Z_{\psi}^{\bm{k}}=1+\delta_{\psi}^{\bm{k}}$, $Z_{c0}=1+\delta_{c0}$, $Z_{\bm{c}}=1+\delta_{\bm{c}}$, $Z_{\Gamma V}=1+\delta_{\Gamma V}$ and $Z_{\Gamma U}=1+\delta_{\Gamma U}$.
B.3 Feynman Rules
In the momentum and frequency space the effective action is written as
$$\displaystyle S[\bar{\psi}^{a},\psi^{a}]=\sum_{p}\bar{\psi}^{a}_{p}\big{(}\not%
{p}+\not{c}\gamma^{5}\big{)}\psi^{a}_{p}-\frac{1}{L^{3}}\sum_{p_{j}}\bigg{[}%
\frac{\Gamma_{V}}{2}(\bar{\psi}^{b}_{p_{1}}\gamma^{0}\psi^{b}_{p_{2}})(\bar{%
\psi}^{c}_{p_{3}}\gamma^{0}\psi^{c}_{p_{4}})+\frac{\Gamma_{U}}{2}(\bar{\psi}^{%
b}_{p_{1}}\psi^{b}_{p_{2}})(\bar{\psi}^{c}_{p_{3}}\psi^{c}_{p_{4}})\bigg{]}%
\delta^{(3)}_{\bm{p}_{1}-\bm{p}_{2},\bm{p}_{3}-\bm{p}_{4}}\delta_{p_{1}^{0}p_{%
2}^{0}}\delta_{p_{3}^{0}p_{4}^{0}},$$
where Feynman rules are given in Fig. 6.
Since there is a chiral gauge field in the kinetic-energy part, the free propagator becomes a little bit complex. Considering the following identity
$$\displaystyle(\not{p}+\not{c}\gamma^{5})(\not{p}-\not{c}\gamma^{5})(p^{2}+c^{2%
}+2p\cdot c\gamma^{5})=(p^{2}+c^{2}-2p\cdot c\gamma^{5})(p^{2}+c^{2}+2p\cdot c%
\gamma^{5})=(p+c)^{2}(p-c)^{2},$$
we obtain an electron Green function
$$\displaystyle G(p)=-(\not{p}+\not{c}\gamma^{5})^{-1}=-\frac{(\not{p}-\not{c}%
\gamma^{5})(p^{2}+c^{2}+2p\cdot c\gamma^{5})}{(p+c)^{2}(p-c)^{2}}.$$
We introduces the following expression with a Feynman parameter for the renormalization group analysis
$$\displaystyle G(p)=-\int^{1}_{0}dx\frac{(\not{p}-\not{c}\gamma^{5})(p^{2}+c^{2%
}+2p\cdot c\gamma^{5})}{\big{[}\big{(}p+(1-2x)c\big{)}^{2}+4x(1-x)c^{2}\big{]}%
^{2}}.$$
For a future use, we rearrange it in terms of $\bm{p}$ as
$$\displaystyle G(p)$$
$$\displaystyle=$$
$$\displaystyle\int^{1}_{0}dx\frac{\bm{p}^{2}p_{i}\gamma^{i}+\bm{p}^{2}(p_{0}%
\gamma^{0}-\not{c}\gamma^{5})+p_{i}p_{j}(-2c^{i}\gamma^{j}\gamma^{5})+p_{i}f_{%
1}^{i}(p_{0})+f_{0}(p_{0})}{\big{[}\big{(}\bm{p}+(1-2x)\bm{c}\big{)}^{2}+%
\Delta_{0}(p_{0};x)\big{]}^{2}},$$
(31)
$$\displaystyle\Delta_{0}(p_{0};x)$$
$$\displaystyle=$$
$$\displaystyle 4x(1-x)\bm{c}^{2}-(p_{0}+c_{0})^{2}+4xp_{0}c_{0},$$
$$\displaystyle f_{1}^{i}(p_{0})$$
$$\displaystyle=$$
$$\displaystyle-\gamma^{i}(p_{0}\gamma^{0}-\not{c}\gamma^{5})(p_{0}\gamma^{0}+%
\not{c}\gamma^{5})+2c^{i}\gamma^{5}(p_{0}\gamma^{0}-\not{c}\gamma^{5}),$$
$$\displaystyle f_{0}(p_{0})$$
$$\displaystyle=$$
$$\displaystyle-(p_{0}\gamma^{0}-\not{c}\gamma^{5})^{2}(p_{0}\gamma^{0}+\not{c}%
\gamma^{5}).$$
Alternatively, we obtain in terms of $\bm{p}^{\prime}=\bm{p}+(1-2x)\bm{c}$
$$\displaystyle G(p)$$
$$\displaystyle=$$
$$\displaystyle\int^{1}_{0}dx\frac{C_{3}^{i}{\bm{p}^{\prime}}^{2}p^{\prime}_{i}+%
C_{2a}{\bm{p}^{\prime}}^{2}+C_{2b}^{ij}p^{\prime}_{i}p^{\prime}_{j}+C_{1}^{i}p%
^{\prime}_{i}+C_{0}}{\big{[}{\bm{p}^{\prime}}^{2}+\Delta_{0}(p_{0};x)\big{]}^{%
2}},$$
(32)
$$\displaystyle C_{3}^{i}$$
$$\displaystyle=$$
$$\displaystyle\gamma^{i},$$
$$\displaystyle C_{2a}$$
$$\displaystyle=$$
$$\displaystyle\not{u}-\not{c}\gamma^{5},$$
$$\displaystyle C_{2b}^{ij}$$
$$\displaystyle=$$
$$\displaystyle-2\gamma^{i}(u^{j}+c^{j}\gamma^{5}),$$
$$\displaystyle C_{1}^{i}$$
$$\displaystyle=$$
$$\displaystyle-\gamma^{i}(\not{u}-\not{c}\gamma^{5})(\not{u}+\not{c}\gamma^{5})%
-2(u^{i}-c^{i}\gamma^{5})(\not{u}-\not{c}\gamma^{5}),$$
$$\displaystyle C_{0}$$
$$\displaystyle=$$
$$\displaystyle-(\not{u}-\not{c}\gamma^{5})^{2}(\not{u}+\not{c}\gamma^{5}),$$
where $u\equiv(p_{0},\tilde{\bm{c}})$ and $\tilde{\bm{c}}_{x}\equiv(2x-1)\bm{c}$. We may use either of these expressions for convenience. Despite their complicated form, they will not be involved much in actual integration procedures.
Appendix C self-energy corrections
C.1 Relevant Feynman’s diagrams
Within the replica trick, we are allowed to perform the perturbative analysis. The full Green function of $\mathbf{G}(p,q)=\left\langle\psi_{p}\bar{\psi}_{q}\right\rangle$ is evaluated up to the $\Gamma^{2}$ order as follows
$$\displaystyle\mathbf{G}(p,q)$$
$$\displaystyle=$$
$$\displaystyle\lim_{R\to 0}\frac{1}{R}\int\mathcal{D}\bar{\psi}\mathcal{D}\psi%
\big{(}\psi^{a}_{p}\bar{\psi}^{a}_{q}\big{)}e^{-S_{0}[\bar{\psi}^{\alpha},\psi%
^{\alpha}]}e^{\frac{1}{L^{3}}\sum_{p_{j}}\big{[}\frac{\Gamma_{V}}{2}(\bar{\psi%
}^{b}_{p_{1}}\gamma^{0}\psi^{b}_{p_{2}})(\bar{\psi}^{c}_{p_{3}}\gamma^{0}\psi^%
{c}_{p_{4}})+\frac{\Gamma_{U}}{2}(\bar{\psi}^{b}_{p_{1}}\psi^{b}_{p_{2}})(\bar%
{\psi}^{c}_{p_{3}}\psi^{c}_{p_{4}})\big{]}\delta^{(3)}_{\bm{p}_{1}-\bm{p}_{2},%
\bm{p}_{3}-\bm{p}_{4}}\delta_{p_{1}^{0}p_{2}^{0}}\delta_{p_{3}^{0}p_{4}^{0}}}$$
$$\displaystyle\simeq$$
$$\displaystyle\lim_{R\to 0}\frac{1}{R}\int\mathcal{D}\bar{\psi}\mathcal{D}\psi e%
^{-S_{0}[\bar{\psi}^{\alpha},\psi^{\alpha}]}\biggl{[}\psi^{a}_{p}\bar{\psi}^{a%
}_{q}+\frac{\Gamma_{V}}{2L^{3}}\sum_{p_{j}}\big{(}\psi^{a}_{p}\bar{\psi}^{a}_{%
q}\big{)}\big{(}\bar{\psi}^{b}_{p_{1}}\gamma^{0}\psi^{b}_{p_{2}}\bar{\psi}^{c}%
_{p_{3}}\gamma^{0}\psi^{c}_{p_{4}}\big{)}\delta^{(3)}_{\bm{p}_{1}-\bm{p}_{2},%
\bm{p}_{3}-\bm{p}_{4}}\delta_{p_{1}^{0}p_{2}^{0}}\delta_{p_{3}^{0}p_{4}^{0}}$$
$$\displaystyle+\frac{\Gamma_{U}}{2L^{3}}\sum_{p_{j}}\big{(}\psi^{a}_{p}\bar{%
\psi}^{a}_{q}\big{)}\big{(}\bar{\psi}^{b}_{p_{1}}\psi^{b}_{p_{2}}\bar{\psi}^{c%
}_{p_{3}}\psi^{c}_{p_{4}}\big{)}\delta^{(3)}_{\bm{p}_{1}-\bm{p}_{2},\bm{p}_{3}%
-\bm{p}_{4}}\delta_{p_{1}^{0}p_{2}^{0}}\delta_{p_{3}^{0}p_{4}^{0}}$$
$$\displaystyle+\frac{\Gamma_{V}^{2}}{8(L^{3})^{2}}\sum_{p_{j}p^{\prime}_{j}}%
\big{(}\psi^{a}_{p}\bar{\psi}^{a}_{q}\big{)}\big{(}\bar{\psi}^{b}_{p_{1}}%
\gamma^{0}\psi^{b}_{p_{2}}\bar{\psi}^{c}_{p_{3}}\gamma^{0}\psi^{c}_{p_{4}}\big%
{)}\big{(}\bar{\psi}^{b^{\prime}}_{p^{\prime}_{1}}\gamma^{0}\psi^{b^{\prime}}_%
{p^{\prime}_{2}}\bar{\psi}^{c^{\prime}}_{p^{\prime}_{3}}\gamma^{0}\psi^{c^{%
\prime}}_{p^{\prime}_{4}}\big{)}\delta^{(3)}_{\bm{p}_{1}-\bm{p}_{2},\bm{p}_{3}%
-\bm{p}_{4}}\delta_{p_{1}^{0}p_{2}^{0}}\delta_{p_{3}^{0}p_{4}^{0}}\delta^{(3)}%
_{\bm{p}^{\prime}_{1}-\bm{p}^{\prime}_{2},\bm{p}^{\prime}_{3}-\bm{p}^{\prime}_%
{4}}\delta_{p_{1}^{{}^{\prime}0}p_{2}^{{}^{\prime}0}}\delta_{p_{3}^{{}^{\prime%
}0}p_{4}^{{}^{\prime}0}}$$
$$\displaystyle+\frac{\Gamma_{U}^{2}}{8(L^{3})^{2}}\sum_{p_{j}p^{\prime}_{j}}%
\big{(}\psi^{a}_{p}\bar{\psi}^{a}_{q}\big{)}\big{(}\bar{\psi}^{b}_{p_{1}}\psi^%
{b}_{p_{2}}\bar{\psi}^{c}_{p_{3}}\psi^{c}_{p_{4}}\big{)}\big{(}\bar{\psi}^{b^{%
\prime}}_{p^{\prime}_{1}}\psi^{b^{\prime}}_{p^{\prime}_{2}}\bar{\psi}^{c^{%
\prime}}_{p^{\prime}_{3}}\psi^{c^{\prime}}_{p^{\prime}_{4}}\big{)}\delta^{(3)}%
_{\bm{p}_{1}-\bm{p}_{2},\bm{p}_{3}-\bm{p}_{4}}\delta_{p_{1}^{0}p_{2}^{0}}%
\delta_{p_{3}^{0}p_{4}^{0}}\delta^{(3)}_{\bm{p}^{\prime}_{1}-\bm{p}^{\prime}_{%
2},\bm{p}^{\prime}_{3}-\bm{p}^{\prime}_{4}}\delta_{p_{1}^{{}^{\prime}0}p_{2}^{%
{}^{\prime}0}}\delta_{p_{3}^{{}^{\prime}0}p_{4}^{{}^{\prime}0}}$$
$$\displaystyle+\frac{\Gamma_{V}\Gamma_{U}}{4(L^{3})^{2}}\sum_{p_{j}p^{\prime}_{%
j}}\big{(}\psi^{a}_{p}\bar{\psi}^{a}_{q}\big{)}\big{(}\bar{\psi}^{b}_{p_{1}}%
\gamma^{0}\psi^{b}_{p_{2}}\bar{\psi}^{c}_{p_{3}}\gamma^{0}\psi^{c}_{p_{4}}\big%
{)}\big{(}\bar{\psi}^{b^{\prime}}_{p^{\prime}_{1}}\psi^{b^{\prime}}_{p^{\prime%
}_{2}}\bar{\psi}^{c^{\prime}}_{p^{\prime}_{3}}\psi^{c^{\prime}}_{p^{\prime}_{4%
}}\big{)}\delta^{(3)}_{\bm{p}_{1}-\bm{p}_{2},\bm{p}_{3}-\bm{p}_{4}}\delta_{p_{%
1}^{0}p_{2}^{0}}\delta_{p_{3}^{0}p_{4}^{0}}\delta^{(3)}_{\bm{p}^{\prime}_{1}-%
\bm{p}^{\prime}_{2},\bm{p}^{\prime}_{3}-\bm{p}^{\prime}_{4}}\delta_{p_{1}^{{}^%
{\prime}0}p_{2}^{{}^{\prime}0}}\delta_{p_{3}^{{}^{\prime}0}p_{4}^{{}^{\prime}0%
}}\biggr{]},$$
where summations on the replica indices are implied. Feynman diagrams whose internal lines are not connected to external lines always vanish due to the replica symmetry (all Green’s functions with different replica indices are identical) and the replica limit ($\lim_{R\to 0}\frac{1}{R}$). For details, we refer to Ref. Kyoung .
We find self-energy corrections in the first-order (Fig. 7),
$$\displaystyle\Sigma^{(1)}(p)=\frac{\Gamma_{V}}{L^{3}}\sum_{q}\gamma^{0}G(p-q)%
\gamma^{0}\delta_{q^{0}0}+\frac{\Gamma_{U}}{L^{3}}\sum_{q}G(p-q)\delta_{q^{0}0%
}\equiv\Sigma^{(1)}_{V}(p)+\Sigma^{(1)}_{U}(p).$$
(33)
Likewise, we find self-energy corrections in the second order (Fig. 8).
$$\displaystyle\Sigma^{(2),r}(p)$$
$$\displaystyle=$$
$$\displaystyle\frac{\Gamma_{V}^{2}}{(L^{3})^{2}}\sum_{q,l}\gamma^{0}G(p-q)%
\gamma^{0}G(p-q-l)\gamma^{0}G(p-q)\gamma^{0}\delta_{q^{0}0}\delta_{l^{0}0}+%
\frac{\Gamma_{V}\Gamma_{U}}{(L^{3})^{2}}\sum_{q,l}\gamma^{0}G(p-q)G(p-q-l)G(p-%
q)\gamma^{0}\delta_{q^{0}0}\delta_{l^{0}0}$$
$$\displaystyle+\frac{\Gamma_{U}\Gamma_{V}}{(L^{3})^{2}}\sum_{q,l}G(p-q)\gamma^{%
0}G(p-q-l)\gamma^{0}G(p-q)\delta_{q^{0}0}\delta_{l^{0}0}+\frac{\Gamma_{U}^{2}}%
{(L^{3})^{2}}\sum_{q,l}G(p-q)G(p-q-l)G(p-q)\delta_{q^{0}0}\delta_{l^{0}0},$$
$$\displaystyle\equiv$$
$$\displaystyle\Sigma^{(2),r}_{VV}(p)+\Sigma^{(2),r}_{VU}(p)+\Sigma^{(2),r}_{UV}%
(p)+\Sigma^{(2),r}_{UU}(p)$$
$$\displaystyle\Sigma^{(2),c}(p)$$
$$\displaystyle=$$
$$\displaystyle\frac{\Gamma_{V}^{2}}{(L^{3})^{2}}\sum_{q,l}\gamma^{0}G(p-q)%
\gamma^{0}G(p-q-l)\gamma^{0}G(p-l)\gamma^{0}\delta_{q^{0}0}\delta_{l^{0}0}+%
\frac{\Gamma_{V}\Gamma_{U}}{(L^{3})^{2}}\sum_{q,l}\gamma^{0}G(p-q)G(p-q-l)%
\gamma^{0}G(p-l)\delta_{q^{0}0}\delta_{l^{0}0}$$
$$\displaystyle+\frac{\Gamma_{U}\Gamma_{V}}{(L^{3})^{2}}\sum_{q,l}\gamma^{0}G(p-%
q)G(p-q-l)\gamma^{0}G(p-l)\delta_{q^{0}0}\delta_{l^{0}0}+\frac{\Gamma_{U}^{2}}%
{(L^{3})^{2}}\sum_{q,l}G(p-q)G(p-q-l)G(p-l)\delta_{q^{0}0}\delta_{l^{0}0}$$
$$\displaystyle\equiv$$
$$\displaystyle\Sigma^{(2),c}_{VV}(p)+\Sigma^{(2),c}_{VU}(p)+\Sigma^{(2),c}_{UV}%
(p)+\Sigma^{(2),c}_{UU}(p).$$
C.2 Evaluation of relevant diagrams
From now on, we evaluate self-energy diagrams one by one. Since there are two types of interactions, we have many diagrams to evaluate, especially, in the two loop-order. Instead of struggling to evaluate them one by one, we’re going to find integration formulae for products of Green functions and make a use of them for the same types of diagrams.
C.2.1 One-loop order: Fock diagrams
First, we evaluate the first-order Fock diagram
$$\displaystyle\Sigma^{(1)}(p)=\Gamma_{V}\gamma^{0}I_{1}(p)\gamma^{0}+\Gamma_{U}%
I_{1}(p),$$
where $I_{1}(p)$ is given by
$$\displaystyle I_{1}(p)=\int\frac{d^{d+1}q}{(2\pi)^{d+1}}2\pi\delta(q_{0})G(p-q%
)=\int\frac{d^{d}\bm{q}}{(2\pi)^{d}}G(p_{0},\bm{p}-\bm{q})=\int\frac{d^{d}\bm{%
q}}{(2\pi)^{d}}G(p_{0},-\bm{q}).$$
With Eq. (32), the Green function is given by
$$\displaystyle G(p_{0},-\bm{q})=\int^{1}_{0}dx\frac{-C_{3}^{i}{\bm{q}^{\prime}}%
^{2}q^{\prime}_{i}+C_{2a}{\bm{q}^{\prime}}^{2}+C_{2b}^{ij}q^{\prime}_{i}q^{%
\prime}_{j}-C_{1}^{i}q^{\prime}_{i}+C_{0}}{\big{[}{\bm{q}^{\prime}}^{2}+\Delta%
_{0}(p_{0};x)\big{]}^{2}},$$
where $\bm{q}^{\prime}=\bm{q}+\tilde{\bm{c}}_{x}$.
Dropping $\bm{q}^{\prime}$-odd terms, we have
$$\displaystyle I_{1}=\int^{1}_{0}dx\int\frac{d^{d}\bm{q}^{\prime}}{(2\pi)^{d}}%
\frac{C_{2a}\bm{q}^{\prime 2}+C_{2b}^{ij}q^{\prime}_{i}q^{\prime}_{j}+C_{0}}{%
\big{[}{\bm{q}^{\prime}}^{2}+\Delta_{0}(p_{0};x)\big{]}^{2}}=\int^{1}_{0}dx%
\Bigg{[}\bigg{(}\frac{d}{2}C_{2a}+\frac{1}{2}C_{2b}^{ii}\bigg{)}\frac{\Gamma(%
\frac{2-d}{2})}{(4\pi)^{\frac{d}{2}}\Delta_{0}^{\frac{2-d}{2}}}+\frac{C_{0}%
\Gamma(\frac{4-d}{2})}{(4\pi)^{\frac{d}{2}}\Delta_{0}^{\frac{4-d}{2}}}\Bigg{]}.$$
(36)
Then, the self-energy is given by
$$\displaystyle\Sigma^{(1)}(p)=\Gamma_{V}\int^{1}_{0}dx\Bigg{[}\frac{\big{(}%
\frac{d}{2}\bar{C}_{2a}+\frac{1}{2}\bar{C}_{2b}^{ii}\big{)}\Gamma(\frac{2-d}{2%
})}{(4\pi)^{\frac{d}{2}}\Delta_{0}^{\frac{2-d}{2}}}+\frac{\bar{C}_{0}\Gamma(%
\frac{4-d}{2})}{(4\pi)^{\frac{d}{2}}\Delta_{0}^{\frac{4-d}{2}}}\Bigg{]}+\Gamma%
_{U}\int^{1}_{0}dx\Bigg{[}\frac{\big{(}\frac{d}{2}C_{2a}+\frac{1}{2}C_{2b}^{ii%
}\big{)}\Gamma(\frac{2-d}{2})}{(4\pi)^{\frac{d}{2}}\Delta_{0}^{\frac{2-d}{2}}}%
+\frac{C_{0}\Gamma(\frac{4-d}{2})}{(4\pi)^{\frac{d}{2}}\Delta_{0}^{\frac{4-d}{%
2}}}\Bigg{]},$$
where we have introduced a bar-notation: $\bar{A}\equiv\gamma^{0}A\gamma^{0}$. Since we perform dimensional regularization in $d=2+\varepsilon$, the term containing $C_{0}$ gives only a finite value. A relevant part for renormalization is
$$\displaystyle\Sigma^{(1)}(p)$$
$$\displaystyle\simeq$$
$$\displaystyle\frac{\Gamma_{V}}{4\pi}\int^{1}_{0}dx\Bigg{[}\bigg{(}\frac{d}{2}%
\big{(}p_{0}\gamma^{0}-\tilde{c}_{xk}\gamma^{k}+c_{0}\gamma^{0}\gamma^{5}-c_{k%
}\gamma^{k}\gamma^{5}\big{)}+\frac{1}{2}\big{(}-2\tilde{c}_{xk}\gamma^{k}+2c_{%
k}\gamma^{k}\gamma^{5}\big{)}\bigg{)}\Gamma\bigg{(}\frac{2-d}{2}\bigg{)}\bigg{%
(}\frac{\Delta_{0}}{4\pi}\bigg{)}^{\frac{d-2}{2}}\Bigg{]}$$
(37)
$$\displaystyle+\frac{\Gamma_{U}}{4\pi}\int^{1}_{0}dx\Bigg{[}\bigg{(}\frac{d}{2}%
\big{(}p_{0}\gamma^{0}+\tilde{c}_{xk}\gamma^{k}-c_{0}\gamma^{0}\gamma^{5}-c_{k%
}\gamma^{k}\gamma^{5}\big{)}+\frac{1}{2}\big{(}2\tilde{c}_{xk}\gamma^{k}+2c_{k%
}\gamma^{k}\gamma^{5}\big{)}\bigg{)}\Gamma\bigg{(}\frac{2-d}{2}\bigg{)}\bigg{(%
}\frac{\Delta_{0}}{4\pi}\bigg{)}^{\frac{d-2}{2}}\Bigg{]}$$
$$\displaystyle=$$
$$\displaystyle-\frac{\Gamma_{V}}{2\pi\varepsilon}\big{(}p_{0}\gamma^{0}+c_{0}%
\gamma^{0}\gamma^{5}\big{)}-\frac{\Gamma_{U}}{2\pi\varepsilon}\big{(}p_{0}%
\gamma^{0}-c_{0}\gamma^{0}\gamma^{5}\big{)}+\mathcal{O}(1),$$
where $\tilde{c}_{xk}$-terms vanish after the integration over $x$.
Based on this result, we find propagator counter terms in the following way
$$\displaystyle-\frac{\Gamma_{V}}{2\pi\varepsilon}\big{(}p_{0}\gamma^{0}+c_{0}%
\gamma^{0}\gamma^{5}\big{)}-\frac{\Gamma_{U}}{2\pi\varepsilon}\big{(}p_{0}%
\gamma^{0}-c_{0}\gamma^{0}\gamma^{5}\big{)}+\mathcal{O}(1)+(\delta_{\psi}^{%
\omega}p_{0}\gamma^{0}+\delta_{\psi}^{\bm{k}}p_{k}\gamma^{k}+\delta_{c0}c_{0}%
\gamma^{0}\gamma^{5}+\delta_{\bm{c}}c_{k}\gamma^{k}\gamma^{5})=\textrm{finite}.$$
As a result, propagator counter terms up to the one-loop level are obtained as
$$\delta_{\psi}^{\omega}=\frac{\Gamma_{V}}{2\pi\varepsilon}+\frac{\Gamma_{U}}{2%
\pi\varepsilon},~{}~{}~{}~{}~{}\delta_{\psi}^{\bm{k}}=0,~{}~{}~{}~{}~{}\delta_%
{c0}=\frac{\Gamma_{V}}{2\pi\varepsilon}-\frac{\Gamma_{U}}{2\pi\varepsilon},~{}%
~{}~{}~{}~{}\delta_{\bm{c}}=0.$$
(38)
C.2.2 Two-loop order I: Rainbow diagrams
Next, we evaluate the rainbow diagrams
$$\displaystyle\Sigma^{(2),r}(p)$$
$$\displaystyle=$$
$$\displaystyle\Gamma_{V}^{2}I_{3r}(p)[M_{1}=M_{2}=\gamma^{0}]+\Gamma_{V}\Gamma_%
{U}I_{3r}(p)[M_{1}=\gamma^{0},M_{2}=I_{4\times 4}]$$
$$\displaystyle+\Gamma_{U}\Gamma_{V}I_{3r}(p)[M_{1}=I_{4\times 4},M_{2}=\gamma^{%
0}]+\Gamma_{U}^{2}I_{3r}(p)[M_{1}=M_{2}=I_{4\times 4}],$$
where $I_{3r}$ is given by
$$\displaystyle I_{3r}(p)=\int\frac{d^{d+1}q}{(2\pi)^{d+1}}2\pi\delta(q_{0})\int%
\frac{d^{d+1}l}{(2\pi)^{d+1}}2\pi\delta(l_{0})M_{1}G(p-q)M_{2}G(p-q-l)M_{2}G(p%
-q)M_{1}.$$
We may simplify this expression with $I_{1}$ as
$$\displaystyle I_{3r}(p)$$
$$\displaystyle=$$
$$\displaystyle\int\frac{d^{d+1}q}{(2\pi)^{d+1}}2\pi\delta(q_{0})M_{1}G(p-q)M_{2%
}\bigg{[}\int\frac{d^{d+1}l}{(2\pi)^{d+1}}2\pi\delta(l_{0})G(p-q-l)\bigg{]}M_{%
2}G(p-q)M_{1}$$
$$\displaystyle=$$
$$\displaystyle\int\frac{d^{d}\bm{q}}{(2\pi)^{d}}M_{1}G(p_{0},\bm{p}-\bm{q})M_{2%
}I_{1}(p-q)M_{2}G(p_{0},\bm{p}-\bm{q})M_{1}$$
$$\displaystyle=$$
$$\displaystyle\int\frac{d^{d}\bm{q}}{(2\pi)^{d}}M_{1}G(p_{0},-\bm{q})M_{2}I_{1}%
(p_{0})M_{2}G(p_{0},-\bm{q})M_{1},$$
where we used $I_{1}(p_{0},-\bm{q})=I_{1}(p_{0})$.
Taking into account
$$\displaystyle\frac{1}{\big{(}(p_{0}-c_{0})^{2}-(\bm{q}+\bm{c})\big{)}^{2}\big{%
(}(p_{0}+c_{0})^{2}-(\bm{q}-\bm{c})\big{)}^{2}}=\int^{1}_{0}dy\frac{6y(1-y)}{%
\big{[}{\bm{q}^{\prime}}^{2}+\Delta_{0}(p_{0};y)\big{]}^{4}}$$
with $\bm{q}^{\prime}=\bm{q}+\tilde{\bm{c}}_{y}$ and resorting to the representation of Eq. (32), we reach the following expression
$$\displaystyle I_{3r}=\int^{1}_{0}dy6y(1-y)\int\frac{d^{d}\bm{q}^{\prime}}{(2%
\pi)^{d}}\frac{M_{1}(-C_{3}^{i}\bm{q}^{\prime 2}q^{\prime}_{i}+C_{2a}\bm{q}^{%
\prime 2}+C_{2b}^{ij}q^{\prime}_{i}q^{\prime}_{j}-C_{1}^{i}q^{\prime}_{i}+C_{0%
})M_{2}I_{1}M_{2}(-C_{3}^{k}\bm{q}^{\prime 2}q^{\prime}_{k}+C_{2a}\bm{q}^{%
\prime 2}+C_{2b}^{kl}q^{\prime}_{k}q^{\prime}_{l}-C_{1}^{k}q^{\prime}_{k}+C_{0%
})M_{1}}{\big{[}\bm{q}^{\prime 2}+\Delta_{0}(p_{0};y)\big{]}^{4}}.$$
There are many even terms contributing to the integration. However, it turns out that we have to consider the product of $C_{3}^{i}$s only. This is because the divergent part of $I_{1}$ is canceled by the one-loop self-energy diagrams containing the first-order counter term, so only the finite part of $I_{1}$ participates in the remaining calculation.${}^{\ddagger}$ In other words, divergences may arise only by the $q^{6}$-term in the $\bm{q}$-integration. For now, we just assume it (we will be back to this point later).
Keeping this term only, we have
$$\displaystyle I_{3r}(p)$$
$$\displaystyle=$$
$$\displaystyle\int^{1}_{0}dy6y(1-y)\int\frac{d^{d}\bm{q}}{(2\pi)^{d}}\frac{(\bm%
{q}^{2})^{2}q_{i}q_{j}(M_{1}C_{3}^{i}M_{2}I_{1}M_{2}C_{3}^{j}M_{1})}{\big{[}%
\bm{q}^{2}+\Delta_{0}(p_{0};y)\big{]}^{4}}$$
$$\displaystyle=$$
$$\displaystyle\frac{(d+4)(d+2)\Gamma(\frac{2-d}{2})}{8(4\pi)^{\frac{d}{2}}}\int%
^{1}_{0}dy\frac{y(1-y)}{\Delta_{0}^{\frac{2-d}{2}}(y)}(M_{1}C_{3}^{i}M_{2}I_{1%
}M_{2}C_{3}^{i}M_{1}).$$
Then, the second-order self-energy correction for the rainbow diagrams is
$$\displaystyle\Sigma^{(2),r}(p)=\frac{(d+4)(d+2)\Gamma(\frac{2-d}{2})}{8(4\pi)^%
{\frac{d}{2}}}\int^{1}_{0}dy\frac{y(1-y)}{\Delta_{0}^{\frac{2-d}{2}}(y)}\bigg{%
[}\Gamma_{V}^{2}(\gamma^{i}I_{1}\gamma^{i})+2\Gamma_{V}\Gamma_{U}(\gamma^{0}%
\gamma^{i}I_{1}\gamma^{i}\gamma^{0})+\Gamma_{U}^{2}(\gamma^{i}I_{1}\gamma^{i})%
\bigg{]}.$$
When performing the renormalization group analysis in the second order, we should include consistently one-loop self-energy corrections made of a tree-level vertex and a one-loop propagator counter term, given by (Fig. 9)
$$\displaystyle\Sigma^{(1),\delta_{\psi}}(p)$$
$$\displaystyle=$$
$$\displaystyle\Gamma_{V}\int\frac{d^{d+1}q}{(2\pi)^{d+1}}2\pi\delta(q_{0})%
\gamma^{0}G(p-q)(\delta_{\psi}^{\omega}p_{0}\gamma^{0}+\delta_{\psi}^{\bm{k}}p%
_{k}\gamma^{k}+\delta_{c0}c_{0}\gamma^{0}\gamma^{5}+\delta_{\bm{c}}c_{k}\gamma%
^{k}\gamma^{5})G(p-q)\gamma^{0}$$
$$\displaystyle+\Gamma_{U}\int\frac{d^{d+1}q}{(2\pi)^{d+1}}2\pi\delta(q_{0})G(p-%
q)(\delta_{\psi}^{\omega}p_{0}\gamma^{0}+\delta_{\psi}^{\bm{k}}p_{k}\gamma^{k}%
+\delta_{c0}c_{0}\gamma^{0}\gamma^{5}+\delta_{\bm{c}}c_{k}\gamma^{k}\gamma^{5}%
)G(p-q)$$
$$\displaystyle=$$
$$\displaystyle\Gamma_{V}\frac{(d+4)(d+2)\Gamma(\frac{2-d}{2})}{8(4\pi)^{\frac{d%
}{2}}}\int^{1}_{0}dy\frac{y(1-y)}{\Delta_{0}^{\frac{2-d}{2}}(y)}\gamma^{0}%
\gamma^{i}\big{[}-\Gamma_{V}\gamma^{0}\textrm{div}(I_{1})\gamma^{0}-\Gamma_{U}%
\textrm{div}(I_{1})\big{]}\gamma^{i}\gamma^{0}$$
$$\displaystyle+\Gamma_{U}\frac{(d+4)(d+2)\Gamma(\frac{2-d}{2})}{8(4\pi)^{\frac{%
d}{2}}}\int^{1}_{0}dy\frac{y(1-y)}{\Delta_{0}^{\frac{2-d}{2}}(y)}\gamma^{i}%
\big{[}-\Gamma_{V}\gamma^{0}\textrm{div}(I_{1})\gamma^{0}-\Gamma_{U}\textrm{%
div}(I_{1})\big{]}\gamma^{i}$$
$$\displaystyle=$$
$$\displaystyle-\frac{(d+4)(d+2)\Gamma(\frac{2-d}{2})}{8(4\pi)^{\frac{d}{2}}}%
\int^{1}_{0}dy\frac{y(1-y)}{\Delta_{0}^{\frac{2-d}{2}}(y)}\bigg{[}\Gamma_{V}^{%
2}\gamma^{i}\textrm{div}(I_{1})\gamma^{i}+2\Gamma_{V}\Gamma_{U}\gamma^{0}%
\gamma^{i}\textrm{div}(I_{1})\gamma^{i}\gamma^{0}+\Gamma_{U}^{2}\gamma^{i}%
\textrm{div}(I_{1})\gamma^{i}\bigg{]}$$
$$\displaystyle\equiv$$
$$\displaystyle\Sigma^{(1),\delta_{\psi}}_{VV}(p)+\Sigma^{(1),\delta_{\psi}}_{VU%
}(p)+\Sigma^{(1),\delta_{\psi}}_{UV}(p)++\Sigma^{(1),\delta_{\psi}}_{UU}(p),$$
where $\textrm{div}(\cdots)$ means the divergent part of $(\cdots)$. If we add these to the rainbow diagrams, the divergent part of $I_{1}$ in the rainbow diagrams is eliminated and only a finite part participates in the remaining computation (so the remark of $\ddagger$ is proved).
Writing it as $\textrm{fin}(I_{1})\equiv I_{1}-\textrm{div}(I_{1})=\int^{1}_{0}dx\frac{C_{0}%
\Gamma(\frac{4-d}{2})}{(4\pi)^{d/2}\Delta_{0}^{(4-d)/2}}$, we obtain
$$\displaystyle\Sigma^{(2),r}+\Sigma^{(1),\delta_{\psi}}=\frac{(d+4)(d+2)\Gamma(%
\frac{2-d}{2})}{8(4\pi)^{\frac{d}{2}}}\int^{1}_{0}dy\frac{y(1-y)}{\Delta_{0}^{%
\frac{2-d}{2}}(y)}\bigg{[}\Gamma_{V}^{2}\gamma^{i}\textrm{fin}(I_{1})\gamma^{i%
}+2\Gamma_{V}\Gamma_{U}\gamma^{0}\gamma^{i}\textrm{fin}(I_{1})\gamma^{i}\gamma%
^{0}+\Gamma_{U}^{2}\gamma^{i}\textrm{fin}(I_{1})\gamma^{i}\bigg{]}.$$
An expansion about $d=2+\varepsilon$ gives
$$\displaystyle\frac{(d+4)(d+2)\Gamma(\frac{2-d}{2})}{8(4\pi)^{\frac{d}{2}}}\int%
^{1}_{0}dy\frac{y(1-y)}{\Delta_{0}^{\frac{2-d}{2}}(y)}\int^{1}_{0}dx\frac{C_{0%
}(x)\Gamma(\frac{4-d}{2})}{(4\pi)^{\frac{d}{2}}\Delta_{0}^{\frac{4-d}{2}}(x)}=%
-\frac{1}{16\pi^{2}\varepsilon}\int^{1}_{0}dx\frac{C_{0}(x)}{\Delta_{0}(x)}+%
\mathcal{O}(1).$$
As a result, the relevant part for renormalization is given as
$$\displaystyle\Sigma^{(2),r}+\Sigma^{(1),\delta_{\psi}}=-\frac{1}{16\pi^{2}%
\varepsilon}\bigg{[}\Gamma_{V}^{2}\gamma^{i}\int^{1}_{0}dx\frac{C_{0}(x)}{%
\Delta_{0}(x)}\gamma^{i}+2\Gamma_{V}\Gamma_{U}\gamma^{0}\gamma^{i}\int^{1}_{0}%
dx\frac{C_{0}(x)}{\Delta_{0}(x)}\gamma^{i}\gamma^{0}+\Gamma_{U}^{2}\gamma^{i}%
\int^{1}_{0}dx\frac{C_{0}(x)}{\Delta_{0}(x)}\gamma^{i}\bigg{]}+\mathcal{O}(1).$$
The remaining calculation is $\int^{1}_{0}dx\frac{C_{0}(x)}{\Delta_{0}(x)}$. A straightforward calculation gives
$$\displaystyle p_{0}\gamma^{0}\bigg{[}-1-\frac{\alpha}{2}\ln{\bigg{(}\frac{%
\alpha-1}{\alpha+1}\bigg{)}}-\frac{\beta}{2}\ln{\bigg{(}\frac{\beta-1}{\beta+1%
}\bigg{)}}\bigg{]}+c_{0}\gamma^{0}\bigg{[}\frac{1}{2}\ln{\bigg{(}\frac{\alpha-%
1}{\alpha+1}\bigg{)}}+\frac{1}{2}\ln{\bigg{(}\frac{\beta-1}{\beta+1}\bigg{)}}%
\bigg{]}$$
$$\displaystyle+c_{k}\gamma^{k}\bigg{[}-(\alpha+\beta)+\frac{1-\alpha^{2}}{2}\ln%
{\bigg{(}\frac{\alpha-1}{\alpha+1}\bigg{)}}+\frac{1-\beta^{2}}{2}\ln{\bigg{(}%
\frac{\beta-1}{\beta+1}\bigg{)}}\bigg{]}+c_{0}\gamma^{0}\gamma^{5}\bigg{[}1+%
\frac{\alpha}{2}\ln{\bigg{(}\frac{\alpha-1}{\alpha+1}\bigg{)}}+\frac{\beta}{2}%
\ln{\bigg{(}\frac{\beta-1}{\beta+1}\bigg{)}}\bigg{]}$$
$$\displaystyle+p_{0}\gamma^{0}\gamma^{5}\bigg{[}-\frac{1}{2}\ln{\bigg{(}\frac{%
\alpha-1}{\alpha+1}\bigg{)}}-\frac{1}{2}\ln{\bigg{(}\frac{\beta-1}{\beta+1}%
\bigg{)}}\bigg{]}+c_{k}\gamma^{k}\gamma^{5}(-1),$$
where $(\alpha,\beta)\equiv ab\pm\sqrt{(a^{2}-1)(b^{2}-1)}$ and $a\equiv\frac{p_{0}}{\left|\bm{c}\right|}$, $b\equiv\frac{c_{0}}{\left|\bm{c}\right|}$.
Dropping the complex logarithm terms, we have
$$\displaystyle\gamma^{i}\int^{1}_{0}dx\frac{C_{0}(x)}{\Delta_{0}(x)}\gamma^{i}$$
$$\displaystyle\simeq$$
$$\displaystyle\gamma^{i}\Big{(}-p_{0}\gamma^{0}-\frac{2p_{0}c_{0}}{\left|\bm{c}%
\right|^{2}}c_{k}\gamma^{k}+c_{0}\gamma^{0}\gamma^{5}-c_{k}\gamma^{k}\gamma^{5%
}\Big{)}\gamma^{i}$$
$$\displaystyle=$$
$$\displaystyle-dp_{0}\gamma^{0}+(2-d)\frac{2p_{0}c_{0}}{\left|\bm{c}\right|^{2}%
}c_{k}\gamma^{k}-dc_{0}\gamma^{0}\gamma^{5}+(d-2)c_{k}\gamma^{k}\gamma^{5}.$$
As a result, the self-energy correction from rainbow diagrams is
$$\displaystyle\Sigma^{(2),r}(p)+\Sigma^{(1),\delta_{\psi}}(p)=\frac{1}{8\pi^{2}%
\varepsilon}\bigg{[}\Gamma_{V}^{2}(p_{0}\gamma^{0}+c_{0}\gamma^{0}\gamma^{5})+%
\Gamma_{U}^{2}(p_{0}\gamma^{0}+c_{0}\gamma^{0}\gamma^{5})+2\Gamma_{V}\Gamma_{U%
}(p_{0}\gamma^{0}-c_{0}\gamma^{0}\gamma^{5})\bigg{]}+\mathcal{O}(1),$$
(39)
where the result is depicted pictorially in Fig. 10.
C.2.3 Two-loop order II: Crossed diagrams
Last, we evaluate the crossed diagrams
$$\displaystyle\Sigma^{(2),c}(p)$$
$$\displaystyle=$$
$$\displaystyle\Gamma_{V}^{2}I_{3c}(p)[M_{1}=M_{2}=\gamma^{0}]+\Gamma_{V}\Gamma_%
{U}I_{3c}(p)[M_{1}=\gamma^{0},M_{2}=I_{4\times 4}]$$
$$\displaystyle+\Gamma_{U}\Gamma_{V}I_{3c}(p)[M_{1}=I_{4\times 4},M_{2}=\gamma^{%
0}]+\Gamma_{U}^{2}I_{3c}(p)[M_{1}=M_{2}=I_{4\times 4}],$$
where $I_{3c}$ is given by
$$\displaystyle I_{3c}(p)$$
$$\displaystyle=$$
$$\displaystyle\int\frac{d^{d+1}q}{(2\pi)^{d+1}}2\pi\delta(q_{0})\int\frac{d^{d+%
1}l}{(2\pi)^{d+1}}2\pi\delta(l_{0})M_{1}G(p-q)M_{2}G(p-q-l)M_{1}G(p-l)M_{2}$$
$$\displaystyle=$$
$$\displaystyle\int\frac{d^{d}\bm{q}}{(2\pi)^{d}}\int\frac{d^{d}\bm{l}}{(2\pi)^{%
d}}M_{1}G(p_{0},\bm{p}-\bm{q})M_{2}G(p_{0},\bm{p}-\bm{q}-\bm{l})M_{1}G(p_{0},%
\bm{p}-\bm{l})M_{2}$$
$$\displaystyle=$$
$$\displaystyle\int\frac{d^{d}\bm{q}}{(2\pi)^{d}}\int\frac{d^{d}\bm{l}}{(2\pi)^{%
d}}M_{1}G(p_{0},-\bm{q})M_{2}G(p_{0},-\bm{q}-\bm{l})M_{1}G(p_{0},-\bm{l}+\bm{p%
})M_{2}.$$
In this case the loop momenta of $\bm{l}$ and $\bm{q}$ are interwoven and this makes the analysis more complicated.
First, we perform the integration on $\bm{q}$. Using Eq. (32), we have
$$\displaystyle\int^{1}_{0}dx\int^{1}_{0}dy\int\frac{d^{d}\bm{q}}{(2\pi)^{d}}%
\frac{-C_{3}^{i}\bm{q}^{\prime 2}q_{i}+C_{2a}\bm{q}^{\prime 2}+C_{2b}^{ij}q^{%
\prime}_{i}q^{\prime}_{j}-C_{1}^{i}q^{\prime}_{i}+C_{0}}{\big{[}\bm{q}^{\prime
2%
}+\Delta_{0}(p_{0};x)\big{]}^{2}}$$
$$\displaystyle\times M_{2}\frac{-C_{3}^{k}(\bm{q}^{\prime}+\bm{l})^{2}(q^{%
\prime}_{k}+l_{k})+C_{2a}(\bm{q}^{\prime}+\bm{l})^{2}+C_{2b}^{kl}(q^{\prime}_{%
k}+l_{k})(q^{\prime}_{l}+l_{l})-C_{1}^{l}(q^{\prime}_{l}+l_{l})+C_{0}}{\big{[}%
(\bm{q}^{\prime}+\bm{l})^{2}+\Delta_{0}(p_{0};y)\big{]}^{2}}.$$
Denominators are combined as
$$\displaystyle\frac{1}{\big{[}\bm{q}^{\prime 2}+\Delta_{0}(p_{0};x)\big{]}^{2}%
\big{[}(\bm{q}^{\prime}+\bm{l})^{2}+\Delta_{0}(p_{0};y)\big{]}^{2}}=\int^{1}_{%
0}dz\frac{6z(1-z)}{\big{[}(\bm{q}^{\prime}+z\bm{l})^{2}+\Delta_{1}(p_{0},\bm{l%
};x,y,z)\big{]}^{4}},$$
where $\Delta_{1}=z(1-z)\bm{l}^{2}+(1-z)\Delta_{0}(p_{0};x)+z\Delta_{0}(p_{0};y)$. Shifting $\bm{q}^{\prime}\rightarrow\bm{q}^{\prime}-z\bm{l}$ and renaming $\bm{q}^{\prime}$ as $\bm{q}$, we have
$$\displaystyle\int^{1}_{0}dx\int^{1}_{0}dy\int^{1}_{0}dz\int\frac{d^{d}\bm{q}}{%
(2\pi)^{d}}\frac{6z(1-z)}{\big{[}\bm{q}^{2}+\Delta_{1}\big{]}^{4}}\Big{(}-C_{3%
}^{i}(\bm{q}-z\bm{l})^{2}(q_{i}-zl_{i})+C_{2a}(\bm{q}-z\bm{l})^{2}+C_{2b}^{ij}%
(q_{i}-zl_{i})(q_{j}-zl_{j})-C_{1}^{i}(q_{i}-zl_{i})+C_{0}\Big{)}$$
$$\displaystyle\times M_{2}\Big{(}-C_{3}^{k}\big{(}\bm{q}+(1-z)\bm{l}\big{)}^{2}%
\big{(}q_{k}+(1-z)l_{k}\big{)}+C_{2a}\big{(}\bm{q}+(1-z)\bm{l}\big{)}^{2}+C_{2%
b}^{kl}\big{(}q_{k}+(1-z)l_{k}\big{)}\big{(}q_{l}+(1-z)l_{l}\big{)}-C_{1}^{k}%
\big{(}q_{k}+(1-z)l_{k}\big{)}+C_{0}\Big{)}.$$
Despite this complex expression, we need to consider only a few terms for renormalization. This can be understood, considering a simple integral
$$\displaystyle\int\frac{d^{d}\bm{q}}{(2\pi)^{d}}\frac{(\bm{q}^{2})^{m}}{[\bm{q}%
^{2}+\Delta]^{4}}=\frac{\Gamma(\frac{8-d-2m}{2})\Gamma(\frac{d}{2}+m)}{(4\pi)^%
{\frac{d}{2}}\Gamma(\frac{d}{2})\Gamma(4)\Delta^{\frac{8-d-2m}{2}}}.$$
(40)
Since we resort to the dimensional regularization in $d=2+\varepsilon$, an integral for $m$ smaller than $3$ gives a finite value, and it doesn’t participate in renormalization. The product of the $q^{3}$-terms (i.e. $q^{6}$-term) certainly gives renormalization effects. Other than $q^{6}$-term, even terms of $q^{4}l^{2}$, $q^{2}l^{4}$ and $l^{6}$ possibly contribute to renormalization after the $\bm{l}$-integral because there will be an equal number of momentum $l$ in the denominator and numerator (considering the dimension of an integrand, this fact may be easily estimated, because any dimensionful constant in numerator lowers the superficial degree of divergence of the integral). All of those come from the product of the $C_{3}$-terms, so the relevant part is
$$\displaystyle\int^{1}_{0}dx\int^{1}_{0}dy\int^{1}_{0}dz6z(1-z)\int\frac{d^{d}%
\bm{q}}{(2\pi)^{d}}\frac{(\bm{q}-z\bm{l})^{2}(\not{\bm{q}}-z\not{\bm{l}})M_{2}%
\big{(}\bm{q}+(1-z)\bm{l}\big{)}^{2}\big{(}\not{\bm{q}}+(1-z)\not{\bm{l}}\big{%
)}}{\big{[}\bm{q}^{2}+\Delta_{1}(p_{0},\bm{l};x,y,z)\big{]}^{4}},$$
where $\not{\bm{q}}\equiv q_{i}\gamma^{i}(i=1,2,3)$.
The numerator is arranged as
$$\displaystyle N$$
$$\displaystyle=$$
$$\displaystyle(-1)^{1+\frac{1}{4}\textrm{tr}[M_{2}]}M_{2}(\bm{q}-z\bm{l})^{2}%
\big{(}\bm{q}+(1-z)\bm{l}\big{)}^{2}\big{(}\not{\bm{q}}-z\not{\bm{l}}\big{)}%
\big{(}\not{\bm{q}}+(1-z)\not{\bm{l}}\big{)}$$
$$\displaystyle=$$
$$\displaystyle(-1)^{\frac{1}{4}\textrm{tr}[M_{2}]}M_{2}\Big{[}D_{6}(\bm{q}^{2})%
^{3}+D_{4a}(\bm{q}^{2})^{2}+D_{4b}^{ij}\bm{q}^{2}q_{i}q_{j}+D_{2a}\bm{q}^{2}+D%
_{2b}^{ij}q_{i}q_{j}+D_{0}\Big{]}+(\textrm{odd terms}),$$
where the coefficients are given by
$$\displaystyle D_{6}$$
$$\displaystyle=$$
$$\displaystyle 1,$$
$$\displaystyle D_{4a}$$
$$\displaystyle=$$
$$\displaystyle(3z^{2}-3z+1)\bm{l}^{2},$$
$$\displaystyle D_{4b}^{ij}$$
$$\displaystyle=$$
$$\displaystyle(12z^{2}-8z)l^{i}l^{j}+(2-4z)l^{i}\gamma^{j}\not{\bm{l}},$$
$$\displaystyle D_{2a}$$
$$\displaystyle=$$
$$\displaystyle(3z^{4}-6z^{3}+4z^{2}-z)(\bm{l}^{2})^{2},$$
$$\displaystyle D_{2b}^{ij}$$
$$\displaystyle=$$
$$\displaystyle(12z^{4}-20z^{3}+8z^{2})\bm{l}^{2}l^{i}l^{j}+(-4z^{3}+6z^{2}-2z)%
\bm{l}^{2}l^{i}\gamma^{j}\not{\bm{l}},$$
$$\displaystyle D_{0}$$
$$\displaystyle=$$
$$\displaystyle z^{3}(z-1)^{3}(\bm{l}^{2})^{3}.$$
Now, the integral is easily performed to be
$$\displaystyle\int\frac{d^{d}\bm{q}}{(2\pi)^{d}}\frac{(-1)^{\frac{1}{4}\textrm{%
tr}[M_{2}]}M_{2}\big{[}D_{6}(\bm{q}^{2})^{3}+D_{4a}(\bm{q}^{2})^{2}+D_{4b}^{ij%
}\bm{q}^{2}q_{i}q_{j}+D_{2a}\bm{q}^{2}+D_{2b}^{ij}q_{i}q_{j}+D_{0}\big{]}}{%
\big{[}\bm{q}^{2}+\Delta_{1}(\bm{l};x,y,z)\big{]}^{4}}$$
$$\displaystyle=$$
$$\displaystyle\frac{(-1)^{\frac{1}{4}\textrm{tr}[M_{2}]}M_{2}}{(4\pi)^{\frac{d}%
{2}}}\bigg{[}\frac{d(d+4)(d+2)\Gamma\big{(}\frac{2-d}{2}\big{)}}{8\Gamma(4)}%
\frac{D_{6}}{\Delta_{1}^{\frac{2-d}{2}}}+\frac{d(d+2)\Gamma\big{(}\frac{4-d}{2%
}\big{)}}{4\Gamma(4)}\frac{D_{4a}+\frac{D_{4b}^{ii}}{d}}{\Delta_{1}^{\frac{4-d%
}{2}}}+\frac{d\Gamma\big{(}\frac{6-d}{2}\big{)}}{2\Gamma(4)}\frac{D_{2a}+\frac%
{D_{2b}^{ii}}{d}}{\Delta_{1}^{\frac{6-d}{2}}}+\frac{\Gamma\big{(}\frac{8-d}{2}%
\big{)}}{\Gamma(4)}\frac{D_{0}}{\Delta_{1}^{\frac{8-d}{2}}}\bigg{]}.$$
Next, we perform the $\bm{l}$-integral. Using $M_{1}M_{2}M_{1}=M_{2}$ (since the matrices of $M_{1}$ and $M_{2}$ are either $I_{4\times 4}$ or $\gamma^{0}$), we have
$$\displaystyle I_{3c}(p)$$
$$\displaystyle=$$
$$\displaystyle\frac{(-1)^{\frac{1}{4}\textrm{tr}[M_{2}]}}{(4\pi)^{\frac{d}{2}}}%
\int^{1}_{0}dx\int^{1}_{0}dy\int^{1}_{0}dzz(1-z)\int\frac{d^{d}\bm{l}}{(2\pi)^%
{d}}\Bigg{[}\frac{d(d+4)(d+2)\Gamma\big{(}\frac{2-d}{2}\big{)}}{8}\frac{D_{6}}%
{\Delta_{1}^{\frac{2-d}{2}}}$$
$$\displaystyle+\frac{d(d+2)\Gamma\big{(}\frac{4-d}{2}\big{)}}{4}\frac{D_{4a}+%
\frac{D_{4b}^{ii}}{d}}{\Delta_{1}^{\frac{4-d}{2}}}+\frac{d\Gamma\big{(}\frac{6%
-d}{2}\big{)}}{2}\frac{D_{2a}+\frac{D_{2b}^{ii}}{d}}{\Delta_{1}^{\frac{6-d}{2}%
}}+\frac{\Gamma\big{(}\frac{8-d}{2}\big{)}D_{0}}{\Delta_{1}^{\frac{8-d}{2}}}%
\Bigg{]}M_{2}G(p_{0},-\bm{l}+\bm{p})M_{2}+\textrm{(finite parts)}.$$
Taking out $z(1-z)$ from $\Delta_{1}=z(1-z)\bm{l}^{2}+(1-z)\Delta_{0}(x)+z\Delta_{0}(y)$ first, we find that the remaining integrals are such a simple form:
$$\displaystyle\int^{1}_{0}dv\int\frac{d^{d}\bm{l}}{(2\pi)^{d}}\frac{(\bm{l}^{2}%
)^{n}}{\big{[}\bm{l}^{2}+\frac{1}{z}\Delta_{0}(p_{0};x)+\frac{1}{1-z}\Delta_{0%
}(p_{0};y)\big{]}^{n+\frac{2-d}{2}}}$$
$$\displaystyle\times\frac{-(\bm{l}-\bm{p})^{2}(l_{i}-p_{i})\gamma^{i}+(\bm{l}-%
\bm{p})^{2}(p_{0}\gamma^{0}-\not{c}\gamma^{5})+(l_{i}-p_{i})(l_{j}-p_{j})(-2c^%
{i}\gamma^{j}\gamma^{5})-(l_{i}-p_{i})f_{1}^{i}(p_{0})+f_{0}(p_{0})}{\big{[}%
\big{(}\bm{l}-\bm{p}+\tilde{\bm{c}}_{v}\big{)}^{2}+\Delta_{0}(p_{0};v)\big{]}^%
{2}},$$
where the cases of $n=0,1,2,3$ correspond to integrals for $D_{6},~{}D_{4},~{}D_{2}$ and $D_{0}$, respectively. Such integrations result in $\Gamma(n+\frac{2-d}{2}+2-\frac{d}{2}-n-m)=\Gamma(3-d-m)$, where $m=1$ stands for $l^{2}$ (the leading even-term) and $m=0$ for a constant term in the propagator. Within the dimensional regularization in $d=2+\varepsilon$, only the integral of $m=1$ possibly gives a divergent factor of $\Gamma(2-d)$. However, in the $n=0$ case we already got $\Gamma(\frac{2-d}{2})$, and we need to consider the constant ($m=0$) term, which turns out to be important. We first compute this term.
The denominator is transformed as
$$\displaystyle\int^{1}_{0}dv\frac{1}{\big{[}\bm{l}^{2}+\frac{\Delta_{0}(x)}{z}+%
\frac{\Delta_{0}(y)}{1-z}\big{]}^{\frac{2-d}{2}}\big{[}\big{(}\bm{l}-\bm{p}+%
\tilde{\bm{c}}_{v}\big{)}^{2}+\Delta_{0}(v)\big{]}^{2}}=\int^{1}_{0}dv\int^{1}%
_{0}dw\frac{w(1-w)^{-\frac{d}{2}}\Gamma\big{(}\frac{6-d}{2}\big{)}/\Gamma\big{%
(}\frac{2-d}{2}\big{)}}{\big{[}\big{(}\bm{l}-w(\bm{p}-\tilde{\bm{c}}_{v})\big{%
)}^{2}+\Delta_{2}(\bm{p};x,y,z,v,w)\big{]}^{\frac{6-d}{2}}},$$
where $\Delta_{2}=w(1-w)(\bm{p}-\tilde{\bm{c}}_{v})^{2}+\frac{1-w}{z}\Delta_{0}(x)+%
\frac{1-w}{1-z}\Delta_{0}(y)+w\Delta_{0}(v)$. This suggests that we may use Eq. (32) with a slight change. Then, the integral for $m=0$ is
$$\displaystyle\frac{\Gamma\big{(}\frac{6-d}{2}\big{)}}{\Gamma\big{(}\frac{2-d}{%
2}\big{)}}\int^{1}_{0}dv\int^{1}_{0}dww(1-w)^{-\frac{d}{2}}\int\frac{d^{d}\bm{%
l}^{\prime}}{(2\pi)^{d}}\frac{C_{0}(w,v)}{\big{[}\bm{l}^{\prime 2}+\Delta_{2}(%
w,v)\big{]}^{\frac{6-d}{2}}}=\frac{\Gamma(3-d)}{(4\pi)^{\frac{d}{2}}\Gamma\big%
{(}\frac{2-d}{2}\big{)}}\int^{1}_{0}dv\int^{1}_{0}dww(1-w)^{-\frac{d}{2}}\frac%
{C_{0}(w,v)}{\Delta_{2}(w,v)},$$
where $C_{0}(w,v)$ is same with that of Eq. (32) except for $u=(p_{0},(1-w)\bm{p}+w\tilde{\bm{c}}_{v})$. Note $C_{0}(w=1,v)=C_{0}(v)$ (the original definition of $C_{0}$) and $\Delta_{2}(w=1,v)=\Delta_{0}(v)$.
This implies that we may take out a relevant part in the following way
$$\displaystyle\frac{\Gamma(3-d)}{(4\pi)^{\frac{d}{2}}\Gamma\big{(}\frac{2-d}{2}%
\big{)}}\int^{1}_{0}dv\int^{1}_{0}dww(1-w)^{-\frac{d}{2}}\Bigg{[}\frac{C_{0}(v%
)}{\Delta_{0}(v)}+\frac{d}{dw}\left.\frac{C_{0}(w,v)}{\Delta_{2}(w,v)}\right|_%
{w=1}(w-1)+\mathcal{O}(w-1)^{2}\Bigg{]}$$
$$\displaystyle=$$
$$\displaystyle\frac{\Gamma(3-d)}{(4\pi)^{\frac{d}{2}}\Gamma\big{(}\frac{6-d}{2}%
\big{)}}\int^{1}_{0}dv\frac{C_{0}(v)}{\Delta_{0}(v)}-\frac{\frac{4\Gamma(3-d)}%
{(6-d)(4-d)}}{(4\pi)^{\frac{d}{2}}\Gamma\big{(}\frac{2-d}{2}\big{)}}\int^{1}_{%
0}dv\frac{d}{dw}\left.\frac{C_{0}(w,v)}{\Delta_{2}(w,v)}\right|_{w=1}+\cdots.$$
Note that $\Gamma(\frac{2-d}{2})$ in the first term is canceled after the $w$-integral, but $\Gamma\big{(}\frac{2-d}{2}\big{)}$ in the second term is not. Together with $\Gamma\big{(}\frac{2-d}{2}\big{)}$ originating from the $\bm{q}$-integral, the first term contributes to a divergent part while the other higher-order terms give only finite values. In short, the above analysis suggests that we should include $\int^{1}_{0}dv\frac{C_{0}(v)}{\Delta_{0}(v)}$.
Now, we focus on the $m=1$ case. Since $l^{2}$ may arise from $l^{2}$ (surely) and $l^{3}$ (after momentum shift), we’re keeping them. After the similar analysis as the above, we obtain
$$\displaystyle\int^{1}_{0}dv\int\frac{d^{d}\bm{l}}{(2\pi)^{d}}\frac{(\bm{l}^{2}%
)^{n}}{\big{[}\bm{l}^{2}+\frac{1}{z}\Delta_{0}(x)+\frac{1}{1-z}\Delta_{0}(y)%
\big{]}^{n+\frac{2-d}{2}}}\frac{\bm{l}^{2}(\not{p}-\not{c}\gamma^{5})-\bm{l}^{%
2}l_{i}\gamma^{i}+l_{i}l_{j}(-2p^{i}\gamma^{j}-2c^{i}\gamma^{j}\gamma^{5})}{%
\big{[}\big{(}\bm{l}-\bm{p}+\tilde{\bm{c}}_{v}\big{)}^{2}+\Delta_{0}(p_{0};v)%
\big{]}^{2}}$$
$$\displaystyle=$$
$$\displaystyle\int^{1}_{0}dv\int^{1}_{0}dw\frac{w(1-w)^{n-\frac{d}{2}}\Gamma%
\big{(}n+\frac{6-d}{2}\big{)}}{\Gamma(2)\Gamma\big{(}n+\frac{2-d}{2}\big{)}}%
\int\frac{d^{d}\bm{l}}{(2\pi)^{d}}\frac{(\bm{l}^{2})^{n}\big{[}\bm{l}^{2}(\not%
{p}-\not{c}\gamma^{5})-\bm{l}^{2}l_{i}\gamma^{i}+l_{i}l_{j}(-2p^{i}\gamma^{j}-%
2c^{i}\gamma^{j}\gamma^{5})\big{]}}{\big{[}\big{(}\bm{l}-w(\bm{p}-\tilde{\bm{c%
}}_{v})\big{)}^{2}+\Delta_{2}(\bm{p};x,y,z,v,w)\big{]}^{n+\frac{6-d}{2}}}$$
$$\displaystyle\simeq$$
$$\displaystyle\frac{\Gamma\big{(}n+\frac{6-d}{2}\big{)}}{\Gamma\big{(}n+\frac{2%
-d}{2}\big{)}}\int^{1}_{0}dv\int^{1}_{0}dww(1-w)^{n-\frac{d}{2}}\int\frac{d^{d%
}\bm{l}}{(2\pi)^{d}}\frac{(\bm{l}^{2})^{n}\big{[}\bm{l}^{2}(\not{p}-\not{c}%
\gamma^{5}-w(p_{i}-\tilde{c}_{vi})\gamma^{i})+l_{i}l_{j}\big{(}2(n+1)w(p^{i}-%
\tilde{c}_{v}^{i})\gamma^{j}-2p^{i}\gamma^{j}-2c^{i}\gamma^{j}\gamma^{5}\big{)%
}\big{]}}{\big{[}\bm{l}^{2}+\Delta_{2}(\bm{p};x,y,z,v,w)\big{]}^{n+\frac{6-d}{%
2}}},$$
where we have shifted $\bm{l}\rightarrow\bm{l}+w(\bm{p}-\tilde{\bm{c}}_{v})$ and kept only the leading even terms including shifted contributions from $(\bm{l}^{2})^{n}$ and $\bm{l}^{2}l_{i}$.
After the $\bm{l}$-integration, we reach the following expression
$$\displaystyle\frac{\Gamma(2-d)\Gamma\big{(}\frac{d}{2}+n+1\big{)}}{(4\pi)^{%
\frac{d}{2}}\Gamma\big{(}\frac{d}{2}\big{)}\Gamma\big{(}n+\frac{2-d}{2}\big{)}%
}\int^{1}_{0}dv\int^{1}_{0}dww(1-w)^{n-\frac{d}{2}}\Delta_{2}^{d-2}\bigg{(}%
\not{p}-\not{c}\gamma^{5}-w(p_{i}-\tilde{c}_{vi})\gamma^{i}+\frac{1}{d}\big{(}%
-2(n+1)w(p_{i}-\tilde{c}_{vi})\gamma^{i}+2p_{i}\gamma^{i}+2c_{i}\gamma^{i}%
\gamma^{5}\big{)}\bigg{)}.$$
Considering $d=2+\varepsilon$, $\Delta_{2}^{d-2}$ is not involved in the $w$- and the $v$-integral. The effect of the $v$-integral is just to remove $\tilde{c}_{vi}$. The $w$-integral gives
$$\displaystyle\frac{\Gamma(2-d)}{(4\pi)^{\frac{d}{2}}\Gamma\big{(}\frac{d}{2}%
\big{)}}\frac{\Gamma\big{(}\frac{d}{2}+n+1\big{)}}{\Gamma\big{(}n+\frac{6-d}{2%
}\big{)}}\bigg{(}\not{p}-\not{c}\gamma^{5}+\frac{2}{d}\big{(}p_{i}\gamma^{i}+c%
_{i}\gamma^{i}\gamma^{5}\big{)}-\frac{2}{n+\frac{6-d}{2}}\frac{2(n+1)+d}{d}p_{%
i}\gamma^{i}\bigg{)},$$
where $2/(n+\frac{6-d}{2})$ makes up for the difference due to additional $w$. Among the remaining Feynman parameters of $x,y$ and $z$, only $z$ is effective since there are polynomials of $z$ in the $D$s.
The $z$-integrals for each $n$ are performed as (from the first line, $n=0,1,2,3$)
$$\displaystyle\int^{1}_{0}dz\frac{z(1-z)}{z^{\frac{2-d}{2}}(1-z)^{\frac{2-d}{2}%
}}=\frac{\big{[}\Gamma\big{(}\frac{d+2}{2}\big{)}\big{]}^{2}}{\Gamma(d+2)},$$
$$\displaystyle\int^{1}_{0}dzz(1-z)\frac{1-3z(1-z)+\frac{2-12z(1-z)}{d}}{z^{%
\frac{4-d}{2}}(1-z)^{\frac{4-d}{2}}}=\frac{d^{2}+8}{d^{2}}\frac{\big{[}\Gamma%
\big{(}\frac{d+2}{2}\big{)}\big{]}^{2}}{\Gamma(d+2)},$$
$$\displaystyle\int^{1}_{0}dzz(1-z)\frac{z(1-z)\Big{(}-1+3z(1-z)+\frac{-2+12z(1-%
z)}{d}\Big{)}}{z^{\frac{6-d}{2}}(1-z)^{\frac{6-d}{2}}}=-\frac{d^{2}+8}{d^{2}}%
\frac{\big{[}\Gamma\big{(}\frac{d+2}{2}\big{)}\big{]}^{2}}{\Gamma(d+2)},$$
$$\displaystyle\int^{1}_{0}dzz(1-z)\frac{-z^{3}(1-z)^{3}}{z^{\frac{8-d}{2}}(1-z)%
^{\frac{8-d}{2}}}=-\frac{\big{[}\Gamma\big{(}\frac{d+2}{2}\big{)}\big{]}^{2}}{%
\Gamma(d+2)}.$$
As a result, we obtain
$$\displaystyle I_{3c}(p)$$
$$\displaystyle=$$
$$\displaystyle\frac{\Gamma\big{(}\frac{2-d}{2}\big{)}\Gamma(2-d)}{(4\pi)^{d}%
\Gamma\big{(}\frac{d}{2}\big{)}}\frac{\big{[}\Gamma\big{(}\frac{d+2}{2}\big{)}%
\big{]}^{2}}{\Gamma(d+2)}\Bigg{[}(-1)^{\frac{1}{4}\textrm{tr}[M_{2}]}M_{2}%
\bigg{(}\not{p}-\not{c}\gamma^{5}+\frac{2}{d}\big{(}p_{k}\gamma^{k}+c_{k}%
\gamma^{k}\gamma^{5}\big{)}\bigg{)}M_{2}$$
$$\displaystyle\times\Bigg{(}\frac{(d+4)(d+2)d}{8}\frac{\Gamma\big{(}\frac{d+2}{%
2}\big{)}}{\Gamma\big{(}\frac{6-d}{2}\big{)}}+\frac{(d+2)d(2-d)}{8}\frac{%
\Gamma\big{(}\frac{d+2}{2}+1\big{)}}{\Gamma\big{(}\frac{6-d}{2}+1\big{)}}\frac%
{d^{2}+8}{d^{2}}$$
$$\displaystyle-\frac{d(4-d)(2-d)}{8}\frac{\Gamma\big{(}\frac{d+2}{2}+2\big{)}}{%
\Gamma\big{(}\frac{6-d}{2}+2\big{)}}\frac{d^{2}+8}{d^{2}}-\frac{(6-d)(4-d)(2-d%
)}{8}\frac{\Gamma\big{(}\frac{d+2}{2}+3\big{)}}{\Gamma\big{(}\frac{6-d}{2}+3%
\big{)}}\Bigg{)}$$
$$\displaystyle-\frac{2}{d}(-1)^{\frac{1}{4}\textrm{tr}[M_{2}]}M_{2}(p_{k}\gamma%
^{k})M_{2}\Bigg{(}\frac{(d+4)(d+2)d}{8}\frac{2+d}{\frac{6-d}{2}}\frac{\Gamma%
\big{(}\frac{d+2}{2}\big{)}}{\Gamma\big{(}\frac{6-d}{2}\big{)}}+\frac{(d+2)d(2%
-d)}{8}\frac{4+d}{\frac{6-d}{2}+1}\frac{\Gamma\big{(}\frac{d+2}{2}+1\big{)}}{%
\Gamma\big{(}\frac{6-d}{2}+1\big{)}}\frac{d^{2}+8}{d^{2}}$$
$$\displaystyle-\frac{d(4-d)(2-d)}{8}\frac{6+d}{\frac{6-d}{2}+2}\frac{\Gamma\big%
{(}\frac{d+2}{2}+2\big{)}}{\Gamma\big{(}\frac{6-d}{2}+2\big{)}}\frac{d^{2}+8}{%
d^{2}}-\frac{(6-d)(4-d)(2-d)}{8}\frac{8+d}{\frac{6-d}{2}+3}\frac{\Gamma\big{(}%
\frac{d+2}{2}+3\big{)}}{\Gamma\big{(}\frac{6-d}{2}+3\big{)}}\Bigg{)}\Bigg{]}$$
$$\displaystyle+\frac{\Gamma\big{(}\frac{2-d}{2}\big{)}\Gamma(3-d)}{(4\pi)^{d}%
\Gamma\big{(}\frac{6-d}{2}\big{)}}\frac{\big{[}\Gamma\big{(}\frac{d+2}{2}\big{%
)}\big{]}^{2}}{\Gamma(d+2)}\frac{(d+4)(d+2)d}{8}(-1)^{\frac{1}{4}\textrm{tr}[M%
_{2}]}M_{2}\bigg{[}\int^{1}_{0}dv\frac{C_{0}(v)}{\Delta_{0}(v)}\bigg{]}M_{2}$$
$$\displaystyle=$$
$$\displaystyle\bigg{(}\frac{1}{8\pi^{2}\varepsilon^{2}}+\frac{5+6\gamma-6\ln{4%
\pi}}{8\pi^{2}\varepsilon}\bigg{)}(-1)^{\frac{1}{4}\textrm{tr}[M_{2}]}(p_{0}%
\gamma^{0})+\frac{1}{16\pi^{2}\varepsilon}(p_{k}\gamma^{k})+\bigg{(}\frac{1}{8%
\pi^{2}\varepsilon^{2}}+\frac{5+6\gamma-6\ln{4\pi}}{8\pi^{2}\varepsilon}\bigg{%
)}(c_{0}\gamma^{0}\gamma^{5})$$
$$\displaystyle-\frac{1}{16\pi^{2}\varepsilon}(-1)^{\frac{1}{4}\textrm{tr}[M_{2}%
]}(c_{k}\gamma^{k}\gamma^{5})-\frac{1}{8\pi^{2}\varepsilon}(-1)^{\frac{1}{4}%
\textrm{tr}[M_{2}]}M_{2}\bigg{[}\int^{1}_{0}dv\frac{C_{0}(v)}{\Delta_{0}(v)}%
\bigg{]}M_{2}+\mathcal{O}(1),$$
where we have used the matrix identities:
$$\displaystyle(-1)^{\frac{1}{4}\textrm{tr}[M_{2}]}M_{2}\gamma^{0}M_{2}=(-1)^{%
\frac{1}{4}\textrm{tr}[M_{2}]}\gamma^{0},~{}~{}(-1)^{\frac{1}{4}\textrm{tr}[M_%
{2}]}M_{2}\gamma^{0}\gamma^{5}M_{2}=-\gamma^{0}\gamma^{5},$$
$$\displaystyle(-1)^{\frac{1}{4}\textrm{tr}[M_{2}]}M_{2}\gamma^{k}M_{2}=-\gamma^%
{k},~{}~{}(-1)^{\frac{1}{4}\textrm{tr}[M_{2}]}M_{2}\gamma^{k}\gamma^{5}M_{2}=(%
-1)^{\frac{1}{4}\textrm{tr}[M_{2}]}\gamma^{k}\gamma^{5}.$$
Finally, the self-energy correction from the crossed diagrams is
$$\displaystyle\Sigma^{(2),c}(p)$$
$$\displaystyle=$$
$$\displaystyle\Gamma_{V}^{2}\bigg{[}\bigg{(}\frac{1}{8\pi^{2}\varepsilon^{2}}+%
\frac{5+6\gamma-6\ln{4\pi}}{48\pi^{2}\varepsilon}\bigg{)}(p_{0}\gamma^{0}+c_{0%
}\gamma^{0}\gamma^{5})+\frac{1}{16\pi^{2}\varepsilon}p_{k}\gamma^{k}-\frac{1}{%
16\pi^{2}\varepsilon}c_{k}\gamma^{k}\gamma^{5}-\frac{1}{8\pi^{2}\varepsilon}%
\int^{1}_{0}dv\frac{\bar{C}_{0}(v)}{\Delta_{0}(v)}\bigg{]}$$
$$\displaystyle+\Gamma_{V}\Gamma_{U}\bigg{[}\bigg{(}\frac{1}{8\pi^{2}\varepsilon%
^{2}}+\frac{5+6\gamma-6\ln{4\pi}}{48\pi^{2}\varepsilon}\bigg{)}(-p_{0}\gamma^{%
0}+c_{0}\gamma^{0}\gamma^{5})+\frac{1}{16\pi^{2}\varepsilon}p_{k}\gamma^{k}+%
\frac{1}{16\pi^{2}\varepsilon}c_{k}\gamma^{k}\gamma^{5}+\frac{1}{8\pi^{2}%
\varepsilon}\int^{1}_{0}dv\frac{C_{0}(v)}{\Delta_{0}(v)}\bigg{]}$$
$$\displaystyle+\Gamma_{U}\Gamma_{V}\bigg{[}\bigg{(}\frac{1}{8\pi^{2}\varepsilon%
^{2}}+\frac{5+6\gamma-6\ln{4\pi}}{48\pi^{2}\varepsilon}\bigg{)}(p_{0}\gamma^{0%
}+c_{0}\gamma^{0}\gamma^{5})+\frac{1}{16\pi^{2}\varepsilon}p_{k}\gamma^{k}-%
\frac{1}{16\pi^{2}\varepsilon}c_{k}\gamma^{k}\gamma^{5}-\frac{1}{8\pi^{2}%
\varepsilon}\int^{1}_{0}dv\frac{\bar{C}_{0}(v)}{\Delta_{0}(v)}\bigg{]}$$
$$\displaystyle+\Gamma_{U}^{2}\bigg{[}\bigg{(}\frac{1}{8\pi^{2}\varepsilon^{2}}+%
\frac{5+6\gamma-6\ln{4\pi}}{48\pi^{2}\varepsilon}\bigg{)}(-p_{0}\gamma^{0}+c_{%
0}\gamma^{0}\gamma^{5})+\frac{1}{16\pi^{2}\varepsilon}p_{k}\gamma^{k}+\frac{1}%
{16\pi^{2}\varepsilon}c_{k}\gamma^{k}\gamma^{5}+\frac{1}{8\pi^{2}\varepsilon}%
\int^{1}_{0}dv\frac{C_{0}(v)}{\Delta_{0}(v)}\bigg{]}+\mathcal{O}(1).$$
When we take into account the vertex renormalization, we should introduce consistently self-energy corrections made of a vertex counter term, given by (Fig. 11)
$$\displaystyle\Sigma^{(1),\delta_{\Gamma}}(p)=\delta_{\Gamma V}\Gamma_{V}\gamma%
^{0}I_{1}(p)\gamma^{0}+\delta_{\Gamma U}\Gamma_{U}I_{1}(p)\equiv\Sigma^{(1),%
\delta_{\Gamma}}_{VV}(p)+\Sigma^{(1),\delta_{\Gamma}}_{VU}(p)+\Sigma^{(1),%
\delta_{\Gamma}}_{UV}(p)+\Sigma^{(1),\delta_{\Gamma}}_{UU}(p).$$
Recall $I_{1}=\int^{1}_{0}dx\Big{(}\frac{d}{2}(p_{0}\gamma^{0}-\not{c}\gamma^{5})+%
\frac{1}{2}(2\tilde{c}_{xk}\gamma^{k}+2c_{k}\gamma^{k}\gamma^{5})\Big{)}\frac{%
\Gamma(\frac{2-d}{2})}{(4\pi)^{d/2}\Delta_{0}^{(2-d)/2}}+\int^{1}_{0}dx\frac{C%
_{0}\Gamma(\frac{4-d}{2})}{(4\pi)^{d/2}\Delta_{0}^{(4-d)/2}}$ in Eq. (36).
Expanding $I_{1}$ about $d=2+\varepsilon$ and inserting $\delta_{\Gamma V}=\frac{\Gamma_{V}}{2\pi\varepsilon}+\frac{\Gamma_{U}}{2\pi\varepsilon}$ and $\delta_{\Gamma U}=-\frac{\Gamma_{U}}{2\pi\varepsilon}-\frac{\Gamma_{V}}{2\pi\varepsilon}$ into the above expression, which will be computed in the next section, we obtain
$$\displaystyle\Sigma^{(1),\delta_{\Gamma}}(p)$$
$$\displaystyle=$$
$$\displaystyle\Gamma_{V}^{2}\bigg{[}-\bigg{(}\frac{1}{4\pi^{2}\varepsilon^{2}}+%
\frac{1+\gamma-\ln{4\pi}}{8\pi^{2}\varepsilon}\bigg{)}\big{(}p_{0}\gamma^{0}+c%
_{0}\gamma^{0}\gamma^{5}\big{)}+\frac{1}{8\pi^{2}\varepsilon}\int^{1}_{0}dx%
\frac{\bar{C}_{0}(x)}{\Delta_{0}(x)}\bigg{]}$$
$$\displaystyle+\Gamma_{V}\Gamma_{U}\bigg{[}-\bigg{(}\frac{1}{4\pi^{2}%
\varepsilon^{2}}+\frac{1+\gamma-\ln{4\pi}}{8\pi^{2}\varepsilon}\bigg{)}\big{(}%
p_{0}\gamma^{0}+c_{0}\gamma^{0}\gamma^{5}\big{)}+\frac{1}{8\pi^{2}\varepsilon}%
\int^{1}_{0}dx\frac{\bar{C}_{0}(x)}{\Delta_{0}(x)}\bigg{]}$$
$$\displaystyle+\Gamma_{U}\Gamma_{V}\bigg{[}-\bigg{(}\frac{1}{4\pi^{2}%
\varepsilon^{2}}+\frac{1+\gamma-\ln{4\pi}}{8\pi^{2}\varepsilon}\bigg{)}\big{(}%
-p_{0}\gamma^{0}+c_{0}\gamma^{0}\gamma^{5}\big{)}-\frac{1}{8\pi^{2}\varepsilon%
}\int^{1}_{0}dx\frac{C_{0}(x)}{\Delta_{0}(x)}\bigg{]}$$
$$\displaystyle+\Gamma_{U}^{2}\bigg{[}-\bigg{(}\frac{1}{4\pi^{2}\varepsilon^{2}}%
+\frac{1+\gamma-\ln{4\pi}}{8\pi^{2}\varepsilon}\bigg{)}\big{(}-p_{0}\gamma^{0}%
+c_{0}\gamma^{0}\gamma^{5}\big{)}-\frac{1}{8\pi^{2}\varepsilon}\int^{1}_{0}dx%
\frac{C_{0}(x)}{\Delta_{0}(x)}\bigg{]}+\mathcal{O}(1).$$
Adding these contributions to the crossed diagrams, we finally obtain
$$\displaystyle\Sigma^{(2),c}(p)+\Sigma^{(1),\delta_{\Gamma}}(p)$$
$$\displaystyle=$$
$$\displaystyle\Gamma_{V}^{2}\bigg{[}\bigg{(}-\frac{1}{8\pi^{2}\varepsilon^{2}}-%
\frac{1}{48\pi^{2}\varepsilon}\bigg{)}(p_{0}\gamma^{0}+c_{0}\gamma^{0}\gamma^{%
5})+\frac{1}{16\pi^{2}\varepsilon}p_{k}\gamma^{k}-\frac{1}{16\pi^{2}%
\varepsilon}c_{k}\gamma^{k}\gamma^{5}\bigg{]}$$
(41)
$$\displaystyle+\Gamma_{U}^{2}\bigg{[}\bigg{(}-\frac{1}{8\pi^{2}\varepsilon^{2}}%
-\frac{1}{48\pi^{2}\varepsilon}\bigg{)}(-p_{0}\gamma^{0}+c_{0}\gamma^{0}\gamma%
^{5})+\frac{1}{16\pi^{2}\varepsilon}p_{k}\gamma^{k}+\frac{1}{16\pi^{2}%
\varepsilon}c_{k}\gamma^{k}\gamma^{5}\bigg{]}$$
$$\displaystyle+2\Gamma_{V}\Gamma_{U}\bigg{[}\bigg{(}-\frac{1}{8\pi^{2}%
\varepsilon^{2}}-\frac{1}{48\pi^{2}\varepsilon}\bigg{)}c_{0}\gamma^{0}\gamma^{%
5}+\frac{1}{16\pi^{2}\varepsilon}p_{k}\gamma^{k}\bigg{]}+\mathcal{O}(1).$$
This result is depicted pictorially in Fig. 12.
Appendix D vertex correction
D.1 Relevant Feynman diagrams
The vertex renormalization can be found from a four-point function of $\mathbf{D}(p,p^{\prime},q,q^{\prime})=\left\langle\psi_{p}\bar{\psi}_{p^{%
\prime}}\psi_{q}\bar{\psi}_{q^{\prime}}\right\rangle$. Performing the perturbative analysis up to the $\Gamma^{2}$ order, we obtain
$$\displaystyle\mathbf{D}(p,p^{\prime},q,q^{\prime})$$
$$\displaystyle=$$
$$\displaystyle\lim_{R\to 0}\frac{1}{R}\int\mathcal{D}\bar{\psi}\mathcal{D}\psi%
\big{(}\psi^{a}_{p}\bar{\psi}^{a}_{p^{\prime}}\psi^{a}_{q}\bar{\psi}^{a}_{q^{%
\prime}}\big{)}e^{-S_{0}[\bar{\psi}^{\alpha},\psi^{\alpha}]}e^{\frac{1}{L^{3}}%
\sum_{p_{j}}\big{[}\frac{\Gamma_{V}}{2}(\bar{\psi}^{b}_{p_{1}}\gamma^{0}\psi^{%
b}_{p_{2}})(\bar{\psi}^{c}_{p_{3}}\gamma^{0}\psi^{c}_{p_{4}})+\frac{\Gamma_{U}%
}{2}(\bar{\psi}^{b}_{p_{1}}\psi^{b}_{p_{2}})(\bar{\psi}^{c}_{p_{3}}\psi^{c}_{p%
_{4}})\big{]}\delta^{(3)}_{\bm{p}_{1}-\bm{p}_{2},\bm{p}_{3}-\bm{p}_{4}}\delta_%
{p_{1}^{0}p_{2}^{0}}\delta_{p_{3}^{0}p_{4}^{0}}}$$
$$\displaystyle\simeq$$
$$\displaystyle\lim_{R\to 0}\frac{1}{R}\int\mathcal{D}\bar{\psi}\mathcal{D}\psi e%
^{-S_{0}[\bar{\psi}^{\alpha},\psi^{\alpha}]}\biggr{[}\psi^{a}_{p}\bar{\psi}^{a%
}_{p^{\prime}}\psi^{a}_{q}\bar{\psi}^{a}_{q^{\prime}}+\frac{\Gamma_{V}}{2L^{3}%
}\sum_{p_{j}}\big{(}\psi^{a}_{p}\bar{\psi}^{a}_{p^{\prime}}\psi^{a}_{q}\bar{%
\psi}^{a}_{q^{\prime}}\big{)}\big{(}\bar{\psi}^{b}_{p_{1}}\gamma^{0}\psi^{b}_{%
p_{2}}\bar{\psi}^{c}_{p_{3}}\gamma^{0}\psi^{c}_{p_{4}}\big{)}\delta^{(3)}_{\bm%
{p}_{1}-\bm{p}_{2},\bm{p}_{3}-\bm{p}_{4}}\delta_{p^{0}_{1}p^{0}_{2}}\delta_{p^%
{0}_{3}p^{0}_{4}}$$
$$\displaystyle+\frac{\Gamma_{U}}{2L^{3}}\sum_{p_{j}}\big{(}\psi^{a}_{p}\bar{%
\psi}^{a}_{p^{\prime}}\psi^{a}_{q}\bar{\psi}^{a}_{q^{\prime}}\big{)}\big{(}%
\bar{\psi}^{b}_{p_{1}}\psi^{b}_{p_{2}}\bar{\psi}^{c}_{p_{3}}\psi^{c}_{p_{4}}%
\big{)}\delta^{(3)}_{\bm{p}_{1}-\bm{p}_{2},\bm{p}_{3}-\bm{p}_{4}}\delta_{p^{0}%
_{1}p^{0}_{2}}\delta_{p^{0}_{3}p^{0}_{4}}$$
$$\displaystyle+\frac{\Gamma_{V}^{2}}{8(L^{3})^{2}}\sum_{p_{j}p_{j}^{\prime}}%
\big{(}\psi^{a}_{p}\bar{\psi}^{a}_{p^{\prime}}\psi^{a}_{q}\bar{\psi}^{a}_{q^{%
\prime}}\big{)}\big{(}\bar{\psi}^{b}_{p_{1}}\gamma^{0}\psi^{b}_{p_{2}}\bar{%
\psi}^{c}_{p_{3}}\gamma^{0}\psi^{c}_{p_{4}}\big{)}\big{(}\bar{\psi}^{b^{\prime%
}}_{p^{\prime}_{1}}\gamma^{0}\psi^{b^{\prime}}_{p^{\prime}_{2}}\bar{\psi}^{c^{%
\prime}}_{p^{\prime}_{3}}\gamma^{0}\psi^{c^{\prime}}_{p^{\prime}_{4}}\big{)}%
\delta^{(3)}_{\bm{p}_{1}-\bm{p}_{2},\bm{p}_{3}-\bm{p}_{4}}\delta_{p^{0}_{1}p^{%
0}_{2}}\delta_{p^{0}_{3}p^{0}_{4}}\delta^{(3)}_{\bm{p}^{\prime}_{1}-\bm{p}^{%
\prime}_{2},\bm{p}^{\prime}_{3}-\bm{p}^{\prime}_{4}}\delta_{p^{{}^{\prime}0}_{%
1}p^{{}^{\prime}0}_{2}}\delta_{p^{{}^{\prime}0}_{3}p^{{}^{\prime}0}_{4}}$$
$$\displaystyle+\frac{\Gamma_{U}^{2}}{8(L^{3})^{2}}\sum_{p_{j}p_{j}^{\prime}}%
\big{(}\psi^{a}_{p}\bar{\psi}^{a}_{p^{\prime}}\psi^{a}_{q}\bar{\psi}^{a}_{q^{%
\prime}}\big{)}\big{(}\bar{\psi}^{b}_{p_{1}}\psi^{b}_{p_{2}}\bar{\psi}^{c}_{p_%
{3}}\psi^{c}_{p_{4}}\big{)}\big{(}\bar{\psi}^{b^{\prime}}_{p^{\prime}_{1}}\psi%
^{b^{\prime}}_{p^{\prime}_{2}}\bar{\psi}^{c^{\prime}}_{p^{\prime}_{3}}\psi^{c^%
{\prime}}_{p^{\prime}_{4}}\big{)}\delta^{(3)}_{\bm{p}_{1}-\bm{p}_{2},\bm{p}_{3%
}-\bm{p}_{4}}\delta_{p^{0}_{1}p^{0}_{2}}\delta_{p^{0}_{3}p^{0}_{4}}\delta^{(3)%
}_{\bm{p}^{\prime}_{1}-\bm{p}^{\prime}_{2},\bm{p}^{\prime}_{3}-\bm{p}^{\prime}%
_{4}}\delta_{p^{{}^{\prime}0}_{1}p^{{}^{\prime}0}_{2}}\delta_{p^{{}^{\prime}0}%
_{3}p^{{}^{\prime}0}_{4}}$$
$$\displaystyle+\frac{\Gamma_{V}\Gamma_{U}}{4(L^{3})^{2}}\sum_{p_{j}p_{j}^{%
\prime}}\big{(}\psi^{a}_{p}\bar{\psi}^{a}_{p^{\prime}}\psi^{a}_{q}\bar{\psi}^{%
a}_{q^{\prime}}\big{)}\big{(}\bar{\psi}^{b}_{p_{1}}\gamma^{0}\psi^{b}_{p_{2}}%
\bar{\psi}^{c}_{p_{3}}\gamma^{0}\psi^{c}_{p_{4}}\big{)}\big{(}\bar{\psi}^{b^{%
\prime}}_{p^{\prime}_{1}}\psi^{b^{\prime}}_{p^{\prime}_{2}}\bar{\psi}^{c^{%
\prime}}_{p^{\prime}_{3}}\psi^{c^{\prime}}_{p^{\prime}_{4}}\big{)}\delta^{(3)}%
_{\bm{p}_{1}-\bm{p}_{2},\bm{p}_{3}-\bm{p}_{4}}\delta_{p^{0}_{1}p^{0}_{2}}%
\delta_{p^{0}_{3}p^{0}_{4}}\delta^{(3)}_{\bm{p}^{\prime}_{1}-\bm{p}^{\prime}_{%
2},\bm{p}^{\prime}_{3}-\bm{p}^{\prime}_{4}}\delta_{p^{{}^{\prime}0}_{1}p^{{}^{%
\prime}0}_{2}}\delta_{p^{{}^{\prime}0}_{3}p^{{}^{\prime}0}_{4}}\biggl{]}.$$
Among the first-order contributions, fully-connected diagrams give scattering elements (Fig. 13). The four-point function and the scattering matrix element at the tree level are
$$\displaystyle M^{(0)}(p,p;q)$$
$$\displaystyle\equiv$$
$$\displaystyle M^{(0)}_{V}(p,p;q)+M^{(0)}_{U}(p,p;q)=2\Gamma_{V}(\gamma^{0}%
\otimes\gamma^{0})+2\Gamma_{V}(I_{4\times 4}\otimes I_{4\times 4}).$$
(42)
Among the second order contributions, only diagrams fully connected with the external lines survive in the replica limit of $R\rightarrow 0$ and give scattering matrix elements. Thus, the scattering matrix elements in the second order are given by (Fig. 14)
$$\displaystyle\mathcal{M}^{(1)}_{ph}$$
$$\displaystyle=$$
$$\displaystyle\frac{2\Gamma_{V}^{2}}{L^{3}}\sum_{l}\gamma^{0}G(p-l)\gamma^{0}%
\otimes\gamma^{0}G(p^{\prime}-l-q)\gamma^{0}\delta_{l^{0}0}+\frac{2\Gamma_{V}%
\Gamma_{U}}{L^{3}}\sum_{l}\gamma^{0}G(p-l)\otimes G(p^{\prime}-l-q)\gamma^{0}%
\delta_{l^{0}0}$$
$$\displaystyle+\frac{2\Gamma_{U}\Gamma_{V}}{L^{3}}\sum_{l}G(p-l)\gamma^{0}%
\otimes\gamma^{0}G(p^{\prime}-l-q)\delta_{l^{0}0}+\frac{2\Gamma_{U}^{2}}{L^{3}%
}\sum_{l}G(p-l)\otimes G(p^{\prime}-l-q)\delta_{l^{0}0},$$
$$\displaystyle\equiv$$
$$\displaystyle\mathcal{M}^{ph}_{VV}+\mathcal{M}^{ph}_{VU}+\mathcal{M}^{ph}_{UV}%
+\mathcal{M}^{ph}_{UU}$$
$$\displaystyle\mathcal{M}^{(1)}_{pp}$$
$$\displaystyle=$$
$$\displaystyle\frac{2\Gamma_{V}^{2}}{L^{3}}\sum_{l}\gamma^{0}G(p-l)\gamma^{0}%
\otimes\gamma^{0}G(p^{\prime}+l)\gamma^{0}\delta_{l^{0}0}+\frac{2\Gamma_{V}%
\Gamma_{U}}{L^{3}}\sum_{l}\gamma^{0}G(p-l)\otimes\gamma^{0}G(p^{\prime}+l)%
\delta_{l^{0}0}$$
$$\displaystyle+\frac{2\Gamma_{U}\Gamma_{V}}{L^{3}}\sum_{l}G(p-l)\gamma^{0}%
\otimes G(p^{\prime}+l)\gamma^{0}\delta_{l^{0}0}+\frac{2\Gamma_{U}^{2}}{L^{3}}%
\sum_{l}G(p-l)\otimes G(p^{\prime}+l)\delta_{l^{0}0},$$
$$\displaystyle\equiv$$
$$\displaystyle\mathcal{M}^{pp}_{VV}+\mathcal{M}^{pp}_{VU}+\mathcal{M}^{pp}_{UV}%
+\mathcal{M}^{pp}_{UU}$$
$$\displaystyle\mathcal{M}^{(1)}_{ver}$$
$$\displaystyle=$$
$$\displaystyle\frac{2\Gamma_{V}^{2}}{L^{3}}\sum_{l}\gamma^{0}G(p-l)\gamma^{0}G(%
p+q-l)\gamma^{0}\otimes\gamma^{0}\delta_{l^{0}0}+\frac{2\Gamma_{V}\Gamma_{U}}{%
L^{3}}\sum_{l}\gamma^{0}G(p-l)G(p+q-l)\gamma^{0}\otimes I_{4\times 4}\delta_{l%
^{0}0}$$
$$\displaystyle+\frac{2\Gamma_{U}\Gamma_{V}}{L^{3}}\sum_{l}G(p-l)\gamma^{0}G(p+q%
-l)\otimes\gamma^{0}\delta_{l^{0}0}+\frac{2\Gamma_{U}^{2}}{L^{3}}\sum_{l}G(p-l%
)G(p+q-l)\otimes I_{4\times 4}\delta_{l^{0}0},$$
$$\displaystyle\equiv$$
$$\displaystyle\mathcal{M}^{ver}_{VV}+\mathcal{M}^{ver}_{VU}+\mathcal{M}^{ver}_{%
UV}+\mathcal{M}^{ver}_{UU},$$
where $``ph"$, $``pp"$ and $``ver"$ represent “particle-hole”, “particle-particle” and “vertex”, respectively.
D.2 Evaluation of relevant diagrams
D.2.1 Particle-hole channel
First, we evaluate the particle-hole diagrams (the first line in Fig. 14)
$$\displaystyle\mathcal{M}^{(1)}_{ph}$$
$$\displaystyle=$$
$$\displaystyle 2\Gamma_{V}^{2}I_{2ph}[M_{1}=M_{2}=\gamma^{0}]+2\Gamma_{V}\Gamma%
_{U}I_{2ph}[M_{1}=\gamma^{0},M_{2}=I_{4\times 4}]$$
$$\displaystyle+2\Gamma_{U}\Gamma_{V}I_{2ph}[M_{1}=I_{4\times 4},M_{2}=\gamma^{0%
}]+2\Gamma_{U}^{2}I_{2ph}[M_{1}=M_{2}=I_{4\times 4}],$$
where $I_{2ph}$ is given by ($\bm{k}\equiv\bm{p}-\bm{p}^{\prime}+\bm{q}$)
$$\displaystyle I_{2ph}$$
$$\displaystyle=$$
$$\displaystyle\int\frac{d^{d+1}l}{(2\pi)^{d+1}}2\pi\delta(l_{0})M_{1}G(p-l)M_{2%
}\otimes M_{2}G(p^{\prime}-l-q)M_{1}$$
$$\displaystyle=$$
$$\displaystyle\int\frac{d^{d}\bm{l}}{(2\pi)^{d}}M_{1}G(p_{0},\bm{p}-\bm{l})M_{2%
}\otimes M_{2}G(p_{0}^{\prime}-q_{0},\bm{p}^{\prime}-\bm{l}-\bm{q})M_{1}$$
$$\displaystyle=$$
$$\displaystyle\int\frac{d^{d}\bm{l}}{(2\pi)^{d}}M_{1}G(p_{0},-\bm{l})M_{2}%
\otimes M_{2}G(p_{0}^{\prime}-q_{0},-\bm{l}-\bm{k})M_{1}.$$
Using Eq. (31), we have
$$\displaystyle I_{2ph}$$
$$\displaystyle=$$
$$\displaystyle\int^{1}_{0}dx\int^{1}_{0}dy\int\frac{d^{d}\bm{l}}{(2\pi)^{d}}M_{%
1}\frac{-\bm{l}^{2}l_{i}\gamma^{i}+\bm{l}^{2}(p_{0}\gamma^{0}-\not{c}\gamma^{5%
})+l_{i}l_{j}(-2c^{i}\gamma^{j}\gamma^{5})-l_{i}f_{1}^{i}(p_{0})+f_{0}(p_{0})}%
{\big{[}\big{(}\bm{l}-(1-2x)\bm{c}\big{)}^{2}+\Delta_{0}(p_{0};x)\big{]}^{2}}M%
_{2}\otimes M_{2}$$
$$\displaystyle\frac{-(\bm{l}+\bm{k})^{2}(l_{j}+k_{j})\gamma^{j}+(\bm{l}+\bm{k})%
^{2}((p^{\prime}_{0}-q_{0})\gamma^{0}-\not{c}\gamma^{5})+(l_{i}+k_{i})(l_{j}+k%
_{j})(-2c^{i}\gamma^{j}\gamma^{5})-(l_{i}+k_{i})f_{1}^{i}(p^{\prime}_{0}-q_{0}%
)+f_{0}(p^{\prime}_{0}-q_{0})}{\big{[}\big{(}\bm{l}+\bm{k}-(1-2y)\bm{c}\big{)}%
^{2}+\Delta_{0}(p^{\prime}_{0}-q_{0};y)\big{]}^{2}}M_{1}.$$
Despite this complicated expression, only the product of the $l$-cubic terms contributes to renormalization by the same reason that we considered in Eq. (40). Keeping this term only, we obtain
$$\displaystyle I_{2ph}$$
$$\displaystyle\simeq$$
$$\displaystyle\int^{1}_{0}dx\int^{1}_{0}dy\int\frac{d^{d}\bm{l}}{(2\pi)^{d}}%
\frac{\bm{l}^{2}(\bm{l}+\bm{k})^{2}l_{i}(l_{j}+k_{j})(M_{1}\gamma^{i}M_{2}%
\otimes M_{2}\gamma^{j}M_{1})}{\big{[}\big{(}\bm{l}-(1-2x)\bm{c}\big{)}^{2}+%
\Delta_{0}(x)\big{]}^{2}\big{[}\big{(}\bm{l}+\bm{k}-(1-2y)\bm{c}\big{)}^{2}+%
\Delta_{0}(y)\big{]}^{2}}$$
$$\displaystyle=$$
$$\displaystyle\int^{1}_{0}dx\int^{1}_{0}dy\int^{1}_{0}dz6z(1-z)\int\frac{d^{d}%
\bm{l}}{(2\pi)^{d}}\frac{\bm{l}^{2}(\bm{l}+\bm{k})^{2}l_{i}(l_{j}+k_{j})(M_{1}%
\gamma^{i}M_{2}\otimes M_{2}\gamma^{j}M_{1})}{\big{[}{\bm{l}^{\prime}}^{2}+%
\Delta_{1}(\bm{k};x,y,z)\big{]}^{4}},$$
where $\Delta_{1}=z(1-z)\big{(}\bm{k}+2(y-x)\bm{c}\big{)}^{2}+(1-z)\Delta_{0}(x)+z%
\Delta_{0}(y)$ and $\bm{l}^{\prime}=\bm{l}+z\bm{k}-z(1-2y)\bm{c}-(1-z)(1-2x)\bm{c}$.
Renaming momentum as $\bm{l}^{\prime}\rightarrow\bm{l}$ and keeping only a relevant term again, we reach the following expression
$$\displaystyle I_{2ph}$$
$$\displaystyle=$$
$$\displaystyle(M_{1}\gamma^{i}M_{2}\otimes M_{2}\gamma^{j}M_{1})\int^{1}_{0}dx%
\int^{1}_{0}dy\int^{1}_{0}dz6z(1-z)\int\frac{d^{d}\bm{l}}{(2\pi)^{d}}\frac{(%
\bm{l}^{2})^{2}l_{i}l_{j}}{\big{[}\bm{l}^{2}+\Delta_{1}\big{]}^{4}}$$
$$\displaystyle=$$
$$\displaystyle(M_{1}\gamma^{i}M_{2}\otimes M_{2}\gamma^{i}M_{1})\int^{1}_{0}dx%
\int^{1}_{0}dy\int^{1}_{0}dzz(1-z)\frac{(d+4)(d+2)}{32\pi}\Gamma\bigg{(}\frac{%
2-d}{2}\bigg{)}\bigg{(}\frac{\Delta_{1}}{4\pi}\bigg{)}^{\frac{d-2}{2}}$$
$$\displaystyle=$$
$$\displaystyle-\frac{1}{4\pi\varepsilon}(M_{1}\gamma^{i}M_{2}\otimes M_{2}%
\gamma^{i}M_{1})+\mathcal{O}(1).$$
Thus, the scattering matrix element for the particle-hole diagrams is
$$\mathcal{M}^{(1)}_{ph}=-\frac{\Gamma_{V}^{2}}{2\pi\varepsilon}(\gamma^{i}%
\otimes\gamma^{i})+\frac{\Gamma_{V}\Gamma_{U}}{\pi\varepsilon}(\gamma^{0}%
\gamma^{i}\otimes\gamma^{0}\gamma^{i})-\frac{\Gamma_{U}^{2}}{2\pi\varepsilon}(%
\gamma^{i}\otimes\gamma^{i})+\mathcal{O}(1).$$
(43)
D.2.2 Particle-particle channel
Next, we evaluate the particle-particle diagrams (the second line in Fig. 14)
$$\displaystyle\mathcal{M}^{(1)}_{pp}$$
$$\displaystyle=$$
$$\displaystyle 2\Gamma_{V}^{2}I_{2pp}[M_{1}=M_{2}=\gamma^{0}]+2\Gamma_{V}\Gamma%
_{U}I_{2pp}[M_{1}=\gamma^{0},M_{2}=I_{4\times 4}]$$
$$\displaystyle+2\Gamma_{U}\Gamma_{V}I_{2pp}[M_{1}=I_{4\times 4},M_{2}=\gamma^{0%
}]+2\Gamma_{U}^{2}I_{2pp}[M_{1}=M_{2}=I_{4\times 4}],$$
where $I_{2pp}$ is given by ($\bm{k}\equiv\bm{p}+\bm{p}^{\prime}$)
$$\displaystyle I_{2pp}$$
$$\displaystyle=$$
$$\displaystyle\int\frac{d^{d+1}l}{(2\pi)^{d+1}}2\pi\delta(l_{0})M_{1}G(p-l)M_{2%
}\otimes M_{1}G(p^{\prime}+l)M_{2}$$
$$\displaystyle=$$
$$\displaystyle\int\frac{d^{d}\bm{l}}{(2\pi)^{d}}M_{1}G(p_{0},\bm{p}-\bm{l})M_{2%
}\otimes M_{1}G(p^{\prime}_{0},\bm{p}^{\prime}+\bm{l})M_{2}$$
$$\displaystyle=$$
$$\displaystyle\int\frac{d^{d}\bm{l}}{(2\pi)^{d}}M_{1}G(p_{0},-\bm{l})M_{2}%
\otimes M_{1}G(p^{\prime}_{0},\bm{l}+\bm{k})M_{2}.$$
The analysis is quite similar with that of the particle-hole channel. Keeping only a relevant term, we have
$$\displaystyle I_{2pp}$$
$$\displaystyle\simeq$$
$$\displaystyle\int^{1}_{0}dx\int^{1}_{0}dy\int\frac{d^{d}\bm{l}}{(2\pi)^{d}}%
\frac{-\bm{l}^{2}(\bm{l}+\bm{k})^{2}l_{i}(l_{j}+k_{j})(M_{1}\gamma^{i}M_{2}%
\otimes M_{1}\gamma^{j}M_{2})}{\big{[}\big{(}\bm{l}-(1-2x)\bm{c}\big{)}^{2}+%
\Delta_{0}(x)\big{]}^{2}\big{[}\big{(}\bm{l}+\bm{k}+(1-2y)\bm{c}\big{)}^{2}+%
\Delta_{0}(y)\big{]}^{2}}$$
$$\displaystyle=$$
$$\displaystyle-\int^{1}_{0}dx\int^{1}_{0}dy\int^{1}_{0}dz6z(1-z)\int\frac{d^{d}%
\bm{l}}{(2\pi)^{d}}\frac{\bm{l}^{2}(\bm{l}+\bm{k})^{2}l_{i}(l_{j}+k_{j})(M_{1}%
\gamma^{i}M_{2}\otimes M_{1}\gamma^{j}M_{2})}{\big{[}{\bm{l}^{\prime}}^{2}+%
\Delta_{1}(\bm{k};x,y,z)\big{]}^{4}},$$
where $\Delta_{1}=z(1-z)\big{(}\bm{k}+2(1-x-y)\bm{c}\big{)}^{2}+(1-z)\Delta_{0}(x)+z%
\Delta_{0}(y)$ and $\bm{l}^{\prime}=\bm{l}+z\bm{k}+z(1-2y)\bm{c}-(1-z)(1-2x)\bm{c}$. Note a minus sign in front of the integral that essentially originates from the opposite sign in the loop-momentum of the two propagators. Due to this sign difference, the contribution from the pp-diagram will cancel that of the ph-diagram.
The remaining calculation is the same as before. As a result, we reach the following expression
$$\displaystyle I_{2pp}=+\frac{1}{4\pi\varepsilon}(M_{1}\gamma^{i}M_{2}\otimes M%
_{1}\gamma^{i}M_{2})+\mathcal{O}(1).$$
Thus, the scattering matrix elements for the particle-particle diagrams is
$$\mathcal{M}^{(1)}_{pp}=\frac{\Gamma_{V}^{2}}{2\pi\varepsilon}(\gamma^{i}%
\otimes\gamma^{i})+\frac{\Gamma_{V}\Gamma_{U}}{\pi\varepsilon}(\gamma^{0}%
\gamma^{i}\otimes\gamma^{0}\gamma^{i})+\frac{\Gamma_{U}^{2}}{2\pi\varepsilon}(%
\gamma^{i}\otimes\gamma^{i})+\mathcal{O}(1).$$
(44)
D.2.3 Vertex channel
Lastly, we evaluate the vertex diagrams (the third line in Fig. 14)
$$\displaystyle\mathcal{M}^{(1)}_{ver}$$
$$\displaystyle=$$
$$\displaystyle 2\Gamma_{V}^{2}I_{2ver}[M_{1}=M_{2}=\gamma^{0}]+2\Gamma_{V}%
\Gamma_{U}I_{2ver}[M_{1}=\gamma^{0},M_{2}=I_{4\times 4}]$$
$$\displaystyle+2\Gamma_{U}\Gamma_{V}I_{2ver}[M_{1}=I_{4\times 4},M_{2}=\gamma^{%
0}]+2\Gamma_{U}^{2}I_{2ver}[M_{1}=M_{2}=I_{4\times 4}],$$
where $I_{2ver}$ is given by
$$\displaystyle I_{2ver}$$
$$\displaystyle=$$
$$\displaystyle\int\frac{d^{d+1}l}{(2\pi)^{d+1}}M_{1}G(p-l)M_{2}G(p+q-l)M_{1}%
\otimes M_{2}\delta_{l^{0}0}$$
$$\displaystyle=$$
$$\displaystyle\int\frac{d^{d}\bm{l}}{(2\pi)^{d}}M_{1}G(p_{0},\bm{p}-\bm{l})M_{2%
}G(p_{0}+q_{0},\bm{p}+\bm{q}-\bm{l})M_{1}\otimes M_{2}$$
$$\displaystyle=$$
$$\displaystyle\int\frac{d^{d}\bm{l}}{(2\pi)^{d}}M_{1}G(p_{0},-\bm{l})M_{2}G(p_{%
0}+q_{0},-\bm{l}+\bm{q})M_{1}\otimes M_{2}.$$
The analysis is also similar with the ph case except for the fact that “$\otimes$” are not located between propagators.
Keeping only a relevant term, we have
$$\displaystyle I_{2ver}$$
$$\displaystyle\simeq$$
$$\displaystyle\int^{1}_{0}dx\int^{1}_{0}dy\int\frac{d^{d}\bm{l}}{(2\pi)^{d}}%
\frac{\bm{l}^{2}(\bm{l}-\bm{q})^{2}l_{i}(l_{j}-q_{j})(M_{1}\gamma^{i}M_{2}%
\gamma^{j}M_{1}\otimes M_{2})}{\big{[}\big{(}\bm{l}-(1-2x)\bm{c}\big{)}^{2}+%
\Delta_{0}(x)\big{]}^{2}\big{[}\big{(}\bm{l}-\bm{q}-(1-2y)\bm{c}\big{)}^{2}+%
\Delta_{0}(y)\big{]}^{2}}$$
$$\displaystyle=$$
$$\displaystyle\int^{1}_{0}dx\int^{1}_{0}dy\int^{1}_{0}dz6z(1-z)\int\frac{d^{d}%
\bm{l}}{(2\pi)^{d}}\frac{\bm{l}^{2}(\bm{l}-\bm{q})^{2}l_{i}(l_{j}-q_{j})(M_{1}%
\gamma^{i}M_{2}\gamma^{j}M_{1}\otimes M_{2})}{\big{[}{\bm{l}^{\prime}}^{2}+%
\Delta_{1}(\bm{q};x,y,z)\big{]}^{4}},$$
where $\Delta_{1}=z(1-z)\big{(}\bm{q}+2(x-y)\bm{c}\big{)}^{2}+(1-z)\Delta_{0}(x)+z%
\Delta_{0}(y)$ and $\bm{l}^{\prime}=\bm{l}-z\bm{q}-z(1-2y)\bm{c}-(1-z)(1-2x)\bm{c}$.
Renaming momentum as $\bm{l}^{\prime}\rightarrow\bm{l}$ and keeping only a relevant term again, we reach the following expression
$$\displaystyle I_{2ver}$$
$$\displaystyle=$$
$$\displaystyle(M_{1}\gamma^{i}M_{2}\gamma^{j}M_{1}\otimes M_{2})\int^{1}_{0}dx%
\int^{1}_{0}dy\int^{1}_{0}dz6z(1-z)\int\frac{d^{d}\bm{l}}{(2\pi)^{d}}\frac{(%
\bm{l}^{2})^{2}l_{i}l_{j}}{\big{[}\bm{l}^{2}+\Delta_{1}\big{]}^{4}}$$
$$\displaystyle=$$
$$\displaystyle(M_{1}\gamma^{i}M_{2}\gamma^{i}M_{1}\otimes M_{2})\int^{1}_{0}dx%
\int^{1}_{0}dy\int^{1}_{0}dzz(1-z)\frac{(d+4)(d+2)}{32\pi}\Gamma\bigg{(}\frac{%
2-d}{2}\bigg{)}\bigg{(}\frac{\Delta_{1}}{4\pi}\bigg{)}^{\frac{d-2}{2}}$$
$$\displaystyle=$$
$$\displaystyle-\frac{M_{1}\gamma^{i}M_{2}\gamma^{i}M_{1}\otimes M_{2}}{4\pi%
\varepsilon}+\mathcal{O}(1).$$
Thus, the scattering matrix element for the vertex-diagrams is
$$\mathcal{M}^{(1)}_{ver}=-\frac{\Gamma_{V}^{2}}{\pi\varepsilon}(\gamma^{0}%
\otimes\gamma^{0})+\frac{\Gamma_{V}\Gamma_{U}}{\pi\varepsilon}(I_{4\times 4}%
\otimes I_{4\times 4})-\frac{\Gamma_{U}\Gamma_{V}}{\pi\varepsilon}(\gamma^{0}%
\otimes\gamma^{0})+\frac{\Gamma_{U}^{2}}{\pi\varepsilon}(I_{4\times 4}\otimes I%
_{4\times 4})+\mathcal{O}(1),$$
(45)
where the result is depicted pictorially in Fig. 15.
Appendix E renormalization group equations
Combining Eq. (37), Eq. (39) and Eq. (41) in the following way
$$\displaystyle\Sigma^{(1)}(p)+\big{(}\Sigma^{(2),r}(p)+\Sigma^{(1),\delta_{\psi%
}}(p)\big{)}+\big{(}\Sigma^{(2),c}(p)+\Sigma^{(1),\delta_{\Gamma}}(p)\big{)}+%
\big{(}\textrm{propagator counterterms}\big{)}$$
$$\displaystyle=$$
$$\displaystyle-\frac{\Gamma_{V}}{2\pi\varepsilon}\big{(}p_{0}\gamma^{0}+c_{0}%
\gamma^{0}\gamma^{5}\big{)}-\frac{\Gamma_{U}}{2\pi\varepsilon}\big{(}p_{0}%
\gamma^{0}-c_{0}\gamma^{0}\gamma^{5}\big{)}$$
$$\displaystyle+\frac{\Gamma_{V}^{2}}{8\pi^{2}\varepsilon}(p_{0}\gamma^{0}+c_{0}%
\gamma^{0}\gamma^{5})+\frac{\Gamma_{U}^{2}}{8\pi^{2}\varepsilon}(p_{0}\gamma^{%
0}+c_{0}\gamma^{0}\gamma^{5})+\frac{2\Gamma_{V}\Gamma_{U}}{8\pi^{2}\varepsilon%
}(p_{0}\gamma^{0}-c_{0}\gamma^{0}\gamma^{5})$$
$$\displaystyle+\Gamma_{V}^{2}\bigg{[}\bigg{(}-\frac{1}{8\pi^{2}\varepsilon^{2}}%
-\frac{1}{48\pi^{2}\varepsilon}\bigg{)}(p_{0}\gamma^{0}+c_{0}\gamma^{0}\gamma^%
{5})+\frac{1}{16\pi^{2}\varepsilon}p_{k}\gamma^{k}-\frac{1}{16\pi^{2}%
\varepsilon}c_{k}\gamma^{k}\gamma^{5}\bigg{]}$$
$$\displaystyle+\Gamma_{U}^{2}\bigg{[}\bigg{(}-\frac{1}{8\pi^{2}\varepsilon^{2}}%
-\frac{1}{48\pi^{2}\varepsilon}\bigg{)}(-p_{0}\gamma^{0}+c_{0}\gamma^{0}\gamma%
^{5})+\frac{1}{16\pi^{2}\varepsilon}p_{k}\gamma^{k}+\frac{1}{16\pi^{2}%
\varepsilon}c_{k}\gamma^{k}\gamma^{5}\bigg{]}$$
$$\displaystyle+2\Gamma_{V}\Gamma_{U}\bigg{[}\bigg{(}-\frac{1}{8\pi^{2}%
\varepsilon^{2}}-\frac{1}{48\pi^{2}\varepsilon}\bigg{)}c_{0}\gamma^{0}\gamma^{%
5}+\frac{1}{16\pi^{2}\varepsilon}p_{k}\gamma^{k}\bigg{]}+\mathcal{O}(1)+(%
\delta_{\psi}^{\omega}p_{0}\gamma^{0}+\delta_{\psi}^{\bm{k}}p_{k}\gamma^{k}+%
\delta_{c0}c_{0}\gamma^{0}\gamma^{5}+\delta_{\bm{c}}c_{k}\gamma^{k}\gamma^{5}),$$
we find propagator counter terms in Eq. (LABEL:r.counterterms)
$$\displaystyle\delta_{\psi}^{\omega}=\frac{\Gamma_{V}+\Gamma_{U}}{2\pi%
\varepsilon}-\frac{(\Gamma_{V}+\Gamma_{U})^{2}}{8\pi^{2}\varepsilon}+\frac{%
\Gamma_{V}^{2}-\Gamma_{U}^{2}}{48\pi^{2}\varepsilon},~{}~{}~{}~{}~{}~{}~{}~{}%
\delta_{\psi}^{\bm{k}}=-\frac{(\Gamma_{V}+\Gamma_{U})^{2}}{16\pi^{2}%
\varepsilon},$$
$$\displaystyle\delta_{c0}=\frac{\Gamma_{V}-\Gamma_{U}}{2\pi\varepsilon}-\frac{(%
\Gamma_{V}-\Gamma_{U})^{2}}{8\pi^{2}\varepsilon}+\frac{(\Gamma_{V}+\Gamma_{U})%
^{2}}{48\pi^{2}\varepsilon},~{}~{}~{}~{}\delta_{\bm{c}}=\frac{\Gamma_{V}^{2}-%
\Gamma_{U}^{2}}{16\pi^{2}\varepsilon}.$$
Similarly, combining Eq.(43), Eq.(44) and Eq.(45) as follows
$$\displaystyle\mathcal{M}^{(1)}_{ph}+\mathcal{M}^{(1)}_{pp}+\mathcal{M}^{(1)}_{%
ver}+4\times\delta_{\Gamma V}\frac{\Gamma_{V}}{2}(\gamma^{0}\otimes\gamma^{0})%
+4\times\delta_{\Gamma U}\frac{\Gamma_{U}}{2}(I_{4\times 4}\otimes I_{4\times 4%
})+4\times\frac{\delta_{\Gamma T}}{2}(\gamma^{0}\gamma^{i}\otimes\gamma^{0}%
\gamma^{i})$$
$$\displaystyle=$$
$$\displaystyle-\frac{\Gamma_{V}^{2}}{2\pi\varepsilon}(\gamma^{i}\otimes\gamma^{%
i})+\frac{\Gamma_{V}\Gamma_{U}}{\pi\varepsilon}(\gamma^{0}\gamma^{i}\otimes%
\gamma^{0}\gamma^{i})-\frac{\Gamma_{U}^{2}}{2\pi\varepsilon}(\gamma^{i}\otimes%
\gamma^{i})+\frac{\Gamma_{V}^{2}}{2\pi\varepsilon}(\gamma^{i}\otimes\gamma^{i}%
)+\frac{\Gamma_{V}\Gamma_{U}}{\pi\varepsilon}(\gamma^{0}\gamma^{i}\otimes%
\gamma^{0}\gamma^{i})+\frac{\Gamma_{U}^{2}}{2\pi\varepsilon}(\gamma^{i}\otimes%
\gamma^{i})$$
$$\displaystyle-\frac{\Gamma_{V}^{2}}{\pi\varepsilon}(\gamma^{0}\otimes\gamma^{0%
})+\frac{\Gamma_{V}\Gamma_{U}}{\pi\varepsilon}(I_{4\times 4}\otimes I_{4\times
4%
})-\frac{\Gamma_{U}\Gamma_{V}}{\pi\varepsilon}(\gamma^{0}\otimes\gamma^{0})+%
\frac{\Gamma_{U}^{2}}{\pi\varepsilon}(I_{4\times 4}\otimes I_{4\times 4})+%
\mathcal{O}(1)$$
$$\displaystyle+2\delta_{\Gamma V}\Gamma_{V}(\gamma^{0}\otimes\gamma^{0})+2%
\delta_{\Gamma U}\Gamma_{U}(I_{4\times 4}\otimes I_{4\times 4})+2\delta_{%
\Gamma T}(\gamma^{0}\gamma^{i}\otimes\gamma^{0}\gamma^{i}),$$
we find vertex counter terms in Eq. (LABEL:r.counterterms)
$$\displaystyle\delta_{\Gamma V}=\frac{\Gamma_{V}}{2\pi\varepsilon}+\frac{\Gamma%
_{U}}{2\pi\varepsilon},~{}~{}~{}~{}~{}\delta_{\Gamma U}=-\frac{\Gamma_{U}}{2%
\pi\varepsilon}-\frac{\Gamma_{V}}{2\pi\varepsilon},~{}~{}~{}~{}~{}\delta_{%
\Gamma T}=-\frac{\Gamma_{V}\Gamma_{U}}{\pi\varepsilon}.$$
As a result, we obtain the renormalization factors:
$$\displaystyle Z_{\psi}^{\omega}$$
$$\displaystyle\simeq$$
$$\displaystyle\exp{\Big{[}-\frac{\Gamma_{V}+\Gamma_{U}}{2\pi}\ln{M}+\frac{5%
\Gamma_{V}^{2}+12\Gamma_{V}\Gamma_{U}+7\Gamma_{U}^{2}}{48\pi^{2}}\ln{M}\Big{]}},$$
$$\displaystyle Z_{\psi}^{\bm{k}}$$
$$\displaystyle\simeq$$
$$\displaystyle\exp{\Big{[}\frac{(\Gamma_{V}+\Gamma_{U})^{2}}{16\pi^{2}}\ln{M}%
\Big{]}},$$
$$\displaystyle Z_{c0}$$
$$\displaystyle\simeq$$
$$\displaystyle\exp{\Big{[}-\frac{\Gamma_{V}-\Gamma_{U}}{2\pi}\ln{M}+\frac{5%
\Gamma_{V}^{2}-14\Gamma_{V}\Gamma_{U}+5\Gamma_{U}^{2}}{48\pi^{2}}\ln{M}\Big{]}},$$
$$\displaystyle Z_{\bm{c}}$$
$$\displaystyle\simeq$$
$$\displaystyle\exp{\Big{[}-\frac{\Gamma_{V}^{2}-\Gamma_{U}^{2}}{16\pi^{2}}\ln{M%
}\Big{]}},$$
$$\displaystyle Z_{\Gamma V}$$
$$\displaystyle\simeq$$
$$\displaystyle\exp{\Big{[}-\frac{\Gamma_{V}+\Gamma_{U}}{2\pi}\ln{M}\Big{]}},$$
$$\displaystyle Z_{\Gamma U}$$
$$\displaystyle\simeq$$
$$\displaystyle\exp{\Big{[}\frac{\Gamma_{V}+\Gamma_{U}}{2\pi}\ln{M}\Big{]}},$$
(46)
where we have replaced $\frac{1}{\varepsilon}$ with a cut-off scale, $\ln{\frac{1}{M}}$, and approximated the renormalization factor as $Z=1+\delta\simeq\exp{(\delta)}$.
Recall the relations between the bare and renormalized quantities: $\Gamma_{V}=M^{d-2}(Z_{\psi}^{\omega})^{2}(Z_{\Gamma V})^{-1}\Gamma_{BV},~{}%
\Gamma_{U}=M^{d-2}(Z_{\psi}^{\omega})^{2}(Z_{\Gamma U})^{-1}\Gamma_{U},~{}v_{R%
}=Z^{\omega}_{\psi}(Z^{\bm{k}}_{\psi})^{-1}v_{B}~{}c_{R0}=M^{-1}Z_{\psi}^{%
\omega}(Z_{c0})^{-1}c_{B0}$, and $c_{Rk}=M^{-1}Z_{\psi}^{\omega}(Z_{\bm{c}})^{-1}c_{Bk}$. Based on these equations, it is straightforward to find the renormalization group equations
$$\displaystyle\frac{d\ln{\Gamma_{V}}}{d\ln{M}}$$
$$\displaystyle=$$
$$\displaystyle d-2+2\frac{d\ln{Z_{\psi}^{\omega}}}{d\ln{M}}-\frac{d\ln{Z_{%
\Gamma V}}}{d\ln{M}},$$
$$\displaystyle\frac{d\ln{\Gamma_{U}}}{d\ln{M}}$$
$$\displaystyle=$$
$$\displaystyle d-2+2\frac{d\ln{Z_{\psi}^{\omega}}}{d\ln{M}}-\frac{d\ln{Z_{%
\Gamma U}}}{d\ln{M}},$$
$$\displaystyle\frac{d\ln{v}}{d\ln{M}}$$
$$\displaystyle=$$
$$\displaystyle\frac{d\ln{Z_{\psi}^{\omega}}}{d\ln{M}}-\frac{d\ln{Z_{\psi}^{\bm{%
k}}}}{d\ln{M}},$$
$$\displaystyle\frac{d\ln{c_{0}}}{d\ln{M}}$$
$$\displaystyle=$$
$$\displaystyle-1+\frac{d\ln{Z_{\psi}^{\omega}}}{d\ln{M}}-\frac{d\ln{Z_{c0}}}{d%
\ln{M}},$$
$$\displaystyle\frac{d\ln{c_{k}}}{d\ln{M}}$$
$$\displaystyle=$$
$$\displaystyle-1+\frac{d\ln{Z_{\psi}^{\omega}}}{d\ln{M}}-\frac{d\ln{Z_{\bm{c}}}%
}{d\ln{M}}.$$
(47)
Substituting the results of (46) into Eq. (47), we obtain the renormalization group equations [Eq.(15)]
$$\displaystyle\frac{d\Gamma_{V}}{d\ln{M}}$$
$$\displaystyle=$$
$$\displaystyle\Gamma_{V}\bigg{[}1-\frac{\Gamma_{V}+\Gamma_{U}}{2\pi}+\frac{(%
\Gamma_{V}+\Gamma_{U})(5\Gamma_{V}+7\Gamma_{U})}{24\pi^{2}}\bigg{]},$$
$$\displaystyle\frac{d\Gamma_{U}}{d\ln{M}}$$
$$\displaystyle=$$
$$\displaystyle\Gamma_{U}\bigg{[}1-\frac{3(\Gamma_{V}+\Gamma_{U})}{2\pi}+\frac{(%
\Gamma_{V}+\Gamma_{U})(5\Gamma_{V}+7\Gamma_{U})}{24\pi^{2}}\bigg{]},$$
$$\displaystyle\frac{dv}{d\ln{M}}$$
$$\displaystyle=$$
$$\displaystyle v\bigg{[}-\frac{\Gamma_{V}+\Gamma_{U}}{2\pi}+\frac{(\Gamma_{V}+%
\Gamma_{U})(\Gamma_{V}+2\Gamma_{U})}{24\pi^{2}}\bigg{]},$$
$$\displaystyle\frac{dc_{0}}{d\ln{M}}$$
$$\displaystyle=$$
$$\displaystyle c_{0}\bigg{[}-1-\frac{\Gamma_{U}}{\pi}+\frac{\Gamma_{U}(\Gamma_{%
U}+13\Gamma_{V})}{24\pi^{2}}\bigg{]},$$
$$\displaystyle\frac{dc_{k}}{d\ln{M}}$$
$$\displaystyle=$$
$$\displaystyle c_{k}\bigg{[}-1-\frac{\Gamma_{V}+\Gamma_{U}}{2\pi}+\frac{(\Gamma%
_{V}+\Gamma_{U})(2\Gamma_{V}+\Gamma_{U})}{12\pi^{2}}\bigg{]}.$$
We notice that $\Gamma_{V}$ and $\Gamma_{U}$ affect renormalization of the other parameters, but the reverse way is not the case. In other words, $\Gamma_{V}$ and $\Gamma_{U}$ determine renormalization effects of all parameters, including themselves. In this respect we focus first on the equations for $\Gamma_{V}$ and $\Gamma_{U}$:
$$\displaystyle\frac{d\Gamma_{V}}{d\ln{M}}$$
$$\displaystyle=$$
$$\displaystyle\Gamma_{V}\bigg{[}1-\frac{\Gamma_{V}+\Gamma_{U}}{2\pi}+\frac{(%
\Gamma_{V}+\Gamma_{U})(5\Gamma_{V}+7\Gamma_{U})}{24\pi^{2}}\bigg{]},$$
$$\displaystyle\frac{d\Gamma_{U}}{d\ln{M}}$$
$$\displaystyle=$$
$$\displaystyle\Gamma_{U}\bigg{[}1-\frac{3(\Gamma_{V}+\Gamma_{U})}{2\pi}+\frac{(%
\Gamma_{V}+\Gamma_{U})(5\Gamma_{V}+7\Gamma_{U})}{24\pi^{2}}\bigg{]}.$$
It turns out that despite their structural similarity of these equations the fates of two types of disorders are very distinct as depicted in Fig. 16. If we include one-loop corrections only (Left), there appear two critical lines each for $\Gamma_{V}$ and $\Gamma_{U}$. Over the red line $\Gamma_{U}$ starts to increase and over the blue line $\Gamma_{V}$ does, too. However, the total gradient is overwhelmed by that of $\Gamma_{U}$, i.e. almost upward. This means that the anti-screening of $\Gamma_{V}$ is much weaker than that of $\Gamma_{U}$. If we include two-loop corrections also that give rise to screening in both disorders (Right), there appears another critical line for $\Gamma_{U}$ while the critical line for $\Gamma_{V}$ disappears, so $\Gamma_{V}$ becomes irrelevant. As a result, we have two nonzero fixed points on the line of $\Gamma_{V}=0$ as shown in this figure and the first figure in Fig. 3.
This observation suggests that $\Gamma_{U}$ has dominant effects over $\Gamma_{V}$ for the low-energy physics. Since we are interested in the renormalization of $c_{k}$, we need to consider two equations at $\Gamma_{V}=0$:
$$\displaystyle\frac{d\Gamma_{U}}{d\ln{M}}$$
$$\displaystyle=$$
$$\displaystyle\Gamma_{U}\Big{[}1-a_{\Gamma}\Gamma_{U}+b_{\Gamma}\Gamma_{U}^{2}%
\Big{]},$$
$$\displaystyle\frac{dc_{k}}{d\ln{M}}$$
$$\displaystyle=$$
$$\displaystyle c_{k}\Big{[}-1-a_{\bm{c}}\Gamma_{U}+b_{\bm{c}}\Gamma_{U}^{2}\Big%
{]},$$
where the positive numerical constants are given by
$$\displaystyle a_{\Gamma}=\frac{3}{2\pi},~{}b_{\Gamma}=\frac{7}{24\pi^{2}},~{}a%
_{\bm{c}}=\frac{1}{2\pi},~{}b_{\bm{c}}=\frac{1}{12\pi^{2}}.$$
In the first equation for $\Gamma_{U}$, there are three fixed points: $\Gamma_{0}=0,\Gamma_{1}=\frac{a_{\Gamma}-\sqrt{a_{\Gamma}^{2}-4b_{\Gamma}}}{2b%
_{\Gamma}}$, and $\Gamma_{2}=\frac{a_{\Gamma}+\sqrt{a_{\Gamma}^{2}-4b_{\Gamma}}}{2b_{\Gamma}}$. Two stable fixed points of $\Gamma_{0}$ and $\Gamma_{2}$ are identified as a clean Weyl metal state and a diffusive Weyl metal phase, respectively. An unstable fixed point of $\Gamma_{1}$ is identified as the phase transition point from the clean Weyl metal state to the diffusive Weyl metal phase.
Let’s move on the second equation for $c_{k}$. The formal solution is given by
$$\displaystyle c_{k}(T)=c_{k}(T_{0})\exp{\bigg{[}-\int^{\ln{T}}_{\ln{T_{0}}}d%
\ln{M}-a_{\bm{c}}\int^{\ln{T}}_{\ln{T_{0}}}d\ln{M}~{}\Gamma_{U}(M)+b_{\bm{c}}%
\int^{\ln{T}}_{\ln{T_{0}}}d\ln{M}~{}\Gamma_{U}^{2}(M)\bigg{]}},$$
where $T_{0}$ is a UV cutoff. Inserting the solution of $\Gamma_{U}(M)$ into the above, we find that the distance between the pair of Weyl points shows a power-law divergent behavior
$$c_{k}(T)=c_{k}(T_{0})\bigg{(}\frac{T_{0}}{T}\bigg{)}^{\lambda_{\bm{c},fn}},$$
(48)
where $\lambda_{\bm{c},fn}$ is a critical exponent around each fixed point, given by
$$\displaystyle\lambda_{\bm{c},f0}$$
$$\displaystyle=$$
$$\displaystyle 1+a_{\bm{c}}\Gamma_{0}-b_{\bm{c}}\Gamma_{0}^{2}=1,$$
$$\displaystyle\lambda_{\bm{c},f1}$$
$$\displaystyle=$$
$$\displaystyle 1+a_{\bm{c}}\Gamma_{1}-b_{\bm{c}}\Gamma_{1}^{2}\simeq 1.34,$$
$$\displaystyle\lambda_{\bm{c},f2}$$
$$\displaystyle=$$
$$\displaystyle 1+a_{\bm{c}}\Gamma_{2}-b_{\bm{c}}\Gamma_{2}^{2}\simeq 1.60.$$
Disorder scattering changes the temperature-dependent exponent of $c_{k}$ (see Fig. 17).
References
(1)
A. O. Gogolin, A. A. Nersesyan, and A. Tsvelik, Bosonization and Strongly Correlated Systems, (Cambridge University Press, New York, 1998).
(2)
T. Senthil, A. Vishwanath, L. Balents, S. Sachdev and M. P. A. Fisher, Science 303, 1490 (2004); T. Senthil, L. Balents, S. Sachdev, A. Vishwanath and M. P. A. Fisher,
Phys. Rev. B 70, 144407 (2004).
(3)
A. Tanaka and X. Hu, Phys. Rev. Lett. 95, 036402 (2005).
(4)
E. Fradkin, Field Theories of Condensed Matter Physics, (Cambridge University Press, New York, 2013).
(5)
F. Evers and A. D. Mirlin, Rev. Mod. Phys. 80, 1355 (2008).
(6)
A. M. M. Pruisken, Nucl. Phys. B 235, 277 (1984); A. M. M. Pruisken, Nucl. Phys. B 240, 30 (1984).
(7)
L. Fu and C. Kane, Phys. Rev. Lett. 109, 246605 (2012).
(8)
For a review, see Ki-Seok Kim, Heon-Jung Kim, M. Sasaki, J.-F. Wang, L. Li, Sci. Technol. Adv. Mater. 15, 064401 (2014).
(9)
M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory, (Addison Wesley, New York, 1995).
(10)
F. D. M. Haldane, Phys. Rev. Lett. 93, 206602 (2004).
(11)
S. Murakami, New J. Phys. 9, 356 (2007).
(12)
A. A. Burkov and L. Balents, Phys. Rev. Lett. 107, 127205 (2011).
(13)
Kyoung-Min Kim, Yong-Soo Jho, and Ki-Seok Kim, Phys. Rev. B 91, 115125 (2015).
(14)
H. B. Nielsen and M. Ninomiya, Phys. Lett. 130B, 389 (1983).
(15)
Heon-Jung Kim, Ki-Seok Kim, J.-F. Wang, M. Sasaki, N. Satoh, A. Ohnishi, M. Kitaura, M. Yang, and L. Li, Phys. Rev. Lett. 111, 246603 (2013).
(16)
D. T. Son and B. Z. Spivak, Phys. Rev. B 88, 104412 (2013).
(17)
Ki-Seok Kim, Heon-Jung Kim, and M. Sasaki, Phys. Rev. B 89, 195137 (2014).
(18)
A. Altland, D. Bagrets, L. Fritz, A. Kamenev, and H. Schmiedt, Phys. Rev. Lett. 112, 206602 (2014); A. Altland, D. Bagrets, and A. Kamenev, Phys. Rev. B 91, 085429 (2015).
(19)
Yong-Soo Jho, Jae-Ho Han, and Ki-Seok Kim, arXiv:1409.0414.
(20)
A. Altland and B. Simons, Condensed Matter Field Theory, (Cambridge University Press, New York, 2010).
(21)
Ki-Seok Kim, Phys. Rev. B 90, 121108(R) (2014).
(22)
D. Xiao, M.-C. Chang, and Q. Niu, Rev. Mod. Phys. 82, 1959 (2010).
(23)
N. Nagaosa, J. Sinova, S. Onoda, A. H. MacDonald, and N. P. Ong, Rev. Mod. Phys. 82, 1539 (2010).
(24)
P. Goswami and Sumanta Tewari, Phys. Rev. B 88, 245107 (2013).
(25)
A. A. Zyuzin and A. A. Burkov, Phys. Rev. B 86, 115133 (2012).
(26)
Y. Chen, D. L. Bergman, and A. A. Burkov, Phys. Rev. B 88, 125110 (2013).
(27)
Kenji Fukushima, Dmitri E. Kharzeev, and Harmen J. Warringa, Phys. Rev. D 78, 074033 (2008).
(28)
D. T. Son and N. Yamamoto, Phys. Rev. Lett. 109, 181602 (2012).
(29)
M. A. Stephanov and Y. Yin, Phys. Rev. Lett. 109, 162001 (2012).
(30)
Gokce Basar, Dmitri E. Kharzeev, and Ho-Ung Yee, Phys. Rev. B 89, 035142 (2014).
(31)
K. Landsteiner, E. Megias, and F. Pena-Benitez, Phys. Rev. Lett. 107, 021601 (2011).
(32)
Y. Chen, Si Wu, and A. A. Burkov, Phys. Rev. B 88, 125105 (2013).
(33)
D. T. Son and N. Yamamoto, Phys. Rev. D 87, 085016 (2013).
(34)
C. Manuel and Juan M. Torres-Rincon, Phys. Rev. D 90, 076007 (2014).
(35)
J.-Y. Chen, D. T. Son, M. A. Stephanov, Ho-Ung Yee, and Yi Yin, Phys. Rev. Lett. 113, 182302 (2014).
(36)
I. Zahed, Phys. Rev. Lett. 109, 091603 (2012).
(37)
G. Basar, D. E. Kharzeev, and I. Zahed, Phys. Rev. Lett. 111, 161601 (2013).
(38)
C. Duval and P. A. Horvathy, Phys. Rev. D 91, 045013 (2015).
(39)
M. Stone, V. Dwivedi, and T. Zhou, Phys. Rev. D 91, 025004 (2015).
(40)
Yong-Soo Jho and Ki-Seok Kim, Phys. Rev. B 87, 205133 (2013).
(41)
Dongwoo Shin, Yong-Woo Park, M. Sasaki, Heon-Jung Kim, Yoon-Hee Jeong, Ki-Seok Kim, and Jeehoon Kim, in preparation.
(42)
R. D. Peccei and H. R. Quinn, Phys. Rev. Lett. 38, 1440 (1977); R. D. Peccei and H. R. Quinn, Phys. Rev. D 19, 1791 (1977).
(43)
https://web.science.uu.nl/drstp/SHELL/2009/Theses/thesisDenDunnen.pdf. |
Evolution of coronal mass ejections and the corresponding Forbush decreases: modelling vs multi-spacecraft observations
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Hvar Observatory, Faculty of Geodesy, University of Zagreb, Kačićeva 26, HR-10000 Zagreb, Croatia
CAS Key Laboratory of Geospace Environment, School of Earth and Space Sciences, University of Science and Technology of China, Hefei, China
CAS Center for Excellence in Comparative Planetology, USTC, Hefei, 230026, China
Department of Extraterrestrial Physics, Christian-Albrechts University in Kiel, Liebnitzstrasse 11, 24098, Kiel, Germany
Institute of Physics, University of Graz, Universitätsplatz 5, A-8010 Graz, Austria
Dpto. de Fisica y Matematicas, Universidad de Alcala, 28805 Alcala de Henares, Madrid, Spain
Kanzelhöhe Observatory for Solar and Environmental Research, University of Graz, Kanzelhöhe 19, 9521 Treffen, Austria
Skolkovo Institute of Science and Technology, Bolshoy Boulevard 30, bld. 1, Moscow 121205, Russia
Space Research Institute, Austrian Academy of Sciences, Schmiedlstraße 6, 8042 Graz, Austria
Karlovac University of Applied Sciences, Karlovac, Croatia
keywords: Coronal Mass Ejections, Interplanetary; Cosmic Rays, Galactic
\setlastpage\inarticletrue{opening}
addressref=aff1,corref,email=mateja.dumbovic@geof.unizg.hr]Mateja Dumbovićaddressref=aff1]Bojan Vršnakaddressref=aff2,aff3]Jingnan Guoaddressref=aff4]Bernd Heberaddressref=aff5]Karin Dissaueraddressref=aff6]Fernando Carcabosoaddressref=aff5]Manuela Temmeraddressref=aff5,aff7]Astrid Veronigaddressref=aff8]Tatiana Podladchikovaaddressref=aff9]Christian Möstladdressref=aff9]Tanja Amerstorferaddressref=aff10]Anamarija Kirin
1 Introduction
\ilabel
iintro
Coronal mass ejections (CMEs) are magnetic structures that erupt from the solar corona and interact with the ambient plasma and energetic particles (electrons, protons and ions ranging from suprathermal to GeV energies and beyond), as they evolve and propagate through the interplanetary (IP) space. The propagation of CMEs in the IP space is dominated by the emission of MHD waves in the collisionless solar wind environment, that acts to adjust the CME speed to the ambient solar wind, i.e. MHD drag (Cargill et al., 1996; Vršnak
et al., 2013). As they propagate through the IP space, CMEs expand due to the pressure imbalance (e.g. Klein and Burlaga, 1982; Démoulin and Dasso, 2009, and references therein), where, consequently, as the size of the magnetic structure increases, its magnetic field weakens (e.g. Bothmer and Schwenn, 1998; Leitner et al., 2007; Démoulin
et al., 2008; Gulisano et al., 2012; Vršnak
et al., 2019). CMEs with leading fronts moving faster than the ambient solar wind will compress and deflect the upstream plasma producing a so-called sheath region, and if their relative speed is greater than the fast-mode wave speed, a shock will form (e.g. Russell and Mulligan, 2002; Owens et al., 2005). The sheath plasma can be composed of different types of material (coronal/heliospheric, shocked/compressed) and its size can also change, as the CME evolves and propagates (Masías-Meza
et al., 2016; Janvier et al., 2019; Lugaz, Winslow, and
Farrugia, 2020).
Interplanetary coronal mass ejections (ICMEs), according to the standard nomenclature, encompass both a shock/sheath region and a CME magnetic structure (e.g. Rouillard, 2011; Kilpua, Koskinen, and
Pulkkinen, 2017), each showing a number of specific in situ properties. The shock/sheath region is typically characterised by increased density, temperature, magnetic field fluctuations and plasma beta, whereas the CME magnetic structure is typically characterised by a smoothly rotating field, low plasma beta and temperature and a linearly decreasing speed profile indicative of expansion (Zurbuchen and
Richardson, 2006; Kilpua, Koskinen, and
Pulkkinen, 2017). These properties indicate magnetic structures with field lines winding helicoidally around the central axis, therefore, the most commonly assumed simple magnetic structure of a CME is a force-free flux rope with circular cross section (e.g. Lundquist, 1951; Gold and Hoyle, 1960) that expands self-similarly (Démoulin
et al., 2008). It should be noted however, that observed magnetic structures can substantially deviate from this highly ideal concept (e.g. Nieves-Chinchilla
et al., 2018), and might not even be flux ropes, but simply writhed structures (Al-Haddad
et al., 2019). Additional indication of the CME magnetic structure can be increased abundance of high charge states, reflecting the temperature history of the CME and/or its origin (Lepri et al., 2001; Zurbuchen
et al., 2016), or bi-directional suprathermal (60-1000eV) electrons, indicative of a magnetically closed structure (Gosling et al., 1987). ICMEs can also show distinctive signatures in galactic cosmic ray (GCR) count rates, in detectors which have count rates high enough to provide sufficient statistical accuracy, depressions (Forbush decreases, FDs) can be observed.
FDs can be observed throughout the heliosphere (e.g. Paularena
et al., 2001; Witasse et al., 2017; Winslow et al., 2018) using detectors such as ground-based neutron monitors and muon detectors, spacecraft particle detectors and dosimeters and have recently been substantially utilised as ICME signatures at Mars (e.g. Freiherr von Forstner
et al., 2018; Guo et al., 2018b; Freiherr von Forstner
et al., 2019; Papaioannou, 2019). FD properties such as the magnitude, shape, duration and sub-structuring depend on the properties of the corresponding interplanetary transient (see e.g. Richardson, 2004; Cane, 2000; Belov, 2009; Dumbović
et al., 2012). FDs caused by ICMEs often show a two-step profile, one associated with the shock/sheath region and the other with the CME magnetic structure (Barnden, 1973; Cane, 2000). The two regions were found to be roughly equally effective in producing the depression (Richardson and
Cane, 2011a), although because of their different physical properties, the mechanism through which they produce the depression is different and thus should be modelled differently (Wibberenz
et al., 1998). The shock acts as a discontinuity which can reflect particles (Kirin et al., 2020), resulting in a pre-increase (Cane, 2000), whereas the sheath can be described as a diffusive barrier (Wibberenz, Cane, and
Richardson, 1997; Wibberenz
et al., 1998) and the shock/sheath-related FD can be described by solving a full transport (Parker, 1965) equation (e.g. Le Roux and Potgieter, 1991; Wawrzynczak and
Alania, 2010; Alania et al., 2013). On the other hand, a CME magnetic structure can be described as a very slowly filling (and expanding) particle trap where the particles can enter via perpendicular diffusion (e.g. Cane, Richardson, and
Wibberenz, 1995; Munakata et al., 2006; Subramanian
et al., 2009; Dumbović
et al., 2018), or guiding center drifts (e.g. Krittinatham and
Ruffolo, 2009; Tortermpun, Ruffolo, and
Bieber, 2018). Recent efforts in FD modelling also include full trajectory integration using CME flux-rope type models (Petukhova, Petukhov, and
Petukhov, 2019a, b) or CME magnetic field reconstructions from in situ measurements (Benella et al., 2019), as well as describing FDs via the change in the single GCR spectrum modulation parameter attributed with a CME (Guo et al., 2020).
Recently, Dumbović
et al. (2019) utilised FD signatures at Mars to indicate inhibited expansion of a CME magnetic structure and Freiherr von Forstner
et al. (2020) related different FD properties at Earth and Mars to the possible evolution of the ICME sheath. Therefore, there are strong indications that FDs reflect the evolutionary properties of ICMEs and thus FD models not only offer an opportunity to understand the variability of FDs detected in the heliosphere, but also to gain insight into ICME evolution. For that purpose we consider a text-book example of a two-step FD and combine two analytical models, the propagative diffusive barrier (PDB, Wibberenz
et al., 1998) model for the sheath region and the diffusion-expansion Forbush decrease (ForbMod, Dumbović
et al., 2018) model for the CME magnetic structure to produce a generic two-step FD profile (Section \irefiprofile). We adapt these models for energy dependence (Section \irefienergy), since in their current form they are monoenergetic, whereas detectors which have enough statistics to detect FDs detect an energy range (usually $E>E_{\mathrm{cutoff}}$). Finally, using modelling, we analyse an actual multi-spacecraft event recently studied from the observational perspective by Winslow et al. (2018) at 4 different radially aligned heliospheric distances (Section \irefiobs).
2 A generic two-step Forbush decrease profile
\ilabel
iprofile
A two-step Forbush decrease (FD) is often regarded as a textbook FD, which owes its specific morphology to the fact that the measuring instrument passed through the ICME head-on (see e.g. Richardson and
Cane, 2011a, and references therein). In this case, the instrument first encounters the shock front (if developed), then the sheath and finally the CME magnetic structure. The corresponding depression has a sharp onset, sometimes preceded by a small pre-increase, where this first drop is then suddenly interrupted by the second onset, i.e. the second decrease. The first decrease is usually attributed to the joint shock/sheath region, and the second decrease to the magnetic ejecta. The GCR count in a two-step FD does not return to the pre-decrease level immediately after the ICME passage, but recovers slowly over the course of the next couple of days or even weeks. In order to understand the mechanisms which govern the formation of the depression, we need to analyse the interplanetary structures that cause them and how they influence the GCRs.
2.1 The shock-related effect
\ilabel
ishock
Since the magnetic field is compressed in the downstream region of the interplanetary shock it can be regarded as a fast mode MHD shock. It was shown by Kirin et al. (2020) that due to the change of the magnetic field component normal to the shock front, particles coming from the upstream region can be reflected. This can cause or contribute to the first decrease in the two-step FD, but also explain the pre-increase observed in some events, namely, the interaction of the GCR with a shock is a result of the complex interplay of the shock orientation and strength vs GCR energy and direction. Test particle simulations show that a subset of the GCR particles may be reflected between the upstream and downstream regions and reenter the shock region via helical motion (for details see Kirin et al., 2020). Therefore, a small population of particles will linger for some time near the shock front. Although the ’shock’ effect is produced in the extremely small spatial extent of the shock front (typically $\sim 10^{3}$km; Pinter, 1980) due to reflection of particles, it has influence over a much broader spatial scale. Very strong shocks are often associated with so-called precursors, pre-decreases and pre-increases in the CR intensity accompanied by the changes in the first harmonic of the anisotropy at the ecliptic plane, appearing hours before the FD onset (see e.g. Belov et al., 1995; Papailiou
et al., 2012; Lingri et al., 2019, and references therein).
Moreover, there are strong indications that a prolonged recovery phase of the FD is due to the shock front moving away from the observer. As it propagates away the shock reflects the upstream particles across its front, which can be regarded as “casting a shadow” upon the observer. As the shock front propagates away the “shadow” becomes smaller, thus the shadow effect weakens (i.e. decays) resulting in an exponential recovery (for more details see Lockwood, Webber, and
Jokipii, 1986; Dumbović
et al., 2011). The shadow effect is sketched in Figure \irefifig1c. In several studies the recovery phase of FDs was successfully fitted by an exponential decay function (e.g. Penna and Quillen, 2005; Jämsén
et al., 2007; Usoskin et al., 2008; Zhao and Zhang, 2016; Munini et al., 2018), but it should be noted that the fit is not always applicable (Dumbović
et al., 2011). This might be related to the interruption by another interplanetary structure or the definition of the recovery phase.
The recovery phase, as defined by most studies, starts at the minimum of the depression, i.e. in the two-step FD at the center of the CME magnetic structure. The first part of the recovery phase is therefore governed by the interaction with magnetic ejecta, whereas the exponential decay, related to the shock propagating away, starts after the passage of the magnetic ejecta. Adopting the “shadow effect”, we assume that the GCR density will recover at an approximately exponential rate, i.e. that the GCR density amplitude in this part of the recovery phase can be written as the modified exponential function:
$$A(r)=\frac{U(r)-U_{0}}{U_{0}}=A_{0}(r)\mathrm{e}^{-\frac{r-r_{0}}{\lambda}},%
\ilabel{ieq1}$$
(1)
where $A(r)$ is the relative amplitude of the GCR count, $r$ is the radial (heliospheric) distance, $r_{0}$ is the distance at which the recovery starts, $U(r)$ is the GCR phase space density at distance $r$, $U_{0}$ is the unperturbed GCR phase space density, $A_{0}(r)$ is the function which determines the amplitude of the recovery, and $\lambda$ is a constant. Although it has been shown that there is a radial gradient of GCRs of about 3%/au (e.g. Webber and Lockwood, 1999; Gieseler and Heber, 2016; Lawrence et al., 2016), $U_{0}$ is assumed to be constant for simplicity reasons. This assumption was used and tested for flux rope-related FDs by Dumbović
et al. (2018), however, the radial gradient of GCRs might have more significant contribution to the recovery phase of a FD, possibly affecting the recovery rate. Note that $A_{0}(r)$ cannot be constant because $A(r)$ has to satisfy the boundary condition $A(r_{\mathrm{end}})=0$ (i.e. that GCR count fully recovers at some distance $r_{\mathrm{end}}$) and the domain of the exponential function is not restricted. The transformation of Equation \irefieq1 into the time-series scale can be performed with the substitution $r\rightarrow v_{\mathrm{shock}}t$ (assuming constant shock speed, $v_{\mathrm{shock}}$) yielding the expression:
$$A(t)=A_{0}(t)\mathrm{e}^{-\frac{t-t_{0}}{\tau}}\,,\,\,\,\,A_{0}(t)=A_{\mathrm{%
sh,max}}+(t-t_{0})/\tau,\ilabel{ieq2}$$
(2)
where $A(t)$ is the relative amplitude of the GCR count which starts from its maximum value equal to the sheath-related FD amplitude, $A_{\mathrm{sh,max}}$ (see Figure \irefifig2), $A_{0}(t)$ is the function which determines the amplitude of the recovery, $t_{0}$ the start time of the recovery, and $\tau$ is the characteristic recovery time defined by $\lambda$, $\tau=\lambda/v$. The $A_{0}(t)$ functional form is chosen arbitrarily, as the simplest form which can satisfy the initial and final state conditions ($A(t_{0})=A_{\mathrm{sh,max}}$ and $A(t_{\mathrm{end}})=0$, respectively). We note that the characteristic recovery time, $\tau$, can be treated as a free parameter, although it should be related to the speed and spatial extent of the shock, and include other possible influences, such as the radial gradient of GCRs.
2.2 The sheath-related effect
\ilabel
isheath
The sheath region, on the other hand, has a much larger spatial extent compared to the shock (typically $\sim 10$ hours at Earth according to Russell and Mulligan, 2002) and is characterised by a highly compressed and fluctuating magnetic field, as well as increased plasma flow. Wibberenz
et al. (1998) describes the sheath-related FD model first proposed by Chih and Lee (1986), assuming that the sheath acts as a “propagating diffusive barrier” (PDB). In this 1D model the transport equation (Parker, 1965; Jokipii, 1971) is reduced to the convection-diffusion equation by adopting the so-called force field approximation. The force field approximation assumes a steady state without sources of cosmic rays and neglects the adiabatic energy loss, resulting in a solution given by a parameter describing the rigidity loss called the force field potential (Gleeson and Axford, 1968; Caballero-Lopez and
Moraal, 2004). In the force field approximation the change in the GCR distribution function is given by the exponential of the modulation function, which depends on the flow speed and radial diffusion coefficient (Equations 9 and 11 in Caballero-Lopez and
Moraal, 2004). In the PDB model the sheath is represented by a shell where the flow speed is increased and the diffusion coefficient decreased and both have constant values across the shell (see Figure \irefifig1a). The corresponding relative GCR density drop in the sheath (normalised to the onset value) is a linear function of the distance to the border of the shell (for more details see Wibberenz
et al., 1998):
$$A(r)=\frac{U(r)-U_{0}}{U_{0}}=-\frac{V^{\prime}}{D^{\prime}}r\,,\ilabel{ieq3}$$
(3)
where $U(r)$ is the GCR phase space density at distance $r$ from the border of the shell, $U_{0}$ is the GCR phase space density at the shell border (the onset value), and $V^{\prime}$ and $D^{\prime}$ are the flow speed and the radial diffusion coefficient within the shell, respectively. The maximum amplitude of the relative GCR density drop in the sheath, i.e. FD magnitude $A_{\mathrm{sh,max}}$ for the shell of thickness $L$ is then given by $A_{\mathrm{sh,max}}=-V^{\prime}L/D^{\prime}$. Assuming that the diffusion coefficient relates to the magnetic field strength $D^{\prime}\approx 1/B^{\prime}$ (see e.g. Potgieter, 2013, and references therein), the sheath-related FD magnitude will depend on the flow speed in the sheath, the magnetic field strength in the sheath and the sheath thickness. The FD drop rate $\mathrm{d}A(r)/\mathrm{d}r$ is given by the slope of the linear function in Equation \irefieq3 and can be easily shown to be related to the FD magnitude (Freiherr von Forstner
et al., 2020):
$$\frac{\mathrm{d}A(r)}{\mathrm{d}r}=-\frac{A_{\mathrm{sh,max}}}{L}\,.\ilabel{%
ieq4}$$
(4)
Since $A(r)$ is a linear function of $r$, which can be written as $r=V^{\prime}t$, one can easily obtain the time-evolving FD drop rate, $\mathrm{d}A(t)/\mathrm{d}t=\mathrm{d}A(r)/\mathrm{d}r\cdot V^{\prime}$ (Freiherr von Forstner
et al., 2020). Note that this is the steady-state solution, therefore, the sheath evolution might also lead to changes in the FD profile and/or magnitude. We note that Wibberenz
et al. (1998) used PDB to explain the recovery phase of FDs as well, noting that unlike the main phase, the recovery is determined by the global propagation conditions, i.e. by the changes at the shock front. However, such an approach implicitly assumes that the disturbed conditions of the transport parameters persist well after the CME passage, whereas we would expect the interplanetary space to return to its undisturbed state. Another strong argument against such approach is the fact that the recovery is well-represented by the exponential term which is not necessarily energy-dependent (Lockwood, Webber, and
Jokipii, 1986; Jämsén
et al., 2007; Usoskin et al., 2008). This indicates that the recovery phase primarily depends on the decay of the disturbance and only secondarily on the transport parameters, favouring the so-called “shadow effect” (although the energy-dependence could be introduced by allowing the shield-effect to be different for particles of different energies). Nevertheless, assuming that the “shadow effect” is indeed energy-independent, the energy-dependence found in some studies (e.g. Jämsén
et al., 2007; Usoskin et al., 2008; Zhao and Zhang, 2016; Munini et al., 2018) might be related to the fact that the exponential recovery phase in those studies was defined to start at the minimum of the depression, i.e. in the two-step FD at the center of the CME magnetic structure. The first part of the recovery phase is therefore governed by the particle interaction with the magnetic ejecta, which is energy-dependent. Usoskin et al. (2008) found that all largest ($>10\%$) FDs demonstrate an energy dependence of the recovery time, while smaller events can demonstrate either energy dependence or the lack thereof. Since the largest FDs are most prominently caused by shock-associated ICMEs, where both shock/sheath and CME magnetic structure are encountered (Richardson and
Cane, 2011a), the sample of energy dependent recovery events might involve an energy dependent part (due to the CME magnetic structure) and an energy independent part (due to the shadow effect of the shock).
2.3 The CME magnetic structure-related effect
\ilabel
iFR
Finally, we regard the second step of FDs corresponding to the CME magnetic structure. In Dumbović
et al. (2018) an analytical diffusion–expansion Forbush decrease (FD) model ForbMod was presented. The model is restricted to explaining the depression caused by the CME magnetic structure, i.e. the flux rope (FR), and the interaction between the particles and the FR, which is described via diffusion, while taking into account the fact that the FR expands self-similarly (see Figure \irefifig1b). Several representative expansion options, related to the effective change of the axial magnetic flux were considered. In particular, in a force free model (e.g. Lundquist, 1951) for a circular cross-section FR the axial magnetic flux can be written as $\Phi_{\mathrm{ax}}\sim B_{c}a^{2}$, where $B_{c}$ is the magnetic field in the FR center and $a$ is the FR radius. Assuming that $B_{c}$ and $a$ change with heliospheric distance following a power-law with indices $-n_{B}$ and $n_{a}$, respectively, it is trivial to see that in the case when $n_{B}=2n_{a}$ the magnetic flux is conserved. A more general expression can be written in the form $\Phi_{\mathrm{ax}}\sim f(t,x)$, where $x=n_{B}-2n_{a}$ determines whether the magnetic flux is conserved ($x=0$), increasing ($x<0$) or decreasing ($x>0$). In Dumbović
et al. (2018) solutions for 4 specific expansion types ($x=0$, $x=0.5$, $x=-0.5$, and $x=-1$) were provided and analysed. Based on Equations 12 and 15 in Dumbović
et al. (2018) it is possible to find a general solution for an arbitrary expansion type $x$, providing $x\neq-1$ (due to integration rules). The GCR phase space density for particles of a specific rigidity (without energy dependence) can then be written as:
$$U(\hat{r},t)=U_{0}\Bigg{(}1-J_{0}(\alpha_{1}\hat{r})\mathrm{e}^{-\alpha_{1}^{2%
}f(t)}\Bigg{)}\,,\,\,\,\,\,\,\,\,\,\,\,\,f(t)=\frac{D_{0}}{a_{0}^{2}}\cdot\Big%
{(}\frac{v}{R_{0}}\Big{)}^{x}\cdot\frac{t^{x+1}}{x+1}\,,\ilabel{ieq5}$$
(5)
where $U_{0}$ is the GCR phase space density at the FR surface, $J_{0}$ is a Bessel function (of the first kind) of the order 0, $\alpha_{1}$ is a first positive root of $J_{0}$ (tabulated in tables of Bessel functions), $\hat{r}$ is the radial distance from the FR axis to the outer border of the FR, scaled to the FR radius ($\hat{r}=r(t)/a(t)$), $D_{0}$, $a_{0}$ and $R_{0}$ are the initial diffusion coefficient, radius and height, $v$ is the CME speed (assumed to be constant), and $x=n_{B}-2n_{a}$ is the expansion type ($n_{B}$ and $n_{a}$ are power-law indices for $B_{c}$ drop and $a$ increase, respectively). Note that it is assumed that $D\sim 1/B_{c}$ and thus also has power-law behaviour with index $n_{B}$. The decrease is symmetric and constrained within the borders of the flux rope, with the maximum depression in the center of the flux rope, $A_{\mathrm{FR,max}}=-e^{-\alpha_{1}^{2}f(T)}$, where $f(T)$ is given by Equation \irefieq5 and $T$ is the transit time to the observer. Since $A(r)$ for the flux rope part of the FD is given by the Bessel function, the drop rate is not constant across the flux rope:
$$\frac{\mathrm{d}A(\hat{r})}{\mathrm{d}\hat{r}}=\alpha_{1}J_{1}(\alpha_{1}\hat{%
r})e^{-\alpha_{1}^{2}f(t)}\,,\ilabel{ieq6}$$
(6)
where we used the Bessel function property $-\mathrm{d}/\mathrm{d}r(J_{\mu}(r)/r^{\mu})=J_{\mu+1}(r)/r^{\mu}$ (for $\mu\geq 0$). It can be easily shown that the maximum drop rate depends linearly on the FD magnitude, $(\mathrm{d}A(\hat{r})/\mathrm{d}\hat{r})_{\mathrm{max}}\sim A_{\mathrm{FR,max}}$.
Based on these modelling efforts it can be concluded that different interplanetary structures will interact differently with GCRs resulting in different ’stages’ of the FD. These stages can be combined together, by superimposing different effects in order to obtain a ’generic profile’ of a two-step FD. This is shown in Figure \irefifig2, along with a sketch of the assumed solar wind plasma parameters. The values of each parameter were selected for each region separately based on typically observed values for magnetic clouds (e.g. Zurbuchen and
Richardson, 2006; Richardson and Cane, 2010; Richardson and
Cane, 2011b). This includes FD amplitudes of the shock/sheath and FR regions, which were normalised to 3%, following the statistical study of Richardson and
Cane (2011b) for particle energies $>60$ MeV. The values of the physical quantities in Figure \irefifig2 are therefore not necessarily interrelated. Nevertheless, this aims to show the shape of the FD based on the modelling described above. Quantitative analysis should involve real events and will be given in the second part of this paper.
The first stage of the FD is the first step, which starts with the shock arrival and is constrained to the spatial extent of the sheath. The relative amplitude in this region is given by Equation \irefieq3 and reaches its maximum at the end of the sheath region, with corresponding relative amplitude $A_{\mathrm{sh,max}}$. For simplicity we assume that there is no pre-increase or contributions to the drop due to the shock, i.e. that the whole drop in the sheath region is related to the sheath effect and is given by the PDB. The second stage of the FD corresponds to the CME magnetic structure. It includes not only the second step of FD, but also the first part of its recovery. It is constrained within the spatial extent of the CME magnetic structure. The relative amplitude is given by Equation \irefieq5 and reaches its maximum at the center of the CME magnetic structure, i.e. the flux rope, with the corresponding relative amplitude $A_{\mathrm{FR,max}}$. The third and final stage is the exponential recovery, related to the decay of the shock ’shadow’. It starts with the end of the CME magnetic structure, where the relative amplitude is given by Equation \irefieq2 and the duration, determined from the condition $A(t_{\mathrm{end}})=0$ is given by $t_{\mathrm{end}}=t_{0}-\tau A_{\mathrm{sh,max}}$. The total amplitude of the FD is given by $A_{\mathrm{TOT}}=A_{\mathrm{sh,max}}+A_{\mathrm{FR,max}}$.
Note that the change of the relative amplitude for the sheath and the CME magnetic structures is given across spatial scales, and therefore needs to be transformed into the time series; that can easily be achieved using the speed profile. Due to the expansion, the speed profile across the FR is linearly decreasing, resulting in an asymmetry in the time series of the FR-related FD. This asymmetry somewhat ’smears out’ the transition points between different regions, even in this highly idealised representation of the FD. Therefore, it is not surprising that the FD was often considered to be a homologous phenomenon, especially given that a number of additional aspects may influence the measured GCR count (e.g. time-resolution, energy range, external influences, Clem and Dorman, 2000), and even the characteristics of the structure (e.g. faster ICMEs produce more asymmetric depressions, Belov et al., 2015).
3 Adapting Forbush decrease models for energy-dependence
\ilabel
ienergy
The models used in Section \irefiprofile to reproduce the shape of the two-step FD are not explicitly energy-dependent, however, this is an important feature of FDs. In Dumbović
et al. (2018) the case of fixed arbitrary particle energy with a diffusion coefficient which was only a function of time was considered. While this allowed us to qualitatively assess whether the model fits the observations, for a more proper and precise quantitative analysis energy dependence must be considered. This is due to the FD detection limits, as well as additional modulation by e.g. planetary magnetic fields, atmospheres or the detectors themselves. The FD magnitude is typically around several percent, therefore, large statistics are needed, i.e. high particle count rates. These are easily provided by instruments which count all particles that enter from all directions, regardless of their energy, such as large ground based neutron or muon monitors, as well as single counters onboard spacecrafts.
Detectors which measure particles of a specific energy (or narrow energy intervals) typically provide smaller count rates and thus, in order to observe the effect of several percent, the time resolution needs to be decreased. Moreover, the effect in the low-energy detectors is often masked by the increased flux of low-energy solar energetic particles (SEPs). For example, Munini et al. (2018) used PAMELA (Payload for Antimatter-Matter Exploration and Light-nuclei Astrophysics) data to analyse the FD recovery in 9 different narrow energy ranges between 0.4 and 20 GV and reported (1) that the statistics allowed time resolution of the proton flux of 3 or 6 hours up to 5 GV and one day above 5 GV; (2) that the main phase of FD was not visible in $<2$ GV protons due to SEPs. For comparison, during solar minimum SOHO/EPHIN detector F and a typical neutron monitor at the pole have a counting rate of more than 1000 counts/minute (Moraal, Belov, and
Clem, 2000; Kühl et al., 2015), providing sufficient statistics to observe FDs at a minute resolution. We do note that the long integration time needed for PAMELA data is related to its orbit, because the instrument is located at a low-orbiting satellite spending most of the time inside the geomagnetic field and can thus measure low-energy particles only when traversing (sub)polar regions, while SOHO/EPHIN and polar NMs are exposed to low-energy particles at all times. We also note that due to the low cutoff energy (50 MeV), the SOHO/EPHIN detector F also has a problem with the increased flux of SEPs masking the effect, whereas for neutron monitors this will be the case only for very rare, most energetic SEPs which produce ground level enhancements. Since FDs are typically measured with instruments which observe particles of a specific type at all energies above some specific energy/rigidity cutoff, any quantitative comparison of the modelled and observed FDs should consider energy-dependence.
We introduce energy dependence by allowing the diffusion coefficient, $D$, to be a function of rigidity as well as time, which can be expressed through a typical empirical expression as used in numerical models fitted to GCR measurements, as given by Potgieter (2013) and discussed in detail in Appendix \irefidiff_coeff. While this expression might be suitable for the diffusion coefficient within a flux rope at Earth, the flux rope will have a different diffusion coefficient at other heliospheric distances, because it is a function of time. It can be extrapolated back and forth in time assuming that the diffusion coefficient time dependence is defined by that of the magnetic field (power-law), i.e. $D_{0}=D(R(t)/R_{0})^{-n_{B}}$, where $D_{0}$ is the initial diffusion coefficient, $R_{0}$ and $R(t)$ are different heliospheric distances and $n_{B}$ is the expansion factor (power-law index) of the magnetic field strength.
It can be easily shown that allowing the diffusion coefficient to be energy-dependent does not affect the radial dependence in either PDB or ForbMod (because the diffusion coefficient appears only in the time-dependent part), therefore, the initial assumptions are not violated with the introduction of an energy-dependent diffusion coefficient. Addition of the energy-loss effect (adiabatic cooling) on the other hand would change the starting assumptions in both models and is thus not formally taken into account within the models. We note that on the heliospheric scale adiabatic cooling is relevant mostly for lower energy particles ($E<100$ MeV; Gleeson and Urch, 1971), although it might have larger impact for highly expanding CMEs. In general, the energy-loss is expected to introduce additional modulation effects (see e.g. Lockwood, 1971, and references therein), but the exact quantitative contribution is not trivial to estimate. From the qualitative aspect it is expected that the energy–loss term acts to balance out the inward diffusion of particles (Munakata et al., 2006) and therefore acts to increase FD amplitude.
Note that the solution given in Equation \irefieq5 is only valid for a specific rigidity $P$, where a more general GCR phase space density would be given as $U(\hat{r},t,P)$, with $U_{0}\rightarrow U_{0}(P)$, $D_{0}\rightarrow D_{0}(P)$ and consequently $A(r,t)\rightarrow A(r,t,P)$. The total GCR phase space density in the FR after time $t$ and at distance $\hat{r}$ is obtained by integrating over all available rigidities. Since we are interested only in detected particles, the phase space is constrained by the detector cutoff, $P>P_{\mathrm{cut}}$. Recognising that $\int_{{}_{P>P_{\mathrm{CUT}}}}U_{0}(P)dP$ represents the total GCR phase space density at the FR surface for all rigidities that can be measured by the detector, i.e. the total quiet-time GCR phase space density $U_{0,TOT}$, the total FD amplitude can be written as:
$$A(\hat{r},t)=\int\frac{U(r,t,P)-U_{0}(P)}{U_{0}(P)}\mathrm{d}P=-\frac{\int_{{}%
_{P>P_{\mathrm{CUT}}}}U_{0}(P)J_{0}(\alpha_{1}\hat{r})\mathrm{e}^{-\alpha_{1}^%
{2}f(P,t)}\mathrm{d}P}{U_{0,TOT}}\,,\ilabel{ieq7}$$
(7)
where $f(P,t)$ is given by the same expression as in Equation \irefieq5, except that $D_{0}$ is a function of rigidity, $D_{0}(P)$. On the other hand, from Equation \irefieq5 it can easily be derived that the FD amplitude for a particle of specific rigidity can be written as $A(P)=-J_{0}(\alpha_{1}\hat{r})\mathrm{e}^{-\alpha_{1}^{2}f(P,t)}$. Therefore, the total FD amplitude can be expressed as:
$$A(\hat{r},t)=\int_{P>P_{\mathrm{CUT}}}U_{\mathrm{FRAC}}(P)\cdot A(P)\mathrm{d}%
P\,,\,\,\,\,\,\,\,\,\,\,\,\,U_{\mathrm{FRAC}}(P)=\frac{U_{0}(P)}{U_{0,TOT}}\,,%
\ilabel{ieq8}$$
(8)
where $U_{\mathrm{FRAC}}(P)$ can be interpreted as the contribution of different energy particles to the total FD. It can be easily shown that Equation \irefieq8 is valid for the sheath-related FD as well, where FD amplitude for a particle of specific rigidity can be written as $A(P)=-V^{\prime}r/D^{\prime}(P)$.
The expression given in Equation \irefieq8 refers to the amplitude calculated based on the particle phase space density, whereas our aim is to compare it with the amplitude measured by the detector which relates to the GCR count rate in the direct space. For a specific detector, the GCR count rate will depend on the particle intensity, i.e. the GCR spectrum and the yield function of the detector, $C=\int J(E)\cdot Y(E)\mathrm{d}E$ (see e.g. Sullivan, 1971; Clem and Dorman, 2000), where $J(E)$ is the GCR spectral intensity and $Y(E)$ is the function describing the detection response of the detector, as well as any other influence (e.g. from the planetary atmosphere). Note that due to simplicity, in the analytical models presented in Section \irefiprofile and thus also here, we regard only one particle species, i.e. protons. Since energy-dependent intensity spectra are the same as rigidity-dependent density spectra (up to a normalisation factor, see Moraal, 2013) Equation \irefieq8 can be rewritten as:
$$A(\hat{r},t)=\int_{E>E_{\mathrm{CUT}}}\xi(E)\cdot A(E)\mathrm{d}E\,,\,\,\,\,\,%
\,\,\,\,\,\,\,\xi(E)=\frac{J(E)\cdot Y(E)}{\int_{E>E_{\mathrm{CUT}}}J(E)\cdot Y%
(E)\mathrm{d}E}\,\ilabel{ieq9}$$
(9)
where $\xi(E)$ represents the fractional contribution of different energy particles to the total FD.
The GCR spectral intensity can be obtained using the so–called “force–field” approximation (e.g. Gleeson and Axford, 1968; Caballero-Lopez and
Moraal, 2004; Herbst et al., 2010; Usoskin, Bazilevskaya, and
Kovaltsov, 2011; Gieseler, Heber, and
Herbst, 2017, and references therein), as discussed in detail in Appendix \irefiforce_field. Figure \irefifig3a shows the GCR spectral intensity obtained using the force–field approximation (see Appendix \irefiforce_field for details on the calculation) for February 2014 (the timeframe is chosen to be in line with the real event analysed in Section \irefiobs). In the same figure we overlay plots of the calculated GCR fractional contribution, $\xi$, for a “perfect detector” and a real detector (the single detector F of the SOHO/EPHIN instrument). A “perfect detector” responds to all energies above 0.05GeV equally (i.e. $Y(E)=1$), and therefore simply gives a scaled spectral intensity, whereas SOHO/EPHIN responds better to high-energy particles (for details see Appendix \irefiephin). We can see that not all energies contribute equally to the observed GCR count rate and that the main contribution is coming from a quite narrow energy range, as was discussed earlier by Rodari et al. (2018) and Dumbović
et al. (2019) (note that $J$ and $\xi$ are not given in logarithmic scale). This is the case even for the “perfect detector” and even more so in the case of the SOHO/EPHIN, due to its energy-dependent response.
In Figures \irefifig3b and c we show the rigidity dependence of the FD amplitude and FD profiles for three selected energies, respectively, calculated using ForbMod. As a demonstration of the theory we use the event shown in Figure \irefifig2, representing a typical magnetic cloud example with the following values needed as input for ForbMod: propagation speed $v=500$ $\mathrm{~{}km~{}s}^{-1}$, FR radius $a=0.15$ au, central magnetic field strength at Earth, $B=20$ nT. The size expansion factor, $n_{a}=0.91$, was calculated from the linear speed profile similar to Démoulin and Dasso (2009), yielding an initial FR radius $a_{0}=2.9$ $~{}\mathrm{R}_{\odot}$ for the initial heliospheric distance of $R_{0}=15$ $~{}\mathrm{R}_{\odot}$ (with corresponding transit time $TT=77$ h). The magnetic field expansion factor $n_{B}=1.8$ was chosen as a typically expected value (e.g. Leitner et al., 2007). We see a quite strong rigidity dependence of the FD amplitude, although it should be noted that this rigidity dependence, is calculated for monoenergetic cases and is therefore not directly comparable to the observed FD rigidity dependence which is measured for different energy ranges (for details on the observation of FD rigidity dependence see overviews by Lockwood, 1971; Cane, 2000). Finally, in Figure \irefifig3d we show the total FD for the exemplary event, integrated over all possible energies for the “perfect detector” and SOHO/EPHIN. We can see that the FD magnitude for SOHO/EPHIN is almost two times smaller than for the “perfect detector”, the difference coming from the fact that SOHO/EPHIN, unlike the “perfect detector”, has an energy-dependent response. The FD magnitude calculated for the “perfect detector” can be therefore taken as a very rough upper limit (i.e. we always expect to observe smaller FD magnitude in a real detector compared to the “perfect detector”). The FD magnitude calculated for the “perfect detector” ($\sim 3.5\%$) is somewhat smaller, but still comparable to an average total FD amplitude observed by the IMP 8 spacecraft near Earth (4.3%, Richardson and
Cane, 2011b). It should be noted that the total FD amplitude reported by Richardson and
Cane (2011b) includes both the sheath and FR contribution, indicating that the value calculated for SOHO/EPHIN ($\sim 2\%$) is relatively close to the average FR-related FD amplitude observed by the IMP 8 spacecraft near Earth. However, it should be taken into account that the exemplary event is a strong magnetic cloud, where FDs larger than average are usually observed (Richardson and
Cane, 2011b).
4 Multi-spacecraft observation of the February 2014 event
\ilabel
iobs
4.1 In situ measurements at Earth
\ilabel
iinsitu
In order to fully understand and test the analytical models related to CME-GCR interaction, multi-spacecraft measurements are needed, obtained from radially aligned spacecraft at different heliospheric distances (corresponding to different evolutionary stages of a CME). For that purpose we utilise a study by Winslow et al. (2018) who studied a single CME, launched from the Sun on 2014 February 12, and its corresponding in situ signatures at Mercury, Venus, Earth and Mars, including GCR measurements at Mercury, Earth and Mars. In order to analyse this event at Earth, in addition to the measurements used by Winslow et al. (2018), we include spacecraft ion composition and suprathermal electron measurements, as well as spacecraft GCR measurements. These can help us to determine more reliably different sub-structures within one ICME event. For that purpose we use data from the following spacecraft/instruments:
•
Magnetic Field Instrument (MFI, Lepping et al., 1995) onboard Wind: GSE components of the magnetic field, $B_{i}$, magnetic field strength, $B$ and fluctuations, $dB$
•
Solar Wind Experiment (SWE, Ogilvie et al., 1995) onboard Wind: GSE components of the flow speed, $v_{i}$, and total flow speed, $v$, plasma density, $N$ and temperature $T$ (with expected temperature calculated according to Lopez (1987) and Richardson and Cane (1995)), azimuthal flow angle and plasma beta, and ion composition
•
Solar Wind Ion Composition Spectrometer (SWICS, Gloeckler
et al., 1998) onboard the Advanced Composition Explorer (ACE): plasma composition
•
South Pole Neutron Monitor GCR count measurements (relative counts) obtained from the Neutron Monitor Database (NMDB) search tool http://www.nmdb.eu/nest/
•
GCR count measurements obtained from the Electron Proton Helium Instrument (EPHIN, Müller-Mellin
et al., 1995) onboard the Solar and Heliospheric Observatory (SOHO)
In Figure \irefifig4 we present in situ measurements for the February 15 ICME observed at Earth, which is the interplanetary counterpart of the February 12 CME. We note that inclusion of additional in situ measurements compared to that used by Winslow et al. (2018), especially of the SOHO/EPHIN instrument, somewhat changes the perspective on the event. We highlight three distinct regions we observe in SOHO/EPHIN. The first is the region where the F-detector of the SOHO/EPHIN instrument is dominated by low-energy particles (we remove this data so that the measurement scale is suitable for the small depression to be observed). The second region is characterised by a small, relatively symmetric depression constrained in a time period of linearly declining flow speed profile, increased levels of Fe charge states, as well as O7/O6 and alpha-to-proton ratios, and increased magnetic field. The back of region 2, shows ordered and smooth magnetic field properties and counterstreaming electrons indicating a well defined, twisted and closed magnetic structure. In the front part of region 2 we observe increased magnetic fluctuations, plasma beta, temperature and density, all of which indicate a sheath region, however, the composition and flow speed seem to be connected with the smooth and ordered structure in the back of the region 2 indicating that they belong to the same structure. These frontal mixed plasma/ICME signatures can be interpreted as a consequence of flux rope erosion at its frontal part by the interaction with the solar wind. However, from the GCR point of view, it would seem that although the interaction disturbed one part of the structure, GCRs still perceive it as a single global structure. Region 3, highlighted yellow in Figure \irefifig4, presents a new, compact structure as seen from the GCR behaviour in SOHO/EPHIN. However, we note that other in situ measurements show complex signatures. Low plasma beta, smooth magnetic field and composition indicate a CME-like magnetic structure, but other parameters show a complex structure typical for complex ejectas/compound streams (Burlaga et al., 2003; Lugaz et al., 2017).
Winslow et al. (2018) identified somewhat different borders for the ICME, but we note that they used cosmic ray measurements from Cosmic Ray Telescope for Environmental Radiation (CRaTER) on the Lunar Reconnaissance Orbiter (LRO; Spence et al., 2010) and the South Pole neutron monitor (SoPo), where only one drop of prolonged duration is observed, without fine substructures as observed by SOHO /EPHIN. This is probably due to the fact that SOHO/EPHIN is sensitive to lower energy particles than CRaTER and SoPo. Moreover, since they did not observe a second decrease in the GCR data, Winslow et al. (2018) concluded that the ejecta is probably already filled with GCRs by the time it reaches Earth. Our observations are in agreement with their conclusion, as we only observe a second decrease in a detector responsive to lower energies (SOHO/EPHIN), whereas no additional decrease is observed in SoPo, which is probably directly related to the energy dependence shown in Section \irefienergy.
4.2 Remote sensing
\ilabel
iremote
In order to better understand the observed in situ measurements, we perform a more detailed study of its solar sources. To this aim we investigate coronagraph images of the CME using the SOHO/LASCO (Brueckner
et al., 1995) coronagraphs C2 and C3, and STEREO/SECCHI (Howard et al., 2008) coronagraphs COR1 and COR2, as well as its on-disk low coronal signatures using the Atmospheric Imaging Assembly (AIA, Lemen et al., 2012) EUV imager onboard the Solar Dynamics Observatory (SDO). We analyse the eruption as seen in SDO AIA 211 $\mathrm{\AA}$ and the corresponding coronal dimmings which are detected based on a thresholding technique applied to logarithmic base-ratio images (Dissauer
et al., 2018). The eruption site is active region AR11974, a quite large and complex AR, where several smaller eruptions can be observed just before a CME is detected in the STEREO-A COR1 field of view. The first eruption occurs around 04:20 UT, where the eruptive loops are observed moving in south-west direction away from AR11974 (see upper panels of Figure \irefifig5). Two core dimmings can be associated with this eruption; these presumably mark the footpoints of the twisted magnetic structure (Hudson, Acton, and
Freeland, 1996; Mandrini et al., 2005) with the axis approximately aligned with the solar equator. Thereafter, a secondary dimming appears south-west of the eruption site, in the direction in which the eruptive loops were observed to propagate. At $\sim$ 04:30 UT a second eruption is observed with the eruptive loops moving away from the eruption site in north-west direction, followed by the appearance of two new dimmings whose connecting line is tilted by $45^{\circ}$ with respect to the solar equator. Finally, at around 04:55 UT a third eruption is observed with the eruptive loops moving away from the AR in the north-east direction and a pronounced dimming is detected north of the AR, whereas the dimming area to the south-west is growing. The three eruptions are followed by a CME detected in STEREO-A COR1 at around 05:05, moving in the south-east direction and expanding as a bean-shaped front (left upper panel in Figure \irefifig6). There are two additional faint fronts appearing at 05:40 UT and 06:35 UT around the same position angle, but directed slightly more to the north (middle and right panel in Figure \irefifig6). These three fronts are very likely to correspond to the three eruptions observed by SDO. In COR2 and LASCO the three components are not distinguishable, especially in LASCO where a single halo CME is observed. The three eruptions are most likely interrelated and possibly form a single CME. However, it is reasonable to assume that such a CME would not show a nice ordered flux rope structure in in situ measurements and would very likely result in compound stream signatures.
We perform a 3D CME reconstruction using the Graduated Cylindrical Shell model (GCS, Thernisien, Howard, and
Vourlidas, 2006; Thernisien, Vourlidas, and
Howard, 2009; Thernisien, 2011) which assumes that geometrically the magnetic structure of a CME can be represented as a hollow croissant with its origin at Sun center, i.e. with conical legs, circular cross section and pseudo-circular front. At a specific time the croissant is fully defined by the position of its apex (latitude, stonyhurst/carrington longitude, height), tilt (orientation of its axis with respect to solar equator), half-angle (the angle between the central axis of the legs) and ratio (parameter defining the thickness of the conical legs). We fit the projection of the croissant to coronagraphic images from the STEREO-A and -B/COR2 and LASCO/C3 coronagraphs (i.e. from three different vantage points) recorded at approximately the same times to better constrain the fit. The fits are done manually with the main constraint being the structure observed in ST-A as a continuation of the first eruption detected in ST-A/COR1 (left upper panel in Figure \irefifig6). We obtain the best fit for the following GCS parameters: $11^{\circ}$ longitude, $-6^{\circ}$ latitude, $-70^{\circ}$ tilt, 0.3, ratio and $28^{\circ}$ halfangle. The GCS reconstruction of the CME at a height of 17$~{}\mathrm{R}_{\odot}$is given by the green mesh in the middle plots of Figure \irefifig6, whereas the yellow mesh represents the reconstruction of the corresponding shock, obtained assuming it has a spherical shape and similar tilt/source position as the CME. We perform GCS at several different time-steps to estimate the speed of the CME apex ($v=800$$\mathrm{~{}km~{}s}^{-1}$).
We compare the orientation obtained from the GCS reconstruction with the orientation of the magnetic structure corresponding to the 2nd step of the FD observed by SOHO/EPHIN, as obtained by Grad-Shafranov (GS) reconstruction (Hu and Sonnerup, 2002; Möstl et al., 2009; Hu et al., 2017). GS reconstruction is a 2.5D numerical method to obtain flux rope orientation and magnetic field based on solving of the GS equation in the De Hoffmann-Teller frame (frame of static flux rope) and optimal fitting to the measurement data. As a result one derives orientation and radius of the flux rope, as well as the magnetic field contour plot in the xy-plane of the spacecraft, where the goodness of the fit is determined by the minimum of fit residuals, $R_{f}$ (with $R_{f}<0.2$ being the condition for a satisfactory solution, see Hu et al., 2017). We note that although the best reconstruction ($R_{f}=0.11$) is obtained only for the very inner part of the FR where a clear rotation is observed (we do not present these results here), for the borders defined based on the 2nd step of the FD observed by SOHO/EPHIN we find a borderline ($R_{f}=0.2$) solution.
The comparison of GCS and GS reconstruction is shown in the bottom panels of Figure \irefifig6. The left bottom panel of Figure \irefifig6 shows the GCS reconstruction projected onto the solar equatorial plane, with the black dashed line showing the direction of the apex, the red line showing the Sun-Earth line and the red semicircle outlining the cross-section of the croissant in the solar equatorial plane. The GCS results are overlaid on the GS reconstruction image, as determined by the GS method, showing a magnetic field contour plot in the plane of the flux rope cross section which is practically perpendicular to the solar equatorial plane (in the right bottom panel of Figure \irefifig6). The agreement between the orientation of the two reconstructions, can be seen from the tilt angle agreement (GCS tilt angle is $-70^{\circ}$ and GS tilt angle is $-75^{\circ}$). This agreement indicates that the GCS reconstruction fits the main part of the CME, which shows ordered magnetic structure in the in situ measurements corresponding to the 2nd step of FD observed by SOHO/EPHIN. We note that both the GCS and GS reconstruction suggest that the Sun-Earth direction is not perfectly aligned along the diameter of the flux rope, however, the diameter obtained by GS reconstruction ($0.15$ au) is in good agreement with the measured FR size ($0.16$ au, see Table \irefitab1).
4.3 Multi-spacecraft in situ measurements
\ilabel
imulti-spacecraft
Around the time of the CME liftoff several spacecraft at different heliospheric distances were approximately radially aligned with Earth (see Figure \irefifig7). We next analyse the in situ measurements at other heliospheric distances to try to identify the sheath and the structure corresponding to the 2nd step of FD observed by SOHO/EPHIN (hereafter referred to as flux rope, FR). For that purpose we use the MErcury Surface, Space ENvironment, GEochemistry, and Ranging (MESSENGER, MES) magnetometer (MAG, Anderson et al., 2007) and Neutron Spectrometer (NS, Goldsten et al., 2007), the Venus Express (VEX) magnetometer (Zhang et al., 2006) and the Radiation Assessment Detector (RAD, Hassler et al., 2012), on board Mars Science Laboratory’s (MSL) rover Curiosity (Grotzinger
et al., 2012). The multi-spacecraft in situ measurements that we compare are shown in Figure \irefifig7. At MES magnetic field and cosmic ray counts are available, at VEX only magnetic field and at MSL/RAD only cosmic ray counts. At MES a two-step FD is observed (Winslow et al., 2018), which helps us to identify the corresponding sheath. It can be seen in Figure \irefifig7 that the profile of the total magnetic field in the sheath region at MES is remarkably similar to that at Earth. MES was inside Mercury’s magnetosphere during the passage of the CME magnetic structure, thus this data was removed. Nevertheless, the end of the rotation is visible after the data gap, as is the recovery phase of the FD, allowing us to set borders to the magnetic structure, presumably corresponding to the FR. At VEX the similarity to MES and Wind data is not obvious and there are no cosmic ray measurements. Nevertheless, we set the borders of the presumed FR based on the rotation of the magnetic field, assuming that the distorted leading part of the FR as observed at Earth is already present at VEX. Finally, at Mars we observe only one decrease, the second step is not observed in FD. Therefore, similarly as Winslow et al. (2018), we assume that the entire main phase of the FD at Mars (from onset to the minimum) corresponds to the shock/sheath region. This allows us to mark the shock arrival and set borders for the sheath region. However, we cannot identify the FR.
Based on the identification of sheath and FR at different heliospheric locations as shown in Figure \irefifig7 we analyse the evolution of the size and magnetic field in the sheath and FR. The size of the sheath was estimated based on the measured sheath duration and average flow speed. The flow speed at MES, VEX and Mars was estimated using the method described by Vršnak
et al. (2019), where two speed measurements at two different locations are used to extrapolate the propagation speed at a third location (for that purpose we used the CME initial speed and the flow speed observed at Earth). The FR size was similarly estimated based on the measured FR duration and average flow speed, where the average flow speed was estimated based on the expansion speed. At Earth, the expansion speed was estimated from the linearly decreasing speed profile. We then assumed constant expansion speed and at a given location (Mercury, Venus, Mars) subtracted the expansion speed from the sheath flow speed in order to estimate the average FR flow speed (i.e. we assumed the leading edge of the FR has the same speed as the flow in the sheath). In order to add an additional datapoint for the analysis, we also calculated the diameter of the croissant obtained from the GCS fit as a FR size proxy at the Sun, and we used the distance between FR and shock apex obtained from the GCS fit as a proxy of the sheath size at the Sun. The magnetic field strength in the sheath and FR was estimated manually by the observer (M.D.) based on the plot of the time series in the specific region. Although this introduces a certain subjectivity, calculation of e.g. average value or simply taking a peak value might yield a misleading result due to changes on fine time scales, which can substantially deviate from the smooth models we use to understand the global structure.
Next, in order to add an additional datapoint for the analysis of the magnetic field, we estimate the magnetic field in the FR at the Sun using the value of the dimming flux (i.e. total unsigned magnetic flux involved in the mapped coronal dimming region) calculated at the time of the first eruption observed in SDO (upper plots of Figure \irefifig5). We use the dimming flux as a proxy of the magnetic flux contained within the CME magnetic structure (Dissauer
et al., 2018), similarly as was done by Scolini et al. (2020), where the magnetic field inside the structure was estimated using the GCS-reconstructed radius to calculate the cross-section area (we estimate a dimming flux of $\approx 4\times 10^{21}$Mx). The results are summarized in Table \irefitab1 and shown in Figure \irefifig8, where power-law fits are applied to characterise the evolutionary properties. It can be seen that the power-law index for the increase of the FR size is at the lower end of the typical observational range (Gulisano et al., 2012). We also estimate the size expansion factor, i.e. the size power-law index, based on the relation presented by Gulisano et al. (2012), which applies to non-perturbed magnetic clouds expanding self-similarly ($n_{a}=\Delta vR/\Delta t\langle v\rangle^{2}$, where $n_{a}$ is the size power-law index, $\Delta v$ is the difference between the flow speeds of the leading and trailing edge, $R$ is heliospheric distance, $\Delta t$ is duration and $v$ the flow speed). Thus, the obtained size power-law index is somewhat larger ($n_{a}=0.8$) than the one obtained in Figure \irefifig8 ($n_{a}=0.6$). In addition, we find that the size of the sheath also increases with heliospheric distance and moreover at a faster rate ($n_{a}=1$) than the size of the FR. This is in agreement with the study of Janvier et al. (2019), who found that the ratio of the duration of the magnetic ejecta over that of the sheath in general decreases from MESSENGER to ACE. The magnetic field power-law index for the FR ($n_{B}=1.9$) is well within the observationally expected range (see Gulisano et al., 2012), whereas for the sheath it is somewhat larger ($n_{B}=2.4$).
4.4 CME evolutionary properties and Forbush decreases
\ilabel
iFD
We next study how the FR evolutionary properties correspond to the GCR profiles. Due to the complex nature of the event, which deviates substantially from the generic profile discussed in Section \irefiprofile, we do not analyse the recovery phase. Furthermore, the PDB model for the sheath region in its current form is not suitable for quantitative analysis since it is a steady-state model in which the evolution of the sheath is not taken into account and it is not properly normalised by initial and boundary conditions, so that the allowed input yields an FD result in the range [0%,100%]. Therefore, we limit ourselves to the analysis of the FR evolutionary properties and compare them to the ForbMod results. We calculate the expected FD amplitude at SOHO/EPHIN using the observationally constrained FR properties and energy dependence adapted for the SOHO/EPHIN response function, estimated based on the simulations performed by Kühl et al. (2015) (for details see Appendix \irefiephin). As initial FR input we use the results of the GCS reconstruction ($a_{0}=4$ $~{}\mathrm{R}_{\odot}$ at $R=17$ $~{}\mathrm{R}_{\odot}$), whereas the initial diffusion coefficient is calculated based on the FR magnetic field measured at Earth ($B=15$ nT) using the procedure explained in Appendix \irefidiff_coeff. We use the ICME transit time to Earth ($TT=74$h) as the diffusion/expansion time, and observationally obtained expansion indices ($n_{B}=1.9$, $n_{a}=0.8$; note that we use the size expansion index as obtained by the Gulisano et al. (2012) method). The results are presented in Table \irefitab1. The calculated FD amplitude ($1.7\%$) is in good agreement with the observed one ($2\%$).
We next test the assumption that in ForbMod the energy dependence can be simulated with the monoenergetic model, if one conveniently assumes the rigidity of the particles that the detector is mostly sensitive to, as was previously applied by Rodari et al. (2018) and Dumbović
et al. (2019). This is similar to the concept of effective rigidity to characterise the detector’s response (Kalugin and Kabin, 2015; Asvestari
et al., 2017; Koldobskiy
et al., 2019), where the effective rigidity is defined as the rigidity level at which the variations in CR flux are the same as the variations integrated over the entire energy range. In the monoenergetic ForbMod model we assume that the effective rigidity for a specific event observed in SOHO/EPHIN is defined by the peak of the GCR fractional contribution function, taking into account the SOHO/EPHIN response function. The GCR fractional contribution, calculated based on Equation \irefieq9 has a rigidity peak at $1.3$GV. The monoenergetic ForbMod, using the diffusion coefficient for $1.3$GV particles yields an FD amplitude of $2\%$, which is somewhat larger than the result for the energy-dependent model, but interestingly matches the observations. A possible explanation might lie in the fact that the adiabatic cooling was not included, which, if included, might shift both calculated FD amplitudes to higher values.
We now consider the FD amplitude measured in GCR fluxes with different energy ranges (or cutoffs). In order to calculate FD amplitudes measured with other instruments and at other locations, we calculate the FD amplitude assuming a “perfect detector”, i.e. a detector which responds to all energies above the cutoff equally. We note that for many detectors this assumption is invalid, particularly for neutron monitors (e.g. Clem and Dorman, 2000; Mishev et al., 2020). However, as shown in Section \irefienergy, the FD magnitude calculated for the “perfect detector” can be taken as a very rough upper limit. The response of some detectors used in the study to GCRs (MES/NS, MSL/RAD) are quite complex and it is not trivial to obtain their response function in a form that can be easily combined with ForbMod using the procedure explained in Section \irefienergy. On the other hand, a cutoff energy of each detector used in the study is known: 0.125 GeV for MES/NS (Winslow et al., 2018), 0.05 GeV for SOHO/EPHIN (Kühl et al., 2015), 0.5 GeV for SoPo (Clem and Dorman, 2000), and 0.15 GeV for MSL/RAD (Guo et al., 2018a). The calculated FD amplitude for the SOHO/EPHIN cutoff using the “perfect detector” approximation is $3.1\%$, which is almost double the value obtained using its response function ($1.7\%$). The calculated FD amplitude at MES distance (i.e. using the transit time to MES as diffusion/expansion time) for the MES/NS cutoff using the “perfect detector” approximation is $16\%$, which is larger than the observed value by about a factor of 2 - similarly as is for SOHO/EPHIN. The calculations at Earth and Mars, using the SoPo and MSL/RAD cutoffs respectively, in the “perfect detector” approximation, yield FD amplitudes $<1\%$, which is basically within the observational error (given the daily GCR variations).
We note that the “perfect detector” approximation by default increases the FD amplitude, as it assumes the same contribution of low and high-energy particles, where the fractional contribution to the total FD amplitude is substantially lower for high-energy particles. Including the response function in the calculation, which reflects lower sensitivity to low-energy particles, would thus further decrease the FD amplitude. Therefore, ForbMod calculations for SoPo and Mars are in agreement with observations, since no notable FR-related FD amplitude was measured. Applying the monoenergetic approximation to the “perfect detector” yields quite unrealistic results, as can be seen in Table \irefitab1, related to the fact that the peak of the GCR fractional contribution, is shifted to lower rigidity. We stress that the uncertainties of the observational methodology used are large. Nevertheless, we point out that ForbMod gives a quite reasonable agreement with measurements at two locations at Earth, and at Mars. In Figure \irefifig9 we show the comparison of the SoPo and SOHO/EPHIN measurements with the corresponding energy-dependent ForbMod FD profiles. We note that similar comparison cannot be performed for Mars, because we do not know the duration of the FR.
5 Conclusion and Summary
\ilabel
iconclusion
We analyse whether, and how, FDs reflect the evolutionary properties of CMEs. In order to understand the mechanisms which govern the formation of the depression, we analyse separately the interplanetary structures that are the sources of FDs and how they influence the GCRs. We first produce a generic profile of the text-book example two-step FD using three different models for three different regions (sheath, flux rope and post-CME FD recovery region), to see how well it matches the observed FD profiles. We combine two analytical models for the FD main phase (from onset to the minimum), the propagative diffusive barrier (PDB) model for the sheath region and the diffusion-expansion Forbush decrease (ForbMod) model for the CME magnetic structure. The recovery phase of the FD is modelled combining the ForbMod model for the CME magnetic structure and an exponential recovery caused by the shadow effect of the propagating shock. We find that the modelled generic FD profile describes well the observed two-step decreases, and is, in addition, also able to explain why two-step decreases are not very frequently observed. Our modelling efforts show that the transition points between different regions can be smeared out, making the whole FD appear as a homogeneous phenomenon.
We next adapt the analytical models describing the FD main phase for energy dependence, in order to compare the modelled results quantitatively with measurements. This is achieved by allowing the diffusion coefficient to be a function of rigidity as well as time. It is shown that with this new adaptation, the modelled FDs are rigidity dependent in agreement with the observations, i.e. the depression will be larger for lower energy particles. Moreover, it is shown that the contribution to the total FD from particles of different energies comes from a quite constrained energy-range of particles, due to folding of the FD energy dependence and the GCR spectrum. The distribution of the GCR fractional contribution to the total FD shows bell-curve behaviour with a distinct peak, therefore, allowing the FD calculation to be made mono-energetically, that is, by making the approximation that the main contribution to the total FD comes from particles of specific rigidities. This assumption was used for ForbMod analysis in previous studies with reasonable arguments. However, here we tested it for the first time. Comparison of the ForbMod results with full energy integration and with monoenergetic approximation for SOHO/EPHIN shows 15% difference between calculated FD amplitudes. This indicates that using the effective rigidity corresponding to the peak of the GCR fractional contribution can only be taken as a very rough approximation in calculating the total FD amplitude.
Finally, we perform an in-depth study of the multi-spacecraft event to analyse and characterise the CME evolution and apply CME magnetic structure observational characteristics to simulate the corresponding FD. We note that one of the key factors in understanding the inner structure of the CME/ICME event was the FD substructure observed by SOHO/EPHIN. Our modelling results show reasonable agreement with measurements near Earth, at Earth, and at Mars, indicating that FD models not only offer an opportunity to understand the variability of FDs detected in the heliosphere, but also to gain insight into the CME evolution.
Appendix A Diffusion coefficient
\ilabel
idiff_coeff
In order to introduce energy dependence we allow the diffusion coefficient, $D$, to be a function of rigidity as well as time, which can be expressed through an empirical formula as used in numerical models fitted to GCR measurements, as given by e.g. Potgieter (2013):
$$D_{E}(P)=0.02\cdot 10^{22}\cdot k_{\mathrm{||},0}\cdot\beta\frac{P^{a}}{B}%
\Bigg{[}\frac{P^{c}+(P_{k})^{c}}{1+(P_{k})^{c}}\Bigg{]}^{(\frac{b-a}{c})}\,,%
\ilabel{ieqA}$$
(10)
where $D_{E}$ is given in units $\mathrm{cm}^{2}\mathrm{s}^{-1}$, $P$ is rigidity in units GV, $B$ is the magnetic field in units nT, and $k_{\mathrm{||},0}$, $a$, $b$, $c$ and $P_{k}$ are parameters obtained empirically from the observation of the GCR spectrum using instruments such as Payload for Antimatter Matter Exploration and Light-nuclei Astrophysics (PAMELA, Adriani et al., 2011) on board the Russian Resurs-DK1 satellite as by (Potgieter
et al., 2014) or the Alpha Magnetic Spectrometer (AMS-02, Aguilar et al., 2013) experiment on board the International Space Station as by e.g. Corti et al. (2019). We note that the parameters and the dependence in Equation \irefieqA is slightly different for these two studies involving PAMELA and AMS. It should be noted that Potgieter
et al. (2014) studied the period of solar minimum 2006–2009, whereas Corti et al. (2019) studied the period around and after the solar maximum, 2011–2017. It is reasonable to assume that the perpendicular diffusion coefficient changes periodically with time (i.e. solar activity) not only due to the change of the IMF strength, but also the time-varying orientation and complexity of the IMF, closely related to the time-varying speeds and densities of the solar wind and reflected in the change of the parameters in Equation \irefieqA. In Figure \irefifigAa we combine the results for the perpendicular diffusion coefficient at Earth, $D_{E}$, calculated based on Potgieter
et al. (2014) and Corti et al. (2019) for magnetic field strength $B=5$ nT and rigidity $P=1$ GV, where it can be seen that the resulting $D_{E}$ varies periodically in rough anti-correlation to the solar activity indicating that the two empirical formulas corresponding to these two different time-periods can be combined. These two studies therefore provide a calculating frame for the energy-dependent diffusion coefficients. In Figure \irefifigAb we show the rigidity dependence of the diffusion coefficient at Earth in 2014 calculated in this way, as well as the rigidity dependece of the initial diffusion coefficient, estimated at $R_{0}=15\mathrm{R_{SUN}}$ based on $D_{0}=D(R(t)/R_{0})^{-n_{B}}$ assuming a magnetic field expansion factor $n_{B}=1.8$.
Appendix B The GCR spectral intensity
\ilabel
iforce_field
The “force–field” approximation is used to describe the long-term GCR modulation and is typically valid for quiet-time periods. However, Usoskin et al. (2015) have shown that the same approximation can be used to describe the GCR spectrum during an FD. In this approximation all GCR modulation mechanisms are gathered into a single parameter called the modulation potential, $\Phi$, which influences the unmodulated local interstellar spectrum $J_{\mathrm{LIS}}$ to yield the time–dependent differential energy spectrum of GCRs as observed near Earth:
$$J(E,\Phi)=J_{\mathrm{LIS}}(E+\Phi)\frac{(E)(E+2m_{0})}{(E+\Phi)(E+\Phi+2m_{0})%
}\,,\ilabel{ieqB1}$$
(11)
$$J_{\mathrm{LIS}}(E)=\frac{1.9\times 10^{4}\cdot P(E)^{-2.78}}{1+0.4866P(E)^{-2%
.51}}\,,\ilabel{ieqB2}$$
(12)
where we assume all GCRs are protons, $E$ is their kinetic energy, $m_{0}$ their rest mass, $P(E)=\sqrt{E(E+2m_{0})}$ the rigidity and $\Phi$ is the modulation potential which is time-dependent and can be obtained empirically based on GCR measurements (Usoskin, Bazilevskaya, and
Kovaltsov, 2011; Usoskin et al., 2017). However, it was shown by Gieseler, Heber, and
Herbst (2017) that it is not sufficient to describe GCR intensities at Earth by only one rigidity-independent parameter $\Phi$, as it also depends on the energy range of interest and there are severe limitations at lower energies. Therefore, we use a modified force field approach by Gieseler, Heber, and
Herbst (2017) in which the rigidity-dependent modulation parameter is given by:
$$\Phi(P)=\begin{cases}\frac{\Phi_{\mathrm{USO11}}-\Phi_{\mathrm{PP}}}{P_{%
\mathrm{USO11}}-P_{\mathrm{PP}}}\cdot(P-P_{\mathrm{PP}})+\Phi_{\mathrm{PP}},P<%
P_{\mathrm{USO11}}\\
\Phi_{\mathrm{USO11}},P\geq P_{\mathrm{USO11}}\end{cases}\ilabel{ieqB3}$$
(13)
where $\Phi_{\mathrm{USO11}}$ is the solar modulation potential obtained for neutron monitors empirically by Usoskin, Bazilevskaya, and
Kovaltsov (2011), $\Phi_{\mathrm{PP}}$ is the solar modulation potential derived from the 1.28 GV proton proxies IMP-8 helium and ACE/CRIS carbon by Gieseler, Heber, and
Herbst (2017), and $P_{\mathrm{USO11}}=13.83\pm 4.39$GV and $P_{\mathrm{PP}}=1.28\pm 0.01$GV are the corresponding mean rigidities. Equations \irefieqB1 –\irefieqB3 therefore provide a scheme to calculate GCR spectrum for a given event. We note that for the event presented in Section \irefimulti-spacecraft the uncorrected and corrected solar modulation potentials are 0.681 and 0.97 GV, respectively (Gieseler, Heber, and
Herbst, 2017).
Appendix C The SOHO/EPHIN response function
\ilabel
iephin
In order to obtain an analytical form of the response function of the single detector F of the SOHO/EPHIN instrument, we use the GEANT 4 Monte Carlo simulation of the instrument performed by Kühl et al., 2015 for the omnidirectional isotropic flux of protons, given in Figure 1 of Kühl et al. (2015). The data points representing the GEANT 4 simulation are presented in Figure \irefifigC, where a fitting function is applied in order to derive an approximate analytical form of the functional dependency:
$$R=6.4\ln^{3}(E)-0.42\ln^{2}(E)+36.8\ln(E)+288.8\ilabel{ieqC1}$$
(14)
Acknowledgments
The research leading to these results has received funding from the European Unions Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 745782 (ForbMod). B.V. and M.D. acknowledge a support by the Croatian Science Foundation under the project 7549 (MSOC). J.G. acknowledge the Strategic Priority Program of the Chinese Academy of Sciences (Grant No. XDB41000000 and XDA15017300) and the CNSA pre-research Project on Civil Aerospace Technologies (Grant No. D020104). B. H. acknowledges the discussions from the HEROIC team at the International Space Science Institute. K.D. and A.M.V. acknowledge funding by the Austrian Space Applications Programme of the Austrian Research Promotion Agency FFG: projects ASAP-11 4900217 and ASAP-14 865972. F.C. acknowledges the financial support by MINECO-FPI-2016 predoctoral grant with FSE, and its project FEDER/MCIU-AEEI/Proyecto ESP2017-88436-R. C.M. and T.A. thank the Austrian Science Fund (FWF): P31659-N27, P31521-N27, P31265-N27. We acknowledge the NMDB database (http://www.nmdb.eu) founded under the European Unions FP7 programme (contract No. 213007), and the PIs of individual neutron monitors for providing SoPo data. MESSENGER and MSL data are available on the Planetary Data System (https://pds.jpl.nasa.gov). SOHO/EPHIN is supported by the Ministry of Economics via DLR grant 50OG1702. We thank Ewan Dickson, PhD for improving the readability of the paper. Finally, we thank the anonymous reviewer whose thorough revision and insightful comments significantly improved the quality of the paper.
Disclosure of Potential Conflicts of Interest
The authors declare that they have no conflicts of interest.
References
Adriani et al. (2011)
Adriani, O.,
Barbarino, G.C.,
Bazilevskaya, G.A.,
Bellotti, R.,
Boezio, M.,
Bogomolov, E.A.,
Bonechi, L.,
Bongi, M.,
Bonvicini, V.,
Borisov, S.,
Bottai, S.,
Bruno, A.,
Cafagna, F.,
Campana, D.,
Carbone, R.,
Carlson, P.,
Casolino, M.,
Castellini, G.,
Consiglio, L.,
De Pascale, M.P.,
De Santis, C.,
De Simone, N.,
Di Felice, V.,
Galper, A.M.,
Gillard, W.,
Grishantseva, L.,
Jerse, G.,
Karelin, A.V.,
Koldashov, S.V.,
Krutkov, S.Y.,
Kvashnin, A.N.,
Leonov, A.,
Malakhov, V.,
Malvezzi, V.,
Marcelli, L.,
Mayorov, A.G.,
Menn, W.,
Mikhailov, V.V.,
Mocchiutti, E.,
Monaco, A.,
Mori, N.,
Nikonov, N.,
Osteria, G.,
Palma, F.,
Papini, P.,
Pearce, M.,
Picozza, P.,
Pizzolotto, C.,
Ricci, M.,
Ricciarini, S.B.,
Rossetto, L.,
Sarkar, R.,
Simon, M.,
Sparvoli, R.,
Spillantini, P.,
Stozhkov, Y.I.,
Vacchi, A.,
Vannuccini, E.,
Vasilyev, G.,
Voronov, S.A.,
Yurkin, Y.T.,
Wu, J.,
Zampa, G.,
Zampa, N.,
Zverev, V.G.:
2011,
PAMELA Measurements of Cosmic-Ray Proton and Helium Spectra.
Science
332(6025),
69.
DOI.
ADS.
Aguilar et al. (2013)
Aguilar, M.,
Alberti, G.,
Alpat, B.,
Alvino, A.,
Ambrosi, G.,
Andeen, K.,
Anderhub, H.,
Arruda, L.,
Azzarello, P.,
Bachlechner, A.,
Barao, F.,
Baret, B.,
Barrau, A.,
Barrin, L.,
Bartoloni, A.,
Basara, L.,
Basili, A.,
Batalha, L.,
Bates, J.,
Battiston, R.,
Bazo, J.,
Becker, R.,
Becker, U.,
Behlmann, M.,
Beischer, B.,
Berdugo, J.,
Berges, P.,
Bertucci, B.,
Bigongiari, G.,
Biland, A.,
Bindi, V.,
Bizzaglia, S.,
Boella, G.,
de Boer, W.,
Bollweg, K.,
Bolmont, J.,
Borgia, B.,
Borsini, S.,
Boschini, M.J.,
Boudoul, G.,
Bourquin, M.,
Brun, P.,
Buénerd, M.,
Burger, J.,
Burger, W.,
Cadoux, F.,
Cai, X.D.,
Capell, M.,
Casadei, D.,
Casaus, J.,
Cascioli, V.,
Castellini, G.,
Cernuda, I.,
Cervelli, F.,
Chae, M.J.,
Chang, Y.H.,
Chen, A.I.,
Chen, C.R.,
Chen, H.,
Cheng, G.M.,
Chen, H.S.,
Cheng, L.,
Chernoplyiokov, N.,
Chikanian, A.,
Choumilov, E.,
Choutko, V.,
Chung, C.H.,
Clark, C.,
Clavero, R.,
Coignet, G.,
Commichau, V.,
Consolandi, C.,
Contin, A.,
Corti, C.,
Costado Dios, M.T.,
Coste, B.,
Crespo, D.,
Cui, Z.,
Dai, M.,
Delgado, C.,
Della Torre, S.,
Demirkoz, B.,
Dennett, P.,
Derome, L.,
Di Falco, S.,
Diao, X.H.,
Diago, A.,
Djambazov, L.,
Díaz, C.,
von Doetinchem, P.,
Du, W.J.,
Dubois, J.M.,
Duperay, R.,
Duranti, M.,
D’Urso, D.,
Egorov, A.,
Eline, A.,
Eppling, F.J.,
Eronen, T.,
van Es, J.,
Esser, H.,
Falvard, A.,
Fiandrini, E.,
Fiasson, A.,
Finch, E.,
Fisher, P.,
Flood, K.,
Foglio, R.,
Fohey, M.,
Fopp, S.,
Fouque, N.,
Galaktionov, Y.,
Gallilee, M.,
Gallin-Martel, L.,
Gallucci, G.,
García, B.,
García, J.,
García-López, R.,
García-Tabares, L.,
Gargiulo, C.,
Gast, H.,
Gebauer, I.,
Gentile, S.,
Gervasi, M.,
Gillard, W.,
Giovacchini, F.,
Girard, L.,
Goglov, P.,
Gong, J.,
Goy-Henningsen, C.,
Grandi, D.,
Graziani, M.,
Grechko, A.,
Gross, A.,
Guerri, I.,
de la Guía, C.,
Guo, K.H.,
Habiby, M.,
Haino, S.,
Hauler, F.,
He, Z.H.,
Heil, M.,
Heilig, J.,
Hermel, R.,
Hofer, H.,
Huang, Z.C.,
Hungerford, W.,
Incagli, M.,
Ionica, M.,
Jacholkowska, A.,
Jang, W.Y.,
Jinchi, H.,
Jongmanns, M.,
Journet, L.,
Jungermann, L.,
Karpinski, W.,
Kim, G.N.,
Kim, K.S.,
Kirn, T.,
Kossakowski, R.,
Koulemzine, A.,
Kounina, O.,
Kounine, A.,
Koutsenko, V.,
Krafczyk, M.S.,
Laudi, E.,
Laurenti, G.,
Lauritzen, C.,
Lebedev, A.,
Lee, M.W.,
Lee, S.C.,
Leluc, C.,
León Vargas, H.,
Lepareur, V.,
Li, J.Q.,
Li, Q.,
Li, T.X.,
Li, W.,
Li, Z.H.,
Lipari, P.,
Lin, C.H.,
Liu, D.,
Liu, H.,
Lomtadze, T.,
Lu, Y.S.,
Lucidi, S.,
Lübelsmeyer, K.,
Luo, J.Z.,
Lustermann, W.,
Lv, S.,
Madsen, J.,
Majka, R.,
Malinin, A.,
Mañá, C.,
Marín, J.,
Martin, T.,
Martínez, G.,
Masciocchi, F.,
Masi, N.,
Maurin, D.,
McInturff, A.,
McIntyre, P.,
Menchaca-Rocha, A.,
Meng, Q.,
Menichelli, M.,
Mereu, I.,
Millinger, M.,
Mo, D.C.,
Molina, M.,
Mott, P.,
Mujunen, A.,
Natale, S.,
Nemeth, P.,
Ni, J.Q.,
Nikonov, N.,
Nozzoli, F.,
Nunes, P.,
Obermeier, A.,
Oh, S.,
Oliva, A.,
Palmonari, F.,
Palomares, C.,
Paniccia, M.,
Papi, A.,
Park, W.H.,
Pauluzzi, M.,
Pauss, F.,
Pauw, A.,
Pedreschi, E.,
Pensotti, S.,
Pereira, R.,
Perrin, E.,
Pessina, G.,
Pierschel, G.,
Pilo, F.,
Piluso, A.,
Pizzolotto, C.,
Plyaskin, V.,
Pochon, J.,
Pohl, M.,
Poireau, V.,
Porter, S.,
Pouxe, J.,
Putze, A.,
Quadrani, L.,
Qi, X.N.,
Rancoita, P.G.,
Rapin, D.,
Ren, Z.L.,
Ricol, J.S.,
Riihonen, E.,
Rodríguez, I.,
Roeser, U.,
Rosier-Lees, S.,
Rossi, L.,
Rozhkov, A.,
Rozza, D.,
Sabellek, A.,
Sagdeev, R.,
Sandweiss, J.,
Santos, B.,
Saouter, P.,
Sarchioni, M.,
Schael, S.,
Schinzel, D.,
Schmanau, M.,
Schwering, G.,
Schulz von Dratzig, A.,
Scolieri, G.,
Seo, E.S.,
Shan, B.S.,
Shi, J.Y.,
Shi, Y.M.,
Siedenburg, T.,
Siedling, R.,
Son, D.,
Spada, F.,
Spinella, F.,
Steuer, M.,
Stiff, K.,
Sun, W.,
Sun, W.H.,
Sun, X.H.,
Tacconi, M.,
Tang, C.P.,
Tang, X.W.,
Tang, Z.C.,
Tao, L.,
Tassan-Viol, J.,
Ting, S.C.C.,
Ting, S.M.,
Titus, C.,
Tomassetti, N.,
Toral, F.,
Torsti, J.,
Tsai, J.R.,
Tutt, J.C.,
Ulbricht, J.,
Urban, T.,
Vagelli, V.,
Valente, E.,
Vannini, C.,
Valtonen, E.,
Vargas Trevino, M.,
Vaurynovich, S.,
Vecchi, M.,
Vergain, M.,
Verlaat, B.,
Vescovi, C.,
Vialle, J.P.,
Viertel, G.,
Volpini, G.,
Wang, D.,
Wang, N.H.,
Wang, Q.L.,
Wang, R.S.,
Wang, X.,
Wang, Z.X.,
Wallraff, W.,
Weng, Z.L.,
Willenbrock, M.,
Wlochal, M.,
Wu, H.,
Wu, K.Y.,
Wu, Z.S.,
Xiao, W.J.,
Xie, S.,
Xiong, R.Q.,
Xin, G.M.,
Xu, N.S.,
Xu, W.,
Yan, Q.,
Yang, J.,
Yang, M.,
Ye, Q.H.,
Yi, H.,
Yu, Y.J.,
Yu, Z.Q.,
Zeissler, S.,
Zhang, J.G.,
Zhang, Z.,
Zhang, M.M.,
Zheng, Z.M.,
Zhuang, H.L.,
Zhukov, V.,
Zichichi, A.,
Zuccon, P.,
Zurbach, C.:
2013,
First Result from the Alpha Magnetic Spectrometer on the
International Space Station: Precision Measurement of the Positron Fraction
in Primary Cosmic Rays of 0.5-350 GeV.
Phys. Rev. Lett
110(14),
141102.
DOI.
ADS.
Al-Haddad
et al. (2019)
Al-Haddad, N.,
Poedts, S.,
Roussev, I.,
Farrugia, C.J.,
Yu, W.,
Lugaz, N.:
2019,
The Magnetic Morphology of Magnetic Clouds: Multi-spacecraft
Investigation of Twisted and Writhed Coronal Mass Ejections.
Astrophys. J.
870(2),
100.
DOI.
ADS.
Alania et al. (2013)
Alania, M.V.,
Wawrzynczak, A.,
Sdobnov, V.E.,
Kravtsova, M.V.:
2013,
Temporal Changes in the Rigidity Spectrum of Forbush Decreases Based
on Neutron Monitor Data.
Sol. Phys.
286,
561.
DOI.
ADS.
Anderson et al. (2007)
Anderson, B.J.,
Acuña, M.H.,
Lohr, D.A.,
Scheifele, J.,
Raval, A.,
Korth, H.,
Slavin, J.A.:
2007,
The Magnetometer Instrument on MESSENGER.
Space Sci. Rev.
131(1-4),
417.
DOI.
ADS.
Asvestari
et al. (2017)
Asvestari, E.,
Gil, A.,
Kovaltsov, G.A.,
Usoskin, I.G.:
2017,
Neutron Monitors and Cosmogenic Isotopes as Cosmic Ray
Energy-Integration Detectors: Effective Yield Functions, Effective Energy,
and Its Dependence on the Local Interstellar Spectrum.
J. Geophys. Res.
122(10),
9790.
DOI.
ADS.
Barnden (1973)
Barnden, L.R.:
1973,
The Large-Scale Magnetic Field Configuration Associated With Forbush
Decreases.
In: Int. Cosmic Ray Conf.,
International Cosmic Ray Conference
2,
1277.
ADS.
Belov (2009)
Belov, A.V.:
2009,
Forbush effects and their connection with solar, interplanetary and
geomagnetic phenomena.
In: Gopalswamy, N.,
Webb, D.F. (eds.)
IAU Symp.,
IAU Symp.
257,
439.
DOI.
ADS.
Belov et al. (1995)
Belov, A.V.,
Dorman, L.I.,
Eroshenko, E.A.,
Iucci, N.,
Villoresi, G.,
Yanke, V.G.:
1995,
Search for Predictors of Forbush Decreases.
In: International Cosmic Ray Conference,
International Cosmic Ray Conference
4,
888.
ADS.
Belov et al. (2015)
Belov, A.,
Abunin, A.,
Abunina, M.,
Eroshenko, E.,
Oleneva, V.,
Yanke, V.,
Papaioannou, A.,
Mavromichalaki, H.:
2015,
Galactic Cosmic Ray Density Variations in Magnetic Clouds.
Sol. Phys.
290,
1429.
DOI.
ADS.
Benella et al. (2019)
Benella, S.,
Grimani, C.,
Laurenza, M.,
Consolini, G.:
2019,
Grad-Shafranov reconstruction of a magnetic cloud: Effects of the
magnetic-field topology on the galactic cosmic-ray intensity.
Nuovo Cimento C Geophysics Space Physics C
42(1),
44.
DOI.
ADS.
Bothmer and Schwenn (1998)
Bothmer, V.,
Schwenn, R.:
1998,
The structure and origin of magnetic clouds in the solar wind.
Ann. Geophys.
16,
1.
DOI.
ADS.
Brueckner
et al. (1995)
Brueckner, G.E.,
Howard, R.A.,
Koomen, M.J.,
Korendyke, C.M.,
Michels, D.J.,
Moses, J.D.,
Socker, D.G.,
Dere, K.P.,
Lamy, P.L.,
Llebaria, A.,
Bout, M.V.,
Schwenn, R.,
Simnett, G.M.,
Bedford, D.K.,
Eyles, C.J.:
1995,
The Large Angle Spectroscopic Coronagraph (LASCO).
Sol. Phys.
162,
357.
DOI.
ADS.
Burlaga et al. (2003)
Burlaga, L.,
Berdichevsky, D.,
Gopalswamy, N.,
Lepping, R.,
Zurbuchen, T.:
2003,
Merged interaction regions at 1 AU.
J. Geophys. Res.
108,
1425.
DOI.
ADS.
Caballero-Lopez and
Moraal (2004)
Caballero-Lopez, R.A.,
Moraal, H.:
2004,
Limitations of the force field equation to describe cosmic ray
modulation.
J. Geophys. Res.
109,
1101.
DOI.
ADS.
Cane (2000)
Cane, H.V.:
2000,
Coronal Mass Ejections and Forbush Decreases.
Space Sci. Rev.
93,
55.
DOI.
ADS.
Cane, Richardson, and
Wibberenz (1995)
Cane, H.V.,
Richardson, I.G.,
Wibberenz, G.:
1995,
The Response of Energetic Particles to the Presence of Ejecta
Material.
Int. Cosmic Ray Conf.
4,
377.
ADS.
Cargill et al. (1996)
Cargill, P.J.,
Chen, J.,
Spicer, D.S.,
Zalesak, S.T.:
1996,
Magnetohydrodynamic simulations of the motion of magnetic flux tubes
through a magnetized plasma.
J. Geophys. Res.
101,
4855.
DOI.
ADS.
Chih and Lee (1986)
Chih, P.P.,
Lee, M.A.:
1986,
A perturbation approach to cosmic ray transients in interplanetary
space.
J. Geophys. Res.
91,
2903.
DOI.
ADS.
Clem and Dorman (2000)
Clem, J.M.,
Dorman, L.I.:
2000,
Neutron Monitor Response Functions.
Space Sci. Rev.
93,
335.
DOI.
ADS.
Corti et al. (2019)
Corti, C.,
Potgieter, M.S.,
Bindi, V.,
Consolandi, C.,
Light, C.,
Palermo, M.,
Popkow, A.:
2019,
Numerical Modeling of Galactic Cosmic-Ray Proton and Helium Observed
by AMS-02 during the Solar Maximum of Solar Cycle 24.
Astrophys. J.
871(2),
253.
DOI.
ADS.
Démoulin and Dasso (2009)
Démoulin, P.,
Dasso, S.:
2009,
Causes and consequences of magnetic cloud expansion.
Astron. Astrophys.
498(2),
551.
DOI.
ADS.
Démoulin
et al. (2008)
Démoulin, P.,
Nakwacki, M.S.,
Dasso, S.,
Mandrini, C.H.:
2008,
Expected in Situ Velocities from a Hierarchical Model for Expanding
Interplanetary Coronal Mass Ejections.
Sol. Phys.
250,
347.
DOI.
ADS.
Dissauer
et al. (2018)
Dissauer, K.,
Veronig, A.M.,
Temmer, M.,
Podladchikova, T.,
Vanninathan, K.:
2018,
On the Detection of Coronal Dimmings and the Extraction of Their
Characteristic Properties.
Astrophys. J.
855(2),
137.
DOI.
ADS.
Dumbović
et al. (2011)
Dumbović, M.,
Vršnak, B.,
Čalogović, J.,
Karlica, M.:
2011,
Cosmic ray modulation by solar wind disturbances.
Astron. Astrophys.
531,
A91+.
DOI.
ADS.
Dumbović
et al. (2012)
Dumbović, M.,
Vršnak, B.,
Čalogović, J.,
Župan, R.:
2012,
Cosmic ray modulation by different types of solar wind
disturbances.
Astron. Astrophys.
538,
A28.
DOI.
ADS.
Dumbović
et al. (2018)
Dumbović, M.,
Heber, B.,
Vršnak, B.,
Temmer, M.,
Kirin, A.:
2018,
An Analytical Diffusion-Expansion Model for Forbush Decreases Caused
by Flux Ropes.
Astrophys. J.
860,
71.
DOI.
ADS.
Dumbović
et al. (2019)
Dumbović, M.,
Guo, J.,
Temmer, M.,
Mays, M.L.,
Veronig, A.,
Heinemann, S.G.,
Dissauer, K.,
Hofmeister, S.,
Halekas, J.,
Möstl, C.,
Amerstorfer, T.,
Hinterreiter, J.,
Banjac, S.,
Herbst, K.,
Wang, Y.,
Holzknecht, L.,
Leitner, M.,
Wimmer–Schweingruber, R.F.:
2019,
Unusual Plasma and Particle Signatures at Mars and STEREO-A Related
to CME–CME Interaction.
Astrophys. J.
880,
18.
DOI.
ADS.
Freiherr von Forstner
et al. (2018)
Freiherr von Forstner, J.L.,
Guo, J.,
Wimmer-Schweingruber, R.F.,
Hassler, D.M.,
Temmer, M.,
Dumbović, M.,
Jian, L.K.,
Appel, J.K.,
Čalogović, J.,
Ehresmann, B.,
Heber, B.,
Lohf, H.,
Posner, A.,
Steigies, C.T.,
Vršnak, B.,
Zeitlin, C.J.:
2018,
Using Forbush Decreases to Derive the Transit Time of ICMEs
Propagating from 1 AU to Mars.
J. Geophys. Res.
123,
39.
DOI.
ADS.
Freiherr von Forstner
et al. (2019)
Freiherr von Forstner, J.L.,
Guo, J.,
Wimmer-Schweingruber, R.F.,
Temmer, M.,
Dumbović,
, M.,
Veronig, A.,
Möstl, C.,
Hassler, D.M.,
Zeitlin, C.J.,
Ehresmann, B.:
2019,
Tracking and Validating ICMEs Propagating Toward Mars Using STEREO
Heliospheric Imagers Combined With Forbush Decreases Detected by MSL/RAD.
Space Weather
17(4),
586.
DOI.
ADS.
Freiherr von Forstner
et al. (2020)
Freiherr von Forstner, J.L.,
Guo, J.,
Wimmer-Schweingruber, R.F.,
Dumbović, M.,
Janvier, M.,
Démoulin, P.,
Veronig, A.,
Temmer, M.,
Papaioannou, A.,
Dasso, S.,
Hassler, D.M.,
Zeitlin, C.J.:
2020,
Comparing the Properties of ICME-Induced Forbush Decreases at Earth
and Mars.
J. Geophys. Res.
125(3),
e27662.
DOI.
ADS.
Gieseler and Heber (2016)
Gieseler, J.,
Heber, B.:
2016,
Spatial gradients of GCR protons in the inner heliosphere derived
from Ulysses COSPIN/KET and PAMELA measurements.
Astron. Astrophys.
589,
A32.
DOI.
ADS.
Gieseler, Heber, and
Herbst (2017)
Gieseler, J.,
Heber, B.,
Herbst, K.:
2017,
An Empirical Modification of the Force Field Approach to Describe the
Modulation of Galactic Cosmic Rays Close to Earth in a Broad Range of
Rigidities.
J. Geophys. Res.
122(11),
10,964.
DOI.
ADS.
Gleeson and Axford (1968)
Gleeson, L.J.,
Axford, W.I.:
1968,
Solar Modulation of Galactic Cosmic Rays.
Astrophys. J.
154,
1011.
DOI.
ADS.
Gleeson and Urch (1971)
Gleeson, L.J.,
Urch, I.H.:
1971,
Energy losses and modulation of galactic cosmic rays.
Astrophys. & Space Sci.
11,
288.
DOI.
ADS.
Gloeckler
et al. (1998)
Gloeckler, G.,
Cain, J.,
Ipavich, F.M.,
Tums, E.O.,
Bedini, P.,
Fisk, L.A.,
Zurbuchen, T.H.,
Bochsler, P.,
Fischer, J.,
Wimmer-Schweingruber, R.F.,
Geiss, J.,
Kallenbach, R.:
1998,
Investigation of the composition of solar and interstellar matter
using solar wind and pickup ion measurements with SWICS and SWIMS on the ACE
spacecraft.
Space Sci. Rev.
86,
497.
DOI.
ADS.
Gold and Hoyle (1960)
Gold, T.,
Hoyle, F.:
1960,
On the origin of solar flares.
Mon. Not. R. Astron. Soc.
120,
89.
DOI.
ADS.
Goldsten et al. (2007)
Goldsten, J.O.,
Rhodes, E.A.,
Boynton, W.V.,
Feldman, W.C.,
Lawrence, D.J.,
Trombka, J.I.,
Smith, D.M.,
Evans, L.G.,
White, J.,
Madden, N.W.,
Berg, P.C.,
Murphy, G.A.,
Gurnee, R.S.,
Strohbehn, K.,
Williams, B.D.,
Schaefer, E.D.,
Monaco, C.A.,
Cork, C.P.,
Del Eckels, J.,
Miller, W.O.,
Burks, M.T.,
Hagler, L.B.,
Deteresa, S.J.,
Witte, M.C.:
2007,
The MESSENGER Gamma-Ray and Neutron Spectrometer.
Space Sci. Rev.
131(1-4),
339.
DOI.
ADS.
Gosling et al. (1987)
Gosling, J.T.,
Baker, D.N.,
Bame, S.J.,
Feldman, W.C.,
Zwickl, R.D.,
Smith, E.J.:
1987,
Bidirectional solar wind electron heat flux events.
J. Geophys. Res.
92(A8),
8519.
DOI.
ADS.
Grotzinger
et al. (2012)
Grotzinger, J.P.,
Crisp, J.,
Vasavada, A.R.,
Anderson, R.C.,
Baker, C.J.,
Barry, R.,
Blake, D.F.,
Conrad, P.,
Edgett, K.S.,
Ferdowski, B.,
Gellert, R.,
Gilbert, J.B.,
Golombek, M.,
Gómez-Elvira, J.,
Hassler, D.M.,
Jandura, L.,
Litvak, M.,
Mahaffy, P.,
Maki, J.,
Meyer, M.,
Malin, M.C.,
Mitrofanov, I.,
Simmonds, J.J.,
Vaniman, D.,
Welch, R.V.,
Wiens, R.C.:
2012,
Mars Science Laboratory Mission and Science Investigation.
Space Sci. Rev.
170,
5.
DOI.
ADS.
Gulisano et al. (2012)
Gulisano, A.M.,
Démoulin, P.,
Dasso, S.,
Rodriguez, L.:
2012,
Expansion of magnetic clouds in the outer heliosphere.
Astron. Astrophys.
543,
A107.
DOI.
ADS.
Guo et al. (2018a)
Guo, J.,
Zeitlin, C.,
Wimmer-Schweingruber, R.F.,
McDole, T.,
Kühl, P.,
Appel, J.C.,
Matthiä, D.,
Krauss, J.,
Köhler, J.:
2018a,
A Generalized Approach to Model the Spectra and Radiation Dose Rate
of Solar Particle Events on the Surface of Mars.
Astron. J.
155(1),
49.
DOI.
ADS.
Guo et al. (2018b)
Guo, J.,
Dumbović, M.,
Wimmer-Schweingruber, R.F.,
Temmer, M.,
Lohf, H.,
Wang, Y.,
Veronig, A.,
Hassler, D.M.,
Mays, L.M.,
Zeitlin, C.,
Ehresmann, B.,
Witasse, O.,
Freiherr von Forstner, J.L.,
Heber, B.,
Holmström, M.,
Posner, A.:
2018b,
Modeling the Evolution and Propagation of 10 September 2017 CMEs and
SEPs Arriving at Mars Constrained by Remote Sensing and In Situ
Measurement.
Space Weather
16,
1156.
DOI.
ADS.
Guo et al. (2020)
Guo, J.,
Wimmer-Schweingruber, R.F.,
Dumbović, M.,
Heber, B.,
Wang, Y.:
2020,
A new model describing forbush decreases at mars: combining the
heliospheric modulation and the atmospheric influence.
Earth and Planetary Physics
4(1),
62.
DOI.
https://agupubs.onlinelibrary.wiley.com/doi/abs/10.26464/epp2020007.
Hassler et al. (2012)
Hassler, D.M.,
Zeitlin, C.,
Wimmer-Schweingruber, R.F.,
Böttcher, S.,
Martin, C.,
Andrews, J.,
Böhm, E.,
Brinza, D.E.,
Bullock, M.A.,
Burmeister, S.,
Ehresmann, B.,
Epperly, M.,
Grinspoon, D.,
Köhler, J.,
Kortmann, O.,
Neal, K.,
Peterson, J.,
Posner, A.,
Rafkin, S.,
Seimetz, L.,
Smith, K.D.,
Tyler, Y.,
Weigle, G.,
Reitz, G.,
Cucinotta, F.A.:
2012,
The Radiation Assessment Detector (RAD) Investigation.
Space Sci. Rev.
170,
503.
DOI.
ADS.
Herbst et al. (2010)
Herbst, K.,
Kopp, A.,
Heber, B.,
Steinhilber, F.,
Fichtner, H.,
Scherer, K.,
Matthiä, D.:
2010,
On the importance of the local interstellar spectrum for the solar
modulation parameter.
J. Geophys. Res.
115,
0.
DOI.
ADS.
Howard et al. (2008)
Howard, R.A.,
Moses, J.D.,
Vourlidas, A.,
Newmark, J.S.,
Socker, D.G.,
Plunkett, S.P.,
Korendyke, C.M.,
Cook, J.W.,
Hurley, A.,
Davila, J.M.,
Thompson, W.T.,
St Cyr, O.C.,
Mentzell, E.,
Mehalick, K.,
Lemen, J.R.,
Wuelser, J.P.,
Duncan, D.W.,
Tarbell, T.D.,
Wolfson, C.J.,
Moore, A.,
Harrison, R.A.,
Waltham, N.R.,
Lang, J.,
Davis, C.J.,
Eyles, C.J.,
Mapson-Menard, H.,
Simnett, G.M.,
Halain, J.P.,
Defise, J.M.,
Mazy, E.,
Rochus, P.,
Mercier, R.,
Ravet, M.F.,
Delmotte, F.,
Auchere, F.,
Delaboudiniere, J.P.,
Bothmer, V.,
Deutsch, W.,
Wang, D.,
Rich, N.,
Cooper, S.,
Stephens, V.,
Maahs, G.,
Baugh, R.,
McMullin, D.,
Carter, T.:
2008,
Sun Earth Connection Coronal and Heliospheric Investigation
(SECCHI).
Space Sci. Rev.
136,
67.
DOI.
ADS.
Hu and Sonnerup (2002)
Hu, Q.,
Sonnerup, B.U.Ö.:
2002,
Reconstruction of magnetic clouds in the solar wind: Orientations and
configurations.
J. Geophys. Res.
107(A7),
1142.
DOI.
ADS.
Hu et al. (2017)
Hu, Q.,
Linton, M.G.,
Wood, B.E.,
Riley, P.,
Nieves-Chinchilla, T.:
2017,
The Grad-Shafranov Reconstruction of Toroidal Magnetic Flux Ropes:
First Applications.
Sol. Phys.
292(11),
171.
DOI.
ADS.
Hudson, Acton, and
Freeland (1996)
Hudson, H.S.,
Acton, L.W.,
Freeland, S.L.:
1996,
A Long-Duration Solar Flare with Mass Ejection and Global
Consequences.
Astrophys. J.
470,
629.
DOI.
ADS.
Jämsén
et al. (2007)
Jämsén, T.,
Usoskin, I.G.,
Räihä, T.,
Sarkamo, J.,
Kovaltsov, G.A.:
2007,
Case study of Forbush decreases: Energy dependence of the recovery.
Adv. Space Res.
40,
342.
DOI.
ADS.
Janvier et al. (2019)
Janvier, M.,
Winslow, R.M.,
Good, S.,
Bonhomme, E.,
Démoulin, P.,
Dasso, S.,
Möstl, C.,
Lugaz, N.,
Amerstorfer, T.,
Soubrié, E.,
Boakes, P.D.:
2019,
Generic Magnetic Field Intensity Profiles of Interplanetary Coronal
Mass Ejections at Mercury, Venus, and Earth From Superposed Epoch Analyses.
J. Geophys. Res.
124(2),
812.
DOI.
ADS.
Jokipii (1971)
Jokipii, J.R.:
1971,
Propagation of cosmic rays in the solar wind.
Reviews of Geophysics and Space Physics
9,
27.
DOI.
ADS.
Kalugin and Kabin (2015)
Kalugin, G.,
Kabin, K.:
2015,
Estimation of index of power law rigidity spectrum of cosmic rays
using effective rigidity of multidirectional muon detector.
Earth Planet. & Space
67,
149.
DOI.
ADS.
Kilpua, Koskinen, and
Pulkkinen (2017)
Kilpua, E.,
Koskinen, H.E.J.,
Pulkkinen, T.I.:
2017,
Coronal mass ejections and their sheath regions in interplanetary
space.
Living Reviews in Solar Physics
14,
5.
DOI.
ADS.
Kirin et al. (2020)
Kirin, A.,
Vršnak, B.,
Dumbović, M.,
Heber, B.:
2020,
On the Interaction of Galactic Cosmic Rays with Heliospheric Shocks
During Forbush Decreases.
Sol. Phys.
295,
28.
DOI.
Klein and Burlaga (1982)
Klein, L.W.,
Burlaga, L.F.:
1982,
Interplanetary magnetic clouds at 1 AU.
J. Geophys. Res.
87(A2),
613.
DOI.
ADS.
Koldobskiy
et al. (2019)
Koldobskiy, S.A.,
Kovaltsov, G.A.,
Mishev, A.L.,
Usoskin, I.G.:
2019,
New Method of Assessment of the Integral Fluence of Solar Energetic
(> 1 GV Rigidity) Particles from Neutron Monitor Data.
Sol. Phys.
294(7),
94.
DOI.
ADS.
Krittinatham and
Ruffolo (2009)
Krittinatham, W.,
Ruffolo, D.:
2009,
Drift Orbits of Energetic Particles in an Interplanetary Magnetic
Flux Rope.
Astrophys. J.
704,
831.
DOI.
ADS.
Kühl et al. (2015)
Kühl, P.,
Banjac, S.,
Heber, B.,
Labrenz, J.,
Müller-Mellin, R.,
Terasa, C.:
2015,
Extended Measurement Capabilities of the Electron Proton Helium
INstrument aboard SOHO - Understanding single detector count rates.
Cent. European Astrophys. Bull.
39,
119.
ADS.
Lawrence et al. (2016)
Lawrence, D.J.,
Peplowski, P.N.,
Feldman, W.C.,
Schwadron, N.A.,
Spence, H.E.:
2016,
Galactic cosmic ray variations in the inner heliosphere from solar
distances less than 0.5 AU: Measurements from the MESSENGER Neutron
Spectrometer.
J. Geophys. Res.
121,
7398.
DOI.
ADS.
Le Roux and Potgieter (1991)
Le Roux, J.A.,
Potgieter, M.S.:
1991,
The simulation of Forbush decreases with time-dependent cosmic-ray
modulation models of varying complexity.
Astron. Astrophys.
243,
531.
ADS.
Leitner et al. (2007)
Leitner, M.,
Farrugia, C.J.,
MöStl, C.,
Ogilvie, K.W.,
Galvin, A.B.,
Schwenn, R.,
Biernat, H.K.:
2007,
Consequences of the force-free model of magnetic clouds for their
heliospheric evolution.
J. Geophys. Res.
112,
A06113.
DOI.
ADS.
Lemen et al. (2012)
Lemen, J.R.,
Title, A.M.,
Akin, D.J.,
Boerner, P.F.,
Chou, C.,
Drake, J.F.,
Duncan, D.W.,
Edwards, C.G.,
Friedlaender, F.M.,
Heyman, G.F.,
Hurlburt, N.E.,
Katz, N.L.,
Kushner, G.D.,
Levay, M.,
Lindgren, R.W.,
Mathur, D.P.,
McFeaters, E.L.,
Mitchell, S.,
Rehse, R.A.,
Schrijver, C.J.,
Springer, L.A.,
Stern, R.A.,
Tarbell, T.D.,
Wuelser, J.-P.,
Wolfson, C.J.,
Yanari, C.,
Bookbinder, J.A.,
Cheimets, P.N.,
Caldwell, D.,
Deluca, E.E.,
Gates, R.,
Golub, L.,
Park, S.,
Podgorski, W.A.,
Bush, R.I.,
Scherrer, P.H.,
Gummin, M.A.,
Smith, P.,
Auker, G.,
Jerram, P.,
Pool, P.,
Soufli, R.,
Windt, D.L.,
Beardsley, S.,
Clapp, M.,
Lang, J.,
Waltham, N.:
2012,
The Atmospheric Imaging Assembly (AIA) on the Solar Dynamics
Observatory (SDO).
Sol. Phys.
275,
17.
DOI.
ADS.
Lepping et al. (1995)
Lepping, R.P.,
Acũna, M.H.,
Burlaga, L.F.,
Farrell, W.M.,
Slavin, J.A.,
Schatten, K.H.,
Mariani, F.,
Ness, N.F.,
Neubauer, F.M.,
Whang, Y.C.,
Byrnes, J.B.,
Kennon, R.S.,
Panetta, P.V.,
Scheifele, J.,
Worley, E.M.:
1995,
The Wind Magnetic Field Investigation.
Space Sci. Rev.
71,
207.
DOI.
ADS.
Lepri et al. (2001)
Lepri, S.T.,
Zurbuchen, T.H.,
Fisk, L.A.,
Richardson, I.G.,
Cane, H.V.,
Gloeckler, G.:
2001,
Iron charge distribution as an identifier of interplanetary coronal
mass ejections.
J. Geophys. Res.
106(A12),
29231.
DOI.
ADS.
Lingri et al. (2019)
Lingri, D.,
Mavromichalaki, H.,
Belov, A.,
Abunina, M.,
Eroshenko, E.,
Abunin, A.:
2019,
An Extended Study of the Precursory Signs of Forbush Decreases: New
Findings over the Years 2008 - 2016.
Sol. Phys.
294(6),
70.
DOI.
ADS.
Lockwood (1971)
Lockwood, J.A.:
1971,
Forbush decreases in the cosmic radiation.
Space Sci. Rev.
12,
658.
10.1007/BF00173346.
http://dx.doi.org/10.1007/BF00173346.
Lockwood, Webber, and
Jokipii (1986)
Lockwood, J.A.,
Webber, W.R.,
Jokipii, J.R.:
1986,
Characteristic recovery times of Forbush-type decreases in the cosmic
radiation. I - Observations at earth at different energies.
J. Geophys. Res.
91,
2851.
DOI.
ADS.
Lopez (1987)
Lopez, R.E.:
1987,
Solar cycle invariance in solar wind proton temperature
relationships.
J. Geophys. Res.
92(A10),
11189.
DOI.
ADS.
Lugaz, Winslow, and
Farrugia (2020)
Lugaz, N.,
Winslow, R.M.,
Farrugia, C.J.:
2020,
Evolution of a Long-Duration Coronal Mass Ejection and Its Sheath
Region Between Mercury and Earth on 9-14 July 2013.
Journal of Geophysical Research (Space Physics)
125(1),
e27213.
DOI.
ADS.
Lugaz et al. (2017)
Lugaz, N.,
Temmer, M.,
Wang, Y.,
Farrugia, C.J.:
2017,
The Interaction of Successive Coronal Mass Ejections: A Review.
Sol. Phys.
292,
64.
DOI.
ADS.
Lundquist (1951)
Lundquist, S.:
1951,
On the Stability of Magneto-Hydrostatic Fields.
Physical Review
83,
307.
DOI.
ADS.
Mandrini et al. (2005)
Mandrini, C.H.,
Pohjolainen, S.,
Dasso, S.,
Green, L.M.,
Démoulin, P.,
van Driel-Gesztelyi, L.,
Copperwheat, C.,
Foley, C.:
2005,
Interplanetary flux rope ejected from an X-ray bright point. The
smallest magnetic cloud source-region ever observed.
Astron. Astrophys.
434(2),
725.
DOI.
ADS.
Masías-Meza
et al. (2016)
Masías-Meza, J.J.,
Dasso, S.,
Démoulin, P.,
Rodriguez, L.,
Janvier, M.:
2016,
Superposed epoch study of ICME sub-structures near Earth and their
effects on Galactic cosmic rays.
Astron. Astrophys.
592,
A118.
DOI.
ADS.
Mishev et al. (2020)
Mishev, A.L.,
Koldobskiy, S.A.,
Kovaltsov, G.A.,
Gil, A.,
Usoskin, I.G.:
2020,
Updated Neutron-Monitor Yield Function: Bridging Between In Situ and
Ground-Based Cosmic Ray Measurements.
J. Geophys. Res.
125(2),
e27433.
DOI.
ADS.
Moraal (2013)
Moraal, H.:
2013,
Cosmic-Ray Modulation Equations.
Space Sci. Rev.
176,
299.
DOI.
ADS.
Moraal, Belov, and
Clem (2000)
Moraal, H.,
Belov, A.,
Clem, J.M.:
2000,
Design and co-Ordination of Multi-Station International Neutron
Monitor Networks.
Space Sci. Rev.
93,
285.
DOI.
ADS.
Möstl et al. (2009)
Möstl, C.,
Farrugia, C.J.,
Biernat, H.K.,
Leitner, M.,
Kilpua, E.K.J.,
Galvin, A.B.,
Luhmann, J.G.:
2009,
Optimized Grad - Shafranov Reconstruction of a Magnetic Cloud Using
STEREO- Wind Observations.
Sol. Phys.
256(1-2),
427.
DOI.
ADS.
Müller-Mellin
et al. (1995)
Müller-Mellin, R.,
Kunow, H.,
Fleißner, V.,
Pehlke, E.,
Rode, E.,
Röschmann, N.,
Scharmberg, C.,
Sierks, H.,
Rusznyak, P.,
McKenna-Lawlor, S.,
Elendt, I.,
Sequeiros, J.,
Meziat, D.,
Sanchez, S.,
Medina, J.,
Del Peral, L.,
Witte, M.,
Marsden, R.,
Henrion, J.:
1995,
COSTEP - Comprehensive Suprathermal and Energetic Particle
Analyser.
Sol. Phys.
162,
483.
DOI.
ADS.
Munakata et al. (2006)
Munakata, K.,
Yasue, S.,
Kato, C.,
Kota, J.,
Tokumaru, M.,
Kojima, M.,
Darwish, A.A.,
Kuwabara, T.,
Bieber, J.W.:
2006,
On the Cross-Field Diffusion of Galactic Cosmic Rays into an ICME.
Advances in Geosciences, Volume 2: Solar Terrestrial (ST)
2,
115.
DOI.
ADS.
Munini et al. (2018)
Munini, R.,
Boezio, M.,
Bruno, A.,
Christian, E.C.,
de Nolfo, G.A.,
Di Felice, V.,
Martucci, M.,
Merge’, M.,
Richardson, I.G.,
Ryan, J.M.,
Stochaj, S.,
Adriani, O.,
Barbarino, G.C.,
Bazilevskaya, G.A.,
Bellotti, R.,
Bongi, M.,
Bonvicini, V.,
Bottai, S.,
Cafagna, F.,
Campana, D.,
Carlson, P.,
Casolino, M.,
Castellini, G.,
De Santis, C.,
Galper, A.M.,
Karelin, A.V.,
Koldashov, S.V.,
Koldobskiy, S.,
Krutkov, S.Y.,
Kvashnin, A.N.,
Leonov, A.,
Malakhov, V.,
Marcelli, L.,
Mayorov, A.G.,
Menn, W.,
Mikhailov, V.V.,
Mocchiutti, E.,
Monaco, A.,
Mori, N.,
Osteria, G.,
Panico, B.,
Papini, P.,
Pearce, M.,
Picozza, P.,
Ricci, M.,
Ricciarini, S.B.,
Simon, M.,
Sparvoli, R.,
Spillantini, P.,
Stozhkov, Y.I.,
Vacchi, A.,
Vannuccini, E.,
Vasilyev, G.,
Voronov, S.A.,
Yurkin, Y.T.,
Zampa, G.,
Zampa, N.,
Potgieter, M.S.:
2018,
Evidence of Energy and Charge Sign Dependence of the Recovery Time
for the 2006 December Forbush Event Measured by the PAMELA Experiment.
Astrophys. J.
853(1),
76.
DOI.
ADS.
Nieves-Chinchilla
et al. (2018)
Nieves-Chinchilla, T.,
Vourlidas, A.,
Raymond, J.C.,
Linton, M.G.,
Al-haddad, N.,
Savani, N.P.,
Szabo, A.,
Hidalgo, M.A.:
2018,
Understanding the Internal Magnetic Field Configurations of ICMEs
Using More than 20 Years of Wind Observations.
Sol. Phys.
293(2),
25.
DOI.
ADS.
Ogilvie et al. (1995)
Ogilvie, K.W.,
Chornay, D.J.,
Fritzenreiter, R.J.,
Hunsaker, F.,
Keller, J.,
Lobell, J.,
Miller, G.,
Scudder, J.D.,
Sittler, E.C. Jr.,
Torbert, R.B.,
Bodet, D.,
Needell, G.,
Lazarus, A.J.,
Steinberg, J.T.,
Tappan, J.H.,
Mavretic, A.,
Gergin, E.:
1995,
SWE, A Comprehensive Plasma Instrument for the Wind Spacecraft.
Space Sci. Rev.
71,
55.
DOI.
ADS.
Owens et al. (2005)
Owens, M.J.,
Cargill, P.J.,
Pagel, C.,
Siscoe, G.L.,
Crooker, N.U.:
2005,
Characteristic magnetic field and speed properties of interplanetary
coronal mass ejections and their sheath regions.
J. Geophys. Res.
110(A1),
A01105.
DOI.
ADS.
Papailiou
et al. (2012)
Papailiou, M.,
Mavromichalaki, H.,
Belov, A.,
Eroshenko, E.,
Yanke, V.:
2012,
Precursor Effects in Different Cases of Forbush Decreases.
Sol. Phys.
276(1-2),
337.
DOI.
ADS.
Papaioannou (2019)
Papaioannou, A.e.a.:
2019,
A catalogue of Forbush decreases recorded on the surface of Mars from 2012
until 2016: comparison with terrestrial FDs.
In prep..
Parker (1965)
Parker, E.N.:
1965,
The passage of energetic charged particles through interplanetary
space.
Planet. Space. Sci.
13,
9.
DOI.
ADS.
Paularena
et al. (2001)
Paularena, K.I.,
Wang, C.,
von Steiger, R.,
Heber, B.:
2001,
An ICME observed by Voyager 2 at 58 AU and by Ulysses at 5 AU.
Geophys. Res. Lett.
28(14),
2755.
DOI.
ADS.
Penna and Quillen (2005)
Penna, R.F.,
Quillen, A.C.:
2005,
Decay of interplanetary coronal mass ejections and Forbush decrease
recovery times.
J. Geophys. Res.
110(A9),
9.
DOI.
ADS.
Petukhova, Petukhov, and
Petukhov (2019a)
Petukhova, A.S.,
Petukhov, I.S.,
Petukhov, S.I.:
2019a,
Image of Forbush Decrease in a Magnetic Cloud by Three Moments of
Cosmic Ray Distribution Function.
J. Geophys. Res.
124(1),
19.
DOI.
ADS.
Petukhova, Petukhov, and
Petukhov (2019b)
Petukhova, A.S.,
Petukhov, I.S.,
Petukhov, S.I.:
2019b,
Theory of the Formation of Forbush Decrease in a Magnetic Cloud:
Dependence of Forbush Decrease Characteristics on Magnetic Cloud
Parameters.
Astrophys. J.
880(1),
17.
DOI.
ADS.
Pinter (1980)
Pinter, S.:
1980,
The Thickness of Interplanetary Collisionless Shock Waves.
Bulletin of the Astronomical Institutes of Czechoslovakia
31,
368.
ADS.
Potgieter (2013)
Potgieter, M.S.:
2013,
Solar Modulation of Cosmic Rays.
Living Reviews in Solar Physics
10,
3.
DOI.
ADS.
Potgieter
et al. (2014)
Potgieter, M.S.,
Vos, E.E.,
Boezio, M.,
De Simone, N.,
Di Felice, V.,
Formato, V.:
2014,
Modulation of Galactic Protons in the Heliosphere During the Unusual
Solar Minimum of 2006 to 2009.
Sol. Phys.
289,
391.
DOI.
ADS.
Richardson (2004)
Richardson, I.G.:
2004,
Energetic Particles and Corotating Interaction Regions in the Solar
Wind.
Space Sci. Rev.
111,
267.
DOI.
ADS.
Richardson and Cane (1995)
Richardson, I.G.,
Cane, H.V.:
1995,
Regions of abnormally low proton temperature in the solar wind
(1965-1991) and their association with ejecta.
J. Geophys. Res.
100(A12),
23397.
DOI.
ADS.
Richardson and Cane (2010)
Richardson, I.G.,
Cane, H.V.:
2010,
Near-Earth Interplanetary Coronal Mass Ejections During Solar Cycle
23 (1996 - 2009): Catalog and Summary of Properties.
Sol. Phys.
264,
189.
DOI.
ADS.
Richardson and
Cane (2011a)
Richardson, I.G.,
Cane, H.V.:
2011a,
Galactic Cosmic Ray Intensity Response to Interplanetary Coronal Mass
Ejections/Magnetic Clouds in 1995 - 2009.
Sol. Phys.
270,
609.
DOI.
ADS.
Richardson and
Cane (2011b)
Richardson, I.G.,
Cane, H.V.:
2011b,
Geoeffectiveness (Dst and Kp) of interplanetary coronal mass
ejections during 1995-2009 and implications for storm forecasting.
Space Weather
9,
7005.
DOI.
ADS.
Rodari et al. (2018)
Rodari, M.,
Dumbović, M.,
Temmer, M.,
Holzknecht, L.,
Veronig, A.:
2018,
3D reconstruction and interplanetary expansion of the 2010 April
$3^{\mathrm{rd}}$ CME.
Central European Astrophysical Bulletin
42,
11.
ADS.
Rouillard (2011)
Rouillard, A.P.:
2011,
Relating white light and in situ observations of coronal mass
ejections: A review.
J. Atmos. Sol. Terr. Phys.
73,
1201.
DOI.
ADS.
Russell and Mulligan (2002)
Russell, C.T.,
Mulligan, T.:
2002,
On the magnetosheath thicknesses of interplanetary coronal mass
ejections.
Planet. Space. Sci.
50,
527.
DOI.
ADS.
Scolini et al. (2020)
Scolini, C.,
Chané, E.,
Temmer, M.,
Kilpua, E.K.J.,
Dissauer, K.,
Veronig, A.M.,
Palmerio, E.,
Pomoell, J.,
Dumbović, M.,
Guo, J.,
Rodriguez, L.,
Poedts, S.:
2020,
CME–CME interactions as sources of CME
geoeffectiveness: The formation of the complex ejecta and intense geomagnetic
storm in 2017 early september.
Astrophys. J. Suppl.
247(1),
21.
DOI.
https://doi.org/10.3847%2F1538-4365%2Fab6216.
Spence et al. (2010)
Spence, H.E.,
Case, A.W.,
Golightly, M.J.,
Heine, T.,
Larsen, B.A.,
Blake, J.B.,
Caranza, P.,
Crain, W.R.,
George, J.,
Lalic, M.,
Lin, A.,
Looper, M.D.,
Mazur, J.E.,
Salvaggio, D.,
Kasper, J.C.,
Stubbs, T.J.,
Doucette, M.,
Ford, P.,
Foster, R.,
Goeke, R.,
Gordon, D.,
Klatt, B.,
O’Connor, J.,
Smith, M.,
Onsager, T.,
Zeitlin, C.,
Townsend, L.W.,
Charara, Y.:
2010,
CRaTER: The Cosmic Ray Telescope for the Effects of Radiation
Experiment on the Lunar Reconnaissance Orbiter Mission.
Space Sci. Rev.
150(1-4),
243.
DOI.
ADS.
Subramanian
et al. (2009)
Subramanian, P.,
Antia, H.M.,
Dugad, S.R.,
Goswami, U.D.,
Gupta, S.K.,
Hayashi, Y.,
Ito, N.,
Kawakami, S.,
Kojima, H.,
Mohanty, P.K.,
Nayak, P.K.,
Nonaka, T.,
Oshima, A.,
Sivaprasad, K.,
Tanaka, H.,
Tonwar, S.C.,
The Grapes-3 Collaboration:
2009,
Forbush decreases and turbulence levels at coronal mass ejection
fronts.
Astron. Astrophys.
494,
1107.
DOI.
ADS.
Sullivan (1971)
Sullivan, J.D.:
1971,
Geometric factor and directional response of single and multi-element
particle telescopes.
Nuclear Instruments and Methods
95(1),
5.
DOI.
http://www.sciencedirect.com/science/article/pii/0029554X71900334.
Thernisien (2011)
Thernisien, A.:
2011,
Implementation of the Graduated Cylindrical Shell Model for the
Three-dimensional Reconstruction of Coronal Mass Ejections.
Astrophys. J. Suppl.
194,
33.
DOI.
ADS.
Thernisien, Vourlidas, and
Howard (2009)
Thernisien, A.,
Vourlidas, A.,
Howard, R.A.:
2009,
Forward Modeling of Coronal Mass Ejections Using STEREO/SECCHI
Data.
Sol. Phys.
256,
111.
DOI.
ADS.
Thernisien, Howard, and
Vourlidas (2006)
Thernisien, A.F.R.,
Howard, R.A.,
Vourlidas, A.:
2006,
Modeling of Flux Rope Coronal Mass Ejections.
Astrophys. J.
652,
763.
DOI.
ADS.
Tortermpun, Ruffolo, and
Bieber (2018)
Tortermpun, U.,
Ruffolo, D.,
Bieber, J.W.:
2018,
Galactic Cosmic-Ray Anistropy During the Forbush Decrease Starting
2013 April 13.
Astrophys. J. Lett.
852(2),
L26.
DOI.
ADS.
Usoskin, Bazilevskaya, and
Kovaltsov (2011)
Usoskin, I.G.,
Bazilevskaya, G.A.,
Kovaltsov, G.A.:
2011,
Solar modulation parameter for cosmic rays since 1936 reconstructed
from ground-based neutron monitors and ionization chambers.
J. Geophys. Res.
116,
2104.
DOI.
ADS.
Usoskin et al. (2008)
Usoskin, I.G.,
Braun, I.,
Gladysheva, O.G.,
HöRand el, J.R.,
JäMséN, T.,
Kovaltsov, G.A.,
Starodubtsev, S.A.:
2008,
Forbush decreases of cosmic rays: Energy dependence of the recovery
phase.
jgr
113(A7),
A07102.
DOI.
ADS.
Usoskin et al. (2015)
Usoskin, I.G.,
Kovaltsov, G.A.,
Adriani, O.,
Barbarino, G.C.,
Bazilevskaya, G.A.,
Bellotti, R.,
Boezio, M.,
Bogomolov, E.A.,
Bongi, M.,
Bonvicini, V.,
Bottai, S.,
Bruno, A.,
Cafagna, F.,
Campana, D.,
Carbone, R.,
Carlson, P.,
Casolino, M.,
Castellini, G.,
De Donato, C.,
De Santis, C.,
De Simone, N.,
Di Felice, V.,
Formato, V.,
Galper, A.M.,
Karelin, A.V.,
Koldashov, S.V.,
Koldobskiy, S.,
Krutkov, S.Y.,
Kvashnin, A.N.,
Leonov, A.,
Malakhov, V.,
Marcelli, L.,
Martucci, M.,
Mayorov, A.G.,
Menn, W.,
Mergé, M.,
Mikhailov, V.V.,
Mocchiutti, E.,
Monaco, A.,
Mori, N.,
Munini, R.,
Osteria, G.,
Palma, F.,
Panico, B.,
Papini, P.,
Pearce, M.,
Picozza, P.,
Pizzolotto, C.,
Ricci, M.,
Ricciarini, S.B.,
Rossetto, L.,
Sarkar, R.,
Scotti, V.,
Simon, M.,
Sparvoli, R.,
Spillantini, P.,
Stozhkov, Y.I.,
Vacchi, A.,
Vannuccini, E.,
Vasilyev, G.I.,
Voronov, S.A.,
Yurkin, Y.T.,
Zampa, G.,
Zampa, N.,
Zverev, V.G.:
2015,
Force-field parameterization of the galactic cosmic ray spectrum:
Validation for Forbush decreases.
Adv. Space Res.
55,
2940.
DOI.
ADS.
Usoskin et al. (2017)
Usoskin, I.G.,
Gil, A.,
Kovaltsov, G.A.,
Mishev, A.L.,
Mikhailov, V.V.:
2017,
Heliospheric modulation of cosmic rays during the neutron monitor
era: Calibration using PAMELA data for 2006-2010.
J. Geophys. Res.
122(4),
3875.
DOI.
ADS.
Vršnak
et al. (2013)
Vršnak, B.,
Žic, T.,
Vrbanec, D.,
Temmer, M.,
Rollett, T.,
Möstl, C.,
Veronig, A.,
Čalogović, J.,
Dumbović, M.,
Lulić, S.,
Moon, Y.-J.,
Shanmugaraju, A.:
2013,
Propagation of Interplanetary Coronal Mass Ejections: The Drag-Based
Model.
Sol. Phys.
285,
295.
DOI.
ADS.
Vršnak
et al. (2019)
Vršnak, B.,
Amerstorfer, T.,
Dumbović, M.,
Leitner, M.,
Veronig, A.M.,
Temmer, M.,
Möstl, C.,
Amerstorfer, U.V.,
Farrugia, C.J.,
Galvin, A.B.:
2019,
Heliospheric Evolution of Magnetic Clouds.
Astrophys. J.
877(2),
77.
DOI.
ADS.
Wawrzynczak and
Alania (2010)
Wawrzynczak, A.,
Alania, M.V.:
2010,
Modeling and data analysis of a Forbush decrease.
Adv. Space Res.
45,
622.
DOI.
ADS.
Webber and Lockwood (1999)
Webber, W.R.,
Lockwood, J.A.:
1999,
A new look at the $<$70 Mev cosmic ray radial gradients in the
heliosphere measured by spacecraft.
J. Geophys. Res.
104,
2487.
DOI.
ADS.
Wibberenz, Cane, and
Richardson (1997)
Wibberenz, G.,
Cane, H.V.,
Richardson, I.G.:
1997,
Two-step Forbush Decreases in thecInner Solar System and their
Relevance for Models of Transient Disturbances.
Int. Cosmic Ray Conf.
1,
397.
ADS.
Wibberenz
et al. (1998)
Wibberenz, G.,
Le Roux, J.A.,
Potgieter, M.S.,
Bieber, J.W.:
1998,
Transient Effects and Disturbed Conditions.
Space Sci. Rev.
83,
309.
ADS.
Winslow et al. (2018)
Winslow, R.M.,
Schwadron, N.A.,
Lugaz, N.,
Guo, J.,
Joyce, C.J.,
Jordan, A.P.,
Wilson, J.K.,
Spence, H.E.,
Lawrence, D.J.,
Wimmer-Schweingruber, R.F.,
Mays, M.L.:
2018,
Opening a Window on ICME-driven GCR Modulation in the Inner Solar
System.
Astrophys. J.
856,
139.
DOI.
ADS.
Witasse et al. (2017)
Witasse, O.,
Sánchez-Cano, B.,
Mays, M.L.,
Kajdič, P.,
Opgenoorth, H.,
Elliott, H.A.,
Richardson, I.G.,
Zouganelis, I.,
Zender, J.,
Wimmer-Schweingruber, R.F.,
Turc, L.,
Taylor, M.G.G.T.,
Roussos, E.,
Rouillard, A.,
Richter, I.,
Richardson, J.D.,
Ramstad, R.,
Provan, G.,
Posner, A.,
Plaut, J.J.,
Odstrcil, D.,
Nilsson, H.,
Niemenen, P.,
Milan, S.E.,
Mandt, K.,
Lohf, H.,
Lester, M.,
Lebreton, J.-P.,
Kuulkers, E.,
Krupp, N.,
Koenders, C.,
James, M.K.,
Intzekara, D.,
Holmstrom, M.,
Hassler, D.M.,
Hall, B.E.S.,
Guo, J.,
Goldstein, R.,
Goetz, C.,
Glassmeier, K.H.,
Génot, V.,
Evans, H.,
Espley, J.,
Edberg, N.J.T.,
Dougherty, M.,
Cowley, S.W.H.,
Burch, J.,
Behar, E.,
Barabash, S.,
Andrews, D.J.,
Altobelli, N.:
2017,
Interplanetary coronal mass ejection observed at STEREO-A, Mars,
comet 67P/Churyumov-Gerasimenko, Saturn, and New Horizons en route to Pluto:
Comparison of its Forbush decreases at 1.4, 3.1, and 9.9 AU.
J. Geophys. Res.
122,
7865.
DOI.
ADS.
Zhang et al. (2006)
Zhang, T.L.,
Baumjohann, W.,
Delva, M.,
Auster, H.-U.,
Balogh, A.,
Russell, C.T.,
Barabash, S.,
Balikhin, M.,
Berghofer, G.,
Biernat, H.K.,
Lammer, H.,
Lichtenegger, H.,
Magnes, W.,
Nakamura, R.,
Penz, T.,
Schwingenschuh, K.,
Vörös, Z.,
Zambelli, W.,
Fornacon, K.-H.,
Glassmeier, K.-H.,
Richter, I.,
Carr, C.,
Kudela, K.,
Shi, J.K.,
Zhao, H.,
Motschmann, U.,
Lebreton, J.-P.:
2006,
Magnetic field investigation of the Venus plasma environment:
Expected new results from Venus Express.
Planet. Space. Sci.
54(13-14),
1336.
DOI.
ADS.
Zhao and Zhang (2016)
Zhao, L.-L.,
Zhang, H.:
2016,
Transient Galactic Cosmic-ray Modulation during Solar Cycle 24: A
Comparative Study of Two Prominent Forbush Decrease Events.
Astrophys. J.
827(1),
13.
DOI.
ADS.
Zurbuchen and
Richardson (2006)
Zurbuchen, T.H.,
Richardson, I.G.:
2006,
In-Situ Solar Wind and Magnetic Field Signatures of Interplanetary
Coronal Mass Ejections.
Space Sci. Rev.
123,
31.
DOI.
ADS.
Zurbuchen
et al. (2016)
Zurbuchen, T.H.,
Weberg, M.,
von Steiger, R.,
Mewaldt, R.A.,
Lepri, S.T.,
Antiochos, S.K.:
2016,
Composition of Coronal Mass Ejections.
Astrophys. J.
826(1),
10.
DOI.
ADS.
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\lastpagegivenfalse\inarticlefalse |
Si-compatible candidates for \hk dielectrics with the Pbnm
perovskite structure
Sinisa Coh
sinisa@physics.rutgers.edu
Department of Physics and Astronomy, Rutgers University, Piscataway,
NJ 08854-8019, USA
Tassilo Heeg
Department of Materials Science and Engineering,
Cornell University,
Ithaca, NY 14853, USA
J. H. Haeni
Department of Materials Science and Engineering,
Pennsylvania State University,
University Park, PA 16802, USA
M. D. Biegalski
Center for Nanophase Materials Science,
Oak Ridge National Laboratory,
Oak Ridge, TN 37830, USA
J. Lettieri
[
Department of Materials Science and Engineering,
Pennsylvania State University,
University Park, PA 16802, USA
L. F. Edge
Department of Materials Science and Engineering,
Pennsylvania State University,
University Park, PA 16802, USA
K. E. O’Brien
Department of Materials Science and Engineering,
Pennsylvania State University,
University Park, PA 16802, USA
M. Bernhagen
Leibniz Institute for Crystal Growth,
Max-Born-Straße 2, D-12489 Berlin
(Adlershof), Germany
P. Reiche
Leibniz Institute for Crystal Growth,
Max-Born-Straße 2, D-12489 Berlin
(Adlershof), Germany
R. Uecker
Leibniz Institute for Crystal Growth,
Max-Born-Straße 2, D-12489 Berlin
(Adlershof), Germany
S. Trolier-McKinstry
Department of Materials Science and Engineering,
Pennsylvania State University,
University Park, PA 16802, USA
Darrell G. Schlom
Department of Materials Science and Engineering,
Cornell University,
Ithaca, NY 14853, USA
David Vanderbilt
Department of Physics and Astronomy, Rutgers University, Piscataway,
NJ 08854-8019, USA
(December 7, 2020)
Abstract
We analyze both experimentally (where possible) and theoretically
from first-principles the dielectric tensor components and crystal
structure of five classes of $Pbnm$ perovskites. All of these
materials are believed to be stable on silicon and are therefore
promising candidates for high-K dielectrics. We also analyze the
structure of these materials with various simple models,
decompose the lattice contribution to the dielectric tensor into
force constant matrix eigenmode contributions, explore
a peculiar correlation between structural and dielectric
anisotropies in these compounds and give phonon frequencies and
infrared activities of those modes that are infrared-active.
We find that CaZrO${}_{3}$, SrZrO${}_{3}$,
LaHoO${}_{3}$, and LaYO${}_{3}$ are among the most promising candidates
for high-K dielectrics among the compounds we considered.
pacs: 77.22.-d,77.55.df,85.50.-n
]Deceased.
I Introduction
As a result of the ongoing down-scaling of complementary
metal-oxide-semiconductor (CMOS) integrated circuits, the SiO${}_{2}$ gate
oxide of field effect transistors is getting thinner and thinner in
every new generation of devices. Moore (2003) Therefore the leakage
current due to quantum-mechanical tunneling through the dielectric
interface is increasing. One way to reduce this current is to replace
SiO${}_{2}$ with a material that has a higher dielectric constant.
Such a high-K dielectric layer with the same effective dielectric
thickness (i.e., providing the same capacitance) could be physically
thicker and thus reduce the gate leakage.
In order for this replacement material to be useful in practical
applications on silicon, it also needs to be stable in contact with
silicon up to $\sim$1000 ${}^{\circ}$C, and among other things it must
also have an appropriate band alignment with silicon.
Schlom et al. (2005, 2008); Robertson (2008)
Currently, a hafnia-based dielectric
is used as a replacement to SiO${}_{2}$ in advanced CMOS transistors in
production. Mistry et al. (2007); Hicks et al. (2008); Scansen (21 Jan. 2008)
There are, however, drawbacks to this material too, e.g., the
limited K that it provides and undesirable threshold
voltage shifts arising from highly mobile
oxygen vacancies. Guha and Narayanan (2007)
This brings up the natural question: which other materials exist that would satisfy these requirements
and would enable the scaling of MOSFETs to continue beyond today’s
hafnia-based dielectrics?
The stability of single component oxides on silicon has been
demonstrated both experimentally and from thermodynamic
analysis,Schlom et al. (2008) and a candidate list of
multicomponent oxide materials has been compiled. Schlom et al. (2005) A
promising group of these materials consists of perovskite oxides having a
$Pbnm$ (or closely related $P2_{1}/c$) space group.
These compounds are at the focus of the present work. Some of them have been
studied in thin-film form,
Christen et al. (2006); Afanas’ev et al. (2004); Zhao et al. (2005); Heeg et al. (2006); Edge et al. (2006a); Cicerrella et al. (2005); Afanas’ev et al. (2006); Edge et al. (2006b); Sivasubramani et al. (2006); Lopes et al. (2007); Wang et al. (2007); Edge et al. (2008); Özben et al. (2008); Roeckerath et al. (2009); Wagner et al. (2006); Myllymaki et al. (Sept. 2007); Kim et al. (2006); Thomas et al. (2007); Heeg et al. (2007) but the full dielectric tensor of these materials
has not yet been established, making the selection of materials best
suited for high-K applications difficult.
Some of these materials could also be of interest for microwave
dielectric applications.Wersing (1996); Vanderah (2002)
Thus we decided to study,
both theoretically and experimentally, the structural and dielectric
properties of these compounds. The calculations are carried out using
density-functional theory, and we compare the results with experimental
data where we could obtain suitable samples for measurements. To our
knowledge, previous theoretical calculations have been carried out
in only a few cases.Delugas et al. (2007); Vali (2008, 2009)
The paper is organized as follows.
Explanations of both the experimental and theoretical methods used in this
work are given in Sec. II. The main results on the
structural and dielectric properties are given and discussed in
Sec. III.
There we also discuss the correlations between the structural and dielectric
properties of these perovskites, decompose the ionic contribution of the
dielectric tensor into components arising from various force constant matrix
eigenvectors, and discuss the effect of
B${}_{\it A}$ antisite defects on
the dielectric properties.
We finish with a brief summary in Sec. IV.
Supplementary material EPA contains the results of our
calculations of the zone-center phonon frequencies, as well as the
infrared activities for those modes that are infrared-active.
I.1 Compounds under consideration
In this work, we consider the following five groups of
perovskites having the $Pbnm$ space group.
The first group are rare-earth scandates having formula AScO${}_{3}$ where A is a rare-earth atom. In
Sec. III.3.1 we report experimental measurements of
the full dielectric tensors for PrScO${}_{3}$, NdScO${}_{3}$, SmScO${}_{3}$, GdScO${}_{3}$ and
DyScO${}_{3}$. Calculations were done on these and also on LaScO${}_{3}$ and TbScO${}_{3}$.
Note that
HoScO${}_{3}$, ErScO${}_{3}$, TmScO${}_{3}$, YbScO${}_{3}$, LuScO${}_{3}$, and YScO${}_{3}$ do not form
single crystals with the perovskite structure from the melt at
atmospheric pressure. Rather, they form solid solutions of A${}_{2}$O${}_{3}$ and Sc${}_{2}$O${}_{3}$, i.e., (A,Sc)${}_{2}$O${}_{3}$, with the
bixbyite structure.Badie (1978); Badie and Foex (1978); Coutures et al. (1980) Nevertheless,
LuScO${}_{3}$ Heeg et al. (2007) and YbScO${}_{3}$ sch have been formed in
perovskite form as thin films via epitaxial stabilization, and others
might be made in the same way. To analyze trends within this group of
compounds, we also did the calculations of dielectric tensors on
LuScO${}_{3}$ and YScO${}_{3}$ in the $Pbnm$ perovskite structure;
see Sec. III.3.1 for the details.
The second group consists of rare-earth yttrates with formula AYO${}_{3}$.
Only one such compound,
LaYO${}_{3}$, is known to form a perovskite,Berndt et al. (1975) but to analyze
trends the dielectric tensor of DyYO${}_{3}$ in the perovskite structure was
also calculated.
In the third group we consider CaZrO${}_{3}$, SrZrO${}_{3}$ and SrHfO${}_{3}$ perovskites. Experimentally, we find that SrZrO${}_{3}$ and SrHfO${}_{3}$ do
not form single crystals, but instead are rather heavily twinned.
The fourth group of compounds have the formula La${}_{2}$BB${}^{\prime}$O${}_{6}$ where the
B atom is either Mg or Ca and B${}^{\prime}$ is either Zr or Hf.
Little is known experimentally
about these compounds, and single crystals of these
compounds have not been made.Rabenau (1956)
The last group of compounds we considered have the formula AA’O${}_{3}$,
where both A and A’ are rare-earth atoms. These include
the 11 of such compounds that are known to form the perovskite structure
with space group $Pbnm$ at atmospheric pressure: LaHoO${}_{3}$, LaErO${}_{3}$,
LaTmO${}_{3}$, LaYbO${}_{3}$, LaLuO${}_{3}$, CeTmO${}_{3}$, CeYbO${}_{3}$, CeLuO${}_{3}$, PrYbO${}_{3}$,
PrLuO${}_{3}$, and NdLuO${}_{3}$.Berndt et al. (1975); Ito et al. (2001); Bharathy et al. (2009) We calculated the
dielectric and structural properties of all of these compounds.
The experimental determination, however, of the dielectric tensor in
this group of compounds was done only for LaLuO${}_{3}$; the results
will be published elsewhere. hee
II Preliminaries
II.1 Structure of $\bf Pbnm$ perovskites
The ideal cubic perovskite ABO${}_{3}$ consists of a network of
corner-shared octahedra, each with an oxygen on its vertices and a B atom at its center, and A ions that are 12-fold coordinated
in the spaces between octahedra. It is well
known that perovskites having sufficiently small A-site
ions (i.e., a small Goldschmidt tolerance
factorGoldschmidt (1926); Woodward (1997)) often allow for a distorted
perovskite structure that has a rotated framework of oxygen octahedra
and displaced A-site ions. This lowers the space group symmetry
from cubic $Pm\bar{3}m$ ($O_{h}^{1}$) to orthorhombic $Pbnm$ ($D_{2h}^{16}$), and the
number of ABO${}_{3}$ formula units per primitive cell increases from
$Z=1$ to $Z=4$, as shown in Fig. 2.
The rotations of the octahedra in the $Pbnm$ space group
can be decomposed into two steps. The first step is the rotation
around the $[110]$ direction of the original cubic frame (the cubic
frame is rotated by 45${}^{\circ}$ around the $z$ axis with respect to
the $Pbnm$ frame) by an angle $\theta_{\rm R}$ as in Fig. 3(a), and
the second step is a rotation around $[001]$ by $\theta_{\rm M}$ as in
Fig. 3(b). The rotations must be done in that order to
prevent distortions of the octahedra. The
pattern of neighboring octahedral rotations is denoted by ($a^{-}a^{-}c^{+}$) in
Glazer notation Glazer (1972) (or see the directions of the arrows in
Fig. 3). These rotations also allow for the displacement
of A-site ions in the $x$-$y$ plane without further lowering of the
space-group symmetry.
II.2 Computational methods
The main computational method we are using is the density-functional
theory as implemented in the Quantum-Espresso package.Giannozzi et al. (2009)
The exchange-correlation functional was approximated using a
generalized gradient approximation (GGA) of the Perdew-Burke-Ernzerhof
type Perdew et al. (1996) and ultrasoft pseudopotentials were
employed.Vanderbilt (1985) The electronic wavefunctions were
expanded in a basis of plane waves with kinetic energy up to 40 Ry,
while the charge density was expanded up to 300 Ry. The Brillouin
zone was sampled using a $4\times 4\times 4$ Monkhorst-Pack
grid.Monkhorst and Pack (1976)
A new set of ultrasoft pseudopotentialsVanderbilt (1985)
for the lanthanoide series of rare earths, from La to Lu, were generated
for the present project. In all cases the $f$-shell filling was
chosen as appropriate for the 3+ valence state: one $f$ electron for
Ce, two for Pr, etc. The $f$ electrons were then considered to be
in the core (and un-spin-polarized) for the proposes of generating
the pseudopotentials. Thus, the $f$ electrons are not explicitly
included in the solid-state calculations. Such an approximation can
be justified whenever the strong on-site Coulomb interactions of
electrons in the $f$ shell drive the occupied $f$
states well below, and the unoccupied states well above, the energy
range of interest for $spd$ bonding in the crystal. Of course, this
will not be a good approximation for some heavy-fermion or
mixed-valent systems, and in any case our approach is obviously unable
to describe phenomena involving magnetic ordering of $f$ electrons at
low temperature. Nevertheless, we believe that this approach is quite
reasonable for the present purposes.
The artificial nature of the scattering in the $f$ channel did, however,
pose some problems in the pseudopotential construction. In particular, we
found that the lattice constant of a perovskite containing the rare-earth
atom in question could differ for two pseudopotentials having different
scattering properties in the $f$ channel; this causes problems since
the usual approach of matching to the all-electron $f$ scattering is
not appropriate in the present case. To ameliorate this problem, the
$f$-channel parameters of these pseudopotentials were optimized so that
resulting pseudopotentials would give the “correct” cell volumes for
simple rare-earth compounds. Since the GGA typically overestimates
crystal volumes by about 1-2%,Staroverov et al. (2004) the optimization
was actually done in order to produce a corresponding overestimate
in a consistent fashion.
For this procedure, our compounds of choice
were the rare-earth nitrides with the simple rock-salt structure. The
experimentalPies and Weiss (2006) and calculated volumes of these
nitrides are indicated in Fig. 4.
Note that the volumes of CeN, PrN, and GdN show an anomalous behavior
that is presumably due to strong correlation effects associated with
the proximity to a mixed-valent regime, gdn and
therefore they will not be correctly treated by our GGA calculation.
To avoid this problem we first carried out a smoothened fit of the
experimental volumes versus atomic number over the lanthanoide
nitride series, but with CeN, PrN, and GdN omitted from the fit, as
shown by the solid line in Fig. 4. We then
used these fitted values to set the target volumes for the optimization
of the pseudopotentials.
We used density-functional perturbation theoryBaroni et al. (2001) to
calculate the dielectric response. Both purely electronic
$\epsilon^{\rm el}$ and ionic $\epsilon^{\rm ion}$ contributions were
calculated.Cockayne and Burton (2000) The electronic part is defined as
$$\displaystyle\epsilon_{\alpha\beta}^{\rm el}=\delta_{\alpha\beta}+4\pi\frac{%
\partial P_{\alpha}}{\partial{\cal E}_{\beta}}\bigg{|}_{u=0},$$
(1)
where $P_{\alpha}$ is the polarization induced by the electric field
${\cal E}_{\beta}$ while all ions are held fixed ($u=0$). The
remaining component of the dielectric response is by definition
the ionic contribution $\epsilon^{\rm ion}$.
This ionic part can be calculated from the force-constant matrix
$\Phi_{i\alpha,j\beta}$ and the Born effective charge matrix
$Z_{i,\alpha\beta}$. The force-constant matrix is defined as
$$\displaystyle\Phi_{i\alpha,j\beta}=\frac{\partial^{2}E}{\partial u_{i\alpha}%
\partial u_{j\beta}},$$
(2)
where $E$ is the total energy of the system and $u_{i\alpha}$ is the
displacement of the $i$-th atom along the direction $\alpha$. We will
denote the $n$-th normalized eigenvector of this matrix as
$\xi^{n}_{i\alpha}$ and its eigenvalue as $\mu_{n}$.
The Born effective
charge matrix is defined as
$$\displaystyle Z_{i,\alpha\beta}=\frac{V}{e}\frac{\partial P_{\alpha}}{\partial
u%
_{i\beta}},$$
(3)
where $P_{\alpha}$ is the polarization induced in a crystal by the
displacement of the $i$-th atom in the direction $\beta$. $V$ is the
volume of the unit cell and $e$ is the electron charge.
Finally, the ionic part of the dielectric tensor can be written as
$$\displaystyle\epsilon^{\rm ion}_{\alpha\beta}=\frac{4\pi e^{2}}{V}%
\displaystyle\sum_{n}\frac{1}{\mu_{n}}Q_{\alpha}^{n}Q_{\beta}^{n},$$
(4)
where the charge $Q_{\alpha}^{n}$ of the $n$-th eigenmode is defined
through the effective charge matrix as $Q_{\alpha}^{n}=\displaystyle\sum_{i\beta}Z_{i,\alpha\beta}\xi_{i\beta}^{n}$.
II.3 Experimental methods
II.3.1 Crystal growth
PrScO${}_{3}$, NdScO${}_{3}$, SmScO${}_{3}$, GdScO${}_{3}$, TbScO${}_{3}$, and DyScO${}_{3}$ single crystals were grown using an automated Czochralski technique
with RF-induction heating. Uecker et al. (2006, 2008) Pre-dried powders
of Sc${}_{2}$O${}_{3}$, Pr${}_{6}$O${}_{11}$, Nd${}_{2}$O${}_{3}$, Sm${}_{2}$O${}_{3}$,
Gd${}_{2}$O${}_{3}$, Tb${}_{4}$O${}_{7}$, and Dy${}_{2}$O${}_{3}$, were mixed in the
stoichiometric ratio, pressed, and sintered at about 1400 ${}^{\circ}$C
for 15 h. Due to the high melting temperature of PrScO${}_{3}$,
NdScO${}_{3}$, SmScO${}_{3}$, GdScO${}_{3}$, TbScO${}_{3}$, and DyScO${}_{3}$
($\sim$2100 ${}^{\circ}$C), a crucible (cylindrical with 40 mm
or 60 mm diameter
and 40 mm or 60 mm height, depending on the crystal diameter)
and an active afterheater made of iridium were used.
Flowing nitrogen or argon was used as the growth atmosphere.
Those of these rare-earth scandates for which the radiative heat
transport via the crystal is hindered by absorption suffer from a
serious problem of bulk crystal growth, namely that they tend to
exhibit a spiral growth which distinctly decreases the yield. Uecker et al. (2006)
Due to
the lack of seed crystals, the initial growth experiments were
performed with an iridium seed rod. Because these materials tend to
grow as large single-crystalline grains, suitable seeds could be
selected at a very early stage.
All rare-earth scandate crystals were grown along the
$\left[110\right]$ orientation.
The pulling rate was $0.8$-$2$ mm$\cdot$h${}^{-1}$
and the rotation rate was $8$-$15$ min${}^{-1}$
(depending on the crystal diameter). The crystals were 35-50 mm in
length and 18 or 32 mm in diameter. The PrScO${}_{3}$ crystals have a green
color, the NdScO${}_{3}$ crystals are dark purple,
the SmScO${}_{3}$ and DyScO${}_{3}$ crystals are light yellow,
the GdScO${}_{3}$ crystals are colorless, and the TbScO${}_{3}$ crystals
are nearly colorless (see Fig. 1).
In addition
to the Czochralski growth of PrScO${}_{3}$, NdScO${}_{3}$, SmScO${}_{3}$,
GdScO${}_{3}$, TbScO${}_{3}$, and DyScO${}_{3}$, the single-crystal
growth of HoScO${}_{3}$, YScO${}_{3}$, and solid solutions of rare-earth
scandates that would be expected to have smaller lattice constants
than DyScO${}_{3}$ (e.g., Dy${}_{0.5}$Lu${}_{0.5}$ScO${}_{3}$),
was also attempted using the floating-zone technique. In all cases,
pre-dried powders of the rare-earth oxides (R${}_{2}$O${}_{3}$)
and Sc${}_{2}$O${}_{3}$ were ground,
pressed into a rod, and sintered at 1400 ${}^{\circ}$C for 6 hours. The
floating-zone machine was a Gero (model SPO), and heating was
performed with two bulbs at the focal point of ellipsoidal mirrors.
Unfortunately, the 1 kW
rated output power of the quartz-halogen lamps used in this
floating-zone system was close to the melting temperature of the compounds,
limiting the size of the crystals that could be grown to $<2$ mm
diameter.
SrZrO${}_{3}$, SrHfO${}_{3}$, and LaScO${}_{3}$ single crystals grown by the floating
zone technique were also studied. srz
The available R${}_{2}$O${}_{3}$-Sc${}_{2}$O${}_{3}$ binary phase diagrams show a
distinct transition between the Dy${}_{2}$O${}_{3}$-Sc${}_{2}$O${}_{3}$ and the
Ho${}_{2}$O${}_{3}$-Sc${}_{2}$O${}_{3}$ systems (Dy and Ho are neighboring elements
in the periodic table).
According to published phase diagrams, the rare-earth elements of La,
Nd, Sm, Gd, and Dy all form RScO${}_{3}$
compounds that melt congruently. In contrast
the oxides of Ho, Er, Tm, Yb, and Lu form complete
solid solutions with Sc${}_{2}$O${}_{3}$ at their melting
points. Badie (1978); Badie and Foex (1978); Coutures et al. (1980) (The same happens with Y as
well. Badie (1978); Badie and Foex (1978); Coutures et al. (1980))
Although HoScO${}_{3}$ has been synthesized with solid-state
techniques at temperatures (well below the melting point) at which the
perovskite polymorph of HoScO${}_{3}$ is stable, our results confirm
that the perovskite polymorph of HoScO${}_{3}$ is not stable at its melting
point in agreement with
existing phase diagrams. Analysis of our Y-Sc-O and Ho-Sc-O single
crystals revealed that they were solid solution mixtures of
Ho${}_{2}$O${}_{3}$ and Sc${}_{2}$O${}_{3}$, and Y${}_{2}$O${}_{3}$ and Sc${}_{2}$O${}_{3}$, not the
desired perovskite polymorphs of HoScO${}_{3}$ and YScO${}_{3}$. The only
compound that did not behave according to its phase diagram was
LaScO${}_{3}$. Attempts to grow LaScO${}_{3}$ resulted in crystals that were
a mixture of three different phases including LaScO${}_{3}$, Sc${}_{2}$O${}_{3}$,
and a third, unidentified phase. srz X-ray analysis of the
post-annealed polycrystalline feed rod showed single-phase LaScO${}_{3}$,
consistent with the existing phase diagram. The reason for the
introduction of second phases during melting is not clear and requires
more study.
II.3.2 Electrical characterization
Because PrScO${}_{3}$, NdScO${}_{3}$, SmScO${}_{3}$, GdScO${}_{3}$, and DyScO${}_{3}$ are
orthorhombic at room temperature, their dielectric tensor contains
three independent coefficients that can be measured along the three
principle crystal axes. Newnham (2005) From the grown PrScO${}_{3}$,
NdScO${}_{3}$, SmScO${}_{3}$, GdScO${}_{3}$, and DyScO${}_{3}$ single crystals, slices
were cut in different orientations for electrical characterization.
For GdScO${}_{3}$, samples were cut from the as-grown single
crystals along the $\left[100\right]$, $\left[010\right]$, and
$\left[001\right]$ directions, and the dielectric tensor coefficients
were measured using a parallel-plate capacitor configuration. For
PrScO${}_{3}$, NdScO${}_{3}$, SmScO${}_{3}$, and DyScO${}_{3}$, several additional
orientations were prepared, so that a least-squares fit technique
could be applied using standard methods Newnham (2005) to calculate
the dielectric tensor, using their known lattice
parameters.Uecker et al. (2008); Gesing et al. (2009)
Unfortunately, the SrZrO${}_{3}$ and SrHfO${}_{3}$ crystals were heavily
twinned. The formation of twins in the three pseudocubic orientations
is facilitated in these two orthorhombic systems because the deviation from
the cubic symmetry is very small. As a result, only an average dielectric
constant could be measured.
To obtain capacitors, gold or platinum electrodes were
evaporated onto both sides of the approximately 0.5 mm thick slabs
with areas ranging from 15 to 100 mm${}^{2}$, and capacitance measurements
were made at room temperature
with an HP4284A using a 16034E test fixture at 10 kHz, 100 kHz, and
1 MHz. No edge capacitance corrections were used because of the large
ratio of the electrode area to the thickness of the samples. The room
temperature dielectric loss of the samples was generally very small
($<0.1\%$ for most samples) and the frequency dispersion was negligible in
the measured range.
The temperature dependent dielectric measurements on GdScO${}_{3}$ and DyScO${}_{3}$
were made using a HP 4284a LCR meter with a dipstick cryostat.
III Results and discussion
III.1 Structural properties
We focus first on the structural properties of these systems.
The structure of the $Pbnm$ perovkites is described by three orthorhombic
lattice constants plus two A-site and five oxygen Wyckoff
parameters. Figure 5 shows graphically
the most important structural parameters of these systems, while
Table 1 gives detailed information on all the structural
parameters. Rotational angles in Table 1 and in
Fig. 5 were calculated by fitting the structural
parameters to a model in which the octahedra are perfectly rigid (see
Sec. III.2 for the details of this model).
Overall we find good agreement with experimental values for the
structural parameters. The Wyckoff coordinates in particular are in
excellent agreement with experiments, with the average error being on the
order of $2\cdot 10^{-3}$. The volume of the unit cell, on the other
hand, is consistently overestimated by 1-2%, as is usually expected
from the GGA exchange-correlation functional, and as we would expect
from our construction of the rare-earth pseudopotentials.
All structures show an angle $\theta_{\rm R}$ that is about $\sqrt{2}$
times larger than $\theta_{\rm M}$. Therefore, consecutive rotations by $\theta_{\rm R}$
and $\theta_{\rm M}$ can be considered approximately as a single rotation around
a $\{111\}$ axis in the cubic frame. That is, the actual ($a^{-}a^{-}c^{+}$) pattern
of rotations is very nearly ($a^{-}a^{-}a^{+}$) in the Glazer notation.Glazer (1972)
See Sec. III.6 for a more
detailed discussion.
III.1.1 Rare-earth scandates
The rare-earth scandates AScO${}_{3}$ show a decrease in volume by
$\sim$9% while going along the series from A=La to A=Dy
(the calculated primitive unit cell volume is 271.40 Å${}^{3}$ for
LaScO${}_{3}$ and 249.81 Å${}^{3}$ for DyScO${}_{3}$). On the other hand, the Sc-O
distance remains nearly constant along the series (2.12 Å for
LaScO${}_{3}$ and 2.11 Å for DyScO${}_{3}$), which means that the change in
volume is almost entirely due to the larger octahedral rotation angles
for DyScO${}_{3}$ as compared to LaScO${}_{3}$.
Our calculations also show that the same trend continues
all the way to LuScO${}_{3}$.
III.1.2 Rare-earth yttrates
The rare-earth yttrates have a very similar behavior as the
rare-earth scandates. The main quantitative structural difference
between the two comes from the fact that yttrium is a larger ion
than scandium. This leads to a larger volume for the yttrates, and
also a larger rotation angle due to a smaller tolerance factor.
III.1.3 CaZrO${}_{3}$, SrZrO${}_{3}$, and SrHfO${}_{3}$
SrZrO${}_{3}$ and SrHfO${}_{3}$ have quite similar structural
properties. The main difference can be traced to the fact that
Hf is a smaller ion than Zr. Therefore, the calculated average Hf-O
distance is 2.07 Å, while the average Zr-O distance is 2.11 Å.
Furthermore, their octahedral rotation angles are about 1.7 times
smaller than in the rare-earth scandates.
In CaZrO${}_{3}$ the average Zr-O distance is 2.10 Å, which is very close
to the corresponding
distance in SrZrO${}_{3}$ and SrHfO${}_{3}$. Thus, the main reason why
CaZrO${}_{3}$ has a smaller volume than SrZrO${}_{3}$ is because of the larger
rotation angles in CaZrO${}_{3}$.
III.1.4 La${}_{2}$BB${}^{\prime}$O${}_{6}$ compounds
We consider La${}_{2}$BB${}^{\prime}$O${}_{6}$ compounds with B=Mg or Ca and
B${}^{\prime}$=Zr or Hf. These compounds are expected to exhibit rock-salt
ordering of the B-site ions as a result of the difference in
charge and ionic radius between the B and
B${}^{\prime}$ ions.Anderson et al. (1993) This ordering reduces
the symmetry from the orthorhombic $Pbnm$ to
the monoclinic $P2_{1}/c$ ($C_{2h}^{5}$) space group.
The structural properties for these systems are reported in
Fig. 5 and in Table 2. The
rotational angles are obtained by a fit to the rigid-octahedra model
in which we have allowed for different sizes of B- and
B${}^{\prime}$-centered octahedra. (See the end of Sec. III.2
for details.)
The unit cell volume is larger by about 5 Å${}^{3}$ per primitive cell
for the compounds containing Zr than for those containing Hf. On
the other hand, compounds with Ca are larger by about 28 Å${}^{3}$
than those containing Mg. Similarly, the rotation angles are larger in
compounds containing Ca than in those with Mg. The discrepancy between
octahedral sizes is largest for La${}_{2}$CaHfO${}_{6}$ (12% linear increase)
and smallest for La${}_{2}$MgZrO${}_{6}$ (0.4% linear increase).
III.1.5 Rare-earth rare-earth perovskites
We now briefly analyze the structural properties of $Pbnm$ perovskites of type AA’O${}_{3}$ where both A and A’ are
rare-earth atoms. All eleven compounds we considered are known
experimentally to form the perovskite structure in the $Pbnm$ space
group. Berndt et al. (1975); Ito et al. (2001); Bharathy et al. (2009)
Among these 11 compounds, the largest unit-cell volume of
311.58 Å${}^{3}$ is found in LaHoO${}_{3}$, and the smallest
of 292.32 Å${}^{3}$ is in NdLuO${}_{3}$,
Oxygen oxtahedral rotation angles are quite large in
all of these compounds and show very little variation from one
compound to another. The trends of the rotation angles are as expected
from a tolerance-factor analysis: perovskites with smaller
A-site ions but the same B-site ions have larger oxygen
octahedral rotation angles, and the opposite is true for the
B-site ions.
III.2 Comparison with model of perfectly rigid octahedra
In $Pbnm$ perovskites, a rigid rotation of the oxygen
octahedra by $\theta_{\rm R}$ followed by another rigid rotation by $\theta_{\rm M}$ (see
Fig. 3) leads to Wyckoff parameters given by
$$\displaystyle x_{2}$$
$$\displaystyle=-\frac{1}{2\sqrt{2}}\tan\theta_{\rm R},$$
(5)
$$\displaystyle y_{2}$$
$$\displaystyle=-\frac{1}{2\sqrt{2}}\sin\theta_{\rm R}\tan\theta_{\rm M},$$
(6)
$$\displaystyle x_{3}$$
$$\displaystyle=\frac{1}{4}\left(1-\frac{\tan\theta_{\rm M}}{\cos\theta_{\rm R}}%
\right),$$
(7)
$$\displaystyle y_{3}$$
$$\displaystyle=\frac{1}{4}\left(1+\cos\theta_{\rm R}\tan\theta_{\rm M}\right),$$
(8)
$$\displaystyle z_{3}$$
$$\displaystyle=\frac{1}{4\sqrt{2}}\tan\theta_{\rm R},$$
(9)
Here we have denoted the Wyckoff coordinates of the oxygen atoms at the
$4c$ Wyckoff point with $x_{2}$ and $y_{2}$, while those of the
remaining oxygen atoms at the $8d$ point are denoted with $x_{3}$,
$y_{3}$, and $z_{3}$. The Wyckoff coordinates of the A-site ion at
the $4c$ point are denoted by $x_{1}$ and $y_{1}$, but these are left
unspecified in our rigid-octahedra model.
It also leads to orthorhombic lattice constants given by
$$\displaystyle a$$
$$\displaystyle=\sqrt[3]{\frac{V_{0}}{\sqrt{2}}}\cos\theta_{\rm R}\cos\theta_{%
\rm M},$$
(10)
$$\displaystyle b$$
$$\displaystyle=\sqrt[3]{\frac{V_{0}}{\sqrt{2}}}\cos\theta_{\rm M},$$
(11)
$$\displaystyle c$$
$$\displaystyle=\sqrt[3]{2V_{0}}\cos\theta_{\rm R}$$
(12)
where $V_{0}$ is the volume the structure would have if the octahedra
were rotated rigidly back to $\theta_{\rm R}=\theta_{\rm M}=0$.
The Wyckoff parameters and unit-cell ratios from our calculations can
be well fitted by Eqs. 5-12 (see
Table 1 and Fig. 5 for the values of
the fitted angles).
By far the largest discrepancy is found for
Wyckoff parameter $y_{2}$. For a typical system (e.g., LaScO${}_{3}$) the
discrepancy between calculated and fitted $y_{2}$ values is about 0.016, or
50% with respect to the difference from the cubic case. For the
remaining oxygen Wyckoff coefficients, the discrepancy averages about
0.003, or $\sim$5%.
The rotation angles for the La${}_{2}$BB${}^{\prime}$O${}_{6}$ systems were obtained by fitting their
structural parameters to a slightly more complicated model of rigid
octahedra than the one given in Eqs. 5-12. In
this model, we first change the relative sizes of B- and
B${}^{\prime}$-centered octahedra. The ratio of their linear sizes is denoted by
$d/d^{\prime}$. We then proceed with the rotation by an angle $\theta_{\rm R}$
around the $[110]$ axis in the cubic frame. Finally, we perform
a rotation of the B-centered octahedra around $[001]$ by an angle
$\theta_{\rm M}$, and of the B${}^{\prime}$-centered octahedra by an angle $\theta_{\rm M}^{\prime}$ around the same
axis. The resulting fitted values of these parameters are given in
Table 2.
III.3 Dielectric properties
In this section we discuss the dielectric properties of the
materials included in our study. $Pbnm$ perovskites
are orthorhombic and thus have diagonal dielectric tensors, with
$\epsilon_{xx}\neq\epsilon_{yy}\neq\epsilon_{zz}$ in general. In addition to reporting these
components, we also focus on analyzing the results in terms of
the three linear combinations
$$\displaystyle\bar{\epsilon}$$
$$\displaystyle=\displaystyle\frac{1}{3}\left(\epsilon_{xx}+\epsilon_{yy}+%
\epsilon_{zz}\right),$$
(13)
$$\displaystyle\Delta\epsilon_{\parallel}$$
$$\displaystyle=\epsilon_{xx}-\epsilon_{yy},$$
(14)
$$\displaystyle\Delta\epsilon_{\perp}$$
$$\displaystyle=\epsilon_{zz}-\displaystyle\frac{1}{2}\left(\epsilon_{xx}+%
\epsilon_{yy}\right),$$
(15)
representing the average dielectric tensor, a measure of the
$x$-$y$ anisotropy, and a measure of $z$ anisotropy, respectively.
This choice of parameters was made to simplify the analysis of trends of
dielectric properties of these compounds.
The theoretical – and where available, experimental – results
for the dielectric-tensor components
are reported in Figure 7 and in Table 3.
The theoretical values are further decomposed in Table 3
into purely electronic or frozen-ion contributions $\epsilon^{\rm el}$
and lattice-mediated contributions $\epsilon^{\rm ion}$. We find that
the electronic contribution is roughly five times smaller than the
ionic one, is nearly isotropic, and does not show a dramatic variation
from one perovskite to another. Thus, it is clear that the
lattice-mediated ionic contributions play by far the dominant role
in the observed dielectric tensors and their anisotropies.
Our calculations of the zone-center phonon frequencies as well as the
infrared activities for those modes that are infrared-active are given
in the supplementary material. EPA
We now consider each of our chosen classes of $Pbnm$ perovskites
in turn, orienting the presentation from the point of view of the
theoretical calculations, but mentioning the comparison with
experiment where appropriate.
III.3.1 Rare-earth scandates
All rare-earth scandates AScO${}_{3}$ have rather similar values for
their isotropically-averaged dielectric constants, falling between about
$\bar{\epsilon}=26$ and
$\bar{\epsilon}=28$. The $xx$ component for all these systems is larger than the
$yy$ component by about $\Delta\epsilon_{\parallel}=4$. On the other hand, the $zz$ component
changes significantly from LaScO${}_{3}$ to DyScO${}_{3}$. In LaScO${}_{3}$ the average
of the $xx$ and $yy$ components is almost as large as the $zz$
component ($\Delta\epsilon_{\perp}$=-1), while in DyScO${}_{3}$, the $zz$ component is larger
by about $\Delta\epsilon_{\perp}$=9 than the average of $xx$ and $yy$ components.
These results are in good agreement with experiment, especially for
$\bar{\epsilon}$ and $\Delta\epsilon_{\parallel}$. On the other hand, $\Delta\epsilon_{\perp}$ is consistently larger in
experiments by about 3-5, but the trend of increasing $\Delta\epsilon_{\perp}$ is present
in both theory and experiment.
As was mentioned earlier, rare-earth atoms heavier than Dy (i.e., Ho-Lu) and
Y itself do not form single-crystal scandates. Nevertheless, at least
some (YbScO${}_{3}$ sch and LuScO${}_{3}$ Heeg et al. (2007)) can form
$Pbnm$ perovskites in thin-film form. In order to establish the
trends of the dielectric properties for these materials, we calculated the
dielectric tensors of LuScO${}_{3}$ and YScO${}_{3}$. The dielectric tensor of
LuScO${}_{3}$ shows the continuation of the trend from LaScO${}_{3}$ to DyScO${}_{3}$.
Both $xx$ and $yy$ components are slightly smaller than for
DyScO${}_{3}$, their numerical values being 23.5 and 21.4 respectively.
On the other hand, the $zz$ component (44.8) is larger than for
DyScO${}_{3}$ (32.6) and for LaScO${}_{3}$ (27.4). YScO${}_{3}$ has
dielectric tensor components of 26.9, 23.0, and 37.7 for its
$xx$, $yy$, and $zz$ components, respectively.
The experimentally measured dependence of the dielectric tensor (and loss)
components
on temperature is shown in Fig. 6 for two compounds,
GdScO${}_{3}$ and DyScO${}_{3}$. In both cases we find that the dielectric tensor
properties
do not change significantly with temperature over the examined
range ($4.2$-$470$ K). The $\epsilon_{zz}$ component slightly
decreases with temperature while the $\epsilon_{xx}$ and $\epsilon_{yy}$ components show
the opposite
behavior. We expect that similar trends will be observed in all other
rare-earth scandates.
As our calculated dielectric tensor is at 0 K and our measured
dielectric tensor (Table 3 and 4)
is at room temperature, the absence of a
significant temperature dependence of the dielectric tensor is
important to our ability to make a meaningful comparison between the
calculated and measured coefficients.
Anomalies were observed in the dielectric loss along the
$z$ axis ($\tan\delta_{zz}$)
for both DyScO${}_{3}$ and GdScO${}_{3}$ below 50 K. The origin of these is
unknown, but their presence suggests the possibility of a
low-temperature phase transition.
III.3.2 Rare-earth yttrates
We now consider the rare-earth yttrates, i.e.,
AYO${}_{3}$ where A is one of the rare-earth
atoms. These are similar to the rare-earth scandates, but with yttrium on
the B site instead of scandium. Only one such compound, LaYO${}_{3}$,
is known to form a perovskite,Berndt et al. (1975) but others might form in
thin films via epitaxial stabilization. We find that LaYO${}_{3}$ has a
larger $zz$ component than does LaScO${}_{3}$. In LaYO${}_{3}$ the $zz$
component of the dielectric tensor is 38.0, while in LaScO${}_{3}$ it is
27.4. On the other hand, the $xx$ and $yy$ components are almost
unchanged with respect to LaScO${}_{3}$. In LaYO${}_{3}$ the $xx$ component is
30.6 and the $yy$ component is 25.7. We also find that heavier
rare-earth atoms on the A-site tend to destabilize this $Pbnm$ structure even further. For example, we find that DyYO${}_{3}$ in the
$Pbnm$ structure has an
unstable mode at $i70$ cm${}^{-1}$ that is IR-active along the $z$
direction.
III.3.3 CaZrO${}_{3}$, SrZrO${}_{3}$, and SrHfO${}_{3}$
According to our calculations, SrZrO${}_{3}$ and SrHfO${}_{3}$ show large and
rather isotropic dielectric tensors. The average dielectric tensor $\bar{\epsilon}$
is 40.9 and 32.8 in SrZrO${}_{3}$ and SrHfO${}_{3}$, respectively.
Their $x$-$y$ anisotropies have
an opposite sign as compared to all of the other compounds we analyzed.
Unfortunately, because of twinning (see section II.3.2), we
could only measure an average dielectric constant for these systems,
and therefore we could not directly compare our full calculated
dielectric tensors with experiment. Still, if we make a comparison
between theory and experiment for the average dielectric tensor $\bar{\epsilon}$,
the agreement is reasonable.
Our calculations suggest that the CaZrO${}_{3}$ compound, on the other hand, has
a very high value of the $z$-anisotropy of 28.1. Its $x$-$y$ anisotropy
of 1.1, on the other hand, is quite small. The average dielectric tensor
$\bar{\epsilon}=43.6$ is the highest among the all compounds we considered, mostly
because of the very large $\epsilon_{zz}$ component of the dielectric tensor.
Very similar results were also obtained in other theoretical
studies. ben ; Bennett et al. (2008)
III.3.4 La${}_{2}$BB${}^{\prime}$O${}_{6}$ compounds
The La${}_{2}$BB${}^{\prime}$O${}_{6}$ systems show a small, non-zero off-diagonal $\epsilon_{xz}$
component, $-$0.4 for La${}_{2}$MgZrO${}_{3}$ and 4 for La${}_{2}$CaZrO${}_{3}$.
$\epsilon_{xz}$ is allowed because the space group is reduced from
orthorhombic ($Pbnm$) to monoclinic ($P2_{1}/c$) for these compounds.
Their isotropically-averaged
dielectric tensors are larger for systems containing
Ca than for those with Mg, and a bit larger for those with Zr than for
those with Hf. Therefore, the dielectric response in this class of
materials is largest for La${}_{2}$CaZrO${}_{6}$, with $\bar{\epsilon}=28.5$, and smallest
for La${}_{2}$MgHfO${}_{6}$, with $\bar{\epsilon}=23.6$. All computed dielectric tensor
components for these systems are given in Table 4.
III.3.5 Rare-earth rare-earth perovskites
The 11 rare-earth–rare-earth perovskites we considered show a bigger
variation in the isotropically-averaged dielectric constant $\bar{\epsilon}$ than
do the rare-earth scandates (LaScO${}_{3}$ - DyScO${}_{3}$).
The largest average dielectric constant among them is 32.9 in
LaHoO${}_{3}$. The largest component of a dielectric tensor is also
found in LaHoO${}_{3}$, whose $\epsilon_{zz}$ is 41.7.
The measure $\Delta\epsilon_{\parallel}$ of $x$-$y$ anisotropy shows little variation among the
components in this series. The anisotropy is of the same sign as in
the rare-earth scandates.
Finally, the $z$ anisotropy $\Delta\epsilon_{\perp}$ once more shows a larger variation
than in the rare-earth scandates. This anisotropy is largest for
LaHoO${}_{3}$ and smallest for LaLuO${}_{3}$.
III.4 Decomposition of the ionic contribution to the dielectric
tensor
As already mentioned, the ionic contribution to the dielectric tensor
dominates in all of the systems we considered. The expression
for the ionic contribution given in Eq. (4) provides a
decomposition into contributions coming from eigenmodes of the
force-constant matrix.
The $Pbnm$ symmetry in perovskites, which is also approximately satisfied
in La${}_{2}$BB${}^{\prime}$O${}_{6}$ compounds, allows a given eigenmode to contribute only to a
single component ($\epsilon_{xx}$, $\epsilon_{yy}$, or $\epsilon_{zz}$) of the dielectric tensor.
This decomposition is given in Fig. 8 for all three components.
In rare-earth scandates, all three directions are evidently very
different. The $\epsilon_{xx}$ component is dominated by a low-lying mode
whose contribution
is almost constant along the series (it contributes to $\epsilon_{xx}$ by 9.6
for LaScO${}_{3}$ and 11.1 for DyScO${}_{3}$). The $\epsilon_{yy}$ component, on the other
hand, has sizable contributions coming from several modes. Finally, the
$\epsilon_{zz}$ component comes mostly from a single low-lying mode. Unlike
for the $\epsilon_{xx}$ component, the contribution from the mode responsible for
the $\epsilon_{zz}$ component changes dramatically across the series, varying from
6.9 for LaScO${}_{3}$ to 16.3 for DyScO${}_{3}$. This explains the large
value of the $z$ anisotropy in DyScO${}_{3}$ as compared to LaScO${}_{3}$ that
is visible in Fig. 7.
A behavior similar to that of the rare-earth scandates is also observed
in LaYO${}_{3}$ and in the rare-earth rare-earth perovskites.
The SrZrO${}_{3}$ and SrHfO${}_{3}$ compounds show a quite similar behavior to each
other. The $\epsilon_{xx}$ component has contributions coming from many modes,
the $\epsilon_{yy}$ component is dominated by a single low-lying mode, and
$\epsilon_{zz}$ is dominated by two low-lying modes. On the other hand, $\epsilon_{zz}$
component in CaZrO${}_{3}$ shows very large contribution coming from a
single low-lying mode.
Finally, we note that the La${}_{2}$BB${}^{\prime}$O${}_{6}$ compounds containing Ca have stronger contributions to
$\epsilon_{xx}$ and $\epsilon_{zz}$ from low-lying modes than do those containing Mg.
III.5 Compounds with $\it R\bar{3}c$ symmetry
At room temperature the ground state of BiFeO${}_{3}$
is ferroelectric with polar space group $R3c$, the pattern of
octahedral rotations being ($a^{-}a^{-}a^{-}$) in the Glazer notation.
At higher temperature, however, BiFeO${}_{3}$ undergoes a phase transition in
which the ferroelectricity and the ($a^{-}a^{-}a^{-}$) pattern of octahedral
rotations disappear simultaneously.Arnold et al. (2009); Haumont et al. (2008)
This observations led us to hypothesize that rotations of octahedra
around the pseudocubic $[111]$ axis, as in the ($a^{-}a^{-}a^{-}$) pattern, tend to
be energetically compatible with the presence of a ferroelectric
distortion along the same axis. This would tend to suggest that
perovskites that adopt the centrosymmetric $R\bar{3}c$ group, which
also exhibits the ($a^{-}a^{-}a^{-}$) pattern of oxygen octahedra, might be
close to a ferroelectric instability leading to the lower-symmetry
$R3c$ space group, and thus that such compounds might have an
especially large component of the dielectric tensor along the
pseudocubic $[111]$ axis.
To test this hypothesis, we have carried out a series of calculations
on SrZrO${}_{3}$ and GdScO${}_{3}$ in which structural relaxation was allowed
while maintaining the $R\bar{3}c$ symmetry. In both compounds we find
some IR-active phonon modes that either have very low or
imaginary frequency, indicating a near or actual instability.
In the case of SrZrO${}_{3}$ we find
a mode that is active along the $[111]$ pseudocubic
direction and has an extremely small frequency of only 6 cm${}^{-1}$,
while for GdScO${}_{3}$ we find that the corresponding mode
is unstable with an imaginary frequency of i142 cm${}^{-1}$.
These calculations show that imposing the $R\bar{3}c$ structure on
SrZrO${}_{3}$ and GdScO${}_{3}$ make them nearly or actually ferroelectric,
thus confirming our hypothesis.
Incidentally, the observation that SrZrO${}_{3}$ is more likely than
GdScO${}_{3}$ to be stabilized in the $R\bar{3}c$ structure is consistent with
the fact that
perovskite structures that prefer smaller rotation angles are more
likely to form $R\bar{3}c$ than $Pbnm$ structures, as discussed by
Woodward Woodward (1997). We find that the rotational angles
for SrZrO${}_{3}$ in the $Pbnm$ space group are $\theta_{\rm R}=11.6^{\circ}$ and
$\theta_{\rm M}=7.7^{\circ}$, while in GdScO${}_{3}$ they are substantially larger,
$\theta_{\rm R}=19.0^{\circ}$ and $\theta_{\rm M}=13.3^{\circ}$.
More directly, we also find that the ground-state energy of SrZrO${}_{3}$ having the $R\bar{3}c$ structure is only higher by 33.6 meV per
formula unit than in the $Pbnm$ structure. On the other
hand, in GdScO${}_{3}$ the $R\bar{3}c$ is higher in energy by a much larger
increment of 386 meV.
Finally, we note that LaAlO${}_{3}$, NdAlO${}_{3}$, and BaTbO${}_{3}$ may also be
of interest, as these all have the $R\bar{3}c$ space-group symmetry
and should also be chemically stable on silicon.
III.6 Correlation between structural and dielectric properties
The heuristic observation about BiFeO${}_{3}$ mentioned in the previous
section (Sec. III.5) led us to make a more detailed
analysis of the correlation between structural and dielectric anisotropies
in all five groups of $Pbnm$ perovskites.
As can be seen from Fig. 3, the presence of the octahedral
rotations breaks the symmetry among the three Cartesian directions in
the $Pbnm$ perovskites. One would therefore naively expect that the
anisotropy in the dielectric tensor component should also be correlated
with the size of these rotation angles, but this is not what we
observe. For
example, LaScO${}_{3}$ and DyScO${}_{3}$ both have rather substantial
octahedral rotation angles ($\theta_{\rm R}$ is 14.9${}^{\circ}$ in LaScO${}_{3}$ and
19.7${}^{\circ}$ in DyScO${}_{3}$), but they have very different values of the
dielectric $z$ anisotropy ($\Delta\epsilon_{\perp}$ is $-$1.0 in LaScO${}_{3}$ and 8.6 in
DyScO${}_{3}$). An even more extreme behavior can be seen in the case of
rare-earth perovskites between, e.g., LaHoO${}_{3}$ and LaLuO${}_{3}$.
Thus, we find no simple correlation between the dielectric tensor
anisotropies and the values of the octahedral rotation angles.
Instead, we find a correlation between the dielectric tensor
anisotropies and the mismatch of the two rotation angles
$\theta_{\rm R}$ and $\theta_{\rm M}$, as we explain next.
While the $Pbnm$ symmetry does not impose any relationship between
the two octahedral rotation angles $\theta_{\rm R}$ and $\theta_{\rm M}$, we find in
practice that all the compounds we studied obey the heuristic
relationship $\theta_{\rm R}\simeq\sqrt{2}\theta_{\rm M}$. This means that the oxygen
octahedra are rotated about the three Cartesian axes by almost the
same rotation angle, or equivalently, that the rotation axis is
nearly $\langle 111\rangle$. In the Glazer language, these
$Pbnm$ perovskites having ($a^{-}a^{-}c^{+}$) rotations can be said to be
very close to an ($a^{-}a^{-}a^{+}$) pattern.
We can measure the mismatch between the actual
($a^{-}a^{-}a^{+}$) and the hypothetical ($a^{-}a^{-}c^{+}$) rotation pattern by the
quantity $\theta_{\rm M}-\theta_{\rm R}/\sqrt{2}$, and it is this quantity that we
find to be strongly correlated with the dielectric anisotropy
$\Delta\epsilon_{\perp}$.
This is shown in Fig. 9, where $\Delta\epsilon_{\perp}$ is plotted versus
$\theta_{\rm M}-\theta_{\rm R}/\sqrt{2}$ for all of the compounds considered in this work.
It is apparent that the III-III–valent perovskites have a different
behavior than the II-IV–valent ones. Nevertheless, we conclude
that in both cases there is a strong correlation between the mismatch
angle and the dielectric tensor anisotropy. The sign of the correlation
is such that a deviation from the ($a^{-}a^{-}a^{+}$) pattern having an increased
rotation angle around the $z$ axis gives a larger dielectric tensor
component along the $z$ axis, and thus a larger $z$ anisotropy $\Delta\epsilon_{\perp}$.
III.7 Antisite substitutions
Experimentally
the compositions of the perovskites that we have been describing up to
now by their nominal compositions, e.g., LaLuO${}_{3}$, are in fact slightly
different from the compositions of the single crystals on which the
dielectric tensors were measured. This is because our crystals are
grown at the congruently melting compositions, e.g.,
La${}_{0.94}$Lu${}_{1.06}$O${}_{3}$,
which differ from the nominal compositions described up to now. The
congruently melting compositions of all relevant $Pbnm$ perovskites
studied have been found to be poor in the A-site cation and rich in
the B-site cation composition. Berkstresser et al. (1993); Ovanesyan et al. (1999); Gesing et al. (2009)
For this reason, we decided to carry out a theoretical analysis of
the effects of B atoms substituting at the A site on
the structural and dielectric properties of the material. Detailed
calculations were done only for the case of LaLuO${}_{3}$, but we expect
that similar trends will be observed in the remaining rare-earth
rare-earth perovskites as well as in the rare-earth scandates and
yttrates. Of course, other kinds of compositional disorder might
also be present, but such possibilities are not analyzed here.
III.7.1 Analysis of antisite defects in LaLuO${}_{3}$
We studied Lu${}_{\rm La}$ antisite defects in LaLuO${}_{3}$ using a
supercell approach. Specifically, in order to model a situation
in which one of every 16 La atoms is substituted by Lu, which is
about a 6% substitution, we constructed an 80-atom supercell
containing a single antisite defect. The supercell is enlarged
with respect to the primitive 20-atom $Pbnm$ primitive cell by
doubling along both the orthorhombic $a$ and $b$ lattice vectors.
The resulting stoichiometry is
$$\displaystyle({\rm La}_{0.9375}{\rm Lu}_{0.0625}){\rm Lu}{\rm O}_{3}{\rm\quad
or%
\quad}{\rm La}_{0.9375}{\rm Lu}_{1.0625}{\rm O}_{3}.$$
The presence of the Lu${}_{\rm La}$ antisite in this particular 80-atom
supercell reduces the crystal symmetry from orthorhombic $Pbnm$ to
monoclinic $Pm$.
After full relaxation of the crystal structure in this
space group, we find that the $a$, $b$, and $c$ lattice vectors
are reduced by 0.3%, 0.2% ,and 0.1%, respectively, while the
monoclinic angle between $a$ and $b$ lattice vectors of
$90.03^{\circ}$ deviates only very slightly from $90^{\circ}$.
The influence of the Lu${}_{\rm La}$ substitution on the dielectric
properties is more complex. Evaluated in the same coordinate frame as
in the $Pbnm$ unit cell, the $\epsilon_{xx}$ and $\epsilon_{yy}$ dielectric tensor components
remain almost unchanged, and the new $\epsilon_{xy}$ component allowed
by the monoclinic symmetry is quite small, only $2.1$. On the other
hand, the $\epsilon_{zz}$ component is drastically altered by the presence
of Lu atom on the La site. In fact, we find that the 80-atom
supercell is actually just barely unstable in the $Pm$ space group,
as indicated by the presence of a phonon mode with a very small
imaginary frequency of $i16$ cm${}^{-1}$. spo
The contribution of this phonon mode to the $\epsilon_{zz}$ component (evaluated
in the unstable $Pm$ structure) is therefore negative, specifically,
$-33.6$. Since this phonon frequency is so close to zero, we expect
that it would get renormalized to positive frequency at
room temperature. For this reason, we did not follow the structural
relaxation of our 80-atom supercell along the direction of the
unstable mode, and a realistic estimate of the dielectric
response of the system is difficult. Nevertheless, we conclude that
Lu${}_{\rm La}$ substitutions in LaLuO${}_{3}$ have the potential to
increase the $\epsilon_{zz}$ dielectric tensor component substantially.
III.7.2 Discussion
As can be clearly seen in Fig. 7, our calculated $z$
anisotropy ($\Delta\epsilon_{\perp}$) is consistently larger than the measured one
for all the rare-earth scandates for which we have experimental
measurements. In view of the calculations reported for LaLuO${}_{3}$ above, we
tentatively attribute this discrepancy to the generic tendency
of B atoms to substitute on the A site in these compounds.
This observation is consistent with the fact that smaller B ions
that substitute for larger A ions will reside in a relatively
larger cage, providing room to rattle and thereby contribute to an
enhanced dielectric response.
IV Summary
The main focus of this work has been the application of both
computational and experimental methods to study the structural and
dielectric properties of various $Pbnm$ perovskites that have
potentially large dielectric tensor components and are chemically
stable on silicon
up to $\sim$1000 ${}^{\circ}$C. Schlom et al. (2005)
Such compounds might be good candidates for
future use as high-K dielectrics in microelectronics applications,
e.g., as a possible replacement of hafnia-based high-K dielectrics
currently used in the CMOS transistors in integrated circuits.
Of the compounds we have considered, CaZrO${}_{3}$, SrZrO${}_{3}$, LaHoO${}_{3}$, and LaYO${}_{3}$
appear to be especially promising. CaZrO${}_{3}$ has the largest calculated
average dielectric tensor ($\bar{\epsilon}=43.6$) among the compounds we
considered, and SrZrO${}_{3}$ is a close second with $\bar{\epsilon}=40.9$. The dielectric
tensor in CaZrO${}_{3}$ is very anisotropic, with its $\epsilon_{zz}$ component
almost twice as large as $\epsilon_{xx}$ or $\epsilon_{yy}$, while on the other hand
SrZrO${}_{3}$ has an almost isotropic dielectric tensor. Unfortunately,
the full
dielectric tensors of these compounds have not yet been measured due to
lack of single crystals.
Of the rare-earth rare-earth $Pbnm$ perovskites, only LaLuO${}_{3}$ has
had its dielectric tensor measured to date. Our results on this
compound will be presented in detail elsewhere. hee
The theoretical calculations, however, indicate that other
compounds in this series should have even larger dielectric
tensor components, with LaHoO${}_{3}$, having $\bar{\epsilon}=32.9$, being the
most promising among these. LaYO${}_{3}$ is expected to behave very
similar to LaHoO${}_{3}$ since Y and Ho have almost the same ionic
radii, so it may be promising as well ($\bar{\epsilon}=31.4$).
Thus, this series of
compounds clearly deserves additional scrutiny.
Of course, there are good reasons for preferring amorphous over
single-crystalline materials for such high-K applications.
Certainly the ability of amorphous SiO${}_{2}$ to conform to the
substrate and to eliminate electrical traps played a central role
in its dominance as the gate dielectric
of choice for 40 years for silicon-based metal-oxide-semiconductor
field-effect transistors.
The present
hafnia-based high-K dielectrics are amorphous or
nanocrystalline. mul
For this reason, any eventual application of these materials for
high-K applications would presumably require the adoption of one of
two strategies. The first is the possibility of growing crystalline
epitaxial oxides directly on silicon, which clearly would require
a very high level of control of interface chemistry and morphology
before it could become a practical solution. The second is the
possibility that some of the compounds investigated here could be
synthesized in amorphous or nanocrystalline form. We have not
investigated these issues here, nor have we tried to calculate
what (possibly very substantial) changes in the dielectric properties would
occur in the amorphous counterparts, as this would take us far
beyond the scope of the present study. Nevertheless,
these are important questions for future investigations.
Acknowledgements.
The work of S. C. and D. V. was supported in part by NSF Grant
DMR-0545198.
The work of T.H. was supported by the Pennsylvania State University
Materials Research Institute Nanofabrication Lab, the National
Science Foundation Cooperative Agreement No. 0335765 and National
Nanotechnology Infrastructure Network, with Cornell University.
S. T. M. acknowledge support from NSF DMR-0602770.
The research conducted by M. D. B. at the Center for Nanophase
Materials Sciences, is sponsored at Oak Ridge National Laboratory by
the Division of Scientific User Facilities, U.S. Department of
Energy.
D. G. S. would like to acknowledge support from the Semiconductor
Research Corporation and Intel.
References
Moore (2003)
G. Moore, in
2003 IEEE International Solid-State Circuits
Conference. (Piscataway, NJ, USA, 2003),
vol. 1, pp. 20–23.
Schlom et al. (2005)
D. G. Schlom,
C. A. Billman,
J. H. Haeni,
J. Lettieri,
P. H. Tan,
R. R. M. Held,
S. Völk, and
K. J. Hubbard, in
Thin Films and Heterostructures for Oxide
Electronics, edited by S. B.
Ogale (Springer, 2005),
pp. 31–78.
Schlom et al. (2008)
D. G. Schlom,
S. Guha, and
S. Datta,
Mater. Res. Bull. 33,
1017 (2008).
Robertson (2008)
J. Robertson,
J. Appl. Phys. 104,
124111 (2008).
Mistry et al. (2007)
K. Mistry,
C. Allen,
C. Auth,
B. Beattie,
D. Bergstrom,
M. Bost,
M. Brazier,
M. Buehler,
A. Cappellani,
R. Chau, et al.,
in Electron Devices Meeting, 2007. IEDM 2007. IEEE
International (2007), pp. 247–250.
Hicks et al. (2008)
J. Hicks,
D. Bergstrom,
M. Hattendorf,
J. Jopling,
J. Maiz,
S. Pae,
C. Prasad, and
J. Wiedemer,
Intel Technol. J. 12,
131 (2008).
Scansen (21 Jan. 2008)
D. Scansen,
Under the hood: 45 nm: What intel didn’t tell you
(21 Jan. 2008),
www.techonline.com/product/underthehood/205918004.
Guha and Narayanan (2007)
S. Guha and
V. Narayanan,
Phys. Rev. Lett. 98,
196101 (2007).
Christen et al. (2006)
H. M. Christen,
J. G. E. Jellison,
I. Ohkubo,
S. Huang,
M. E. Reeves,
E. Cicerrella,
J. L. Freeouf,
Y. Jia, and
D. G. Schlom,
Appl. Phys. Lett. 88,
262906 (2006).
Afanas’ev et al. (2004)
V. V. Afanas’ev,
A. Stesmans,
C. Zhao,
M. Caymax,
T. Heeg,
J. Schubert,
Y. Jia,
D. G. Schlom,
and G. Lucovsky,
Appl. Phys. Lett. 85,
5917 (2004).
Zhao et al. (2005)
C. Zhao,
T. Witters,
B. Brijs,
H. Bender,
O. Richard,
M. Caymax,
T. Heeg,
J. Schubert,
V. V. Afanas’ev,
A. Stesmans,
et al., Appl. Phys. Lett.
86, 132903 (2005).
Heeg et al. (2006)
T. Heeg,
J. Schubert,
C. Buchal,
E. Cicerrella,
J. Freeouf,
W. Tian,
Y. Jia, and
D. Schlom,
Appl. Phys. A 83,
103 (2006).
Edge et al. (2006a)
L. Edge,
D. Schlom,
S. Stemmer,
G. Lucovsky, and
J. Luning,
Radiat. Phys. Chem. 75,
1608 (2006a).
Cicerrella et al. (2005)
E. Cicerrella,
J. L. Freeouf,
L. F. Edge,
D. G. Schlom,
T. Heeg,
J. Schubert, and
S. A. Chambers,
J. Vac. Sci. Technol., A 23,
1676 (2005).
Afanas’ev et al. (2006)
V. V. Afanas’ev,
A. Stesmans,
L. F. Edge,
D. G. Schlom,
T. Heeg, and
J. Schubert,
Appl. Phys. Lett. 88,
032104 (2006).
Edge et al. (2006b)
L. F. Edge,
D. G. Schlom,
S. Rivillon,
Y. J. Chabal,
M. P. Agustin,
S. Stemmer,
T. Lee,
M. J. Kim,
H. S. Craft,
J.-P. Maria,
et al., Appl. Phys. Lett.
89, 062902
(2006b).
Sivasubramani et al. (2006)
P. Sivasubramani,
T. H. Lee,
M. J. Kim,
J. Kim,
B. E. Gnade,
R. M. Wallace,
L. F. Edge,
D. G. Schlom,
F. A. Stevie,
R. Garcia,
et al., Appl. Phys. Lett.
89, 242907 (2006).
Lopes et al. (2007)
J. Lopes,
M. Roeckerath,
T. Heeg,
U. Littmark,
J. Schubert,
S. Mantl,
Y. Jia, and
D. Schlom,
Microelectron. Eng. 84,
1890 (2007).
Wang et al. (2007)
M. Wang,
W. He,
T. P. Ma,
L. F. Edge, and
D. G. Schlom,
Appl. Phys. Lett. 90,
053502 (2007).
Edge et al. (2008)
L. Edge,
W. Tian,
V. Vaithyanathan,
T. Heeg,
D. Schlom,
D. Klenov,
S. Stemmer,
J. Wang, and
M. Kim, in
ECS Trans. (Honolulu, HI,
United states, 2008), vol. 16, pp.
213 – 227.
Özben et al. (2008)
E. D. Özben,
J. M. J. Lopes,
M. Roeckerath,
S. Lenk,
B. Holländer,
Y. Jia,
D. G. Schlom,
J. Schubert, and
S. Mantl,
Appl. Phys. Lett. 93,
052902 (2008).
Roeckerath et al. (2009)
M. Roeckerath,
J. Lopes,
E. Durgun Ozben,
C. Sandow,
S. Lenk,
T. Heeg,
J. Schubert, and
S. Mantl,
Appl. Phys. A 94,
521 (2009).
Wagner et al. (2006)
M. Wagner,
T. Heeg,
J. Schubert,
C. Zhao,
O. Richard,
M. Caymax,
V. Afanas’ev,
and S. Mantl,
Solid-State Electron. 50,
58 (2006).
Myllymaki et al. (Sept. 2007)
P. Myllymaki,
M. Roeckerath,
M. Putkonen,
S. Lenk,
J. Schubert,
L. Niinisto, and
S. Mantl,
Appl. Phys. A A88,
633 (Sept. 2007).
Kim et al. (2006)
K. H. Kim,
D. B. Farmer,
J.-S. M. Lehn,
P. V. Rao, and
R. G. Gordon,
Appl. Phys. Lett. 89,
133512 (2006).
Thomas et al. (2007)
R. Thomas,
P. Ehrhart,
M. Roeckerath,
S. van Elshocht,
E. Rije,
M. Luysberg,
M. Boese,
J. Schubert,
M. Caymax, and
R. Waser, J.
Electrochem. Soc. 154, 147
(2007).
Heeg et al. (2007)
T. Heeg,
M. Roeckerath,
J. Schubert,
W. Zander,
C. Buchal,
H. Y. Chen,
C. L. Jia,
Y. Jia,
C. Adamo, and
D. G. Schlom,
Appl. Phys. Lett. 90,
192901 (2007).
Wersing (1996)
W. Wersing,
Curr. Opin. Solid State Mater. Sci.
1, 715 (1996).
Vanderah (2002)
T. A. Vanderah,
Science 298,
1182 (2002).
Delugas et al. (2007)
P. Delugas,
V. Fiorentini,
A. Filippetti,
and G. Pourtois,
Phys. Rev. B 75,
115126 (2007).
Vali (2008)
R. Vali, Solid
State Commun. 145, 497
(2008).
Vali (2009)
R. Vali, Solid
State Commun. 149, 519
(2009).
(33)
See
www.physics.rutgers.edu/~sinisa/highk/supp.pdf or
http://link.aps.org/supplemental/10.1103/PhysRevB.82.064101 for zone-center
phonon frequencies, as well as the infrared activities for those modes that
are infrared-active.
Badie (1978)
J. Badie, Rev.
Int. Hautes Temp. Réfact., Fr 15,
183 (1978).
Badie and Foex (1978)
J. Badie and
M. Foex, J.
Solid State Chem. 26, 311
(1978).
Coutures et al. (1980)
J.-P. Coutures,
J. Badie,
R. Berjoan,
J. Coutures,
R. Flamand, and
A. Rouanet,
High Temp. Sci. 13,
331 (1980).
(37)
J. Schubert (private communication).
Berndt et al. (1975)
U. Berndt,
D. Maier, and
C. Keller,
J. Solid State Chem. 13,
131 (1975).
Rabenau (1956)
A. Rabenau, Z.
Anorg. Allg. Chem. 288, 221
(1956).
Ito et al. (2001)
K. Ito,
K. Tezuka, and
Y. Hinatsu,
J. Solid State Chem. 157,
173 (2001).
Bharathy et al. (2009)
M. Bharathy,
A. Fox,
S. Mugavero, and
H.-C. zur Loye,
Solid State Sci. 11,
651 (2009).
(42)
T. Heeg, K. Wiedenmann, M. Roeckerath, S. Coh, D. Vanderbilt, J.
Schubert and D. G. Schlom, unpublished.
Goldschmidt (1926)
V. Goldschmidt,
Naturwissenschaften 21,
477 (1926).
Woodward (1997)
P. M. Woodward,
Acta Crystallogr. Sect. B 53,
32 (1997).
Glazer (1972)
A. M. Glazer,
Acta Crystallogr. Sect. B 28,
3384 (1972).
Giannozzi et al. (2009)
P. Giannozzi,
S. Baroni,
N. Bonini,
M. Calandra,
R. Car,
C. Cavazzoni,
D. Ceresoli,
G. L. Chiarotti,
M. Cococcioni,
I. Dabo, et al.,
Journal of Physics: Condensed Matter
21, 395502 (19pp)
(2009), URL http://www.quantum-espresso.org.
Perdew et al. (1996)
J. P. Perdew,
K. Burke, and
M. Ernzerhof,
Phys. Rev. Lett. 77,
3865 (1996).
Vanderbilt (1985)
D. Vanderbilt,
Phys. Rev. B 32,
8412 (1985).
Monkhorst and Pack (1976)
H. J. Monkhorst
and J. D. Pack,
Phys. Rev. B 13,
5188 (1976).
Staroverov et al. (2004)
V. N. Staroverov,
G. E. Scuseria,
J. Tao, and
J. P. Perdew,
Phys. Rev. B 69,
075102 (2004).
Pies and Weiss (2006)
W. Pies and
A. Weiss,
Landolt-Börnstein - Group III Condensed Matter
(Springer-Verlag, 2006), vol.
7c1, chap. VI.1.5.1, pp.
14–34.
(52)
In the case of GdN the anomaly is more likely the result of the
large spin splitting and strong ferromagnetism associated with the huge
magnetic moment of the $4f^{7}$ configuration.
Baroni et al. (2001)
S. Baroni,
S. de Gironcoli,
A. Dal Corso,
and
P. Giannozzi,
Rev. Mod. Phys. 73,
515 (2001).
Cockayne and Burton (2000)
E. Cockayne and
B. P. Burton,
Phys. Rev. B 62,
3735 (2000).
Uecker et al. (2006)
R. Uecker,
H. Wilke,
D. Schlom,
B. Velickov,
P. Reiche,
A. Polity,
M. Bernhagen,
and M. Rossberg,
J. Cryst. Growth 295,
84 (2006).
Uecker et al. (2008)
R. Uecker,
B. Velickov,
D. Klimm,
R. Bertram,
M. Bernhagen,
M. Rabe,
M. Albrecht,
R. Fornari, and
D. Schlom,
J. Cryst. Growth 310,
2649 (2008).
(57)
The SrZrO${}_{3}$, SrHfO${}_{3}$, and LaScO${}_{3}$ single crystals were
grown by float-zone by Dima Souptel and Anatoly Balbashov of the Moscow Power
Engineering Institute, Moscow, Russia. For details on the growth parameters
see D. Souptel, G. Behr, and A.M. Balbashov, J. Cryst. Growth 236, 583
(2002).
Newnham (2005)
R. Newnham,
Properties of Materials: Anisotropy, Symmetry,
Structure (Oxford University Press, Oxford,
2005).
Gesing et al. (2009)
T. M. Gesing,
R. Uecker, and
J.-C. Buhl,
Z. Kristallogr. NCS 224,
365 (2009).
Anderson et al. (1993)
M. T. Anderson,
K. B. Greenwood,
G. A. Taylor,
and K. R.
Poeppelmeier, Prog. Sol. St. Chem.
22, 197 (1993).
Liferovich and Mitchell (2004)
R. P. Liferovich
and R. H.
Mitchell, J. Phys. Chem. Solids
177, 2188
(2004).
Velickov et al. (2007)
B. Velickov,
V. Kahlenberg,
R. Bertram, and
M. Bernhagen,
Z. Kristallogr. 9,
466 (2007).
Kennedy
et al. (1999a)
B. J. Kennedy,
C. J. Howard,
and B. C.
Chakoumakos, Phys. Rev. B
59, 4023
(1999a).
Kennedy
et al. (1999b)
B. J. Kennedy,
C. J. Howard,
and B. C.
Chakoumakos, Phys. Rev. B
60, 2972
(1999b).
Velickov et al. (2008)
B. Velickov,
V. Kahlenberg,
R. Bertram, and
R. Uecker,
Acta Crystallogr. Sect. E 64,
i79 (2008).
Ruiz-Trejo et al. (1999)
E. Ruiz-Trejo,
M. S. Islam, and
J. A. Kilner,
Solid State Ionics 123,
121 (1999).
Levin et al. (2003)
I. Levin,
T. G. Amos,
S. M. Bell,
L. Farber,
T. A. Vanderah,
R. S. Roth, and
B. H. Toby,
J. Solid State Chem. 175,
170 (2003), ISSN 0022-4596.
(68)
J. W. Bennett, private communication.
Bennett et al. (2008)
J. W. Bennett,
I. Grinberg, and
A. M. Rappe,
Chem. Mater 20,
5134 (2008).
Arnold et al. (2009)
D. C. Arnold,
K. S. Knight,
F. D. Morrison,
and
P. Lightfoot,
Phys. Rev. Lett. 102,
027602 (2009).
Haumont et al. (2008)
R. Haumont,
I. A. Kornev,
S. Lisenkov,
L. Bellaiche,
J. Kreisel, and
B. Dkhil,
Phys. Rev. B 78,
134108 (2008).
Berkstresser et al. (1993)
G. W. Berkstresser,
A. J. Valentino,
and C. D.
Brandle, J. Cryst. Growth
128, 684 (1993).
Ovanesyan et al. (1999)
K. L. Ovanesyan,
A. Petrosyan,
G. O. Shirinyan,
C. Pedrini, and
L. Zhang, J.
Cryst. Growth 198, 497
(1999).
(74)
The eigendisplacement of this unstable mode is very close to
that of the $S$-point $(\frac{1}{2}\frac{1}{2}0)$ phonon of the 20-atom
$Pbnm$ structure having a frequency of 39 cm${}^{-1}$. However, this phonon
remains inactive in the $Pbnm$ structure because it does not appear at the
$\Gamma$ point.
(75)
D. A. Muller, private communication. |
A high resolution scintillating fiber tracker with SiPM readout for
the PEBS experiment
H. Gast
R. Greim
T. Kirn
G. Roper Yearwood${}^{*}$
S. Schael
I. Physikalisches Institut B, Rheinisch-Westfälische Technische
Hochschule Aachen,
Aachen, 52062, Germany
${}^{*}$E-mail: roper@physik.rwth-aachen.de
www.physik.rwth-aachen.de
Abstract
Using thin scintillating fibers with Silicon Photomultiplier (SiPM) readout a modular high-resolution charged-particle tracking detector has been designed. The fiber modules consist of 2 x 5 layers of 128 round multiclad scintillating fibers of 0.250mm diameter. The fibers are read out by four SiPM arrays (8mm x 1mm) each on either end of the module.
The basic features of this detector concept have been evaluated in a test beam in October 2006 using novel SiPM detectors with improved photon detection efficiency. This detector has been developed for a balloon borne spectrometer (PEBS) to measure the comsic ray positron- and electron flux with high precision.
This particle detection concept is also very interesting for future applications, for example as an outer layer of an ILC detector or for other astroparticle physics experiments.
keywords: Scintillating fiber tracker, silicon photomultiplier arrays, balloon experiments
\bodymatter
1 Introduction
Scintillating fiber trackers have already been realized using
Visible-Light Photon Counters[1] or Multi-Anode
Photomultiplier Tubes[2] as photodectors. Multi-anode PMTs
do not exhibit a sufficiently high photodetection efficiency (generally around
$20\%$ to $30\%$) to encourage their use for a high precision SciFi tracker
using very thin scintillating fibers which yield only a small number of photons
for a traversing minimal ionizing particle. VLPCs on the other hand require a
significant overhead since they require operating temperatures of $7K$.
SiPMs achieve high photodetection efficiencies (PDE) of over $60\%$[3]
and allow for operation at room temperature ($>20^{\circ}C$). Furthermore
novel SiPM arrays with very densely packed channels allow for
a compact design (see fig. 1).
2 SiPM
A SiPM is a multipixel semiconductor photodiode that achieves a high intrinsic gain of $10^{5}$ to $10^{6}$ when being operated above its breakdown voltage[4]. In addition, they scale to small dimensions, allowing for a compact readout of thin scintillating fibers.
Commercial distributors of SiPMs are Hamamatsu[5], Japan and Photonique[6], Switzerland. Photonique SiPMs of type SSPM-0606EXP ($40\%$ PDE) and SSPM-050701GR ($25\%$ PDE) were used during the beamtest of a SciFi/SiPM tracker prototype (fig. 2). New SiPMs of type Hamamatsu MPPC S10362-100C ($65\%$ PDE) and 32 channel SiPM arrays produced by the INFN Perugia are already available and currently tested.
3 Beamtest of the first Prototype
The prototype consisted of $300\mu m\times 300\mu m$ square, multiclad fibers of type Bicron[7] BCF-20 with white EMA coating and Photonique SiPMs of type SSPM-050701GR and of type SSPM-0606EXP. The peak emission wavelength of BCF-20 fibers is at $492nm$, matching the peak sensitivity of Photonique SSPM-0606EXP SiPMs.
The scintillating fibers were arranged in two ribbons of $3\times 10$ fibers. The fiber ribbons were stabilized using glue as an adhesive. Both ends of the 3-fiber ribbons were glued into a plastic connector and polished. One end was connected to a SiPM by mounting it into a copper block and held in place by an aluminum frame and a spring.
The SiPMs were mounted into the copper-block to allow for a temperature control. During part of the beam test the opposing end of the fibers was covered by a reflective foil to increase the light output for the SiPMs.
A beam telescope with four silicon strip modules from the CMS tracker project was used to measure the position of the incident particles.
The beamtest of the prototype took place in a 10GeV proton beam at PS, CERN. During the beamtest, 1.5 million events were recorded and about 800,000 particle tracks were reconstructed with the beam telescope. The position of each fiber column was determined by reconstructing the position of particles that produced a high signal within the fiber. The average measured distance between two fiber columns was $309\mu m$ with a precision of $10\mu m$. The spatial resolution for particles of perpendicular incidence that pass through all three fibers of one fiber column was about $90\mu m$ which matches the expected intrinsic resolution of $\frac{d}{\sqrt{12}}$ where $d$ is the fiber pitch.
Knowing the positions of the fibers, we determined the average photoelectron yield for particles that passed through one of the fiber columns. For particles with perpendicular incidence the average photoelectron yield for both types of SiPMs with and without reflective foil on the opposing fiber end was measured. The SSPM-050701GR signal was $1.9\pm 0.3$ photoelectrons without and $3.2\pm 0.2$ photoelectrons with reflective foil. The SSPM-0606EXP achieved an average photoelectron yield per MIP of $3.6\pm 0.2$ without reflective foil and $5.4\pm 0.3$ with reflective foil (omitting one SiPM that actually showed a reduced photoelectron yield after adding the reflective foil).
The mean efficiency for perpendicular incidence, setting a cut at 0.5 photoelectrons, was $96\%$ for the ribbon read out by the SSPM 0606EXP and $91\%$ for the SSPM 050701GR (fig. 2).
4 Tracker Design
The tracker design for the PEBS detector (fig. 3) is modular. It consists of several layers of tracker modules, each module consisting of 10 layers of $0.25mm$ thin, round scintillating fibers with 128 fibers in every layer. 5 layers of fibers are glued to each side of a $10mm$ thick module core of Rohacell foam covered by $100\mu m$ thin carbon fiber skins on either side of the foam. Neighboring layers are shifted by one half of the fiber pitch with respect to each other to improve the spatial resolution.
SiPM arrays with an area of $8mm\times 1.1mm$ and 32 readout cells, each sensitive cell covering an area of $0.23mm\times 1.1mm$, are used for column-wise fiber readout. The SiPM arrays are mounted on alternating ends of the fiber modules along with an integrated preamplifier and digitization solution. The respective opposing ends of the fibers are covered by a reflective coating.
A dedicated Monte Carlo simulation, using the GEANT4[11] package, has been developed for comparison to and generalization of the testbeam results[10]. A key question to be answered was the spatial resolution obtained with a fiber module as a function of the mean photoelectron yield $n_{p.e.}$ of the fiber-SiPM chain (see fig. 4). For the photo electron yield achieved in the testbeam, a spatial resolution of $72\,\mu{}m$ is obtained at the mean projected angle of incidence, which is $\bar{\alpha}=11^{\circ}$ for the PEBS geometry.
5 Conclusion
The testbeam results indicate that this concept for a high-resolution SciFi/SiPM tracker is technically feasible. The average yield of 5.4 photoelectrons with a reflective foil on one fiber end and the SSPM-0606EXP SiPMs is acceptable. SiPMs with a reduced pixel density and a $50\%$ higher PDE are commercially available from Hamamatsu and will be tested thoroughly during the next test beam in fall 2007.
The light output from scintillating fibers can be improved by $20\%-40\%$ using fibers without white coating as measurements with different fiber coatings conducted for the CREAM experiment have shown[12].
A spatial resolution as good as $40\mu m$ is possible, depending on the granularity of the readout, the quality of the SiPMs, the qualtity of the optical coupling of the fibers to the SiPMs and the type of fibers used. For the final tracker an average spatial resolution of $60\mu m$ is expected.
References
[1]
DØ Collaboration (NIM A565, p. 463-537, 2006).
[2]
E. C. Aschenauer, J. Baehr, V. Gapienko, B. Hoffmann, A. Kharchilava, H. Luedecke, R. Nahnhauer, R. Shanidze, Development of scintillating fiber detector technology for high rate particle tracking (arXiv:hep-ex/9710001v1).
[3]
K. Yamamoto, K. Yamamura, K. Sato, T. Ota, H. Suzuki, S. Ohsuka (Nuclear Science Symposium Conference Record, p. 1094-1097, 2006).
[4]
Dolgoshein, B. et al. (NIM A504, p. 48-52, 2003).
[5]
Hamamatsu Photonics, K.K., Japan (URL: http://sales.hamamatsu.com).
[6]
Photonique SA, Switzerland (URL: http://www.photonique.ch).
[7]
Saint-Gobain Crystals, Paris, France (URL: http://www.bicron.com).
[8]
The CMS Collaboration (NIM A517, p. 77-93, 2004).
[9]
Musienko, Y., Measurements for the PEBS project at APDlab, CERN, Switzerland, 2007.
[10]
Gast, H. et al. (arXiv:astro-ph/0702567v1).
[11]
A toolkit for the simulation of the passage of particles through matter (URL: www.cern.ch/geant4/).
[12]
Young Soo Yoon et al. (29th ICRC Pune, p. 101-104, 2005). |
ViTKD: Practical Guidelines for ViT feature knowledge distillation
Zhendong Yang${}^{1,2}$ Zhe Li Ailing Zeng${}^{2}$ Zexian Li${}^{3}$ Chun Yuan$\dagger$${}^{1}$ Yu Li$\dagger$${}^{2}$
${}^{1}$Tsinghua Shenzhen International Graduate School
${}^{2}$International Digital Economy Academy (IDEA) ${}^{3}$Beihang University
{yangzd21@mails, yuanc@sz}.tsinghua.edu.cn axel.li@outlook.com
{zengailing, liyu}@idea.edu.cn lizexian0427@gmail.com
Abstract
Knowledge Distillation (KD) for Convolutional Neural Network (CNN) is extensively studied as a way to boost the performance of a small model. Recently, Vision Transformer (ViT) has achieved great success on many computer vision tasks and KD for ViT is also desired. However, besides the output logit-based KD, other feature-based KD methods for CNNs cannot be directly applied to ViT due to the huge structure gap. In this paper, we explore the way of feature-based distillation for ViT. Based on the nature of feature maps in ViT, we design a series of controlled experiments and derive three practical guidelines for ViT’s feature distillation. Some of our findings are even opposite to the practices in the CNN era. Based on the three guidelines, we propose our feature-based method ViTKD which brings consistent and considerable improvement to the student. On ImageNet-1k, we boost DeiT-Tiny from $74.42\%$ to $76.06\%$, DeiT-Small from $80.55\%$ to $81.95\%$, and DeiT-Base from $81.76\%$ to $83.46\%$. Moreover, ViTKD and the logit-based KD method are complementary and can be applied together directly. This combination can further improve the performance of the student. Specifically, the student DeiT-Tiny, Small, and Base achieve $77.78\%$, $83.59\%$, and $85.41\%$, respectively.
***The code is available at https://github.com/yzd-v/cls_KD.
22footnotetext: Corresponding authors
1 Introduction
Knowledge Distillation (KD) (Hinton et al., 2015) utilizes the output of the teacher model as soft labels to supervise the student model, bringing the lightweight models impressive improvements without extra costs for inference. It has been consistently explored for Convolutional Neural Network (CNN) models and applied successfully to many vision tasks successfully, including image classification (Zhou et al., 2020; Yang et al., 2020; Chen et al., 2021; Zhao et al., 2022; Lin et al., 2022), object detection (Li et al., 2022a; Yang et al., 2022d; Zheng et al., 2022; Yang et al., 2022a; Wang et al., 2022), semantic segmentation (Liu et al., 2019; He et al., 2019; Shu et al., 2021; Yang et al., 2022b).
Recently, Vision Transformer (ViT) (Dosovitskiy et al., 2021) has achieved great success for image classification and inspired various transformers (Yuan et al., 2021; Han et al., 2021; Touvron et al., 2021b; Liu et al., 2021). Compared with CNN-based models, the ViT-based methods generally need more parameters but can achieve better performance, making them harder to be deployed. Therefore, boosting the performance of small ViT models using KD is of great value.
In this work, we look into how to apply KD to ViT-based models. A direct thought would be directly transferring the KD methods for CNN to ViT. In fact, some fundamental distillation works (Hinton et al., 2015; Romero et al., 2014) are inherently structure-independent. For example, the classic logit-based distillation directly use the model’s final output logit, thus it can apply for both CNNs and ViTs. DeiT (Touvron et al., 2021a) verifies this for ViT’s distillation.
However, the rest KD methods are mostly specially designed for CNN-based models and many of them work on the intermediate features. They are inapplicable to ViT-based models as there is a huge gap between these two architectures. The recent MiniViT (Zhang et al., 2022) adopts Self-Attention distillation and Hidden-State Distillation for feature-based distillation. Compared with the logit based distillation, its improvement is still quite limited.
Before developing new feature-based KD for ViT, we first conduct simple studies of transferring the knowledge from the last layer of a teacher (DeiT-Small) following the CNN’s distillation, from the last 6 layers like PKD (Sun et al., 2019) for BERT’s (Devlin et al., 2018) distillation, and from the whole 12 layers. Surprisingly, the results for all the intuitive feature distillations shown in Table 1 are not satisfactory which consistently degrade the performance of the student (DeiT-Tiny). Specifically, the Top-1 accuracy of the student is just $73.36\%$ when distilling on the last layer. This distillation on the last layer is widely used for CNN’s distillation, but here it causes a $1.06\%$ accuracy drop.
To further explore the features in ViT, we visualize the attention maps of the student and the teacher across different layers in Figure 1. For the shallow layers (e.g., layers 0 and 1), the attention appears mainly on the diagonal, which indicates they focus on themselves. Both the student and teacher have similar patterns. While for the deep layers (e.g., layers 10 and 11), the difference between student and teacher’s attention is greater. Their attention is decided by several sparse key tokens. Besides, they focus on completely different tokens. Such gap makes it hard for the student to mimic the teacher’s final feature directly. This phenomenon suggests different layers may need different methods.
Accordingly, we perform a series of controlled experiments to examine the effects of different distillation methods, different layers, and different modules. As a consequence, we derive three practical guidelines for ViT’s feature distillation in Section 2. Based on these principles, we propose a nontrivial way for feature-based ViT distillation, named ViTKD, and describe the details in Section 3. Extensive experiments demonstrate its effectiveness in Section 4. For instance, we boost the student DeiT-Tiny from $74.42\%$ to $76.06\%$, DeiT-Small from $80.55\%$ to $81.95\%$ and DeiT-Base from $81.76\%$ to $83.46\%$ on ImageNet-1K. Besides, when combining ViTKD with the logit-based distillation, we can further advance their Top-1 accuracy to $77.78\%$, $83.59\%$ and $85.41\%$. The comparison is shown in Figure 2. We also demonstrate the models trained with distillation are beneficial to other vision tasks like object detection.
2 Practical Guidelines for ViT’s Feature Distillation
To explore the practical guidelines, we take larger DeiT (Touvron et al., 2021a) and DeiT III (Touvron et al., 2022) models as the teacher to distill lighter DeiT models on ImageNet-1k (Deng et al., 2009). The DeiT teacher is trained from scratch on ImageNet-1K, and DeiT III teacher is pre-trained on ImageNet-21K. As shown in Section 1, the attention maps vary greatly from different layers. Based on this observation, we analyze where and how to distill the student effectively and propose three practical guidelines for ViT’s feature distillation. Specifically, we conduct distillation experiments on features in different layers of DeiT with two strategies, namely mimicking and generation. When using mimicking, we align the embedding dimensions of the student and the teacher by a linear layer and correlation matrix, respectively. As for generation, we randomly mask the student’s tokens and utilize a generative block to restore the feature. Furthermore, we choose different generative blocks, including cross-attention block (Chen et al., 2022), self-attention block (He et al., 2022), and convolutional projector (Yang et al., 2022e). For both mimicking and generation, we calculate the square of $L_{2}$ distance as the distillation loss. The details about mimicking and generation are elaborated in Subsection 3.1 and 3.2, respectively.
G1) For distillation on the deep layer, generation is more suitable than mimicking . For CNN’s feature distillation, many works (Park et al., 2019; Tian et al., 2019; Yang et al., 2022e) transfer teachers’ semantic information from the last-stage feature. Most feature-based distillation methods (Romero et al., 2014; Zagoruyko & Komodakis, 2016; Heo et al., 2019) aim at making students get similar feature maps to the teacher. While MGD (Yang et al., 2022e) forces the student to generate the teacher’s full feature instead of mimicking it directly.
As our results shown in Table 2, the way to mimic the last layer feature of the teacher surprisingly impair the student’s performance noticeably. Specifically, the student’s Top-1 accuracy drops about $2\%$ when mimicking the teacher’s correlation matrix. This trend is completely different from distillation for CNN-based models. Instead, the generation methods can improve the accuracy of the student mostly. The largest gains are obtained by using the convolutional projector as the generative block. These results reveal that generation is more suitable than mimicking for the deep layer.
G2) Distillation on the shallow layers also works for ViT with mimicking. For the CNN-based model’s feature distillation, the feature of shallow layers has a small receptive field and lacks semantic information, making it unsuitable for distillation. As the attention map shown in Figure 1, the shallow feature of DeiT also has a small receptive. That is, the tokens in the first two layers just have responses to themselves. Still, we believe such incipient attention knowledge is useful for distillation because it can teach the student how to form a better attention map at the beginning.
We pick the first two layers for distillation by either mimicking or generation in Table 3. Interestingly, the conclusion for feature distillation on the shallow layers and deep layers is the opposite. The relations of different tokens and semantic information from the shallow layers are so weak that the student can not utilize its masked feature to generate the full feature from the teacher. It makes the generation way just bring a little improvement for the shallow-layer distillation.
Moreover, different from distillation for CNN-based models, transferring the knowledge from teacher’s shallow layer by directly mimicking makes great progress.
Mimicking by ‘correlation matrix’ performs a little better than the ‘linear layer’ way when using DeiT-S as the teacher. When the teacher performs better, the ‘linear layer’ way benefits the student much more than the ‘correlation matrix’ way.
As we described above, the results validate that the shallow layer matters for distillation by mimicking. We fix to use the ‘linear layer’ strategy to distill the shallow layers.
G3) The FFN-out features are better than the MHA-out features for distillation.
The ViT-based models are built by stacking several encoder layers. Each encoder layer consists of a multi-head attention (MHA) module and a feed-forward network (FFN) module. Based on the findings from G1 and G2, we further conduct experiments on the first two layers of the student to explore how to choose the modules for ViT’s feature distillation.
We use the ‘linear layer’ way for the shallow-layer distillation on the MHA-out feature and FFN-out feature, respectively.
Table 4 demonstrates that distilling on the MHA-out feature or FFN-out both bring the student improvements and transferring the knowledge from the FFN-out feature is better than the MHA-out feature.
3 Methodology
As mentioned in our guidelines in Section 2, we apply ‘linear layer’ and ‘correlation matrix’ for mimicking. For generation, we use ‘cross-attention’, ‘self-attention’, and ‘conv. projector’. In this section, we describe the details of these methods and the final formulation of our ViTKD. To begin, we recall the feature distillation of CNN is based on the $L_{2}$ distances between the feature maps. We also follow this paradigm. The general form of CNN’s feature distillation loss is as following:
$$\mathcal{L}_{fea}=\sum_{k=1}^{C}\sum_{i=1}^{H}\sum_{j=1}^{W}\big{(}\mathcal{F}_{k,i,j}^{T}-f(\mathcal{F}_{k,i,j}^{S})\big{)}^{2},$$
(1)
where $\mathcal{F}^{T}$ and $\mathcal{F}^{S}$ denote the feature from the teacher and student, respectively, and $f(\cdot)$ is the adaptation layer to reshape the $\mathcal{F}^{S}$ to the same dimension as $\mathcal{F}^{T}$. $H$ and $W$ denote the height and width of the feature, and $C$ is the channel length. In the next, we shift to feature distillation for ViT.
3.1 Mimicking for shallow layers
For each sample, we can denote student’s and teacher’s feature as $\mathcal{F}^{S}\in\mathcal{R}^{N\times D_{S}}$ and $\mathcal{F}^{T}\in\mathcal{R}^{N\times D_{T}}$, respectively. For the mimicking method, we utilize a linear layer to align the dimension of the student’s $D_{S}$ and the teacher’s $D_{T}$. We term the strategy as ‘linear layer’ and summarize it as:
$$\mathcal{L}_{lr}=\sum_{i=1}^{N}\sum_{j=1}^{D}\big{(}\mathcal{F}_{i,j}^{T}-fc(\mathcal{F}^{S})_{i,j}\big{)}^{2},$$
(2)
where $fc(\cdot)$ is a linear layer to reshape the $\mathcal{F}^{S}$ to the same dimension as $\mathcal{F}^{T}$. $N,D$ denote the number of patch tokens and the embedding dimension of the teacher’s feature.
Besides, we use a correlation matrix to describe the response among different patch tokens and force the student to learn the correlation matrix of the teacher’s features. In this case, we do not need the adaption layer to align the embedding dimension. The correlation matrix for each sample can be calculated as:
$$\mathcal{M}=\frac{\mathcal{FF}^{Tr}}{\sqrt{D}},$$
(3)
where $\mathcal{F}\in\mathcal{R}^{N\times D}$ denotes the student or teacher’s feature. $D$ is their embedding dimension and $Tr$ denotes transposition for the feature, so $\mathcal{F}^{Tr}\in\mathcal{R}^{D\times N}$. In this case, student’s and teacher’s relation matrices have the same shape $\mathcal{M}\in\mathcal{R}^{N\times N}$ and describe the response between different patch tokens. With ‘correlation matrix’, we calculate the distillation loss as:
$$\mathcal{L}_{rm}=\sum_{i=1}^{N}\sum_{j=1}^{N}\big{(}\mathcal{M}_{i,j}^{T}-\mathcal{M}_{i,j}^{S}\big{)}^{2}.$$
(4)
3.2 Generation for deep layers
For generation, we first use a linear layer to align the feature dimension of the student and teacher. Then, we set a random mask $Mask\in\mathcal{R}^{N\times 1}$ and use masked tokens to replace the student’s original tokens, which can be formulated as:
$$\mathcal{\hat{F}}_{i}^{S}=\begin{cases}masked~{}~{}token,&\text{if}\ r_{i}<\lambda\\
original~{}~{}token,&\text{Otherwise},\end{cases}$$
(5)
$$Mask_{i}=\begin{cases}1,&\text{if}\ r_{i}<\lambda\\
0,&\text{Otherwise},\end{cases}$$
(6)
where $r_{i}$ is a random number uniformly distributed in $[0,1]$ and $i\in[0,N-1]$ is the coordinates of the tokens dimension. $\lambda$ is a hyper-parameter that is set as 0.5 for all the experiments. The $masked~{}~{}token$ is the parameter to learn during training.
Finally, we use the new masked feature $\mathcal{\hat{F}}_{i}^{S}$ to generate teacher’s the full feature through a generative block $\mathcal{G}$, which can be formulated as follows:
$$\mathcal{G}(\mathcal{\hat{F}}^{S})\longrightarrow\mathcal{F}^{T}.$$
(7)
We choose three ways to set the generative block $\mathcal{G}$. The first way is a ‘cross-attention’ block from CAE (Chen et al., 2022), which includes 6 transformer layers. The second way is a ‘self-attention’ block from MAE (He et al., 2022), which also includes 6 transformer layers. The difference between them is that cross-attention uses masked tokens as the query tokens. The third way is a ‘convolutional projector’ from MGD (Yang et al., 2022e), which includes two conventional layers. For all the three ways, we only calculate the distillation loss of masked tokens.
For generation method, we design the distillation loss $\mathcal{L}_{gen}$ as:
$$\mathcal{L}_{gen}=\sum_{i=1}^{N}\sum_{j=1}^{D}Mask_{i}\big{(}\mathcal{F}_{i,j}^{T}-\mathcal{G}(\mathcal{\hat{F}}_{i,j}^{S})\big{)}^{2}.$$
(8)
3.3 ViTKD
Based on the findings from G1, G2 and G3, we finally propose our method ViTKD. We first use the ‘linear layer’ approach for the first two layers’ distillation, where the distillation loss is $\mathcal{L}_{lr}$. As for the last layer, we apply the ‘conv. projector’ for the generation distillation, where the distillation loss is $\mathcal{L}_{gen}$. The ViTKD we propose is shown in Figure 3. To sum up, we train the student model with the total loss as follows:
$$\mathcal{L}=\mathcal{L}_{ori}+\alpha\mathcal{L}_{lr}+\beta\mathcal{L}_{gen},$$
(9)
where $\mathcal{L}_{ori}$ is the original loss for the models, e.g., the cross-entropy loss in DeiT-Tiny. $\alpha$ and $\beta$ are two hyper-parameters to balance the loss.
4 Experiment
4.1 Settings
Datasets. We explore the feature distillation for ViT-based models on ImageNet-1k (Deng et al., 2009), which contains 1000 object categories. We use the 1.2 million images to train the model and 50k images to evaluate the performance. For the downstream task, we evaluate our model on the COCO dataset (Lin et al., 2014), which contains 80 object classes. We use the 120k train images for training and 5k validation images for testing.
Implementation details. ViTKD uses the hyper-parameters $\alpha$ and $\beta$ to balance the distillation loss in Equation 9. Another hyper-parameter $\lambda$ is used to adjust the masked ratio for deep layer distillation in Equation 5. We adopt the hyper-parameters $\{\alpha=3\times 10^{-5},\beta=3\times 10^{-6},\lambda=0.5\}$ for all the experiments. As for the logit distillation, we apply the distillation method NKD (Yang et al., 2022c) and set the hyper-parameters $\{\alpha=1,temperature=1\}$. Besides, to keep the model to be the same for the feature and logit distillation, we remove the extra distillation token which is used for logit distillation in DeiT. The image resolution for all the experiments is 224$\times$224. The other training details for distillation follow the setting from the baseline training setting in MMClassification (Contributors, 2020). All the experiments are conducted on 8 GPUs with MMClassification in Pytorch (Paszke et al., 2019). Unless specified, we evaluate the model with the performance of the last epoch.
4.2 Main results
To evaluate our methods for ViT-based models, we utilize different teachers to distill different students. We train the student with the proposed ViTKD, the classic KD, the state-of-the-art logit method NKD (Yang et al., 2022c). We also combine ViTKD and NKD to explore the upper bound of the student’s performance. In Table 5, all the teachers bring the students remarkable performance improvements, e.g., the DeiT III-Small teacher boosts the student’s Top-1 accuracy from $74.42\%$ to $76.06\%$ with our ViTKD method. The results of ViTKD even surpass the classic logit-based KD method. Comparing the results between different teachers, we find the student achieves better performance with a stronger teacher, e.g., the student DeiT-Tiny achieves $75.40\%$ and $76.06\%$ Top-1 accuracy with teh DeiT-Small and DeiT III-Small teacher, respectively. Furthermore, we also apply our method to a stronger student DeiT-Small and DeiT-Base. ViTKD can also bring them significant improvements, helping it to achieve $81.95\%$ and $83.46\%$, respectively.
Besides, ViTKD is a feature-based knowledge distillation method and can be combined with other logit-based methods for image classification. Therefore, we try to add the state-of-the-art logit-based distillation loss NKD to our ViTKD. In this way, the students with different teachers all get another significant accuracy improvement, e.g., the student DeiT-Small gets another $1.64\%$ gains and achieves $83.59\%$ Top-1 accuracy with a DeiT III-Base teacher. Surprisingly, the student DeiT-Small is just trained on ImageNet-1K, but its performance surpasses DeiT III-Small, which needs to be pre-trained on ImageNet-21k and then finetuned on ImageNet-1k.
4.3 Downstream task
The model with ViTKD achieves significant improvements for the classification task on ImageNet. To further evaluate the effectiveness of the model with distillation, we try to apply the model to object detection. We use Mask-RCNN (He et al., 2017) as the detector and follow the training setting from ViTDet (Li et al., 2022b) on detectron2 (Wu et al., 2019).
As the results shown in Table 6, the backbone DeiT-Small trained with ViTKD brings the detector 1.21 mAP gains. When the backbone is trained with ViTKD and NKD, the mAP improvement can be boosted to 1.62. The results demonstrate the model trained with distillation has not only better performance for image classification but also stronger semantic information for the downstream task.
5 More Analyses
5.1 Do we need to distill the middle layers?
We have discussed the distillation strategies for the shallow and deep layers of ViT-based models. As shown in Figure 1 and 4, the attention distributions of the middle layers are similar to that of shallow layers. Accordingly, we explore the effects of distillation on the middle layers (e.g., the $6_{th}$ layer) in Table 8. In general, distillation on either the shallow or middle layers can benefit the student. The first layer’s knowledge boosts the student most. Besides comparing the improvements from different distillation layers, we find that the knowledge from the shallow layers is much more helpful than that from the middle layer for distillation. Furthermore, when combing the shallow and middle layers together, the accuracy improvement is just $0.02\%$. Considering the trade-offs between time consumption and performance, we do not distill on the middle layers eventually.
5.2 Teachers with the same architecture as the student are appropriate.
We have chosen teachers with the same architecture to guide the student in Table 5, achieving significant improvements.
In this subsection, we explore whether a teacher with different architecture is still suitable for distillation. Here we choose CaiT-S24 (Touvron et al., 2021b) as the teacher, which has a high performance of $83.37\%$ and a different architecture with DeiT.
Following the previous guidance, we utilize CaiT-S24’s shallow layers to distill the student’s shallow layers, which leads to a $1.29\%$ accuracy drop (shown in Table 8). To further analyze what causes the degradation, we visualize the attention maps of the three used models in Figure 4. Interestingly, the shallow and deep layer’s attention distributions of DeiT and CaiT are quite different, making it hard for the student to learn such attention. This phenomenon is not consistent with the assumptions of our proposed ViTKD, which causes inevitable performance degradation. In contrast, when using a teacher with the same architecture for distillation, the student gains noticeably. This observation indicates that a teacher with the same architecture who generates similar attention, is more suitable for ViTKD.
5.3 Effects of different losses
As described in the practical guidance, we distill the shallow layers and deep layers by mimicking and generation, respectively. In this subsection, we conduct experiments of Mimicking loss $\mathcal{L}_{lr}$ and Generation loss $\mathcal{L}_{gen}$ to investigate their influences on the student with DeiT-Tiny. As shown in Table 9, both the knowledge from shallow and deep layers are helpful for the student. When just applying a single loss, the $\mathcal{L}_{lr}$ on shallow layers benefits the student much more than $\mathcal{L}_{gen}$ on the deep layer. This phenomenon shows that incipient attention knowledge really matters for ViT’s feature distillation, which is completely different from the CNN-based model’s feature distillation. Furthermore, these two losses are complementary to each other. For example, when combing $\mathcal{L}_{lr}$ and $\mathcal{L}_{gen}$ together, the student with a DeiT III-Small teacher achieve $76.06\%$ Top-1 Accuracy, which is much higher than just applying $\mathcal{L}_{lr}$’s $75.31\%$ and $\mathcal{L}_{gen}$’s $74.79\%$.
5.4 Sensitivity study of hyper-parameters
In ViTKD, we use $\alpha$ and $\beta$ in Equation 9 to balance the shallow layer’s distillation loss $\mathcal{L}_{lr}$ and the deep layer’s distillation loss $\mathcal{L}_{gen}$, respectively. To explore the sensitivity of the hyper-parameters, we conduct experiments by adopting DeiT III-Small to distill DeiT-Tiny on ImageNet-1K.
As shown in Figure 5, ViTKD is not sensitive to the hyper-parameters $\alpha$ or $\beta$ which is just used for balancing the distillation loss. Specifically, when $\alpha$ varies from 2 to 6, the student’s worst accuracy is $76.03\%$, which is just $0.12\%$ lower than the highest accuracy. Besides, it is still $1.61\%$ higher than the baseline model, demonstrating our ViTKD is not sensitive to the hyper-parameters.
6 Conclusion
In this paper, we explore a feature-based distillation method for ViT-based models. To this end, we design a series of experiments and discuss the effects of different distillation methods, layers, and modules. From the results, We derive three practical guidelines for ViT’s feature distillation. We propose our method ViTKD based on the guidelines, which includes the distillation on shallow layers via mimicking and deep layers via generation. ViTKD brings the student significant improvements on the image classification task and also benefits other downstream task. Besides, ViTKD is truly a feature-based method that can be easily combined with logit-based distillation methods to further improve the student.
Limitations
We use the mimicking method for the shallow layer’s distillation and the generation method for the deep layer’s distillation. However, the way to achieve mimicking and generation is still simple and needs further exploration. Moreover, it is still interesting to transfer the feature knowledge to the student from a teacher with different architecture.
References
Chen et al. (2021)
Pengguang Chen, Shu Liu, Hengshuang Zhao, and Jiaya Jia.
Distilling knowledge via knowledge review.
In Proceedings of the IEEE/CVF Conference on Computer Vision
and Pattern Recognition, pp. 5008–5017, 2021.
Chen et al. (2022)
Xiaokang Chen, Mingyu Ding, Xiaodi Wang, Ying Xin, Shentong Mo, Yunhao Wang,
Shumin Han, Ping Luo, Gang Zeng, and Jingdong Wang.
Context autoencoder for self-supervised representation learning.
arXiv preprint arXiv:2202.03026, 2022.
Contributors (2020)
MMClassification Contributors.
Openmmlab’s image classification toolbox and benchmark.
https://github.com/open-mmlab/mmclassification, 2020.
Deng et al. (2009)
Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei.
Imagenet: A large-scale hierarchical image database.
In Proceedings of the IEEE/CVF Conference on Computer Vision
and Pattern Recognition, pp. 248–255, 2009.
Devlin et al. (2018)
Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova.
Bert: Pre-training of deep bidirectional transformers for language
understanding.
arXiv preprint arXiv:1810.04805, 2018.
Dosovitskiy et al. (2021)
Alexey Dosovitskiy, Lucas Beyer, Alexander Kolesnikov, Dirk Weissenborn,
Xiaohua Zhai, Thomas Unterthiner, Mostafa Dehghani, Matthias Minderer, Georg
Heigold, Sylvain Gelly, Jakob Uszkoreit, and Neil Houlsby.
An image is worth 16x16 words: Transformers for image recognition at
scale.
In International Conference on Learning Representations, 2021.
Han et al. (2021)
Kai Han, An Xiao, Enhua Wu, Jianyuan Guo, Chunjing Xu, and Yunhe Wang.
Transformer in transformer.
Advances in Neural Information Processing Systems,
34:15908–15919, 2021.
He et al. (2017)
Kaiming He, Georgia Gkioxari, Piotr Dollár, and Ross Girshick.
Mask r-cnn.
In Proceedings of the IEEE international conference on computer
vision, pp. 2961–2969, 2017.
He et al. (2022)
Kaiming He, Xinlei Chen, Saining Xie, Yanghao Li, Piotr Dollár, and Ross
Girshick.
Masked autoencoders are scalable vision learners.
In Proceedings of the IEEE/CVF Conference on Computer Vision
and Pattern Recognition, pp. 16000–16009, 2022.
He et al. (2019)
Tong He, Chunhua Shen, Zhi Tian, Dong Gong, Changming Sun, and Youliang Yan.
Knowledge adaptation for efficient semantic segmentation.
In Proceedings of the IEEE/CVF Conference on Computer Vision
and Pattern Recognition, pp. 578–587, 2019.
Heo et al. (2019)
Byeongho Heo, Jeesoo Kim, Sangdoo Yun, Hyojin Park, Nojun Kwak, and Jin Young
Choi.
A comprehensive overhaul of feature distillation.
In Proceedings of the IEEE/CVF International Conference on
Computer Vision, pp. 1921–1930, 2019.
Hinton et al. (2015)
Geoffrey Hinton, Oriol Vinyals, Jeff Dean, et al.
Distilling the knowledge in a neural network.
arXiv preprint arXiv:1503.02531, 2(7), 2015.
Li et al. (2022a)
Gang Li, Xiang Li, Yujie Wang, Shanshan Zhang, Yichao Wu, and Ding Liang.
Knowledge distillation for object detection via rank mimicking and
prediction-guided feature imitation.
In Proceedings of the AAAI Conference on Artificial
Intelligence, pp. 1306–1313, 2022a.
Li et al. (2022b)
Yanghao Li, Hanzi Mao, Ross Girshick, and Kaiming He.
Exploring plain vision transformer backbones for object detection.
arXiv preprint arXiv:2203.16527, 2022b.
Lin et al. (2022)
Sihao Lin, Hongwei Xie, Bing Wang, Kaicheng Yu, Xiaojun Chang, Xiaodan Liang,
and Gang Wang.
Knowledge distillation via the target-aware transformer.
In Proceedings of the IEEE/CVF Conference on Computer Vision
and Pattern Recognition, pp. 10915–10924, 2022.
Lin et al. (2014)
Tsung-Yi Lin, Michael Maire, Serge Belongie, James Hays, Pietro Perona, Deva
Ramanan, Piotr Dollár, and C Lawrence Zitnick.
Microsoft coco: Common objects in context.
In European conference on computer vision, pp. 740–755,
2014.
Liu et al. (2019)
Yifan Liu, Ke Chen, Chris Liu, Zengchang Qin, Zhenbo Luo, and Jingdong Wang.
Structured knowledge distillation for semantic segmentation.
In Proceedings of the IEEE/CVF Conference on Computer Vision
and Pattern Recognition, pp. 2604–2613, 2019.
Liu et al. (2021)
Ze Liu, Yutong Lin, Yue Cao, Han Hu, Yixuan Wei, Zheng Zhang, Stephen Lin, and
Baining Guo.
Swin transformer: Hierarchical vision transformer using shifted
windows.
In Proceedings of the IEEE/CVF International Conference on
Computer Vision, pp. 10012–10022, 2021.
Park et al. (2019)
Wonpyo Park, Dongju Kim, Yan Lu, and Minsu Cho.
Relational knowledge distillation.
In Proceedings of the IEEE/CVF Conference on Computer Vision
and Pattern Recognition, pp. 3967–3976, 2019.
Paszke et al. (2019)
Adam Paszke, Sam Gross, Francisco Massa, Adam Lerer, James Bradbury, Gregory
Chanan, Trevor Killeen, Zeming Lin, Natalia Gimelshein, Luca Antiga, et al.
Pytorch: An imperative style, high-performance deep learning library.
Advances in neural information processing systems,
32:8026–8037, 2019.
Romero et al. (2014)
Adriana Romero, Nicolas Ballas, Samira Ebrahimi Kahou, Antoine Chassang, Carlo
Gatta, and Yoshua Bengio.
Fitnets: Hints for thin deep nets.
arXiv preprint arXiv:1412.6550, 2014.
Shu et al. (2021)
Changyong Shu, Yifan Liu, Jianfei Gao, Zheng Yan, and Chunhua Shen.
Channel-wise knowledge distillation for dense prediction.
In Proceedings of the IEEE/CVF International Conference on
Computer Vision, pp. 5311–5320, 2021.
Sun et al. (2019)
Siqi Sun, Yu Cheng, Zhe Gan, and Jingjing Liu.
Patient knowledge distillation for bert model compression.
In EMNLP/IJCNLP (1), 2019.
Tian et al. (2019)
Yonglong Tian, Dilip Krishnan, and Phillip Isola.
Contrastive representation distillation.
In International Conference on Learning Representations, 2019.
Touvron et al. (2021a)
Hugo Touvron, Matthieu Cord, Matthijs Douze, Francisco Massa, Alexandre
Sablayrolles, and Hervé Jégou.
Training data-efficient image transformers & distillation through
attention.
In International Conference on Machine Learning, pp. 10347–10357, 2021a.
Touvron et al. (2021b)
Hugo Touvron, Matthieu Cord, Alexandre Sablayrolles, Gabriel Synnaeve, and
Hervé Jégou.
Going deeper with image transformers.
In Proceedings of the IEEE/CVF International Conference on
Computer Vision, pp. 32–42, 2021b.
Touvron et al. (2022)
Hugo Touvron, Matthieu Cord, and Hervé Jégou.
Deit iii: Revenge of the vit.
arXiv preprint arXiv:2204.07118, 2022.
Wang et al. (2022)
Luting Wang, Xiaojie Li, Yue Liao, Zeren Jiang, Jianlong Wu, Fei Wang, Chen
Qian, and Si Liu.
Head: Hetero-assists distillation for heterogeneous object detectors.
arXiv preprint arXiv:2207.05345, 2022.
Wu et al. (2019)
Yuxin Wu, Alexander Kirillov, Francisco Massa, Wan-Yen Lo, and Ross Girshick.
Detectron2.
https://github.com/facebookresearch/detectron2, 2019.
Yang et al. (2022a)
Chenhongyi Yang, Mateusz Ochal, Amos Storkey, and Elliot J Crowley.
Prediction-guided distillation for dense object detection.
arXiv preprint arXiv:2203.05469, 2022a.
Yang et al. (2022b)
Chuanguang Yang, Helong Zhou, Zhulin An, Xue Jiang, Yongjun Xu, and Qian Zhang.
Cross-image relational knowledge distillation for semantic
segmentation.
In Proceedings of the IEEE/CVF Conference on Computer Vision
and Pattern Recognition, pp. 12319–12328, 2022b.
Yang et al. (2020)
Jing Yang, Brais Martinez, Adrian Bulat, and Georgios Tzimiropoulos.
Knowledge distillation via softmax regression representation
learning.
In International Conference on Learning Representations, 2020.
Yang et al. (2022c)
Zhendong Yang, Zhe Li, Yuan Gong, Tianke Zhang, Shanshan Lao, Chun Yuan, and
Yu Li.
Rethinking knowledge distillation via cross-entropy.
arXiv preprint arXiv:2208.10139, 2022c.
Yang et al. (2022d)
Zhendong Yang, Zhe Li, Xiaohu Jiang, Yuan Gong, Zehuan Yuan, Danpei Zhao, and
Chun Yuan.
Focal and global knowledge distillation for detectors.
In Proceedings of the IEEE/CVF Conference on Computer Vision
and Pattern Recognition, pp. 4643–4652, 2022d.
Yang et al. (2022e)
Zhendong Yang, Zhe Li, Mingqi Shao, Dachuan Shi, Zehuan Yuan, and Chun Yuan.
Masked generative distillation.
arXiv preprint arXiv:2205.01529, 2022e.
Yuan et al. (2021)
Li Yuan, Yunpeng Chen, Tao Wang, Weihao Yu, Yujun Shi, Zi-Hang Jiang,
Francis EH Tay, Jiashi Feng, and Shuicheng Yan.
Tokens-to-token vit: Training vision transformers from scratch on
imagenet.
In Proceedings of the IEEE/CVF International Conference on
Computer Vision, pp. 558–567, 2021.
Zagoruyko & Komodakis (2016)
Sergey Zagoruyko and Nikos Komodakis.
Paying more attention to attention: Improving the performance of
convolutional neural networks via attention transfer.
arXiv preprint arXiv:1612.03928, 2016.
Zhang et al. (2022)
Jinnian Zhang, Houwen Peng, Kan Wu, Mengchen Liu, Bin Xiao, Jianlong Fu, and
Lu Yuan.
Minivit: Compressing vision transformers with weight multiplexing.
In Proceedings of the IEEE/CVF Conference on Computer Vision
and Pattern Recognition, pp. 12145–12154, 2022.
Zhao et al. (2022)
Borui Zhao, Quan Cui, Renjie Song, Yiyu Qiu, and Jiajun Liang.
Decoupled knowledge distillation.
In Proceedings of the IEEE/CVF Conference on Computer Vision
and Pattern Recognition, pp. 11953–11962, 2022.
Zheng et al. (2022)
Zhaohui Zheng, Rongguang Ye, Ping Wang, Dongwei Ren, Wangmeng Zuo, Qibin Hou,
and Ming-Ming Cheng.
Localization distillation for dense object detection.
In Proceedings of the IEEE/CVF Conference on Computer Vision
and Pattern Recognition, pp. 9407–9416, 2022.
Zhou et al. (2020)
Helong Zhou, Liangchen Song, Jiajie Chen, Ye Zhou, Guoli Wang, Junsong Yuan,
and Qian Zhang.
Rethinking soft labels for knowledge distillation: A bias–variance
tradeoff perspective.
In International Conference on Learning Representations, 2020. |
A remark on the geometric interpretation of the A3w condition from optimal transport
Cale Rankin
The Fields Institute for Research in Mathematical Sciences
cale.rankin@utoronto.ca
cale.rankin@gmail.com
Abstract.
We provide a geometric interpretation of the well known A3w condition for regularity of optimal transport maps.
This research is supported by ARC DP 200101084 and the Fields Institute for Research in Mathematical Sciences.
1. Introduction
In optimal transport a condition known as A3w is necessary for regularity of the optimal transport map. Here we provide a geometric interpretation of A3w. We’ll use freely the notation from [4]. Let $c\in C^{2}(\mathbf{R}^{n}\times\mathbf{R}^{n})$ satisfy A1 and A2 (see §2). Keeping in mind the prototypical case $c(x,y)=|x-y|^{2}$, we fix $x_{0},y_{0}\in\mathbf{R}^{n}$ and perform a linear transformation so $c_{xy}(x_{0},y_{0})=-I$. Define coordinates
(1)
$$\displaystyle q(x)$$
$$\displaystyle:=-c_{y}(x,y_{0}),$$
(2)
$$\displaystyle p(y)$$
$$\displaystyle:=-c_{x}(x_{0},y),$$
and the inverse transformations by $x(q),y(p)$. Write $c(q,p)=c(x(q),y(p))$ and let $q_{0}=q(x_{0})$ and $p_{0}=p(y_{0})$.
We prove A3w is satisfied if and only if whenever these transformations are performed
$$\displaystyle(q-q_{0})\cdot(p-p_{0})\geq 0\implies$$
$$\displaystyle c(q,p)+c(q_{0},p_{0})\leq c(q,p_{0})+c(q_{0},p).$$
Heuristically, A3w implies when $q-q_{0}$ “points in the same direction” as $p-p_{0}$ it is cheaper to transport $q$ to $p$ and $q_{0}$ to $p_{0}$ than the alternative $q$ to $p_{0}$ and $q_{0}$ to $p$. Thus, A3w implies compatibility between directions in the cost-convex geometry and the cost of transport.
A3w first appeared (in a stronger form) in [4]. It was weakened in [6] and a new interpretation given in [2]. The impetus for the above interpretation is Lemma 2.1 in [1]. Our result can also be realised by a particular choice of c-convex function in the unpublished preprint [5].
Acknowledgements. My thanks to Jiakun Liu and Robert McCann for helpful comments and discussion.
2. Proof of result
Let $c\in C^{2}(\mathbf{R}^{n}\times\mathbf{R}^{n})$ satisfy following the well known conditions
A1. For each $x_{0},y_{0}\in\mathbf{R}^{n}$ the following mappings are injective
$$\displaystyle x\mapsto c_{y}(x,y_{0}),\quad\text{and}\quad y\mapsto c_{x}(x_{0},y).$$
A2. For each $x_{0},y_{0}\in\mathbf{R}^{n}$ we have $\det c_{i,j}(x_{0},y_{0})\neq 0$.
Here, and throughout, subscripts before a comma denote differentiation with respect to the first variable, subscripts after a comma denote differentiation with respect to the second variable.
By A1 we define on $\mathcal{U}:=\{(x,c_{x}(x,y));x,y\in\mathbf{R}^{n}\}$ a mapping $Y:\mathcal{U}\rightarrow\mathbf{R}^{n}$ by
$$c_{x}(x,Y(x,p))=p.$$
The A3w condition, usually expressed with fourth derivatives but written here as in [3], is the following.
A3w. Fix $x$. The function
$$p\mapsto c_{ij}(x,Y(x,p))\xi_{i}\xi_{j},$$
is concave along line segments orthogonal to $\xi$.
To verify A3w it suffices to verify midpoint concavity, that is whenever $\xi\cdot\eta=0$ there holds
(3)
$$0\geq[c_{ij}(x,Y(x,p+\eta))-2c_{ij}(x,Y(x,p))+c_{ij}(x,Y(x,p-\eta))]\xi_{i}\xi_{j}.$$
Finally, we recall $A\subset\mathbf{R}^{n}$ is called $c$-convex with respect to $y_{0}$ provided $c_{y}(A,y_{0})$ is convex. When A3w is satisfied and $y,y_{0}\in\mathbf{R}^{n}$ are given the section $\{x\in\mathbf{R}^{n};c(x,y)>c(x,y_{0})\}$ is $c$-convex with respect to $y_{0}$ [3].
Now fix $(x_{0},p_{0})\in\mathcal{U}$ and $y_{0}=Y(x_{0},p_{0})$. To simplify the proof we assume $x_{0},y_{0},q_{0},p_{0}=0$. Up to an affine transformation (replace $y$ with $\tilde{y}:=-c_{xy}(0,0)y$) we assume $c_{xy}(0,0)=-I$. Note with $q,p$ as defined in (1), (2), this implies $\frac{\partial q}{\partial x}(0)=I$. Put
$$\displaystyle\tilde{c}(x,y)$$
$$\displaystyle:=c(x,y)-c(x,0)-c(0,y)+c(0,0),$$
$$\displaystyle\overline{c}(q,p)$$
$$\displaystyle:=\tilde{c}(x(q),y(p)).$$
Theorem 1.
The A3w condition is satisfied if and only if whenever the above transformations are applied the following implication holds
(4)
$$q\cdot p\geq 0\implies\overline{c}(q,p)\leq 0.$$
Proof.
Observe by a Taylor series
(5)
$$\overline{c}(q,p)=-(q\cdot p)+\overline{c}_{ij}(\tau q,p)q_{i}q_{j},$$
for some $\tau\in(0,1)$. First, assume A3w and let $q\cdot p>0$. By (5) we have $\overline{c}(-tq,p)>0>\overline{c}(tq,p)$ for $t>0$ sufficiently small. If $\overline{c}(q,p)>0$ then the $c$-convexity (in our coordinates, convexity) of the section
$$\{q\ ;\ \overline{c}(q,p)>\overline{c}(q,0)=0\},$$
is violated. By continuity $\overline{c}(q,p)\leq 0$ whenever $q\cdot p\geq 0$.
In the other direction, take nonzero $q$ with $q\cdot p=0$ and small $t$. By (4) and (5)
$$0\geq\overline{c}(tq,p)/t^{2}=\overline{c}_{ij}(t\tau q,p)q_{i}q_{j}.$$
This inequality also holds with $-p$. Moreover $\overline{c}_{ij}(t\tau q,0)=0$. Thus
$$0\geq[\overline{c}_{ij}(t\tau q,p)-2\overline{c}_{ij}(t\tau q,0)+\overline{c}_{ij}(t\tau q,-p)]q_{i}q_{j}.$$
Sending $t\rightarrow 0$ and returning to our original coordinates we obtain (3).
∎
Remarks. (1) On a Riemannian manifold with $c(x,y)=d(x,y)^{2}$, for $d$ the distance function, Loeper [2] proved A3w implies nonnegative sectional curvature. Our result expedites his proof. Let $x_{0}=y_{0}\in M$ and $u,v\in T_{x_{0}}M$ satisfy $u\cdot v=0$ with $x=\exp_{x_{0}}(tu),y=\exp_{x_{0}}(tv)$. Working in a sufficiently small local coordinate chart our previous proof implies if A3w is satisfied
(6)
$$d(x,y)^{2}\leq d(x_{0},y)^{2}+d(x_{0},x)^{2}=2t.$$
The sectional curvature in the plane generated by $u,v$ is the $\kappa$ satisfying
(7)
$$d(\exp_{x_{0}}(tu),\exp_{x_{0}}(tv))=\sqrt{2}t\big{(}1-\frac{\kappa}{12}t^{2}+O(t^{3})\big{)}\text{ as }t\rightarrow 0,$$
whereby comparison with (6) proves the result (see [7, eq. 1] for (7)). We note Loeper proved his result using an infinitesimal version of (6).
References
[1]
Shibing Chen and Xu-Jia Wang.
Strict convexity and $C^{1,\alpha}$ regularity of potential
functions in optimal transportation under condition A3w.
J. Differential Equations, 260(2):1954–1974, 2016.
[2]
Grégoire Loeper.
On the regularity of solutions of optimal transportation problems.
Acta Math., 202(2):241–283, 2009.
[3]
Grégoire Loeper and Neil S. Trudinger.
Weak formulation of the MTW condition and convexity properties of
potentials.
Methods Appl. Anal., 28(1):53–60, 2021.
[4]
Xi-Nan Ma, Neil S. Trudinger, and Xu-Jia Wang.
Regularity of potential functions of the optimal transportation
problem.
Arch. Ration. Mech. Anal., 177(2):151–183, 2005.
[5]
Neil S. Trudinger and Xu-Jia Wang.
On convexity notions in optimal transportation.
preprint, 2008.
[6]
Neil S. Trudinger and Xu-Jia Wang.
On the second boundary value problem for Monge-Ampère type
equations and optimal transportation.
Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 8(1):143–174, 2009.
[7]
Cédric Villani.
Synthetic theory of Ricci curvature bounds.
Jpn. J. Math., 11(2):219–263, 2016. |
Context dependent preferential attachment model for complex networks
Pradumn Kumar Pandey
and Bibhas Adhikari
Centre for System Science,
IIT Jodhpur, India, E-mail: pg201283006@iitj.ac.inCentre for System Science,
IIT Jodhpur, India, E-mail: bibhas@iitj.ac.in
Abstract
In this paper, we propose a growing random complex network model, which we call context dependent preferential attachment model (CDPAM), when the preference of a new node to get attached to old nodes is determined by the local and global property of the old nodes. We consider that local and global properties of a node as the degree and relative average degree of the node respectively. We prove that the degree distribution of complex networks generated by CDPAM follow power law with
exponent lies in the interval [2, 3] and the expected diameter grows logarithmically with the size of new nodes added in the initial small network. Numerical results show that the expected diameter stabilizes when alike weights to the local and global properties are assigned by the new nodes. Computing various measures including clustering coefficient, assortativity, number of triangles, algebraic connectivity, spectral radius, we show that the proposed model replicates properties of real networks better than BA model for all these measures when alike weights are given to local and global property. Finally, we observe that the BA model is a limiting case of CDPAM when new nodes tend to give large weight to the local property compared to the weight given to the global property during link formation.
context dependent preferential attachment, degree, relative average degree, clustering coefficient, assortativity, number of triangles, algebraic connectivity, spectral radius, diameter.
I Introduction
Modelling complex networks has been an active area of research in literature due to its applications in various field of science and technology [1][2][3][4]. Several attempts have been made to generate deterministic and random complex network models which can capture the spirit of several large scale real world networks such as social networks [5], biological networks [6], technological networks [7]etc. Two prime characteristics of a large class of real networks that have been observed and established by leading scholars in the area of complex networks are power-law degree distribution of the nodes and small-world behavior of the networks [8] [9][10][11][12]. The Erd$\ddot{o}$s-R$\acute{e}$nyi (ER) model [13] is one of the first initiatives to generate random networks where the links are made by following a random procedure when a fixed number of nodes is chosen at the initial stage of the network formation. However, later it has been observed that ER model fails to represent the essence of real networks, for example, degree distribution is not a power-law. Consequently, a lot of interest has been generated to produce networks having power-law degree distributions.
One of the insightful growing random complex network models is proposed by Barabasi et al. in 1999, also called BA-model [11]. In this model, a small network is chosen in the beginning of the method, then new nodes appear and get linked with the existing nodes in a probabilistic fashion which is decided by the property (degree) of the existing nodes [11, 14]. The philosophy adopted here is that at each iteration, the new nodes prefer to get attached with an old node which has high degree (among all existing ones) which sometimes represent the importance of a node in social context. Interestingly, the network generated by this model has power-law degree distribution and thus the concept of scale-free networks emerged. In his seminal paper [11], Barabasi et al. have also predicted that the growth and preferential attachment are jointly responsible for the emergence of the scale-free property in real networks. It has also been shown that the diameter grows approximately logarithmically with the size of the network.
Does a new node always wish to form links with important (high degree) nodes or the choice get influenced by other factors also? Moreover, if the choice gets influenced by other properties of the existing nodes, will the network be having power-law degree distribution? An evidence of a phenomena that peoples choice does not depend on only one property is given in [15] supported by an empirical data (see [16][17][18] also). The data shows that at the time of purchasing a product, a buyer considers the background (history) of the product and relative attractiveness of the product with respect to other products in the same reference. Thus, the concept of context preferential attachment was introduced in [15].
In this paper, we propose a growing random complex network model where the probability of link formation is determined by weighted local and global property of the existing nodes. We consider that local and global properties of a node are given by the degree and relative average degree of the node in a network. Thus, we call the proposed model, the context dependent preferential attachment model (CDPAM) for complex networks. We prove that the degree distribution of complex networks generated by CDPAM follow power law $P(k)=L(k)k^{-\gamma}$ where $2\leq\gamma\leq 3$ and $L(k)\rightarrow\alpha$ (a constant which depends on the weights given on local and global property of the nodes) as $k\rightarrow\infty.$ We also prove that the expected diameter grows logarithmically with the size of the new nodes added in the network, however the growth of the expected diameter is slower than that of the BA model. However, our numerical simulations show that the expected diameter stabilizes when alike weights are given to the local and global property which determine the preference of link formation. In contrast to the conventional wisdom that diameter shows as a function of $\ln(\ln N)$ or $\ln N$ in real networks, the authors in [19] observed that the diameter stabilizes or shrinks as a network grows. The proposed model reveals how shrinking and increasing of diameter are related to the weights on local and global property of the nodes during expansion of the network.
A variety of mathematical and statistical measures have been proposed in the literature in order to characterize global and local structure of complex networks. We derived clustering coefficient, assortativity, number of triangles, algebraic connectivity, spectral radius for different complex networks generated by CDPAM and compare them with the same obtained from the complex network generated by BA model. We show that our model replicates properties of real networks better than BA model for all these measures when alike weights are given to local and global property. Finally, we observe that the BA model is a limiting case of CDPAM when new nodes tend to give large weight to the local property compared to the weight given on the global property during link formation.
II Context dependent preferential attachment model (CDPAM)
In this section, we propose a random complex network model which relies on the fact that the network is open i.e. a network continuously grows in time with the addition of new nodes in to a fixed small network chosen in the beginning of the process [20]. It is important to notice that the link formation in BA model is biased as the link formation depends only on the high degree (importance) of the existing nodes. However, in real life we prefer to form relationship (link) with important (global property) people in society but also give importance to background (local property) of the people before making the relation. Inspired by this thought, we introduce the model as follows.
1.
Growth: Starting with a small network having $m_{0}$ nodes, at every timestep we add a new node with $m\leq m_{0}$ edges and the new nodes get linked with the nodes already present in the network.
2.
Context preferential attachment: Assume that $N(t)$ denotes the node set of the network after $t$-time step. When a new node $j$ appears at time $t+1$ would get connected to node $i\in N(t)$ with probability $p_{j}^{i}(t+1)$ given by
$$p_{j}^{i}(t+1)=\frac{\beta\,f_{B}(i)+\theta\,g(i,N(t))}{\sum_{i\in N(t)}(\beta%
\,f_{B}(i)+\theta\,g(i,N(t)))}$$
(1)
where $f_{B}(i)$ quantifies the background (local context) of node $i$, $g(i,N(t))$ determines the relative advantage (global context) of a nodes over others in the network $N(t),$ and $\beta,\theta(<\beta)$ are the positive control parameters for the property of the nodes in $N(t)$.
In order to simplify the model, we consider
$$f_{B}(i)=k_{i}\,\mbox{and}\,g(i,N(t))=\frac{\sum_{l\in N(t)}k_{i}-k_{l}}{|N(t)|}$$
where $k_{i}$ denotes the degree of a node $i$ and $|N(t)|$ is the number of nodes in $N(t).$ As we consider that a single node appears at each timestep, after time t there will be $t+m_{0}$ nodes in the network and for a large value of $t(\gg m_{0}),$ $|N(t)|\approx t.$ Consequently, we have
$$\begin{split}\displaystyle p_{j}^{i}(t+1)&\displaystyle\approx\frac{\beta k_{i%
}+\theta\sum_{l\in N(t)}\frac{k_{i}-k_{l}}{t}}{\sum_{l\in N(t)}\beta k_{l}+%
\sum_{l\in N(t)}\frac{(t+m_{0})k_{l}-2mt-m_{0}(m_{0}-1)}{t}}\\
&\displaystyle\approx\frac{\beta k_{i}+\theta(k_{i}-2m)}{2m\beta t}\end{split}$$
for a very small value of $m_{0}.$ Assuming $k_{i}$ to be a continuous real variable function and the rate of change of $k_{i}$ is proportional to $p_{i}^{j}(t),$ we have
$$\frac{\partial k_{i}}{\partial t}=m\frac{\beta k_{i}+\theta(k_{i}-2m)}{2m\beta
t}$$
(2)
by applying mean field theory.
The degree distribution of the network generated by the CDPAM is provided in the following theorem.
Theorem II.1
The degree distribution of a complex network generated by CDPAM described above exhibits a power law in their tail given by $P(k)=L(k)k^{-\gamma}$ where $L(k)\rightarrow(\gamma-1)(m-c)^{(\gamma-1)}$ as $k\rightarrow\infty$ and $\gamma=1+\frac{2\beta}{\beta+\theta},c=\frac{2m\theta}{\beta+\theta}.$ In particular, $\gamma\approx 2$ if $\beta\approx\theta$ and $\gamma\approx 3$ if $\beta\gg\theta.$
Proof: From (2) we have
$$\frac{\partial k_{i}}{\partial t}=m\frac{\beta k_{i}+\theta(k_{i}-2m)}{2m\beta
t%
}=\frac{k_{i}-c}{(\gamma-1)t}$$
solving which we obtain
$$k_{i}(t)=(m-c)\left(\frac{t}{t_{i}}\right)^{1/(\gamma-1)}+c$$
(3)
when the initial condition is given by $k_{i}(t_{0})=m.$ This yields
$$P(k_{i}(t)<k)=P(t_{i}>(m-c)^{\gamma-1}(k-c)^{1-\gamma}t).$$
Assuming $k_{i}(t)<k,$ we have $t_{i}>(m-c)^{\gamma-1}(k-c)^{1-\gamma}t.$ Further, since it is assumed that a single node gets added at each timestep, it is equivalent to a uniform distribution of $t_{i}$, given by $P(t_{i})=1/(m_{0}+t).$ Consequently,
$$\displaystyle P(k_{i}(t)<k)$$
$$\displaystyle=P(t_{i}>(m-c)^{\gamma-1}(k-c)^{1-\gamma}t)$$
$$\displaystyle=1-\frac{t}{t+m_{0}}(m-c)^{\gamma-1}(k-c)^{1-\gamma}$$
The degree distribution is obtained by
$$P(k)=\frac{\partial P(k_{i}(t)<k)}{\partial k}=\frac{t}{t+m_{0}}(\gamma-1)(m-c%
)^{\gamma-1}(k-c)^{-\gamma}.$$
The desired result for degree distribution follows from the fact that $t\rightarrow\infty.$
Setting the initial network the complete network with $7$ nodes, i.e. $m_{0}=7$ and $m=5$, we plot degree distributions of complex networks generated by CDPAM for different values of $\beta$ and $\gamma$ in Fig 1. We also calculate the $p$-value which is a measure of goodness-of-fit based on KS statistics, to validate the power-law degree distribution of the networks [9]. The numerical simulations show that the exponent $\gamma$ is an increasing function of $\beta$ when $\theta$ is fixed.
In order to show that the diameter of complex network constructed by the CDPAM is small, we proceed as follows. Let the node $i$ and $j$ appeared in the network at time $t_{i}$ and $t_{j}$ respectively. Assume that $t_{i}<t_{j}.$ Then the probability of the node $j$ to be linked with the node $i$ is given by
$$p_{j}^{i}=m\frac{\beta k_{i}(t_{j})+\theta(k_{i}(t_{j})-2m)}{2m\beta t_{j}}$$
where $k_{i}(t_{j})=(m-c)\left(\frac{t_{j}}{t_{j}}\right)^{1/(\gamma-1)}+c$ (see (3)) is the degree of the node $i$ at time $t_{j}.$ Thus,
$$p_{j}^{i}=\frac{m-c}{(\gamma-1)t_{i}^{1/(\gamma-1)}t_{j}^{1-1/(\gamma-1)}}+%
\frac{m(1-2\theta)}{2\beta t_{j}}.$$
(4)
Remark II.2
It is evident from the above derivation that the control parameters $\beta$ and $\theta$ which represent weights to the local and global property of the existing nodes respectively, determine the topology of the network generated by CDPAM. A natural question would be: Does there exist a functional relation between these parameters? To investigate how different values of these parameters affect the topology of the network, we fix the parameter $\theta$ and vary $\beta$ in the sequel. Thus, now onward we set $\theta=0.5.$
We recall the following lemma from [21].
Lemma II.3
If $A_{1},A_{2},...A_{n}$ are mutually independent events and their probabilities full fill the relations $P(A_{i})\leq\epsilon$ for all $i$ then
$$P\left(\bigcup_{i=1}^{n}A_{i}\right)=1-exp\left(-\sum_{i=1}^{n}P(A_{i})\right)-Q$$
where $0\leq Q<\sum_{j=0}^{n+1}(n\epsilon)^{j}/j!-(1+\epsilon)^{n}.$
Assume that $N(t)$ denotes the set of all nodes which have been added in the network up to timestep $t.$ In the network generated by CDPAM, assume that the nodes $i,j\in N(t)$ are connected by a path $(i,v_{1},v_{2},\ldots,v_{l-1},j)$ of length $l$ where $v_{k}\in N(t)$ for all $k=1:l-1.$ Consider that this sequence is a single event $A_{k}.$ The total number of such events possible is $|N(t)|^{l-1}.$ Thus, as given in [21], the probability of the existence of a path between $i$ and $j$ of length not more than $l$ is given by
$$\begin{split}\displaystyle P_{ij}(l)&\displaystyle=P\left(\bigcup_{k=1}^{|N(t)%
|^{l-1}}A_{k}\right)\\
&\displaystyle=1-exp\left[-\sum_{v_{1}=1}^{|N(t)|}\ldots\sum_{v_{l-1}=1}^{|N(t%
)|}p_{i}^{v_{1}}\ldots p_{v_{l-1}}^{j}\right]\end{split}$$
(5)
We use this result to obtain the following corollary.
Corollary II.4
The probability of the existence of a path between two vertices $i,j\in N(t)$ of length not more than $l$ is given by
$$P_{ij}(l)=1-exp\left[-\dfrac{K^{l}H_{n}^{l-1}}{t_{i}^{1/(\gamma-1)}t_{j}^{1-1/%
(\gamma-1)}}\right]$$
where $K=\frac{(\beta+0.5)(m-c)}{2\beta},H_{n}=\sum_{k=1}^{|N(t)|}\frac{1}{k}$ and $c$ is given in theorem II.1.
Proof: Using (4) and (5) the result follows.
Corollary II.5
The expected value $l_{ij}$ of the distance between two nodes $i,j\in N(t)$ is given by
$$l_{ij}=\dfrac{\left(1-\dfrac{1}{\gamma-1}\right)\ln t_{j}+\dfrac{1}{\gamma-1}%
\ln t_{i}-\log K-r}{\ln(KH_{n})}+\dfrac{1}{2}.$$
Proof: The result follows from the fact that
$$l_{ij}=\sum_{l=0}^{\infty}F(l)$$
where $F(l)=1-P_{ij}(l)$ (see [21]).
Observe in Corollary II.5 that the expected distance $l_{ij}$ between two nodes $i,j\in N(t)$ is an increasing function of $t_{i}$ and $t_{j}$ when other parameters are fixed. This implies that the diameter of the network is the expected distance between the first node and the last node added in the network. Hence, setting $t_{j}=|N(t)|$ and $t_{i}=1$ we obtain the following result.
Corollary II.6
The expected diameter of a complex network generated by CDPAM is given by
$$D=\dfrac{\left(1-\dfrac{1}{\gamma-1}\right)\ln|N(t)|-\ln K-r}{\ln(KH_{n})}+%
\dfrac{1}{2}.$$
Thus it follows from the above corollary that the expected diameter of the network depends on the logarithmic value of the size of new nodes added in the network.
In Fig.2, we calculated the expected diameter for CDPAM and the approximate diameter given by BA model $(\sim\ln N/\ln\ln N)$ [22]. However, numerical simulations show that the expected diameter of CDPAM stabilizes when alike weights are assigned to both the local and global properties which determine the preference of link formation. In contrast to the conventional wisdom that diameter is a function of $\ln(\ln N)$ or $\ln N$ in real networks, the authors in [19] observed that the diameter stabilizes or shrinks as a network grows. The CDPAM reveals how shrinking and increasing of diameter are related to the weights on local and global property of the nodes during expansion of the network.
III Property of complex networks generated by CDPAM
In this section, we numerically calculate various measures which include clustering coefficient, assortativity, algebraic connectivity, and spectral radius for the complex networks generated by CDPAM. These measures determine various topological features of the network and enable to compare how the proposed model captures the property of different real networks. We also compare values of these measures with that of complex network generated by BA model. We have used MATLAB R2012a for the numerical simulations.
III-A Clustering coefficient
Clustering coefficient (CC) of a node signifies the local edge density among the neighbors of the node. The CC of a network is the average of CC of all the nodes. Thus, for a network $N,$
$$CC(i)=\frac{2|E_{i}|}{k_{i}(k_{i}-1)}\,\,\mbox{and}\,\,CC(N)=\frac{1}{|N|}\sum%
_{i}CC(i)$$
where $|E_{i}|$ denotes the number of links adjacent to a node $i$ of a network [8]. It is evident that $0\leq CC(N)\leq 1$ for any network $N.$ In Fig. 3, we plot the CC of different size of complex networks generated by CDPAM with different values of $\beta$ and $\theta=0.5.$ It shows that as the value of $\beta$ increases the CC of the network decreases and eventually when $\beta$ is very large, the CC is close to the CC of the network generated by BA model. The Fig 4 shows that the CC gets close to $0.8$ as $\log\beta$ gets close to zero. Thus, we conclude that, in CDPAM model, if link is formed by giving equal weights to local and global properties of the existing node then the CC gets close to $0.8$ which is a property of a large class of real networks like ego-Facebook network, ego-Gplus network, ego-Twitter [5].
III-B Assortativity index
The Assortative Index (AI) of a network $N$ is defined by
$$AI(N)=\frac{\sum_{ij}(a_{ij}-\frac{k_{i}k_{j}}{2m})k_{i}k_{k}}{\sum_{ij}(k_{i}%
\delta_{ij}-\frac{k_{i}k_{j}}{2m})k_{i}k_{j}}$$
where $a_{ij}$ is the $ij$-th entry of the adjacency matrix associated with $N,$ $\delta_{ij}$ is the Kronecker delta function [23]. Obviously $-1\leq AI(N)\leq 1.$ A positive value of $AI(N)$ signifies nodes with similar degree nodes are linked whereas a negative value of $AI(N)$ implies that similar degree nodes are not linked.
Consider the network $N$ after addition of $t$ nodes to the given small network.Then by (3), it follows that degree of a not is a decreasing function of timestep of its appearance. Further, if a node $j$ which appeared in the network at the $t_{j}$ timestep has probability $p^{i}_{j}$ to get linked with an existing node $i$ appeared at $t_{i}<t_{j}$, is a decreasing function in both $t_{i}$ and $t_{j}$, see (4). These indicate, the probability of having a link between high degree nodes is larger compared to the probability of having a link in between low degree nodes. Therefore, we conclude that the network is assortative for higher degree nodes and disassortative for low degree nodes. Since the network has a few high degree nodes, overall the network is disassortative. The plots given in Fig 5 assert the same for different values of $\beta$ and $\theta=0.5.$ We mention here disassortative phenomena of networks occur in a large class of real networks including World-Wide-Web [11], Marine food web [24], freshwater food web [25].
III-C Number of triangles
A triangle is a cycle with three nodes. The number of triangles is a fundamental building block for many real networks. In a social network, if nodes are human beings and links are described by friendship relation, then the a triangle means friends of a friend are friends. Often real networks consists of a huge number of triangles which could be both homogeneous and heterogeneous [26]. In Fig 6, we show that the proposed complex networks by CDPAM contain huge number of triangles compared to a network constructed by the BA model for example ego-Facebook network, ego-Gplus network, ego-Twitter [5].
III-D Algebraic connectivity
Algebraic connectivity of a network $N$ is the second largest eigenvalue of the Laplacian matrix $L(N)=D(N)-A(N)$ associated with the network where $D(N)=\mathrm{diag}\{k_{1},\ldots,k_{n}\}$ denotes the degree matrix and $A(N)$ is the adjacency matrix of the network [27]. Obviously, $L(N)$ is a symmetric positive semi-definite matrix. It is well known that the second eigenvalue $\lambda_{2}$ of $L(N)$ is positive if and only if $N$ is connected. More importantly, $\lambda_{2}$ determines the robustness of a network, i.e. larger the value of $\lambda_{2},$ the more difficult to make the network disconnected by removal of nodes or edges [27]. In particular, if $\mu(N)$ and $\eta(N)$ denote the vertex and edge connectivity of a network $N$ respectively, then $\lambda_{2}\leq\mu(N)\leq\eta(N).$ We show in Fig 7 that if a complex network is produced by CDPAM after setting $\beta\approx\theta,$ that is giving almost equal weighage to both local and global property of the existing nodes, then the network has higher algebraic connectivity than that of a network produced by the BA model.
III-E Spectral radius
Spectral radius of a network is the maximum modulus of eigenvalues of the network. In [28] it has been shown that the reciprocal of the spectral radius decides the threshold of virus propagation in the network. The smaller the spectral radius is, the larger the robustness of a network against the spread of viruses [28].
In Fig. 8 we plot the spectral radius of networks generated by CDPAM and compared with BA model.
Real world networks show considerable larger spectral radius compared to BA model. CDPAM is capable to inherit large spectral radius as many real world networks including Dutch soccer team network [28], Dutch roadmap network [29], Internet graph at the IP-level [30] and the Autonomous System level [31].
IV Conclusion
In the literature of social choice theory and management science it has been established that the choice of a person get influenced by a given offered set and ultimately, the choice is determined by the local and global contexts of the items in the offered set. Inspired by this concept, we introduced a preferential attachment model for generating growing complex networks when the preference of a new node to get linked with old nodes in a network is determined by local and global properties of the old nodes. We call the model, the context dependent preferential attachment model (CDPAM) and the local property is given by the degree of a node, the global property is given by the relative average degree of the old nodes. We proved that the complex networks generated by CDPAM have power law degree distribution and expected diameter depends logarithmically with the size of new nodes added in the network. In contrast to the general intuition that diameter grows with the addition of new nodes, we numerically showed that, in the CDPAM model, the expected diameter stabilizes when the new nodes get linked by giving alike importance (weight) to both local and global property of the old nodes.
In order to investigate how the complex networks generated by CDPAM and BA models are related, we calculated clustering coefficient, assortativity, number of triangles, algebraic connectivity, spectral radius for both the models. We compared these measures and concluded that BA model is a limiting case of CDPAM when new nodes tend to give large weight to the local property compared to the weight given to the global property during link formation. By using these measures, we showed that the CDPAM captures the properties of real networks better than BA model.
An interesting question is: can communities emerge in CDPAM? We believe that communities will also emerge when the weights to the local and global properties will not be constant for all new nodes but vary with the new nodes. We plan to investigate this phenomenon in future.
References
[1]
F. Murray, “Innovation as co-evolution of scientific and technological
networks: exploring tissue engineering,” Research Policy, vol. 31,
no. 8, pp. 1389–1403, 2002.
[2]
W.-X. Wang, B.-H. Wang, B. Hu, G. Yan, and Q. Ou, “General dynamics of
topology and traffic on weighted technological networks,” Physical
review letters, vol. 94, no. 18, p. 188702, 2005.
[3]
J. Balthrop, S. Forrest, M. E. Newman, and M. M. Williamson, “Technological
networks and the spread of computer viruses,” arXiv preprint
cs/0407048, 2004.
[4]
E. Bullmore and O. Sporns, “Complex brain networks: graph theoretical analysis
of structural and functional systems,” Nature Reviews Neuroscience,
vol. 10, no. 3, pp. 186–198, 2009.
[5]
J. McAuley and J. Leskovec, “Learning to discover social circles in ego
networks (2012),” in Neural Information Processing Systems (NIPS).
[6]
S. Maere, K. Heymans, and M. Kuiper, “Bingo: a cytoscape plugin to assess
overrepresentation of gene ontology categories in biological networks,”
Bioinformatics, vol. 21, no. 16, pp. 3448–3449, 2005.
[7]
R. Albert, I. Albert, and G. L. Nakarado, “Structural vulnerability of the
north american power grid,” Physical review E, vol. 69, no. 2, p.
025103, 2004.
[8]
D. J. Watts and S. H. Strogatz, “Collective dynamics of
âsmall-worldânetworks,” nature, vol. 393, no. 6684, pp. 440–442,
1998.
[9]
A. Clauset, C. R. Shalizi, and M. E. Newman, “Power-law distributions in
empirical data,” SIAM review, vol. 51, no. 4, pp. 661–703, 2009.
[10]
M. E. Newman, “Power laws, pareto distributions and zipf’s law,”
Contemporary physics, vol. 46, no. 5, pp. 323–351, 2005.
[11]
A.-L. Barabási and R. Albert, “Emergence of scaling in random networks,”
science, vol. 286, no. 5439, pp. 509–512, 1999.
[12]
A.-L. Barabási, Z. Dezsö, E. Ravasz, S.-H. Yook, and Z. Oltvai,
“Scale-free and hierarchical structures in complex networks,” in
Modeling of Complex Systems: Seventh Granada Lectures, vol. 661,
no. 1. AIP Publishing, 2003, pp.
1–16.
[13]
P. Erdös and A. Rényi, “On random graphs i.” Publ. Math.
Debrecen, vol. 6, pp. 290–297, 1959.
[14]
R. Albert, H. Jeong, and A.-L. Barabási, “Internet: Diameter of the
world-wide web,” Nature, vol. 401, no. 6749, pp. 130–131, 1999.
[15]
A. Tversky and I. Simonson, “Context-dependent preferences,” Management
science, vol. 39, no. 10, pp. 1179–1189, 1993.
[16]
A. Tversky, “Elimination by aspects: A theory of choice.” Psychological
review, vol. 79, no. 4, p. 281, 1972.
[17]
J. Huber, J. W. Payne, and C. Puto, “Adding asymmetrically dominated
alternatives: Violations of regularity and the similarity hypothesis,”
Journal of consumer research, pp. 90–98, 1982.
[18]
I. Simonson, “Choice based on reasons: The case of attraction and compromise
effects,” Journal of consumer research, pp. 158–174, 1989.
[19]
J. Leskovec, J. Kleinberg, and C. Faloutsos, “Graph evolution: Densification
and shrinking diameters,” ACM Transactions on Knowledge Discovery from
Data (TKDD), vol. 1, no. 1, p. 2, 2007.
[20]
A.-L. Barabási, R. Albert, and H. Jeong, “Mean-field theory for scale-free
random networks,” Physica A: Statistical Mechanics and its
Applications, vol. 272, no. 1, pp. 173–187, 1999.
[21]
A. Fronczak, P. Fronczak, and J. A. Hołyst, “Average path length in random
networks,” Physical Review E, vol. 70, no. 5, p. 056110, 2004.
[22]
R. Cohen and S. Havlin, “Scale-free networks are ultrasmall,” Phys.
Rev. Lett., vol. 90, p. 058701, Feb 2003. [Online]. Available:
http://link.aps.org/doi/10.1103/PhysRevLett.90.058701
[23]
M. E. Newman, “Assortative mixing in networks,” Physical review
letters, vol. 89, no. 20, p. 208701, 2002.
[24]
M. Huxham, S. Beaney, and D. Raffaelli, “Do parasites reduce the chances of
triangulation in a real food web?” Oikos, pp. 284–300, 1996.
[25]
N. D. Martinez, “Artifacts or attributes? effects of resolution on the little
rock lake food web,” Ecological Monographs, pp. 367–392, 1991.
[26]
N. Durak, A. Pinar, T. G. Kolda, and C. Seshadhri, “Degree relations of
triangles in real-world networks and graph models,” in Proceedings of
the 21st ACM international conference on Information and knowledge
management. ACM, 2012, pp.
1712–1716.
[27]
M. Fiedler, “Algebraic connectivity of graphs,” Czechoslovak
Mathematical Journal, vol. 23, no. 2, pp. 298–305, 1973.
[28]
A. Jamakovic, “Characterization of complex networks: application to robustness
analysis,” 2008.
[29]
A. Jamakovic, H. Wang, and P. Van Mieghem, “Topological characteristics of the
dutch road infrastructure,” in Seminar Infrastructure Reliability TU
Delft, 2006.
[30]
A. Jamakovic and P. Van Mieghem, “The laplacian spectrum of complex
networks,” in European Conference on Complex Systems, 2006.
[31]
W. Mühlbauer, A. Feldmann, O. Maennel, M. Roughan, and S. Uhlig, “Building
an as-topology model that captures route diversity,” in ACM SIGCOMM
Computer Communication Review, vol. 36, no. 4. ACM, 2006, pp. 195–206. |
Orientation-dependent C${}_{60}$ electronic structures revealed by photoemission
V. Brouet${}^{1,2}$, W.L. Yang${}^{1,2}$, X.J. Zhou${}^{2}$, H.J. Choi${}^{3,4}$, S.G. Louie${}^{3,5}$,
M.L. Cohen${}^{3,5}$, A. Goldoni${}^{6}$, F. Parmigiani${}^{6}$,
Z. Hussain${}^{1}$, and Z.X. Shen${}^{2}$
${}^{1}$Advanced Light Source, Lawrence Berkeley
National Laboratory, Berkeley, California 94720
${}^{2}$ Stanford Synchrotron Radiation Laboratory and Department of
Applied Physics,
Stanford university, Stanford, California 94305
${}^{3}$ Department of Physics, University of California at
Berkeley,
Berkeley, California 94720
${}^{4}$ Korea Institute for Advanced Study, 207-43 Cheongryangri
Dongdaemun,
Seoul 130-722, Korea
${}^{5}$Materials Science Divisions, Lawrence Berkeley
National Laboratory, Berkeley, California 94720
${}^{6}$ Sincrotone Trieste S.C.p.A., S.S. 14 Km 163.5, in Area
Science Park, 34012 Trieste, Italy
(November 25, 2020)
Abstract
We observe, with angle-resolved photoemission, a dramatic change
in the electronic structure of two C${}_{60}$ monolayers, deposited
respectively on Ag (111) and (100) substrates, and similarly doped
with potassium to half-filling of the C${}_{60}$ lowest unoccupied
molecular orbital. The Fermi surface symmetry, the bandwidth, and
the curvature of the dispersion at $\Gamma$ point are different.
Orientations of the C${}_{60}$ molecules on the two substrates are
known to be the main structural difference between the two
monolayers, and we present new band-structure calculations for
some of these orientations. We conclude that orientations play a
key role in the electronic structure of fullerides.
In a standard formulation of quantum theory of solids, the
emphasis is on the periodic nature of the lattice structure and
the internal degrees of freedom are usually ignored. As the
frontier of condensed matter physics moves to more complex solids,
such issues become more and more important. Complexity often
arises from situations where interactions with similar energy
scales are competing, and no degrees of freedom can be safely
ignored. Fullerides offer one very interesting example of such a
situation. They are challenging standard approximations in solid
state physics, because electronic correlations in the 3-fold
degenerate band are strong and electrons are coupled to high
frequency phonons GunnarssonRMP . The primary reason for
physicists to study them is to understand how these parameters
might lead to new behaviors. However, they are also archetypical
molecular systems, and many degrees of freedom associated with the
C${}_{60}$ molecule (e.g. vibrational modes, Jahn-Teller
distortions, orientational order etc.) should be taken into
account, which greatly complicates the analysis. In fact, strong
electronic correlations enhance the sensitivity to these
local scale structures, because they increase the average time
spent by one electron near a C${}_{60}$, so that such a problem is
typically to be expected in a strongly correlated material.
We reveal here an extreme sensitivity of the band structure of
C${}_{60}$ monolayers to one of this internal degree of freedom,
namely the molecular orientations. The role of orientations
in the electronic properties of fullerides has often be
questioned. For example, A${}_{3}$C${}_{60}$ and Na${}_{2}$AC${}_{60}$
(A=K, Rb), which have similar structures but different
orientational states, are both superconducting but with a
different dependence of the transition temperature on the lattice
parameter YildirimPRL96 ; BrownPRB99 .
In the (AC${}_{60}$)${}_{n}$ polymers, a different orientations in C${}_{60}$ chains might
control a transition between 1D and 3D electronic structures
AlloulPRL96 . In TDAE-C${}_{60}$, the orientational order can
be changed by the cooling process, which results in different
magnetic ground states MihailovicScience95 . Nevertheless,
the correlation between electronic properties and orientations has
remained difficult to pinpoint. Recently, we have resolved the
dispersion of a band in a C${}_{60}$ monolayer through
angle-resolved photoemission (ARPES) YangScience03 , which
opens the possibility to monitor directly the changes in band
structure as a function of orientations. A high sensitivity of the
band structure to relative molecular orientations can be expected
because the three degenerate lowest unoccupied molecular orbitals
(LUMOs) that form the conduction band, which are mainly built out
of $p$-orbitals pointing radially at each carbon atom, have high
angular momenta (L=5) LaouiniPRB95 . We present here an
ARPES study of C${}_{60}$ monolayers where structural changes,
including different molecular orientations, are induced by the use
of two different substrates, Ag(111) and Ag(100). We evidence a
complete change of symmetry of the Fermi surface (FS) and of the
band dispersion, and investigate the role of orientations in the
electronic structure with first-principles band-structure
calculations for some of the configurations encountered in these
monolayers.
The growth of C${}_{60}$
monolayers on different substrates is very well documented
ReviewRudolf . We first deposited a C${}_{60}$ multilayer onto
a clean Ag substrate and obtained a monolayer by annealing it at
$\sim$ 650 K, and then we doped the layer by potassium (K)
evaporation. The cleanliness of the substrate was checked by the
observation of Ag surface states AgSurfaceState , which
disappear after C${}_{60}$ deposition. The
structure of the monolayer results from a compromise between the C${}_{60}$-substrate and C${}_{60}$-C${}_{60}$ interactions, which are of
similar strength on noble metal surfaces ReviewRudolf . The
Ag(111) surface offers the best lattice match with C${}_{60}$,
leading to a hexagonal C${}_{60}$ overlayer, very similar to a (111)
plane of the bulk compounds. In case of Ag (100), the hexagonal
packing of C${}_{60}$ is distorted along one of the two equivalent
directions, as illustrated in Fig. 1a. While this
structure has first been described as c(6*4) GoldoniPRB , an
incommensurate structure was proposed more recently
PaiPRB03 . For our monolayer, the low energy electron
diffraction (LEED) pattern, presented in Fig. 1b, is
in better agreement with c(6*4), although some distortion from
this model structure might be present. As for the C${}_{60}$
orientation on top of the Ag(100) substrate, the scanning
tunneling microscopy (STM) GrobisPRB and the X-ray
photoelectron diffraction (XPD) CepekPRB2001 reveal the
coexistence of two orientations, with either a single (5-6) bond
(between a pentagon and a hexagon) or a double (6-6) bond (between
two hexagons) facing the substrate and being aligned with the
[110] or [1-10] direction
(see examples on Fig. 1a). These orientations contrast with most noble metal (111)
surfaces where a hexagon of C${}_{60}$ faces the substrate
FaselPRL96 , as for Ag(111) Osterwalder .
The main ARPES results of the present study are summarized in
Fig. 2, which compares the electronic structure of two
monolayers. In both cases, the number of electrons per C${}_{60}$ is
estimated from the integrated area of the LUMO peak to be near 3,
i.e. the band is half filled. All data were collected at the
Advanced Light Source with a 35 eV photon beam in grazing
incidence and polarized nearly perpendicularly to the sample
surface YangScience03 . In the top of Fig. 2, we
show with dotted lines the contour of the FS that reflects the
different symmetry of the structure of the C${}_{60}$ monolayers,
induced by the substrate. The FS is almost circular in the case of
C${}_{60}$/Ag(111), while it is rather rectangular for
C${}_{60}$/Ag(100). The complete analysis of the FS symmetry was
given in ref. YangScience03 for Ag(111) and will be given
below for Ag(100). In the bottom of Fig. 2, we further
compare the dispersion along high
symmetry lines, which reveals a more unexpected contrast. Most notably, the $\Gamma$ point is unoccupied in C${}_{60}$/Ag(111),
while it is occupied for at least one of the three LUMO sub-bands in C${}_{60}$/Ag(100). As the $\Gamma$ point is common to the two
C${}_{60}$/Ag(100) domains, this behavior directly establishes a
significant difference in band structures, regardless of
any further analysis or structural details. We argue below that
this change is related to the different orientations.
Figure 3 presents a larger view of the reciprocal
space in the case of the Ag(100) substrate. The map was obtained
by integration of the spectral intensity between 0.01 eV and
-0.05 eV from the Fermi level (E${}_{F}$). As a result of the
molecular nature of C${}_{60}$-based compounds, the photoemission
cross-sections are strongly energy- and angle-dependent SchiesslingPRB03 , and particular attention has to be given to
the meaning of the measured intensities. Here, dispersion images
show that each high intensity region of the map corresponds to a
band dispersing towards E${}_{F}$, like in Fig. 2. This
rules out a simple modulation of the intensity due to
cross-section or photoelectron diffraction effects
Photoelectron . Furthermore, the map presents the
periodicity of the C${}_{60}$ reciprocal lattice, whereas such
modulations would be expected over a much larger angular range.
The determination of the FS is complicated in Ag(100) by the
presence of two domains, but their respective contribution can be
distinguished by sampling a large area of momentum space, as in
Fig. 3, because this covers many Brillouin Zones (BZ)
with inequivalent contributions from the two domains. The map is
characterized by a clear symmetry with respect to the diagonals
(black dashed lines), which is actually expected from the
superposition of the two domains (see inset at left of Fig. 3). Furthermore, the high intensity regions (yellow to
black color) are concentrated along regularly spaced
vertical and horizontal lines, shown in Fig. 3 as
blue and red dotted lines, respectively. As there is no 4-fold
symmetry for one domain, this regular pattern must originate from
a well-defined axial symmetry within each domain, which
will appear as a squaring after superposition. There are two
symmetry axes in the BZ that could play this role, $\Gamma M$ and
$\Gamma K^{\prime}$ (see Fig. 1c), but the spacing between the
dotted lines is only consistent with segments oriented along
$\Gamma K^{\prime}$. This means that the vertical segments of
high spectral intensity arise from the domain drawn in blue, and
the horizontal ones from the one in red. For each domain, the
dotted lines define a “stripe”, reported on the inset of
Fig. 3, which must contain most of the FS. It can be
worked out that it is the overlap between the bands of the two
domains that blurs the intensity at E${}_{F}$ in some regions of the
map (e.g. along the blue line at ky=0).
The clarity of the square pattern implies a simple Fermi surface
geometry, which is in fact surprising when one considers that the
C${}_{60}$ LUMOs are triply degenerate, suggesting more than one
piece
of FS. To create a simple pattern, all LUMO-derived sub-bands must have
similar FS contours mostly following the dotted lines. In fact, we
could only distinguish sub-bands if/when their FS contours
deviate from the dotted lines. There is such a region near $(k_{x}=0.4\,,$ $k_{y}=-0.4)$, indicated by black triangles. The dispersion image at k${}_{y}$=-0.4 Å${}^{-1}$(not shown) reveals two bands crossing the Fermi level with
opposite slopes. This allows to refine the contour of the FS for
these two different sub-bands and the result is shown in
Fig. 3. The larger, more rectangular, contour
corresponds to one (or possibly two) sub-band(s) empty at $\Gamma$, while $\Gamma$ is filled for the other contour and
corresponding band(s). For clarity, we have reported only the
average contour of these two pieces of Fermi surface in
Fig. 2.
With this knowledge about the FS of C${}_{60}$/Ag(100), we can
return in more details to the comparison of Fig. 2.
For C${}_{60}$/Ag(111), the monolayer is a single domain and all
directions look roughly the same due to the high hexagonal
symmetry. The comparison with the theoretical band structure
indicates that the observed dispersion corresponds to two
unresolved sub-bands and that the third one remains totally empty
YangScience03 . For C${}_{60}$/Ag(100), we present the
dispersion along the diagonal, where it is the clearest because it
is the same direction, roughly corresponding to $\Gamma M^{\prime}$, in
both domains. In addition to the difference in curvature at
$\Gamma$, it is interesting to note that the dispersion is
significantly larger for Ag(100) compared to Ag(111), namely 135 meV $\pm$ $15$ meV compared to about 100 meV. Naively, one would expect the
opposite because the distances between C${}_{60}$ are larger on
Ag(100) than Ag(111), which should reduce the bandwidth. This
further proves that the band structure is not simply
“rescaled” according to the new lattice but much more deeply
modified and that all parameters must be considered before
comparing two different C${}_{60}$ systems.
Two factors come to mind to explain the change in band structure,
either the interaction with the substrate or the orientations of
the C${}_{60}$ molecules. It is difficult to estimate possible
contribution from the substrate, but the electronic structure for
the K-doped C${}_{60}$/Ag(111) monolayer was found very similar to
that of the bulk YangScience03 , suggesting only a marginal
influence, as also concluded in Ref. HoogenboomPRB98 for
doped monolayers on noble metal surfaces.
To get a better understanding of the possible role of the
orientations, we now take a closer look at the contact geometry
between two neighboring molecules in the different cases. On
Ag(111), the contact is always through two single bonds, as
sketched in Fig.
4A. Note that this ordering of the C${}_{60}$ molecules is very close to that found in the (111) plane of the disordered $fcc$ structure of A${}_{3}$C${}_{60}$. On Ag(100), more different
contact geometries are encountered, depending on the respective
orientations of two neighboring molecules. A molecule oriented
along a 6-6 bond on top can present either a single bond or a
pentagon to its neighbor. This results in three possible contact
geometries, for this 6-6 orientation only (see
Fig. 4) : two single bonds face to face (B), a
single bond towards a pentagon (C), and two pentagons face to face
(D). To investigate the impact of these changes on the electronic
structure, we have calculated the band structure Cohen in
cases B and D and found a large difference, as shown on
Fig. 4, with bands
reaching much lower energies at $\Gamma$ for case D. This demonstrates the ability of the
orientations to change the electronic structure.
It is difficult to make a full realistic calculation for Ag(100)
because of the coexistence of different orientations and also of
some uncertainty in the 5-6 orientation (it differs by a few
degrees between STM and XPD GrobisPRB ; CepekPRB2001 ).
However, the previous analysis supports the idea that it is the
type of contact geometry that defines the general shape of the
band structure. Indeed, for cases A and B, which are in a similar
(but not identical) configuration, the dispersion is maximum at
$\Gamma$ for at least two bands, in sharp contrast with the very
different configuration represented in D, where two bands show a
deep minimum. STM images show frequent alternation between
different orientations, although in a random way. The structure in
Ag(100) is then likely to be dominated by the “bond vs polygon”
type of configuration (case C), which is obtained each time two
neighboring molecules have a different orientation. In fact, this
type of configuration is the most favorable energetically because
an electron-rich bond faces an electron-poor polygon. This is what
stabilizes the orientationally ordered $sc$ structure
LaunoisC60 (that of Na${}_{2}$CsC${}_{60}$ at low
temperatures), where four different orientations alternate in the
(111) plane to create the “bond vs polygon” situation. We
believe that it is these identical contact geometries that allow
the development of a well defined dispersive structure despite the
disorder in orientations. Furthermore, our ARPES result in
C${}_{60}$/Ag(100) is in qualitative agreement with the calculation
for the $sc$ structure, based on these contact geometries, where
$\Gamma$ is occupied for two bands LaouiniPRB95 .
In conclusion, we evidence here for the first time the impact of a
change in C${}_{60}$ orientations on the band structure of these
materials. We find that relative orientations can be more
important in defining the band structure than the distances
between molecules. Very early, calculations have suggested that
the band structure could be very sensitive to relative
orientations SGLouie ; GelfandPRL92 ; LaouiniPRB95 . We present
here new calculations for two different arrangements of the
C${}_{60}$ molecules, related to those found in our monolayers,
which reveal differences in band structures in qualitative
agreement with our experimental observation. We show
that the difference in orientations observed in C${}_{60}$/Ag(111) and C${}_{60}$/Ag(100) closely correspond to the different arrangement of the
molecules in a (111) plane of the $fcc$ and $sc$ structures,
respectively. Then, our study gives an experimental basis to the
relevance of orientations in band-structure calculations, not only
for these monolayers, but also to approach real situations in the
bulk, like the difference between A${}_{3}$C${}_{60}$ and
Na${}_{2}$CsC${}_{60}$. More generally, this study gives an example of
how the internal structure of the building block of complex
systems can affect their macroscopic properties.
We would like to thank M. Grobis and X. Lu for useful discussion
of their STM data. The SSRL’s effort is supported by DOE’s Office
of Basic Energy Sciences, Division of Materials Science with
contract DE-FG03-01ER45929-A001. The work at Stanford was
supported by ONR grant N00014-98-1-0195-P0007 and NSF grant DMR-
0071897. The computational work was supported by NSF Grant No.
DMR00-87088 and BES’s Office of the DOE under Contract
DE-AC03-76SF00098. Computational resources have been provided by
NSF at NCSA and by NERSC.
References
(1)
O. Gunnarsson, Rev. of Modern Physics 69,
575 (1997)
(2)
T. Yildirim et al., Phys. Rev. Lett.
77, 167 (1996)
(3)
C.M. Brown et al., Phys. Rev. B 59,
4439 (1999)
(4)
H. Alloul et al., Phys. Rev. Lett. 76, 2922 (1996)
(5)
D. Mihailovic et al., Science
268, 400 (1995)
(6)
W.L. Yang et al., Science 300,
303 (2003)
(7)
N. Laouini, O.K. Andersen and O. Gunnarsson, Phys.
Rev. B 51, 17446 (1995)
(8)
For a review, see P. Rudolf Fullerenes and
Fullerene Nanostructures, edited by World Scientific (Singapore),
p. 263 (1996)
(9)
A. Goldman and E. Bartels, Surface
Science Letters 122, L629 (1982)
(10)
A. Goldoni et al., Phys. Rev. B 58,
2228 (1998)
(11)
W.W. Pai and C.L. Hsu, Phys. Rev. B 68, 121403
(2003)
(12)
M. Grobis, X. Lu and M.F. Crommie, Phys. Rev. B 66, 161408 (2002)
(13)
C. Cepek et al., Phys. Rev. B 63,
125406 (2001)
(14)
R. Fasel et al., Phys. Rev. Lett. 76, 4733 (1996)
(15)
A. Tamai, J. Osterwalder et al., to be published
(16)
J. Schliessling et al., Phys. Rev. B
68, 205405 (2003)
(17)
S. Hüffner, Photoelectron spectroscopy, Springer-Verlag
(18)
B.W. Hoogenboom, R. Hesper, L.H. Tjeng and G.A.
Sawatzky, Phys. Rev. B 57, 11939 (1998)
(19)
M. L. Cohen, Phys. Scr. T1, 5 (1982); N. Troullier and J. L. Martins, Phys. Rev. B 43,
1993 (1991); D. Sánchez-Portal, P. Ordejón, E. Artacho,
and J. M. Soler, Int. J. Quantum Chem. 65, 453 (1997).
(20)
P. Launois, S. Ravy and R. Moret, Phys. Rev. B 55, 2651 (1997) and references therein.
(21)
E.L. Shirley and S.G. Louie, Phys. Rev. Lett. 71, 133 (1993)
(22)
M.P. Gelfand and J.P. Lu, Phys. Rev. Lett. 68, 1050 (1992) |
Direct evaluation of attachment and detachment rate factors of atoms in crystallizing supercooled liquids
Dinar T. Yarullin
YarullinDT@gmail.com
Bulat N. Galimzyanov
bulatgnmail@gmail.com
Anatolii V. Mokshin
anatolii.mokshin@mail.ru
Kazan Federal University, 420008 Kazan, Russia
Udmurt Federal Research Center of the Ural Branch of the Russian Academy of Sciences, 426067 Izhevsk, Russia
Abstract
Kinetic rate factors of crystallization have a direct effect on formation and growth of an ordered solid phase in supercooled liquids and glasses. Using crystallizing Lennard-Jones liquid as an example, in the present work we perform a direct quantitative estimation of values of the key crystallization kinetic rate factors – the rate $g^{+}$ of particle attachments to a crystalline nucleus and the rate $g^{-}$ of particle detachments from a nucleus. We propose a numerical approach, according to which a statistical treatment of the results of molecular dynamics simulations was performed without using any model functions and/or fitting parameters. This approach allows one to accurately estimate the critical nucleus size $n_{c}$. We find that for the growing nuclei, whose sizes are larger than the critical size $n_{c}$, the dependence of these kinetic rate factors on the nucleus size $n$ follows a power law. In the case of the subnucleation regime, when the nuclei are smaller than $n_{c}$, the $n$-dependence of the quantity $g^{+}$ is strongly determined by the inherent microscopic properties of a system and this dependence cannot be described in the framework of any universal law (for example, a power law). It has been established that the dependence of the growth rate of a crystalline nucleus on its size goes into the stationary regime at the sizes $n>3n_{c}$ particles.
keywords:
crystallization kinetics, crystal growth, nucleation, supercooled liquids, glasses
1 Introduction
Crystallization and condensation are processes in which the rates of attachment and detachment of monomers (atoms, molecules) to and from nuclei play an important role in the nucleation and growth kinetics Kelton_Greer_1986 ; Yasuoka_Matsumoto_1998 . The details of the condensation kinetics have been well studied by experimental and molecular dynamics simulation methods Schaaf_Senger_2001 ; Diemand_Angelil_2013 . Although crystallization of supercooled liquids has also been the subject of extensive studies, the crystallization kinetics is not well understood, especially for deep levels of supercooling Sosso_Chen_2016 . One of the main reasons for this is the absence of studies, which are focused on accurate quantification and theoretical description of the monomer gain and loss processes during the crystallization of supercooled liquids and glasses Kashchiev_Nucleation_2000 .
The transition rate $g^{+}$ of particles from a liquid to a crystalline phase and the detachment rate $g^{-}$ of particles from a crystalline nucleus are the main kinetic rate factors in the theory of nucleation Kelton_Greer_1986 ; Kashchiev_Nucleation_2000 . These kinetic factors are required to determine the rate characteristics of the crystal nucleation and crystal growth processes Agrawal_2014 ; Mura_Zaccone_2016 ; Baidakov_2019 ; Mokshin_Galimzyanov_PCCP_2017 ; Rodrigues_2018 . Therefore, the quantities $g^{+}$ and $g^{-}$ are included in the master equations of well-known kinetic theories and theoretical models that describe the nucleation and growth of crystals. The Wilson-Frenkel theory Wilson_1900 ; Frenkel_1932 , the Turnbull-Fisher model Turnbull_Fisher_1949 , the Kelton-Greer model Kelton_Greer_1986 and the gain-loss theory Weinberg_2002 are among such the kinetic theories and models.
Direct experimental measurement of the kinetic rate factors $g^{+}$ and $g^{-}$ for a crystallizing bulk system is a very complex task. This is due to difficulties in tracking the trajectories of individual atoms of nano-sized scales. The quantity $g^{+}$ can be calculated indirectly by methods based on Turnbull or Kelton equations Turnbull_1961 ; Kelton_1991 ; Baidakov_2019 ; MG_JETPLett_2019 by using experimentally measured self-diffusion coefficient or viscosity. Note that the accuracy of these methods is usually insignificant, especially, for the systems at deep supercooling. On the other hand, classical molecular dynamics simulations are an excellent tool to extract complete information about a crystallizing system as well as for the study of nucleation and growth processes Alder_MD ; Orava_Greer_2014 ; Kirova_Pisarev_2019 ; Galenko_Salhoumi_2019 ; Kamaeva_Ryltsev_2020 ; Ryltsev_Chtchelkatchev_2020 ; Inogamov_Khokhlov_2020 . In this regard, the results of molecular dynamics simulations can be used to evaluate the kinetic rate factors.
Statistical treatment of information obtained from molecular dynamics simulations can be performed using the mean first-passage time (MFPT) method Wedekind_2007 ; Mokshin_Galimzyanov_2012 ; Mokshin_Galimzyanov_PCCP_2017 . The MFPT-method is straightforward to implement in a simulation and this method allows one to determine the activation barrier, growth curves, and lag times Wedekind_2007 . As shown before, this method can be used to estimate the value of the kinetic rate factor $g_{n_{c}}^{+}$ for the critically-sized nucleus [Ref. Wedekind_2008 ; Lundrigan_2009 ; Mendelev_2018 ]. Another method proposed by Auer and Frenkel Auer_Frenkel_2004 to compute the quantity $g_{n_{c}}^{+}$ for nuclei with the critical size $n_{c}$ is also based on the statistical treatment of trajectories of growing crystalline nuclei. In this method, it is assumed that the size of a nucleus fluctuates around its critical value and this nucleus grows via the diffusive attachment of single particles.
In the present work, we propose a simple and accurate approach for direct evaluation of the kinetic rate factors $g^{+}(n)$ and $g^{-}(n)$. According to this approach, the calculations can be performed as the size-dependent quantities without using model functions and fitting parameters. We demonstrate the efficiency of the approach for the case of study of the crystallization kinetics of supercooled Lennard-Jones (LJ) liquid.
2 Approach for direct evaluation of the kinetic rate factors
Let us consider an idealized situation of mononuclear crystallization when a single crystalline nucleus grows isotropically in a supercooled liquid. The growth of this nucleus occurs due to the local rearrangements of parent (disordered) phase particles, which are located near the surface of a crystalline nucleus. Then, the quantity $k^{+}$ will determine the number of particles (monomers) attached to the nucleus surface in a unit time. At the same time, a crystalline nucleus can decay due to the detachment of particles from its surface. The number of detached particles we denote as $k^{-}$. Thus, a nucleus grows when $k^{+}>k^{-}$. On the other hand, the nucleus size remains unchanged when $k^{+}=k^{-}$; whereas the nucleus size decreases in the case $k^{+}<k^{-}$.
Let us suppose that the trajectories $\vec{r}_{i}(t)$ [$i=1,2,...,N$, $N$ is the number of particles] of all particles in the system are known. Each particle has a unique label number, which is assigned during cluster analysis of simulation results. Then, a nucleus of the size $n(t)$ at the time $t$ can be represented as a one-dimensional array that consists of the labels of nucleus particles. Changes of label numbers in this array are tracked at each simulation time step [see scheme in Fig. 1]. So, the appearance of new labels (particles) in this array is the manifestation of the so-called gain process. The number of such the labels will determine the value of the quantity $k^{+}(n)$. The disappearance of labels from the array corresponds to the loss process and the number of such the labels defines value of the quantity $k^{-}(n)$.
The mononuclear crystallization scenario considered above is very specific and it implements rarely. The polynuclear scenario is more common at the crystallization of supercooled liquids and glasses. This scenario involves formation of stable crystalline nuclei in a system JETP_2018 . In the case of a high concentration of crystalline nuclei, these nuclei grow mainly due to coalescence with each other JCG_2019 . Therefore, for evaluate the quantity $k^{+}$ we take into account only the particles, which transfer to a the crystalline phase directly from a disordered phase. On the other hand, a crystalline domain can decay into separate parts when incomplete coalescence of nuclei occurs or when a nucleus has a highly ramified shape. Such the decay can lead to a rapid increase in the number of detached particles, $k^{-}$. Therefore, at calculation the value of the quantity $k^{-}$ we consider the particles that transfer from the crystalline phase to the disordered phase. Exclusion from consideration of the nuclei coalescence and nucleus decay processes allows one to determine values of the kinetic rate factors $g^{+}$ and $g^{-}$ with high accuracy. Thus, the proposed approach is not limited to any supercooling regime since these kinetic rate factors are calculated from their basic definitions.
According to the basic definition Kashchiev_Nucleation_2000 , the kinetic rate factor $g^{+}$ characterizes the number of particles (monomers) attached to a crystalline nucleus of size $n$ over the shortest time step $\Delta t$. The kinetic rate factor $g^{-}$ determines the number of particles detached from $n$-sized crystalline nucleus and transferred to the parent phase over the time step $\Delta t$. Thus, the expressions for estimation the values of the quantities $g^{+}$ and $g^{-}$ will have the following forms
$$g^{+}(n)=\frac{\left\langle k^{+}(n)\right\rangle}{\Delta t},$$
(1)
$$g^{-}(n)=\frac{\left\langle k^{-}(n)\right\rangle}{\Delta t}.$$
(2)
In the present work, the simulation time step is $\Delta t=0.01\,\tau$ [in the case of argon with the parameters $m=6.63\times 10^{-26}$ kg, $\sigma=0.341$ nm, $\epsilon/k_{B}=119.8$ K and $\tau=\sigma\sqrt{m/\epsilon}$, this time step is $\simeq 0.0215$ ps]. The brackets $\langle...\rangle$ denote averaging over various molecular dynamics iterations. Statistical treatment of our results is carried out over $50$ independent trajectories $n(t)$ of the largest growing crystalline nucleus.
3 Results and Discussion
3.1 Estimation of the nucleus critical size
In the present work, we consider the crystallization of the supercooled Lennard-Jones liquid at the temperature $T=0.5\,\epsilon/k_{B}$ and the pressure $p=2.0\,\epsilon/\sigma^{3}$. The simulation details and applied methods are given in the Appendix. A well-known peculiarity of one-component LJ-system is that this system is a poor glass-former and it crystallizes rapidly after cooling below the melting temperature $T_{m}$ [Ref. Baidakov_Protsenko_2019 ; Stephan_Thol_2019 ]. This means that it is possible to observe the crystal nucleation and crystal growth processes in this crystallizing system at time scales available for simulation. For the considered thermodynamic ($p$, $T$)-state, the stable crystalline nuclei appear at the times $t>10$ $\tau$ [see Figs. 2(a) and 2(b)]. The high concentration of crystalline nuclei leads to their coalescence at the times $t>50\,\tau$ [see Fig. 2(c)]. Fig. 2(d) shows that the system forms a polycrystal that is the typical final structure appeared due to the crystallization of a liquid of moderate supercooling levels JETP_2018 . To minimize the effects related with the nuclei coalescence and nucleus decay processes on values of the kinetic rate factors, we will consider the nuclei that grow only within the time range $t\in[0;\,50]\,\tau$.
Fig. 3(a) shows the dependence of the mean first-passage time $\bar{\tau}$ of the largest crystalline nucleus on its size $n$ [Ref. Mokshin_Galimzyanov_2012 ; Mokshin_Galimzyanov_PCCP_2017 ]. As can be seen from Fig. 3(a), the curve $\bar{\tau}(n)$ contains the pronounced inflection point Wedekind_2008 ; Lundrigan_2009 ; Gunawardana_2018 . The derivative of $\bar{\tau}(n)$ over variable $n$ has one pronounced maximum, as seen from Fig. 3(b). From the location of this maximum on $n$-scale, we find the critical nucleus size $n_{c}\simeq(50\pm 3)$ particles. This value of the critical size is typical for spontaneously crystallizing LJ-system at the considered ($p$, $T$)-state Baidakov_2019 . The average waiting time for the critically-sized nucleus is $\tau_{c}\simeq(11.5\pm 1.2)\,\tau$ [in the case of argon with the interatomic potential parameter $\sigma=0.341$ nm and $\epsilon/k_{B}=119.8$ K, this time $\tau_{c}$ corresponds to the value $(24.7\pm 2.58)$ ps]. Such the relatively small value of the waiting time $\tau_{c}$ indicates on an extremely high crystal nucleation rate in the supercooled LJ-system Baidakov_Protsenko_2019 .
Based on the obtained MFPT-curve shown in Fig. 3, we have calculated the height of the nucleation barrier Wedekind_2007 ; Mokshin_Galimzyanov_2012
$$\beta\Delta G^{*}=\frac{3\pi}{4}\left(\frac{n_{c}}{\tau_{c}}\right)^{2}\left[\frac{\partial\bar{\tau}(n)}{\partial n}\Bigg{|}_{n=n_{c}}\right]^{2},$$
(3)
which takes the value $\beta\Delta G^{*}\approx(2.6\pm 0.4)$ (where $\beta=(k_{B}T)^{-1}$) for the LJ-system in the considered ($p$, $T$)-state. In our study, the system at the high pressure $p=2.0\leavevmode\nobreak\ \epsilon/\sigma^{3}$ is considered. As we know, high pressures can accelerate crystallization Wolde_Frenkel_1996 ; Koperwas_Affouard_2017 . As a result, we have such the small nucleation barrier, which provides the fact that nucleation events occur in time scales available for molecular dynamics simulation.
3.2 Size-dependence of the attachment rate
Fig. 4(a) shows the dependence of the reduced kinetic rate factor $g^{+}/g^{+}_{n_{c}}$ on the reduced nucleus size $n/n_{c}$, where $g^{+}_{n_{c}}$ is the rate of particles attachment to the critically-sized nucleus $n_{c}$ [see Table 1]. In this dependence, two regimes can be distinguished, which are typical for the quantity $g^{+}(n)$ [Ref. Kashchiev_Nucleation_2000 ]. The first regime covers the size range $n/n_{c}\in[0;1]$ corresponding to subcritically-sized crystalline nuclei. The second regime at the sizes $n/n_{c}>1$ corresponds to growing supercritical nuclei.
The found ($n/n_{c}$)-dependence of the reduced kinetic rate factor $g^{+}/g^{+}_{n_{c}}$ is compared with the simulation data obtained for LJ-system at the supercooling $\Delta T/T_{m}=0.49$ [at the number density $\rho=0.95\,\sigma^{-3}$] Lundrigan_2009 , as well as with the data obtained for pure Ni and binary NiAl-alloy at the supercooling $\Delta T/T_{m}=0.3$ [Ref. Mendelev_2018 ]. The values of the critical size $n_{c}$ and the kinetic rate factor $g_{n_{c}}^{+}$ estimated for the considered systems are shown in Table 1. As seen from Fig. 4(a), all the dependencies are similar and have a common trend: the larger the nucleus size $n$, the greater the attachment rate $g^{+}$ of particles to the nucleus surface. Here, the values of the kinetic rate factor $g^{+}$ estimated by our approach are obtained for nuclei with sizes up to $10n_{c}$ particles. As far as we know, the quantity $g^{+}$ has not been previously evaluated for nuclei with sizes more than $2n_{c}$.
In addition, we have computed the attachment rate by the mean first-passage time $\tau(n)$ using Wedekind and Reguera method Wedekind_2008 :
$$g^{+}(n)=B(n)/\frac{\partial\bar{\tau}(n)}{\partial n},$$
(4)
where
$$B(n)=-\frac{1}{P(n)}\left[\int_{n}^{2n_{c}}P(n^{\prime})dn^{\prime}-\frac{2\tau_{c}-\bar{\tau}(n)}{2\tau_{c}}\right].$$
(5)
Here $P(n)$ is the probability of the formation of the largest nucleus with size $n$ in the system. As seen from Fig. 4, the approach for direct estimation and Eq. (4) have a similar tendency to increase $g^{+}$ with increasing nucleus size. The agreement between the two approaches is good despite some noise from Eq. (4). The $n$-dependence of the quantity $g^{+}$ found by Eq. (4) is also in good agreement with the data of Lundrigam’s et al. obtained for crystallizing LJ-system with the supercooling $\Delta T/T_{m}=0.49$ [Ref. Lundrigan_2009 ].
The ($n/n_{c}$)-dependencies of the kinetic rate factor $g^{+}$ shown on Fig. 4(a) can be well fitted by the power-law Mokshin_Galimzyanov_PCCP_2017
$$g^{+}(n)=g_{n_{c}}^{+}\left(\frac{n}{n_{c}}\right)^{(3-\xi)/3},\,\,0<\xi\leq 3.$$
(6)
Here, the exponent $\xi$ characterizes the growth regime of crystalline nuclei. If the exponent $\xi=3$, then the quantity $g^{+}$ is independent of the nucleus size, whereas at the value $\xi=0$ this kinetic rate factor changes with increasing nucleus size according to $g^{+}(n)\sim n$. For LJ-system at the considered ($p$, $T$)-state, the value of the exponent is $\xi\simeq 1.0$. This value indicates that the attachment rate is proportional to the nucleus surface Volterra_Cooper_1985 . This corresponds to the so-called ballistic model with $g^{+}(n)\sim n^{2/3}$ in Eq. (6) Mokshin_Galimzyanov_PCCP_2017 . In particular, this scenario is most often realized in the case of fluid droplet growth in a supersaturated vapor Volterra_Cooper_1985 ; Diemand_Angelil_2013 . As we found, the power-law (6) is valid only for nuclei with sizes $n_{c}<n\leq 5n_{c}$ that is seen from Fig. 4(a). From the results of cluster analysis, it follows that the nuclei with the sizes $n>5n_{c}$ interact with other small nuclei. The largest nucleus often grows through the mechanism of restructuration and absorption of a small nucleus. In particular, such a mechanism was observed during the crystallization of single-component metal melts at low and deep levels of supercooling JCG_2019 . Due to the small difference in the structure and free energy of the absorbing and absorbed nuclei, the attachment of particles of the absorbed nucleus occurs faster than the transition of particles from the parent phase. Note that this coalescence mechanism is not excluded from consideration in the proposed approach for the evaluation of kinetic rate factors. Thus, the deviation of the size dependence of the attachment rate from the power-law $\sim n^{2/3}$ for the nuclei with the sizes $n>5n_{c}$ is mainly due to that the largest nucleus interacts with small nuclei. Fig. 4(b) shows that at the sizes $n<n_{c}$, the dependence between $\log[g^{+}/g_{n_{c}}^{+}]$ and $\log[n/n_{c}]$ deviates from the straight line with the slope $\xi\simeq 1.0$. The $n$-dependence of the quantity $g^{+}$ calculated for subcritical sized nuclei strongly depends on the type of the system, and this dependence is not reproducible by a power law of form (6).
3.3 Competition between particle attachment and detachment processes
Fig. 5(a) shows the ($n/n_{c}$)-dependences of the kinetic rate factors $g^{+}$ and $g^{-}$ in log-log scale.
As can be seen, the crossover of these dependences occurs in the neighborhood of the critical size $n_{c}\simeq 50$ particles. Presence of such the crossover point is in agreement with the general ideas of the classical nucleation theory Kashchiev_Nucleation_2000 . Obviously, for growing crystalline nuclei, the value of the kinetic rate factor $g^{+}$ will prevail over the value of the quantity $g^{-}$. Then, the difference between $g^{+}$ and $g^{-}$ will determine the growth rate of nuclei in terms of the number of particles Mydlarz_Jones_1993
$$\vartheta(n)=g^{+}(n)-g^{-}(n).$$
(7)
The $n$-dependent growth rate $\vartheta$ is negative for subcritically-sized nuclei, whereas the growth rate $\vartheta$ takes positive values for nuclei with the critical and supercritical sizes [see Fig. 5(b)]. The found $n$-dependence of the quantity $\vartheta$ reaches saturation at the sizes $n>3n_{c}$. The growth of nuclei with the sizes $n>3n_{c}$ occurs due to predominance of the kinetic rate factor $g^{+}$ over the quantity $g^{-}$. This finding is in agreement with the data obtained before for crystallizing Dzugutov system and binary LJ-system, where the transition to stationary growth regime occurs at the nucleus size $n\simeq[1.7;3.0]n_{c}$ [Ref. Mokshin_Galimzyanov_PCCP_2017 ]. As a rule, such the stable growth ceases either after transition of the crystallization process to the coalescence regime or when the system is fully crystallized JETP_2018 ; JCG_2019 .
The found ($n/n_{c}$)-dependence of the growth rate $\vartheta$ is reproduced by equation Mokshin_Galimzyanov_PCCP_2017
$$\vartheta(n)=g^{+}_{n_{c}}\left(\frac{n}{n_{c}}\right)^{2/3}\left\{1-\exp\left(-\beta\Delta\mu\left[\left(\frac{n}{n_{c}}\right)^{1/3}-1\right]\right)\right\}.$$
(8)
In Eq. (8), the quantity $\Delta\mu$ is the difference between the chemical potentials of the disordered and crystalline phases. As seen from Fig. 5(b), the theoretical curve obtained from Eq. (8) reproduces only the subcritical nuclei appearance regime, when we use $\beta\Delta\mu\approx(0.85\pm 0.1)$. The estimated value of the difference between the chemical potentials is $\Delta\mu\approx 0.43\leavevmode\nobreak\ \epsilon$ for LJ-system at the considered ($p$,$T$)-state. For comparison, this value is close to the value $\Delta\mu\approx 0.36\leavevmode\nobreak\ \epsilon$ calculated before for the supercooled LJ-system under close thermodynamic conditions Gunawardana_2018 ; Pedersen_2013 . In the saturation regime, the ($n/n_{c}$)-dependence of the growth rate $\vartheta$ is reproduced by the exponential function
$$\vartheta(n)=\vartheta_{st}\left\{1-\exp\left(-a\left[\frac{n}{n_{c}}\right]\right)\right\}.$$
(9)
The growth model (9) was proposed earlier by Mydlarz and Jones to describe the crystal growth in a stable growth regime (see Ref. Mydlarz_Jones_1993 ). As follows from this growth model, the growth rate is close to the constant value $\vartheta_{st}\approx 32\leavevmode\nobreak\ \tau^{-1}$ for nuclei with the sizes $n>n_{c}$. In the case of argon, this value of $\vartheta_{st}$ is $14.8$ ps${}^{-1}$. It is interesting to note that the value of the empirical parameter $a$ coincides with the value of the chemical potential, $a=\beta\Delta\mu\approx 0.85$. This indicates that for nuclei with the sizes $n>n_{c}$, the nature of the relationship between the growth rate $\vartheta$ and the nucleus size $n$ is completely determined by the difference of the chemical potentials $\Delta\mu$.
Inset on Fig. 5(b) shows the size-dependence of the Gibbs free energy $\beta\Delta G(n)$ obtained at known values of the kinetic rate factors $g^{+}$ and $g^{-}$. We use the expression Kelton_Greer_2010
$$\beta\Delta G(n)=-\int_{1}^{n}\ln\left[\frac{g^{+}(n^{\prime})}{g^{-}(n^{\prime}+1)}\right]dn^{\prime},$$
(10)
which is obtained from the detailed balance condition
$$g^{-}(n+1)N^{eq}(n+1)=g^{+}(n)N^{eq}(n)$$
(11)
and Gibbs distribution for crystalline nucleus sizes
$$N^{eq}(n)=N_{0}\exp[-\beta\Delta G(n)].$$
(12)
Here $\Delta G(n)$ is the work required to form the $n$-sized nucleus; $N_{0}$ is the pre-exponential constant; $N^{eq}$ is the equilibrium size distribution for the largest nucleus. The position of the maximum in the $n$-dependence of the quantity $\beta\Delta G$ corresponds to the critically-sized nucleus containing $52$ particles. This value almost coincides with the critical size calculated through the derivative of the MFPT-curve (see Table 1). Moreover, Eq. (10) yields the nucleation barrier $\beta\Delta G^{*}\approx(4.85\pm 0.35)$ that is comparable with the value $\beta\Delta G^{*}\approx(2.6\pm 0.4)$ found also through the MFPT analysis.
4 Concluding Remarks
Thus, direct evaluation of the kinetic rate factors $g^{+}$ and $g^{-}$ was performed for crystalline nuclei that grow in supercooled Lennard-Jones liquid. The size-dependences of these kinetic rate factors were determined. Calculations were performed for nuclei with sizes up to $10n_{c}$ without using model functions and fitting parameters. As far as we know, such the calculations have not been performed before. The scaled kinetic rate factors were applied to compare our results with other known simulation data obtained for supercooled LJ-liquid as well as for supercooled Ni and NiAl melts. We found that the ($n/n_{c}$)-dependence of the reduced kinetic rate factor $g^{+}/g^{+}_{n_{c}}$ follows a power law at the nucleus sizes $n\geq n_{c}$. It has been found that in the case of subcritically-sized nuclei (for $n<n_{c}$) this power law is not observed. This finding indicates on a mixing different nucleus growth regimes. Moreover, the dependence of the nucleus growth rate $\vartheta$ on the nucleus size $n$ was calculated. In this dependence, the transition to the stationary growth regime occurs for the nuclei with the sizes $n>3n_{c}$ particles. These results are in good agreement with theoretical calculations.
In conclusion, we note that the results of the present work can be applied to solve the following topical tasks: development of more accurate methods for evaluation of the rate characteristics of structural transformations in systems with different physical and chemical properties (ionic liquids, molecular liquids, polymer systems, colloidal solutions) Huang_Ruan_2017 ; Khusnutdinoff_2019 ; Brazhkin_2019 ; development of severe model and/or theory for describe the nucleus size dependence of the kinetic rate factors $g^{+}$ and $g^{-}$; quantitative characterization and theoretical description of the decay process of crystalline structures in supercooled liquids and glassy materials.
Acknowledgement
This work is supported by the Russian Science Foundation (project 19-12-00022).
Appendix: Parameters of the considered system and applied methods
The crystallization process of the supercooled Lennard-Jones liquid is considered, where the interaction between particles is determined by the pair potential:
$$U(r)=4\epsilon\left[\left(\frac{\sigma}{r}\right)^{12}-\left(\frac{\sigma}{r}\right)^{6}\right].$$
(13)
Here $r$ is the distance between particles, $\sigma$ is the effective diameter of a particle, $\epsilon$ is the parameter that characterizes the depth of the potential well. Units of physical quantities are expressed in the terms of the potential parameters $\sigma$ and $\epsilon$: the temperature $T$ in the unit $\epsilon/k_{B}$, the pressure $p$ in the unit $\epsilon/\sigma^{3}$, the kinetic rate factors $g^{+}$ and $g^{-}$ in the unit $\tau^{-1}$, where $\tau=\sigma\sqrt{m/\epsilon}$ is the time unit, $m$ is the particle mass, $k_{B}$ is the Boltzmann constant. For argon with the parameters $m=6.63\times 10^{-26}$ kg, $\sigma=0.341$ nm and $\epsilon/k_{B}=119.8$ K we have the value $\tau\simeq 2.15$ ps.
The considered system contains $N=13500$ particles located inside the simulation cubic cell with periodic boundary conditions in all three directions. We use the standard velocity-Verlet integrator and the time-step $\Delta t=0.01\leavevmode\nobreak\ \tau$ for integrate Newton’s equations of motion and calculate the trajectories of particles Verlet_1967 . The simulation was performed in the isothermal-isobaric ensemble. Pressure and temperature were controlled via the Nose-Hoover barostat and thermostat, respectively. The damping thermostat and barostat constants were taken to be $Q_{T}=100\Delta t$ and $Q_{p}=1000\Delta t$, respectively. These values of the quantities $Q_{T}$ and $Q_{p}$ are optimal for the system at the considered ($p$,$T$)-state. The initial system is a crystal with the face-centered cubic lattice at the temperature $T=0\,$K. Further, the system was heated to the temperature $T=2.5\,\epsilon/k_{B}$ at the constant pressure $p=2.0\,\epsilon/\sigma^{3}$ and, then, the system was brought to equilibrium. To prepare a supercooled sample, the equilibrated melt was rapidly cooled with the rate $0.04\,\epsilon/(k_{B}\tau)$ to the temperature $T=0.5\,\epsilon/k_{B}$. This temperature corresponds to the supercooling level $\Delta T/T_{m}=0.43$, where $\Delta T=T_{m}-T$ and the melting temperature is $T_{m}=0.88\,\epsilon/k_{B}$ on the isobar $p=2.0\,\epsilon/\sigma^{3}$. The numerical density $\rho$ of the system at this pressure is $\simeq 0.92\,\sigma^{-3}$ [see phase diagram of the system in Ref. LJ_PD ]. Detection of structural changes in the system starts immediately after receiving a supercooled liquid state.
The centers of the crystalline phase are identified using cluster analysis based on estimation of the local orientational order parameters Steinhardt_1983 . Particles involved in the formation of the crystalline structure are detected by ten Wolde-Frenkel condition Wolde_Frenkel_1996
$$0.5<\left|\sum_{m=-6}^{6}\bar{q}_{6m}(i)\bar{q}_{6m}^{*}(j)\right|\leq 1,$$
(14)
where
$$\bar{q}_{6m}(i)=q_{6m}(i)\bigg{/}\sqrt{\sum_{m=-6}^{6}\left|q_{6m}(i)\right|^{2}}.$$
(15)
Here, the $6$-fold bond order parameter is calculated through the expression
$$q_{6m}(i)=\frac{1}{n_{b}(i)}\sum_{j=1}^{n_{b}(i)}Y_{6m}(\theta_{ij},\phi_{ij}),$$
(16)
where $Y_{6m}(\theta_{ij},\phi_{ij})$ are the spherical harmonics with the polar $\theta_{ij}$ and azimuthal $\phi_{ij}$ angles, $n_{b}(i)$ is the number
of neighbors for the $i$th particle. According to condition (14), the $i$th particle is considered as involved in a crystalline phase if this particle has four or more crystal-like bonds with their own “neighbors”. The most probable growth trajectory – the time dependence of the nucleus size $n(t)$ – of the largest crystalline nucleus is determined through the statistic treatment of the cluster analysis results. Here the $50$ independent trajectories $n(t)$ were used. The values of the crystal nucleation characteristics such as the critical size $n_{c}$ and the waiting time $\tau_{c}$ of the critically-sized nucleus are estimated by the method of inverted averaging of nucleus growth trajectories Mokshin_Galimzyanov_2012 ; Mokshin_Galimzyanov_PCCP_2017 .
References
(1)
K. F. Kelton and A. L. Greer, J. Non-Cryst. Solids 79, 295 (1986).
(2)
K. Yasuoka, M. Matsumoto, Chem. Phys. 109, 8451 (1998).
(3)
P. Schaaf, B. Senger, J.-C. Voegel, R. K. Bowles, H. Reiss, J. Chem. Phys. 114, 8091 (2001).
(4)
J. Diemand, R. Angelil, K. K. Tanaka, H. Tanaka, J. Chem. Phys. 139, 074309 (2013).
(5)
G. C. Sosso, J. Chen, J. Cox, M. Fitzner, P. Pedevilla, A. Zen, A. Michaelides, Chem. Rev. 116, 7078 (2016).
(6)
D. Kashchiev, Nucleation: Basic Theory with Applications (Butterworth-Heinemann, Oxford, 2000).
(7)
V. Agrawal, B. Peters, Advances in Chemical Physics 155, 97 (2014).
(8)
F. Mura and A. Zaccone, Phys. Rev. E 93, 042803 (2016).
(9)
A. V. Mokshin, B. N. Galimzyanov, Phys. Chem. Chem. Phys. 19, 11340 (2017).
(10)
V. G. Baidakov, A. O. Tipeev, J. Non-Cryst. Solids 503–504, 302 (2019).
(11)
A. M. Rodrigues, D. R. Cassar, V. M. Fokin, E. D. Zanotto, J. Non-Cryst. Solids 479, 55 (2018).
(12)
H. A. Wilson, Philos. Mag. 50, 238–250 (1900).
(13)
J. Frenkel, Phys. Z. Sowjetunion 1, 498 (1932).
(14)
D. Turnbull and J. C. Fisher, J. Chem. Phys. 17, 71 (1949).
(15)
M. C. Weinberg, W. Howard Poisl, L. Granasy, C. R. Chimie 5, 765 (2002).
(16)
D. Turnbull and R. L. Cormia, J. Chem. Phys. 34, 820 (1961).
(17)
K. F. Kelton, Crystal Nucleation in Liquids and Glasses (Academic, Boston, 1991).
(18)
A. V. Mokshin, B. N. Galimzyanov, D. T. Yarullin, JETP Lett. 110, 511 (2019)
(19)
B. J. Alder, T. E. Wainwright, J. Chem. Phys. 27, 1208 (1957).
(20)
R. E. Ryltsev, N. M. Chtchelkatchev, J. Cryst. Growth 531, 125374 (2020).
(21)
E. Kirova, V. Pisarev, J. Cryst. Growth 528, 125266 (2019).
(22)
N. A. Inogamov, V. A. Khokhlov, Y. V. Petrov, VV. Zhakhovsky, Optical and Quantum Electronics 52, 63 (2020).
(23)
L. V. Kamaeva, R. E. Ryltsev, V. I. Lad’yanov, N. M. Chtchelkatchev, J. Mol. Liquids 299, 112207 (2020).
(24)
P. K. Galenko, A. Salhoumi, and V. Ankudinov, IOP Conf. Series: Materials Science and Engineering 529, 012035 (2019).
(25)
J. Orava and A. L. Greer, J. Chem. Phys. 140, 214504 (2014).
(26)
J. Wedekind, R. Strey and D. Reguera, J. Chem. Phys. 126, 134103 (2007).
(27)
A. V. Mokshin, B. N. Galimzyanov, J. Phys. Chem. B 116, 11959 (2012).
(28)
J. Wedekind, D. Reguera, J. Phys. Chem. B 112, 11060 (2008).
(29)
S. E. M. Lundrigan and I. Saika-Voivod, J. Chem. Phys. 131, 104503 (2009).
(30)
H. Song, Y. Sun, F. Zhang, C. Z. Wang, K. M. Ho, and M. I. Mendelev, Phys. Rev. Materials 2, 023401 (2018).
(31)
S. Auer and D. Frenkel, J. Chem. Phys. 120, 3015 (2004).
(32)
B. N. Galimzyanov, D. T. Yarullin, and A. V. Mokshin, JETP Lett. 107, 629 (2018).
(33)
B. N. Galimzyanov, V. I. Ladyanov, A. V. Mokshin, J. Cryst. Growth. 526, 125214 (2019).
(34)
V. G. Baidakov, K. R. Protsenko, J. Phys. Chem. B 123, 8103 (2019).
(35)
S. Stephan, M. Thol, J. Vrabec, and H. Hasse, J. Chem. Inf. Model. 59, 4248 (2019).
(36)
K. G. S. H. Gunawardana, X. Song, J. Chem. Phys. 148, 204506 (2018).
(37)
P. R. ten Wolde, M. J. Ruiz-Montero, D. Frenkel, J. Chem. Phys. 104, 9932 (1996).
(38)
K. Koperwas, F. Affouard, J. Gerges, L.-C. Valdes, K. Adrjanowicz, and M. Paluch, Phys. Rev. B 96, 224106 (2017).
(39)
H. Heinz, R. A. Vaia, B. L. Farmer, and R. R. Naik, J. Phys. Chem. C 112, 17281 (2008).
(40)
M. Sanati, R. C. Albers and F. J. Pinski, J. Phys. Condens. Matter 13, 5387 (2001).
(41)
V. Volterra, A. R. Cooper, J. Non-Cryst. Solids 74, 85 (1985).
(42)
U. R. Pedersen, J. Chem. Phys. 139, 104102 (2013).
(43)
J. Mydlarz and A. G. Jones, Chem. Eng. J. 53, 125 (1993).
(44)
K. F. Kelton, A. L. Greer, Nucleation in Condensed Matter: Applications in Materials and Biology (Elsevier’s Science and Technology, Oxford, UK, 2010).
(45)
C. Huang, S. Ruan, T. Cai, and L. Yu, J. Phys. Chem. B 121, 9463 (2017).
(46)
R. M. Khusnutdinoff, A. V. Mokshin, J. Cryst. Growth 524, 125182 (2019).
(47)
V. V. Brazhkin, Phys. Usp. 62, 623 (2019).
(48)
L. Verlet, Phys. Rev. 159, 98 (1967).
(49)
A. Travesset, J. Chem. Phys. 141, 164501 (2014).
(50)
P. J. Steinhardt, D. R. Nelson, M. Ronchetti, Phys. Rev. B 28, 784 (1983). |
Partially isometric matrices: a brief and selective survey
Stephan Ramon Garcia
Department of Mathematics,
Pomona College,
Claremont, California
91711, USA
Stephan.Garcia@pomona.edu
,
Matthew Okubo Patterson
Department of Mathematics,
Pomona College,
Claremont, California
91711, USA
matthewpatterson477@gmail.com
and
William T. Ross
Department of Mathematics and Computer Science, University of Richmond, Richmond, VA 23173, USA
wross@richmond.edu
Abstract.
We survey a variety of results about partially isometric matrices. We focus primarily on results that are distinctly finite-dimensional. For example, we cover a recent solution to the similarity problem for partial isometries. We also discuss the unitary similarity problem and several other results.
Key words and phrases:Partial isometry, unitary matrix, partially isometric matrix, compressed shift, singular value decomposition, polar decomposition, numerical range, Moore–Penrose inverse, pseudoinverse, characteristic function, similarity, unitary similarity
2010 Mathematics Subject Classification: 15B10, 15B99, 15A23, 15A60, 15A18
The first author was partially supported by NSF grant DMS-1800123.
1. Introduction
This paper is a selective survey about partially isometric matrices.
These matrices are characterized by the equation $AA^{*}A=A$, in which $A^{*}$ denotes the conjugate transpose of $A$. We refer to such a matrix as a partial isometry.
Much of this material dates back to early work of
Erdélyi [3, 4, 5], Halmos & McLaughlin [17], and Hearon [20], among others.
Some of our results will be familiar to many readers. Others are more recent or perhaps not so well known.
The study of partial isometries on infinite-dimensional spaces is much richer and more difficult.
A proper account of the infinite-dimensional setting would occupy a large volume and we
therefore restrict ourselves here to the finite-dimensional case.
For the sake of simplicity, and because we are interested in topics such as similarity and unitary similarity,
we further narrow our attention to square matrices. However, many of the following results hold for nonsquare matrices if the indices and subscripts
are adjusted appropriately.
This survey is organized as follows.
Section 2 introduces the basic properties of partial isometries.
In particular, the connection between partial isometries, orthogonal projections, and subspaces
is considered.
Section 3 covers the algebraic structure of partial isometries. For example, we consider the singular value and polar decompositions, the Moore–Penrose pseudoinverse, and products of partial isometries.
In Section 4 we study the similarity problem for partial isometries and characterize their spectra and Jordan canonical forms.
Section 5 concerns various topics connected to unitary similarity.
For example, partial isometric extensions of contractions, the Livšic characteristic function, and the
Halmos–McLaughlin characterization of defect-one partial isometries are covered.
We conclude in Section 6 with a brief treatment of the compressed shift operator,
a concrete realization of certain partial isometries in terms of operators on spaces of rational functions.
Notation
In what follows,
$\mathsf{M}_{m\times n}$ denotes the set of $m\times n$ complex matrices. We write $\mathsf{M}_{n}$ for the set of $n\times n$ complex matrices.
A convenient shorthand for the $n\times n$ diagonal matrix with diagonal entries $\lambda_{1},\lambda_{2},\ldots,\lambda_{n}$ is
$\operatorname{diag}(\lambda_{1},\lambda_{2},\ldots,\lambda_{n})$.
The spectrum of $A\in\mathsf{M}_{n}$ (the set of eigenvalues of $A$) is denoted $\sigma(A)$ and its characteristic polynomial is
$p_{A}(z)=\det(zI-A)$. The open unit disk $|z|<1$ and unit circle $|z|=1$ are denoted $\mathbb{D}$ and $\mathbb{T}$, respectively.
We write $I_{n}$ and $0_{n}$ for the $n\times n$ identity and zero matrices, respectively. Occasionally $0$ denotes
a zero matrix whose size is to be inferred from context.
Boldface letters, such as ${\bf x}$, denote column vectors. Zero vectors are written as ${\bf 0}$ and their lengths
determined from context. A row vector is the transpose ${\bf x}^{\mathsf{T}}$ of a column vector.
The range (or column space) and kernel (or nullspace) of $A\in\mathsf{M}_{n}$ are denoted $\operatorname{ran}A$ and $\ker A$,
respectively. By $\|A\|$ we mean the operator norm of $A$, the maximum of $\|A{\bf x}\|$ for $\|{\bf x}\|=1$.
2. Preliminaries
Although partially isometric matrices enjoy several equivalent definitions,
we choose a distinctively algebraic approach because of its intrinsic nature.
This suits the matrix-theoretic perspective adopted in this article and permits us to phrase things
mostly in terms of matrices (as opposed to subspaces).
Definition 2.1.
$A\in\mathsf{M}_{n}$ is a partially isometric matrix (or partial isometry) if $AA^{*}A=A$.
The preceding definition is concise. However, it does not provide much intuition about what a partial isometry is,
although it does hint at potential relationships with unitary matrices, orthogonal projections, and the Moore–Penrose pseudoinverse. All of these suggestions
are fruitful and relevant.
Before proceeding, we require a brief review of two important topics.
We say that $A,B\in\mathsf{M}_{n}$ are unitarily similar (denoted $A\cong B$)
if there is a unitary $U\in\mathsf{M}_{n}$ such that $A=UBU^{*}$.
As the notation $\cong$ suggests, unitary similarity is an equivalence relation on $\mathsf{M}_{n}$.
Recall that $P\in\mathsf{M}_{n}$ is an orthogonal projection if
$P$ is Hermitian and idempotent ($P=P^{*}$ and $P^{2}=P$).
The spectrum of an orthogonal projection
is contained in $\{0,1\}$ and the spectral theorem ensures that
$P\cong I_{r}\oplus 0_{n-r}$, in which $r=\operatorname{rank}P$. We permit $P=0$ and $P=I$ in the degenerate cases
$r=0$ and $r=n$, respectively. In particular,
$\mathbb{C}^{n}=\ker P\oplus\operatorname{ran}P$,
in which $\ker P$ and $\operatorname{ran}P$, the eigenspaces corresponding
to $0$ and $1$, respectively, are orthogonal.
We now investigate several consequences of Definition 2.1 and identify a few distinguished
classes of partial isometries.
Proposition 2.2.
(a)
$A$ is a partial isometry if and only if $A^{*}$ is a partial isometry.
(b)
If $P\in\mathsf{M}_{n}$ is an orthogonal projection, then $P$ is a partial isometry.
(c)
If $A\in\mathsf{M}_{n}$ is a partial isometry and $U,V\in\mathsf{M}_{n}$ are unitary, then $UAV$ is a partial isometry.
(d)
A matrix that is unitarily similar to a partial isometry is a partial isometry.
(e)
If $U\in\mathsf{M}_{n}$ is unitary, then $U$ is a partial isometry.
(f)
If $A$ is a normal partial isometry, then $A$ is unitarily similar to the
direct sum of a zero matrix and a unitary matrix (either factor may be omitted).
(g)
An invertible partial isometry is unitary.
Proof.
(a) $AA^{*}A=A$ and $A^{*}(A^{*})^{*}A^{*}=A^{*}$ are adjoints of each other.
(b) If $P\in\mathsf{M}_{n}$ is an orthogonal projection, then
$PP^{*}P=P^{3}=P$.
(c) If $A\in\mathsf{M}_{n}$ is a partial isometry and $B=UAV$, in which $U,V\in\mathsf{M}_{n}$ are unitary,
then $BB^{*}B=(UAV)(UAV)^{*}(UAV)=UAA^{*}AV=UAV=B$.
(d) Let $V=U^{*}$ in (c).
(e) Let $A=V=I$ in (c).
(f) Suppose that $A\in\mathsf{M}_{n}$ is a normal partial isometry.
In light of the spectral theorem and (d), we may assume that $A$ is diagonal. Then
$A=AA^{*}A$ implies that $\lambda=\lambda|\lambda|^{2}$ for all $\lambda\in\sigma(A)$.
Thus, $\sigma(A)\subseteq\{0\}\cup\mathbb{T}$ and hence $A$ is the
direct sum of a zero matrix and a unitary matrix (either factor may be omitted).
(g) If $A\in\mathsf{M}_{n}$ is invertible and $AA^{*}A=A$, then $A^{*}A=I$. Thus, $A$ is unitary.
∎
An important relationship between partial isometries and orthogonal projections
is contained in the following theorem.
Theorem 2.3.
For $A\in\mathsf{M}_{n}$ the following conditions are equivalent.
(a)
$A$ is a partial isometry.
(b)
$A^{*}A$ is an orthogonal projection (in fact, the projection onto $(\ker A)^{\perp}$).
(c)
$AA^{*}$ is an orthogonal projection (in fact, the projection onto $\operatorname{ran}A$).
Proof.
(a) $\Rightarrow$ (b) If $AA^{*}A=A$, then $(A^{*}A)^{2}=A^{*}A$. Since
$A^{*}A$ is selfadjoint and idempotent, it is an orthogonal projection. Since111First observe that $\ker A\subseteq\ker A^{*}A$. For the converse, note that if ${\bf x}\in\ker A^{*}A$, then $\|A{\bf x}\|^{2}=\langle A{\bf x},A{\bf x}\rangle=\langle A^{*}A{\bf x},{\bf x%
}\rangle=0$ and hence ${\bf x}\in\ker A$. Thus, $\ker A^{*}A\subseteq\ker A$.
$\ker A^{*}A=\ker A$, it follows that $A^{*}A$ is the orthogonal projection onto $(\ker A)^{\perp}$.
(b) $\Rightarrow$ (a) If $A^{*}A$ is an orthogonal projection, then it is the orthogonal projection onto $(\ker A)^{\perp}$.
For ${\bf x}\in\ker A$, we have $A{\bf x}={\bf 0}=AA^{*}A{\bf x}$.
If ${\bf x}\in(\ker A)^{\perp}$, then ${\bf x}=A^{*}A{\bf x}$ and hence
$A{\bf x}=A(A^{*}A{\bf x})=(AA^{*}A){\bf x}$.
Thus, $A=AA^{*}A$.
(b) $\Leftrightarrow$ (c)
Proposition 2.2 and the equivalence (a) and (b) ensure that
$A^{*}A$ is an orthogonal projection $\Leftrightarrow$ $A$ is a partial isometry
$\Leftrightarrow$ $A^{*}$ is a partial isometry $\Leftrightarrow$ $(A^{*})^{*}(A^{*})=AA^{*}$ is a partial isometry.
∎
Corollary 2.4.
If $A\in\mathsf{M}_{n}$ is a partial isometry and $A\neq 0$, then $\|A\|=1$.
Proof.
If $A\in\mathsf{M}_{n}$ is a partial isometry and $A\neq 0$,
then $\|A\|^{2}=\|A^{*}A\|=1$ since $A^{*}A$ is a nonzero orthogonal projection.
∎
Example 2.5.
The matrices
$$A=\begin{bmatrix}0&0\\
\frac{\sqrt{3}}{2}&\frac{1}{2}\end{bmatrix},\qquad B=\begin{bmatrix}0&1\\
0&0\end{bmatrix},\quad\text{and}\quad C=\frac{1}{3}\begin{bmatrix}2&-1&0\\
2&2&0\\
-1&2&0\\
\end{bmatrix}$$
are partial isometries since
$$A^{*}A=\begin{bmatrix}\frac{3}{4}&\frac{\sqrt{3}}{4}\\
\frac{\sqrt{3}}{4}&\frac{1}{4}\end{bmatrix},\qquad B^{*}B=\begin{bmatrix}0&{{%
\color[rgb]{.75,.75,.75}0}}\\
{{\color[rgb]{.75,.75,.75}0}}&1\end{bmatrix},\qquad\quad\text{and}\quad C^{*}C%
=\begin{bmatrix}1&{{\color[rgb]{.75,.75,.75}0}}&{{\color[rgb]{.75,.75,.75}0}}%
\\
{{\color[rgb]{.75,.75,.75}0}}&1&{{\color[rgb]{.75,.75,.75}0}}\\
{{\color[rgb]{.75,.75,.75}0}}&{{\color[rgb]{.75,.75,.75}0}}&0\\
\end{bmatrix}$$
are orthogonal projections. Also note that
$$AA^{*}=\begin{bmatrix}0&{{\color[rgb]{.75,.75,.75}0}}\\
{{\color[rgb]{.75,.75,.75}0}}&1\end{bmatrix},\qquad BB^{*}=\begin{bmatrix}1&{{%
\color[rgb]{.75,.75,.75}0}}\\
{{\color[rgb]{.75,.75,.75}0}}&0\end{bmatrix},\quad\text{and}\quad CC^{*}=\frac%
{1}{9}\begin{bmatrix}5&2&-4\\
2&8&2\\
-4&2&5\\
\end{bmatrix}$$
are orthogonal projections.
Example 2.6.
If $N\in\mathsf{M}_{n\times r}$ has orthonormal columns, then $A=[N\,\,0]\in\mathsf{M}_{n}$
is a partial isometry since $N^{*}N=I\in\mathsf{M}_{r}$ and hence
$$A^{*}A=\begin{bmatrix}N^{*}\\
0\end{bmatrix}[N\,\,0]=\begin{bmatrix}N^{*}N&0\\
0&0\end{bmatrix}=\begin{bmatrix}I_{r}&0\\
0&0_{n-r}\end{bmatrix}$$
is an orthogonal projection.
Definition 2.7.
If $A\in\mathsf{M}_{n}$ is a partial isometry, then $(\ker A)^{\perp}$
is the initial space of $A$ and $\operatorname{ran}A$ is the final space of $A$.
If $A\in\mathsf{M}_{n}$ is a partial isometry, then $A^{*}A$ and $AA^{*}$ are orthogonal projections.
We can be more specific: they are the orthogonal projections onto the initial and final spaces of $A$, respectively.
The following proposition indicates the origin of the term “partial isometry.”
Proposition 2.8.
If $A\in\mathsf{M}_{n}$ is a partial isometry, then $A$ maps $(\ker A)^{\perp}$ isometrically onto $\operatorname{ran}A$.
Proof.
If $A\in\mathsf{M}_{n}$ is a partial isometry, then $A^{*}A$ is the orthogonal projection
onto $(\ker A)^{\perp}$ (Theorem 2.3). For ${\bf x}\in(\ker A)^{\perp}$,
we have $\|A{\bf x}\|^{2}=\langle A{\bf x},A{\bf x}\rangle=\langle A^{*}A{\bf x},{\bf x%
}\rangle=\langle{\bf x},{\bf x}\rangle=\|{\bf x}\|^{2}$.
Thus, $A$ maps $(\ker A)^{\perp}$ isometrically into $\operatorname{ran}A$. Since
$$\dim\ker A+\dim(\ker A)^{\perp}=n=\dim\ker A+\dim\operatorname{ran}A,$$
we see that $\dim\operatorname{ran}A=\dim(\ker A)^{\perp}$, so
the image of $(\ker A)^{\perp}$ under $A$ is $\operatorname{ran}A$.
∎
Example 2.9.
For the partial isometries in Example 2.5,
$$\displaystyle(\ker A)^{\perp}$$
$$\displaystyle=\operatorname{span}\left\{\begin{bmatrix}1\\
-\sqrt{3}\end{bmatrix}\right\},$$
$$\displaystyle\operatorname{ran}A$$
$$\displaystyle=\operatorname{span}\left\{\begin{bmatrix}0\\
1\end{bmatrix}\right\},$$
$$\displaystyle(\ker B)^{\perp}$$
$$\displaystyle=\operatorname{span}\left\{\begin{bmatrix}0\\
1\end{bmatrix}\right\},$$
$$\displaystyle\operatorname{ran}B$$
$$\displaystyle=\operatorname{span}\left\{\begin{bmatrix}1\\
0\end{bmatrix}\right\},$$
$$\displaystyle(\ker C)^{\perp}$$
$$\displaystyle=\operatorname{span}\left\{\begin{bmatrix}1\\
0\\
0\end{bmatrix},\begin{bmatrix}0\\
1\\
0\end{bmatrix}\right\},$$
$$\displaystyle\operatorname{ran}C$$
$$\displaystyle=\operatorname{span}\left\{\begin{bmatrix}2\\
2\\
-1\end{bmatrix},\begin{bmatrix}-1\\
2\\
2\end{bmatrix}\right\}.$$
3. Algebraic properties and factorizations
In this section we survey a few algebraic results about partial isometries.
Section 3.1 concerns singular value decompositions of a partial isometry.
A characterization of partial isometries in terms of the Moore–Penrose pseudoinverse is
discussed in Section 3.2. The role of partial isometries in the polar decomposition of a square matrix is covered in
Section 3.3. We wrap up with a study of products of partial isometries in Section 3.4.
3.1. Singular value decomposition
A singular value decomposition (SVD) of $A\in\mathsf{M}_{n}$ is a factorization of the form
$A=U\Sigma V^{*}$, in which $U,V\in\mathsf{M}_{n}$ are unitary and
$$\Sigma=\operatorname{diag}(\sigma_{1},\sigma_{2},\ldots,\sigma_{n})$$
with
$$\sigma_{1}\geqslant\sigma_{2}\geqslant\cdots\sigma_{n}\geqslant 0;$$
see [6, Thm. 14.1.4].
Singular value decompositions always exist, but they are never unique (for example, replace $U,V$ with $-U,-V$, respectively).
For a general $A\in\mathsf{M}_{m\times n}$, a similar decomposition holds with $U\in\mathsf{M}_{m}$, $V\in\mathsf{M}_{n}$, and $\Sigma\in\mathsf{M}_{m\times n}$.
The nonnegative numbers $\sigma_{i}$ above are the singular values of $A$; they are the square roots of the eigenvalues of
$A^{*}A$ and $AA^{*}$ since
$$V^{*}(A^{*}A)V=\Sigma^{2}=U^{*}(AA^{*})U.$$
In particular, $\sigma_{1}=\|A\|$ and $\sigma_{r+1}=\cdots=\sigma_{n}=0$, in which $\operatorname{rank}A=r$.
For $r=1,2,\ldots,n-1$, define
$$X_{r}=I_{r}\oplus 0_{n-r}.$$
(3.1)
By convention, we let $\Sigma_{0}=0$ and $\Sigma_{n}=I$. The following theorem characterizes
singular value decompositions of partial isometries.
Theorem 3.2.
For $A\in\mathsf{M}_{n}$ with $\operatorname{rank}A=r$, the following are equivalent.
(a)
$A\in\mathsf{M}_{n}$ is a partial isometry.
(b)
$A=UX_{r}V^{*}$ for some unitary $U,V\in\mathsf{M}_{n}$.
(c)
$U^{*}AV$ is a partial isometry for some unitary $U,V\in\mathsf{M}_{n}$.
Proof.
(a) $\Rightarrow$ (b)
If $A\in\mathsf{M}_{n}$ is a partial isometry with singular value decomposition
$A=U\Sigma V^{*}$, then $A=AA^{*}A$ implies
$\Sigma^{3}=\Sigma$. Thus, the diagonal entries of $\Sigma$ belong to $\{0,1\}$.
Since $\operatorname{rank}A=\operatorname{rank}\Sigma$, we have $\Sigma=X_{r}$, in which $r=\operatorname{rank}A$.
(b) $\Rightarrow$ (c)
Since $X_{r}$ is an orthogonal projection, this follows from Proposition 2.2.
(c) $\Rightarrow$ (a)
If $B=U^{*}AV$ is a partial isometry for some unitary $U,V\in\mathsf{M}_{n}$,
then $A=UBV^{*}$ is a partial isometry by Proposition 2.2.
∎
Example 3.3.
The rank-$2$ partial isometry
$$\begin{bmatrix}1&{{\color[rgb]{.75,.75,.75}0}}&{{\color[rgb]{.75,.75,.75}0}}\\
{{\color[rgb]{.75,.75,.75}0}}&\frac{\sqrt{3}}{2}&{{\color[rgb]{.75,.75,.75}0}}%
\\
{{\color[rgb]{.75,.75,.75}0}}&\frac{1}{2}&{{\color[rgb]{.75,.75,.75}0}}\\
\end{bmatrix}$$
has singular value decomposition
$$\begin{bmatrix}{{\color[rgb]{.75,.75,.75}0}}&1&{{\color[rgb]{.75,.75,.75}0}}\\
\frac{\sqrt{3}}{2}&{{\color[rgb]{.75,.75,.75}0}}&-\frac{1}{2}\\
\frac{1}{2}&{{\color[rgb]{.75,.75,.75}0}}&\frac{\sqrt{3}}{2}\\
\end{bmatrix}\begin{bmatrix}1&{{\color[rgb]{.75,.75,.75}0}}&{{\color[rgb]{%
.75,.75,.75}0}}\\
{{\color[rgb]{.75,.75,.75}0}}&1&{{\color[rgb]{.75,.75,.75}0}}\\
{{\color[rgb]{.75,.75,.75}0}}&{{\color[rgb]{.75,.75,.75}0}}&0\\
\end{bmatrix}\begin{bmatrix}{{\color[rgb]{.75,.75,.75}0}}&1&{{\color[rgb]{%
.75,.75,.75}0}}\\
1&{{\color[rgb]{.75,.75,.75}0}}&{{\color[rgb]{.75,.75,.75}0}}\\
{{\color[rgb]{.75,.75,.75}0}}&{{\color[rgb]{.75,.75,.75}0}}&1\\
\end{bmatrix}.$$
The characterization of partial isometries in terms of the singular value decomposition
leads to a standard presentation of a partial isometry, up to unitary similarity.
Theorem 3.4.
For $A\in\mathsf{M}_{n}$ with $\operatorname{rank}A=r$ the following are equivalent.
(a)
$A$ is a partial isometry.
(b)
$A\cong[N\,\,0]$, in which $N\in\mathsf{M}_{n\times r}$ has orthonormal columns.
(c)
$A\cong\begin{bmatrix}B&0\\
C&0\end{bmatrix}$, in which $B\in\mathsf{M}_{r}$, $C\in\mathsf{M}_{(n-r)\times r}$,
and $B^{*}B+C^{*}C=I_{r}$.
Proof.
(a) $\Rightarrow$ (b) Let $A\in\mathsf{M}_{n}$ be a partial isometry.
Theorem 3.2 ensures that
$A=UX_{r}V^{*}$ for some unitary $U,V\in\mathsf{M}_{n}$.
Thus, $V^{*}AV=(V^{*}U)X_{r}=[N\,\,0]$, in which $N\in\mathsf{M}_{n\times r}$ is comprised
of the first $r$ columns (necessarily orthonormal) of the unitary matrix $V^{*}U$.
(b) $\Rightarrow$ (c)
Suppose that $A\cong[N\,\,0]$, in which $N\in\mathsf{M}_{n\times r}$ has orthonormal columns.
Then Proposition 2.2 ensures that
$$\qquad\qquad\qquad[N\,\,0]=\begin{bmatrix}B&0\\
C&0\end{bmatrix},\qquad B\in\mathsf{M}_{r},\,C\in\mathsf{M}_{(n-r)\times r},$$
is a partial isometry. Since $N^{*}N=I_{r}$,
$$\begin{bmatrix}I_{r}&0\\
0&0_{n-r}\end{bmatrix}=\begin{bmatrix}N^{*}\\
0\end{bmatrix}[N\,\,0]=\begin{bmatrix}B^{*}&C^{*}\\
0&0\end{bmatrix}\begin{bmatrix}B&0\\
C&0\end{bmatrix}=\begin{bmatrix}B^{*}B+C^{*}C&0\\
0&0\end{bmatrix}$$
(3.5)
and hence $B^{*}B+C^{*}C=I_{r}$.
(c) $\Rightarrow$ (a)
The computation (3.5) and Theorem 2.3
ensure that $A$ is unitarily similar to a partial isometry.
∎
Example 3.6.
The matrix
$$A=\frac{1}{9}\begin{bmatrix}8&2&2\\
2&5&-4\\
-2&4&-5\\
\end{bmatrix}$$
is a partial isometry. Indeed, $A=UX_{2}V^{*}$, in which
$$U=\begin{bmatrix}\frac{2}{\sqrt{5}}&\frac{2}{3\sqrt{5}}&\frac{1}{3}\\
0&\frac{\sqrt{5}}{3}&-\frac{2}{3}\\
-\frac{1}{\sqrt{5}}&\frac{4}{3\sqrt{5}}&\frac{2}{3}\\
\end{bmatrix}\qquad\text{and}\qquad V=\begin{bmatrix}\frac{2}{\sqrt{5}}&\frac{%
2}{3\sqrt{5}}&-\frac{1}{3}\\
0&\frac{\sqrt{5}}{3}&\frac{2}{3}\\
\frac{1}{\sqrt{5}}&-\frac{4}{3\sqrt{5}}&\frac{2}{3}\\
\end{bmatrix}$$
are unitary. Following the proof of (a) $\Rightarrow$ (b) in Theorem 3.4 we find
$$V^{*}AV=\begin{bmatrix}\frac{3}{5}&\frac{8}{15}&{{\color[rgb]{.75,.75,.75}0}}%
\\
\frac{8}{15}&\frac{13}{45}&{{\color[rgb]{.75,.75,.75}0}}\\
-\frac{4}{3\sqrt{5}}&\frac{16}{9\sqrt{5}}&{{\color[rgb]{.75,.75,.75}0}}\\
\end{bmatrix}=[N\,\,0],$$
in which $N$ has orthonormal columns. Moreover, $B^{*}B+C^{*}C=I_{2}$, in which
$$B=\begin{bmatrix}\frac{3}{5}&\frac{8}{15}\\
\frac{8}{15}&\frac{13}{45}\\
\end{bmatrix}\qquad\text{and}\qquad C=\left[-\tfrac{4}{3\sqrt{5}}\,\,\,\,%
\tfrac{16}{9\sqrt{5}}\right],$$
as suggested by Theorem 3.4c.
3.2. Pseudoinverses
Let $A\in\mathsf{M}_{n}$ with $\operatorname{rank}A=r$ and let $\sigma_{1}\geqslant\sigma_{2}\geqslant\cdots\geqslant\sigma_{r}$
be the nonzero singular values of $A$. Let $A=U\Sigma V^{*}$ be a singular
value decomposition of $A$, in which
$$\Sigma=\operatorname{diag}(\sigma_{1},\sigma_{2},\ldots,\sigma_{r},0,0,\ldots,%
0).$$
Then
$$\Sigma^{+}=\operatorname{diag}(\sigma_{1}^{-1},\sigma_{2}^{-1},\ldots,\sigma_{%
r}^{-1},0,0,\ldots,0)$$
satisfies
$$\Sigma\Sigma^{+}=\Sigma^{+}\Sigma=X_{r},$$
in which $X_{r}=I_{r}\oplus 0_{n-r}$, as defined in (3.1).
The pseudoinverse of $A$ is
$$A^{+}=V\Sigma^{+}U^{*},$$
which satisfies
(a)
$AA^{+}A=A$,
(b)
$A^{+}AA^{+}=A^{+}$,
(c)
$(AA^{+})^{*}=AA^{+}$, and
(d)
$(A^{+}A)^{*}=A^{+}A$.
In particular, $A^{+}=A^{-1}$ if $A$ is invertible.
The matrix $A^{+}$ is uniquely determined by the conditions (a)-(d) above and is often alternately
referred to as the Moore–Penrose generalized inverse of $A$.
The pseudoinverse satisfies $(AB)^{+}=B^{+}A^{+}$ for $A,B\in\mathsf{M}_{n}$.
Theorem 3.7.
$A\in\mathsf{M}_{n}$ is a partial isometry if and only if $A^{+}=A^{*}$.
Proof.
Let $A\in\mathsf{M}_{n}$ have singular value decomposition $A=U\Sigma V^{*}$.
If $A$ is a partial isometry, then $\Sigma=X_{r}$, in which $\operatorname{rank}A=r$ (Theorem 3.2).
Thus, $A^{+}=V\Sigma^{+}U^{*}=VX_{r}U^{*}=A^{*}$.
Conversely, suppose that $A^{+}=A^{*}$. Then
$V\Sigma^{+}U^{*}=V\Sigma U^{*}$ and hence $\Sigma^{+}=\Sigma$.
The definition of $\Sigma^{+}$ ensures that
each nonzero singular value of $A$ is $1$ and hence $A=UX_{r}V^{*}$
is a partial isometry (Theorem 3.2).
∎
3.3. Polar decomposition
The singular value decomposition leads to a matrix analogue of the polar
form of a complex number $z$, in which partial isometries play a critical role.
We first consider a closely-related factorization of partial isometries.
Theorem 3.8.
For $A\in\mathsf{M}_{n}$ the following are equivalent.
(a)
$A$ is a partial isometry.
(b)
$A=WP$, in which $P$ is an orthogonal projection and $W$ is unitary.
(c)
$A=QW$, in which $Q$ is an orthogonal projection and $W$ is unitary.
Proof.
(a) $\Rightarrow$ (b)
Let $A\in\mathsf{M}_{n}$ be a partial isometry with singular value decomposition
$A=UX_{r}V^{*}$ (Theorem 3.2). Then
$A=WP$, in which $W=UV^{*}$ is unitary and
$P=VX_{r}V^{*}$ is an orthogonal projection.
(b) $\Rightarrow$ (c)
Let $A=WP$, in which $W$ is unitary $P$ is an orthogonal projection. Then
$A=QW$, in which $Q=WPW^{*}$ is an orthogonal projection.
(c) $\Rightarrow$ (a)
If $A=QW$, in which $Q$ is an orthogonal projection and $W$ is unitary, then
$AA^{*}A=(QW)(QW)^{*}(QW)=QWW^{*}Q^{*}QW=Q^{3}W=QW=A$
since $Q$ is Hermitian and idempotent.
∎
Theorem 3.8 permits one to extend a non-unitary
partial isometry to a unitary matrix. If $A$ is a partial isometry and $A=WP$, in which $W$ is
unitary and $P=A^{*}A$ is an orthogonal projection, then $W$ agrees with
$A$ on the initial space $(\ker A)^{\perp}$ and acts on $\ker A$ such that
$\|W{\bf x}\|=\|{\bf x}\|$ for all ${\bf x}\in\mathbb{C}^{n}$. We regard $W$ as a unitary extension of $A$.
Example 3.9.
The rank-$2$ partial isometry
from Example 3.3 factors as
$$\underbrace{\begin{bmatrix}1&{{\color[rgb]{.75,.75,.75}0}}&{{\color[rgb]{%
.75,.75,.75}0}}\\
{{\color[rgb]{.75,.75,.75}0}}&\frac{\sqrt{3}}{2}&{{\color[rgb]{.75,.75,.75}0}}%
\\
{{\color[rgb]{.75,.75,.75}0}}&\frac{1}{2}&{{\color[rgb]{.75,.75,.75}0}}\\
\end{bmatrix}}_{A}=\underbrace{\begin{bmatrix}1&{{\color[rgb]{.75,.75,.75}0}}&%
{{\color[rgb]{.75,.75,.75}0}}\\
{{\color[rgb]{.75,.75,.75}0}}&\frac{\sqrt{3}}{2}&-\frac{1}{2}\\
{{\color[rgb]{.75,.75,.75}0}}&\frac{1}{2}&\frac{\sqrt{3}}{2}\\
\end{bmatrix}}_{W}\underbrace{\begin{bmatrix}1&{{\color[rgb]{.75,.75,.75}0}}&{%
{\color[rgb]{.75,.75,.75}0}}\\
{{\color[rgb]{.75,.75,.75}0}}&1&{{\color[rgb]{.75,.75,.75}0}}\\
{{\color[rgb]{.75,.75,.75}0}}&{{\color[rgb]{.75,.75,.75}0}}&0\end{bmatrix}}_{P%
}=\underbrace{\begin{bmatrix}1&{{\color[rgb]{.75,.75,.75}0}}&{{\color[rgb]{%
.75,.75,.75}0}}\\
{{\color[rgb]{.75,.75,.75}0}}&\frac{3}{4}&\frac{\sqrt{3}}{4}\\
{{\color[rgb]{.75,.75,.75}0}}&\frac{\sqrt{3}}{4}&\frac{1}{4}\\
\end{bmatrix}}_{Q}\underbrace{\begin{bmatrix}1&{{\color[rgb]{.75,.75,.75}0}}&{%
{\color[rgb]{.75,.75,.75}0}}\\
{{\color[rgb]{.75,.75,.75}0}}&\frac{\sqrt{3}}{2}&-\frac{1}{2}\\
{{\color[rgb]{.75,.75,.75}0}}&\frac{1}{2}&\frac{\sqrt{3}}{2}\\
\end{bmatrix}}_{W},$$
in which $W$ is unitary and $P,Q$ are orthogonal projections.
For each $A\in\mathsf{M}_{n}$, the positive semidefinite matrix $A^{*}A$ has a unique
positive semidefinite square root $(A^{*}A)^{1/2}$, usually denoted $|A|$. In fact, $|A|=p(A^{*}A)$
for any polynomial $p$ with the property that $p(\lambda)=\lambda^{1/2}$
for each $\lambda\in\sigma(A^{*}A)\subseteq[0,\infty)$.
Theorem 3.10.
If $A\in\mathsf{M}_{n}$, then there is a unique partial isometry $E\in\mathsf{M}_{n}$
and positive semidefinite $R\in\mathsf{M}_{n}$
so that $A=ER$ and $\ker E=\ker R$. In fact, $R=|A|$.
Proof.
Let $A\in\mathsf{M}_{n}$ and $r=\operatorname{rank}A$.
Write a singular value decomposition $A=U\Sigma V^{*}$ and observe that
$A^{*}A=V\Sigma^{2}V^{*}$ and hence $|A|=V\Sigma V^{*}$. Then
$A=ER$, in which $E=UX_{r}V^{*}$ is a partial isometry
and $R=|A|$. Moreover, $\ker E=\ker R$ by construction.
This establishes the existence of the desired factorization.
Now suppose that $A=FS$, in which $F\in\mathsf{M}_{n}$ is a partial isometry, $S\in\mathsf{M}_{n}$ is positive semidefinite,
and $\ker F=\ker S$. Then
$A^{*}A=S^{*}F^{*}FS=S^{*}S$
since $F^{*}F$ is the orthogonal projection onto $(\ker F)^{\perp}=\operatorname{ran}S$. The uniqueness of the positive semidefinite
square root of a positive semidefinite matrix ensures that $S=|A|$. In particular, $\ker F=\ker|A|=\ker E$.
Let ${\bf y}\in(\ker F)^{\perp}=\operatorname{ran}|A|$. Then ${\bf y}=|A|{\bf x}$ for some ${\bf x}\in\mathbb{C}^{n}$
and hence $F{\bf y}=F|A|{\bf x}=A{\bf x}=E|A|{\bf x}=E{\bf y}$. Thus, $E=F$.
∎
3.4. Products of partial isometries
The set of partial isometries is not closed under multiplication. For example,
$$\begin{bmatrix}0&1\\
0&0\end{bmatrix}\begin{bmatrix}0&\frac{1}{\sqrt{2}}\\
0&\frac{1}{\sqrt{2}}\end{bmatrix}=\begin{bmatrix}0&\frac{1}{\sqrt{2}}\\
0&0\end{bmatrix}$$
is the product of partial isometries but is not a partial isometry.
The main result of this section (Theorem 3.13) is a criterion
for when the product of two partial isometries is a partial isometry.
The proof requires two preparatory lemmas.
Lemma 3.11.
If $A\in\mathsf{M}_{n}$ is idempotent and $\|A\|\leqslant 1$, then $A$ is an orthogonal projection.
Proof.
Suppose that $A\in\mathsf{M}_{n}$ is idempotent and $\|A\|\leqslant 1$. For ${\bf x}\in\mathbb{C}^{n}$,
$$\displaystyle\|A{\bf x}-A^{*}A{\bf x}\|^{2}$$
$$\displaystyle=\|A{\bf x}\|^{2}+\|A^{*}A{\bf x}\|^{2}-2\operatorname{Re}\langle
A%
{\bf x},A^{*}A{\bf x}\rangle$$
$$\displaystyle\leqslant\|A{\bf x}\|^{2}+\|A^{*}\|^{2}\|A{\bf x}\|^{2}-2%
\operatorname{Re}\langle A^{2}{\bf x},A{\bf x}\rangle$$
$$\displaystyle\leqslant\|A{\bf x}\|^{2}+\|A{\bf x}\|^{2}-2\operatorname{Re}%
\langle A{\bf x},A{\bf x}\rangle$$
$$\displaystyle=\|A{\bf x}-A{\bf x}\|^{2}$$
$$\displaystyle=0.$$
Thus, $A=A^{*}A$ is Hermitian and hence $A$ is an orthogonal projection.
∎
Lemma 3.12.
Let $P,Q\in\mathsf{M}_{n}$ be orthogonal projections. Then $PQ$ is a partial isometry
if and only if it is an orthogonal projection.
Proof.
Let $P,Q\in\mathsf{M}_{n}$ be orthogonal projections.
If $A=PQ$ is a partial isometry, then $\|A\|=\|PQ\|\leqslant\|P\|\|Q\|\leqslant 1$ and
$A=AA^{*}A=(PQ)(QP)(PQ)=(PQ)(PQ)=A^{2}$,
so Lemma 3.11 ensures that $A$ is an orthogonal projection.
Conversely, if $A=PQ$ is an orthogonal projection, then it is a partial isometry.
∎
With the preceding two lemmas, we can prove the following result [19, Thm. 5].
Theorem 3.13.
Let $A,B\in\mathsf{M}_{n}$ be partial isometries. Then $AB$ is a partial isometry
if and only if $A^{*}A$ and $BB^{*}$ commute.
Proof.
Let $A,B\in\mathsf{M}_{n}$ be partial isometries.
Write $A=UP$ and $B=QV$, in which $U,V\in\mathsf{M}_{n}$ are unitary and $P=A^{*}A$
and $Q=AA^{*}$ are orthogonal projections.
($\Rightarrow$)
If $AB\in\mathsf{M}_{n}$ is a partial isometry,
then $AB=UPQV$ is a partial isometry, so $PQ$ is a partial isometry (Proposition 2.2).
Lemma 3.12 ensures that $PQ$ is an orthogonal
projection, so $PQ=(PQ)^{*}=Q^{*}P^{*}=QP$. Thus, $A^{*}A$ and $BB^{*}$ commute.
($\Leftarrow$)
If $P=A^{*}A$ and $Q=BB^{*}$ commute, then $PQ$ is a partial isometry since
$(PQ)(PQ)^{*}(PQ)=PQQ^{*}P^{*}PQ=PQPQ=PQ$.
Thus, $AB=(UP)(QV)=U(PQ)V$ is a partial isometry.
∎
Example 3.14.
The partial isometries
$$A=\begin{bmatrix}1&0&{{\color[rgb]{.75,.75,.75}0}}\\
0&\frac{1}{2}&{{\color[rgb]{.75,.75,.75}0}}\\
0&\frac{\sqrt{3}}{2}&{{\color[rgb]{.75,.75,.75}0}}\\
\end{bmatrix}\qquad\text{and}\qquad B=\begin{bmatrix}{{\color[rgb]{.75,.75,.75%
}0}}&{{\color[rgb]{.75,.75,.75}0}}&{{\color[rgb]{.75,.75,.75}0}}\\
\frac{2}{3}&\frac{2}{3}&\frac{1}{3}\\
\frac{1}{3}&-\frac{2}{3}&\frac{2}{3}\\
\end{bmatrix}$$
satisfy $A^{*}A=\operatorname{diag}(1,1,0)$ and $BB^{*}=\operatorname{diag}(0,1,1)$. Since $A^{*}A$ and $B^{*}B$ commute,
Theorem 3.13 implies that
$$AB=\begin{bmatrix}0&0&0\\
\frac{1}{3}&\frac{1}{3}&\frac{1}{6}\\
\frac{1}{\sqrt{3}}&\frac{1}{\sqrt{3}}&\frac{1}{2\sqrt{3}}\\
\end{bmatrix}$$
is a partial isometry (it is a partial isometry of rank one).
Theorem 3.7 ensures that $A\in\mathsf{M}_{n}$ is a partial isometry if and only if $A^{*}=A^{+}$.
This yields the following result of Erdélyi [5, Thm. 3] (this paper contains
several other results concerning products of partial isometries).
Proposition 3.15.
Let $A_{1},A_{2},\ldots,A_{k}\in\mathsf{M}_{n}$ be partial isometries.
Then $A_{1}A_{2}\cdots A_{k}$ is a partial isometry if and only if
$(A_{1}A_{2}\cdots A_{n})^{+}=A_{n}^{+}A_{n-1}^{+}\cdots A_{1}^{+}$.
Any product of partial isometries is a contraction. Which contractions are products of partial isometries?
A precise answer was provided by Kuo and Wu [23].
Theorem 3.16.
For a contraction $A\in\mathsf{M}_{n}$ the following are equivalent.
(a)
$A$ is the product of $k$ partial isometries.
(b)
$\operatorname{rank}(I-A^{*}A)\leqslant k\dim\ker A$.
(c)
$|A|=(A^{*}A)^{1/2}$ is the product of $k$ idempotent matrices.
Since the proof of the Kuo–Wu theorem is long and somewhat computational, we do not include it here.
Their theorem provides the following interesting corollary.
Corollary 3.17.
(a)
Any contraction $A\in\mathsf{M}_{n}$ can be factored into a finite product of partial isometries if and only if $A$ is unitary or singular.
(b)
Any singular contraction can be factored as a product of $n$ partial isometries.
(c)
There are singular contractions that cannot be factored as a product of $n-1$ partial isometries.
See Theorem 5.6 for another problem concerning products of partial isometries.
Although the matrix product of two partial isometries need not be a partial isometry, their Kronecker product is.
Proposition 3.18.
Let $A\in\mathsf{M}_{m}$ and $B\in\mathsf{M}_{n}$.
Then $A\otimes B$ is a partial isometry if and only if $A$ and $B$ are partial isometries.
Proof.
This follows from the fact that
$AA^{*}A\otimes BB^{*}B=(A\otimes B)(A\otimes B)^{*}(A\otimes B)$;
see [6, Sect. 3.6] for properties of the Kronecker product.
∎
4. Similarity
In this section we consider similarity invariants, such as the spectrum, characteristic polynomial,
and Jordan canonical form, of partial isometries. Among other things, we discuss a recent result of the first author
and David Sherman, who solved the similarity problem for partially isometric matrices [11].
4.1. Spectrum and characteristic polynomial
In this section we describe the spectrum and characteristic polynomial of a partial isometry.
Proposition 4.1.
If $A\in\mathsf{M}_{n}$ is a partial isometry, then
$\sigma(A)\subseteq\mathbb{D}^{-}$. Moreover,
$0\in\sigma(A)$ if and only if $A$ is not unitary.
Proof.
If $A{\bf z}=\lambda{\bf z}$ and $\|{\bf z}\|=1$, then
$|\lambda|=\|A{\bf z}\|\leqslant\|A\|\|{\bf z}\|\leqslant 1$
by Corollary 2.4. Thus, $\sigma(A)\subseteq\mathbb{D}^{-}$.
For the second statement, observe that $0\notin\sigma(A)$ if and only if
$A$ is an invertible partial isometry, that is, $A$ is unitary.
∎
Not every finite subset of $\mathbb{D}^{-}$ is the spectrum of a partial isometry. Proposition 4.1
ensures that $0$ is an eigenvalue of every non-unitary partial isometry.
Halmos and McLaughlin proved that this is essentially the only restriction [17, Thm. 3].
Theorem 4.2.
Every monic polynomial whose roots lie in $\mathbb{D}^{-}$ and include zero
is the characteristic polynomial of a (non-unitary) partial isometry.
Proof.
We proceed by induction on the degree $n$ of the polynomial.
The base case $n=1$ concerns the polynomial $z$, which is the characteristic polynomial of the
$1\times 1$ partial isometry $[0]$.
For our induction hypothesis, suppose that every monic polynomial of degree $n-1$ whose roots lie in $\mathbb{D}^{-}$
and include zero is the characteristic polynomial of a partial isometry.
Suppose that $p$ is a polynomial of degree $n$ whose roots lie in $\mathbb{D}^{-}$ and include $0$.
There are two possibilities.
(a)
If the other $n-1$ roots of $p$ lie on $\mathbb{T}$, then there is a unitary $U\in\mathsf{M}_{n-1}$ with these roots as eigenvalues,
repeated according to multiplicity. The characteristic polynomial of the partial isometry $U\oplus[0]\in\mathsf{M}_{n}$ is $p(z)$, as desired.
(b)
If $p(z)/z$ has a root $\lambda\in\mathbb{D}$, then $p(z)=(z-\lambda)q(z)$, in which $q$ is monic, has zero as a root, and $\deg q=n-1$.
The induction hypothesis give a partial isometry $A$ with characteristic polynomial $q(z)$.
Since $0\in\sigma(A)$, it follows that $\operatorname{rank}A\leqslant n-1$
and hence there is a ${\bf z}\in(\operatorname{ran}A)^{\perp}$ with $\|{\bf z}\|^{2}=1-|\lambda|^{2}$. Now verify that
$$\begin{bmatrix}A&{\bf z}\\
{\bf 0}^{\mathsf{T}}&\lambda\end{bmatrix}\in\mathsf{M}_{n+1}$$
is a partial isometry with characteristic polynomial $(z-\lambda)q(z)=p(z)$.
This completes the induction.
∎
Example 4.3.
$z(z-\frac{1}{2})$ is the characteristic polynomial of the partial isometry
$$\begin{bmatrix}0&\frac{\sqrt{3}}{2}\\
0&\frac{1}{2}\end{bmatrix}.$$
On the other hand, $(z-\frac{1}{2})^{2}$ is not the characteristic polynomial of a partial isometry.
If it were, then the partial isometry would be invertible ($0$ is not an eigenvalue) and hence unitary.
Thus, its eigenvalues would lie on $\mathbb{T}$, which is not the case.
There is another proof, which appeared in [11], of Theorem 4.2 that is of independent interest
because of its critical use of the Weyl–Horn inequalities [21, 32].222The Horn
in question is Alfred Horn, not the Roger A. Horn of Matrix Analysis fame [22].
Theorem 4.4.
There is an $n\times n$ matrix with singular values
$\sigma_{1}\geqslant\sigma_{2}\geqslant\cdots\geqslant\sigma_{n}\geqslant 0$ and eigenvalues $\lambda_{1},\lambda_{2},\ldots,\lambda_{n}$,
indexed so that $|\lambda_{1}|\geqslant|\lambda_{2}|\geqslant\cdots\geqslant|\lambda_{n}|$, if and only if
$$\sigma_{1}\sigma_{2}\cdots\sigma_{n}=|\lambda_{1}\lambda_{2}\cdots\lambda_{n}|%
\qquad\text{and}\qquad\sigma_{1}\sigma_{2}\cdots\sigma_{k}\geqslant|\lambda_{1%
}\lambda_{2}\cdots\lambda_{k}|$$
for $k=1,2,\ldots,n-1$.
Suppose that $\lambda_{1},\lambda_{2},\ldots,\lambda_{n}\in\mathbb{D}$ are indexed so that
$$|\lambda_{1}|\geqslant|\lambda_{2}|\geqslant\cdots\geqslant|\lambda_{r}|>|%
\lambda_{r+1}|=\cdots=|\lambda_{n}|=0;$$
that is, the final $n-r$ terms in the sequence are $0$.
If we let
$$\sigma_{1}=\sigma_{2}=\cdots=\sigma_{r}=1\qquad\text{and}\qquad\sigma_{r+1}=%
\cdots=\sigma_{n}=0,$$
then Theorem 4.4 provides an $A\in\mathsf{M}_{n}$ with singular values
$\sigma_{1},\sigma_{2},\ldots,\sigma_{n}$ and eigenvalues $\lambda_{1},\lambda_{2},\ldots,\lambda_{n}$.
The singular values of $A$ are in $\{0,1\}$, so $A$ is a partial isometry whose characteristic polynomial
has the prescribed roots.
4.2. Similarity and Jordan form
Theorem 4.2 describes the possible
characteristic polynomials of partial isometries. The following examples show that this does not settle the similarity problem for the class.
Example 4.5.
The partial isometries
$$\begin{bmatrix}0&1\\
0&0\end{bmatrix}\qquad\text{and}\qquad\begin{bmatrix}0&0\\
0&0\end{bmatrix}$$
have the same characteristic polynomial, namely $z^{2}$, but they are not similar
since their ranks differ.
A complicating issue is that the property “similar to a partial isometry” is
not inherited by direct summands. Consider the next example.
Example 4.6.
The $1\times 1$ matrix $[\frac{1}{2}]$ is a direct summand of $\operatorname{diag}(0,\frac{1}{2})$,
which is similar to the partial isometry
$$\begin{bmatrix}0&\frac{\sqrt{3}}{2}\\
0&\frac{1}{2}\end{bmatrix}.$$
However, $[\frac{1}{2}]$ is not similar to a partial isometry since the spectrum of a non-unitary partial isometry
must include $0$ (Proposition 4.1).
The following theorem is due to the first author and David Sherman [11].
The proof requires several lemmas and is deferred until the end of this section.
In what follows, let $J_{n}(\lambda)$ denote the $n\times n$ Jordan block with eigenvalue $\lambda$.
Recall that every $n\times n$ matrix is similar to a direct
sum of Jordan blocks [6, Thm. 11.2.14].
The nullity of $A-\lambda I$ equals the number of Jordan blocks for the eigenvalue $\lambda$.
Theorem 4.7.
$A\in\mathsf{M}_{n}$ is similar to a partial isometry if and only if the following conditions hold.
(a)
$\sigma(A)\subseteq\mathbb{D}^{-}$.
(b)
If $\zeta\in\sigma(A)\cap\mathbb{T}$, then its algebraic and geometric multiplicities are equal.
(c)
$\dim\ker A\geqslant\dim\ker(A-\lambda I)$ for each $\lambda\in\sigma(A)\cap\mathbb{D}$.
Condition (b) ensures that the Jordan blocks
for each eigenvalue of unit modulus are all $1\times 1$ and (c) tells us that no eigenvalue in $\mathbb{D}$
can give rise to more Jordan blocks than $0$ does. Consequently,
$$\begin{bmatrix}\frac{1}{2}&&&\\
&\frac{1}{2}&&\\
&&0&\\
&&&0\\
\end{bmatrix},\qquad\begin{bmatrix}\frac{1}{2}&1&&\\
&\frac{1}{2}&&\\
&&0&\\
&&&0\\
\end{bmatrix}\quad\text{and}\quad\begin{bmatrix}\frac{1}{2}&1&&\\
&\frac{1}{2}&&\\
&&0&1\\
&&&0\\
\end{bmatrix}$$
are possible Jordan forms for a partial isometry, while
$$\begin{bmatrix}\frac{1}{2}&&&\\
&\frac{1}{2}&&\\
&&0&1\\
&&&0\\
\end{bmatrix}$$
is not.
The first lemma that we need is a
variation of Theorem 4.2.
To prescribe the Jordan canonical form of the resulting upper-triangular partial isometry,
we need to control the entries on its first superdiagonal.
Lemma 4.8.
For any $\xi_{1},\xi_{2},\ldots,\xi_{n-1}\in\mathbb{D}$, there
exists an upper-triangular partial isometry
$V\in\mathsf{M}_{n}$ such that
(a)
the diagonal of $V$ is $(0,\xi_{1},\xi_{2},\ldots,\xi_{n-1})$,
(b)
the final $n-1$ columns of $V$ are are orthonormal, and
(c)
each entry of $V$ on the first superdiagonal is nonzero.
Proof.
We proceed by induction on $n$. For the base case $n=2$,
$$V=\begin{bmatrix}0&\sqrt{1-|\xi_{1}|^{2}}\\
0&\xi_{1}\end{bmatrix}$$
is a partial isometry with the desired properties. For the induction hypothesis, suppose that the lemma holds for some $n$.
Suppose that $\xi_{1},\xi_{2},\ldots,\xi_{n}\in\mathbb{D}$ are given and apply the induction hypothesis to
$\xi_{1},\xi_{2},\ldots,\xi_{n-1}$ to obtain an upper-triangular partial isometry $V\in\mathsf{M}_{n}$ that satisfies (a), (b), and (c).
Since the first column of $V$ is ${\bf 0}$, there is a
${\bf v}\in(\operatorname{ran}V)^{\perp}$ with $\|{\bf v}\|=\sqrt{1-|\xi_{n}|^{2}}$. Define
$$V^{\prime}=\begin{bmatrix}V&{\bf v}\\
{\bf 0}^{\mathsf{T}}&\xi_{n}\end{bmatrix}\in\mathsf{M}_{n+1}.$$
Then $V^{\prime}$ has diagonal $(0,\xi_{1},\xi_{2},\ldots,\xi_{n})$. Its first column is ${\bf 0}$ and its final $n$ columns are orthonormal,
so $V^{\prime}$ is a partial isometry.
The entries of $V$ on the first superdiagonal are nonzero, so all of the entries of $V^{\prime}$ on its first superdiagonal are
nonzero, except possibly the $(n,n+1)$ entry. Suppose toward a contradiction that
$$V^{\prime}=\left[\begin{array}[]{c|ccccc}0&v_{1,2}&\cdots&v_{1,n-1}&v_{1,n}&v_%
{1,n+1}\\
0&\xi_{1}&\cdots&v_{2,n-1}&v_{2,n}&v_{2,n+1}\\
0&0&\ddots&\vdots&\vdots&\vdots\\
0&0&\cdots&\xi_{n-1}&v_{n-1,n}&v_{n-1,n+1}\\
\hline 0&0&\cdots&0&\xi_{n-1}&0\\
0&0&\cdots&0&0&\xi_{n}\end{array}\right].$$
Then the upper right $(n-2)\times(n-1)$ submatrix has $n-1$ orthogonal nonzero columns, which is impossible.
Thus, each entry on the first superdiagonal of $V^{\prime}$ is nonzero. This completes the induction.
∎
Lemma 4.9.
If $T\in\mathsf{M}_{n}$ is upper triangular with $\sigma(T)=\{\lambda\}$,
and the entries on the first superdiagonal of $T$ are all nonzero, then $T\sim J_{n}(\lambda)$.
Proof.
The superdiagonal condition ensures that $\operatorname{rank}(T-\lambda I)=n-1$ since the reduced row echelon form of $T-\lambda I$
has exactly $n-1$ leading ones.
Thus, the Jordan canonical form of $T$
is $J_{n}(\lambda)$.
∎
The following lemma is [22, Theorem 2.4.6.1]:
Lemma 4.10.
Suppose that $T=[T_{ij}]_{i,j}^{d}\in\mathsf{M}_{n}$ is block upper triangular,
and each $T_{ii}\in\mathsf{M}_{n_{i}}(\mathbb{C})$ is upper triangular with all diagonal entries equal to $\lambda_{i}$.
If $\lambda_{i}\neq\lambda_{j}$
for $i\neq j$, then $T\sim T_{11}\oplus T_{22}\oplus\cdots\oplus T_{dd}$.
We are now ready for the proof of Theorem 4.7.
Proof of Theorem 4.7.
($\Rightarrow$) Since conditions (a), (b), and (c) of Theorem 4.7
are preserved by similarity, it suffices to show that all three conditions are satisfied by any partial isometry.
Conditions (a) and (b) are implied by Proposition 4.1 and
Theorem 5.7, respectively, so we focus on (c).
Suppose that $A\in\mathsf{M}_{n}$ is a partial isometry with $\operatorname{rank}A=r$. Then $A=UP$, in which $U$ is unitary and $P$ is an orthogonal
projection of rank $r$ (Theorem 3.8). If $\lambda\in\mathbb{D}$, then the unitarity of $U$ ensures that $U-\lambda I$ is invertible and hence
$$\displaystyle n$$
$$\displaystyle=\operatorname{rank}(U-\lambda I)$$
$$\displaystyle=\operatorname{rank}\big{(}(UP-\lambda I)+U(I-P)\big{)}$$
$$\displaystyle\leqslant\operatorname{rank}(A-\lambda I)+\operatorname{rank}(I-P)$$
$$\displaystyle=\operatorname{rank}(A-\lambda I)+n-r.$$
Thus, $\operatorname{rank}A\leqslant\operatorname{rank}(A-\lambda I)$ from which (c) follows.
($\Leftarrow$)
Suppose that $A\in\mathsf{M}_{n}$ satisfies conditions (a), (b), and (c) of Theorem 4.7.
Condition (b) ensures that
$A\sim A^{\prime}\oplus U$, in which $\sigma(A^{\prime})\subseteq\mathbb{D}$ and
$U$ is a diagonal matrix whose eigenvalues are on $\mathbb{T}$ (either summand may be vacuous).
Then $U$ is unitary, so $A$ is similar to a partial isometry if and only if $A^{\prime}$ is.
Without loss of generality, we may assume that $\sigma(A)\subseteq\mathbb{D}$.
Proposition 4.1 ensures that $0\in\sigma(A)$.
Then $m=\operatorname{nullity}A$ is the number of Jordan blocks for the eigenvalue $0$
in the Jordan canonical form of $A$. Moreover, condition (c) implies that the Jordan canonical form
of $A$ has at most $m$ Jordan blocks for any nonzero eigenvalue of $A$.
Thus, it suffices to show that any matrix of the form
$$B=J_{n_{0}}(0)\oplus\bigoplus_{i=1}^{d}J_{n_{i}}(\lambda_{i}),\qquad d%
\geqslant 0,\quad 0<n_{i}\leqslant n,$$
(4.11)
in which
$\lambda_{1},\lambda_{2},\ldots,\lambda_{d}\in\mathbb{D}\backslash\{0\}$ are distinct, is similar to a partial
isometry. This is because $A$ is similar to a direct sum of matrices of the form (4.11).
Lemma 4.8 ensures that there exists
a partial isometry $V\in\mathsf{M}_{n}$
with nonzero entries on its first superdiagonal and
whose diagonal entries are
$$\underbrace{0,0,\ldots,0}_{\text{$n_{0}$ times}},\underbrace{\lambda_{1},%
\lambda_{1},\ldots,\lambda_{1}}_{\text{$n_{1}$ times}},\underbrace{\lambda_{2}%
,\lambda_{2},\ldots,\lambda_{2}}_{\text{$n_{2}$ times}},\ldots,\underbrace{%
\lambda_{d},\lambda_{d},\ldots,\lambda_{d}}_{\text{$n_{d}$ times}},$$
in that order.
Partition $V$ with respect to $B$, that is, so that $V_{i,j}\in\mathsf{M}_{n_{i}\times n_{j}}$.
Then Lemma 4.9 implies that $V_{i,i}\sim J_{n_{i}}(\lambda_{i})$ and hence
$V\sim B$ by Lemma 4.10.
∎
5. Unitary similarity
In this section we consider several questions
connected to partial isometries and unitary similarity. Recall that $A,B\in\mathsf{M}_{n}$ are unitarily similar if
$A=UBU^{*}$ for some unitary $U\in\mathsf{M}_{n}$. This relationship is denoted $A\cong B$.
5.1. Partial isometric extension of a contraction
Suppose that $A\in\mathsf{M}_{n}$ is a contraction, that is, $\|A\|\leqslant 1$.
Then $I-A^{*}A$ is positive semidefinite and has a unique positive semidefinite
square root, denoted $(I-A^{*}A)^{1/2}$. We follow Halmos and McLaughlin [17] and define
$$M(A)=\begin{bmatrix}A&(I-AA^{*})^{1/2}\\
0&0\end{bmatrix}\in\mathsf{M}_{2n},$$
which is a partial isometry since
$$M(A)M(A)^{*}=\begin{bmatrix}A&(I-AA^{*})^{1/2}\\
0&0\end{bmatrix}\begin{bmatrix}A^{*}&0\\
(I-AA^{*})^{1/2}&0\end{bmatrix}=\begin{bmatrix}I&0\\
0&0\end{bmatrix}$$
(5.1)
is an orthogonal projection. Thus, every contraction is the restriction of a partial isometry
to an invariant subspace.
The matrix $M(A)$ is relevant to the unitary similarity problem for contractions.
Theorem 5.2.
Let $A,B\in\mathsf{M}_{n}$ be contractions. Then $A\cong B$ if and only if $M(A)\cong M(B)$.
Proof.
Let $A,B\in\mathsf{M}_{n}$ be contractions.
($\Rightarrow$) Suppose that $A\cong B$.
Then $UA=BU$ for some unitary $U\in\mathsf{M}_{n}$ and hence
$U(AA^{*})=(BB^{*})U$. In particular, $AA^{*}$ and $BB^{*}$
have the same eigenvalues, all of them nonnegative, with the same multiplicities.
If $p$ is a polynomial such that $p(\lambda)=(1-\lambda)^{1/2}$ for
each such eigenvalue, then
$$p(AA^{*})=(I-AA^{*})^{1/2}\qquad\text{and}\qquad p(BB^{*})=(I-BB^{*})^{1/2}.$$
Thus,
$$U(I-AA^{*})^{\frac{1}{2}}=Up(AA^{*})=p(UAA^{*})=p(BB^{*}U)=p(BB^{*})U=(I-BB^{*%
})^{\frac{1}{2}}U.$$
A computation then confirms that $(U\oplus U)M(A)=M(B)(U\oplus U)$.
($\Leftarrow$)
If $M(A)\cong M(B)$, then (5.1) ensures that
$$A\oplus 0_{n}=M(A)^{2}M(A)^{*}\cong M(B)^{2}M(B)^{*}=B\oplus 0_{n}.$$
For any word $w(x,y)$ in two noncommuting variables,
$$\displaystyle\operatorname{tr}w(A,A^{*})$$
$$\displaystyle=\operatorname{tr}w\big{(}A\oplus 0_{n},(A\oplus 0_{n})^{*}\big{)}$$
$$\displaystyle=\operatorname{tr}w\big{(}B\oplus 0_{n},(B\oplus 0_{n})^{*}\big{)}$$
$$\displaystyle=\operatorname{tr}w(B,B^{*}).$$
A well-known theorem of Specht ensures that $A\cong B$ [31].333Pearcy showed that it suffices to consider
words of total degree $2n^{2}$ [29]. For $n=3$ and $n=4$, much better results are known [2, 28, 30]; see Section 5.3.
∎
Remark 5.3.
The proof that $M(A)\cong M(B)$ implies $A\cong B$ provided in Theorem 5.2
is inherently finite dimensional because of its reliance on Specht’s theorem [31]. This is to be expected,
since the result fails for operators on infinite-dimensional Hilbert spaces: let $A=I$ and $B=I\oplus 0$.
Then $M(A)\cong I\oplus 0\cong M(B)$, but $A$ and $B$ are not unitarily similar.
The forward implication of Theorem 5.2 is due to Halmos and McLaughlin [17, Thm. 1].
They proved the converse under the assumption that $A$ or $B$ is invertible. A similar method
applies if $A$ or $B$ is a strict contraction. Here is the argument.
Let $M(A)\cong M(B)$ and
suppose without loss of generality that $A$ is invertible or a strict contraction.
Then $UM(A)=M(B)U$ for some unitary
$$U=\begin{bmatrix}X&Y\\
Z&W\end{bmatrix}\in\mathsf{M}_{2n},$$
in which $X,Y,Z,W\in\mathsf{M}_{n}$. Thus,
$$\begin{bmatrix}XA&X(I-AA^{*})^{\frac{1}{2}}\\
ZA&Z(I-AA^{*})^{\frac{1}{2}}\end{bmatrix}=\begin{bmatrix}BX+(I-BB^{*})^{\frac{%
1}{2}}Z&BY+(I-BB^{*})^{\frac{1}{2}}W\\
0&0\end{bmatrix}.$$
(5.4)
If $A$ is invertible, we see from the $(2,1)$ entry above
that $Z=0$. If $A$ is a strict contraction, then $(I-AA^{*})^{1/2}$ is invertible
and we see from the $(2,2)$ entry in (5.4) that $Z=0$.
However,
$$\begin{bmatrix}I&0\\
0&I\end{bmatrix}=I=U^{*}U=\begin{bmatrix}X^{*}&0\\
Y^{*}&W^{*}\end{bmatrix}\begin{bmatrix}X&Y\\
0&W\end{bmatrix}=\begin{bmatrix}X^{*}X&X^{*}Y\\
Y^{*}X&Y^{*}Y+W^{*}W\end{bmatrix}$$
(5.5)
and hence $X^{*}X=I$, that is, $X$ is unitary.
Since $Z=0$ in (5.4), we see that $XA=BX$, so $A\cong B$.
A related result about products of matrices is due to Erdélyi [5, Thm. 5].
Theorem 5.6.
Let $A,B\in\mathsf{M}_{n}$. Then $M(A)M(B)$ is a partial isometry if and only if $A$ is a partial isometry.
Proof.
Let $A,B\in\mathsf{M}_{n}$ and define
$$M=M(A)M(B)=\begin{bmatrix}AB&A(I-BB^{*})^{\frac{1}{2}}\\
0&0\end{bmatrix}.$$
Then
$$MM^{*}=\begin{bmatrix}AB&A(I-BB^{*})^{\frac{1}{2}}\\
0&0\end{bmatrix}\begin{bmatrix}B^{*}A^{*}&0\\
(I-BB^{*})^{\frac{1}{2}}A^{*}&0\end{bmatrix}=\begin{bmatrix}AA^{*}&0\\
0&0\end{bmatrix}$$
is an orthogonal projection if and only if $AA^{*}$ is an orthogonal projection.
Thus, $M$ is a partial isometry if and only if $A$ is (Proposition 2.2).
∎
5.2. Unitary and completely non-unitary parts
The spectrum of a partial isometry is contained in $\mathbb{D}^{-}$ (Proposition 4.1).
The following theorem concerns a useful decomposition of a partial isometry $A\in\mathsf{M}_{n}$ that corresponds to the partition
$\sigma(A)=(\sigma(A)\cap\mathbb{D})\cup(\sigma(A)\cap\mathbb{T})$ of its spectrum.
Theorem 5.7.
Let $A\in\mathsf{M}_{n}$ be a partial isometry. Then $A\cong T\oplus U$, in which
$T$ is an upper-triangular partial isometry with $\sigma(T)\subseteq\mathbb{D}$ and $U$ is unitary
(either summand may be absent).
Proof.
If $A\in\mathsf{M}_{n}$ is a partial isometry, then Schur’s theorem on unitary triangularization
implies that
$$A\cong\begin{bmatrix}T&B\\
0&U\end{bmatrix},$$
in which $\sigma(T)\subseteq\mathbb{D}$ and $U$ is upper-triangular with $\sigma(U)\subseteq\mathbb{T}$ [6, Thm. 10.1.1].
Since $A$ is a contraction, each of its columns has norm at most $1$. Since every entry on the main diagonal of $U$
has unit modulus, it follows that $U$ is diagonal and hence $B=0$. Thus, $A\cong T\oplus U$, in which
$U$ is unitary and $\sigma(T)\subseteq\mathbb{D}$. Since $AA^{*}A=A$, we conclude that $TT^{*}T=T$, so $T$ is a partial isometry.
∎
The summand $U$ in Theorem 5.7 is the unitary part of $A$ and the summand $T$ is the completely non-unitary (cnu)
part of $A$. The latter name arises from the fact that there is no reducing subspace upon which $T$ acts unitarily.
Indeed, otherwise $T\cong T^{\prime}\oplus U^{\prime}$, in which $U^{\prime}$ is unitary (and hence $\sigma(U^{\prime})\subseteq\mathbb{T}$),
and this violates the hypothesis that $\sigma(T)\subseteq\mathbb{D}$. In particular, a partial isometry is completely non-unitary if and
only if its spectrum lies in $\mathbb{D}$.
Example 5.8.
The partial isometry
$$\begin{bmatrix}0&\frac{\sqrt{3}}{2}&{{\color[rgb]{.75,.75,.75}0}}&{{\color[rgb%
]{.75,.75,.75}0}}\\
0&\frac{1}{2}&{{\color[rgb]{.75,.75,.75}0}}&{{\color[rgb]{.75,.75,.75}0}}\\
{{\color[rgb]{.75,.75,.75}0}}&{{\color[rgb]{.75,.75,.75}0}}&-1&0\\
{{\color[rgb]{.75,.75,.75}0}}&{{\color[rgb]{.75,.75,.75}0}}&0&1\\
\end{bmatrix}=\underbrace{\begin{bmatrix}0&\frac{\sqrt{3}}{2}\\
0&\frac{1}{2}\\
\end{bmatrix}}_{T}\oplus\underbrace{\begin{bmatrix}1&0\\
0&-1\end{bmatrix}}_{U}$$
is a direct sum of a completely non-unitary partial isometry $N$
and a unitary $U$.
Corollary 5.9.
Let $A\in\mathsf{M}_{n}$ be a partial isometry.
Then $A^{n}\to 0$ if and only if $A$ is completely non-unitary.
Proof.
One can use the Jordan canonical form of a matrix to show that $A^{n}\to 0$ if and only if $\sigma(A)\subseteq\mathbb{D}$
[6, Thm. 11.6.6].
∎
5.3. Low dimensions
In low dimensions there are simple conditions
to determine when two partial isometries are unitarily similar.
Although the two-dimensional situation is rather straightforward, we include
it for completeness because it suggests a similar approach in dimensions three and four.
Recall that $p_{A}(z)=\det(zI-A)$ is the characteristic polynomial of $A\in\mathsf{M}_{n}$.
Theorem 5.10.
Partial isometries $A,B\in\mathsf{M}_{2}$ are unitarily similar if and only if
$p_{A}=p_{B}$ and $\operatorname{rank}A=\operatorname{rank}B$.
Proof.
$A,B\in\mathsf{M}_{2}$ are unitarily similar if and only if $\Phi(A)=\Phi(B)$, in which
$\Phi(x)=(\operatorname{tr}x,\operatorname{tr}x^{2},\operatorname{tr}x^{*}x)$ [25].
Since the trace of a matrix is the sum of its eigenvalues, counted with multiplicity,
$\Phi(A)=\Phi(B)$ occurs for partial isometries $A,B\in\mathsf{M}_{n}$ if and only if $p_{A}=p_{B}$ and $\operatorname{rank}A=\operatorname{tr}A^{*}A=\operatorname{tr}B^{*}B=%
\operatorname{rank}B$.
∎
The $3\times 3$ case is slightly more complicated and involves a lemma of
Sibirskiĭ [30] that
streamlines an earlier result of Pearcy [28].
Lemma 5.11.
$A,B\in\mathsf{M}_{3}$ are unitarily similar if and only if $\Phi(A)=\Phi(B)$,
in which $\Phi:M_{3}(\mathbb{C})\to\mathbb{C}^{7}$ is
$$\Phi(x)=(\operatorname{tr}x,\,\operatorname{tr}x^{2},\,\operatorname{tr}x^{3},%
\,\operatorname{tr}x^{*}x,\,\operatorname{tr}x^{*}x^{2},\,\operatorname{tr}{x^%
{*}}^{2}x^{2},\,\operatorname{tr}x^{*}x^{2}{x^{*}}^{2}x).$$
(5.12)
If $A$ is a partial isometry, then $P_{A^{*}}=A^{*}A$ and $P_{A}=AA^{*}$ are the orthogonal projections
onto the initial and final spaces of $A$, respectively. That is,
$\operatorname{ran}P_{A}=\operatorname{ran}A$ and $\operatorname{ran}P_{A^{*}}=\operatorname{ran}A^{*}$. To extend
Theorem 5.10 to $3\times 3$ matrices, we require an additional condition
concerning such projections.
Theorem 5.13.
Partial isometries $A,B\in\mathsf{M}_{3}$ are unitarily similar if and only if
(a)
$p_{A}=p_{B}$,
(b)
$\operatorname{rank}A=\operatorname{rank}B$, and
(c)
$\operatorname{tr}P_{A}P_{A^{*}}=\operatorname{tr}P_{B}P_{B^{*}}$.
Proof.
Suppose that $x\in\mathsf{M}_{3}$ is a partial isometry.
Observe that $(\operatorname{tr}x,\operatorname{tr}x^{2},\operatorname{tr}x^{3})$ is uniquely determined by the characteristic polynomial of $x$
and that $\operatorname{rank}x=\operatorname{tr}x^{*}x$. The invariance of the trace under cyclic permutations of its argument
ensures that
$$\displaystyle\operatorname{tr}x^{*}x^{2}$$
$$\displaystyle=\operatorname{tr}xx^{*}x=\operatorname{tr}x,$$
$$\displaystyle\operatorname{tr}{x^{*}}^{2}x^{2}$$
$$\displaystyle=\operatorname{tr}(x^{*}x)(xx^{*}),\quad\text{and}$$
$$\displaystyle\operatorname{tr}x^{*}x^{2}{x^{*}}^{2}x$$
$$\displaystyle=\operatorname{tr}(x^{*}x)(xx^{*})(x^{*}x)=\operatorname{tr}(x^{*%
}x)^{2}(xx^{*})=\operatorname{tr}(x^{*}x)(xx^{*}).$$
Thus, partial isometries $A,B\in\mathsf{M}_{3}$ are unitarily similar if and only if
conditions (a), (b), and (c) holds.
∎
In 2007, Djoković extended the
Pearcy-Sibirskiĭ trace conditions to four dimensions and obtained a complete unitary invariant [2, Thm. 4.4].
Lemma 5.14.
$A,B\in\mathsf{M}_{4}$ are unitarily similar if and only if
$$\operatorname{tr}w_{i}(A,A^{*})=\operatorname{tr}w_{i}(B,B^{*})$$
for $i=1,2,\ldots,20$, in which the words $w_{i}(x,y)$ are
(1)
$x$
(2)
$x^{2}$
(3)
$xy$
(4)
$x^{3}$
(5)
$x^{2}y$
(6)
$x^{4}$
(7)
$x^{3}y$
(8)
$x^{2}y^{2}$
(9)
$xyxy$
(10)
$x^{3}y^{2}$
(11)
$x^{2}yx^{2}y$
(12)
$x^{2}y^{2}xy$
(13)
$y^{2}x^{2}yx$
(14)
$x^{3}y^{2}xy$
(15)
$x^{3}y^{2}x^{2}y$
(16)
$x^{3}y^{3}xy$
(17)
$y^{3}x^{3}yx$
(18)
$x^{3}yx^{2}yxy$
(19)
$x^{2}y^{2}xyx^{2}y$
(20)
$x^{3}y^{3}x^{2}y^{2}$.
Using with Djoković’s result, we obtain a complete unitary invariant for $4\times 4$ partial isometries.
Theorem 5.15.
Partial isometries $A,B\in\mathsf{M}_{4}$ are unitarily similar if and only if
(a)
$p_{A}=p_{B}$,
(b)
$\operatorname{rank}A=\operatorname{rank}B$,
(c)
$\operatorname{tr}w(A,A^{*})=\operatorname{tr}w(B,B^{*})$ for the six words $w(x,y)$ given by
$$x^{2}y^{2},\quad x^{3}y^{2},\quad x^{4}y^{2},\quad x^{3}y^{3},\quad x^{4}y,%
\quad x^{3}y^{3}x^{2}y^{2}.$$
(5.16)
Proof.
Suppose that $A,B\in\mathsf{M}_{4}$ are partial isometries, $p_{A}=p_{B}$, and $\operatorname{rank}A=\operatorname{rank}B$.
Then $\operatorname{tr}A^{k}=\operatorname{tr}B^{k}$ for $k=1,2,3,4$ and hence we need not check words $w_{1}(x,y)=x$, $w_{2}(x,y)=x^{2}$,
$w_{4}(x,y)=x^{3}$, and $w_{6}(x,y)=x^{4}$ on Djoković’s list.
Since $\operatorname{tr}A^{*}A=\operatorname{rank}A=\operatorname{rank}B=%
\operatorname{tr}B^{*}B$, we can also ignore $w_{3}(x,y)=xy$. More words can be proved
redundant when we add the relations $xyx=x$ and $yxy=y$:
$$\displaystyle w_{9}(x,y)$$
$$\displaystyle=xyxy=xy=w_{3}(x,y),$$
$$\displaystyle w_{11}(x,y)$$
$$\displaystyle=x(xyx)xy=x^{3}y=w_{7}(x,y),$$
$$\displaystyle w_{12}(x,y)$$
$$\displaystyle=x^{2}y^{2}xy=x^{2}y(yxy)=x^{2}y^{2}=w_{8}(x,y),\quad\text{and}$$
$$\displaystyle w_{14}(x,y)$$
$$\displaystyle=x^{3}y^{2}xy=x^{3}y(yxy)=x^{3}y^{2}=w_{10}(x,y).$$
The invariance of the trace under cyclic permutations yields
$$\displaystyle\operatorname{tr}w_{5}(x,y)$$
$$\displaystyle=\operatorname{tr}x^{2}y=\operatorname{tr}xyx=\operatorname{tr}x=%
\operatorname{tr}w_{1}(x,y),$$
$$\displaystyle\operatorname{tr}w_{7}(x,y)$$
$$\displaystyle=\operatorname{tr}x^{3}y=\operatorname{tr}x(xyx)=\operatorname{tr%
}x^{2}=\operatorname{tr}w_{2}(x,y),$$
$$\displaystyle\operatorname{tr}w_{13}(x,y)$$
$$\displaystyle=\operatorname{tr}y^{2}x^{2}yx=\operatorname{tr}x^{2}(yxy)y=%
\operatorname{tr}x^{2}y^{2}=\operatorname{tr}w_{8}(x,y),$$
$$\displaystyle\operatorname{tr}w_{17}(x,y)$$
$$\displaystyle=\operatorname{tr}y^{3}x^{3}yx=\operatorname{tr}x^{3}(yxy)y^{2}=%
\operatorname{tr}x^{3}y^{3}=\operatorname{tr}w_{16}(x,y),\quad\text{and}$$
$$\displaystyle\operatorname{tr}w_{19}(x,y)$$
$$\displaystyle=\operatorname{tr}x^{2}y^{2}xyx^{2}y=\operatorname{tr}xy(yxy)x(%
xyx)=\operatorname{tr}xy^{2}x^{2}=\operatorname{tr}x^{3}y^{2}=\operatorname{tr%
}w_{10}(x,y).$$
Thus, $w_{i}(x,y)$ is redundant for $i=1,2,3,4,5,6,7,9,11,12,13,14,17,19$ and we need only consider
the six words $w_{i}(x,y)$ for $i=8,10,15,16,18,20$. For some of these, we have simplifications:
$$\displaystyle\operatorname{tr}w_{15}(x,y)$$
$$\displaystyle=\operatorname{tr}x^{3}y^{2}x^{2}y=\operatorname{tr}x^{2}y^{2}x(%
xyx)=\operatorname{tr}x^{2}y^{2}x^{2}=\operatorname{tr}x^{4}y^{2},$$
$$\displaystyle\operatorname{tr}w_{16}(x,y)$$
$$\displaystyle=\operatorname{tr}x^{3}y^{3}xy=\operatorname{tr}x^{3}y^{2}(yxy)=%
\operatorname{tr}x^{3}y^{3},$$
$$\displaystyle\operatorname{tr}w_{18}(x,y)$$
$$\displaystyle=\operatorname{tr}x^{3}yx^{2}yxy=\operatorname{tr}x^{2}(xyx)x(yxy%
)=\operatorname{tr}x^{4}y.$$
This yields the list (5.16).
∎
The words in (5.16) can be interpreted more concretely in terms of orthogonal projections. For example,
if $A\in\mathsf{M}_{4}$ is a partial isometry, then the trace corresponding to the second word in (5.16) is
$\operatorname{tr}A^{3}{A^{*}}^{2}=\operatorname{tr}A(AA^{*})(A^{*}A)=%
\operatorname{tr}P_{A^{*}}AP_{A}$. The interested reader might pursue this further.
5.4. Defect index one
Let $A\in\mathsf{M}_{n}$ be a partial isometry. Then
$\dim\ker A$
is the defect index of $A$. It measures, in a crude sense, the extent to which
$A$ fails to be unitary. Indeed, if $A$ is unitary, then its defect index is $0$.
The following theorem of Halmos and McLaughlin provides a criterion
to determine whether two partial isometries with defect index one are unitarily similar
[17].
Theorem 5.17.
Let $A,B\in\mathsf{M}_{n}$ be partial isometries with one-dimensional kernels.
Then $A\cong B$ if and only if they have the same characteristic polynomial.
Proof.
$(\Rightarrow)$
If $A,B\in\mathsf{M}_{n}$ are partial isometries and
$A\cong B$, then $A$ and $B$ have the same characteristic polynomials [6, Thm. 9.3.1].
$(\Leftarrow)$
We proceed by induction on $n$.
The base case $n=1$ is true because every
$1\times 1$ partial isometry with one-dimensional kernel is $[0]$.
Suppose for our induction hypothesis that two $n\times n$ partial isometries with defect index one are unitarily similar
whenever they have the same characteristic polynomial.
Let $A,B\in\mathsf{M}_{n+1}$ be partial isometries with defect index one and suppose that $p_{A}=p_{B}$. Note that $0$ is an eigenvalue of both $A$ and $B$.
In light of Schur’s theorem on unitary triangularization [6, Thm. 10.1.1], we may assume that
$$A=\begin{bmatrix}A^{\prime}&{\bf a}\\
{\bf 0}^{*}&\alpha\\
\end{bmatrix}\qquad\text{and}\qquad B=\begin{bmatrix}B^{\prime}&{\bf b}\\
{\bf 0}^{*}&\alpha\\
\end{bmatrix},$$
in which $\alpha\in\mathbb{C}$, ${\bf a},{\bf b}\in\mathbb{C}^{n}$, and
$A^{\prime},B^{\prime}\in\mathsf{M}_{n}$ are upper-triangular matrices with
$p_{A^{\prime}}=p_{B^{\prime}}$ and $0$ as their $(1,1)$ entries.
Because $A$ and $B$ have one-dimensional kernels and $A^{\prime},B^{\prime}$ have first column ${\bf 0}$,
we have $\ker A=\ker B=\operatorname{span}\{{\bf e}_{1}\}$ and
$$A=[{\bf 0}\,\,X]\qquad\text{and}\qquad B=[{\bf 0}\,\,Y],$$
(5.18)
in which $X,Y\in\mathsf{M}_{(n+1)\times n}$. Use (5.18) to compute
$A^{*}A=B^{*}B$, the orthogonal projection onto
$\{{\bf e}_{1}\}^{\perp}$, and deduce that
$$X^{*}X=Y^{*}Y=I_{n}$$
and $\|{\bf a}\|=\|{\bf b}\|$. Then $X$
has orthonormal columns and hence the final $n$ columns of $A$ are orthonormal.
Consequently, the final $n-1$ columns of $A^{\prime}$ are orthonormal and its first column is ${\bf 0}$.
This implies that $A^{\prime}$ is a partial isometry with one-dimensional kernel. The same reasoning
applies to $B^{\prime}$. Since $p_{A^{\prime}}=p_{B^{\prime}}$, the induction hypothesis provides
a unitary $W^{\prime}\in\mathsf{M}_{n}$ such that
$W^{\prime}A^{\prime}=B^{\prime}W^{\prime}$.
Let $\xi\in\mathbb{T}$ be such that $\xi W^{\prime}{\bf a}={\bf b}$.
Then $W=W^{\prime}\oplus[\xi]\in\mathsf{M}_{n+1}$ is unitary and $WAW^{*}=B$ since
$$\begin{bmatrix}W&{\bf 0}\\
{\bf 0}^{*}&\xi\end{bmatrix}\begin{bmatrix}A^{\prime}&{\bf a}\\
{\bf 0}^{*}&\alpha\end{bmatrix}\begin{bmatrix}W^{*}&{\bf 0}\\
{\bf 0}^{*}&\overline{\xi}\end{bmatrix}=\begin{bmatrix}WA^{\prime}W^{*}&%
\overline{\xi}W{\bf a}\\
{\bf 0}^{*}&\alpha\end{bmatrix}=\begin{bmatrix}B^{\prime}&{\bf b}\\
{\bf 0}^{*}&\alpha\end{bmatrix}.$$
This completes the induction.
∎
Example 5.19.
The matrices
$$A=\begin{bmatrix}0&1&0&{{\color[rgb]{.75,.75,.75}0}}\\
0&0&1&{{\color[rgb]{.75,.75,.75}0}}\\
0&0&0&{{\color[rgb]{.75,.75,.75}0}}\\
{{\color[rgb]{.75,.75,.75}0}}&{{\color[rgb]{.75,.75,.75}0}}&{{\color[rgb]{%
.75,.75,.75}0}}&0\\
\end{bmatrix}\qquad\text{and}\qquad B=\begin{bmatrix}0&1&{{\color[rgb]{%
.75,.75,.75}0}}&{{\color[rgb]{.75,.75,.75}0}}\\
0&0&{{\color[rgb]{.75,.75,.75}0}}&{{\color[rgb]{.75,.75,.75}0}}\\
{{\color[rgb]{.75,.75,.75}0}}&{{\color[rgb]{.75,.75,.75}0}}&0&1\\
{{\color[rgb]{.75,.75,.75}0}}&{{\color[rgb]{.75,.75,.75}0}}&0&0\\
\end{bmatrix}$$
(5.20)
are partial isometries, $p_{A}=p_{B}$, and $\dim\ker A=\dim\ker B=2$.
However, $A$ and $B$ are not similar since they
have different Jordan canonical forms. In particular, $A$ and $B$ are not unitarily similar.
Thus, the one-dimensional kernel condition in Theorem 5.17 cannot be ignored.
We will see another unitary invariant that rectifies this in Section 5.6.
5.5. The transpose of a partial isometry
Although every $A\in\mathsf{M}_{n}$ is similar to $A^{\mathsf{T}}$ [6, Thm. 11.8.1],
it is not always the case that $A\cong A^{\mathsf{T}}$ [18, Pr. 159]. In this section
we tackle the question of when $A\cong A^{\mathsf{T}}$ for a partial isometry $A\in\mathsf{M}_{n}$.
Proposition 5.21.
If $A\in\mathsf{M}_{n}$ is a partial isometry with one-dimensional kernel, then $A\cong A^{\mathsf{T}}$.
Proof.
Suppose that $A\in\mathsf{M}_{n}$ is a partial isometry with one-dimensional kernel.
Then $A^{\mathsf{T}}$ is a partial isometry with one-dimensional kernel and $p_{A}=p_{A^{\mathsf{T}}}$,
so Theorem 5.17 ensures that $A\cong A^{\mathsf{T}}$.
∎
Example 5.25 below demonstrates that a partial isometry with
two-dimensional kernel need not be unitarily similar to its transpose. Our next lemma
provides a simple condition that ensures a matrix is unitarily similar to its transpose.
Lemma 5.22.
If $A\in\mathsf{M}_{n}$ is unitarily similar to a complex symmetric (self-transpose) matrix, then $A\cong A^{\mathsf{T}}$.
Proof.
If $S=UAU^{*}$, in which $S=S^{\mathsf{T}}$ and $U$ is unitary, then
$UAU^{*}=S=S^{\mathsf{T}}=\overline{U}A^{\mathsf{T}}U^{\mathsf{T}}$
and hence $VA=A^{\mathsf{T}}V$, in which $V=U^{\mathsf{T}}U$ is unitary.
∎
The condition in Lemma 5.22
is sufficient but not necessary. The first author and James Tener showed that in dimensions eight and above
$A\cong A^{\mathsf{T}}$ may hold while $A$ is not unitarily similar to a complex
symmetric matrix [12]. On the other hand, if $A\in\mathsf{M}_{n}$ for some $n\leqslant 7$ and $A\cong A^{\mathsf{T}}$,
then $A$ is unitarily similar to a complex symmetric matrix.
The following theorem, whose proof depends upon the theory of
complex symmetric operators [9, 10, 14],
is due to the first author and Warren Wogen [13].
Theorem 5.23.
A partial isometry
$$\begin{bmatrix}X&0\\
Y&0\end{bmatrix},$$
in which $X$ is square and $X^{*}X+Y^{*}Y=I$,
is unitarily similar to a complex symmetric matrix
if and only if $X$ is.
Theorem 3.4 ensures that any partial isometry is unitarily similar to one of the form
in Theorem 5.23. Thus, a partial isometry is unitarily similar to a complex symmetric matrix if and only if its restriction to its initial space
has that property.
Proposition 5.24.
If $A\in\mathsf{M}_{n}$ is a partial isometry and $1\leqslant n\leqslant 4$, then $A\cong A^{\mathsf{T}}$.
Proof.
For $n=1$ the result is obvious.
If $A\in\mathsf{M}_{2}$ is a partial isometry, it is either $0$, unitary, or has a one-dimensional kernel.
In all three cases, $A\cong A^{\mathsf{T}}$. An alternate approach is to use Lemma 5.22
after noting that every
$2\times 2$ matrix is unitarily similar to a complex symmetric matrix [14, Cor. 1].
Suppose that $A\in\mathsf{M}_{3}$ is a partial isometry. If $\operatorname{rank}A=0$, then $A=0$ and we are done.
If $\operatorname{rank}A=1$, then $A$ is unitarily similar to a complex symmetric matrix [14, Cor. 5] and
we may apply Lemma 5.22.
If $\operatorname{rank}A=2$, then $A$ has a one-dimensional kernel and hence Proposition 5.21 implies that $A\cong A^{\mathsf{T}}$.
If $\operatorname{rank}A=3$, then $A$ is unitary and therefore $A\cong A^{\mathsf{T}}$.
Suppose that $A\in\mathsf{M}_{4}$ is a partial isometry. Proceeding as before leaves
only the case $\operatorname{rank}A=2$ unsettled. Then $A$ is unitarily similar to
$$\begin{bmatrix}B&0\\
C&0\end{bmatrix},$$
in which $B,C\in\mathsf{M}_{2}$ and $B^{*}B+C^{*}C=I$, by Theorem 3.4.
Since every $2\times 2$ matrix is unitarily similar to a complex symmetric matrix [14, Cor. 1],
Theorem 5.23 ensures that $A$ is unitarily similar to a complex symmetric. Thus, $A\cong A^{\mathsf{T}}$.
∎
Example 5.25.
The conditions in Propositions 5.21 and
5.24 are best possible. The partial isometry
$$A=\begin{bmatrix}0&1&0&{{\color[rgb]{.75,.75,.75}0}}&{{\color[rgb]{.75,.75,.75%
}0}}\\
0&0&\frac{1}{2}&{{\color[rgb]{.75,.75,.75}0}}&{{\color[rgb]{.75,.75,.75}0}}\\
0&0&0&{{\color[rgb]{.75,.75,.75}0}}&{{\color[rgb]{.75,.75,.75}0}}\\
1&0&0&{{\color[rgb]{.75,.75,.75}0}}&{{\color[rgb]{.75,.75,.75}0}}\\
0&0&\frac{\sqrt{3}}{2}&{{\color[rgb]{.75,.75,.75}0}}&{{\color[rgb]{.75,.75,.75%
}0}}\\
\end{bmatrix}$$
is $5\times 5$ and has a two-dimensional kernel.
Although $A$ and $A^{\mathsf{T}}$ are similar, they are not unitarily similar since the
unitarily invariant function
$f(x)=\operatorname{tr}x^{3}{x^{*}}^{3}x^{2}{x^{*}}^{2}$ assumes the values
$f(A)=\frac{1}{4}$ and $f(A^{\mathsf{T}})=\frac{1}{16}$.
The peculiar choice of trace is motivated by
Djoković [2, Thm. 4.4] (see Section 5.3).
In fact, $f(x)$ is Djoković’s twentieth unitary invariant; the first nineteen are unable to distinguish $A$ from $A^{\mathsf{T}}$.
5.6. Livšic characteristic functions
Theorem 5.17 provides a simple criterion to determine whether two partial
isometries of defect one are unitarily similar. Example 5.19 illustrates that
the defect-one condition cannot be overlooked. A suitable replacement of Theorem
5.17 for higher defect is due to Livšic [24].
Let $A\in\mathsf{M}_{n}$ be a partial isometry with defect $r\geqslant 1$ whose spectrum contained in $\mathbb{D}$ (so $A$ is completely non-unitary; see Section 5.2).
Let ${\bf v}_{1},{\bf v}_{2},\ldots,{\bf v}_{r}$ be an orthonormal basis for $\ker A$.
Theorem 3.8 ensures that $A$ has a unitary extension $U$ (in fact many of them).
For $z\in\mathbb{D}$, define
$$w_{A}(z)=z\big{[}\langle(U-zI)^{-1}{\bf v}_{i},{\bf v}_{j}\rangle\big{]}\big{[%
}\langle(U-zI)^{-1}U{\bf v}_{i},{\bf v}_{j}\rangle\big{]}^{-1}\in\mathsf{M}_{r}.$$
(5.26)
Livšic showed that $w_{A}$ is an analytic, contractive $\mathsf{M}_{r}$-valued function on $\mathbb{D}$ such that $w_{A}(\zeta)$ is unitary for $\zeta\in\mathbb{T}$.
He showed that different choices of
$\{{\bf v}_{1},{\bf v}_{2},\ldots,{\bf v}_{r}\}$ and $U$ result in
$Q_{1}w_{A}Q_{2}$, where $Q_{1},Q_{2}$ are constant unitary matrices. The function $w_{A}$ (more precisely, the family of functions) is
the Livšic characteristic function of $A$ and
it is a unitary invariant for the partial isometries with defect $r$ [24].
Theorem 5.27.
Let $A,B\in\mathsf{M}_{n}$ be partial isometries with defect $r\geqslant 1$ and whose spectra are contained in $\mathbb{D}$.
Then $A$ and $B$ are unitarily similar if and only if there are unitary $Q_{1},Q_{2}\in\mathsf{M}_{r}$ such that
$$w_{A}(z)=Q_{1}w_{B}(z)Q_{2}$$
for all $z\in\mathbb{D}$.
Proof.
The details are technical so we only sketch the proof in the case $r=1$.
Let $A,B\in\mathsf{M}_{n}$ be partial isometries with defect $1$ and whose spectra are contained in $\mathbb{D}$.
In this case,
$\ker A=\operatorname{span}\{{\bf v}\}$ for some unit vector ${\bf v}$. Then
$$w_{A}=\frac{z\langle(U-zI)^{-1}{\bf v},{\bf v}\rangle}{\langle(U-zI)^{-1}U{\bf
v%
},{\bf v}\rangle},$$
in which $U\in\mathsf{M}_{n}$ is a unitary extension of $A$. One can verify that $w_{A}$ only changes by a unimodular constant if one selects
a different $U$.
If $B=VAV^{*}$ for some unitary $V\in\mathsf{M}_{n}$, then $VUV^{*}$ is a unitary extension of $B$ and
$\ker B=\operatorname{span}\{V{\bf v}\}$. Thus,
$$\displaystyle w_{B}(z)$$
$$\displaystyle=\frac{z\langle(VUV^{*}-zI)^{-1}V{\bf v},V{\bf v}\rangle}{\langle%
(VUV^{*}-zI)^{-1}VUV^{*}V{\bf v},V{\bf v}\rangle}=\frac{z\langle(U-zI)^{-1}{%
\bf v},{\bf v}\rangle}{\langle(U-zI)^{-1}U{\bf v},{\bf v}\rangle}=w_{A}(z).$$
For the other direction, we first give an alternate formula for $w_{A}$.
Let $U\in\mathsf{M}_{n}$ be a unitary extension of $A$ and write
$$U=Q\operatorname{diag}(\xi_{1},\xi_{2},\ldots,\xi_{n})Q^{*},$$
in which $Q\in\mathsf{M}_{n}$ is unitary and $|\xi_{1}|=|\xi_{2}|=\cdots=|\xi_{n}|=1$.
Denote the $i$th entry of ${\bf q}=Q^{*}{\bf v}$ by $q_{i}$. Then
$$\langle(U-zI)^{-1}{\bf v},{\bf v}\rangle=\sum_{j=1}^{n}\frac{1}{\xi_{j}-z}|q_{%
i}|^{2}\quad\text{and}\quad\langle(U-zI)^{-1}U{\bf v},{\bf v}\rangle=\sum_{j=1%
}^{n}\frac{\xi_{j}}{\xi_{j}-z}|q_{i}|^{2}.$$
From here we see that
$$\displaystyle w_{A}(z)$$
$$\displaystyle=\frac{z\langle(U-zI)^{-1}{\bf v},{\bf v}\rangle}{\langle(U-zI)^{%
-1}U{\bf v},{\bf v}\rangle}$$
$$\displaystyle=\frac{\displaystyle\sum_{i=1}^{n}\frac{z+\xi_{i}+z-\xi_{i}}{z-%
\zeta_{i}}|q_{i}|^{2}}{\displaystyle\sum_{i=1}^{n}\frac{z+\xi_{i}-z+\xi_{i}}{z%
-\xi_{i}}|q_{i}|^{2}}$$
$$\displaystyle=\frac{\displaystyle\sum_{i=1}^{n}\frac{z+\xi_{i}}{z-\xi_{i}}|q_{%
i}|^{2}+\sum_{i=1}^{n}|q_{i}|^{2}}{\displaystyle\sum_{i=1}^{n}\frac{z+\xi_{i}}%
{z-\xi_{i}}|q_{i}|^{2}-\sum_{i=1}^{n}|q_{i}|^{2}}$$
$$\displaystyle=\frac{\displaystyle\sum_{i=1}^{n}\frac{z+\xi_{i}}{z-\xi_{i}}|q_{%
i}|^{2}+1}{\displaystyle\sum_{i=1}^{n}\frac{z+\xi_{i}}{z-\xi_{i}}|q_{i}|^{2}-1}$$
since
$$\sum_{i=1}^{n}|q_{i}|^{2}=\|Q^{*}{\bf v}\|^{2}=\|{\bf v}\|^{2}=1.$$
A similar formula holds for $w_{B}(z)$. Now suppose that $w_{A}=\xi w_{B}$ for some $\xi\in\mathbb{T}$. By adjusting the unitary extension of either $A$ or $B$ we can assume that $w_{A}=w_{B}$ (an equality of rational functions). Thus,
$U_{A}$ and $U_{B}$ have the same eigenvalues and multiplicities, so
they are unitarily similar: $XU_{A}X^{*}=U_{B}$ for some unitary matrix $X$.
Since $X(\ker A)^{\perp}=(\ker B)^{\perp}$ and $U_{A}|_{(\ker A)^{\perp}}=U_{B}|_{(\ker B)^{\perp}}$, it follows that $XAX^{*}=B$ and hence $A$ and $B$ are unitarily similar.
∎
Example 5.28.
The partial isometry
$$A=\begin{bmatrix}0&-\frac{1}{\sqrt{2}}&\frac{1}{2}\\
0&\frac{1}{\sqrt{2}}&\frac{1}{2}\\
0&0&\frac{1}{\sqrt{2}}\\
\end{bmatrix}$$
is completely non-unitary since $\sigma(A)=\{0,\frac{1}{\sqrt{2}}\}\subseteq\mathbb{D}$.
A unitary extension for $A$ is
$$U=\begin{bmatrix}-\frac{1}{2}&-\frac{1}{\sqrt{2}}&\frac{1}{2}\\
-\frac{1}{2}&\frac{1}{\sqrt{2}}&\frac{1}{2}\\
\frac{1}{\sqrt{2}}&0&\frac{1}{\sqrt{2}}\\
\end{bmatrix}$$
and $\ker A=\operatorname{span}\{(1,0,0)\}$.
A computation using (5.26) shows that
$$w_{A}(z)=-z\bigg{(}\frac{z-1/\sqrt{2}}{1-z/\sqrt{2}}\bigg{)}^{2}.$$
This is a finite Blaschke product whose zeros are the eigenvalues of $A$ with the corresponding multiplicities.
Example 5.29.
Consider the partially isometric matrix
$$A=\begin{bmatrix}0&0&0&0\\
\frac{1}{\sqrt{2}}&-\frac{1}{\sqrt{2}}&0&0\\
\frac{1}{2}&\frac{1}{2}&0&0\\
\frac{1}{2}&\frac{1}{2}&0&0\\
\end{bmatrix}.$$
Since
$\sigma(A)=\{-\frac{1}{\sqrt{2}},0\}\subseteq\mathbb{D}$, we may apply Livšic’s theorem.
Noting that $\ker A=\operatorname{span}\{(0,0,0,1),(0,0,1,0)\}$, we see that
$A$ has unitary extension
$$U=\begin{bmatrix}0&0&1&0\\
\frac{1}{\sqrt{2}}&-\frac{1}{\sqrt{2}}&0&0\\
\frac{1}{2}&\frac{1}{2}&0&-\frac{1}{\sqrt{2}}\\
\frac{1}{2}&\frac{1}{2}&0&\frac{1}{\sqrt{2}}\\
\end{bmatrix}.$$
A computation with (5.26) yields the $2\times 2$ matrix-valued function
$$w_{A}(z)=\begin{bmatrix}\frac{z}{\sqrt{2}}&\frac{z^{2}\left(2z+\sqrt{2}\right)%
}{2\left(z+\sqrt{2}\right)}\\
-\frac{z}{\sqrt{2}}&\frac{z^{2}\left(2z+\sqrt{2}\right)}{2\left(z+\sqrt{2}%
\right)}\\
\end{bmatrix}.$$
In particular, $w_{A}(\zeta)$ is unitary for every $\zeta\in\mathbb{T}$.
Example 5.30.
Consider the partial isometry
$$A=\begin{bmatrix}0&\frac{1}{2}&0&\frac{1}{2}&0\\
0&0&0&0&0\\
0&\frac{1}{2}&0&\frac{1}{2}&0\\
0&\frac{1}{\sqrt{2}}&0&-\frac{1}{\sqrt{2}}&0\\
0&0&0&0&0\\
\end{bmatrix},$$
which has unitary extension
$$U=\begin{bmatrix}0&\frac{1}{2}&-\frac{1}{\sqrt{2}}&\frac{1}{2}&0\\
1&0&0&0&0\\
0&\frac{1}{2}&\frac{1}{\sqrt{2}}&\frac{1}{2}&0\\
0&\frac{1}{\sqrt{2}}&0&-\frac{1}{\sqrt{2}}&0\\
0&0&0&0&1\\
\end{bmatrix}.$$
We have
$\sigma(A)=\{-\frac{1}{\sqrt{2}},0\}\subseteq\mathbb{D}$
and
$$\ker A=\operatorname{span}\{(0,0,0,0,1),(0,0,1,0,0),(1,0,0,0,0)\}.$$
A computation with (5.26) yields the $3\times 3$ matrix-valued function
$$w_{A}(z)=\begin{bmatrix}z&0&0\\
0&\frac{z}{\sqrt{2}}&\frac{z^{2}\left(2z+\sqrt{2}\right)}{2\left(z+\sqrt{2}%
\right)}\\
0&-\frac{z}{\sqrt{2}}&\frac{z^{2}\left(2z+\sqrt{2}\right)}{2\left(z+\sqrt{2}%
\right)}\\
\end{bmatrix}.$$
As expected, $w_{A}(\zeta)$ is unitary for every $\zeta\in\mathbb{T}$.
Example 5.31.
For the two matrices $A$ and $B$ from (5.20),
$$w_{A}(z)=\begin{bmatrix}z&0\\
0&z^{3}\\
\end{bmatrix}\qquad\text{and}\qquad w_{B}=\begin{bmatrix}z^{2}&0\\
0&z^{2}\\
\end{bmatrix}.$$
There are no unitaries $Q_{1},Q_{2}$ such that $w_{A}(z)=Q_{1}w_{B}(z)Q_{2}$ for all $z\in\mathbb{D}$.
If there were, then $|z|=\|w_{A}(z)\|=\|w_{B}(z)\|=|z|^{2}$ for all $z\in\mathbb{D}$, which is impossible.
6. The compressed shift
If a partial isometry $A\in\mathsf{M}_{n}$ satisfies
$$\sigma(A)=\{0,\lambda_{1},\lambda_{2},\ldots\lambda_{n-1}\}\subseteq\mathbb{D}%
\qquad\text{and}\qquad\operatorname{\dim}\ker A=1,$$
(6.1)
then there is a tangible representation of $A$ as a certain operator on a Hilbert space of rational functions.
What follows is a highly abbreviated treatment.
See [7, 8] for the basics and [27, 26] for an encyclopedic treatment.
6.1. A concrete model
For a partial isometry $A\in\mathsf{M}_{n}$ that satisfies (6.1),
the model space corresponding to $A$ is
$$\mathcal{K}_{A}=\bigg{\{}\frac{a_{0}+a_{1}z+a_{2}z^{2}+\cdots+a_{n-1}z^{n-1}}{%
(1-\overline{\lambda_{1}}z)(1-\overline{\lambda_{2}}z)\cdots(1-\overline{%
\lambda_{n-1}}z)}:a_{j}\in\mathbb{C}\bigg{\}}.$$
We endow $\mathcal{K}_{A}$ with a Hilbert-space structure by regarding it as a subspace of $L^{2}=L^{2}(\mathbb{T})$, with inner product
$$\langle f,g\rangle=\int_{\mathbb{T}}f(\zeta)\overline{g(\zeta)}dm(\zeta),$$
in which $dm(\zeta)=|d\zeta|/2\pi$ is normalized Lebesgue measure on the unit circle $\mathbb{T}$.
Let $H^{2}$ denote the Hardy space of analytic functions $f:\mathbb{D}\to\mathbb{C}$ with square-summable Taylor coefficients at the origin.
It can be viewed as a subspace of $L^{2}$ by considering boundary values on $\mathbb{T}$ (see [7] and the references therein).
We associate to the partial isometry $A$, with data (6.1), the $n$-fold Blaschke product
$$B_{A}(z)=z\prod_{j=1}^{n-1}\frac{z-\lambda_{j}}{1-\overline{\lambda_{j}}z}.$$
(6.2)
Then an exercise with the Cauchy integral formula confirms that $\mathcal{K}_{A}=H^{2}\ominus B_{A}H^{2}$.
Moreover, $\mathcal{K}_{A}$ is a reproducing kernel Hilbert space with kernel
$$k^{A}_{\lambda}(z)=\frac{1-\overline{B_{A}(\lambda)}B_{A}(z)}{1-\overline{%
\lambda}z}.$$
A convenient orthonormal basis for $\mathcal{K}_{A}$ is the Takenaka basis [7, Prop. 5.9.2].
Proposition 6.3.
Let $A\in\mathsf{M}_{n}$ be a partial isometry that satisfies (6.1). Then
$$\displaystyle v_{1}(z)$$
$$\displaystyle=1,$$
$$\displaystyle v_{2}(z)$$
$$\displaystyle=z\frac{\sqrt{1-|\lambda_{1}|^{2}}}{1-\overline{\lambda_{1}}z},$$
$$\displaystyle v_{3}(z)$$
$$\displaystyle=z\frac{z-\lambda_{1}}{1-\overline{\lambda_{1}}z}\frac{\sqrt{1-|%
\lambda_{2}|^{2}}}{1-\overline{\lambda_{2}}z},$$
$$\displaystyle v_{4}(z)$$
$$\displaystyle=z\frac{z-\lambda_{1}}{1-\overline{\lambda_{1}}z}\frac{z-\lambda_%
{2}}{1-\overline{\lambda_{2}}z}\frac{\sqrt{1-|\lambda_{3}|^{2}}}{1-\overline{%
\lambda_{3}}z},$$
$$\displaystyle\,\,\,\vdots$$
$$\displaystyle v_{n}(z)$$
$$\displaystyle=z\bigg{(}\prod_{j=1}^{n-2}\frac{z-\lambda_{j}}{1-\overline{%
\lambda_{j}}z}\bigg{)}\frac{\sqrt{1-|\lambda_{n-1}|^{2}}}{1-\overline{\lambda_%
{n-1}}z},$$
is an orthonormal basis for $\mathcal{K}_{A}$.
The orthogonal projection $P_{A}:L^{2}\to L^{2}$ with range $\mathcal{K}_{A}$ is
$$(P_{A}f)(z)=\sum_{i=1}^{n}\langle f,v_{i}\rangle v_{i}(z).$$
This permits us to define the following operator.
Definition 6.4.
Let $A\in\mathsf{M}_{n}$ be a partial isometry that satisfies (6.1).
The compressed shift $S_{A}:\mathcal{K}_{A}\to\mathcal{K}_{A}$ is
$$(S_{A}f)(z)=\sum_{i=1}^{n}\langle zf,v_{i}\rangle v_{i}(z).$$
The operator $S_{A}$ enjoys the following properties; see [8] for details.
It is a completely non-unitary partial isometry on $\mathcal{K}_{A}$ (that is, $S_{A}S_{A}^{*}S_{A}=S_{A}$).
Moreover, $\sigma(S_{A})=\{0,\lambda_{1},\lambda_{2},\ldots,\lambda_{n-1}\}$ and
$\ker S_{A}=\operatorname{span}\{B_{A}(z)/z\}$ is one dimensional.
The matrix representation of $S_{A}$ with respect to the Takenaka basis is
$$\begin{bmatrix}0&&&&\\
&\lambda_{1}&&&\\
&&\ddots&\\
&q_{i,j}&&\lambda_{n-2}&\\
&&&&\lambda_{n-1}\\
\end{bmatrix},$$
(6.5)
in which
$$q_{i,j}=\left(\prod_{k=i+1}^{j-1}(-\overline{\lambda_{k}})\right)\sqrt{1-|%
\lambda_{i}|^{2}}\sqrt{1-|\lambda_{j}|^{2}}.$$
In particular, $A$ is unitarily similar to (6.5) because they are partial isometries with one-dimensional kernels
and the same characteristic polynomials (Theorem 5.17).
Proposition 6.6.
If $A\in\mathsf{M}_{n}$ is a partial isometry that satisfies (6.1), then $A$ is unitarily similar to (6.5).
Thus, the compressed shift is a model for certain types of partial isometries.
6.2. Numerical range
The numerical range of $A\in\mathsf{M}_{n}$ is
$$W(A)=\big{\{}\langle A{\bf x},{\bf x}\rangle:\|{\bf x}\|=1\big{\}}.$$
The continuity of $f({\bf x})=\langle A{\bf x},{\bf x}\rangle$, the compactness of the unit ball in $\mathbb{C}^{n}$, and the
Cauchy–Schwarz inequality ensure that $W(A)$ is a compact subset of $\{z\in\mathbb{C}:|z|\leqslant\|A\|\}$.
The Toeplitz–Hausdorff theorem says that $W(A)$ is convex [8, Thm. 10.3.9].
The numerical range is unitarily invariant: if $A\cong B$, then $W(A)=W(B)$. This permits
us to characterize the numerical range of a normal matrix using the following notions.
A convex combination of
$\xi_{1},\xi_{2},\ldots,\xi_{n}\in\mathbb{C}$ is an expression
$$c_{1}\xi_{1}+c_{2}\xi_{2}+\cdots+c_{n}\xi_{n},$$
in which
$$c_{1},c_{2},\ldots,c_{n}\in[0,1]\qquad\text{and}\qquad c_{1}+c_{2}+\cdots+c_{n%
}=1.$$
The convex hull of $\{\xi_{1},\xi_{2},\ldots,\xi_{n}\}$
is the set of all convex combinations of $\xi_{1},\xi_{2},\ldots,\xi_{n}$. It is the smallest filled polygon that contains
the points $\xi_{1},\xi_{2},\ldots,\xi_{n}$.
Proposition 6.7.
The numerical range of a normal matrix is the convex hull of its eigenvalues.
Proof.
Let $N\in\mathsf{M}_{n}$ be normal. The spectral theorem ensures that $N$ is unitarily similar to
a diagonal matrix $D=\operatorname{diag}(\xi_{1},\xi_{2},\ldots,\xi_{n})$. Thus,
$$\displaystyle W(N)$$
$$\displaystyle=W(D)=\Big{\{}\langle D{\bf x},{\bf x}\rangle:\|{\bf x}\|^{2}=%
\sum_{i=1}^{n}|x_{i}|^{2}=1\Big{\}}$$
$$\displaystyle=\Big{\{}\sum_{i=1}^{n}\xi_{i}|x_{i}|^{2}:\sum_{i=1}^{n}|x_{i}|^{%
2}=1\Big{\}}$$
$$\displaystyle=\Big{\{}\sum_{i=1}^{n}c_{i}\xi_{i}:\sum_{i=1}^{n}c_{i}=1\Big{\}}$$
is the convex hull of $\{\xi_{1},\xi_{2},\ldots,\xi_{n}\}$.
∎
Since the eigenvalues of a unitary matrix have unit modulus, the numerical range of a unitary matrix
is a polygon inscribed in the unit circle (Proposition 6.7); see Figure 1.
For a partial isometry there is some beautiful geometry behind the scenes;
see the recent book of Daepp, Gorkin, Shaffer, and Voss [1].
A partial isometry $A\in\mathsf{M}_{n}$ with spectrum
$\{0,\lambda_{1},\lambda_{2},\ldots,\lambda_{n-1}\}$ and one-dimensional kernel, in which
$\lambda_{1},\lambda_{2},\ldots,\lambda_{n-1}\in\mathbb{D}$, is unitarily similar to (6.5).
The numerical range of $S_{A}$, and hence $W(A)$, can be computed as follows.
First consider the $(n+1)$-fold Blaschke product
$$b_{A}(z)=z^{2}\prod_{j=1}^{n-1}\frac{z-\lambda_{j}}{1-\overline{\lambda_{j}}z}.$$
For each $\xi\in\mathbb{T}$, there are $n+1$ distinct points
$\zeta_{1},\zeta_{2},\ldots,\zeta_{n+1}\in\mathbb{T}$
such that $b_{A}(\zeta_{i})=\xi$ for $1\leqslant i\leqslant n+1$ [8, p. 48]. Let
$Q_{\xi}$ denote the convex hull of $\operatorname{conv}(\{\zeta_{1},\zeta_{2},\ldots,\zeta_{n+1}\})$,
which is a $(n+1)$-gon whose vertices are on $\mathbb{T}$. Then,
$$W(A)=\bigcap_{\xi\in\mathbb{T}}Q_{\xi}.$$
(6.8)
Example 6.9.
The partial isometry
$$A=\begin{bmatrix}0&0\\
\frac{\sqrt{3}}{2}&\frac{1}{2}\\
\end{bmatrix}$$
has $\sigma(A)=\{0,\frac{1}{2}\}$. Then
$$b_{A}(z)=z^{2}\bigg{(}\frac{z-1/2}{1-z/2}\bigg{)}.$$
The sets $Q_{\xi}$ are filled triangles; see Figure 1(a).
The numerical range $W(A)$ of $A$ is the intersection of the $Q_{\xi}$,
which is an ellipse; see Figure 1(b). See [1] for more on the specifics about the ellipse.
Example 6.10.
Consider the compressed shift $A$ with
$$\sigma(A)=\{0,\tfrac{1}{\sqrt{2}},\tfrac{1}{\sqrt{3}},\tfrac{1}{\sqrt{5}}\}.$$
The associated five-fold Blaschke product is
$$b_{A}(z)=z^{2}\bigg{(}\frac{z-1/\sqrt{2}}{1-z/\sqrt{2}}\bigg{)}\bigg{(}\frac{z%
-1/\sqrt{3}}{1-z/\sqrt{3}}\bigg{)}\bigg{(}\frac{z-1/\sqrt{5}}{1-z/\sqrt{5}}%
\bigg{)}.$$
The sets $Q_{\xi}$ are (irregular) pentagons; see Figure 2(a).
The numerical range $W(A)$ of $A$ is the intersection of the $Q_{\xi}$, which is an ellipse; see Figure 2(b).
A result that relates Corollary 6.6 and
the Halmos–McLaughlin theorem on defect-one partial isometries
(Theorem 5.17) is the following [15, 16].
Theorem 6.11.
If $A,B\in\mathsf{M}_{n}$ are partial isometries with spectra contained in $\mathbb{D}$ and one-dimensional kernels, then
$A\cong B$ if and only if $W(A)=W(B)$.
References
[1]
U. Daepp, P. Gorkin, A. Shaffer, and K. Voss.
Finding Ellipses, volume 34 of Carus Mathematical
Monographs.
American Mathematical Society, 2018.
[2]
Dragomir Ž. Djoković.
Poincaré series of some pure and mixed trace algebras of two
generic matrices.
J. Algebra, 309(2):654–671, 2007.
[3]
Ivan Erdélyi.
On partial isometries in finite-dimensional Euclidean spaces.
SIAM J. Appl. Math., 14:453–467, 1966.
[4]
Ivan Erdélyi.
Partial isometries and generalized inverses.
In Proc. Sympos. Theory and Application of Generalized
Inverses of Matrices (Lubbock, Texas, 1968), pages 203–217. Texas
Tech. Press, Lubbock, Tex., 1968.
[5]
Ivan Erdélyi.
Partial isometries closed under multiplication on Hilbert spaces.
J. Math. Anal. Appl., 22:546–551, 1968.
[6]
Stephan Ramon Garcia and Roger A. Horn.
A second course in linear algebra.
Cambridge University Press, Cambridge, 2017.
[7]
Stephan Ramon Garcia, Javad Mashreghi, and William T. Ross.
Introduction to model spaces and their operators, volume 148 of
Cambridge Studies in Advanced Mathematics.
Cambridge University Press, Cambridge, 2016.
[8]
Stephan Ramon Garcia, Javad Mashreghi, and William T. Ross.
Finite Blaschke products and their connections.
Springer, Cham, 2018.
[9]
Stephan Ramon Garcia and Mihai Putinar.
Complex symmetric operators and applications.
Trans. Amer. Math. Soc., 358(3):1285–1315 (electronic), 2006.
[10]
Stephan Ramon Garcia and Mihai Putinar.
Complex symmetric operators and applications. II.
Trans. Amer. Math. Soc., 359(8):3913–3931 (electronic), 2007.
[11]
Stephan Ramon Garcia and David Sherman.
Matrices similar to partial isometries.
Linear Algebra Appl., 526:35–41, 2017.
[12]
Stephan Ramon Garcia and James E. Tener.
Unitary equivalence of a matrix to its transpose.
J. Operator Theory, 68(1):179–203, 2012.
[13]
Stephan Ramon Garcia and Warren R. Wogen.
Complex symmetric partial isometries.
J. Funct. Anal., 257(4):1251–1260, 2009.
[14]
Stephan Ramon Garcia and Warren R. Wogen.
Some new classes of complex symmetric operators.
Trans. Amer. Math. Soc., 362(11):6065–6077, 2010.
[15]
Hwa-Long Gau and Pei Yuan Wu.
Numerical ranges and compressions of $S_{n}$-matrices.
Oper. Matrices, 7(2):465–476, 2013.
[16]
Hwa-Long Gau and Pei Yuan Wu.
Structures and numerical ranges of power partial isometries.
Linear Algebra Appl., 440:325–341, 2014.
[17]
P. R. Halmos and J. E. McLaughlin.
Partial isometries.
Pacific J. Math., 13:585–596, 1963.
[18]
Paul R. Halmos.
Linear algebra problem book, volume 16 of The Dolciani
Mathematical Expositions.
Mathematical Association of America, Washington, DC, 1995.
[19]
John Z. Hearon.
Partially isometric matrices.
J. Res. Nat. Bur. Standards Sect. B, 71B:225–228, 1967.
[20]
John Z. Hearon.
Polar factorization of a matrix.
J. Res. Nat. Bur. Standards Sect. B, 71B:65–67, 1967.
[21]
A. Horn.
On the eigenvalues of a matrix with prescribed singular values.
Proc. Amer. Math. Soc., 5:4–7, 1954.
[22]
Roger A. Horn and Charles R. Johnson.
Matrix analysis.
Cambridge University Press, Cambridge, second edition, 2013.
[23]
Kung Hwang Kuo and Pei Yuan Wu.
Factorization of matrices into partial isometries.
Proc. Amer. Math. Soc., 105(2):263–272, 1989.
[24]
M. S. Livšic.
Isometric operators with equal deficiency indices, quasi-unitary
operators.
Amer. Math. Soc. Transl. (2), 13:85–103, 1960.
[25]
Francis D. Murnaghan.
On the unitary invariants of a square matrix.
Anais Acad. Brasil. Ci., 26:1–7, 1954.
[26]
N. Nikolski.
Operators, functions, and systems: an easy reading. Vol. 2,
volume 93 of Mathematical Surveys and Monographs.
American Mathematical Society, Providence, RI, 2002.
Model operators and systems, Translated from the French by Andreas
Hartmann and revised by the author.
[27]
Nikolai K. Nikolski.
Operators, functions, and systems: an easy reading. Vol. 1,
volume 92 of Mathematical Surveys and Monographs.
American Mathematical Society, Providence, RI, 2002.
Hardy, Hankel, and Toeplitz, Translated from the French by Andreas
Hartmann.
[28]
Carl Pearcy.
A complete set of unitary invariants for $3\times 3$ complex
matrices.
Trans. Amer. Math. Soc., 104:425–429, 1962.
[29]
Carl Pearcy.
A complete set of unitary invariants for operators generating finite
$W^{*}$-algebras of type I.
Pacific J. Math., 12:1405–1416, 1962.
[30]
K. S. Sibirskiĭ.
A minimal polynomial basis of unitary invariants of a square matrix
of order three.
Mat. Zametki, 3:291–295, 1968.
[31]
Wilhelm Specht.
Zur Theorie der Matrizen. II.
Jber. Deutsch. Math. Verein., 50:19–23, 1940.
[32]
H. Weyl.
Inequalities between the two kinds of eigenvalues of a linear
transformation.
Proc. Nat. Acad. Sci. U. S. A., 35:408–411, 1949. |
Microscopic description of the pygmy and giant electric dipole resonances
in stable Ca isotopes.
G.Tertychny${}^{a,b}$, V.Tselyaev${}^{a,c}$,
S.Kamerdzhiev${}^{a,b}$
F.Grümmer${}^{a}$,
S.Krewald${}^{a}$, J.Speth${}^{a}$,
A. Avdeenkov${}^{a,b}$, E.Litvinova${}^{b}$,
${}^{a}$Institut für Kernphysik, Forschungszentrum Jülich,
52425 Jülich, Germany
${}^{b}$Institute of Physics and Power Engineering,
249020 Obninsk, Russia
${}^{c}$Institute of Physics S.Petersburg University, Russia
()
Abstract
The properties of the pygmy (PDR) and giant dipole resonance
(GDR)in the stable ${}^{40}Ca$,${}^{44}Ca$ and ${}^{48}Ca$ isotopes have
been calculated within the Extended Theory of Finite Fermi
Systems(ETFFS). This approach is based on the random phase
approximation (RPA) and includes the single particle continuum as
well as the coupling to low-lying collectives states which are
considered in a consistent microscopic way. For ${}^{44}Ca$ we also
include pairing correlations. We obtain good agreement with the
experimental data for the gross properties of both resonances. It
is demonstrated that the recently measured A-dependence of the
strength of the PDR below 10 MeV is well understood in our
model:due to the phonon coupling some of the strength in ${}^{48}Ca$
is simply shifted beyond 10 MeV. The predicted fragmentation of
the PDR can be investigated in $(e,e^{\prime})$ and $(\gamma,\gamma^{\prime})$ experiments. Whereas the isovector dipole strength of the PDR
is small in all Ca isotopes, we find in this region surprisingly
strong isoscalar dipole states, in agreement with an $(\alpha,\alpha^{\prime}\gamma)$ experiment.
We conclude that for the detailed understanding of the
structure of excited nuclei e.g. the PDR and GDR an approach like
the present one is absolutely necessary.
PACS: 24.30.Cz; 21.60.Ev; 27.40.+z
Keywords:
Microscopic theory,
Pygmy and giant dipole resonances,
Single-particle continuum,
Transition densities.
I Introduction
The pygmy dipole resonance (PDR) or ”soft” electric dipole
resonances
have attracted the interest of several research groups in the past
decade palit04 ; kneisl96 . The PDR is defined as a part of
the low-energy tail of the isovector dipole resonance (GDR) and
dwells well below the GDR maximum.
In phenomenological
models these resonances are described as vibrations of the excess
neutrons (neutron skin) against an inert core with N=Z
ikeda . Therefore, one expects little strength in N=Z nuclei
and increasing strength with increasing neutron excess. In exotic
nuclei with extreme neutron excess these PDR should be especially
pronounced. As the dipole strength near the nucleon separation
energy strongly influences the r-process in astrophysics, the PDR
in such nuclei is of importance for the nuclear synthesis.
Another important reason of the interest in the PDR are modern
experimental techniques which allow to distinguish between
1${}^{-}$,2${}^{+}$ and 1${}^{+}$ states kneisl96 ; las85 . In
recent experiments new data have been obtained for several groups
of isotopes (see palit04 ; hartmann02 ; we04 ) including the
stable Ca isotopes hartmann02 ; we04 . The latter results for
the three nuclei ${}^{40}Ca$,${}^{44}Ca$ and ${}^{48}Ca$, which are
the bridge between light and medium weight nuclei, are especially
interesting. The experiment shows a large difference in the summed
E1 strengths of the PDR below 10 MeV for ${}^{40}Ca$ and ${}^{44}Ca$.
Surprisingly the EWSR of the PDR does not rise with the neutron
excess but is roughly the same for ${}^{44}Ca$ and ${}^{48}Ca$. This
contradicts the phenomenological models as well as microscopic RPA
calculations chamb94 .
Clearly, such an analysis should be performed within microscopic
theoretical approaches which simultaneously describes the PDR and
GDR and other nuclear structure properties. The models should not
have free fitting parameters for each nucleus in order to obtain
reliable results which allow to understand the underlying physics.
Such an analysis has been performed in ref. chamb94 for the
three Ca isotopes within the RPA based on the self-consistent
density functional method and for heavier nuclei within the
relativistic self-consistent RPA or QRPA based on the relativistic
Hartree-Bogoliubov model ring05 . In both approaches the
self-consistency ensured a reliable comparison between different
nuclei. Goriely and Khan goriely02 calculated within the
self-consistent QRPA the E1 strength distributions
goriely02 for all nuclei with $8\leq Z\leq 110$ between
the proton and neutron drip lines using known Skyrme forces. In
their calculation the low-lying E1 strength was located
systematically higher by some 3 MeV compared with the available
data.
In the past 15 years Kamerdzhiev, Speth, Tertychny and Tselyaev
have developed
the Extended Theory of Finite Fermi Systems(ETFFS) which
uses the Green function formalism, is based on the RPA and
includes the single-particle continuum as well as the coupling to
the low-lying phonons. It also considers the effect of the phonons
in the ground state correlations. The authors have shown in
numerous publications (see rev and references therein),
that this approach allows to calculate simultaneously low-lying as
well as high lying nuclear structure properties in quantitative
agreement with the experimental data. The model has been applied
to all closed shell nuclei. As far as the electric dipole states
are concerned the complete spectrum has always been calculated,
but as the PDR are weak compared to the GDR they appear only as
small fluctuations at the lower end of the GDR.
The first calculations of the PDR within this model including
pairing have been performed for Ca we04 and Sn
kaev04 isotopes.
Here the characteristics of the PDR are in much better
agreement with the data than the above mentioned self consistent
calculations. This is due to coupling to the phonons, first of all,
and due to another single-particle scheme.
There exists a basic difference between the present approach and
self-consistent (Q)RPA calculations e.g. chamb94 ; goriely02 ; ring05
as we will discuss below.
Long time ago the conventional (1p-1h) RPA and QRPA have been
extended to include the effects of phonons. Soloviev et al.
introduced 30 years ago the Quasiparticle Phonon Model
(QPM)soloviev ; vg92 , the present model $(ETFFS)$ was first
applied nearly 15 years ago in ref.kst93 ; kstw93 and around
the same time Broglia et al. developed the Phonon Coupling
Model(PCM) colo94 . However, the extended models
rev ; colo94 have been used only recently to calculate the
properties of PDR in non-magic nuclei because an additional work
had to be done to include pairing to these models . In addition to
the above mentioned results, the PDR calculations have been
performed for ${}^{208}Pb$ resaeva02 and for a long Sn
isotopes chain tsoneva04 within the QPM, for
${}^{120}Sn$,${}^{132}Sn$ , ${}^{208}Pb$ (together with the giant E1
resonances)sarchi04 and for ${}^{18}$O,${}^{20}$O,${}^{22}$O
colo01 within the PCM generalized to include pairing. Very
recently, the E1 photo absorption cross sections have been
calculated in ${}^{116}Sn$, ${}^{120}Sn$ and ${}^{124}Sn$ with a version
of the ETFFS (see below) lt05 ,which also considers pairing.
So far the single-particle continuum for non-magic nuclei has been
only taken into account in the present approach (ETFFS)
we04 ; kaev04 ; lt05 .
In the recent $ETFFS$ calculations for the PDR in Ca isotopes
we04 , several approximations (for details see
kaev04 ) have been made which were responsible that the
theoretical results agreed only qualitatively with the data. The
aim of the present investigation is threefold: firstly, we repeat
the previous calculations without the approximations and apply for
${}^{44}Ca$ the generalized version of the $ETFFS$ which includes
the pairing effects, secondly we clarify the role of the continuum
for PDR and GDR, which has attracted a great interest now
richter05 , and finally we investigate in detail the
microscopic structure of the PDR, especially the role of the
isoscalar dipole components. The result of the present calculation
agrees well with the experimental
data hartmann02 ; we04 and it explains in a natural way
the somewhat surprising result that the strength of the PDR does
(seemingly) not scale with the neutron excess.
II Method
We have used the generalized version of the ETFFS with pairing
which differs from the one used in we04 ; kaev04 by taking
into account the self-energy and the induced ph- and pp-
interaction graphs on the same footing, i.e. by a consistent
summations of the so-called g${}^{2}$ diagrams where g is the
amplitude for low-lying phonon creation. The formalism developed
for nuclei with pairing, is called Quasiparticle Time
Blocking Approximation (QTBA), and is described in detail in
tselyaevQTBA and lt05 .(However, the ground state
correlations induced by the phonon coupling have not been included
to the present calculations ). The single-particle continuum has
been included on the RPA level where it is taken into account
correctly within our Green function technique in the coordinate
representation. For details see rev . Therefore in our
approach we consider the three mechanisms which create the width
of giant resonance, namely (I) the Landau damping ((Q)RPA
configurations),(II) the escape width (the single-particle
continuum) and (III) the spreading width (phonon coupling, or
complex configurations). As in all our previous calculations
within the ETFFS, we include the most collective low-lying
phonons, namely 3${}^{-}_{1}$, 5${}^{-}_{1}$ phonons for ${}^{40}$Ca,
and 2${}^{+}_{1}$,3${}^{-}_{1}$,5${}^{-}_{1}$ phonons for ${}^{44}$Ca and
${}^{48}$Ca. The collective phonons have been microscopically
calculated within the RPA for ${}^{40}$Ca and ${}^{48}$Ca. In the case
of ${}^{44}$Ca we used the corresponding QRPA with pairing. The
standard BCS equation have been applied using the
particle-particle Landau-Migdal forces sap .
In the present and in our previous calculations we used
the special ”forced consistency” procedure kl98
to obtain the energy of the spurious E1 state to be exactly equal
to zero without the procedure of fitting force parameters.
As in our previous calculations we used the effective
particle-hole Landau-Migdal interaction:
$$\displaystyle F({\bf r},{\bf r^{\prime}})=C_{0}[f(r)+f^{\prime}(r){\bf{\mbox{%
\boldmath$\tau$}}_{1}}\cdot{\bf{\mbox{\boldmath$\tau$}}_{2}}+$$
$$\displaystyle(g+g^{\prime}{\bf{\mbox{\boldmath$\tau$}}_{1}}\cdot{\bf{\mbox{%
\boldmath$\tau$}}_{2}}){\bf{\mbox{\boldmath$\sigma$}}_{1}}\cdot{\bf{\mbox{%
\boldmath$\sigma$}}_{2}}]\;{\delta}({\bf r}-{\bf r^{\prime}}),$$
(2.1)
with the conventional interpolation formula, for example, for the
parameter f
$$f(r)=f_{ex}+(f_{in}-f_{ex})\rho_{0}(r)/\rho_{0}(0)$$
(2.2)
and similarly for the other $r$-dependent parameters.
Here $\rho_{0}(r)$ is the
density distribution of the ground state of the nucleus under
consideration and $f_{in}$ and $f_{ex}$ are the force parameters
inside and outside of the nucleus. The standard values of the
parameters, which have been already used for all the nuclei under
consideration rev ; we04 ; kaev04 , are as follows
$$\displaystyle f_{in}=-0.002,\;f_{ex}=-1.4,\;f_{ex}^{\prime}=2.30,\;f_{in}^{%
\prime}=0.76,$$
$$\displaystyle g=0.05,\;g^{\prime}=0.96,\;C_{0}=300\;{\rm MeVfm^{3}}.$$
(2.3)
For the nuclear density $\rho_{0}(r)$ in the interpolation formula
we chose the theoretical ground state density distribution
of the corresponding nucleus,
$$\rho_{0}(r)=\sum_{\epsilon_{i}\leq\epsilon_{F}}\frac{1}{4\pi}(2j_{i}+1)R^{2}_{%
i}(r),$$
(2.4)
which is more consistent than the previously used Woods-Saxon
distribution. For that reason we had to readjust $f_{ex}$ and
obtained the value of $f_{ex}$ = -1.4 used kst97 .
Here $R_{i}(r)$ are the single-particle radial wave functions of
the single-particle model used. For other details of the
calculations, see rev ; we04 ; kaev04 .
III PDR Results
The main results of our calculations for the gross properties PDR
are given in Fig.1 and Table 1.
Within the experimental errors, our theoretical results agree with
the data as shown in $TableI$. This holds for the total B(E1)
strength, EWSR and the EWSR share in the energy interval up to 10
MeV. As for the
PDR mean energies, which were defined as $\overline{E}=\Sigma E_{i}B_{i}(E1)/\Sigma B_{i}(E1)$,
the agreement is rather good for ${}^{44}Ca$ and ${}^{48}Ca$ but the
theoretical result 8.68 MeV is too high
in ${}^{40}Ca$. In all the
cases the results of the extended theory, i.e. inclusion of phonon
coupling and single-particle continuum, give rise to a
considerable improvement compared to the (Q)RPA results.
and (especially for
${}^{44}$Ca) our results of ref. we04 . The latter
is due to an improvement of the theory but even more importantly
to more careful calculations of the single-particle levels and of
the (Q)RPA phonons, including the use of eq.(2.4) for them.
In Fig.1 we compare the
theoretical EWSR of ${}^{40}Ca$,${}^{44}Ca$ and ${}^{48}Ca$ for the
isovector dipole strength up to 10 MeV with the experimental
data. The dash-dotted line is the (Q)RPA result which rises with
neutron excess in agreement with the phenomenological model. The
experiment shows only a strong increase for ${}^{44}Ca$ compared to
${}^{40}Ca$ but the strength of ${}^{48}Ca$ and ${}^{44}Ca$ does not
change within the experimental errors. The ETFFS result reproduces
this somewhat surprising result.
The explanation is given in Fig.2 where we compare the theoretical
PDR-spectrum of ${}^{44}Ca$ and ${}^{48}Ca$. Whereas the (Q)RPA result
is in both cases around 10 MeV and below, the ETFFS result differs
in the two nuclei. Due to the phonon splitting
an appreciable fraction of the strength in ${}^{48}Ca$ is shifted
to about 11 MeV and has therefore not be measured in the cited
experiment. If one adds this part ( the 10 MeV limit is somewhat
arbitrary ) the total PDR strength increase also in ETFFS with the
neutron excess. One also realizes that the PDR for the open shell
nucleus ${}^{44}Ca$ is somewhat lower compared to the PDR in the closed
shell nucleus ${}^{48}Ca$.
In Fig.3 the transition densities for protons and neutrons of the
electric dipole isovector strength up to 10 MeV are shown for the
three Ca isotopes. In ${}^{40}Ca$ the density is of purely isoscalar
nature. With increasing neutron excess the neutron density is
slightly shifted to a larger radius, indicating some an isovector
admixture, but the isoscalar character remains. This is easily
understood in the framework of microscopic models. The isovector
ph-interaction is repulsive and shifts the low-lying isovector
strength into the region of the GDR. On the other hand, the
isoscalar ph-interaction is attractive and shifts some of the high
lying isoscalar dipole strength to lower energies and gives rise
to some isoscalar collectivity below 10 MeV in contrast to the
isovector case.
In Fig.4 the isoscalar strength distributions (with the operator
${\bf r}^{3}Y_{10}$) in ${}^{40}Ca$ and ${}^{48}Ca$ calculated within
the ETFFS and RPA are shown. In ${}^{40}Ca$ the influence of the
phonons are small compared to ${}^{48}Ca$ especially for the low
energy spectrum. Whereas in ${}^{40}Ca$ the phonons only give rise
to an energy shift one observes in ${}^{48}Ca$ also some
fragmentation of the low lying strength. This fragmentation is due
to the low-lying $2^{+}$ state which can be coupled to 1p1h 1${}^{-}$
states to produce the necessary 1${}^{-}$ complex configurations. The
corresponding states do not exist in ${}^{40}Ca$. The two peaks in
${}^{40}Ca$ at 5.8 MeV and 7.8 MeV are in an agreement with the
experimental finding hvw01 .
IV The giant resonance results
For consistency we also calculated the giant isovector resonances in the three calcium isotopes. Results are presented in
Table 2
and in Figs.5 and 6. In Table 2 the results obtained in three different
approximations are shown: (Q)RPA calculations with the
single-particle continuum as well as
the ETFFS(QTBA)
calculations with and without the continuum.
The theoretical values are extracted from the calculated strength
distribution under the condition that the three lowest
energy-weighted
moments of the Lorentzian and the theoretical curve should
coincide in the (10-40) MeV interval kst93 ; tselyaev00 .
The averaging parameter $\Delta$ which was used in addition to
the single-particle continuum the giant resonance calculation was
400 keV. The differences between the (Q)RPA and full ETFFS(QTBA)
are most striking for the width $\Gamma$ and the maximum of the
photo absorption cross section $\sigma_{max}$, where the results
differ nearly by a factor of two. We obtain good agreement with
the experiments only within our full model, i.e. including the
phonons and the single-particle continuum. Here we want to stress
that we calculate GDR and the low-lying states simultaneously
within the same configuration space and with the same forces.
As compared with our earlier results for ${}^{40}Ca$ and ${}^{48}Ca$
kst93 ; kst97 , here
the ground state correlations induced by the phonon coupling
were not taken into account and we have chosen a different energy
interval for the summation. In addition, a different
definition for the mean energy and another ”refining” procedure
tselyaevQTBA have been used.
Like in the previous calculations of giant multipole resonances
in closed shell nuclei rev
the inclusion of phonon coupling , i.e. the calculation
within the full ETFFS (QTBA), increases the widths by a factor of 2
and (except for ${}^{40}$Ca) shifts the mean energies by 1.0-1.5 MeV
to higher energy.
In Fig.5 the results for the GDR in ${}^{44}Ca$ are shown,
calculated again within the three approximations. In the full
model an appreciable fraction of the strength is shifted to much
higher energies compared to the QRPA result which give rise to an
asymmetric shape. Finally in Fig.6 the transition density for the
GDR in ${}^{44}Ca$ is plotted. It has the expected isovector shape.
In order to study the role of the single-particle continuum
we have calculated the giant resonance (see Table 2 and Fig.5)
and PDR characteristics without the continuum. It should be
expected that the role of the single-particle continuum is
important in light nuclei. It is of special interest to consider
the question within the same calculational scheme for the Ca
isotopes, including the non-magic ${}^{44}$Ca, because there is no
information about the role of the continuum in the approaches of
similar kind in non-magic nuclei. One can see from Table 2 and
Fig.5 a very noticeable influence of the continuum for the
integral characteristics in all the Ca isotopes as well as for
the strength distribution. Moreover, the continuum’s influence is
noticeable even for the PDR : our full calculations have given
0.38% of the EWSR in the (5.0-10.0) MeV interval for ${}^{44}$Ca
whereas the calculation without continuum gave 0.19%. The
exclusion of the continuum also results in a considerable
redistribution of the PDR strength. These features can be studied
in experiments with electrons and gamma-rays.
V Summary
We have investigated the low-lying (PDR)
and high-lying (GDR) electric dipole states
in the magic ${}^{40}$Ca and ${}^{48}$Ca and non-magic ${}^{44}$Ca
within the ETFFS(QTBA) which, in addition to the standard (Q)RPA,
takes into account the single-particle continuum and phonon
coupling. In the present approach we corrected for the double
counting due to the phonons in our generalized propagator. Good
agreement with experiment available for integral characteristics
has been found.
The agreement for the PDR is much better as compared with the
previous ETFFS calculations we04 .
The PDR, i.e. the
isovector dipole strength below 10 MeV, is less than 1 percent of
the energy weighted sum rule because the strongly repulsive
isovector force shift the strength into GDR region.
In order to investigate the structure of the PDR further
we calculated the corresponding transition densities. It turned
out that these densities are isoscalar with some isovector
admixture in ${}^{44}$Ca and ${}^{48}$Ca which is due to the strongly
attractive
isoscalar interaction which shifts the high-lying isoscalar strength
below 10 MeV.
In ${}^{40}$Ca we obtain two prominent isoscalar dipole states
(induced by the external field ${\bf r}^{3}Y_{10}$) at 5.8 MeV and
7.8 MeV. The predicted isoscalar strength can be detected in
${(\alpha,\alpha^{\prime}\gamma)}$ experiments, see zielges05 .
We have shown that the role of the single-particle continuum is
very noticeable for the GDR and quantatively important for the
PDR in the Ca isotopes considered. The same should be true for
lighter nuclei too.
Thus, taking into account the new effects as
compared with the standard RPA or QRPA , i.e. the phonon coupling
and single-particle continuum , is necessary to explain the
properties of both these resonances.
VI Acknowledgments
The authors are very thankful to N.
Luytorovich for permanent collaboration in the preparation of this
article and to
A.Zilges for the information
about the experiments zielges05 . S.Kamerdzhiev thanks
the Institute
for Nuclear Theory (Seattle, USA) for partial support during the
participation in the Program INT-05-3 which was very useful for
this work. The work was supported in part by the DFG and RFBR
grants Nos.GZ:432RUS113/806/0-1 and 05-02-04005 and by the INTAS
grant No.03-54-6545.
References
(1)
R. Palit, P. Adrich, T. Aumann, et al., Nucl. Phys. A731 (2004) 235.
(2)
U. Kneisl, H.H. Pilz, and A. Zilges, Progr. Part. Nucl. Phys.
37 (1996) 349.
(3)
Y. Suzuki, K. Ikeda, and H. Sato, Progr. Theor. Phys. 83 (1990) 180.
(4)
R.M.Laszewski, P.Rullhusen, S.D.Hoblit and S.F.LeBrun, Phys. Rev. Lett.
54 (1985) 530.
(5)
T. Hartmann, J. Enders, P.Mohr, et al., Phys. Rev.
C 65 (2002) 034301.
(6)
T. Hartmann, M. Babilon, S. Kamerdzhiev, et al.
, Phys. Rev. Lett. 93 (2004) 192501.
(7)
J. Chambers, E. Zaremba, J.P. Adams, and B.Castel, Phys. Rev. C 50
(1994) 2671R.
(8)
D. Vretenar, A.V. Afanasjev, G.A. Lalazissis, and P.Ring, Phys. Rep.
409 (2005) 101.
(9)
S. Goriely, E. Khan, Nucl.Phys. A 706 (2002) 217.
(10)
S. Kamerdzhiev, J. Speth, G. Tertychny, Phys. Rep. 393 (2004) 1.
(11)
S.P. Kamerdzhiev and E.V. Litvinova, Phys. At. Nucl. 67 (2004) 183; erratum
Phys. At. Nucl. 67 (2004) 1610.
(12)
V.G. Soloviev, Theory of Complex Nuclei (Pergamon Press, Oxford,
1976) [Russ. original, Nauka, 1971]
(13)
V.G. Soloviev, Theory of Atomic Nuclei: Quasiparticlws and Phonons
(Institute of Physics , Bristol and Phyladelphia, USA, 1992)
(14)
S. Kamerdzhiev, J. Speth, G. Tertychny and V.Tselyaev, Nucl.Phys.
A555 (1993) 90.
(15)
S. Kamerdzhiev, J. Speth, G. Tertychny and J.Wambach,Z.Physik
A 336 (1993) 253.
(16)
G. Colo, Nguyen Van Giai, P.F. Bortignon, R.A Broglia, Phys. Rev.
C 50 (1994) 1496.
(17)
N. Ryezaeva, T. Hartmann, Y. Kalmykov, et.al.,
Phys. Rev. Lett. 89 (2002) 272502.
(18)
N. Tsoneva, H. Lenske, Ch. Stoyanov, Phys. Lett. B 586 (2004) 213.
(19)
D.Sarchi, P.F. Bortignon, G. Colo, Phys. Lett. B 601 (2004) 27.
(20)
G. Colo, P.F. Bortignon, Nucl.Phys.A 696 (2001) 427.
(21)
E.V. Litvinova, V.I. Tselyaev, arXiv:nucl-th/0512030 v2 9 Dec
2005.
(22)
A. Richter, Prog. Part. Nucl.Phys. 55 (2005) 387.
(23)
V.I. Tselyaev arXiv:nucl-th/0505031 v1 May 2005
(24)
M.V. Zverev, E.E. Saperstein, Yad.Fiz. 42 (1082) 1985.
(25)
S.Kamerdzhiev,R.J.Liotta,E.Litvinova and V.Tselyaev, Phys.Rev.C 58
(1998) 172
(26)
S. Kamerdzhiev, J. Speth, G. Tertychny, Nucl.Phys. A 624 (1997) 328.
(27)
M.N. Harakeh and A.van der Woude, Giant Resonances
(Clarendon Press, Oxford, 2001).
(28)
V.I. Tselyaev, Bull. Rus. Acad. Sci. Phys. 64 (2000) 434.
(29)
A. Zilges, M. Babilon, T.Hartmann, et al.,Progr. Part. Nucl. Phys.
55 (2005) 408. |
Coupling-assisted quasi-bound states in the continuum in heterogeneous metasurfaces
Wei Huang
Guangxi Key Laboratory of Optoelectronic Information Processing, School of Optoelectronic Engineering, Guilin University of Electronic Technology, Guilin 541004, China
Songyi Liu
Guangxi Key Laboratory of Optoelectronic Information Processing, School of Optoelectronic Engineering, Guilin University of Electronic Technology, Guilin 541004, China
Dehui Zeng
Guangxi Key Laboratory of Optoelectronic Information Processing, School of Optoelectronic Engineering, Guilin University of Electronic Technology, Guilin 541004, China
Quanlong Yang
Nonlinear Physics Centre, School of Physics, Australian National University, Canberra, ACT 2601, Australia
Wentao Zhang
Guangxi Key Laboratory of Optoelectronic Information Processing, School of Optoelectronic Engineering, Guilin University of Electronic Technology, Guilin 541004, China
Shan Yin
syin@guet.edu.cn
Guangxi Key Laboratory of Optoelectronic Information Processing, School of Optoelectronic Engineering, Guilin University of Electronic Technology, Guilin 541004, China
Jiaguang Han
jiaghan@tju.edu.cn
Guangxi Key Laboratory of Optoelectronic Information Processing, School of Optoelectronic Engineering, Guilin University of Electronic Technology, Guilin 541004, China
Center for Terahertz Waves and College of Precision Instrument and Optoelectronics Engineering, Tianjin University, Tianjin 3000072, China
Abstract
In this paper, we present a Bound states in the continuum (BIC) metamaterial in heterogeneous structures based on the universal coupled mode theory.
We find the more general physical parameters to represent BIC, which are the resonant frequencies and corresponding phases of metamaterial structures.
Therefore, BIC metamaterial comes from the equal value of the resonant frequencies and phases of metamaterial structures which are not only for homogeneous structures.
Meanwhile if slightly vary one of metamaterial structure’s resonant frequency and phase by varying geometry, we can obtain the quasi-BIC instead of broken symmetry of homogeneous structures.
In this paper, we provide the BIC and quasi-BIC with one example of two heterogeneous structures which are cut wire (CW) and Split-Ring Resonator (SRR), to widely extends the metamaterial BIC beyond common sense. Furthermore, we demonstrate the simulation results and experimental results to proof our idea.
pacs: 42.82.Et, 42.81.Qb, 42.79.Gn, 32.80.Xx
I Introduction
Bound states in the continuum (BIC) is initially proposed in quantum mechanics Neumann1929 , which trapped or guided modes with their frequencies in the frequency intervals of radiation modes in optical system Hsu2016 ; Marinica2008 ; Plotnik2011 ; Gao2016 . Based on this idea, most recently, the concept of BIC has already been introduced into the metamaterial structure, which can offer the ultra-high Q resonance Azzam2018 ; Koshelev2018 ; Koshelev2019 ; Miroshnichenko2010 ; Kupriianov2019 ; Abujetas2019 ; Cong2015 ; Liang2020 .
Based on the metamaterial BIC, there are many practical applications for photonic systems, such as lasers Kodigala2017 ; Ha2018 , sensors Liu2017 ; Romano2018 , high-sensitive medical devices Khanikaev2013 ; Tittl2018 and filters Foley2014 .
Due to widely applications, the BIC has been attracted many research interests.
The reason of causing metamaterial BIC is that if two lossy fields of metamaterial structures have the same phase and resonant frequency of single metamaterial structure, the far field of two lossy fields have destructive interference with each others. When we slightly vary one of metamaterial structure, the far field of two lossy fields can not destructive interference, but provides high Q Fano-shape resonance instead, so-called broken symmetry of quasi-BIC.
The most intuitive choice and common sense for BIC metamaterial is made of two uniform structures and quasi-BIC comes from the broken symmetry of homogeneous structures, which can obtain the high Q resonance.
The coupled mode theory (CMT) is a prosperous and universal theory and it widely used in many systems Cong2015 ; Niu2021 ; Huang2014 ; Huang2019 ; Ostrovskaya2000 ; Huang2020 ; Koshiba2011 ; Huang20202 ; Huang2018 .
Most recently, CMT implies the more fundamental physical parameters for BIC metamaterial, such as the resonant frequencies and corresponding phases of metamaterial structures Huang2021 . Therefore, we can simply extend this deduction of establishing BIC metamaterial by employing two heterogeneous structures with the same value of resonant frequencies and corresponding phases.
In the other words, even for two heterogeneous metamaterial structures (for example, Cut Wire (CW) and Split-Ring Resonator (SRR)) which have the same resonant frequency and corresponding phase, we can obtain the BIC metamaterial with heterogeneous configuration.
This finding will exceed the common sense of BIC metamaterial with homogeneous structures
Normally, previous researches only realize BIC metamaterial caused by two homogeneous structures (or quasi-BIC caused by two slightly different structures).
In this paper, we obtain the BIC (and quasi-BIC) with two heterogeneous metamaterial structures, as shown in Fig. 1 (CW and SRR) for the first time.
We provide the CW and SRR with the same phase and resonant frequency (see Fig. 2a) for the BIC case as shown in Fig. 2b. Furthermore, we slightly vary the structure of CW to change the phase and resonant frequency of CW (see Fig. 3a and Fig. 3b), to obtain the corresponding quasi-BIC as shown in Fig. 4a and Fig. 4b. Subsequently, we continuously vary the length of CW to find the corresponding Q-value of quasi-BIC, and we give the fitting function according to $\alpha^{-2}$, where $\alpha$ is the asymmetric parameter Azzam2018 , as shown in Fig.4.
The paper is organized as follows. In section II, we firstly introduce the universal coupled mode theory for metamaterial BIC. Section III employs the universal coupled mode theory to predict the BIC with the same phase and resonant frequency of CW and SRR. After that, we slightly change the phase and resonant frequencies of CW, which provides a different Q-value of quasi-BIC. In addition, we give the experimental results to demonstrate our device.
In the section IV, we provide the further discussions on our design. Finally, we conclude in Section V.
II Universal coupled mode theory
The universal coupled mode theory can be described as following Huang2021 ,
$$\left[\begin{matrix}\ w-w_{a}-i\gamma_{a}&\Omega\\
\Omega&\ w-w_{b}-i\gamma_{b}\end{matrix}\right]\left[\begin{matrix}a\\
b\end{matrix}\right]=\left[\begin{matrix}\sqrt{\gamma_{a}}E\\
\sqrt{\gamma_{b}}e^{{}_{i\phi}}E\end{matrix}\right],$$
(1)
where $|a|^{2}$ and $|b|^{2}$ as the energies in each metamaterial structure. $\omega$ is the frequency of input THz wave. $\omega_{a}$ and $\omega_{b}$ represent the frequencies of the metamaterial structures, which is the same as the resonant frequency of the corresponding metamaterial structure ($\omega_{a}$ = $\omega_{1}$; $\omega_{b}$ = $\omega_{b}$). $\Omega$ is the coupling strength with loss, due to the loss of transferring energy from one metamaterial structure to another, with $\Omega=g-i\sqrt{\gamma_{a}\gamma_{b}}e^{{}_{i\phi}}$, where $g$ is the coupling strength between two metamaterial structures. $\gamma_{a}$ and $\gamma_{b}$ are the loss terms of metamaterial structures, which is close to $\gamma_{1}$, $\gamma_{2}$ for each single metamaterial structure ($\gamma_{a}=\gamma_{1}$; $\gamma_{b}=(\gamma_{1}-\gamma_{2})/2$). $\phi$ is the phase information, which can be calculated by the phase $\phi_{1}$ and $\phi_{2}$ for each metamaterial structure ($\phi=(\phi_{1}-\phi_{2})d$, where $d$ is the width of metamaterial structure). $E$ is the amplitude of external exciting THz wave.
Subsequently, we can obtain the energy amplitudes $a$, $b$ of each metamaterial structure by solving the Eq. 1, as shown,
$$a=\frac{((w-w_{b}-i\gamma_{b})\sqrt{\gamma_{a}}-\Omega\sqrt{\gamma_{b}}e^{{}_{i\phi}})E}{(w-w_{b}-i\gamma_{b})(w-w_{a}-i\gamma_{a})-\Omega^{2}};$$
(2)
$$b=\frac{((w-w_{a}-i\gamma_{a})\sqrt{\gamma_{b}}-\Omega\sqrt{\gamma_{a}})E}{(w-w_{b}-i\gamma_{b})(w-w_{a}-i\gamma_{a})-\Omega^{2}}.$$
(3)
Then we calculate the effective susceptibility which is the linear superposition with energy amplitudes $|a|^{2}$ and $|b|^{2}$. the effective electric susceptibility of the metamaterial can be written as meng2012
$$\chi_{\text{eff}}=\frac{\sqrt{\gamma_{a}}a+\sqrt{\gamma_{b}}e^{{}_{i\phi}}b}{\epsilon_{0}E}.$$
(4)
Finally, we obtain the transmission spectrum with $T\approx 1-\text{Im}(\chi_{\text{eff}})$ Cong2015 , as shown,
$$T\approx 1-\text{Im}(\frac{(w-w_{a}-i\gamma_{a})\gamma_{b}e^{{}_{2i\phi}}+((w-w_{b}-i\gamma_{b})\gamma_{a}-2\Omega\sqrt{\gamma_{a}\gamma_{b}}e^{{}_{i\phi}}}{(w-w_{b}-i\gamma_{b})(w-w_{a}-i\gamma_{a})-\Omega^{2}}).$$
(5)
Hence, we can predict the transmission spectrum of the structure Huang2021 , to obtain the BIC and Q-value of quasi-BIC.
Therefore, our theory implies that the BIC only is relative to the phases and resonant frequencies of two adjacent metamaterial structures. Thus, if two metamaterial structures have the same phase and resonant frequency, they should have the BIC phenomenon even for two entirely different structures.
In other words, if we carefully design two heterogeneous structures which have the same value of resonant frequency and corresponding phase, we can establish BIC metamaterial beyond the common sense of homogeneous structures. Furthermore, when we slightly vary resonant frequency and corresponding phase of any one heterogeneous structures by changing the geometrical parameters, the quasi-BIC with two heterogeneous structures will appear.
In this paper, we takes the CW and SRR as the example to demonstrate our idea, but our results are not only for CW and SRR.
III BIC with two different structures
Based on the prediction of our theory, we carefully select the structures of CW and SRR to obtain the BIC or quasi-BIC. In other to easier demonstration, we fix the structure parameters of SRR, where the gap of SRR is 4 $\mu m$; width of SRR is 4 $\mu m$ and the side length of SRR is 60 $\mu m$. Besides, the width of CW is fixed 20 $\mu m$ with the length $L$ of CW, as shown in Fig. 1. Therefore, we just required to change the length $L$ of CW to vary the phase and resonant frequency of CW, to obtain the BIC or quasi-BIC.
For the BIC case, we select the length $L_{0}$ of CW as 85.5 $\mu m$, where the resonant frequency of CW is 0.6097 THz and the corresponding phase at the resonant frequency is 26.125 degrees. Besides, the resonant frequency of SRR is 0.6095 THz and corresponding phase at resonant frequency is 26.051 degrees, as shown in Fig. 2a. Thus, according to our theory, this configuration generates the ultra-high Q resonance or we can call it as BIC.
The transmission spectrum of CW and SRR coupling demonstrates in Fig. 2b. From the results, coupling between two entirely different structures performs the transmission spectrum as the single resonance. Thus, the Q-value at BIC frequency reveals the infinite.
Based on our universal coupled mode theory, if we slightly change the length $L$ which varies the resonant frequencies and the corresponding phases of CW, the resonant frequencies and corresponding phases of CW and SRR are not match. Therefore, the Fano resonant shapes will appear in the transmission spectrum, so-called quasi-BIC cases.
In order to demonstrate the correct prediction of our theory, we propose the full-wave simulations, our theory and experiments by varying the length of CW from $L=67$ $\mu m$ to $110$ $\mu m$, as shown in Fig. 3. The first column and second column of Fig. 3 demonstrate our designed structure of units cells and our experimental devices, respectively. The third, fourth, and fifth columns are the full-wave simulations, our theory and corresponding experimental results, respectively.
In Fig. 3, our theoretical results can be well-fitted to our full-wave simulations. Meanwhile, our experimental results are also well consistent with our theoretical and full-wave simulated results, comparing with the resonance frequencies in our simulations, theoretical and experimental results.
As we can obtain that when we slightly change the length $L$, the Fano resonant shapes do appear in the transmission spectrum as shown in Fig. 3 except (b), because Fig. 3 (b) is the BIC case ($L_{0}=85.5$ $\mu m$) which the resonant frequency and corresponding phase of CW are very closed to SRR, as shown in Fig. 2.
IV Discussion
From the results of Fig. 3, our experimental results well satisfied with full-wave simulation and theoretical results. Therefore, we can conclude that our simulated and theoretical results have very good confidence and we can employ our simulated and theoretical results to obtain more features of our device.
In order to better demonstrate the BIC with different structures, we vary the length of CW $L$ from 66 $\mu m$ to 107 $\mu m$, which crosses the same phase and resonant frequency of CW and SRR ($L_{0}$ = 85.5 $\mu m$). Therefore, from the universal coupled mode theory, we can easily predict that the highest Q-value appears when the same phase and resonant frequency of CW and SRR ,such as the $L_{0}$ = 85.5 $\mu m$. Therefore, when the length of CW is $L_{0}$ = 85.5 $\mu m$, it is the BIC point. When the length $L$ of CW is farther away from the BIC point, the Q-value of resonant will be smaller instead. The relationship between the Q-values and the length $L$ of CW can be fitted by function $Q\propto\alpha^{-2}$, where $\alpha$ is the asymmetric parameter which is given by $\alpha=\frac{L-L_{0}}{L_{0}}$ Azzam2018 . However, in our new type of BIC, the asymmetric parameter $\alpha$ is described by asymmetry between CW and SRR, which means the resonant frequencies between CW and SRR.
As we can see from the Fig. 4, it is very easy to see that the Q-values of quasi-BIC are higher, when the length of CW $L$ is more closer to 85.5 $\mu m$ (BIC point) and the Q-values can be fitted by the asymmetric parameter between CW and SRR with the fitting function $Q\propto\alpha^{-2}$. The fitting functions are given in black line and red line in Fig. 4. Our results of Fig. 4 is consistent with our theory and analysis. Hence, our theory can predict and have a very good fitting with BIC or quasi-BIC with CW and SRR, which is the entirely new metamaterial structure for BIC.
It is worth emphasizing that we employ the CW and SRR as the heterogeneous example to construct BIC and quasi-BIC in this paper, but our conclusions and results can be extended to arbitrary heterogeneous structure to build up the BIC and quasi-BIC, not only for CW and SRR.
V Conclusion
In this paper, we propose the new configuration of metamaterial BIC with two heterogeneous structures (Cut Wire and Split-Ring Resonator). We firstly theoretically and experimentally demonstrate the BIC and quasi-BIC with two heterogeneous structures. Our theory implies that resonant frequencies and corresponding phases of metamaterial structures are more fundamental physical parameters of BIC and we give the simulation and experimental results to proof our idea. This finding will extend the boundary of metamaterial BIC and could be widely used in applications.
Acknowledgements
This work acknowledges funding from National Science and Technology
Major Project (grant no: 2017ZX02101007-003); National
Natural Science Foundation of China (grant no: 61565004;
61965005; 61975038; 62005059). The Science and
Technology Program of Guangxi Province (grant no:
2018AD19058). Innovation Project of GUET Graduate
Education(grant no: 2021YCXS129). W.H. acknowledges funding from Guangxi
oversea 100 talent project; W.Z. acknowledges funding from
Guangxi distinguished expert project.
References
(1)
Neumann J V and Wigner E P 1929 Uber merkw¨urdige diskrete eigenwerte Z. Physik 50 291-3
(2)
Hsu C W, Zhen B, Stone A D et al 2016 Bound states in the continuum Nature Reviews Materials 1 1-13
(3)
Marinica D C, Borisov A G, Shabanov S V. Bound states in the continuum in photonics[J]. Physical review letters, 2008, 100(18): 183902.
(4)
Plotnik Y, Peleg O, Dreisow F, et al. Experimental observation of optical bound states in the continuum[J]. Physical review letters, 2011, 107(18): 183901.
(5)
Gao X, Hsu C W, Zhen B et al 2016 Formation mechanism of guided resonances and bound states in the continuum in photonic crystal slabs Scientific reports 6 1-7
(6)
Azzam S I, Shalaev V M, Boltasseva A et al 2018 Formation of bound states in the continuum in hybrid plasmonic-photonic systems Physical review letters 121 253901
(7)
Koshelev K, Lepeshov S, Liu M et al 2018 Asymmetric metasurfaces with high-Q resonances governed by bound states in the continuum Physical review letters 121 193903
(8)
Koshelev K, Bogdanov A, Kivshar Y. Meta-optics and bound states in the continuum[J]. Science Bulletin, 2019, 64(12): 836-842.
(9)
Miroshnichenko A E, Flach S and Kivshar Y S 2010 Fano resonances in nanoscale structures Reviews of Modern Physics 82 2257
(10)
Kupriianov A S, Xu Y, Sayanskiy A et al 2019 Metasurface engineering through bound states in the continuum Physical Review Applied 12 014024
(11)
Abujetas D R, van Hoof N, ter Huurne S et al 2019 Spectral and temporal evidence of robust photonic bound states in the continuum on terahertz metasurfaces Optica 6 996-1001
(12)
Cong L, Manjappa M, Xu N et al 2015 Fano resonances in terahertz metasurfaces: a figure of merit optimization Advanced Optical Materials 3 1537-43
(13)
Liang Y, Koshelev K, Zhang F et al 2020 Bound states in the continuum in anisotropic plasmonic metasurfaces Nano Letters 20 6351-6
(14)
Kodigala A, Lepetit T, Gu Q et al 2017 Lasing action from photonic bound states in continuum Nature 541 196-9
(15)
Ha S T, Fu Y H, Emani N K et al 2018 Directional lasing in resonant semiconductor nanoantenna arrays Nature nanotechnology 13 1042-7
(16)
Liu Y, Zhou W and Sun Y 2017 Optical refractive index sensing based on high-Q bound states in the continuum in free-space coupled photonic crystal slabs Sensors 17 1861
(17)
Romano S, Lamberti A, Masullo M et al 2018 Optical biosensors based on photonic crystals supporting bound states in the continuum Materials 11 526
(18)
Khanikaev A B, Wu C and Shvets G 2013 Fano-resonant metamaterials and their applications Nanophotonics 2 247-64
(19)
Tittl A, Leitis A, Liu M et al 2018 Imaging-based molecular barcoding with pixelated dielectric metasurfaces Science 360 1105-9
(20)
Foley J M, Young S M and Phillips J D 2014 Symmetry-protected mode coupling near normal incidence for narrow-band transmission filtering in a dielectric grating Physical Review B 89 165111
(21)
Huang W, Rangelov A A and Kyoseva E 2014 Complete achromatic optical switching between two waveguides with a sign flip of the phase mismatch Physical Review A 90 053837
(22)
Huang W, Yin S, Zhang W et al 2019 Robust and broadband integrated terahertz coupler conducted with adiabatic following New Journal of Physics 21 113004
(23)
Ostrovskaya E A, Kivshar Y S, Lisak M, et al. Coupled-mode theory for Bose-Einstein condensates[J]. Physical Review A, 2000, 61(3): 031601.
(24)
Huang W, Qu X, Yin S et al 2020 Quantum Engineering Enables Broadband and Robust Terahertz Surface Plasmon-Polaritons Coupler IEEE Journal of Selected Topics in Quantum Electronics 27 1-7
(25)
Koshiba M, Saitoh K, Takenaga K, et al. Multi-core fiber design and analysis: coupled-mode theory and coupled-power theory[J]. Optics express, 2011, 19(26): B102-B111.
(26)
Huang W, Qu X, Yin S et al 2020 Long-distance adiabatic wireless energy transfer via multiple coils coupling Results in Physics 19 103478
(27)
Niu L et al 2021 Coupling plasmonic system for efficient wavefront control ACS Appl. Mater. Interfaces 13 5844–52
(28)
Huang W, Liang S J, Kyoseva E, et al. Adiabatic control of surface plasmon-polaritons in a 3-layers graphene curved configuration[J]. Carbon, 2018, 127: 187-192.
(29)
Huang W, Liu S, Cheng Y, et al. Universal coupled theory for metamaterial bound states in the continuum[J]. New Journal of Physics, 2021, 23(9): 093017.
(30)
Zhen B, Hsu C W, Igarashi Y et al 2015 Spawning rings of exceptional points out of Dirac cones Nature 525 354-8
(31)
Zhen B, Hsu C W, Lu L et al 2014 Topological nature of optical bound states in the continuum Physical review letters 113 257401
(32)
Limonov M F, Rybin M V, Poddubny A N et al 2017 Fano resonances in photonics Nature Photonics 11 543
(33)
Gallinet B and Martin O J F 2011 Ab initio theory of Fano resonances in plasmonic nanostructures and metamaterials Physical Review B 83 235427
(34)
Zhang F, Huang X C, Zhao Q et al 2014 Fano resonance of an asymmetric dielectric wire pair Applied Physics Letters 105 172901
(35)
Meng F Y, Wu Q, Erni D et al 2012 Polarization-independent metamaterial analog of electromagnetically induced transparency for a refractive-index-based sensor IEEE transactions on microwave theory and techniques 60 3013-22 |
Estimates for the first eigenvalue of Jacobi operator on hypersurfaces
Daguang Chen and Qing-Ming Cheng
Daguang Chen
Department of Mathematical Sciences,
Tsinghua University,
Beijing 100084, P. R. China, dgchen@math.tsinghua.edu.cn
Qing-Ming Cheng
Department of Applied Mathematics, Faculty of Sciences,
Fukuoka University, 814-0180, Fukuoka, Japan, cheng@fukuoka-u.ac.jp
(Date:: )
Abstract.
In this paper, we study the first eigenvalue of Jacobi operator on
an $n$-dimensional non-totally umbilical compact hypersurface with
constant mean curvature $H$ in the unit sphere $S^{n+1}(1)$.
We give an optimal upper bound for the first eigenvalue of Jacobi operator,
which only depends on the mean curvature $H$ and the dimension $n$.
This bound is attained if and only if, $\varphi:\ M\to S^{n+1}(1)$ is isometric to
$S^{1}(r)\times S^{n-1}(\sqrt{1-r^{2}})$ when $H\neq 0$ or $\varphi:\ M\to S^{n+1}(1)$ is isometric to
a Clifford torus $S^{n-k}(\sqrt{\dfrac{n-k}{n}})\times S^{k}(\sqrt{\dfrac{k}{n}})$,
for $k=1,2,\cdots,n-1$ when $H=0$.
2001 Mathematics Subject Classification: 53C42, 58J50
Key words and phrases: hypersurfaces, Jacobi operator,
the mean curvature, the first eigenvalue
The first author is supported by NSFC.
The second author is partially supported by JSPS Grant-in-Aid for Scientific Research (B): No.16H03937.
1. Introduction
Let $\varphi:M\to S^{n+1}(1)$ be an $n$-dimensional compact hypersurface in the unit sphere $S^{n+1}(1)$
of dimension $n+1$. We consider
a variation of the hypersurface $\varphi:M\to S^{n+1}(1)$, for any $t\in(-\varepsilon,\varepsilon)$,
$$\varphi_{t}:M\to S^{n+1}(1)$$
is an immersion with
$\varphi_{0}=\varphi$.
The area of $\varphi_{t}$ is given by
$$A(t)=\int_{M}dA_{t}$$
and the volume of $\varphi_{t}$ is defined by
$$V(t)=\dfrac{1}{n+1}\int_{M}\langle\varphi_{t},N(t)\rangle dA_{t},$$
where $N(t)$ denotes the unit normal of $\varphi_{t}$.
For any $t$, if $V(t)=V(0)$, then the variation
$\varphi_{t}$ is called volume-preserving. If the variational
vector $\dfrac{\partial\varphi_{t}}{\partial t}|_{t=0}=fN$ for a
smooth function $f$, then the variation is called a
normal variation, where $N$ is the unit normal of
$\varphi$.
Let $H$ denote the mean curvature of $\varphi$. The first variation
formula of the area functional $A(t)$ is given by
$$\frac{dA(t)}{dt}|_{t=0}=-\int_{M}nHfdA,$$
where $f=\langle\frac{\partial\varphi_{t}}{\partial t}|_{t=0},N\rangle.$
Hence, we know, for a compact minimal hypersurface, that is, $H=0$
$$\frac{dA(t)}{dt}|_{t=0}=0,$$
namely, compact minimal hypersurfaces are critical points of the area functional
$A(t)$.
The second variation formula of $A(t)$ is given by
$$\dfrac{d^{2}A(t)}{dt^{2}}|_{t=0}=-\int_{M}fJ_{m}fdA$$
and
$$J_{m}f=\Delta f+(S+n)f,$$
where $S$ denotes the squared norm of the second fundamental form
of $\varphi$
and $\Delta$ stands for the Laplace-Beltrami operator.
The $J_{m}$ is called a Jacobi operator or a stability operator of the minimal hypersurface $\varphi$ (cf. [8]).
Let $\lambda^{J_{m}}_{1}$ denote the first eigenvalue of the Jacobi operator $J_{m}$. Then
$$J_{m}u=-\lambda^{J_{m}}_{1}u$$
and
the $\lambda^{J_{m}}_{1}$ is given by
$$\lambda^{J_{m}}_{1}=\inf_{f\not\equiv 0}\dfrac{-\int_{M}fJ_{m}fdA}{\int_{M}f^{%
2}dA}.$$
Simons [9] proves
$$\lambda^{J_{m}}_{1}\leq-n$$
and $\lambda^{J_{m}}_{1}=-n$ if and only if $\varphi:M\to S^{n+1}(1)$ is
totally geodesic.
Furthermore,
Wu [10] proves that for an $n$-dimensional compact non-totally geodesic
minimal hypersurface $\varphi:M\to S^{n+1}(1)$ in $S^{n+1}(1)$,
then $\lambda^{J_{m}}_{1}\leq-2n$ and $\lambda^{J_{m}}_{1}=-2n$ if and only if $\varphi:M\to S^{n+1}(1)$ is
a Clifford torus
$S^{n-k}(\sqrt{\frac{n-k}{n}})\times S^{k}(\sqrt{\frac{k}{n}}),$
for $k=1,2,\cdots,n-1$. Thus, we know that the upper bound for the first eigenvalue
$\lambda^{J_{m}}_{1}$ due to Wu is optimal and it only depends on the dimension $n$, does not depends on the immersion.
On the other hand, if one considers the volume-preserving variation of $\varphi$,
then we have
$$\int_{M}fdA=0.$$
From the first variation formula:
$$\dfrac{dA(t)}{dt}|_{t=0}=-\int_{M}nHfdA,$$
we know that compact hypersurfaces with constant mean curvature
are critical points of the area functional $A(t)$ for the volume-preserving variation
and the second variation formula of $A(t)$ is given by
$$\dfrac{d^{2}A(t)}{dt^{2}}|_{t=0}=-\int_{M}fJ_{m}fdA,$$
where the Jacobi operator $J_{m}$ of compact hypersurfaces with constant mean curvature
is the same as one of compact minimal hypersurfaces ([3]).
Alias, Barros and Brasil [2] study the first eigenvalue of the Jacobi operator
$J_{m}$ of compact hypersurfaces with constant mean curvature. They prove the following:
Theorem ABB. If $\varphi:M\to S^{n+1}(1)$ is an $n$-dimensional compact
hypersurface with non-zero constant mean curvature $H$ in the unit sphere
$S^{n+1}(1)$, then
$\lambda^{J_{m}}_{1}=-n(1+H^{2})$ and $\varphi:M\to S^{n+1}(1)$ is totally umbilical or
$$\displaystyle\lambda^{J_{m}}_{1}\leq$$
$$\displaystyle-2n(1+H^{2})+\dfrac{n(n-2)|H|}{\sqrt{n(n-1)}}\max\sqrt{S-nH^{2}}$$
and the equality holds if and only if $\varphi:M\to S^{n+1}(1)$ is
$S^{1}(r)\times S^{n-1}(\sqrt{1-r^{2}})$, with $r^{2}>\frac{1}{n}$ for $n\geq 3$.
According to this theorem, we know that, for $n=2$, the upper bund of the first eigenvalue
$\lambda^{J_{m}}_{1}$ of the Jacobi operator of non-totally umbilical compact hypersurfaces with constant mean curvature
only depends on the mean curvature $H$ and the dimension. But for $n\geq 3$,
the upper bund of the first eigenvalue
$\lambda^{J_{m}}_{1}$ of the Jacobi operator on non-totally umbilical compact hypersurfaces with constant mean curvature
includes the term $\max\sqrt{S-nH^{2}}$. Hence,
the upper bund of the first eigenvalue
$\lambda^{J_{m}}_{1}$ does not only depend on the mean curvature $H$ and the dimension $n$, but also depends on the immersion $\varphi$.
It is natural and important to propose the following:
Problem 1.1. To find an optimal upper bound for the first eigenvalue
$\lambda^{J_{m}}_{1}$ of the Jacobi operator of non-totally umbilical compact
hypersurfaces with constant mean curvature, which
only depends on the mean curvature $H$ and the dimension $n$.
In this paper, we give an affirmative answer for the above problem 1.1.
Theorem 1.1.
Let $\varphi:M\to S^{n+1}(1)$ be an $n$-dimensional non-totally umbilical compact
hypersurface with constant mean curvature $H$ in the unit sphere $S^{n+1}(1)$.
(1)
If $2\leq n\leq 4$ or $n\geq 5$ and $n^{2}H^{2}<\frac{16(n-1)}{n(n-4)}$,
then the first eigenvalue $\lambda_{1}^{J_{m}}$ of the Jacobi operator $J_{m}$ satisfies
$$\lambda_{1}^{J_{m}}\leq-n(1+H^{2})-\dfrac{n(\sqrt{4(n-1)+n^{2}H^{2}}-(n-2)|H|)%
^{2}}{4(n-1)}$$
and the equality holds if and only if $\varphi:M\to S^{n+1}(1)$ is isometric to
$S^{1}(r)\times S^{n-1}(\sqrt{1-r^{2}})$ with $r>0$ satisfying
$$\begin{cases}1>r^{2}>\dfrac{1}{n}&\ \text{for}\ 2\leq n\leq 4,\\
\dfrac{n}{(n-2)^{2}}>r^{2}>\dfrac{1}{n},&\ \text{for}\ n\geq 5\ \text{and}\ n^%
{2}H^{2}<\dfrac{16(n-1)}{n(n-4)}\end{cases}$$
or $\varphi:M\to S^{n+1}(1)$ is isometric to
a Clifford torus
$S^{n-k}(\sqrt{\frac{n-k}{n}})\times S^{k}(\sqrt{\frac{k}{n}}),$
for $k=1,2,\cdots,n-1$ with $H=0$.
(2)
If $n\geq 5$ and $n^{2}H^{2}\geq\frac{16(n-1)}{n(n-4)}$, the first eigenvalue $\lambda_{1}^{J_{m}}$ of the Jacobi operator $J_{m}$ satisfies
$$\displaystyle\lambda^{J_{m}}_{1}\leq-2(n-1)(1+H^{2})+\dfrac{(n-2)^{4}}{8(n-1)}%
H^{2}$$
and the equality holds if and only if $\varphi:M\to S^{n+1}(1)$ is isometric to
$S^{1}(\frac{\sqrt{n}}{n-2})\times S^{n-1}(\frac{\sqrt{(n-1)(n-4)}}{n-2})$.
Remark 1.1.
Since the first eigenvalue of Jacobi operator $J_{m}$ on
totally umbilical hypersurfaces satisfies $\lambda_{1}^{J_{m}}=-n(1+H^{2})$, according to our theorem, one knows
that for $2\leq n\leq 4$, there are no $n$-dimensional compact hypersurfaces in the unit sphere with constant mean
curvature $H$ so that the first eigenvalue $\lambda_{1}^{J_{m}}$ of Jacobi operator $J_{m}$ takes a value in the internal
$$\biggl{(}-n(1+H^{2})-\dfrac{n(\sqrt{4(n-1)+n^{2}H^{2}}-(n-2)|H|)^{2}}{4(n-1)},%
\ -n(1+H^{2})\biggl{)}.$$
For any $n\geq 2$, there are no $n$-dimensional compact hypersurfaces in the unit sphere with constant mean
curvature $H$ satisfying $n^{2}H^{2}<\frac{16(n-1)}{n(n-4)}$
so that the first eigenvalue $\lambda_{1}^{J_{m}}$ of Jacobi operator $J_{m}$ takes a value in the internal
$$\biggl{(}-n(1+H^{2})-\dfrac{n(\sqrt{4(n-1)+n^{2}H^{2}}-(n-2)|H|)^{2}}{4(n-1)},%
\ -n(1+H^{2})\biggl{)}.$$
One should compare the bound
$$-n(1+H^{2})-\dfrac{n(\sqrt{4(n-1)+n^{2}H^{2}}-(n-2)|H|)^{2}}{4(n-1)}$$
with the pinching constant in the rigidity theorem of Cheng and Nakagawa [6] or Alencar and do Carmo [1].
In the section 2, we shall give some examples of compact hypersurfaces, the first eigenvalue of Jacobi operator is given.
2. Prelimenary
Throughout this paper, all manifolds are assumed to be smooth and
connected without boundary. Let $\varphi:M\to S^{n+1}(1)$ be an
$n$-dimensional hypersurface
in a unit sphere $S^{n+1}(1)$. We choose a local orthonormal frame
$\{{\bf e}_{1},\cdots,{\bf e}_{n},{\bf e}_{n+1}\}$ and the dual
coframe $\{\omega_{1},\cdots,$ $\omega_{n}$, $\omega_{n+1}\}$ in
such a way that $\{{\bf e}_{1},\cdots,{\bf e}_{n}\}$ is a local
orthonormal frame on $M$. Hence, we have
$$\omega_{n+1}=0$$
on $M$. From Cartan’s lemma, we have
(2.1)
$$\omega_{in+1}=\sum_{j=1}^{n}h_{ij}\omega_{j},\ h_{ij}=h_{ji}.$$
The mean curvature $H$ and the second fundamental form
${II}$ of $\varphi:\ M\to S^{n+1}(1)$ are defined, respectively, by
$$H=\frac{1}{n}\sum_{i=1}^{n}h_{ii},\ {II}=\sum_{i,j=1}^{n}h_{ij}\omega_{i}%
\otimes\omega_{j}{\bf e}_{n+1}.$$
When the mean curvature $H$ of $\varphi:M\to S^{n+1}(1)$ is identically zero, we
recall that $\varphi:M\to S^{n+1}(1)$ is by definition a minimal hypersurface.
From the structure equations of $\varphi:M\to S^{n+1}(1)$, Gauss equation is given by
(2.2)
$$R_{ijkl}=(\delta_{ik}\delta_{jl}-\delta_{il}\delta_{jk})+(h_{ik}h_{jl}-h_{il}h%
_{jk}),$$
From
(2.2), we have
$$\displaystyle n(n-1)r=n(n-1)+n^{2}H^{2}-S,$$
where $n(n-1)r$ and $S$ denote the scalar curvature and the squared norm
of the second fundamental form of $\varphi:M\to S^{n+1}(1)$, respectively.
Defining the covariant derivative of $h_{ij}$ by
(2.3)
$$\sum_{k}h_{ijk}\omega_{k}=dh_{ij}+\sum_{k}h_{ik}\omega_{kj}+\sum_{k}h_{kj}%
\omega_{ki},$$
we obtain the Codazzi equations
(2.4)
$$h_{ijk}=h_{ikj}.$$
By taking exterior differentiation of (2.3), and defining
(2.5)
$$\sum_{l}h_{ijkl}\omega_{l}=dh_{ijk}+\sum_{l}h_{ljk}\omega_{li}+\sum_{l}h_{ilk}%
\omega_{lj}+\sum_{l}h_{ijl}\omega_{lk},$$
we have the following Ricci identities:
(2.6)
$$h_{ijkl}-h_{ijlk}=\sum_{m}h_{mj}R_{mikl}+\sum_{m}h_{im}R_{mjkl}.$$
For any $C^{2}$-function $f$ on $M$, we define its gradient and Hessian by
$$df=\sum_{i=1}^{n}f_{i}\omega_{i},$$
$$\sum_{j=1}^{n}f_{ij}\omega_{j}=df_{i}+\sum_{j=1}^{n}f_{j}\omega_{ji}.$$
Thus, the Laplace-Beltrami operator $\Delta$ is given by
$$\Delta f=\sum_{i=1}^{n}f_{ii}.$$
Example 2.1.
For totally umbilical sphere $S^{n}(r)$ of radius $r>0$, the first eigenvalue $\lambda_{1}^{J_{m}}=-n(1+H^{2})$ with $H=\frac{1}{r}$.
Example 2.2.
For Clifford torus $S^{n-k}(\sqrt{\frac{n-k}{n}})\times S^{k}(\sqrt{\frac{k}{n}})$, $k=1,2,\dots,n$, the first eigenvalue $\lambda_{1}^{J_{m}}=-2n$ with $H=0$.
Example 2.3.
For hypersurfaces
$S^{1}(r)\times S^{n-1}(\sqrt{1-r^{2}})$ with $0<r<1$, the principal curvatures are given by
$$k_{1}=-\dfrac{\sqrt{1}-r^{2}}{r},\ \ k_{2}=\cdots=k_{n}=\dfrac{r}{\sqrt{1-r^{2%
}}}.$$
Hence, we know that
$$nH=\dfrac{nr^{2}-1}{r\sqrt{1-r^{2}}},\ \ S=\dfrac{1-2r^{2}+nr^{4}}{r^{2}(1-r^{%
2})}.$$
For $r^{2}\geq\frac{1}{n}$, by a direct computation, we know that the first eigenvalue
$\lambda_{1}^{J_{m}}$ of $S^{1}(r)\times S^{n-1}(\sqrt{1-r^{2}})$ satisfies
$$\lambda_{1}^{J_{m}}=-n(1+H^{2})-\dfrac{n(\sqrt{4(n-1)+n^{2}H^{2}}-(n-2)|H|)^{2%
}}{4(n-1)}.$$
For $n\geq 5$ and $\frac{1}{n}\leq r^{2}<\frac{n}{(n-2)^{2}}$, we know the hypersurface $S^{1}(r)\times S^{n-1}(\sqrt{1-r^{2}})$
satisfies
$$n^{2}H^{2}<\frac{16(n-1)}{n(n-4)}$$
and
$$\lambda_{1}^{J_{m}}=-n(1+H^{2})-\dfrac{n(\sqrt{4(n-1)+n^{2}H^{2}}-(n-2)|H|)^{2%
}}{4(n-1)}.$$
The hypersurface $S^{1}(\frac{\sqrt{n}}{n-2})\times S^{n-1}(\frac{\sqrt{(n-1)(n-4)}}{n-2})$ satisfies
$$\displaystyle\lambda^{J_{m}}_{1}=-2(n-1)(1+H^{2})+\dfrac{(n-2)^{4}}{8(n-1)}H^{2}$$
with $n^{2}H^{2}=\frac{16(n-1)}{n(n-4)}$.
3. Proof of theorem 1.1.
In this section, we give a proof of the theorem 1.1.
Proof of theorem 1.1.
By making use of the Codazzi equations, Ricci identities and a standard computation of Simons’ type formula
(cf. [6], [4, 5], [7] and [9]),
we have
(3.1)
$$\frac{1}{2}\Delta S=\sum_{i,j,k=1}^{n}h_{ijk}^{2}+nS-n^{2}H^{2}+nHf_{3}-S^{2},$$
where $f_{3}=\sum_{i=1}^{n}k_{i}^{3}$ and $k_{i}$, $i=1,2,\dots,n$ denote the principal curvatures.
Putting
$\mu_{i}=k_{i}-H$, we have
(3.2)
$$B:=\sum_{i=1}^{n}\mu_{i}^{2}=S-nH^{2}\geq 0,\ \ f_{3}=B_{3}+3HB+nH^{3},$$
where $B_{3}=\sum_{i=1}^{n}\mu_{i}^{3}$.
The following inequality is known (cf. [6] and [7]):
(3.3)
$$|B_{3}|\leq\dfrac{n-2}{\sqrt{n(n-1)}}B^{\frac{3}{2}},$$
and the equality holds if and only if at least $n-1$ of $k_{i}$, for $i=1,2,\dots,n$, are equal with each other.
Since $H$ is constant, we can assume $H\geq 0$. Thus, from (3.1), (3.2) and (3.3), we have
(3.4)
$$\displaystyle\frac{1}{2}\Delta B=\frac{1}{2}\Delta S\geq\sum_{i,j,k=1}^{n}h_{%
ijk}^{2}+B(n+nH^{2}-B)-nH\dfrac{n-2}{\sqrt{n(n-1)}}B^{\frac{3}{2}}.$$
For any constant $\alpha>0$ and $\varepsilon>0$, we consider a function $f_{\varepsilon}=(B+\varepsilon)^{\alpha}>0$.
Hence, we have, from (3.4),
(3.5)
$$\displaystyle\Delta f_{\varepsilon}$$
$$\displaystyle=\alpha(\alpha-1)(B+\varepsilon)^{\alpha-2}|\nabla B|^{2}+\alpha(%
B+\varepsilon)^{\alpha-1}\Delta B$$
$$\displaystyle\geq\alpha(\alpha-1)(B+\varepsilon)^{\alpha-2}|\nabla B|^{2}$$
$$\displaystyle+2\alpha(B+\varepsilon)^{\alpha-1}\biggl{(}\sum_{i,j,k=1}^{n}h_{%
ijk}^{2}+B(n+nH^{2}-B)-nH\dfrac{n-2}{\sqrt{n(n-1)}}B^{\frac{3}{2}}\biggl{)}.$$
Since $H$ is constant, we have
(3.6)
$$\displaystyle\nabla_{k}(nH)=\sum_{i=1}^{n}h_{iik}=0,\ \ \ h_{kkk}^{2}\leq(n-1)%
\sum_{i\neq k}h_{iik}^{2}$$
$$\displaystyle|\nabla B|^{2}=\sum_{k=1}^{n}(2\sum_{i=1}^{n}\mu_{i}h_{iik})^{2}%
\leq 4B\sum_{i,k=1}^{n}h_{iik}^{2}.$$
Thus, we obtain
(3.7)
$$\displaystyle|\nabla B|^{2}$$
$$\displaystyle\leq 4B\sum_{i,k=1}^{n}h_{iik}^{2}$$
$$\displaystyle=4B\bigl{(}\dfrac{n}{n+2}\sum_{k=1}^{n}h_{kkk}^{2}+\dfrac{2}{n+2}%
\sum_{k=1}^{n}h_{kkk}^{2}+\sum_{i\neq k}h_{iik}^{2}\bigl{)}$$
$$\displaystyle\leq\dfrac{4n}{n+2}B\bigl{(}\sum_{k=1}^{n}h_{kkk}^{2}+3\sum_{i%
\neq k}h_{iik}^{2}\bigl{)}.$$
For any constant $\beta$, we have
$$\displaystyle\lambda_{1}^{J_{m}}\int_{M}f_{\varepsilon}^{2}dA\leq-\int_{M}f_{%
\varepsilon}J_{m}f_{\varepsilon}dA$$
$$\displaystyle=-\beta\int_{M}f_{\varepsilon}\Delta f_{\varepsilon}dA-\int_{M}%
\biggl{(}(1-\beta)f_{\varepsilon}\Delta f_{\varepsilon}+(S+n)f_{\varepsilon}^{%
2}\biggl{)}dA$$
$$\displaystyle=\beta\int_{M}|\nabla f_{\varepsilon}|^{2}dA-\int_{M}f_{%
\varepsilon}\biggl{\{}(1-\beta)\biggl{(}\alpha(\alpha-1)(B+\varepsilon)^{%
\alpha-2}|\nabla B|^{2}$$
$$\displaystyle+\alpha(B+\varepsilon)^{\alpha-1}\Delta B\biggl{)}+(B+nH^{2}+n)f_%
{\varepsilon}\biggl{\}}dA$$
$$\displaystyle=\alpha\int_{M}f_{\varepsilon}\bigl{\{}1+2\alpha\beta-\beta-%
\alpha\bigl{\}}(B+\varepsilon)^{\alpha-2}|\nabla B|^{2}dA$$
$$\displaystyle-\int_{M}f_{\varepsilon}^{2}\biggl{\{}\dfrac{\alpha(1-\beta)}{B+%
\varepsilon}\Delta B+B+nH^{2}+n\biggl{\}}dA.$$
Since
$$\sum_{i,j,k=1}^{n}h_{ijk}^{2}=\sum_{k=1}^{n}h_{kkk}^{2}+3\sum_{i\neq k}h_{iik}%
^{2}+\sum_{i\neq j\neq k\neq i}^{n}h_{ijk}^{2},$$
from (3.7),
for $\alpha$ and $\beta$, which satisfy
$$\alpha>\frac{n-2}{4n},\ \ 1-\beta=\dfrac{2n\alpha}{4n\alpha+2-n},$$
we obtain
(3.8)
$$\displaystyle\bigl{(}1+2\alpha\beta-\beta-\alpha\bigl{)}|\nabla B|^{2}-2(1-%
\beta)(B+\varepsilon)\sum_{i,j,k=1}^{n}h_{ijk}^{2}$$
$$\displaystyle\leq\dfrac{2}{n+2}B\biggl{\{}(n-2)(1-\beta)-4n\alpha(1-\beta)+2n%
\alpha\biggl{\}}\bigl{(}\sum_{k=1}^{n}h_{kkk}^{2}+3\sum_{i\neq k}h_{iik}^{2}%
\bigl{)}=0.$$
Thus,
we infer
$$\displaystyle\lambda_{1}^{J_{m}}\int_{M}f_{\varepsilon}^{2}dA$$
$$\displaystyle\leq\alpha\int_{M}f_{\varepsilon}(B+\varepsilon)^{\alpha-2}\biggl%
{\{}\bigl{(}1+2\alpha\beta-\beta-\alpha\bigl{)}|\nabla B|^{2}-2(1-\beta)(B+%
\varepsilon)\sum_{i,j,k=1}^{n}h_{ijk}^{2}\biggl{\}}dA$$
$$\displaystyle-\int_{M}f_{\varepsilon}^{2}\biggl{\{}\dfrac{2\alpha(1-\beta)B}{B%
+\varepsilon}\biggl{(}(n+nH^{2}-B)-nH\dfrac{(n-2)}{\sqrt{n(n-1)}}B^{\frac{1}{2%
}}\biggl{)}+B+nH^{2}+n\biggl{\}}dA$$
$$\displaystyle\leq-\int_{M}f_{\varepsilon}^{2}\biggl{\{}\dfrac{B}{B+\varepsilon%
}\biggl{(}\bigl{\{}1-2\alpha(1-\beta)\bigl{\}}B-\dfrac{2\alpha(1-\beta)(n-2)}{%
\sqrt{n(n-1)}}nHB^{\frac{1}{2}}+\varepsilon\biggl{)}dA$$
$$\displaystyle-2\alpha(1-\beta)(n+nH^{2})\int_{M}f_{\varepsilon}^{2}\dfrac{B}{B%
+\varepsilon}dA-(n+nH^{2})\int_{M}f_{\varepsilon}^{2}dA.$$
For $1-2\alpha(1-\beta)>0$, we obtain
$$\displaystyle\lambda_{1}^{J_{m}}\int_{M}f_{\varepsilon}^{2}dA$$
$$\displaystyle\leq\int_{M}f_{\varepsilon}^{2}\biggl{\{}\dfrac{B}{B+\varepsilon}%
\biggl{(}\dfrac{\alpha^{2}(1-\beta)^{2}(n-2)^{2}}{(1-2\alpha(1-\beta))n(n-1)}(%
nH)^{2}-\varepsilon\biggl{)}dA$$
$$\displaystyle-2\alpha(1-\beta)(n+nH^{2})\int_{M}f_{\varepsilon}^{2}\dfrac{B}{B%
+\varepsilon}dA-(n+nH^{2})\int_{M}f_{\varepsilon}^{2}dA.$$
Since $\varphi:M\to S^{n+1}(1)$ is not totally umbilical, we have
$$\lim_{\varepsilon\to 0}\int_{M}f_{\varepsilon}^{2}dA=\int_{M}B^{2\alpha}dA>0.$$
Letting $\varepsilon\to 0$,
we derive
(3.9)
$$\displaystyle\lambda^{J_{m}}_{1}\leq-(1+2\alpha(1-\beta))n(1+H^{2})+\dfrac{%
\alpha^{2}(1-\beta)^{2}}{1-2\alpha(1-\beta)}\dfrac{(n-2)^{2}}{n(n-1)}n^{2}H^{2}.$$
If $2\leq n\leq 4$ or $n\geq 5$ and $n^{2}H^{2}<\frac{16(n-1)}{n(n-4)}$,
we have
$$\dfrac{1}{2}\biggl{(}1-\sqrt{\dfrac{(n-2)^{2}H^{2}}{4(n-1)+n^{2}H^{2}}}\biggl{%
)}\geq\dfrac{1}{2}-\dfrac{1}{n}.$$
By taking
$$\alpha(1-\beta)\to\dfrac{1}{2}\biggl{(}1-\sqrt{\dfrac{(n-2)^{2}H^{2}}{4(n-1)+n%
^{2}H^{2}}}\biggl{)},$$
we obtain
$$\lambda^{J_{m}}_{1}\leq-n(1+H^{2})-\dfrac{n}{4(n-1)}(\sqrt{4(n-1)+n^{2}H^{2}}-%
(n-2)|H|)^{2}.$$
If the equality holds and $H=0$,
we have $\lambda_{1}^{J_{m}}=-2n$. Hence
$\varphi:M\to S^{n+1}(1)$ is isometric to
a Clifford torus
$S^{n-k}(\sqrt{\frac{n-k}{n}})\times S^{k}(\sqrt{\frac{k}{n}})$,
for $k=1,2,\cdots,n-1$.
If $H\neq 0$ and the equality holds,
we know that $h_{ijk}=0$, for any $i,j,k=1,2,\dots,n$. Hence, we know that
the second fundamental form is parallel and $S$ is constant. Thus, we know that
$\varphi:M\to S^{n+1}(1)$ is isometric to
$S^{1}(r)\times S^{n-1}(\sqrt{1-r^{2}})$ since, from the (3.3), the $n-1$ of the principal curvatures are equal with each other.
From the examples in the section 2, we know that $r$ satisfies
$$\begin{cases}r^{2}>\dfrac{1}{n}&\ \text{for}\ 2\leq n\leq 4,\\
\dfrac{1}{n}<r^{2}<\dfrac{n}{(n-2)^{2}},&\ \text{for}\ n\geq 5\ \text{and}\ n^%
{2}H^{2}<\dfrac{16(n-1)}{n(n-4)}.\end{cases}$$
If $n\geq 5$ and $n^{2}H^{2}\geq\frac{16(n-1)}{n(n-4)}$, we take
$$\alpha(1-\beta)=\dfrac{1}{2}-\dfrac{1}{n}.$$
Thus,
$$\lambda^{J_{m}}_{1}\leq-2(n-1)(1+H^{2})+\dfrac{(n-2)^{4}}{8(n-1)}H^{2}.$$
If the equality holds,
we know
$$(1-2\alpha(1-\beta))\sqrt{B}=\dfrac{\alpha(1-\beta)(n-2)}{\sqrt{n(n-1)}}nH.$$
Thus, we have
(3.10)
$$S=B+nH^{2}=nH^{2}+\dfrac{(n-2)^{4}}{16n(n-1)}n^{2}H^{2}.$$
because of
$$\alpha(1-\beta)=\dfrac{1}{2}-\dfrac{1}{n}.$$
Since $S$ is constant, the first eigenvalue $\lambda_{1}^{J_{m}}$ of the Jacobi operator is given
by
$$\lambda_{1}^{J_{m}}=-S-n=-2(n-1)(1+H^{2})+\dfrac{(n-2)^{4}}{8(n-1)}H^{2}.$$
Hence, we obtain
(3.11)
$$S=n-2+2(n-1)H^{2}-\dfrac{(n-2)^{4}}{8(n-1)}H^{2}.$$
From (3.10) and (3.11), we get
$$n-2=(2-n)H^{2}+\dfrac{(n-2)^{4}(n+2)}{16(n-1)}H^{2},$$
$$1=\dfrac{n(n-4)}{16(n-1)}n^{2}H^{2},$$
that is,
$$n^{2}H^{2}=\dfrac{16(n-1)}{n(n-4)}.$$
Since, from the (3.3), the $n-1$ of the principal curvatures are equal with each other,
From the examples in the section 2,
we know that $\varphi:M\to S^{n+1}(1)$ is isometric to
$S^{1}(\frac{\sqrt{n}}{n-2})\times S^{n-1}(\frac{\sqrt{(n-1)(n-4)}}{n-2})$.
It completes the proof of theorem 1.1.
$\square$
References
[1]
Alencar, H. and do Carmo, M. hypersurfaces with constant mean curvature,
Proc. Amer. Math. Soc. , 120(1994), 11223-1229.
[2]
Alías, L. J., Barros, A. & Brasil, A. Jr, A spectral characterization of
$H(r)$-torus by the first stability eigenvalue, Proc. Amer. Math. Soc., 133(2005),
875-884.
[3]
Barbosa, J. L., do Carmo, M. & Eschenburg, J., Stability of hypersurfaces
with constant mean curvature in Riemannian Manifolds, Math. Z., 197(1988), 123-138.
[4]
Cheng, Q. -M., The rigidity of Clifford torus
$S^{1}(\sqrt{\frac{1}{n}})\times S^{n-1}(\sqrt{\frac{n-1}{n}})$,
Comment. Math. Helv., 71(1996), 60-69.
[5]
Cheng, Q. -M., Hypersurfaces in a unit sphere $S^{n+1}(1)$
with constant scalar curvature,
J. London Math. Soc., 64(2001), 755-768.
[6]
Cheng, Q. -M. & Nakagawa, H.,
Totally umbilical hypersurfaces, Hiroshima Math. J., 20(1990),
1-10.
[7]
Li, H., Hypersurfaces with constant scalar curvature in space forms, Math. Ann.,
305(1996), 665-672.
[8]
Perdomo, O., First stability eigenvalue characterization of Clifford hypersurfaces,
Proc. Amer. Math. Soc., 130(2002), 3379-3384
[9]
Simons, J., Minimal varieties in Riemannian manifolds, Ann. of Math., 88(1968),
62-105.
[10]
Wu, C., New characterizations of the Clifford tori and the Veronese surface,
Arch. Math. (Basel), 61(1993), 277-284 |
Multiple solutions to logarithmic Schrödinger
equations
with periodic potential
Marco Squassina
Dipartimento di Informatica
Università degli Studi di Verona
Strada Le Grazie 15, 37134 Verona, Italy
marco.squassina@univr.it
and
Andrzej Szulkin
Department of Mathematics
Stockholm University
106 91 Stockholm, Sweden
andrzejs@math.su.se
Abstract.
We study a class of logarithmic Schrödinger equations with periodic potential
which come from physically relevant situations and obtain the existence of infinitely many geometrically distinct solutions.
Key words and phrases:Logarithmic Schrödinger equations, Logarithmic Sobolev inequality,
multiplicity of solutions, nonsmooth critical point theory
2000 Mathematics Subject Classification: 35Q51, 35Q40, 35J20
The first author was supported by the MIUR project:
“Variational and Topological Methods in the Study of Nonlinear Phenomena”.
The work was partially carried out during a stay of M. Squassina in Stockholm.
He would like to express his gratitude to the Department of Mathematics of Stockholm
University for the warm hospitality.
1. Introduction and results
We consider the equation
(1.1)
$$-\Delta u+V(x)u=Q(x)u\log u^{2}\quad\,\,\,\text{in $\mathbb{R}^{N}$},$$
where the external potential $V$ and the term $Q$ are $1$-periodic functions of the variables
$x_{1},\ldots,x_{N}$, $Q\in C^{1}(\mathbb{R}^{N})$, $\min_{\mathbb{R}^{N}}Q>0$ and $\min_{\mathbb{R}^{N}}(V+Q)>0$.
The problem is formally associated with the energy functional $J:H^{1}(\mathbb{R}^{N})\to\mathbb{R}\cup\{+\infty\}$ defined by
$$J(u)=\frac{1}{2}\int_{\mathbb{R}^{N}}\left(|\nabla u|^{2}+(V(x)+Q(x))u^{2}%
\right)dx-\frac{1}{2}\int_{\mathbb{R}^{N}}Q(x)u^{2}\log u^{2}\,dx.$$
Problem (1.1) admits applications related to quantum mechanics,
quantum optics, nuclear physics, transport and diffusion
phenomena, open quantum systems,
effective quantum gravity, theory of
superfluidity and Bose-Einstein condensation
(see e.g. [22] and the references therein).
We stress that, specifically, periodic potentials $V$ can play a significant rôle in
crystals and in artificial crystals formed by light beams.
Although the logarithmic Schrödinger equation has been ruled out as a fundamental quantum wave equation by
very accurate experiments on neutron diffraction, it is currently under discussion if this equation can be adopted
as a simplified model for some physical phenomena.
We refer the reader to [7, 8, 9]
for existence and uniqueness of solutions of the associated Cauchy problem
in a suitable functional framework and to a study of orbital stability, with respect to radial perturbations, of the ground state solution (see [3, 4, 5]).
In light of a simple modification (see formula (2.3) in Section 2)
of the standard logarithmic Sobolev inequality [15]
(1.2)
$$\int_{\mathbb{R}^{N}}u^{2}\log u^{2}\leq\frac{a^{2}}{\pi}\|\nabla u\|_{2}^{2}+%
(\log\|u\|_{2}^{2}-N(1+\log a))\|u\|_{2}^{2},\quad\hbox{for }u\in H^{1}(%
\mathbb{R}^{N})\hbox{ and }a>0,$$
it is easy to see that $J(u)>-\infty$ for all $u\in H^{1}(\mathbb{R}^{N})$,
but there exist $u$ in $H^{1}(\mathbb{R}^{N})$ with
$\int_{\mathbb{R}^{N}}u^{2}\log u^{2}dx=-\infty$.
Thus, in general, $J$ fails to be finite and
$C^{1}$ smooth on $H^{1}(\mathbb{R}^{N})$.
Due to this loss of smoothness, in order to study existence
of solutions, to the best of our knowledge, at least three approaches
were used so far in the literature.
On the one hand, in [7], the idea is
to work in a suitable Banach space $W$ endowed with a Luxemburg type norm
in order to make the functional
$J:W\to\mathbb{R}$ well defined and $C^{1}$ smooth. On the other hand,
in [14] the authors
penalize the nonlinearity around the origin and try to obtain
a priori estimates to get a nontrivial solution
at the limit.
However, the drawback of these indirect approaches is that
the Palais-Smale condition cannot be obtained, due to a loss
of coercivity of the functional $J$,
and, in general, no multiplicity result can be obtained
by the Lusternik-Schnirelmann category theory.
Recently, in [11], in the case of constant potentials $V$ and $Q$,
the existence of infinitely many weak solutions was obtained
by considering the functional $J$ on $H^{1}_{\rm rad}(\mathbb{R}^{N})$ as merely
lower semicontinuous and by applying the nonsmooth critical point theory of
[12], originally formulated to tackle semilinear elliptic equations with one-sided
growth conditions, and based upon the general theory developed in [10, 6]. The restriction to the space of radially symmetric
functions in $H^{1}(\mathbb{R}^{N})$ is related to having the Palais-Smale condition (in a suitable sense)
satisfied at an arbitrary energy level.
In this paper, we shall work in the unrestricted space $H^{1}(\mathbb{R}^{N})$ and we exploit the
fact that the functional $J$, although being nonsmooth, can be decomposed into the
sum of a $C^{1}$ functional and a convex lower semicontinuous functional. If $u$ is a solution
to (1.1), so are the elements of its orbit under the action of $\mathbb{Z}^{N},$
${\mathcal{O}}(u):=\{u(\cdot-k):k\in\mathbb{Z}^{N}\},$ and two solutions are said to be geometrically distinct if
${\mathcal{O}}(u_{1})\cap{\mathcal{O}}(u_{2})=\emptyset.$
By adapting some arguments in [19] and using tools from [18], we prove the following result.
Theorem 1.1.
Equation (1.1) has infinitely many geometrically distinct solutions.
Furthermore, setting:
(1.3)
$${\mathcal{N}}:=\Big{\{}u\in H^{1}(\mathbb{R}^{N})\setminus\{0\}:\int_{\mathbb{%
R}^{N}}\left(|\nabla u|^{2}+V(x)u^{2}\right)dx=\int_{\mathbb{R}^{N}}Q(x)u^{2}%
\log u^{2}\,dx\Big{\}},$$
we have the following existence result for ground state solutions to (1.1).
Theorem 1.2.
The infimum $\inf_{{\mathcal{N}}}J>0$ is attained at a solution $u$ to equation (1.1). Moreover, $u(x)>0$ for all $x\in\mathbb{R}^{N}$ or $u(x)<0$ for all $x\in\mathbb{R}^{N}$.
In the case of constant potentials, the ground state solution is known explicitly
(it is called the Gausson in the physical literature [3, 4, 5]) and, as proved in [11], it is nondegenerate, that is to say the dimension of the
nullspace of the linearized operator is $N$, i.e. smallest possible.
In what follows by a solution to (1.1) we shall always mean a function $u\in H^{1}(\mathbb{R}^{N})$ such that $u^{2}\log u^{2}\in L^{1}(\mathbb{R}^{N})$ and
$$\int_{\mathbb{R}^{N}}(\nabla u\cdot\nabla v+V(x)uv)\,dx=\int_{\mathbb{R}^{N}}Q%
(x)uv\log u^{2}\,dx,\quad\text{for all }v\in C_{0}^{\infty}(\mathbb{R}^{N}).$$
Note that since $|u\log u^{2}|\leq C(1+|u|^{q})$, where $q>1$, we may use local estimates and standard bootstrap
to assert that $u$ is, in fact, a classical solution (cf. [17, Appendix B]).
Notation. $C,C_{1},C_{2}$ etc. will denote positive constants whose exact values are inessential. $\langle.\,,.\rangle$
is the duality pairing between $E^{\prime}$ and $E$, where $E$ is a Hilbert (more generally, Banach) space and $E^{\prime}$ is its dual. $\|\cdot\|_{p}$ is
the norm of the space $L^{p}(\mathbb{R}^{N})$. $2^{*}:=2N/(N-2)$ if $N\geq 3$ and $2^{*}:=\infty$ if $N=1$ or $2$.
$B_{R}(x)$ denotes the open ball centered at $x$ and having radius $R$ and $S_{R}(x):=\partial B_{R}(x)$.
For a functional $J$, we set $J^{b}:=\{u\in E:J(u)\leq b\}$, $J_{a}:=\{u\in E:J(u)\geq a\}$ as well as $J_{a}^{b}:=J_{a}\cap J^{b}$.
Acknowledgements. The first author would like to thank Prof. Iwo Białynicki-Birula
from the Center for Theoretical Physics PAN, Warsaw, for useful comments about
the relevance of equation (1.1) from the physical point of view.
2. Proof of Theorem 1.1
In order to prove Theorem 1.1, we first need to state some preliminary results.
2.1. Preliminary results
We shall denote by $E$ the Hilbert space of functions $u\in H^{1}(\mathbb{R}^{N})$ and we endow it with the norm
$$\|u\|:=\left(\int_{\mathbb{R}^{N}}\left(|\nabla u|^{2}+(V(x)+Q(x))u^{2}\right)%
dx\right)^{1/2}\!\!\!\!\!\!\!.$$
Furthermore, let us set:
$$\displaystyle F_{1}(s):=\left\{\begin{array}[]{ll}-\frac{1}{2}s^{2}\log s^{2},%
&|s|<\delta,\\
\omit\span\@@LTX@noalign{\vskip 3.0pt}\omit\cr-\frac{1}{2}s^{2}(\log\delta^{2}%
+3)+2\delta|s|-\frac{1}{2}\delta^{2},&|s|>\delta,\end{array}\right.$$
$$\displaystyle F_{2}(s):=\left\{\begin{array}[]{ll}0,&|s|<\delta,\\
\omit\span\@@LTX@noalign{\vskip 3.0pt}\omit\cr\frac{1}{2}s^{2}\log(s^{2}/%
\delta^{2})+2\delta|s|-\frac{3}{2}s^{2}-\frac{1}{2}\delta^{2},&|s|>\delta.\end%
{array}\right.$$
Then $F_{2}(s)-F_{1}(s)=\frac{1}{2}s^{2}\log s^{2}$ and $F_{1}$ is convex, provided $\delta>0$ is sufficiently small
which we assume from now on. Moreover, $F_{1},F_{2}\in C^{1}(\mathbb{R})$. Re-write the functional $J$ as
$J(u)=\Phi(u)+\Psi(u)$, $u\in E$, where we have set:
$$\displaystyle\Phi(u)$$
$$\displaystyle:=\frac{1}{2}\|u\|^{2}-\int_{\mathbb{R}^{N}}Q(x)F_{2}(u)\,dx,$$
$$\displaystyle\Psi(u)$$
$$\displaystyle:=\int_{\mathbb{R}^{N}}Q(x)F_{1}(u)\,dx.$$
Choosing $p\in(2,2^{*})$, we have $|F_{2}^{\prime}(s)|\leq C|s|^{p-1}$ for some $C>0$ and all $s\in\mathbb{R}$,
and hence it follows that $\Phi\in C^{1}(E,\mathbb{R})$ [21, Lemma 3.10]. Note that $\Psi$ is convex, $\Psi\geq 0$ and $\Psi(u)=+\infty$ for certain $u\in E$. Moreover, it is easy to see by Fatou’s lemma that $\Psi$ (and therefore also $J$) is lower semicontinuous (cf. [11, Proposition 2.9]).
Remark 2.1.
Theorems 1.1 and 1.2 remain valid for any $F_{1},F_{2}$ such that $F_{1},F_{2}\in C^{1}(\mathbb{R}^{N})$, $F_{1}$ is convex and nonnegative, $F_{1}(0)=0$, $|F_{1}^{\prime}(s)|,|F_{2}^{\prime}(s)|\leq C(1+|s|^{p-1})$ for some $C>0$ and $p\in(2,2^{*})$, and $F_{2}^{\prime}(s)/s\to 0$ as $s\to 0$. Some additional conditions may also be needed in order to ensure that the corresponding functional $J$ satisfies the conclusion of Lemma 2.9. The proof is the same except that small modifications are necessary at some places because $|F_{2}^{\prime}(s)|\leq C|s|^{p-1}$ may not hold. Instead, for each $\varepsilon>0$ there exists $C_{\varepsilon}>0$ such that $|F_{2}^{\prime}(s)|\leq\varepsilon|s|+C_{\varepsilon}|s|^{p-1}$.
Lemma 2.2.
If $\Omega\subset\mathbb{R}^{N}$ is a bounded domain, then $\Psi$ (and hence $J$) is of class $C^{1}$ in $H^{1}(\Omega)$.
Proof.
Since $|F_{1}^{\prime}(s)|\leq C(1+|s|^{p-1})$, the conclusion follows from [21, Lemma 2.16]. In [21] the result is stated in $H^{1}_{0}(\Omega)$ but the argument remains valid in $H^{1}(\Omega)$.
∎
We shall need the following definitions, essentially taken from [18]:
Definitions 2.3.
$(i)$ The set
$$\partial J(u):=\{w\in E^{\prime}:\langle\Phi^{\prime}(u),v-u\rangle+\Psi(v)-%
\Psi(u)\geq\langle w,v-u\rangle,\,\,\,\text{for all $v\in E$}$$
is called the subdifferential of $J$ at $u$.
$(ii)$ $u\in E$ is a critical point of $J$ if $J(u)<+\infty$ and $0\in\partial J(u)$, i.e.
$$\langle\Phi^{\prime}(u),v-u\rangle+\Psi(v)-\Psi(u)\geq 0,\quad\,\,\text{for %
all $v\in E$}.$$
$(iii)$ $(u_{n})$ is a Palais-Smale sequence if $(J(u_{n}))$ is bounded and there exist $\varepsilon_{n}\to 0^{+}$ such that
$$\langle\Phi^{\prime}(u_{n}),v-u_{n}\rangle+\Psi(v)-\Psi(u_{n})\geq-\varepsilon%
_{n}\|v-u_{n}\|,\quad\,\,\text{for all $v\in E$}.$$
$(iv)$ The set $D(J):=\{u\in E:J(u)<+\infty\}$ is called the effective domain of $J$.
Lemma 2.4.
If $u\in D(J)$, then $\partial J(u)\neq\emptyset$, i.e. there exists $w\in E^{\prime}$ such that
$$\langle\Phi^{\prime}(u),v-u\rangle+\Psi(v)-\Psi(u)\geq\langle w,v-u\rangle,%
\quad\,\,\text{for all $v\in E$}.$$
Moreover, this $w$ is unique and satisfies
$$\langle\Phi^{\prime}(u),z\rangle+\int_{\mathbb{R}^{N}}Q(x)F_{1}^{\prime}(u)z\,%
dx=\langle w,z\rangle\quad\text{for all $z\in E$ such that $F_{1}^{\prime}(u)z%
\in L^{1}(\mathbb{R}^{N})$}.$$
Proof.
Assume by contradiction that for each $w\in E^{\prime}$ there exists $v\in E$ such that
$$\langle\Phi^{\prime}(u),v-u\rangle+\Psi(v)-\Psi(u)-\langle w,v-u\rangle<0.$$
Let $u_{n}\in C_{0}^{\infty}(\mathbb{R}^{N})$, $u_{n}\to u$ in $\|\cdot\|$ as $n\to\infty$. Then, by the lower semicontinuity of $\Psi$,
$$\displaystyle\limsup_{n\to\infty}\left(\langle\Phi^{\prime}(u_{n}),v-u_{n}%
\rangle+\Psi(v)-\Psi(u_{n})-\langle w,v-u_{n}\rangle\right)$$
$$\displaystyle\leq\langle\Phi^{\prime}(u),v-u\rangle+\Psi(v)-\Psi(u)-\langle w,%
v-u\rangle<0.$$
So $\partial J(u_{n})=\emptyset$ for almost all $n\geq 1$ which is impossible because $u_{n}\in C_{0}^{\infty}(\mathbb{R}^{N})$ and hence, using Lemma 2.2 and convexity of $\Psi$,
$$\langle\Phi^{\prime}(u_{n}),v-u_{n}\rangle+\Psi(v)-\Psi(u_{n})\geq\langle\Phi^%
{\prime}(u_{n})+\Psi^{\prime}(u_{n}),v-u_{n}\rangle\quad\text{for all }v.$$
Let now $v=u+tz$, where $z\in C_{0}^{\infty}(\mathbb{R}^{N})$ and let $w\in\partial J(u)$. Then
$$\langle\Phi^{\prime}(u),z\rangle+\int_{\mathbb{R}^{N}}Q(x)\,\frac{F_{1}(u+tz)-%
F_{1}(u)}{t}\,dx\geq\langle w,z\rangle.$$
Since the integrand is 0 for $x\not\in\text{supp\,}z$, we can pass to the limit as $t\to 0$ (cf. Lemma 2.2) and we obtain
$$\langle\Phi^{\prime}(u),z\rangle+\int_{\mathbb{R}^{N}}Q(x)F_{1}^{\prime}(u)z\,%
dx\geq\langle w,z\rangle.$$
Since this also holds for $-z$,
$$\langle\Phi^{\prime}(u),z\rangle+\int_{\mathbb{R}^{N}}Q(x)F_{1}^{\prime}(u)z\,%
dx=\langle w,z\rangle,\quad\text{for all $z\in C_{0}^{\infty}(\mathbb{R}^{N})$}.$$
By density of $C_{0}^{\infty}(\mathbb{R}^{N})$ in $E$, $w$ is unique and the equality above holds for all $z\in E$ such that $F_{1}^{\prime}(u)z\in L^{1}(\mathbb{R}^{N})$.
∎
Definition 2.5.
Let $u\in D(J)$. Then the unique element $w\in E^{\prime}$ of $\partial J(u)$ introduced in Lemma 2.4 will
be denoted by $J^{\prime}(u)$.
An immediate consequence of Lemma 2.4 is the following
Corollary 2.6.
Suppose $(J(u_{n}))$ is bounded. Then $(u_{n})\subset E$ is a Palais-Smale sequence if and only if $J^{\prime}(u_{n})\to 0$ in $E^{\prime}$
as $n\to\infty$, or equivalently,
$$\lim_{n\to\infty}\sup\big{\{}\langle J^{\prime}(u_{n}),v\rangle:v\in C_{0}^{%
\infty}(\mathbb{R}^{N}),\|v\|=1\big{\}}=0.$$
It follows from Lemma 2.4 that if $u\in D(J)$, then
$$\langle J^{\prime}(u),v\rangle=\langle\Phi^{\prime}(u),v\rangle+\int_{\mathbb{%
R}^{N}}Q(x)F_{1}^{\prime}(u)v\,dx\quad\text{for all $v\in E$ such that $F_{1}^%
{\prime}(u)v\in L^{1}(\mathbb{R}^{N})$}.$$
Next we construct a vector field of pseudo-gradient type. Denote the set of critical points of the functional $J$ by $K$.
Lemma 2.7.
There exist a locally finite countable covering $(W_{j})$ of $D(J)\setminus K$, a set of points $(u_{j})\subset D(J)\setminus K$ and a locally Lipschitz continuous vector field $H:D(J)\setminus K\to E$ with the following properties:
(i) The diameter of $W_{j}$ and the distance from $u_{j}$ to $W_{j}$ tend to 0 as $j\to\infty$.
(ii) $\|H(u)\|\leq 1$ and $\langle J^{\prime}(u),H(u)\rangle>z(u)$, where $z(u):=\min\frac{1}{2}\|J^{\prime}(u_{j})\|$ for all $j$ such that $u\in W_{j}$.
(iii) $H$ has locally compact support, i.e. for each $u_{0}\in D(J)\setminus K$ there
exists a neighbourhood $U_{0}$ of $u_{0}$ in $D(J)\setminus K$ and $R>0$ such that $\text{supp\,}H(u)\subset B_{R}(0)$ for all $u\in U_{0}$.
(iv) $J(u)>J(u_{j})-\gamma_{j}$ for all $j$ such that $u\in W_{j}$, where $\gamma_{j}>0$ and $\gamma_{j}\to 0$ as $j\to\infty$.
(v) $H$ is odd in $u$.
Remark 2.8.
The special properties of the covering $(W_{j})$ and the field $H$ in Lemma 2.7
will be essential in the proofs of Lemmas 2.13 and 2.14.
Proof.
Since $E$ is separable, there exists a countable dense set of points $(\widetilde{u}_{k})\subset D(J)\setminus K$. For each $k$ we can choose $\widetilde{v}_{k}\in C_{0}^{\infty}(\mathbb{R}^{N})$, $\|\widetilde{v}_{k}\|=1$, such that $\langle J^{\prime}(\widetilde{u}_{k}),\widetilde{v}_{k}\rangle>\frac{1}{2}\|J^%
{\prime}(\widetilde{u}_{k})\|$.
Since $\widetilde{v}_{k}$ has compact support, $u\mapsto\langle J^{\prime}(u),\widetilde{v}_{k}\rangle$ is continuous according to Lemma 2.2 and hence
(2.1)
$$\langle J^{\prime}(u),\widetilde{v}_{k}\rangle>\frac{1}{2}\|J^{\prime}(%
\widetilde{u}_{k})\|.$$
for all $u$ in some neighbourhood $W(\widetilde{u}_{k})$ of $\widetilde{u}_{k}$ in $D(J)\setminus K$. Moreover, we may assume that the diameter of $W(\widetilde{u}_{k})$ tends to 0 as $k\to\infty$ and by the lower semicontinuity of $J$, $W(\widetilde{u}_{k})$ may be chosen so that $J(u)>J(\widetilde{u}_{k})-1/k$ in $W(\widetilde{u}_{k})$. Clearly, $(W(\widetilde{u}_{k}))$ is a covering of $D(J)\setminus K$. Since $E$ is metric and hence paracompact, we can find a locally finite refinement $(W_{j})$ of $(W(\widetilde{u}_{k}))$ and for each $W_{j}$ we choose $u_{j}:=\widetilde{u}_{k_{j}}$ for some $k_{j}$ such that $W_{j}\subset W(\widetilde{u}_{k_{j}})$ (note that $u_{j}$ may not be in $W_{j}$). So $(W_{j})$ and $(u_{j})$ satisfy (i) and by (2.1), the inequality
$$\langle J^{\prime}(u),v_{j}\rangle>\frac{1}{2}\|J^{\prime}(u_{j})\|$$
holds for $u_{j}=\widetilde{u}_{k_{j}}$, $v_{j}=\widetilde{v}_{k_{j}}$ and all $u\in W_{j}\subset W(\widetilde{u}_{k_{j}})$.
Let $\widetilde{\rho}_{j}(u):=\text{dist}(u,E\setminus W_{j})$,
$$\rho_{j}(u):=\widetilde{\rho}_{j}(u)\Big{/}\sum_{j=1}^{\infty}\widetilde{\rho}%
_{j}(u)\quad\text{and}\quad H(u):=\sum_{j=1}^{\infty}\rho_{j}(u)v_{j}.$$
It is easy to see that the properties (ii)-(iv) are satisfied (in (iv) we take $\gamma_{j}=1/k_{j}$). Moreover, as $J$ is even, $H$ may be constructed so that $H(-u)=-H(u)$ (e.g. by taking $\pm W_{j}$, $\pm u_{j}$ etc.). Hence also (v) holds.
∎
Lemma 2.9.
If $(u_{n})$ is a sequence such that $(J(u_{n}))$ is bounded above and $J^{\prime}(u_{n})\to 0$, then $(u_{n})$ is bounded. Moreover, since $\Psi\geq 0$, $(J(u_{n}))$ is also bounded below and hence it is a Palais-Smale sequence.
Proof.
Let $(u_{n})\subset E$ be a sequence with $(J(u_{n}))$ bounded above and $J^{\prime}(u_{n})\to 0$ as $n\to\infty$.
Then, choosing $v=u_{n}$ as test function, we end up with
(2.2)
$$C_{1}\|u_{n}\|_{2}^{2}\leq\int_{\mathbb{R}^{N}}Q(x)u^{2}_{n}\,dx=2J(u_{n})-%
\langle J^{\prime}(u_{n}),u_{n}\rangle\leq C+o(1)\|u_{n}\|,\quad\text{as $n\to%
\infty$.}$$
Replacing $u$ by $\sqrt{Q}u$ in (1.2) yields
$$\int_{\mathbb{R}^{N}}Qu^{2}\log(Qu^{2})\,dx\leq\frac{2a^{2}}{\pi}(\|\sqrt{Q}%
\nabla u\|_{2}^{2}+\|u\nabla\sqrt{Q}\|_{2}^{2})+(\log\|\sqrt{Q}u\|_{2}^{2}-N(1%
+\log a))\|\sqrt{Q}u\|_{2}^{2}.$$
This gives, taking $a>0$ small enough,
(2.3)
$$\int_{\mathbb{R}^{N}}Q(x)u^{2}\log u^{2}\,dx\leq\frac{1}{2}\|\nabla u\|_{2}^{2%
}+C_{2}(\log\|u\|_{2}^{2}+1)\|u\|_{2}^{2}.$$
So using (2.2), we obtain
$$\displaystyle C\geq 2J(u_{n})$$
$$\displaystyle=\|u_{n}\|^{2}-\int_{\mathbb{R}^{N}}Q(x)u_{n}^{2}\log u_{n}^{2}\,%
dx\geq\frac{1}{2}\|u_{n}\|^{2}-C_{2}(\log\|u_{n}\|_{2}^{2}+1)\|u_{n}\|_{2}^{2}$$
$$\displaystyle\geq\frac{1}{2}\|u_{n}\|^{2}-C_{3}(1+\|u_{n}\|^{1+\delta}),\,%
\quad\text{as }n\to\infty,$$
where we can take $\delta<1$.
Hence $(u_{n})$ is bounded, proving the first assertion. Then the second
assertion immediately follows.
∎
Assume throughout the rest of this section that $J$ has only finitely many critical orbits and choose a finite set $\mathcal{F}\subset K$ such that $\mathcal{F}=-\mathcal{F}$ and each critical orbit has a unique representative in $K$.
Lemma 2.10.
$\kappa:=\inf\{\|u-v\|:u,v\in K,\ u\neq v\}>0$.
Proof.
This follows by a straightforward modification of the proof of [19, Lemma 2.13].
∎
In the next three lemmas we adapt some arguments from [19].
Lemma 2.11.
Let $(u_{n}^{1}),(u_{n}^{2})\subset E$ be two Palais-Smale sequences,
Then either $\|u_{n}^{1}-u_{n}^{2}\|\to 0$ as $n\to\infty$ or $\limsup_{n\to\infty}\|u_{n}^{1}-u_{n}^{2}\|\geq\kappa$.
Proof.
By Lemma 2.9, it follows that $(u_{n}^{1})$ and $(u_{n}^{2})$ are bounded in $E$. Choose $p\in(2,2^{*})$.
Then $|F_{2}^{\prime}(s)|\leq C|s|^{p-1}$ for some $C>0$ and all $s\in\mathbb{R}$.
Suppose first $\|u_{n}^{1}-u_{n}^{2}\|_{p}\to 0$ as $n\to\infty$. Then, by Definition 2.3(iii), we obtain
$$\langle\Phi^{\prime}(u_{n}^{1}),u_{n}^{2}-u_{n}^{1}\rangle+\Psi(u_{n}^{2})-%
\Psi(u_{n}^{1})\geq-\varepsilon_{n}\|u_{n}^{2}-u_{n}^{1}\|,$$
and a similar inequality holds with the roles of $u_{n}^{1}$ and $u_{n}^{2}$ interchanged. Hence
$$\displaystyle\|u_{n}^{1}-u_{n}^{2}\|^{2}$$
$$\displaystyle=\langle\Phi^{\prime}(u_{n}^{1}),u_{n}^{1}-u_{n}^{2}\rangle-%
\langle\Phi^{\prime}(u_{n}^{2}),u_{n}^{1}-u_{n}^{2}\rangle$$
$$\displaystyle\quad+\int_{\mathbb{R}^{N}}Q(x)(F_{2}^{\prime}(u_{n}^{1})-F_{2}^{%
\prime}(u_{n}^{2}))(u_{n}^{1}-u_{n}^{2})\,dx$$
$$\displaystyle\leq 2\varepsilon_{n}\|u_{n}^{1}-u_{n}^{2}\|+C\int_{\mathbb{R}^{N%
}}Q(x)(|u_{n}^{1}|^{p-1}+|u_{n}^{2}|^{p-1})|u_{n}^{1}-u_{n}^{2}|\,dx$$
$$\displaystyle\leq 2\varepsilon_{n}\|u_{n}^{1}-u_{n}^{2}\|+D\|u_{n}^{1}-u_{n}^{%
2}\|_{p}.$$
So $\|u_{n}^{1}-u_{n}^{2}\|\to 0$.
Suppose now $\|u_{n}^{1}-u_{n}^{2}\|_{p}\not\to 0$. By Lions’ lemma [16, Lemma I.1], [21, Lemma 1.21], we can find $\varepsilon>0$ and $(y_{n})\subset\mathbb{R}^{N}$ with
$$\int_{B_{1}(y_{n})}(u_{n}^{1}-u_{n}^{2})^{2}\,dx\geq\varepsilon,$$
after passing to a subsequence. Since $J$ is invariant under
translations $u\mapsto u(\cdot-k)$, $k\in\mathbb{Z}^{N}$, we may assume the sequence $(y_{n})$ is bounded. Hence, passing to a subsequence once more, $u_{n}^{1}\rightharpoonup u^{1}$, $u_{n}^{2}\rightharpoonup u^{2}$ and $u^{1}\neq u^{2}$.
The functional $\Psi$ is lower semicontinuous and hence weakly lower semicontinuous (by convexity). So $\Psi(u^{1})<\infty$ and therefore $u^{1}\in D(J)$. Moreover, since $\langle J^{\prime}(u_{n}^{1}),v\rangle\to 0$ for all $v\in C_{0}^{\infty}(\mathbb{R}^{N})$, it is easy to see that $u^{1}\in K$. Similarly, $u^{2}\in K$. Hence
$$\limsup_{n\to\infty}\|u_{n}^{1}-u_{n}^{2}\|\geq\|u^{1}-u^{2}\|\geq\kappa,$$
concluding the proof.
∎
Remark 2.12.
For the purpose of the next section we note that if there are finitely many critical orbits below a certain level $d>0$, then the conclusions of Lemmas 2.10, 2.11 as well as of Lemmas 2.13 and 2.14 below remain valid on $J^{d}$. The proofs go through unchanged except that we need to show that $u^{1},u^{2}\in J^{d}$ in the proof of Lemma 2.11. Since $(u_{n}^{1})$ is bounded and $J^{\prime}(u^{1})=0$, we have
$$\displaystyle d$$
$$\displaystyle\geq J(u_{n}^{1})=J(u_{n}^{1})-\frac{1}{2}\langle J^{\prime}(u_{n%
}^{1}),u_{n}^{1}\rangle+o(1)=\frac{1}{2}\int_{\mathbb{R}^{N}}Q(x)(u_{n}^{1})^{%
2}dx+o(1)$$
$$\displaystyle\geq\frac{1}{2}\int_{\mathbb{R}^{N}}Q(x)(u^{1})^{2}dx+o(1)=J(u^{1%
})-\frac{1}{2}\langle J^{\prime}(u^{1}),u^{1}\rangle+o(1)=J(u^{1})+o(1).$$
So $u^{1}\in J^{d}$ and similarly, $u^{2}\in J^{d}$.
Consider now the flow $\eta$ given by
$$\left\{\begin{array}[]{l}\frac{d}{dt}\eta(t,u)=-H(\eta(t,u)),\\
\@@LTX@noalign{\vskip 3.0pt}\omit\cr\eta(0,u)=u,\ u\in D(J)\setminus K,\end{%
array}\right.$$
and let $(T^{-}(u),T^{+}(u))$ be the maximal existence time for the trajectory $t\mapsto\eta(t,u)$.
Lemma 2.13.
Let $u\in D(J)\setminus K$. Then either $\lim_{t\to T^{+}(u)}\eta(t,u)$ exists and is a critical point of $J$ or $\lim_{t\to T^{+}(u)}J(\eta(t,u))=-\infty$. In the latter case $T^{+}(u)=+\infty$.
Proof.
Since $\eta(s,u)=u-\int_{0}^{s}H(\eta(\tau,u))\,d\tau$ and $H$ has locally compact support, $\tau\mapsto J(\eta(\tau,u))$ is continuously differentiable. To see this, consider
$$\frac{1}{h}(\Psi(\eta(t+h,u))-\Psi(\eta(t,u)))=\frac{1}{h}\int_{\mathbb{R}^{N}%
}Q(x)(F_{1}(\eta(t+h,u))-F_{1}(\eta(t,u)))\,dx.$$
Since $\eta(s,u(x))=u(x)$ for all $s\in[0,t+h]$ and $|x|$ large enough, we can pass to the limit as $h\to 0$ using Lemma 2.2 and we obtain, by Lemma 2.7,
$$\frac{d}{dt}J(\eta(t,u))=-\langle J^{\prime}(\eta(t,u)),H(\eta(t,u))\rangle%
\leq-z(\eta(t,u))<0.$$
So $t\mapsto J(\eta(t,u))$ is decreasing.
Suppose $T^{+}(u)<\infty$ and let $0\leq s<t<T^{+}(u)$. Then
$$\|\eta(t,u)-\eta(s,u)\|\leq\int_{s}^{t}\|H(\eta(\tau,u))\|\,d\tau\leq t-s.$$
Hence the limit exists and if it is not a critical point, then $\eta(\cdot,u)$ can be continued for $t>T^{+}(u)$.
Suppose $T^{+}(u)=+\infty$ and $J(\eta(t,u))$ is bounded below. It suffices to show that
for each $\varepsilon>0$ there exists $t_{\varepsilon}>0$ such that
$\|\eta(t_{\varepsilon},u)-\eta(t,u)\|<\varepsilon$ whenever $t\geq t_{\varepsilon}$.
Assuming the contrary, we can find $\varepsilon\in(0,\kappa/2)$ and $(t_{n})\subset\mathbb{R}^{+}$ with $t_{n}\to+\infty$ and $\|\eta(t_{n},u)-\eta(t_{n+1},u)\|=\varepsilon$ for all $n\geq 1$. Choose the smallest $t_{n}^{1}\in(t_{n},t_{n+1})$ such that $\|\eta(t_{n},u)-\eta(t_{n}^{1},u)\|=\varepsilon/3$ and let $\kappa_{n}:=\min\{z(\eta(s,u)):s\in[t_{n},t_{n}^{1}]\}$. Then $\kappa_{n}>0$ and
$$\displaystyle\frac{\varepsilon}{3}$$
$$\displaystyle=\|\eta(t_{n}^{1},u)-\eta(t_{n},u)\|\leq\int_{t_{n}}^{t_{n}^{1}}%
\|H(\eta(s,u))\|\,ds\leq t_{n}^{1}-t_{n}$$
$$\displaystyle\leq\frac{1}{\kappa_{n}}\int_{t_{n}}^{t_{n}^{1}}\langle J^{\prime%
}(\eta(s,u)),H(\eta(s,u))\rangle\,ds=\frac{1}{\kappa_{n}}\left(J(\eta(t_{n},u)%
)-J(\eta(t_{n}^{1},u))\right).$$
Since $J(\eta(t_{n},u))-J(\eta(t_{n}^{1},u))\to 0$, it follows that $\kappa_{n}\to 0$. Hence we can find $s_{n}^{1}\in[t_{n},t_{n}^{1}]$ such that $z(\eta(s_{n}^{1},u))\to 0$ as $n\to\infty$. So by Lemma 2.7 there exist $u_{n}^{1}$ (where $u_{n}^{1}=u_{j_{n}}$ for some $j_{n}$) such that $J^{\prime}(u_{n}^{1})\to 0$, $J(u_{n}^{1})\leq J(\eta(s_{n}^{1},u))+\gamma_{n}^{1}$ and $\|u_{n}^{1}-\eta(s_{n}^{1},u)\|\to 0$. Here it is important that the diameter of $W_{j_{n}}$ and the distance from $u_{j_{n}}$ to $W_{j_{n}}$ in Lemma 2.7 tend to 0 and that (iv) of this lemma gives a uniform bound from above for $J(u_{n}^{1})$. Similarly we can first find a largest $t_{n}^{2}\in[t_{n}^{1},t_{n+1}]$ for which $\|\eta(t_{n+1},u)-\eta(t_{n}^{2},u)\|=\varepsilon/3$ and then $s_{n}^{2}\in[t_{n}^{1},t_{n+1}]$ and $u_{n}^{2}$ such that $J^{\prime}(u_{n}^{2})\to 0$, $J(u_{n}^{2})\leq J(\eta(s_{n}^{2},u))+\gamma_{n}^{2}$ and $\|u_{n}^{2}-\eta(s_{n}^{2},u)\|\to 0$. Since $(J(u_{n}^{1}))$, $(J(u_{n}^{2}))$ are bounded above, $(u_{n}^{1})$ and $(u_{n}^{2})$ are Palais-Smale sequences according to Lemma 2.9. Hence
$$\frac{\varepsilon}{3}\leq\limsup_{n\to\infty}\|u_{n}^{1}-u_{n}^{2}\|\leq 2%
\varepsilon<\kappa,$$
a contradiction to Lemma 2.11. This completes the proof.
∎
Let $d>0$ and choose $\varepsilon_{0}>0$ such that $J_{d-2\varepsilon_{0}}^{d+2\varepsilon_{0}}\cap K=K_{d}:=\{u\in K:J(u)=d\}$. Denote
$$U_{\delta}(K_{d}):=\{u\in E:\text{dist}(u,K_{d})<\delta\}.$$
Lemma 2.14.
For each $\delta>0$ there exists $\varepsilon\in(0,\varepsilon_{0})$ such that
$$\lim_{t\to T^{+}(u)}J(\eta(t,u))<d-\varepsilon,\quad\text{ whenever }u\in J^{d%
+\varepsilon}\setminus U_{\delta}(K_{d}).$$
Moreover, $\eta(t,u)\notin U_{\delta/2}(K_{d})$ for any $t\in[0,T^{+}(u))$.
Proof.
Assume without loss of generality that $\delta<\kappa$. Let
$$\tau:=\inf\{z(u):u\in J^{d+2\varepsilon_{0}}_{d-2\varepsilon_{0}}\cap U_{%
\delta}(K_{d})\setminus U_{\delta/2}(K_{d})\}.$$
We show that $\tau>0$. Arguing by contradiction, we find a sequence $w_{n}^{1}\in J^{d+2\varepsilon_{0}}_{d-2\varepsilon_{0}}\cap U_{\delta}(K_{d})%
\setminus U_{\delta/2}(K_{d})$ with $z(w_{n}^{1})\to 0$ and then, using Lemma 2.7, $u_{n}^{1}$ (where $u_{n}^{1}=u_{j_{n}}$ for some $j_{n}$) such that $J^{\prime}(u_{n}^{1})\to 0$, $\|u_{n}^{1}-w_{n}^{1}\|\to 0$ and $J(u_{n}^{1})\leq J(w_{n}^{1})+\gamma_{n}^{1}$ ($\gamma_{n}^{1}\to 0$). Hence $J(u_{n}^{1})$ is bounded above, so $(u_{n}^{1})$ is a Palais-Smale sequence by Lemma 2.9. Using Lemma 2.10 and $\mathbb{Z}^{N}$-invariance of $J$ we may assume $w_{n}^{1}\in U_{\delta}(u_{0})\setminus U_{\delta/2}(u_{0})$ for some $u_{0}\in K_{d}$. Set $u_{n}^{2}:=u_{0}$ for all $n$. This is obviously a Palais-Smale sequence and we have
$$\frac{\delta}{2}\leq\limsup_{n\to\infty}\|u_{n}^{1}-u_{n}^{2}\|\leq\delta<\kappa,$$
a contradiction to Lemma 2.11. So $\tau>0$.
If the conclusion of the lemma is false, there exists $w\in K_{d}$ such that $\eta(t,u)$ will enter $U_{\delta/2}(w)$. Set
$$\displaystyle t_{1}:=\sup\{t\in[0,T^{+}(u)):\eta(t,u)\not\in U_{\delta}(w)\},$$
$$\displaystyle t_{2}:=\inf\{t\in(t_{1},T^{+}(u)):\eta(t,u)\in U_{\delta/2}(w)\}.$$
Then
$$\frac{\delta}{2}\leq\|\eta(t_{2},u)-\eta(t_{1},u)\|\leq\int_{t_{1}}^{t_{2}}\|H%
(\eta(s,u))\|\,ds\leq t_{2}-t_{1}$$
and therefore
$$J(\eta(t_{2},u))-J(\eta(t_{1},u))=-\int_{t_{1}}^{t_{2}}\langle J^{\prime}(\eta%
(s,u)),H(\eta(s,u))\rangle\,ds\leq-\tau(t_{2}-t_{1})\leq-\frac{\tau\delta}{2}.$$
So $J(\eta(t_{2},u))\leq d+\varepsilon-\tau\delta/2<d$ for $\varepsilon$ small. Hence $\eta(t,u)$ cannot enter $U_{\delta/2}(w)$.
∎
Lemma 2.15.
There exist $\rho,b>0$ such that $J(u)\geq 0$ for all $u\in B_{\rho}(0)$ and $J(u)\geq b$ for all $u\in S_{\rho}(0)$.
Proof.
Recalling that $\Psi\geq 0$ and $|F_{2}^{\prime}(s)|\leq C|s|^{p-1}$, we obtain $J(u)\geq\Phi(u)=\frac{1}{2}\|u\|^{2}+o(\|u\|^{2})$. Hence the conclusion.
∎
In the proof of Theorem 1.1 we shall need a variant of Benci’s pseudoindex [1, 2] which we now introduce. Let $\Sigma:=\{A\subset D(J):A=-A\text{ and }A\text{ is compact}\}$ and
$$\displaystyle\mathcal{H}:=\{h:D(J)\to E,\,\,\text{$h$ odd homeomorphism onto $%
h(D(J))$}$$
$$\displaystyle\text{and $J(h(u))\leq J(u)$ for all $u\in D(J)$}\}.$$
Denote Krasnoselskii’s genus of $A\in\Sigma$ by $\gamma(A)$ [17] and set
$$i^{*}(A):=\min_{h\in\mathcal{H}}\gamma(h(A)\cap S_{\rho}(0)),$$
where $\rho$ is as in Lemma 2.15.
Lemma 2.16.
Let $A,B\in\Sigma$.
(i) If $A\subset B$, then $i^{*}(A)\leq i^{*}(B)$.
(ii) $i^{*}(A\cup B)\leq i^{*}(A)+\gamma(B)$.
(iii) If $g\in\mathcal{H}$, then $i^{*}(A)\leq i^{*}(g(A))$.
(iv) Let $E_{k}$ be a $k$-dimensional subspace of $D(J)$. Then $i^{*}(E_{k}\cap\overline{B}_{R}(0))\geq k$ whenever $R$ is large enough.
Proof.
(i) follows immediately from the properties of genus [17].
(ii) For each $h\in\mathcal{H}$,
$$i^{*}(A\cup B)\leq\gamma(h(A\cup B)\cap S_{\rho}(0))=\gamma((h(A)\cup h(B))%
\cap S_{\rho}(0))\leq\gamma(h(A)\cap S_{\rho}(0))+\gamma(B).$$
Taking the minimum over all $h\in\mathcal{H}$ on the right-hand side we obtain the conclusion.
(iii) Since $J(g(u))\leq J(u)$ for all $u\in D(J)$, $h\circ g\in\mathcal{H}$ if $h\in\mathcal{H}$. Hence
$$\min_{h\in\mathcal{H}}\gamma(h(A)\cap S_{\rho}(0))\leq\min_{h\in\mathcal{H}}%
\gamma((h\circ g)(A)\cap S_{\rho}(0)).$$
(iv) It is easy to see that $J(u)\to-\infty$ uniformly for $u\in E_{k}$, $\|u\|\to\infty$. So $J(u)<0$ on $E_{k}\setminus B_{R}(0)$ if $R$ is large enough. Let $D:=E_{k}\cap\overline{B}_{R}(0)$. Suppose $i^{*}(D)<k$, choose $h\in\mathcal{H}$ such that $\gamma(h(D)\cap S_{\rho}(0))<k$ and an odd mapping $f:h(D)\cap S_{\rho}(0)\to\mathbb{R}^{k-1}\setminus\{0\}$. Let $U:=h^{-1}(B_{\rho}(0))\cap E_{k}$. Since $J(h(u))\leq J(u)<0$ when $u\in E_{k}\setminus B_{R}(0)$, $U\subset D$ according to Lemma 2.15 and hence $U$ is an open and bounded neighbourhood of 0 in $E_{k}$. If $u\in\partial U$, then $h(u)\in S_{\rho}(0)$ and therefore $f\circ h:\partial U\to\mathbb{R}^{k-1}\setminus\{0\}$, contradicting the Borsuk-Ulam theorem [17, Proposition II.5.2], [21, Theorem D.17]. So $i^{*}(D)\geq k$.
∎
2.2. Proof of Theorem 1.1 completed
Let
$$d_{k}:=\inf_{i^{*}(A)\geq k}\sup_{u\in A}J(u).$$
Since there exist sets of arbitrarily large pseudoindex $i^{*}$, $d_{k}$ is well defined for all $k\geq 1$ and it follows from Lemma 2.15 that $d_{k}\geq b$.
We shall show that $K_{d_{k}}\neq\emptyset$ and $d_{k}<d_{k+1}$ for all $k$, and this contradicts our assumption that there are only finitely many critical orbits.
Put $d\equiv d_{k}$. By Lemma 2.10, $\gamma(K_{d})=0$ (if $K_{d}=\emptyset$) or 1. Let $U:=U_{\delta}(K_{d})$ where $\delta$ is so small that $\gamma(\overline{U})=\gamma(K_{d})$ and choose $\varepsilon>0$ as in Lemma 2.14. Choose $A\in\Sigma$ such that $i^{*}(A)\geq k$ and $\sup_{u\in A}J(u)\leq d+\varepsilon$.
We need to modify the flow $\eta$. Let $\chi_{1}:E\to[0,1]$ be locally Lipschitz continuous and
such that $\chi_{1}=0$ on $J^{d-2\varepsilon_{0}}$, $\chi_{1}>0$ otherwise. Since $\{u\in E:J(u)>d+2\varepsilon_{0}\}$ is an open set, there exists a locally Lipschitz continuous function $\chi_{2}:E\to[0,1]$ such that $\chi_{2}=1$ on $J^{d+2\varepsilon_{0}}\setminus U_{\delta/2}(K_{d})$ and $\chi_{2}=0$ in a neighbourhood of $K\cap J_{d-2\varepsilon_{0}}$. Put $\chi(u)=\chi_{1}(u)\chi_{2}(u)$. The flow $\widetilde{\eta}$ given by
$$\left\{\begin{array}[]{l}\frac{d}{dt}\widetilde{\eta}(t,u)=-\chi(\widetilde{%
\eta}(t,u))H(\widetilde{\eta}(t,u)),\\
\widetilde{\eta}(0,u)=u,\ u\in D(J)\end{array}\right.$$
is defined for all $t>0$ an has the same flow lines on $J_{d-2\varepsilon_{0}}^{d+2\varepsilon_{0}}\setminus U_{\delta/2}(K_{d})$ as $\eta$.
Choose $T$ so that $J(\widetilde{\eta}(T,u))<d-\varepsilon$ for all $u\in A\setminus U$. Such $T$ exists because $A$ is compact and $\lim_{t\to T^{+}(u)}J(\eta(t,u))\leq d-2\varepsilon_{0}$. The properties of pseudoindex give
$$k\leq i^{*}(A)\leq i^{*}(A\setminus U)+\gamma(\overline{U})\leq i^{*}(%
\widetilde{\eta}(T,A\setminus U))+\gamma(\overline{U})\leq k-1+\gamma(K_{d}).$$
So $\gamma(K_{d})\neq 0$ and hence $K_{d}\neq\emptyset$. If $d\equiv d_{k}=d_{k+1}$, then $i^{*}(A)\geq k+1$ and
therefore $\gamma(K_{d})\geq 2$ which is impossible. So $d_{k}<d_{k+1}$ for all $k$. ∎
3. Proof of Theorem 1.2
Let $u\in D(J)\setminus\{0\}$. Then the map
$s\mapsto J(su)$ admits a unique maximum point on $(0,\infty)$. In fact, if $\varphi:(0,\infty)\to\mathbb{R}$
is the map defined by
$$\varphi(s):=J(su)=J(u)s^{2}-s^{2}\log s\int_{\mathbb{R}^{N}}Q(x)u^{2}dx,\quad%
\,\,s>0,$$
we have $\varphi(s)>0$ for $s>0$ sufficiently small and
$\varphi(s)<0$ for all $s>0$ large enough. Moreover,
$\varphi^{\prime}(s)=0$ with $s>0$ if and only if
$$J(u)=\frac{2\log s+1}{2}\int_{\mathbb{R}^{N}}Q(x)u^{2}dx,$$
which proves the claim. Since $\varphi^{\prime}(s)=\langle J^{\prime}(su),u\rangle$,
the ray $\{su:s>0\}$ intersects the Nehari
manifold $\mathcal{N}$ (see definition (1.3)) at exactly one point. Moreover, there exists $s_{0}>0$ such that
for all $u\in D(J)\cap S_{1}(0)$, $s\mapsto\Phi(su)$ is increasing when $0<s<s_{0}$. Since $s\mapsto\Psi(su)$ is increasing for all $s>0$ (by convexity),
$\mathcal{N}$ is bounded away from $0$.
Alternatively one can observe that, if $u\in\mathcal{N}$, then inequality (2.3) yields
$$\int_{\mathbb{R}^{N}}\left(|\nabla u|^{2}+V(x)u^{2}\right)dx\leq\frac{1}{2}\|%
\nabla u\|_{2}^{2}+C_{2}(\log\|u\|_{2}^{2}+1)\|u\|_{2}^{2}.$$
Then, if $\|u\|$ is so small that $C_{2}(\log\|u\|_{2}^{2}+1)\leq\inf V$ one gets the contradiction $u=0$. Let
$$\Gamma:=\{\alpha\in C([0,1],E):\alpha(0)=0,\ J(\alpha(1))<0\}$$
and
$$c:=\inf_{\alpha\in\Gamma}\sup_{s\in[0,1]}J(\alpha(s)).$$
By Lemma 2.15, $c\geq b>0$ (and $c$ is the Mountain Pass level). Clearly, $c\leq c_{\mathcal{N}}:=\inf_{\mathcal{N}}J$. Assume that, for some $\varepsilon_{0}>0$, there exists no nontrivial solution
below the energy level $c+\varepsilon_{0}$. According to Remark 2.12, we can use Lemma 2.14 with $U_{\delta}(K_{c})=\emptyset$ and a
sufficiently small $\varepsilon<\varepsilon_{0}$. Let $\chi:E\to[0,1]$ be a locally Lipschitz continuous function such that $\chi=0$ on $J^{c/2}$, $\chi>0$ otherwise, and consider the flow
$$\left\{\begin{array}[]{l}\frac{d}{dt}\widehat{\eta}(t,u)=-\chi(\widehat{\eta}(%
t,u))H(\widehat{\eta}(t,u)),\\
\widehat{\eta}(0,u)=u,\ u\in J^{c+\varepsilon}.\end{array}\right.$$
Choosing $\alpha\in\Gamma$ such that $\sup_{s\in[0,1]}J(\alpha(s))\leq c+\varepsilon$ and setting $\beta(s):=\widehat{\eta}(T,\alpha(s))$, where $T$ is large enough,
we obtain $\sup_{s\in[0,1]}J(\beta(s))<c$ which is a contradiction because $\beta\in\Gamma$. Hence there exists a sequence of nontrivial solutions $(u_{n})$
with $\limsup_{n\to\infty}J(u_{n})\leq c$ (we do not exclude the possibility that $u_{n}=u$ for all $n$ and some $u$). Since $u_{n}\in\mathcal{N}$ and $c\leq c_{\mathcal{N}}$,
it follows that $c=c_{\mathcal{N}}$ and thus $J(u_{n})\to c$. Obviously, $(u_{n})$ is a Palais-Smale sequence, hence it is bounded according to Lemma 2.9 and we may assume that $u_{n}\rightharpoonup u$ in $E$ as $n\to\infty$.
As we have seen earlier, $u$ is a solution for (1.1). If $\|u_{n}\|_{p}\to 0$ for some $p\in(2,2^{*})$, then
$$0=\langle J^{\prime}(u_{n}),u_{n}\rangle\geq\|u_{n}\|^{2}-C\int_{\{|u_{n}|\geq
1%
\}}|u_{n}|^{p}\,dx,$$
yielding $\|u_{n}\|\to 0$ as $n\to\infty,$ contrary to the fact that $(u_{n})\subset{\mathcal{N}}$. Hence, by means of
Lions’ lemma [16, Lemma I.1], [21, Lemma 1.21], we have
$$\int_{B_{1}(y_{n})}u_{n}^{2}\,dx\geq\varepsilon,$$
for some sequence $(y_{n})\subset\mathbb{R}^{N}$ and some $\varepsilon>0$. As in the proof of Lemma 2.11 we may assume,
making translations if necessary, that $(y_{n})$ is bounded. So for the (translated) sequence $(u_{n})$ we have $u_{n}\rightharpoonup u\neq 0$ as $n\to\infty$.
Notice that $J(u)=\inf_{{\mathcal{N}}}J$. In fact, $J(u)\leq c$ by the same argument as in Remark 2.12 and obviously, $J(u)\geq c$.
Hence $u$ is a ground state solution.
Finally, the solution $u$ has constant sign. In fact, let us write $u=u^{+}-u^{-}$. Then $J(u)=J(u^{+})+J(u^{-})$ and $u^{+},u^{-}\in D(J)$.
Moreover, $0=\langle J^{\prime}(u),u^{+}\rangle=\langle J^{\prime}(u^{+}),u^{+}\rangle$, so either $u^{+}\in{\mathcal{N}}$ or $u^{+}=0$.
A similar conclusion holds for $u^{-}$. Hence, either one of the functions $u^{+}$, $u^{-}$ is equal to $0$ or
$J(u)\geq 2c$, which yields a contradiction. Suppose $u=u^{+}$. Then, by a slight variant of the argument
in [11, Section 3.1] it follows from the maximum principle (see [20, Theorem 1]) that $u(x)>0$, for a.e. $x\in\mathbb{R}^{N}$. ∎
4. A note on the $p$-Laplacian
Our arguments also allow to prove Theorems 1.1 and 1.2 for the equation
(4.1)
$$-\Delta_{p}u+V(x)|u|^{p-2}u=Q(x)|u|^{p-2}u\log|u|^{p},\quad u\in W^{1,p}(%
\mathbb{R}^{N}),$$
where $\Delta_{p}u:=\text{div}(|\nabla u|^{p-2}\nabla u)$ is the $p$-Laplacian ($1<p<N$) and $V,Q$ satisfy the conditions stated at the
beginning of Section 1. Here one needs to make use of a $p$-logarithmic Sobolev inequality, see e.g. [13] and the references therein. The functional corresponding to (4.1) is
$$J(u)=\frac{1}{p}\int_{\mathbb{R}^{N}}(|\nabla u|^{p}+(V(x)+Q(x))|u|^{p})\,dx-%
\frac{1}{p}\int_{\mathbb{R}^{N}}Q(x)|u|^{p}\log|u|^{p}\,dx,\quad u\in W^{1,p}(%
\mathbb{R}^{N}).$$
In order to show boundedness of the sequence $(u_{n})$ with $J(u_{n})$ bounded above and $J^{\prime}(u_{n})\to 0$, one needs to use
[13, formula (3)] with $u$ replaced by $Q^{1/p}u$ and modify the proof of Lemma 2.9 in a suitable way. We omit the easy but somewhat tedious details.
References
[1]
P. Bartolo, V. Benci, D. Fortunato, Abstract critical point theorems and applications to some nonlinear problems with “strong” resonance at infinity, Nonlinear Anal. 7 (1983), 981–1012.
[2]
V. Benci, On critical point theory for indefinite functionals in the presence of symmetries, Trans. Amer. Math. Soc. 274 (1982), 533–572.
[3]
I. Białynicki-Birula, J. Mycielski, Nonlinear wave mechanics, Ann. Physics 100 (1976), 62–93.
[4]
I. Białynicki-Birula, J. Mycielski, Gaussons: solitons of the logarithmic Schrödinger equation, Special issue on solitons in physics, Phys. Scripta 20 (1979), 539–544.
[5]
I. Białynicki-Birula, J. Mycielski, Wave equations with logarithmic nonlinearities, Bull. Acad. Polon. Sci. Cl. III 23, 461 (1975).
[6]
I. Campa, M. Degiovanni, Subdifferential calculus and nonsmooth critical point theory, SIAM J. Optim. 10 (2000), 1020–1048.
[7]
T. Cazenave, Stable solutions of the logarithmic Schrödinger equation, Nonlinear Anal. 7 (1983), 1127–1140.
[8]
T. Cazenave, An introduction to nonlinear
Schrödinger equations, Textos de Métodos Matemáticos
26, Universidade Federal do Rio de Janeiro 1996.
[9]
T. Cazenave, A. Haraux, Équations d’évolution avec non linéarité logarithmique,
Ann. Fac. Sci. Toulouse Math. 2 (1980), 21–51.
[10]
J.N. Corvellec, M. Degiovanni, M. Marzocchi, Deformation properties for continuous functionals and critical point theory, Topol. Methods Nonlinear Anal. 1 (1993), 151–171.
[11]
P. D’Avenia, E. Montefusco, M. Squassina, On the logarithmic Schrödinger equation, Comm. Contemp. Math. (2014), to appear,
[12]
M. Degiovanni, S. Zani, Multiple solutions of semilinear elliptic equations with one-sided growth conditions, Nonlinear operator theory, Math. Comput. Modelling 32 (2000), 1377–1393.
[13]
M. Del Pino, J. Dolbeault, The optimal Euclidean $L^{p}$-Sobolev logarithmic inequality, J. Funct. Anal. 197 (2003), 151–161.
[14]
P. Guerrero, J.L. López, J. Nieto, Global $H^{1}$ solvability of the 3D logarithmic Schrödinger equation, Nonlinear Anal. Real World Appl. 11 (2010), 79–87.
[15]
E.H. Lieb, M. Loss, Analysis, Second edition. Graduate Studies in Math. 14, AMS, Providence, RI, 2001.
[16]
P.L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. II,
Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), 223–283.
[17]
M. Struwe, Variational Methods, Springer, New York, 1990.
[18]
A. Szulkin, Minimax principles for lower semicontinuous functions and applications to nonlinear boundary value problems, Ann. Inst. H. Poincaré, Analyse non linéaire 3 (1986), 77–109.
[19]
A. Szulkin, T. Weth, Ground state solutions for some indefinite problems, J. Funct. Anal. 257 (2009), 3802–3822.
[20]
J.L. Vázquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim. 12 (1984), 191–202.
[21]
M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996.
[22]
K.G. Zloshchastiev, Logarithmic nonlinearity in theories of quantum gravity: origin of time and observational consequences,
Grav. Cosmol. 16 (2010), 288–297. |
The instantaneous shear modulus in the shoving model
Jeppe C. Dyre
dyre@ruc.dk
DNRF Centre “Glass and Time”, IMFUFA, Department of Sciences, Roskilde University, Postbox 260, DK-4000 Roskilde, Denmark
Wei Hua Wang
whw@iphy.ac.cn
Institute of Physics, Chinese Academy of Sciences, Beijing 100190, P.O. Box 603, China
(November 25, 2020)
Abstract
We point out that the instantaneous shear modulus $G_{\infty}$ of the shoving model for the non-Arrhenius temperature dependence of viscous liquids’ relaxation time is the experimentally accessible high-frequency plateau modulus, not the idealized instantaneous affine shear modulus that cannot be measured. Data for a large selection of metallic glasses are compared to three different versions of the shoving model. The original shear-modulus based version shows a slight correlation to the Poisson ratio, which is eliminated by the energy-landscape formulation of the model in which the bulk modulus plays a minor role.
I Introduction
In a recent Communicationpuo12 Puosi and Leporini showed from computer simulations that the relevant high-frequency shear modulus controlling the relaxation is not the idealized shear modulus corresponding to affine deformations at truly infinite frequency. Rather, it is the shear modulus referring to time scales that are on the one hand much shorter than any relaxation time, but on the other hand much longer than typical vibration times. This confirms findings by other groups.sta02 ; rib09 ; rib11 Puosi and Leporini further proposed an extension of the shoving model to allow for heterogeneities and showed that the new model fits simulation data very well. This extension is consistent with previous works by Khronik et al., who introduced the idea of a distribution of local shear moduli to explain sub $T_{g}$ relaxations within the shoving-model framework.kho09 ; mit11
Commenting on the instantaneous shear modulus of the shoving model Puosi and Leporini wrote: “It is quite apparent that $G_{\infty}$, the central quantity of the standard elastic models, poorly correlates with the structural relaxation time.” This conclusion derives from the understanding that the shoving model and related elastic models are based on the idealized affine infinite-frequency shear modulus, not the experimentally measurable high-frequency plateau modulus, which was traditionally used for comparing model to experiment. Unfortunately, theorists and experimentalists have used for many years the same symbol, $G_{\infty}$, for different physical properties. In experiment the term “instantaneous shear modulus” always meant the shear modulus at the highest obtainable frequencies (typically MHz or GHz, or even high kHz, depending on the technique used),bar67 ; har76 ; nos96 ; iiz05 ; rod07 ; sle08 ; pau10 where it generally becomes frequency independent and is usually – though not always – denoted by $G_{\infty}$ .bar67 ; har76 In his excellent text book The dynamic properties of supercooled liquids from 1976 Harrison refers to this quantity as “the high-frequency limiting shear modulus”.har76 Likewise, whenever the Maxwell relaxation time $\tau_{M}=\eta/G_{\infty}$ in the literature has been calculated for a liquid, $G_{\infty}$ was always identified as the experimental limiting high-frequency shear modulus of the plateau, much below phonon frequencies.
In liquid-state theory, $G_{\infty}$ was traditionally the idealized, truly infinite-frequency limit of the fluctuation-dissipation theorem expression,zwa65 ; han06 proportional to the mean-square shear-stress equilibrium fluctuation. Previously, there was no reason to believe that these two quantities should differ in any significant way. It now turns out that for some systems, as temperature changes at constant density, one quantity increases and the other decreases.puo12 ; yos10 The purpose of the present paper is to show that the shoving model derivation assumes $G_{\infty}$ is the quantity, which Puosi and Leporini referred to as the plateau modulus.puo12 ; sta02 ; rib09 ; rib11 We suggest a consistent notation for the two instantaneous shear moduli, which can be used whenever there is a risk of confusing them.puo12 ; yan11 Finally, we reanalyze data for a large selection of metallic glasses in order to compare to predictions of three versions of the shoving model.
Section II reviews the shoving model, Sec. III recalls the model energy-barrier calculation and relates it to the reversible-work theorem of statistical mechanics. Section IV compares model predictions to data, and Sec. V gives a brief summary.
II The shoving model
The shoving model for the non-Arrhenius temperature dependence of the main (alpha) relaxation time $\tau$ of a glass-forming liquid predicts that $\tau(T)=\tau_{0}\exp\left[G_{\infty}(T)V_{c}/(k_{B}T)\right]$.dyr96 ; dyr98a Here $\tau_{0}$ is a prefactor of order 0.1 picosecond, $G_{\infty}$ is the “instantaneous” shear modulus, $V_{c}$ is a characteristic volume of order a molecular volume, $k_{B}$ is Boltzmann’s constant, and $T$ is the temperature. In order to minimize the number of free parameters, the shoving model assumes ad hoc that $V_{c}$ is temperature independent. In this way the model connects directly two experimentally measurable quantities, $\tau(T)$ and $G_{\infty}(T)$. The model does not consider what causes the unusually large temperature dependence of glass-forming liquids’ instantaneous shear moduli. This problem was addressed recently in interesting papers by Brito and Wyart, who proposed that the increase of the instantaneous shear modulus derives from a stiffening of the Boson peak as temperature is lowered.bri09 ; wya10
The shoving model and related elastic models have been confirmed for a number of organic, oxide, chalcogenide, oxynitride, and metallic liquids,dyr96 ; dyr06b ; mag08 ; tor09 ; rou11 ; wan11 ; xu11 ; wan12 , as well as in some computer simulations,puo12 ; sta02 ; rib09 ; rib11 but failures of the model have also been reported.gra98 ; buc09 Most experimental conformations relate to a liquid’s temperature-dependent equilibrium relaxation time, but there are also tests confirming the shoving and related elastic models for aging experiments.dyr98 ; kho08 ; kho09 ; mit11 More data are certainly needed before it is clear whether the shoving and related models account generally for the non-Arrhenius temperature dependence of $\tau$ observed in supercooled liquids. For reviews of elastic models the reader is referred to Refs. dyr06a, and nem06, .
The shoving model is based on the assumption that the main contribution to the activation energy for a “flow event” – a rearrangement from one potential energy minimum to another – is the work done in shoving aside the surroundings in order to increase the volume available for rearranging the molecules. The model assumptions are:dyr96 ; dyr06b ; dyr06a
•
The main contribution to the activation free energy is elastic energy.
•
This elastic energy is located in the surroundings of the rearranging molecules.
•
The elastic energy is shear elastic energy, i.e., not associated with density changes.
To make things simple, the shoving model assumes spherical symmetry of a flow event and calculates the activation free energy as the work done in shoving aside the surroundings (expanding a sphere) in order to create room for a flow event.
Is it reasonable to assume that the main contribution to the activation energy comes from the surroundings? What about contributions from the rearranging molecules themselves? A simple argument shows that the former contribution dominates.dyr98 Suppose that rearranging at constant volume is energetically very costly because the molecules are forced into close contact during the rearrangement process; this is the main physical idea of the shoving model. In this case, allowing for just a slightly larger volume for the rearranging molecules implies a considerable lowering of the energy cost. If the radius change is $\Delta r$ and the energy barrier contribution from the rearranging molecules within the sphere is $f(\Delta r)$, the total energy barrier involves a further quadratic contribution from deforming elastically the surroundings: $\Delta E=f(\Delta r)+A(\Delta r)^{2}$. The fact that the function $f(\Delta r)$ decreases significantly when $\Delta r$ increases slightly above zero is expressed mathematically as $|d\ln f/d\ln\Delta r|\gg 1$. Optimizing $\Delta r$ in order to find the lowest barrier leads to $f^{\prime}(\Delta r)+2A\Delta r=0$. Thus the ratio between the “shoving” work and the “inner” contribution is
$A(\Delta r)^{2}/f(\Delta r)=-f^{\prime}(\Delta r)/[2\Delta rf(\Delta r)]=|d\ln
f%
/d\ln\Delta r|/2\gg 1$.
It is the high-frequency shear elastic constant that enters into the shoving model prediction because the expansion of a sphere in an elastic solid results in a radial displacement in the surroundings, which varies with distance $r$ to the sphere center as $r^{-2}$.lan70 This is a pure shear deformation, i.e., with zero divergence and thus no density changes anywhere (compare to the Coulomb electric field of a point charge $\propto r^{-2}$, which also has zero divergence). Less idealized geometries would result in some density change and thus also involve the bulk modulus, but it has been shown generally that the bulk elastic energy of the far field of an elastic dipole constitutes a most 10% of the total elastic energy.dyr07 This confirms the interesting phenomenon of “shear dominance” noted some time ago in various contexts of condensed matter physics and materials science (see Refs. dyr07, and gra92, and their references).
III Calculating the free energy barrier using the reversible-work theorem of statistical mechanics
The idealized instantaneous affine shear modulus refers to the hypothetical situation where one imposes an instantaneous, perfectly affine shear deformation on the system, corresponding to a time scale so fast that the atoms do not move, i.e., on a femtosecond time scale. This quantity is not experimentally accessible for the following reasons. It is difficult to imagine imposing an affine shear deformation on a system on a femtosecond time scale. Even if this were possible, one would not observe a high-frequency limiting shear modulus above THz frequencies – inertial effects would set in and cause the modulus to go to zero as frequency diverges without limit. On the other hand, in a computer simulation an instantaneous affine shear deformation is easily imposed on a system.
The energy barrier is calculated as the work done in shoving aside the surroundings. According to a fundamental theorem of statistical mechanics, the difference in free energy between two states can be calculated as the reversible isothermal work done to bring the system from one to the other state. Thus, in calculating the energy barrier, the expansion of the sphere must take place so slowly that there is equilibrium in the surroundings throughout the expansion process. On the other hand, since the activation (free) energy refers to the thermally activated creation of extra volume in the fixed, glassy structure of the surrounding molecules, the expansion must be fast enough that no relaxations take place in the surrounding liquid. This means that the time of expansion must be much smaller than the alpha relaxation time. In conclusion, the relevant elastic constants of elastic models refer to high frequencies, still much below phonon frequencies. In this frequency range the bulk and shear elastic constants are usually frequency independent for highly viscous liquids (ignoring possible secondary relaxations).
Over the years we have occasionally met the misunderstanding that the “shoving” process is the work done during the actual barrier transition of the rearranging molecules, which takes place over a few picoseconds. This is not correct; the (free) energy barrier is a difference in free energy between two states – the starting state and the transition state – and as detailed above this quantity is calculated by reference to statistical mechanics and the reversible-work theorem.
Both the experimentally measurable high-frequency plateau shear modulus – referring to frequencies much below phonon frequencies – and the experimentally non-accessible truly instantaneous affine shear modulus have traditionally been denoted by $G_{\infty}$. This would not present a serious problem if the two moduli were more or less identical or, from the shoving model perspective, if they were proportional in their temperature variation throughout the phase diagram. It now appears that neither is the case.puo12 ; yos10 This calls for introducing an unambiguous notation. We suggest denoting the experimental “instantaneous” shear modulus by $G_{\infty,{\rm p}}$ (p for plateau) and the idealized, affine truly instantaneous shear modulus by $G_{\infty,{\rm af}}$ (“af” for affine). In this notation, which need only to be used whenever there is a risk of confusion, the shoving model prediction is
$$\tau(T)\,=\,\tau_{0}\exp\left[\frac{G_{\infty,{\rm p}}(T)V_{c}}{k_{B}T}\right]\,.$$
(1)
Before proceeding we note the fluctuation-dissipation (FD) theorem expressions for the two instantaneous shear moduli. Recall that if one defines $S_{xy}\equiv\sum_{i}x_{i}F_{y,i}$ where $x_{i}$ is the x component of the position vector of particle $i$ and $F_{y,i}$ is the y component on the force on this particle, the FD theorem expression for the frequency-dependent shear viscosity $\eta(\omega)$ is (where $V$ is the system volume)
$$\eta(\omega)\,=\,\frac{\int_{0}^{\infty}\langle S_{xy}(0)S_{xy}(t)\rangle e^{-%
i\omega t}dt}{k_{B}T\,V}\,.$$
(2)
Since the frequency-dependent shear modulus is related to the viscosity by $G(\omega)=i\omega\eta(\omega)$, letting frequency go to infinity in Eq. (2) gives the well-known expression for the idealized affine infinite-frequency shear modulus $G_{\infty,{\rm af}}$,han06
$$G_{\infty,{\rm af}}\,=\,\frac{\langle S_{xy}^{2}\rangle}{k_{B}T\,V}\,.$$
(3)
As mentioned, this quantity cannot be measured in experiment. The quantity that can be measured, the plateau modulus $G_{\infty,{\rm p}}$, is given by the analogous expression where $S_{xy}$ is averaged over a few molecular vibration periods (i.e., over some picoseconds):
$$G_{\infty,{\rm p}}\,=\,\frac{\langle\overline{S}_{xy}^{2}\rangle}{k_{B}T\,V}\,.$$
(4)
We finally note that $G_{\infty,{\rm af}}/(\rho T)$ and $G_{\infty,{\rm p}}/(\rho T)$ are both isomorph invariants. The invariance of the first expression was shown in Ref. gna09, , that of the latter follows by analogous arguments. This shows that the shoving model survives the “isomorph filter” according to which any generally applicable model for the non-Arrhenius temperature dependence of a supercooled liquid’s relaxation time must give this as a function of an isomorph invariant.gna09
IV Comparing the energy-landscape version of the shoving model to metallic glass data
The energy-landscape justification of the shoving modeldyr04 is based on a classical argument, which estimates the barrier height for a jump between two (free) energy minima from the curvature at the minima, leading to $\ln\tau\propto 1/\langle u^{2}\rangle$ where $\langle u^{2}\rangle$ is the vibrational mean-square displacement.fly68 ; hal87 ; koh88 ; buc92 If the high-frequency shear and bulk plateau moduli, $G_{\infty,{\rm p}}$ and $K_{\infty,{\rm p}}$, differ from the ideal, affine moduli $G_{\infty,{\rm af}}$ and $K_{\infty,{\rm af}}$, the former are the relevant ones for the phonon spectra. In fact, one way to probe plateau moduli is to measure the linear (i.e., low-wavevector) parts of the phonon spectra or, equivalently, the shear and longitudinal high-frequency sound velocities. This is also how Ribero et al. probed $G_{\infty}$ in computer simulations.rib09 ; rib11 In this section term “instantaneous” moduli implies the plateau moduli.
In a simple, isotropic elastic model the vibrational mean-square displacement may be estimated from the instantaneous moduli by using the fact that for a given wavevector, there are two transverse and one longitudinal phonon. If $M=K+(4/3)G$ is the longitudinal modulus, this leadsdyr04 to $\langle u^{2}\rangle/T\propto 1/M_{\infty,{\rm p}}+2/G_{\infty,{\rm p}}$. For the temperature dependence of the relaxation time this implies via $\ln\tau\propto 1/\langle u^{2}\rangle$
$$\tau(T)\,=\,\tau_{0}\exp\left[\frac{V_{c}^{\prime}}{k_{B}T}\left(\frac{1}{M_{%
\infty,{\rm p}}(T)}+\frac{2}{G_{\infty,{\rm p}}(T)}\right)^{-1}\right]\,,$$
(5)
where $V_{c}^{\prime}$ is a molecular-sized volume. As shown in Ref. dyr04, , this expression implies shear dominance for the temperature dependence of $\tau$. This was shown by first defining for any quantity $Q$ that increases as $T$ decreases its temperature index as $I_{Q}\equiv-d\ln Q/d\ln T$. Equation (5) implies that the temperature index of $\tau$’s activation energy can be written $(1-\alpha)I_{G_{\infty,{\rm p}}}+\alpha I_{K_{\infty,{\rm p}}}$, where $0<\alpha<0.08$ is obeyed no matter what is the ratio $G_{\infty,{\rm p}}/K_{\infty,{\rm p}}$.dyr04 In other words, at least 92% of the non-Arrhenius temperature dependence of the relaxation time derives from $G_{\infty,{\rm p}}$’s temperature dependence. The instantaneous bulk modulus $K_{\infty,{\rm p}}$ plays only a minor role for three reasons:
•
Two phonons are transversal for each one that is longitudinal.
•
The instantaneous shear modulus $G_{\infty,{\rm p}}$ affects also the longitudinal phonons.
•
Longitudinal phonons are harder than transverse and therefore contribute less than one third to the vibrational mean-square displacement.
It follows from Eq. (5) that the temperature-dependent activation (free) energy $\Delta E(T)$ is given by
$$\Delta E(T)\,\propto\,G_{\infty,{\rm p}}(T)V_{c}^{\prime}\frac{K_{\infty,{\rm p%
}}(T)+4G_{\infty,{\rm p}}(T)/3}{2K_{\infty,{\rm p}}(T)+11G_{\infty,{\rm p}}(T)%
/3}\,,$$
(6)
Traditionally the shoving model is compared to experiment by plotting the logarithm of the relaxation time (or, equivalently, the viscosity) as a function of $G_{\infty,{\rm p}}(T)/T$.dyr96 ; dyr98a ; dyr06b ; mag08 ; tor09 ; rou11 ; buc09 Such a plot checks whether the predicted linear relationship between $\ln\tau$ and $G_{\infty,{\rm p}}(T)/T$ is observed with a physically reasonable prefactor. A convenient way of making this plot is via a generalized Angell plot in which the x coordinate is normalized to unity at the glass transition by defining $X\equiv[G_{\infty,{\rm p}}(T)T_{g}]/[G_{\infty,{\rm p}}(T_{g})T]$. Unfortunately, measuring the high-frequency plateau shear modulus of the equilibrium metastable supercooled liquid is difficult and not too many data are available even today. An alternative way of testing the shoving model is by making use of the facts that 1) for a fixed cooling rate the relaxation time has a certain value at $T_{g}$, 2) the (DC) shear modulus of the glass, $G$, is almost temperature independent and equal to the instantaneous shear modulus of the liquid at $T_{g}$: $G\cong G_{\infty,{\rm p}}(T_{g})$. The glass shear modulus is easy to measure, and the prediction of the model is that $GV_{c}/k_{B}T_{g}\cong{\rm Const.}$, in which it is reasonable to assume that the constant is more or the same for similar systems. For the range of metallic glasses compared below to model predictions we further assume that the characteristic volume $V_{c}$ is proportional to the molar volume $V_{m}$ with a universal proportionality constant. The point of these simplifying assumptions is to eliminate all non-trivial free parameters.
Figure 1 gives bulk and shear modulus data for a range of metallic glasses, as well as their glass transition temperatures ranging from 317K for some of the Ca-based glasses to 930K for some Fe-based ones. Most of these data were discussed and compared to elastic model predictions in Refs. wan11, ; wan12, , but the cupper-based glasses were replaced.
Figure 2 compares data to different versions of the shoving model [Eq. (1)]. Since the glass transition takes place when the relaxation time upon cooling reaches a certain value, the standard shoving model prediction is
$$T_{g}\propto\Delta E(T_{g})\propto G\,V_{m}\,$$
(7)
with a universal proportionality constant. Figure 2(a) tests this prediction based on the data of Fig. 1, where the metallic glasses are sorted according to their Poisson ratio ($R$ is the gas constant). Given the diversity of the glasses, the model’s simplicity, and the simplifying assumption that $V_{c}\propto V_{m}$ with a universal proportionality constant, the data show good agreement with the shoving model. There is, as noted in Refs. wan11, and wan12, , some correlation with Poisson’s ratio.gre11 Before addressing whether the energy-landscape version of the shoving model rectifies this, we show in Fig. 2(b) how data compares to the elastic model in which $G$ is replaced by $K$. Here the correlation with Poisson’s ratio is much stronger. Altogether, Figs. 2(a) and 2(b) show that if an elastic model is to work for metallic glasses with a universal value of the characteristic volume $V_{c}$ in terms of the molar volume $V_{m}$, the bulk modulus can play only a minor role. This constitutes an experimental demonstration of “shear dominance”.
For the energy-landscape version of the shoving model Eq. (6) translates into the prediction
$$T_{g}\propto\Delta E(T_{g})\propto GV_{m}\frac{K+4G/3}{2K+11G/3}\,\,.$$
(8)
This is compared to experiment in Fig. 2(c). The fit is good and there is no correlation to the Poisson ratio.
The three “zero-parameter” elastic models of Fig. 2 have standard deviations, respectively, of 8% for the standard shoving model [Fig. 2(a)], 22% for the bulk modulus elastic model [Fig. 2(b)], and 8% for the energy-landscape version of the shoving model [Fig. 2(c)]. The latter gives no better fit to the data than the original shoving model (the standard deviation of it is 0.6% lower, but this is hardly significant), but has the advantage of eliminating the correlation to the Poisson ratio.
A pragmatic one-parameter version of the elastic models is to allow for an arbitrary combination of $G$ and $K$ by assuming that $T_{g}$ is controlled by $\alpha G+(1-\alpha)K$. Due to shear dominance one expects $\alpha$ to be close to unity. Indeed, Ref. wan12, showed that for $\alpha=10/11$ a very good fit to data is obtained for metallic glasses, a fit which like the energy-landscape shoving model eliminates the correlation to the Poisson ratio. For this one-parameter model the standard deviation is 6%, i.e., somewhat better than the standard shoving model and its energy-landscape version.
V Summary
The instantaneous shear modulus of the shoving model refers to the plateau modulus that is traditionally in experiment denoted by $G_{\infty}$. This quantity is measurable, in contrast to the idealized affine instantaneous shear modulus of liquid state theory, which was also traditionally denoted by $G_{\infty}$. A consistent notation has been suggested for distinguishing between these two quantities whenever there is a risk of confusing them. Data for a range of metallic glasses have been shown to be consistent with shoving model predictions; in particular the energy-landscape version of the model eliminates the correlation of model predictions to Poisson’s ratio.
Acknowledgements.The centre for viscous liquid dynamics “Glass and Time” is sponsored by the Danish National Research Foundation (DNRF).
References
(1)
F. Puosi and D. Leporini, J. Chem. Phys. 136, 041104 (2012).
(2)
F. W. Starr, S. Sastry, J. F. Douglas, and S. C. Glotzer, Phys. Rev. Lett. 89, 125501 (2002).
(3)
M. C. C. Ribeiro, J. Non-Cryst. Solids 355, 1659 (2009).
(4)
M. C. C. Ribeiro, T. Scopigno, and G. Ruocco, J. Chem. Phys. 135, 164510 (2011).
(5)
V. A. Khonik, Y. P. Mitrofanov, S. A. Lyakhov, A. N. Vasiliev, S. V. Khonik, and D. A. Khoviv, Phys. Rev. B 79, 132204 (2009).
(6)
Y. P. Mitrofanov, V. A. Khonik, A. V. Granato, D. M. Joncich, and S. V. Khonik, J. Appl. Phys. 109, 073518 (2011).
(7)
A. J. Barlow, J. Lamb, A. J. Matheson, P. R. K. L. Padmini, and J. Richter, Proc. R. Soc. London, Ser. A 298, 467 (1967).
(8)
G. Harrison, The dynamic properties of supercooled liquids (Academic Press, New York, 1976).
(9)
R. Nossal, Physica A 231, 265 (1996).
(10)
A. Iizuka, S. Tachibana, K. Kawai, and H. Ohta, Soils and Foundations 45, 135 (2005).
(11)
M. N. Rodnikova, J. Mol. Liquids 136, 211 (2007).
(12)
N. H. Sleep and S. Ma, Geochem. Geophys. Geosystems 9, Q03008 (2008).
(13)
S. Pauly, S. Gorantla, G. Wang, U. Kühn, and J. Eckert, Nat. Mater. 9, 473 (2010).
(14)
R. Zwanzig and R. Mountain, J. Chem. Phys. 43, 4464 (1965).
(15)
J.-P. Hansen and I. McDonald, Theory of Simple Liquids, 3rd ed. (Academic, New York, 2006).
(16)
H. Yoshino and M. Mezard, Phys. Rev. Lett. 105, 015504 (2010).
(17)
J. Yang and K. S. Schweizer, J. Chem. Phys. 134, 204909 (2011).
(18)
J. C. Dyre, N. B. Olsen, and T. Christensen, Phys. Rev. B 53, 2171 (1996).
(19)
J. C. Dyre, J. Non-Cryst. Solids 235, 142 (1998).
(20)
B. Brito and M. Wyart, J. Chem. Phys. 131, 024504 (2009).
(21)
M. Wyart, Phys. Rev. Lett. 104, 095901 (2010).
(22)
J. C. Dyre, T. Christensen, and N. B. Olsen, J. Non-Cryst. Solids 352, 4635 (2006).
(23)
C. Maggi, B. Jakobsen, T. Christensen, N. B. Olsen, and J. C. Dyre, J. Phys. Chem. B 112, 16320 (2008).
(24)
D. H. Torchinsky, J. A. Johnson, and K. A. Nelson, J. Chem. Phys. 130, 064502 (2009).
(25)
T. Rouxel, J. Chem. Phys. 135, 184501 (2011).
(26)
J. Q. Wang, W. H. Wang, Y. H. Liu, and H. Y. Bai, Phys Rev. B 83, 012201 (2011).
(27)
B. Xu and G. B. McKenna, J. Chem. Phys. 134, 124902 (2011).
(28)
W. H. Wang, Prog. Mater. Sci. 57, 487 (2012).
(29)
A. V. Granato, Metall. Mater. Trans. A 29, 1837 (1998).
(30)
U. Buchenau, Phys. Rev. B 80, 172201 (2009).
(31)
N. B. Olsen, J. C.
, and T. Christensen, Phys. Rev. Lett. 81, 1031 (1998).
(32)
S. V. Khonik, A. V. Granato, D. M. Joncich, A. Pompe, V. A. Khonik, Phys. Rev. Lett. 100, 065501 (2008).
(33)
J. C. Dyre, Rev. Mod. Phys. 78, 953 (2006).
(34)
S. V. Nemilov, J. Non-Cryst. Solids 352, 2715 (2006).
(35)
L. D. Landau and E. M. Lifshitz, 1970, Theory of Elasticity, 2nd ed. (Pergamon, London, 1970).
(36)
J. C. Dyre, Phys. Rev. B 75, 092102 (2007).
(37)
A. V. Granato, Phys. Rev. Lett. 68, 974 (1992).
(38)
N. Gnan, T. B. Schrøder, U. R. Pedersen, N. P. Bailey, and J. C. Dyre, J. Chem. Phys. 131, 234504 (2009).
(39)
J. C. Dyre and N. B. Olsen, Phys. Rev. E 69, 042501 (2004).
(40)
C. P. Flynn, Phys. Rev. 171, 682 (1968).
(41)
R. W. Hall and P. G. Wolynes, J. Chem. Phys. 86, 2943 (1987).
(42)
U. Köhler and C. Herzig, Philos. Mag. A 58, 769 (1988).
(43)
U. Buchenau, Philos. Mag. B 65, 303 (1992).
(44)
G. N. Greaves, A. L. Greer, R. S. Lakes, and T. Rouxel, Nat. Mater. 10, 823 (2011). |
Isovector splitting of nucleon effective masses, ab-initio benchmarks and
extended stability criteria for Skyrme energy functionals
T. Lesinski
lesinski@ipnl.in2p3.fr
Institut de Physique Nucléaire de Lyon,
CNRS-IN2P3/Université Claude Bernard Lyon 1,
43, bd. du 11 novembre 1918,
F-69622 Villeurbanne Cedex, France
K. Bennaceur
Institut de Physique Nucléaire de Lyon,
CNRS-IN2P3/Université Claude Bernard Lyon 1,
43, bd. du 11 novembre 1918,
F-69622 Villeurbanne Cedex, France
CEA-Saclay DSM/DAPNIA/SPhN, F-91191 Gif-sur-Yvette, France
T. Duguet
National Superconducting Cyclotron Laboratory and
Department of Physics and Astronomy,
Michigan State University, East Lansing, MI 48824, USA
J. Meyer
Institut de Physique Nucléaire de Lyon,
CNRS-IN2P3/Université Claude Bernard Lyon 1,
43, bd. du 11 novembre 1918,
F-69622 Villeurbanne Cedex, France
(November 27, 2020)
Abstract
We study the effect of the splitting of neutron and proton effective masses
with isospin asymmetry on the properties of the Skyrme energy density
functional. We discuss the ability of the latter to predict observable of
infinite matter and finite nuclei, paying particular attention to
controlling the agreement with ab-initio predictions of the spin-isospin
content of the nuclear equation of state, as well as diagnosing the onset of
finite size instabilities, which we find to be of critical importance. We
show that these various constraints cannot be simultaneously fulfilled by
the standard Skyrme force, calling at least for an extension of its
P-wave part.
pacs:
21.30.Fe, 21.60.Jz
I Introduction
In the study of medium to heavy mass nuclei, nuclear Energy Density Functional
(EDF) approaches, based on self-consistent Hartree-Fock (HF)
methods and their extensions, constitute the theoretical tool of
choice Bender et al. (2003). Thanks to the
development of better energy functionals and to the increase of computer
resources, nuclear EDF is on the edge of becoming a predictive theory for all
nuclei but the lightest. This is not only true for ground state properties,
such as binding energies, radii or multipoles of the density, but also for
low-energy spectroscopy and decay probabilities Bender et al. (2003).
However, the accuracy and predictive power needed for unknown regions of the
nuclear chart still leave a lot of room for improvement. The phenomenological
nature of Skyrme functionals makes their ability to faithfully predict
observable or phenomena not linked with those used for their construction quite
weak. Indeed, the limited number of adjustable parameters (compared to the
wealth of nuclear observable to be matched) turns fitting a Skyrme functional
into an overconstrained problem (which, of course, does not prevent some parts
of it from being underconstrained).
As a direct consequence, many properties of existing parameterizations
are biased to the fitting procedure and the limited analytical form of
the Skyrme force, rather than to physical reasoning. A well-known example
is the equation of state (EOS) of Pure Neutron Matter (PNM), which is sometimes
subject to a pathological collapse at high density when not explicitly
constrained. This is problematic insofar as one of the major challenges of
contemporary nuclear theory is to predict properties of very isospin asymmetric
nuclear systems, i.e. neutron rich nuclei and matter in neutron stars.
Experimental data being unavailable in this domain of isospin, one has started
relying on ab-initio theoretical results to constrain isovector properties of
the functional. It has led to the construction of the “Saclay-Lyon” SLy
series of parameterizations Chabanat et al. (1997); Chabanat
et al. (1998a, b) by
fitting (among other quantities) a theoretical equation of state of neutron
matter.
Isovector features of the nuclear EOS are crucial for a good
understanding of neutron stars, exotic nuclear collisions produced at
radioactive beam facilities and to describe the structure of exotic nuclei. For
instance, the density dependence of the volume symmetry energy determines the
proton fraction in $\beta$ equilibrium in neutron stars, which ultimately
drives the cooling rate and neutrino emission Lattimer and Prakash (2004). The
high-density part of the symmetry energy, which happens to be strongly model
dependent, also influences significantly the isospin diffusion in heavy-ion
collisions Chen et al. (2005). Finally, the low-density part of the symmetry energy
is correlated with the size of neutron skins in finite nuclei Typel and Brown (2001).
Beyond global isospin-dependent properties of the EOS, the isovector part of
nucleon-dependent quantities may influence the behavior of the above mentioned
systems. Thus, collision observable depend on the momentum dependence of the
mean-field, in particular on its isovector component Li (2004); Li et al. (2004). Also,
some properties of neutron stars require a precise knowledge of isoscalar and
isovector nucleon effective masses Bethe (1990); Farine et al. (2001). The latter, which
drives the splitting of neutron and proton effective masses with
neutron/proton asymmetry, will serve as a starting point for our study. Indeed,
a lot of efforts has recently been devoted to the microscopic characterization
of neutron and proton effective masses in infinite Asymmetric Nuclear Matter
(ANM) Bombaci and Lombardo (1991); Kubis and Kutschera (1997); Zuo et al. (1999); Greco et al. (2001); Hofmann et al. (2001); Liu et al. (2002); Rizzo et al. (2004); Ma et al. (2004); van Dalen et al. (2005); Satula et al. (2006).
Either in ANM or in nuclei, the two species acquire different effective masses.
This property is quantified by the difference $\Delta m^{\ast}(I)=m^{\ast}_{\mathrm{n}}(I)-m^{\ast}_{\mathrm{p}}(I)$, where
$I=(\rho_{\mathrm{n}}-\rho_{\mathrm{p}})/(\rho_{\mathrm{n}}+\rho_{\mathrm{p}})$ is the isospin asymmetry while $\rho_{\mathrm{n}}$ and
$\rho_{\mathrm{p}}$ denote neutron and proton densities, respectively. Note that the
different effective masses $m^{\ast}$ discussed in the text always refer in
fact to the ratio $m^{\ast}/m$, where $m$ is the bare nucleon mass. The latter
is taken to be the same for neutrons and protons.
This effective-mass splitting, though, is only one of a wealth of observable
which can be subject to comparison between ab-initio predictions and EDF
models. In this work we present results of a classical yet long unused test:
the separation of infinite Symmetric Nuclear Matter (SNM) potential energy per
particle into spin-isospin channels.
We shall also pay particular attention to controlling instabilities
(i.e. non-physical spontaneous breaking of spin, isospin and/or spatial
symmetries), and correlate $\Delta m^{\ast}(I)$ with vector properties of the
functional. We thus investigate the behavior of the latter with respect to the
breaking of time-reversal invariance and the onset of spin polarization,
looking for an overall consistency check of its spin-isospin content. Indeed,
such properties will become more and more important as one attempts to use
full-fledged Skyrme functionals to study odd-mass nuclei, calculate
rotational properties through self-consistent cranking calculations, or use
more general dynamical methods Bender et al. (2002).
This paper is organized as follows: in section II we present
the set of Skyrme parameterizations used in this study and examine basic
properties of nuclear matter and finite nuclei. From then,
in section III we perform a more detailed study of the spin-isospin
content of the functionals and of their stability against finite-size spin
and isospin perturbations using response functions in the Random Phase
Approximation (RPA).
II Constraining the isovector effective mass
The nucleon effective mass $m^{\ast}$ is a key property
characterizing the propagation of (quasi)nucleons through the
nuclear medium Jeukenne et al. (1976). It is a reminder of the
non-locality and energy dependence of the nucleon self-energy
$\Sigma(k,\epsilon)$, themselves originating from the finite
range and non-locality in time and space of the nucleon-nucleon
interaction. Mean-field-like theories of finite nuclei or infinite
matter assume an on-shell propagation of the nucleons. The
effective mass associated with such an on-shell propagation does
not take into account the fragmentation of the single-particle
strength and thus, a limited part of the effects associated with
the energy dependence of $\Sigma(k,\epsilon)$. Finally, the total
energy is calculated by considering the quasi-holes (particles) to
have spectroscopic factors of 1. In this context, either
microscopic Baldo (1999) or making use of phenomenological
interactions or functionals Bender et al. (2003), mean-field methods
do not correspond to a naive Hartree-Fock theory and always amount
to renormalizing a certain class of correlations into the
effective vertex. However, the energy dependence of the
self-energy arising from the correlations only influences the
position of the quasi-particle peak energy. The standard nuclear
EDF methods differ in several respects from Kohn-Sham Density
Functional Theory Kohn and Sham (1964). In particular, the strategy
usually followed in the former is to leave room for improvement
through further inclusions of correlations. This can be done by
performing (Quasiparticle)-Random-Phase-Approximation (QRPA)
calculations Blaizot and Gogny (1977); Severyukhin et al. (2002) or by employing the Generator
Coordinate Method (GCM) on top of symmetry-restored mean-field
states Meyer et al. (1995).
Thus, the effective mass adjusted at the pure mean-field level is not
expected
to generate single-particle spectra matching exactly experimental
data extracted from neighboring odd-mass nuclei. In particular,
the coupling of single-particle motions to surface vibrations in
closed-shell nuclei is known to increase the density of states at
the Fermi surface and thus the effective
mass Bernard and Giai (1980); Goriely et al. (2003). A mean-field isoscalar effective
mass $m^{\ast}_{\mathrm{s}}$ lying in the interval $0.7/0.8$ in SNM, is able to
account for a good reproduction of both isoscalar quadrupole giant
resonances data in doubly closed-shell nuclei Liu and Giai (1976) and of
single-particle spectra in neighboring ones provided
particle-vibration coupling has been properly included. When the
latter coupling is taken into account, the effective mass becomes
greater than one for states near the Fermi surface. Certainly, a
lot remains to be done to understand these features
microscopically in more involved cases Charity et al. (2006). This is
not only true for mid-shell nuclei where the coupling to both
rotational and vibrational states can be important, but also for
exotic nuclei where the coupling to the continuum becomes crucial
and where shape coexistence and/or large amplitude motion appear
more systematically.
In very exotic systems, the isovector behavior of $m^{\ast}_{\mathrm{p}}$ and $m^{\ast}_{\mathrm{n}}$ should
play
an important role. However, so far, no experimental data from finite nuclei has
allowed a determination of the effective mass splitting as a function of
neutron richness. In this context, ab-initio calculations of ANM are
of great help. Non-relativistic Brueckner-Hartree-Fock (BHF) calculations, with
or without three-body force, and, with or without rearrangement terms in the
self-energy, predicted $\Delta m^{\ast}(I)$ to be such that $m^{\ast}_{\mathrm{n}}\geq m^{\ast}_{\mathrm{p}}$ in
neutron-rich matter, that is, for $I\geq 0$. Such a conclusion was also
reached
by calculating the energy dependence of the symmetry potential (the Lane
potential Lane (1962)) within a phenomenological formalism Li (2004).
The latter result was confirmed by microscopic Dirac-Brueckner-Hartree-Fock
(DBHF) calculations Sammarruca et al. (2005). The situation regarding the
prediction of the
effective mass splitting was complexified due to an apparent contradiction
between results obtained from BHF Bombaci and Lombardo (1991); Zuo et al. (1999) and DBHF
calculations Hofmann et al. (2001). However, the situation was finally clarified in
Ref. Ma et al. (2004); van Dalen et al. (2005) where the importance of the energy dependence of the
self-energy and the need to compare the non-relativistic effective mass with
the vector effective mass in the relativistic framework M. Jaminon (1989) were
pointed out.
Thus, the sign of the splitting is rather solidly predicted. However, its
amplitude is subject to a much greater uncertainty. Starting from that
observation, the goal of the present section is to study the impact of
the effective-mass splitting on properties of exotic nuclei predicted by
Skyrme-Hartree-Fock calculations. As far as the effective-mass splitting is
concerned, one expects consequences onto structure properties of neutron-rich
nuclei. As a relatively large asymmetry may be necessary to reveal the
influence
of the splitting, data from nuclei not yet studied experimentally should
provide crucial information in that respect. As the effective mass governs
the density of states at the Fermi surface (together with the spin-orbit and
the tensor forces), the amplitude of the splitting may influence properties
such as masses and single particle properties of exotic nuclei, the evolution
of
isotopic shifts across neutron rich closed-shell nuclei or shell corrections in
superheavy nuclei around the $(N=184,Z=120)$ island of
stability Bender et al. (1999); Kruppa et al. (2000); Bender et al. (2001); Berger et al. (2001). Also, neutron and
proton
correlations beyond the mean-field should develop rather differently depending
on the direction and amplitude of the effective-mass splitting. This could be
true for static and dynamical pairing correlations as well as for the coupling
to vibrational and rotational states. Finally, the effective mass splitting
should leave its fingerprint onto the characteristics of isovector vibrational
states of different sorts in neutron rich
nuclei Paar et al. (2005).
II.1 Fitting protocol
Trying to keep a coherence in the way we construct Skyrme functionals, we take
the fitting protocol used to define the SLy
forces Chabanat et al. (1997); Chabanat
et al. (1998a, b)
as a basis for the present work. Also, we pay attention to the fact that any
improved or complexified functional includes all features validated by the SLy
ones.
The SLy functionals were derived from an effective interaction, that is,
time-odd components of the functional are eventually linked to time-even ones.
However, some terms in the functional given by
Eqs. 26a-26d were dropped for some of the
parameterizations. For instance, time-odd terms of the form $\mathbf{s}_{q}\cdot\triangle\mathbf{s}_{q^{\prime}}$ have not been considered when calculating odd
nuclei Duguet et al. (2002a) or rotational states Rigollet et al. (1999). Also, $\mathbb{J}^{2}$
contributions to the energy functional associated with momentum-dependent terms
in the central Skyrme force have been omitted for some of the SLy
parameterizations, as it has been the case for most of the Skyrme
functionals so far Bender et al. (2003). Rigorously, omitting terms is inconsistent
with the idea of deriving a functional from an interaction. In any case,
the latter approach is only used for simplicity until proper adjustment or
derivation
of time-odd terms is feasible. We refer the reader to
Refs. Chabanat et al. (1997); Chabanat
et al. (1998a, b); Bender et al. (2003) for a more extensive
discussion on the subject.
We presently take the SLy5 parameterization as a starting point. Thus, the
two-body part of the center of mass correction is omitted whereas the
$\mathbb{J}^{2}$ terms are fully kept. The spin-orbit term is the standard one,
with a single parameter adjusted on the splitting of the $3p$ neutron
level in ${}^{208}$Pb.
Within this general scheme, we have built a series of three new Skyrme forces,
denoted hereafter $f_{-},f_{0}$ and $f_{+}$. The departures from the SLy protocol
considered presently are (i) a better control of spin-isospin instabilities via
Landau parameters (ii) the use of two density-dependent zero-range
terms Cochet et al. (2004) (iii) a constraint on the isovector effective mass, such
that, in neutron-rich systems, $m^{\ast}_{\mathrm{n}}<m^{\ast}_{\mathrm{p}}$ for $f_{-}$, $m^{\ast}_{\mathrm{n}}=m^{\ast}_{\mathrm{p}}$ for $f_{0}$ and
$m^{\ast}_{\mathrm{n}}>m^{\ast}_{\mathrm{p}}$ for $f_{+}$.
With two density dependent terms, the compressibility and the isoscalar
effective mass are no longer bound together and can be chosen independently.
However, this is not directly used here and an isoscalar effective mass of
$m^{\ast}_{\mathrm{s}}=0.7$, close to the SLy5 value, is chosen for the three
parameterizations $f_{-},f_{0},f_{+}$. The additional freedom brought about by the second density-dependent
term is only used to adjust more easily the high-density part of the PNM EOS
(see below). In the end, the only parameter subject to variation between $f_{-}$,
$f_{0}$ and $f_{+}$ is the isovector effective mass $m^{\ast}_{\mathrm{v}}$ which, $m^{\ast}_{\mathrm{s}}$ being
constant, drives the splitting $\Delta m^{\ast}(I)$.
In the present work, we use the SLy5 force as a reference, and include a
comparison with the recently proposed LNS parameterization Cao et al. (2006)
which was also built to match the splitting of effective masses and the neutron
matter EOS predicted by BHF calculations. The SkP
force Dobaczewski et al. (1984), initially built for the study of pairing
effects, will be used for a special purpose in the discussion about
instabilities.
II.2 Elementary properties of studied forces
As we focus on the behavior of effective masses $m^{\ast}_{q}$ with isospin
asymmetry, we recall that these quantities are related to the dependence of the
energy density functional, Eqs. 26a–26d, on
kinetic densities $\tau_{q}$, as
$$\displaystyle\frac{\hbar^{2}}{2m^{\ast}_{q}(I)}$$
$$\displaystyle=$$
$$\displaystyle\frac{\partial{\mathcal{H}}}{\partial\tau_{q}}\leavevmode\nobreak%
\ =\leavevmode\nobreak\ \frac{\hbar^{2}}{2m}+C^{\tau}_{0}\leavevmode\nobreak\ %
\rho_{0}+qI\leavevmode\nobreak\ C^{\tau}_{1}\leavevmode\nobreak\ \rho_{0}$$
(1a)
$$\displaystyle\frac{m}{m^{\ast}_{q}(I)}$$
$$\displaystyle\equiv$$
$$\displaystyle\frac{m}{m^{\ast}_{\mathrm{s}}}+qI\leavevmode\nobreak\ \left(%
\frac{m}{m^{\ast}_{\mathrm{s}}}-\frac{m}{m^{\ast}_{\mathrm{v}}}\right)$$
(1b)
where $\rho_{0}$ is the scalar-isoscalar density
and $q=+1,-1$ respectively for neutrons and protons (for the definition of
$C^{\tau}_{t}$ coefficients, see appendix A). The splitting of effective
masses, quantified by
$$\displaystyle\frac{\Delta m^{\ast}(I)}{m}$$
$$\displaystyle=$$
$$\displaystyle\frac{m^{\ast}_{\mathrm{n}}(I)}{m}-\frac{m^{\ast}_{\mathrm{p}}(I)%
}{m},$$
(2)
is governed by the isoscalar and isovector effective masses
$$\displaystyle\frac{m}{m^{\ast}_{\mathrm{s}}}\leavevmode\nobreak\ =$$
$$\displaystyle 1+\frac{2m}{\hbar^{2}}\leavevmode\nobreak\ C^{\tau}_{0}%
\leavevmode\nobreak\ \rho_{0}$$
$$\displaystyle\equiv\leavevmode\nobreak\ 1+\kappa_{\mathrm{s}},$$
(3a)
$$\displaystyle\frac{m}{m^{\ast}_{\mathrm{v}}}\leavevmode\nobreak\ =$$
$$\displaystyle 1+\frac{2m}{\hbar^{2}}\leavevmode\nobreak\ \left(C^{\tau}_{0}-C^%
{\tau}_{1}\right)\leavevmode\nobreak\ \rho_{0}$$
$$\displaystyle\equiv\leavevmode\nobreak\ 1+\kappa_{\mathrm{v}}.$$
(3b)
We use the usual convention for the isovector effective mass, which stems from
its definition through the enhancement factor $\kappa_{\mathrm{v}}$ of the Thomas-Reiche-Kuhn
sum rule Bohigas et al. (1979). However, $m^{\ast}_{\mathrm{v}}$ and $\kappa_{\mathrm{v}}$ are not isovector
quantities in the sense of isovector couplings of the functional.
In the following, we shall discuss the value of $\Delta m^{\ast}(I)$ at $I=1$, which we note
$\Delta m^{\ast}$ in the following, for the sake of brevity. We have
$$\displaystyle\frac{\Delta m^{\ast}}{m}$$
$$\displaystyle=$$
$$\displaystyle\frac{2(\kappa_{\mathrm{v}}-\kappa_{\mathrm{s}})}{(1+\kappa_{%
\mathrm{s}})^{2}-(\kappa_{\mathrm{v}}-\kappa_{\mathrm{s}})^{2}},$$
(4)
such that $\Delta m^{\ast}>0$ for $\kappa_{\mathrm{v}}>\kappa_{\mathrm{s}}$, or equivalently $m^{\ast}_{\mathrm{v}}<m^{\ast}_{\mathrm{s}}$, or
$C^{\tau}_{1}<0$.
Bulk properties of $f_{x}$ parameterizations are displayed in Table 1.
We
note that, while the position of the saturation point varies little between our
forces (SLy5 and $f_{x}$), this consistency is lost in the case of LNS and SkP.
These properties depend on the observable used in the fitting procedure. In the
case of LNS, the saturation point relates to an Extended
Brueckner-Hartree-Fock (EBHF) calculation Zuo et al. (1999), predicting values of
$(E/A)_{\mathrm{sat}}$ and $\rho_{\mathrm{sat}}$ which are larger than empirical ones. A
similar but lesser trend is observed for SkP. In this case it seems to be
correlated with the choice of effective masses and their interplay with other
parameters of the force. Indeed, binding energies computed with SkP compare
satisfactorily with experimental ones, while LNS suffers in this respect from
the
lack of readjustment of the saturation point on nuclear data. As it has been
shown in Ref. F.Bertsch et al. (2005), nuclear binding energies are highly sensitive
to the choice of the energy at saturation, which is therefore constrained to a
very tight interval if one wants to reproduce such quantities. This constraint
is especially tight compared to the uncertainty of ab-initio predictions.
Despite the fit of surface properties ($C^{\Delta\rho}_{0}$ parameter) on a set of
nuclear
data, the accuracy of binding energies predicted by LNS is of the order of 5%,
to be compared with less than 1% for SLy5.
II.3 Properties of the nuclear matter EOS
It is interesting to note that SLy parameterizations were fitted to PNM EOS
with the idea of improving isospin properties of the functionals. One
consequence was to generate functionals with $\Delta m^{\ast}<0$, in opposition to
ab-initio predictions. On the other hand, older functionals such as
SIII Beiner et al. (1975) and SkM${}^{\ast}$ Bartel et al. (1982), which were not fitted
to PNM, had $\Delta m^{\ast}>0$. The same exact situation happens for the Gogny
force Chappert et al. (2006). Thus, improving global isovector properties (EOS)
seems to deteriorate those related to single-particle states ($m^{\ast}_{\mathrm{v}}$) with
currently used functionals. This can be better understood by examining the
expressions for SNM and PNM EOS:
$$\displaystyle\frac{E}{A}(\rho_{0},I=0)$$
$$\displaystyle=$$
$$\displaystyle\frac{3}{5}\frac{\hbar^{2}}{2m}\left(\frac{3\pi^{2}}{2}\right)^{2%
/3}\rho_{0}^{2/3}+C^{\rho}_{0}(\rho_{0})\leavevmode\nobreak\ \rho_{0}$$
(5a)
$$\displaystyle+\leavevmode\nobreak\ C^{\tau}_{0}\frac{3}{5}\left(\frac{3\pi^{2}%
}{2}\right)^{2/3}\rho_{0}^{5/3},$$
$$\displaystyle\frac{E}{A}(\rho_{0},I=1)$$
$$\displaystyle=$$
$$\displaystyle\frac{3}{5}\frac{\hbar^{2}}{2m}\left(3\pi^{2}\right)^{2/3}\rho_{0%
}^{2/3}$$
(5b)
$$\displaystyle+\leavevmode\nobreak\ [C^{\rho}_{0}(\rho_{0})+C^{\rho}_{1}(\rho_{%
0})]\rho_{0}$$
$$\displaystyle+\leavevmode\nobreak\ [C^{\tau}_{0}+C^{\tau}_{1}]\leavevmode%
\nobreak\ \frac{3}{5}\left(3\pi^{2}\right)^{2/3}\rho_{0}^{5/3}\,.$$
If $C^{\rho}_{t}(\rho_{0})$ coefficients only contain one low power of the density
($\propto\rho_{0}^{1/6}$), the latter influences low-density parts of the EOS
more than high-density ones. The effective mass term then determines the
high-density part of the EOS. In SNM, this translates into the well-known
relation between $m^{\ast}_{\mathrm{s}}$ and the incompressibility
$K_{\infty}$ Chabanat et al. (1997); Chabanat
et al. (1998a, b). In
the case of PNM, the EOS above $\rho_{\mathrm{sat}}$ is then mostly fixed by the term
proportional to $C^{\tau}_{0}+C^{\tau}_{1}$ in Eq. (5b), and any attempt to
use the density dependence to counteract its effects, results in a very strong
constraint on the latter. This in turn degrades the behavior of the functional
at and below saturation density and the fit to properties of finite nuclei. We
recall at this point that the condition $\Delta m^{\ast}>0$ corresponds to $C^{\tau}_{1}<0$, which drives the high-density PNM EOS down and explains why usual Skyrme
functionals predict either a collapse of the PNM EOS if $\Delta m^{\ast}>0$, or, like
the SLy functionals fitted to PNM EOS, the wrong sign of the effective mass
splitting in neutron rich matter.
If $C^{\rho}_{t}(\rho_{0})$ coefficients contain an additional density dependence
with a higher
power, the previous discussion does not apply: using two density-dependent
terms in the functional ($\propto\rho_{0}^{1/3};\rho_{0}^{2/3}$)
Cochet et al. (2004) allowed us to construct ($f_{-}$, $f_{0}$, $f_{+}$) with a
good fit to PNM EOS, a free choice of effective masses and satisfactory nuclear
properties.
The previous discussion already shows the type of problems and information
arising from our attempt to improve on the fitting protocol of SLy functionals
by using more inputs from ab-initio calculations.
Now, Fig. 1 shows SNM and PNM EOS as obtained from ($f_{-}$,
$f_{0}$, $f_{+}$, SLy5) and as predicted by Variational Chain Summation (VCS)
methods Akmal et al. (1998). At this point, one can see that the four forces
($f_{-}$, $f_{0}$, $f_{+}$, SLy5) reproduce both microscopic EOS with the same
accuracy. However, it remains to be seen whether or not this translates into
identical global spin-isospin properties and into similar nuclear structure
properties.
II.4 Effects on properties of nuclei
We now study the effects of the variation of the isovector effective mass on
selected properties of spherical nuclei. We start with HF single-particle
energies, then binding energies, ending with a short sum-rule based analysis
of isovector giant resonances.
For computations of open-shell nuclei, we use, in the particle-particle
channel, a zero-range interaction with a density dependent (mixed
surface and volume) form factor defined as:
$$V_{\mathrm{pair}}(\mathbf{R},\mathbf{r})=V_{0}\leavevmode\nobreak\ \delta(%
\mathbf{r})\left[1-\frac{\rho_{0}(\mathbf{R})}{2\rho_{\mathrm{sat}}}\right]\,,$$
(6)
where $\mathbf{R}=(\mathbf{r}_{1}+\mathbf{r}_{2})/2$ and
$\mathbf{r}=\mathbf{r}_{1}-\mathbf{r}_{2}$.
The local HFB equations are renormalized following the procedure developed in
Ref. Bulgac and Yu (2002).
The strength $V_{0}$ is adjusted to the mean pairing gaps of six semi-magic
nuclei
(neutron gaps in ${}^{120}$Sn, ${}^{198}$Pb, ${}^{212}$Pb and proton gaps in
${}^{92}$Mo, ${}^{144}$Sm and ${}^{212}$Rn). In this procedure we compute
theoretical spectral gaps defined as
$\Delta_{th}=\mathrm{Tr}(\tilde{h}\tilde{\rho})/\mathrm{Tr}(\tilde{\rho})$,
where $\tilde{h}$ is the pairing field and $\tilde{\rho}$ the pairing
density Dobaczewski et al. (1984),
and adjust
each of them upon an experimental gap extracted through a five point difference
formula from masses of neighboring nuclei, as suggested in
Ref. Duguet et al. (2002b).
II.4.1 Single-particle energies
Effective masses are known to control the average density of single-particle
states. It is thus interesting to check to what extent such statement applies
to neutron-rich nuclei when varying $m^{\ast}_{\mathrm{v}}$.
In this part of the study, we are mainly interested in evaluating
the change in the single-particle energies generated by the functional
for different splittings and not directly by a comparison with
experimental results.
Single-particle energies in ${}^{132}$Sn and ${}^{208}$Pb are plotted on
Fig. 2. The general trend followed by neutron states with increasing
$\Delta m^{\ast}$ (from force $f_{-}$ to $f_{+}$) corresponds to an increase of the
density of neutron states: they tend to come closer to the Fermi energy
$\varepsilon_{\mathrm{F}}$;
notable exceptions being both
neutron $1i$ levels in ${}^{208}$Pb. The opposite behavior is observed in
proton levels, which spread away from $\varepsilon_{\mathrm{F}}$ with increasing
$\Delta m^{\ast}$ (except for
the proton $1h_{11/2}$ level). However, these trends are rather marginal, which
can be linked with the moderate bulk asymmetry of these nuclei ($I=(N-Z)/A=0.24$ for ${}^{132}$Sn and $0.21$ for ${}^{208}$Pb). This moderate asymmetry
means that the isovector term in the definition of the effective mass
(Eq. (1b)) is weakly probed.
Let us therefore examine similar spectra for more neutron-rich nuclei,
i.e.
${}^{78}$Ni ($I=0.28$, experimentally observed Hosmer et al. (2005)) and
${}^{156}$Sn
($I=0.36$). The ${}^{156}$Sn nucleus is used as an example of an extremely
asymmetric system, even beyond the reach of planned radioactive beam
facilities sp2 (2006). We observe on the lower right panel of Fig. 3
that the effect of $\Delta m^{\ast}$ on proton single-particle energies at $Z=50$
is more pronounced
in ${}^{156}$Sn than it was in ${}^{132}$Sn. The modification of level
densities appears quite clearly in ${}^{78}$Ni also, while neutron levels
around $\varepsilon_{\mathrm{F}}$ in ${}^{156}$Sn are shifted in a slightly more
disordered way.
High-$\ell$/low-$n$ orbitals ($n,\ell$ being respectively the principal
and orbital quantum numbers) are in fact more sensitive to variations of
the spin-orbit field than to $\Delta m^{\ast}$ because of
their spatial localization near the surface of the nucleus. The
spin-orbit field is modified between functionals by the interplay between
$\mathbb{J}^{2}$-term coefficients and effective mass parameters, since these both
depend on the same non-local terms of the Skyrme force Dobaczewski (2006).
The spin-orbit force
($\rho\boldsymbol{\nabla}\cdot\mathbf{J}$ terms in the EDF),
which is subject to a slight readjustment, does affect the spectra as well. We
observed, overall, a marginal increase of the spin-orbit field strength when
going from $f_{-}$ to $f_{+}$. This implies that while the global effect of
modifying the level density is quite clearly observed when we alter the
effective mass parameters, details of the spectroscopy are at least as
sensitive to the terms connected to the spin-orbit field.
II.4.2 Pairing gaps
As an example, neutron spectral gaps are plotted on Fig. 4 for Sn
and Pb series, up to the drip line, against experimental gaps extracted through
five-point mass formulas Duguet et al. (2002a, b). The slight change in the
level density translates into a modification of the pairing gaps: a higher
neutron effective mass ($f_{+}$) corresponds to a denser spectrum and higher
gaps. The effect, which increases with asymmetry, remains however very small,
because of the limited alteration of single-particle levels seen
on Figs. 2 and 3.
In the end, the effect is negligible and would be overwhelmed by any
other modification of the particle-hole part of the functional.
For example the spin-orbit force,
acting on the detailed level scheme, could alter the shape of gaps. The
pairing functional itself is a subject of current debate regarding its density
dependence, regularization scheme and finite-range
corrections Duguet and Bennaceur (2006), while the HFB formalism can itself be
improved (particle number projection) as well as the mere choice of observable
to be compared (definition of theoretical an experimental gaps), although our
choice has been proven to be possibly the most sound for extracting pure
pairing effects Duguet et al. (2002b).
II.4.3 Binding energies
Let us now study the effect of the aforementioned variation of level densities
and pairing gaps on binding energies. On Fig. 5 we show the
binding
energy residuals $E_{\mathrm{th}}-E_{\mathrm{exp}}$ for Sn and Pb isotopes and
$N=50$ and $N=82$
isotones. The evolution of $E_{\mathrm{th}}-E_{\mathrm{exp}}$ along such series
is usually
plagued by an underbinding of open-shell nuclei with respect to closed-shell
ones which translates into an arch
shape of $E$-residual curves. Although the variation of $m^{\ast}_{\mathrm{v}}$ seems to impact
the arches, again, the effect is negligible compared to the absolute value of
deviations from experiment, except in the $N=82$ series where open-shell nuclei
tend to be more underbound in the case of $f_{+}$.
II.4.4 Isovector giant resonances
The isovector effective mass is usually defined from the energy-weighted
sum rule $\mathfrak{m}_{1}$ (the TRK sum rule Bohigas et al. (1979)) of the isovector giant
dipole resonance (IVGDR):
$$\mathfrak{m}_{1}(E1\;;\leavevmode\nobreak\ T=1)=\frac{\hbar^{2}}{2m}\frac{NZ}{%
A}\left(1+\kappa_{\mathrm{v}}\right)=\frac{\hbar^{2}}{2m}\frac{NZ}{A}\frac{m}{%
m^{\ast}_{\mathrm{v}}},$$
(7)
which exhibits its link with the strength distribution of isovector collective
modes. We perform here a schematic study of dynamical properties of $f_{-}$,
$f_{0}$, $f_{+}$ by means of results derived in Ref. Colò et al. (1995). Thanks to RPA
sum rules similar to Eq. (7), it is possible to fit an accurate
parameterization of the energy $E_{1}=\mathfrak{m}_{1}/\mathfrak{m}_{-1}$ of
isovector giant resonances in a given nucleus as a function of Skyrme
parameters. Results for GDR ($L=1$) and isovector giant monopole (IVGMR, $L=0$)
modes in ${}^{208}$Pb are shown in Table 2, compared to experimental
energies (respectively from Refs. Ritman et al. (1993) and Erell et al. (1986) and
corrected, as suggested in Colò et al. (1995), for the shift due to the spreading of
the strength by damping effects – 2 MeV for GMR, 1 MeV for GDR).
While $f_{-}$ predicts both energies lower than experimental ones, values
for $f_{0}$ and $f_{+}$ are compatible with experiment for the $L=0$ mode, and
only $f_{+}$ approaches the experimental value for the $L=1$ mode. This suggests
that values of $\kappa_{\mathrm{v}}$ corresponding to a positive value of $\Delta m^{\ast}$ (equal to
or higher than 0.43 in our case) better describe isovector dynamics than lower
values.
As a summary, the effect of the splitting of neutron and proton effective
masses
with isospin asymmetry on single-particle energies, pairing gaps and binding
energies, is noticeable and consistent, yet limited and thus hardly meaningful
when compared to the overall (in)accuracy of the predictions made by current
nuclear EDF. In fact,
the main reason for not seeing a dramatic modification of EDF predictions when
altering $\Delta m^{\ast}$ is the limited amount of
strongly asymmetric nuclear matter at high enough density in the ground state
of nuclei with realistic isospin as already suggested in Goriely et al. (2003).
This makes the effect of the isovector effective
mass rather marginal. Giant isovector resonances are certainly more fruitful
to seek for an effect of a modification of $\Delta m^{\ast}$. Indeed,
a sum-rule-based analysis of isovector collective modes
allows a slightly more clear-cut conclusion, with a tendency to favor
$\Delta m^{\ast}\gtrsim 0$. The conclusion of the phenomenological study done in this
section is that, while no observable listed here
strongly ask for $\Delta m^{\ast}>0$, there is no reason to omit this constraint in
future functionals, since, as already stated,
ab-initio predictions for the sign of $\Delta m^{\ast}$ are solid.
There remains to check the intrinsic consistency of the functional in terms of
other ab-initio inputs and stability criteria.
III Further study of infinite matter
III.1 Separation of the EOS into $(S,T)$ channels
III.1.1 Introduction
In this section, we discuss the contributions to the potential energy of SNM
from the four two-body spin-isospin $(S,T)$ channels. We compare our results
with those predicted by BHF calculations Baldo (2006) using
AV18 Wiringa et al. (1995) two-body interaction and a three-body force constructed
from meson exchange theory Grangé et al. (1989); Lejeune et al. (2000).
Using projectors on spin singlet and triplet states, respectively
$$\displaystyle\hat{P}_{S=0}\leavevmode\nobreak\ =\leavevmode\nobreak\ \frac{1}{%
2}(1-\hat{P}_{\sigma}),$$
$$\displaystyle\hat{P}_{S=1}\leavevmode\nobreak\ =\leavevmode\nobreak\ \frac{1}{%
2}(1+\hat{P}_{\sigma}),$$
(8)
where $\hat{P}_{\sigma}$ is the spin-exchange operator, and similar expressions for
isospin projectors $\hat{P}_{T}$ using the isospin exchange operator $\hat{P}_{\tau}$, yields
the potential energy in each $(S,T)$ channel
$$\displaystyle E^{ST}_{\mathrm{pot}}$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{2}\sum_{kl}{\left\langle\>{kl}\left|\raisebox{0.0pt}[6.9%
99893pt][3.079953pt]{$V\hat{P}_{S}\hat{P}_{T}$}\right|{\overline{kl}}\>\right%
\rangle}\rho_{kk}\rho_{ll},$$
(9)
where the sum on $k,l$ runs over all HF single-particle eigenstates whereas
$\rho_{kk}$ designates the diagonal one-body density matrix. The notation
$|\overline{kl}\rangle$ denotes a non-normalized but antisymmetrized
two-body state. In order to compare different
many-body approaches (ab-initio or EDF), we use the “potential
energy” which refers to the total binding energy from which is subtracted the
kinetic energy of the non-interacting particle system.
Note that
due to the zero-range character of the Skyrme force, together with at most
second-order derivative terms, only $L=0,1$ partial waves occur explicitly
whereas higher partial waves contribute to ab-initio EOS.
We find, for SNM,
$$\displaystyle\frac{E^{00}_{\mathrm{pot}}}{A}$$
$$\displaystyle=$$
$$\displaystyle\frac{3}{160}t_{2}(1-x_{2})\left(\frac{3\pi^{2}}{2}\right)^{2/3}%
\rho_{0}^{5/3},$$
(10a)
$$\displaystyle\frac{E^{10}_{\mathrm{pot}}}{A}$$
$$\displaystyle=$$
$$\displaystyle\frac{3}{16}\sum_{i=0}^{2}t_{0i}(1+x_{0i})\rho_{0}^{1+i/3}+\frac{%
9}{160}t_{1}(1+x_{1})\left(\frac{3\pi^{2}}{2}\right)^{2/3}\rho_{0}^{5/3},$$
(10b)
$$\displaystyle\frac{E^{01}_{\mathrm{pot}}}{A}$$
$$\displaystyle=$$
$$\displaystyle\frac{3}{16}\sum_{i=0}^{2}t_{0i}(1-x_{0i})\rho_{0}^{1+i/3}+\frac{%
9}{160}t_{1}(1-x_{1})\left(\frac{3\pi^{2}}{2}\right)^{2/3}\rho_{0}^{5/3},$$
(10c)
$$\displaystyle\frac{E^{11}_{\mathrm{pot}}}{A}$$
$$\displaystyle=$$
$$\displaystyle\frac{27}{160}t_{2}(1+x_{2})\left(\frac{3\pi^{2}}{2}\right)^{2/3}%
\rho_{0}^{5/3}\,,$$
(10d)
where $(t_{i},x_{i})$ are usual coefficients of the Skyrme EDF, whereas
$(t_{0i},x_{0i})$ characterize the density-independent zero-range term and the
two density-dependent ones, following the parameterization of
equation (21).
The coefficients occurring in
Eqs. (10a)–(10d) stem from the
antisymmetrization condition $(-)^{L+S+T}=-1$, the relative angular momentum
$L$ being even for $t_{0i}$ and $t_{1}$ ($\mathbf{k}^{2}$) terms and odd for $t_{2}$
($\mathbf{k}^{\prime}\cdot\mathbf{k}$) terms. The expression of the potential energy in channels
$(S,T)=(0,0)$ and $(1,1)$ is very simple since only the $t_{2}$
term contributes.
III.1.2 Force vs. functional
Previous statements, however, apply only to the case where the EDF is computed
as the expectation value of an (antisymmetrized) effective force. In the more
general case, it is still possible to define $(S,T)$ channels starting
from any Hartree-like functional. Indeed, the functional can
always be expressed in terms of an effective non-antisymmetrized vertex and one
can still plug a projector in the calculation of its matrix elements.
In the pure functional case, there is however no more clear definition of
partial waves, and spin-isospin channels emerge from the balance between
coefficients of (iso)scalar/(iso)vector couplings (see appendix B
for a more detailed discussion).
As long as there are not enough inputs to constrain all
degrees of freedom of a general functional, the “force” approach remains
as an acceptable path, and hence shall be used in the following.
III.1.3 Results
Results are plotted against BHF predictions on Fig. 6. First, one
can observe that results are rather scattered. Second, the main source of
binding, from $(S,T)=(0,1)$ and $(1,0)$ channels, is not well described and
the detailed saturation mechanism is not captured. It is clear that, even
though
all four forces reproduce perfectly PNM and SNM EOS, they do not have the
same spin-isospin content, and that the latter is in general rather poor. Thus,
fitting the global EOS is an important element but it does not mean that
spin-isospin properties of the functional are fixed once and for all. One needs
to do more and fitting ab-initio predictions of $E_{\mathrm{pot}}^{(S,T)}$
seems to be a good idea in the near future. However, one needs to make
sure that the theoretical uncertainty of the data used is smaller than the
expected accuracy of the fit to them. This calls for predictions from other
ab-initio methods using the same two-body plus three-body Hamiltonian.
Then, those ab-initio calculations should be repeated using different sets of
two-body plus three-body Hamiltonians in order to provide a theoretical error
bar on those predictions.
The most obvious discrepancy appears in $(0,0)$ and $(1,1)$ channels where
Skyrme and BHF data have opposite signs above saturation density. The SLy5
parameter set shows a particular behavior in channel $(1,1)$ due to the
choice of $x_{2}=-1$ to prevent ferromagnetic instabilities in PNM. Note that
these two channels are taken care of, in the Skyrme functional, by the
density-independent $P$-wave term only. The upper-right panel of
Fig. 6 points out the tendency of Skyrme parameterizations to be
attractive in polarized PNM, and hence to cause a collapse of its EOS at high
density. At lower densities, BHF data show a distinctive behavior, being
slightly attractive below $\rho_{\mathrm{sat}}$ and repulsive above. This feature cannot be
matched by the standard Skyrme functional which exhibits a monotonous behavior
as a function of density in this channel, regardless of the value of
$(t_{2},x_{2})$.
It is also worth noticing that the failure in channel $(1,1)$ becomes more
and more prominent as one makes $\Delta m^{\ast}$ closer to the ab-initio predictions
(force $f_{+}$). The effective masses being governed by the momentum dependent
terms of the force, it is not a surprise that the modification of the former
impacts channels $(0,0)$ and $(1,1)$. What changes in the coefficients
entering Eqs. (10a-10d) stems only from
the variation of $m^{\ast}_{\mathrm{v}}$ and the associated rearrangement of parameters in the
functional, most notably the $C^{\Delta\rho}_{0,1}$ coefficients closely related to
surface and surface-symmetry energies. The relatively tight requirements on the
latter imply that the four parameters of the non-local terms in the standard
Skyrme energy functional would be dramatically overconstrained if we were to
add the $(S,T)$-channel decomposition in the fitting data.
In the end, the rather poor properties of the functional in channels $(0,0)$
and $(1,1)$, the degradation of the latter as the effective mass splitting
is improved, the idea of using ab-initio $(S,T)$ contributions in the fit,
call, at least, for a refinement of the odd-$L$ term in the sense either of a
density dependence or of a higher-order derivative term. The latter being prone
to numerical instabilities and interpretation problems, a density-dependent
$\mathbf{k}^{\prime}\cdot\mathbf{k}$ term remains as one of the next potential enhancements to be
brought to the Skyrme EDF (density-dependent derivative terms have
been considered already, but with a focus on even-$L$
terms of the form $t_{4}(\mathbf{k}^{2}+\mathbf{k}^{\prime 2})\rho_{0}^{\beta}$ Farine and Tondeur (1997)).
Phenomenological constraints on gradient terms are mainly related to the
surface
of nuclei, i.e. low-density regions. One can expect that, to first order, BHF
data in channel $(S,T)=(1,1)$ can be matched with an extended functional while
retaining a good agreement with other (experimental) data. It is less clear in
channel $(0,0)$ but further exploration of the extended parameter space may
bring Skyrme and BHF data in better agreement.
III.2 RPA linear response functions and the diagnosis of instabilities
We attempt here to study general stability conditions of SNM with respect to
finite-size density, spin, isospin and spin-isospin perturbations. Our basic
ingredient is the RPA response function Fetter and Walecka (1971) derived analytically
in Ref. García-Recio
et al. (1992) for the central part of the Skyrme interaction.
Recent work was done to incorporate the effect of the spin-orbit part which was
found to be quite negligible Margueron et al. (2006), and will be omitted in the
present work. One starts by defining a one-body perturbing operator
$$\displaystyle{\cal Q}^{(\alpha)}$$
$$\displaystyle=$$
$$\displaystyle e^{-i\omega t}\leavevmode\nobreak\ \sum_{a}e^{i\mathbf{q}\cdot%
\mathbf{r}_{a}}\leavevmode\nobreak\ \Theta_{a}^{(\alpha)},$$
(11)
where $a$ indexes particles in the system. The one-body spin-isospin
operators $\Theta_{a}^{(\alpha)}$ are defined as
$$\Theta_{a}^{\mathrm{ss}}=1_{a},\leavevmode\nobreak\ \leavevmode\nobreak\ %
\leavevmode\nobreak\ \Theta_{a}^{\mathrm{vs}}=\boldsymbol{\sigma}_{a},%
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \Theta_{a}^{%
\mathrm{sv}}=\vec{\tau}_{a},\leavevmode\nobreak\ \leavevmode\nobreak\ %
\leavevmode\nobreak\ \Theta_{a}^{\mathrm{vv}}=\boldsymbol{\sigma}_{a}\vec{\tau%
}_{a},$$
(12)
where we use the denomination of (iso-)scalar (s) and (iso-)vector (v) channels
in order to distinguish the uncoupled spin-isospin channels from the coupled
two-body $(S,T)$ channels discussed in the previous section. In
Eq. (12) and the following, the first (second) subscripts
denotes the spin (isospin). We then study the response to each type of
perturbation separately through the response functions
$$\displaystyle\chi^{(\alpha)}(\omega,\mathbf{q})$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{\Omega}\sum_{n}|\langle n|{\cal Q}^{(\alpha)}|0\rangle|^%
{2}$$
(13)
$$\displaystyle\times$$
$$\displaystyle\left(\frac{1}{\omega-E_{n0}+i\eta}-\frac{1}{\omega+E_{n0}-i\eta}%
\right),$$
at the RPA level, where $\Omega$ stands for a normalization volume and $|n\rangle$ is an excited state of the system, $E_{n0}$ being the corresponding
energy. Since the central residual interaction does not couple the channels
defined through Eq. (12) in SNM, we can indeed consider each channel
separately.
The response function $\chi^{(\alpha)}$ can be seen as the propagator of the
collective perturbation, i.e. the positions of its poles in the
$(q,\omega)$
plane yield the dispersion relation of the mode. In this formalism, the onset
of an unstable mode is marked by the occurrence of a pole in $\chi^{(\alpha)}$ at
$\omega=0$, corresponding to zero excitation energy. Such a pole marks the
transition between stable ($\chi^{(\alpha)}<0$) and unstable ($\chi^{(\alpha)}>0$)
domains. Unstable modes of infinite wavelength ($q=0$) are those traditionally
discussed in the context of Landau parameters. A pole at finite $q$
characterizes a system which is unstable with respect to the appearance of a
spatial oscillation of a given type (density, spin, isospin or spin-isospin)
with a given wavelength $\lambda=2\pi/q$.
The evaluation of response functions calls for the residual interaction
$V_{\mathrm{p-h}}$,
defined as the second-order functional derivative of the energy with respect to
the density matrix. Its momentum-space matrix elements can be written, using
total momentum conservation, as García-Recio
et al. (1992):
$$\displaystyle V_{\mathrm{p-h}}(\mathbf{q}_{1},\mathbf{q}_{2},\mathbf{q})$$
$$\displaystyle=$$
$$\displaystyle\langle\mathbf{q}_{1}\leavevmode\nobreak\ \mathbf{q}_{2}+\mathbf{%
q}|\leavevmode\nobreak\ V_{\mathrm{p-h}}\leavevmode\nobreak\ |\mathbf{q}_{1}+%
\mathbf{q}\leavevmode\nobreak\ \mathbf{q}_{2}\rangle,$$
(14)
$$\displaystyle=$$
$$\displaystyle\hat{W}_{1}(q)+\hat{W}_{2}(q)\leavevmode\nobreak\ (\mathbf{q}_{1}%
-\mathbf{q}_{2})^{2},$$
(15)
with
$$\displaystyle\hat{W}_{1}(q)$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{4}\leavevmode\nobreak\ [\leavevmode\nobreak\ W_{1}^{%
\mathrm{ss}}(q)+W_{1}^{\mathrm{vs}}(q)\leavevmode\nobreak\ \boldsymbol{\sigma}%
_{1}\cdot\boldsymbol{\sigma}_{2}+W_{1}^{\mathrm{sv}}(q)\leavevmode\nobreak\ %
\vec{\tau}_{1}\circ\vec{\tau}_{2}$$
(16)
$$\displaystyle+\leavevmode\nobreak\ W_{1}^{\mathrm{vv}}(q)\leavevmode\nobreak\ %
\boldsymbol{\sigma}_{1}\cdot\boldsymbol{\sigma}_{2}\leavevmode\nobreak\ \vec{%
\tau}_{1}\circ\vec{\tau}_{2}\leavevmode\nobreak\ ],$$
and a similar expression for $\hat{W}_{2}$. We find:
$\displaystyle\frac{W_{1}^{\mathrm{ss}}(q)}{4}$
$\displaystyle=$
$\displaystyle 2C^{\rho,0}_{0}+\sum_{i=1}^{2}C^{\rho,i}_{0}\frac{(i+6)(i+3)}{9}%
\rho_{0}^{i/3}-2C^{\Delta\rho}_{0}\mathbf{q}^{2},$
(17a)
$\displaystyle\frac{W_{1}^{\mathrm{vs}}(q)}{4}$
$\displaystyle=$
$\displaystyle 2C^{s,0}_{0}+2\sum_{i=1}^{2}C^{s,i}_{0}\rho_{0}^{i/3}-2C^{\Delta
s%
}_{0}\mathbf{q}^{2},$
(17b)
$\displaystyle\frac{W_{1}^{\mathrm{sv}}(q)}{4}$
$\displaystyle=$
$\displaystyle 2C^{\rho,0}_{1}+2\sum_{i=1}^{2}C^{\rho,i}_{1}\rho_{0}^{i/3}-2C^{%
\Delta\rho}_{1}\mathbf{q}^{2},$
(17c)
$\displaystyle\frac{W_{1}^{\mathrm{vv}}(q)}{4}$
$\displaystyle=$
$\displaystyle 2C^{s,0}_{1}+2\sum_{i=1}^{2}C^{s,i}_{1}\rho_{0}^{i/3}-2C^{\Delta
s%
}_{1}\mathbf{q}^{2},$
(17d)
$\displaystyle\frac{W_{2}^{\mathrm{ss}}(q)}{4}$
$\displaystyle=$
$\displaystyle C^{\tau}_{0},$
(18a)
$\displaystyle\frac{W_{2}^{\mathrm{vs}}(q)}{4}$
$\displaystyle=$
$\displaystyle C^{sT}_{0},$
(18b)
$\displaystyle\frac{W_{2}^{\mathrm{sv}}(q)}{4}$
$\displaystyle=$
$\displaystyle C^{\tau}_{1},$
(18c)
$\displaystyle\frac{W_{2}^{\mathrm{vv}}(q)}{4}$
$\displaystyle=$
$\displaystyle C^{sT}_{1}.$
(18d)
With the above expression for the residual interaction, the response
function reads as
$$\displaystyle\chi^{(\alpha)}$$
$$\displaystyle(\omega,\mathbf{q})$$
$$\displaystyle=4\Pi_{0}\left[\raisebox{0.0pt}[20.0pt][10.0pt]{\leavevmode%
\nobreak\ }\right.1-W_{1}^{(\alpha)}\Pi_{0}-2W_{2}^{(\alpha)}k_{\mathrm{F}}^{2%
}\left(\overline{q}^{2}-\frac{\nu^{2}}{1-\frac{m^{\ast}k_{\mathrm{F}}^{3}}{3%
\pi^{2}}\leavevmode\nobreak\ W_{2}^{(\alpha)}}\right)\Pi_{0}$$
$$\displaystyle+$$
$$\displaystyle 2W_{2}^{(\alpha)}k_{\mathrm{F}}^{2}(2\overline{q}^{2}\leavevmode%
\nobreak\ \Pi_{0}-\Pi_{2})+(W_{2}^{(\alpha)}k_{\mathrm{F}}^{2})^{2}\left(\Pi_{%
2}^{2}-\Pi_{0}\Pi_{4}+4\overline{q}^{2}\nu^{2}\Pi_{0}^{2}-\frac{2m^{\ast}k_{%
\mathrm{F}}}{3\pi^{2}}\overline{q}^{2}\Pi_{0}\right)\left.\raisebox{0.0pt}[20.%
0pt][10.0pt]{\leavevmode\nobreak\ }\right]^{-1},$$
where $\overline{q}=q/2k_{\mathrm{F}}$, $\nu=\omega m^{\ast}_{\mathrm{s}}/qk_{\mathrm{F}}$ and $\Pi_{0,2,4}$
are generalized Lindhard functions, see Ref. García-Recio
et al. (1992).
As already said, the limit $\mathbf{q}\rightarrow 0$ corresponds to perturbations
of infinite wavelength, keeping the system homogeneous. There, the residual
interaction is uniquely determined by Landau parameters $F_{l},F^{\prime}_{l},G_{l},G^{\prime}_{l}$,
with $l=0,1$, and well known stability conditions are obtained under the
form Midgal (1967):
$$\displaystyle 1+\frac{X_{l}}{2l+1}>0,$$
(20)
where $X_{l}$ represents any of the Landau parameters. We have used this
criterion in the fit of our forces $f_{x}$, ensuring that no spin or spin-isospin
instability would occur below $2\rho_{\mathrm{sat}}$. We observe that, from the point of
view of Landau parameters, the most critical channel is the vector-isovector
one, with associated instabilities at densities as low as $2\rho_{sat}$
(see the upper-right panel of Fig. 9). This behavior is linked
to the attractive
character of the functional in channel $(S,T)=(1,1)$ which gives rise to
a collapse of spin-polarized PNM, and accordingly, a vanishing spin-isospin
symmetry energy. Therefore, better reproducing the decomposition into $(S,T)$
channels of EOS obtained from ab-initio methods is not only a matter of
microscopic motivation, but also a necessity to avoid unwanted instabilities.
Beyond infinite wavelength instabilities, we also aim at demonstrating that a
more general treatment is needed to fully describe and control unstable modes
which arise in the Skyrme EDF framework. Thus, contributions to the residual
interaction coming from functional terms of the form $\rho\Delta\rho$ are zero
for $\mathbf{q}=0$, whereas such terms drive finite-size instabilities.
Indeed, we have observed that existing (SkP) or new parameterizations built
with a high value of $\kappa_{v}$ in order to reproduce the microscopic splitting
of effective masses, tend to spatially separate protons from neutrons in
spherical mean-field calculations, where enough iterations lead to states with
strongly oscillating densities and a diverging energy. Following a preliminary
phenomenological reasoning, we could relate this effect to the
$C^{\Delta\rho}_{1}{\vec{\rho}}_{1}\cdot\Delta{\vec{\rho}}_{1}$ term in the functional, as this term
can energetically favor strong oscillations of the isovector density ${\vec{\rho}}_{1}$
which arise in the case of such a spatial n-p separation. Moreover,
Eqs. (17a-18d) show that such a term
can yield an attractive contribution to the residual interaction in the case of
a short-wavelength (high $q$) perturbation. We found empirically that parameter
sets for which this instability arises are characterized by a high value of
$C^{\Delta\rho}_{1}$, that is $C^{\Delta\rho}_{1}\gtrsim 30$. As seen from Table 3,
this parameter is strongly correlated with the effective mass splitting
$\Delta m^{\ast}$ in such a way that a positive splitting, as required by ab-initio
predictions, leads to instabilities.
Whereas we were obviously unable to provide a fully converged (and hence
physically meaningful) and clearly unstable force to illustrate the previous
statements, we found that certain forces available in the literature present
the aforementioned behavior. For example, convergence problems
have arisen (and have already been pointed out in another study Terasaki and Engel (2006))
for the SkP parameter set Dobaczewski et al. (1984). The nature of the instabilities
discussed here is illustrated on the lower panels of Fig. 7,
where neutron and proton densities are plotted at various stages of execution
of a self-consistent iterative procedure with SkP in ${}^{56}$Ni. We see that
strong, opposing oscillations of neutron and proton densities are formed, and
steadily increase with iterations. Such a behavior happens after a seemingly
converged situation for which the relative energy variation is small but almost
constant over a large number of iterations and the evolution of the energy
is monotonous.
The study of the linear response function in the scalar-isovector channel
allows us to provide a more quantitative ground to the previous observation. By
plotting critical densities (lowest density of occurrence of a pole in
$\chi^{(\alpha)}(\omega=0,q)$) for a given $q$ on Fig. 8,
we see that these critical densities can be very close to $\rho_{\mathrm{sat}}$ for $q\approx 2.5$ to $3\leavevmode\nobreak\ \mathrm{fm}^{-1}$. This is the case for SkP, which displays the lowest
scalar-isovector critical density of all forces studied in this paper.
Accordingly, it is the most prone to a lack of convergence in HF calculations.
The link between response functions and convergence problems can indeed be
understood by classifying them by their magnitude: in case of a stable but very
soft mode, lack of convergence arises from the existence of a continuum of
quasi-degenerate mean-field states, among which no minimization or
self-consistency algorithm shall be able to decisively find an energy minimum
without a considerable amount of iterations. If the soft mode becomes unstable,
it causes a divergence of the energy and of other observable such as the
densities. We see in the agreement between the RPA study of SNM and the
observation of unstable HF calculations of nuclei a qualitative validation of
our Local Density Approximation (LDA)-based treatment of instabilities: soft
or unstable modes occurring in INM happen for the same parameter sets in finite
nuclei.
The large number of iterations needed for the divergence to occur on
Fig. 7 is a consequence of the limiting case embodied by SkP,
such that the existence of a definite instability is highly dependent on finite
size effects (choice of the nucleus) and discretization details in the
numerical procedure. If SkP is a limiting case, LNS also displays a low
critical
density in the scalar-isovector channel (Fig. 8). In this
case, we observed proton-neutron separation in ${}^{40}$Ca and for small mesh
steps ($0.1$ fm) only (see Fig. 7), while it is more frequent
with SkP. Our force $f_{+}$, with a critical density just slightly higher than
LNS, successfully passed the test of computing a series of 134 spherical
nuclei.
This again demonstrates that testing finite-size instabilities through
response functions constitutes an accurate tool, the critical density (and its
proximity to $\rho_{\mathrm{sat}}$) being a good measure of the gravity of the problems
one might encounter in finite nuclei. Although the actual occurrence of
instabilities is subject to details of the numerical treatment, it is now clear
that their origin can be traced back to the choice of parameters in the
functional itself.
Nevertheless, even if a functional does not display clear instabilities but
only spurious soft collective modes, convergence difficulties shall arise in
mean-field calculations while such a mode will translate into a non-physical
low-lying spectrum in a beyond-mean-field framework. This can then
yield excessive correlation energies if one systematically includes
correlations in the ground state e.g. in (Q)RPA or GCM methods. One
should thus make sure that no spurious (even remotely) soft mode occurs at
saturation density in order to prevent such problems.
Having demonstrated the importance of finite-size instabilities, let us go back
to discussing our original set of forces and perform a generalization to other
spin-isospin channels.
Critical densities are plotted on Fig. 9 for the four channels
defined in Eq. (12). The upper-left panel shows that, while no
unstable mode occurs at $q=0$ thanks to fitting PNM EOS to relatively high
density, scalar-isovector instabilities may happen little above $\rho_{\mathrm{sat}}$ for
$q\approx 2.5$ to $3\leavevmode\nobreak\ \mathrm{fm}^{-1}$. In addition, there is a clear trend for lowering the
critical density when $\Delta m^{\ast}$ is increased, in agreement with the preliminary
phenomenological reasoning on $C^{\Delta\rho}_{1}$.
Spin channels have been taken care of during the fit thanks to Landau
parameters, which describe the residual interaction at $q=0$. The result can
be seen on the right panels of Fig. 9, where the critical
densities of instability are plotted for spin-flip modes (isoscalar and
isovector). As previously stated, the most dangerous $q=0$ instability is found
in the vector-isovector channel. By looking at the upper-right panel of
Fig. 9 one can see that the critical density is even lower
($1.5\rho_{\mathrm{sat}}$) at $q=3\leavevmode\nobreak\ \mathrm{fm}^{-1}$ than at $q=0$, a domain not covered by the
criterion of Eq. (20).
An even more prominent finite-size effect can be observed in the isoscalar
spin-flip channel (lower-right panel of Fig. 9) where, while
no instability occurs at $q=0$ as in the case of most Skyrme forces,
finite-size instabilities occur at low density, even below $\rho_{\mathrm{sat}}$ !
These instabilities are linked to the $C^{\Delta s}_{0}\leavevmode\nobreak\ \mathbf{s}_{0}\cdot\Delta\mathbf{s}_{0}$ term
which makes the vector-isoscalar $V_{\mathrm{p-h}}$ attractive at large $q$ whereas it is
repulsive at $q=0$. Values of $C^{\Delta s}_{0}$, indeed, are as high as $45.85$ and
$47.32$ for SLy5 and $f_{-}$, respectively. As a consequence, one can expect
divergences in calculations of odd or rotating nuclei with the latter forces if
the aforementioned terms are included. In this case, though, increasing
$\Delta m^{\ast}$ pushes the critical density farther from $\rho_{\mathrm{sat}}$: $f_{0}$ and $f_{+}$
functionals are thus the only ones to be free from instabilities at $\rho_{\mathrm{sat}}$,
without being totally satisfactory either.
The previous discussion is valid if the full time-odd functional is
taken into account. This must be stressed since $\mathbf{s}_{0}\cdot\Delta\mathbf{s}_{0}$
terms, which drive the most critical, finite-size instabilities, have never
been included in self-consistent mean field calculations to date. However, RPA
calculations are commonly performed by computing the residual interaction
matrices directly from the antisymmetrized force (plus rearrangement terms),
which amounts to implicitly including the contribution to $V_{\mathrm{p-h}}$ from all
terms in the functional Terasaki et al. (2005).
The latter findings finalize the picture of a competition between spin and
isospin instabilities. All in all, the strong interplay between the
various quantities linked to the four parameters of the non-local terms in
the Skyrme force does not seem to allow for a fully satisfactory compromise
between stability criteria and ab-initio constraints on $\Delta m^{\ast}$.
Again, we see that the non-local part of the Skyrme force is too simplistic to
control all relevant properties. An extension with density- and
momentum-dependent terms, allowing the fine-tuning of the functional at various
densities, combined with the formal checks advocated in this paper, could prove
to significantly improve the predictive power of Skyrme EDF.
IV Conclusion
We have built a series of Skyrme energy density functionals to study the effect
of a variation of the splitting of neutron and proton effective masses with
isospin asymmetry on properties of this mean-field-based model. Thanks to the
use of a second density-dependent term in the underlying effective force, we
could cover a wide range of effective mass splittings ($\Delta m^{\ast}$) with a
satisfactory fit to nuclear properties. Indeed, nuclear observable predicted by
our functionals $f_{-}$, $f_{0}$ and $f_{+}$ show a remarkable similarity, pointing
out that spectra, pairing gaps and masses of bound nuclei are weakly sensitive
to $\Delta m^{\ast}$, mostly due to their relatively low isospin asymmetry. Although
observable were affected in a noticeable and consistent way, no clear
improvement was seen when altering $\Delta m^{\ast}$ either way.
Beyond this phenomenological study, we have compared the splitting of the
equation of state of symmetric infinite matter into spin-isospin channels
provided by our functionals
and by ab-initio Brueckner-Hartree-Fock calculations. Such a comparison showed
an obvious discrepancy in $(S,T)=(0,0)$ and $(1,1)$ channels, where energies
predicted by Skyrme functionals and by BHF calculations have opposite
signs. The inconsistency in channel $(S,T)=(1,1)$, where the Skyrme functional
is attractive, translates into a collapse of polarized neutron matter EOS,
related to the onset of spin-isospin instabilities at quite low density
($2\rho_{\mathrm{sat}}$). In this channel, ab-initio predictions cannot be matched (in the
Skyrme “force” approach) without an extension of the P-wave term. We also
identified finite-size isospin instabilities caused by strong isovector
gradient terms, which prevent the convergence of mean-field calculations. We
were able to provide a firm and quantitative basis to these observations
through an analysis of finite-size instabilities by use of RPA linear response
functions in SNM. The latter showed that finite-size effects in the analysis
of instabilities tend to always dominate.
The present study leads us to propose the systematic inclusion of consistency
checks with ab-initio predictions of spin-isospin properties in the
construction of our future functionals, as well as a systematic diagnosis of
finite-size instabilities.
Acknowledgements.
Work by K. B. was performed within the framework of the Espace de
Structure Nucléaire Théorique (ESNT).
The authors are grateful to D. Lacroix for a careful reading of the manuscript
and helpful comments and also thank M. Bender and B. Cochet for valuable
discussions during the development of this study. Two of us (K. B. and T. L.)
wish to thank the NSCL for its hospitality during final stages of this work.
This work was supported by the U.S. National Science Foundation under Grant No.
PHY-0456903.
Appendix A Skyrme energy functional
We take the particle-hole part of the functional as given by
the expectation value of a Skyrme effective force including
two density-dependent terms:
$$\displaystyle V(\mathbf{R},\mathbf{r})$$
$$\displaystyle=$$
$$\displaystyle\sum_{i=0}^{2}t_{0i}\left(1+x_{0i}\hat{P}_{\sigma}\right)\delta(%
\mathbf{r})\left[\rho_{0}\left(\mathbf{R}\right)\right]^{i/3}$$
(21)
$$\displaystyle+$$
$$\displaystyle\frac{1}{2}t_{1}\left(1+x_{1}\hat{P}_{\sigma}\right)\,\left[%
\delta(\mathbf{r})\leavevmode\nobreak\ \mathbf{k}^{2}+\mathbf{k}^{\prime 2}%
\leavevmode\nobreak\ \delta(\mathbf{r})\right]$$
$$\displaystyle+$$
$$\displaystyle t_{2}\left(1+x_{2}\hat{P}_{\sigma}\right)\mathbf{k}^{\prime}%
\cdot\delta(\mathbf{r})\leavevmode\nobreak\ \mathbf{k}$$
$$\displaystyle+$$
$$\displaystyle iW_{0}\left[\boldsymbol{\sigma}_{1}+\boldsymbol{\sigma}_{2}%
\right]\mathbf{k}^{\prime}\times\delta(\mathbf{r})\leavevmode\nobreak\ \mathbf%
{k},$$
with the usual notations
$$\displaystyle\mathbf{R}$$
$$\displaystyle=$$
$$\displaystyle\left(\mathbf{r}_{1}+\mathbf{r}_{2}\right)/2,$$
(22a)
$$\displaystyle\mathbf{r}$$
$$\displaystyle=$$
$$\displaystyle\mathbf{r}_{1}-\mathbf{r}_{2},$$
(22b)
$$\displaystyle\mathbf{k}$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{2i}\leavevmode\nobreak\ \left(\boldsymbol{\nabla}_{1}-%
\boldsymbol{\nabla}_{2}\right),$$
(22c)
$$\displaystyle\mathbf{k}^{\prime}$$
$$\displaystyle=$$
$$\displaystyle{\mbox{C.C. of $\mathbf{k}$ acting on the left}},$$
(22d)
$$\displaystyle\boldsymbol{\sigma}$$
$$\displaystyle=$$
$$\displaystyle\boldsymbol{\sigma}_{1}+\boldsymbol{\sigma}_{2},$$
(22e)
$$\displaystyle\hat{P}_{\sigma}$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{2}\leavevmode\nobreak\ \left(1+\boldsymbol{\sigma}_{1}%
\cdot\boldsymbol{\sigma}_{2}\right).$$
(22f)
The total binding energy of a nuclear system can
be written as a functional of a local energy density
$$\displaystyle E$$
$$\displaystyle=$$
$$\displaystyle\int\mathrm{d}^{3}\mathbf{r}\leavevmode\nobreak\ {\mathcal{H}}(%
\mathbf{r}),$$
(23)
$$\displaystyle{\mathcal{H}}$$
$$\displaystyle=$$
$$\displaystyle\frac{\hbar^{2}}{2m}\tau_{0}+{\mathcal{H}}_{\mathrm{Skyrme}}+{%
\mathcal{H}}_{\mathrm{Coul.}},$$
(24)
$$\displaystyle{\mathcal{H}}_{\mathrm{Skyrme}}$$
$$\displaystyle=$$
$$\displaystyle{\mathcal{H}}_{0}^{\mathrm{even}}+{\mathcal{H}}_{1}^{\mathrm{even%
}}+{\mathcal{H}}_{0}^{\mathrm{odd}}+{\mathcal{H}}_{1}^{\mathrm{odd}},$$
(25)
where the superscripts in the last equation indicate the behavior with
respect to time reversal of densities occurring in each term, while subscripts
indicate the rank of the densities in isospin space. The
corresponding expressions are
$$\displaystyle{\mathcal{H}}_{0}^{\mathrm{even}}$$
$$\displaystyle=$$
$$\displaystyle C^{\rho}_{0}\leavevmode\nobreak\ \rho_{0}^{2}+C^{\Delta\rho}_{0}%
\leavevmode\nobreak\ \rho_{0}\Delta\leavevmode\nobreak\ \rho_{0}+C^{\tau}_{0}%
\leavevmode\nobreak\ \rho_{0}\leavevmode\nobreak\ \tau_{0}+C^{J}_{0}%
\leavevmode\nobreak\ \mathbb{J}_{0}^{2}+C^{\nabla J}_{0}\leavevmode\nobreak\ %
\rho_{0}\boldsymbol{\nabla}\cdot\mathbf{J}_{0},$$
(26a)
$$\displaystyle{\mathcal{H}}_{1}^{\mathrm{even}}$$
$$\displaystyle=$$
$$\displaystyle C^{\rho}_{1}\leavevmode\nobreak\ {\vec{\rho}}_{1}^{\leavevmode%
\nobreak\ 2}+C^{\Delta\rho}_{1}\leavevmode\nobreak\ {\vec{\rho}}_{1}\cdot%
\Delta{\vec{\rho}}_{1}+C^{\tau}_{1}\leavevmode\nobreak\ {\vec{\rho}}_{1}\cdot{%
\vec{\tau}}_{1}+C^{J}_{1}\leavevmode\nobreak\ {\vec{\mathbb{J}}}_{1}^{%
\leavevmode\nobreak\ 2}+C^{\nabla J}_{1}\leavevmode\nobreak\ {\vec{\rho}}_{1}%
\cdot\boldsymbol{\nabla}\cdot{\vec{\mathbf{J}}}_{1},$$
(26b)
$$\displaystyle{\mathcal{H}}_{0}^{\mathrm{odd}}$$
$$\displaystyle=$$
$$\displaystyle C^{s}_{0}\leavevmode\nobreak\ \mathbf{s}_{0}^{2}+C^{\Delta s}_{0%
}\leavevmode\nobreak\ \mathbf{s}_{0}\cdot\Delta\mathbf{s}_{0}+C^{sT}_{0}%
\leavevmode\nobreak\ \mathbf{s}_{0}\cdot\mathbf{T}_{0}+C^{\nabla s}_{0}%
\leavevmode\nobreak\ (\boldsymbol{\nabla}\cdot\mathbf{s}_{0})^{2}+C^{j}_{0}%
\leavevmode\nobreak\ \mathbf{j}_{0}^{2}+C^{\nabla j}_{0}\leavevmode\nobreak\ %
\mathbf{s}_{0}\cdot(\boldsymbol{\nabla}\times\mathbf{j}_{0}),$$
(26c)
$$\displaystyle{\mathcal{H}}_{1}^{\mathrm{odd}}$$
$$\displaystyle=$$
$$\displaystyle C^{s}_{1}\leavevmode\nobreak\ {\vec{\mathbf{s}}}_{1}^{%
\leavevmode\nobreak\ 2}+C^{\Delta s}_{1}\leavevmode\nobreak\ {\vec{\mathbf{s}}%
}_{1}\cdot\Delta{\vec{\mathbf{s}}}_{1}+C^{sT}_{1}\leavevmode\nobreak\ {\vec{%
\mathbf{s}}}_{1}\cdot{\vec{\mathbf{T}}}_{1}+C^{\nabla s}_{1}\leavevmode%
\nobreak\ (\boldsymbol{\nabla}\cdot{\vec{\mathbf{s}}}_{1})^{2}+C^{j}_{1}%
\leavevmode\nobreak\ {\vec{\mathbf{j}}}_{1}^{\leavevmode\nobreak\ 2}+C^{\nabla
j%
}_{1}\leavevmode\nobreak\ {\vec{\mathbf{s}}}_{1}\cdot(\boldsymbol{\nabla}%
\times{\vec{\mathbf{j}}}_{1}).$$
(26d)
Bold letters denote vector densities and
arrows denote isovector densities. Neutron and proton densities ($q=+1$ for
neutrons, $q=-1$ for protons) are thus given by
$$\displaystyle\rho_{q}\left(\mathbf{r}\right)$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{2}\left(\rho_{0}+q\leavevmode\nobreak\ \rho_{1,3}\right),$$
$$\displaystyle\tau_{q}\left(\mathbf{r}\right)$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{2}\left(\tau_{0}+q\leavevmode\nobreak\ \tau_{1,3}\right),$$
$$\displaystyle\mathbb{J}_{q}\left(\mathbf{r}\right)$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{2}\left(\mathbb{J}_{0}+q\leavevmode\nobreak\ \mathbb{J}_%
{1,3}\right),$$
(27)
with similar expressions for time-odd densities. The spin-current
vector $\mathbf{J}_{t}$ is built from the antisymmetric part of tensor $\mathbb{J}_{t}$.
Let us give the expressions, in terms of Skyrme force parameters, of the
coupling constants entering the HFB calculations which are
altered by the addition of a second density-dependent term
$$\displaystyle C^{\rho}_{0}$$
$$\displaystyle=$$
$$\displaystyle\sum_{i=0}^{2}\leavevmode\nobreak\ C^{\rho,i}_{0}\leavevmode%
\nobreak\ \rho_{0}^{i/3}\leavevmode\nobreak\ =\leavevmode\nobreak\ \sum_{i=0}^%
{2}\leavevmode\nobreak\ \frac{3}{8}\leavevmode\nobreak\ t_{0i}\leavevmode%
\nobreak\ \rho_{0}^{i/3},$$
(28a)
$$\displaystyle C^{\rho}_{1}$$
$$\displaystyle=$$
$$\displaystyle\sum_{i=0}^{2}\leavevmode\nobreak\ C^{\rho,i}_{1}\leavevmode%
\nobreak\ \rho_{0}^{i/3}$$
(28b)
$$\displaystyle=$$
$$\displaystyle\sum_{i=0}^{2}\leavevmode\nobreak\ -\frac{1}{8}\leavevmode%
\nobreak\ t_{0i}\leavevmode\nobreak\ (1+2x_{0i})\leavevmode\nobreak\ \rho_{0}^%
{i/3},$$
as well as the constants related, respectively, to isoscalar and isovector
effective masses,
$$\displaystyle C^{\tau}_{0}$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{16}\left[\leavevmode\nobreak\ 3t_{1}+t_{2}(5+4x_{2})%
\leavevmode\nobreak\ \right],$$
(29a)
$$\displaystyle C^{\tau}_{1}$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{16}\left[\leavevmode\nobreak\ -t_{1}(1-2x_{1})+t_{2}(1+2%
x_{2})\leavevmode\nobreak\ \right],$$
(29b)
and constants multiplying gradient terms discussed in sec. III.2,
$$\displaystyle C^{\Delta\rho}_{0}$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{64}\left[\leavevmode\nobreak\ -9t_{1}+t_{2}(5+4x_{2})%
\leavevmode\nobreak\ \right],$$
(30a)
$$\displaystyle C^{\Delta\rho}_{1}$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{64}\left[\leavevmode\nobreak\ 3t_{1}(1+2x_{1})+t_{2}(1+2%
x_{2})\leavevmode\nobreak\ \right],$$
(30b)
$$\displaystyle C^{\Delta s}_{0}$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{64}\left[\leavevmode\nobreak\ 3t_{1}(1-2x_{1})+t_{2}(1+2%
x_{2})\leavevmode\nobreak\ \right],$$
(30c)
$$\displaystyle C^{\Delta s}_{1}$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{64}\left[\leavevmode\nobreak\ 3t_{1}+t_{2}\leavevmode%
\nobreak\ \right].$$
(30d)
The expressions of all other coupling constants are given in
Ref. Bender et al. (2003).
Some of the above constants are linked through the gauge invariance of the
functional, related to the Galilean invariance of the underlying effective
force:
$$C^{j}_{t}=-C^{\tau}_{t},\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode%
\nobreak\ C^{J}_{t}=-C^{sT}_{t},\leavevmode\nobreak\ \leavevmode\nobreak\ %
\leavevmode\nobreak\ C^{\nabla j}_{t}=C^{\nabla J}_{t}.$$
(31)
The densities used above can be written as functionals of the density matrix
expressed in coordinate space, i.e.
$$\displaystyle\hat{\rho}(\mathbf{x}\sigma q,\mathbf{x}^{\prime}\sigma^{\prime}q%
^{\prime})$$
$$\displaystyle=$$
$$\displaystyle\sum_{k}\langle k|\mathbf{x}^{\prime}\sigma^{\prime}q^{\prime}%
\rangle\langle\mathbf{x}\sigma q|k\rangle\leavevmode\nobreak\ \rho_{kk},$$
(32)
as
$$\displaystyle\rho_{0}(\mathbf{r})$$
$$\displaystyle=$$
$$\displaystyle\!\int\!\mathrm{d}^{3}\mathbf{x}\mathrm{d}^{3}\mathbf{x}^{\prime}%
\sum_{\sigma\sigma^{\prime}qq^{\prime}}\delta(\mathbf{r}-\mathbf{x})\;\delta(%
\mathbf{x}^{\prime}-\mathbf{x})\;\leavevmode\nobreak\ \delta_{q^{\prime}q}\;%
\delta_{\sigma^{\prime}\sigma}$$
(33a)
$$\displaystyle\times$$
$$\displaystyle\hat{\rho}(\mathbf{x}\sigma q,\mathbf{x}^{\prime}\sigma^{\prime}q%
^{\prime}),$$
$$\displaystyle\Delta\rho_{0}(\mathbf{r})$$
$$\displaystyle=$$
$$\displaystyle\!\int\!\mathrm{d}^{3}\mathbf{x}\mathrm{d}^{3}\mathbf{x}^{\prime}%
\sum_{\sigma\sigma^{\prime}qq^{\prime}}\delta(\mathbf{r}-\mathbf{x})\;\delta(%
\mathbf{x}^{\prime}-\mathbf{x})\;\leavevmode\nobreak\ \delta_{q^{\prime}q}\;%
\delta_{\sigma^{\prime}\sigma}$$
(33b)
$$\displaystyle\times$$
$$\displaystyle(\boldsymbol{\nabla}^{\prime 2}\!+2\boldsymbol{\nabla}^{\prime}%
\cdot\boldsymbol{\nabla}+\boldsymbol{\nabla}^{2})\,\hat{\rho}(\mathbf{x}\sigma
q%
,\mathbf{x}^{\prime}\sigma^{\prime}q^{\prime}),$$
$$\displaystyle\tau_{0}(\mathbf{r})$$
$$\displaystyle=$$
$$\displaystyle\!\int\!\mathrm{d}^{3}\mathbf{x}\mathrm{d}^{3}\mathbf{x}^{\prime}%
\sum_{\sigma\sigma^{\prime}qq^{\prime}}\delta(\mathbf{r}-\mathbf{x})\;\delta(%
\mathbf{x}^{\prime}-\mathbf{x})\;\leavevmode\nobreak\ \delta_{q^{\prime}q}\;%
\delta_{\sigma^{\prime}\sigma}$$
(33c)
$$\displaystyle\times$$
$$\displaystyle\boldsymbol{\nabla}^{\prime}\cdot\boldsymbol{\nabla}\leavevmode%
\nobreak\ \hat{\rho}(\mathbf{x}\sigma q,\mathbf{x}^{\prime}\sigma^{\prime}q^{%
\prime}),$$
$$\displaystyle\mathbb{J}_{0}(\mathbf{r})$$
$$\displaystyle=$$
$$\displaystyle\!\int\!\mathrm{d}^{3}\mathbf{x}\mathrm{d}^{3}\mathbf{x}^{\prime}%
\sum_{\sigma\sigma^{\prime}qq^{\prime}}\delta(\mathbf{r}-\mathbf{x})\;\delta(%
\mathbf{x}^{\prime}-\mathbf{x})\;\leavevmode\nobreak\ \delta_{q^{\prime}q}$$
(33d)
$$\displaystyle\times$$
$$\displaystyle\frac{1}{2i}(\boldsymbol{\nabla}-\boldsymbol{\nabla}^{\prime})%
\otimes\boldsymbol{\sigma}_{\sigma^{\prime}\sigma}\;\leavevmode\nobreak\ \hat{%
\rho}(\mathbf{x}\sigma q,\mathbf{x}^{\prime}\sigma^{\prime}q^{\prime}),$$
$$\displaystyle\mathbf{j}_{0}(\mathbf{r})$$
$$\displaystyle=$$
$$\displaystyle\!\int\!\mathrm{d}^{3}\mathbf{x}\mathrm{d}^{3}\mathbf{x}^{\prime}%
\sum_{\sigma\sigma^{\prime}qq^{\prime}}\delta(\mathbf{r}-\mathbf{x})\;\delta(%
\mathbf{x}^{\prime}-\mathbf{x})\;\leavevmode\nobreak\ \delta_{q^{\prime}q}\;%
\delta_{\sigma^{\prime}\sigma}$$
(33e)
$$\displaystyle\times$$
$$\displaystyle\frac{1}{2i}(\boldsymbol{\nabla}-\boldsymbol{\nabla}^{\prime})%
\leavevmode\nobreak\ \hat{\rho}(\mathbf{x}\sigma q,\mathbf{x}^{\prime}\sigma^{%
\prime}q^{\prime}),$$
where $\sigma$, $\sigma^{\prime}$ are indices referring to spin, $q$ and $q^{\prime}$
refer to isospin, $\boldsymbol{\nabla}$ is the gradient operator acting on
the coordinate $x$, $\boldsymbol{\nabla}^{\prime}$ being the same acting on $x^{\prime}$. Isovector and
other time-odd densities can be expressed by replacing, respectively,
$\delta_{q^{\prime}q}\;$ by ${\vec{\tau}}_{q^{\prime}q}\;$ and $\delta_{\sigma^{\prime}\sigma}\;$ by $\boldsymbol{\sigma}_{\sigma^{\prime}\sigma}\;$ where appropriate.
Appendix B Separation of the energy into spin-isospin channels
When the EDF is defined as the expectation value of an
effective Hamiltonian, separating it into spin-isospin channels is
straightforward, as in Eq. (9). However, one can extend this
definition to the case of any Hartree-like functional: let us start by
recalling that in the case of the Skyrme force, the direct and exchange terms
have the same analytical structure; one thus usually uses the expressions
$$\displaystyle E_{\mathrm{pot}}$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{2}\sum_{kl}{\left\langle\>{kl}\left|\raisebox{0.0pt}[6.9%
99893pt][3.079953pt]{$V_{\mathrm{Skyrme}}$}\right|{\overline{kl}}\>\right%
\rangle}\leavevmode\nobreak\ \rho_{kk}\leavevmode\nobreak\ \rho_{ll},$$
(34)
$$\displaystyle\left|\overline{kl}\leavevmode\nobreak\ \right\rangle$$
$$\displaystyle=$$
$$\displaystyle\left|kl\leavevmode\nobreak\ \right\rangle-\left|lk\leavevmode%
\nobreak\ \right\rangle\leavevmode\nobreak\ =\leavevmode\nobreak\ (1-\hat{P}_{%
r}\hat{P}_{\sigma}\hat{P}_{\tau})\left|kl\leavevmode\nobreak\ \right\rangle,$$
(35)
where the last expression uses the position, spin and isospin exchange
operators to define an antisymmetrized and non-normalized two-body state. One
then writes down the antisymmetrized form of Eq. (21) and the EDF by
using the definition of densities entering
Eqs. (26a)-(26d).
Leaving the antisymmetrized Hamiltonian framework, it is always possible to
define the potential part of the functional as the direct term of the
expectation value of a certain operator, as in
$$\displaystyle E_{\mathrm{pot}}$$
$$\displaystyle=$$
$$\displaystyle\sum_{kl}{\left\langle\>{kl}\left|\raisebox{0.0pt}[6.999893pt][3.%
079953pt]{$V_{\mathrm{EDF}}$}\right|{kl}\>\right\rangle}\leavevmode\nobreak\ %
\rho_{kk}\leavevmode\nobreak\ \rho_{ll},$$
(36)
recalling that $V_{\mathrm{EDF}}=V_{\mathrm{Skyrme}}(1-\hat{P}_{r}\hat{P}_{\sigma}\hat{P}_{%
\tau})$ in the Hamiltonian case.
One then defines the energy per channel as
$$\displaystyle E_{\mathrm{EDF}}^{ST}$$
$$\displaystyle=$$
$$\displaystyle\sum_{kl}{\left\langle\>{kl}\left|\raisebox{0.0pt}[6.999893pt][3.%
079953pt]{$V_{\mathrm{EDF}}\leavevmode\nobreak\ \hat{P}_{S}\hat{P}_{T}$}\right%
|{kl}\>\right\rangle}\leavevmode\nobreak\ \rho_{kk}\leavevmode\nobreak\ \rho_{%
ll},$$
(37)
which, with the definitions (26a)-(26d) for coupling
constants, yields (retaining only terms acting in infinite matter)
$$\displaystyle E_{\mathrm{pot}}^{ST}$$
$$\displaystyle=$$
$$\displaystyle\int\mathrm{d}^{3}\mathbf{r}\leavevmode\nobreak\ {\mathcal{H}}^{%
ST}(\mathbf{r})$$
(38)
$$\displaystyle{\mathcal{H}}^{ST}$$
$$\displaystyle=$$
$$\displaystyle\big{[}C^{\rho}_{0}+(4S-3)C^{s}_{0}+(4T-3)C^{\rho}_{1}+(4S-3)(4T-%
3)C^{s}_{1}\big{]}$$
$$\displaystyle\leavevmode\nobreak\ \times\leavevmode\nobreak\ \frac{1}{16}%
\leavevmode\nobreak\ \big{[}(2S+1)(2T+1)\rho_{0}^{2}+(2S-1)(2T+1)\mathbf{s}_{0%
}^{2}\leavevmode\nobreak\ +\leavevmode\nobreak\ (2S+1)(2T-1){\vec{\rho}}_{1}^{%
\leavevmode\nobreak\ 2}+(2S-1)(2T-1){\vec{\mathbf{s}}}_{1}^{\leavevmode%
\nobreak\ 2}\big{]}$$
$$\displaystyle+\leavevmode\nobreak\ \big{[}C^{\tau}_{0}+(4S-3)C^{sT}_{0}+(4T-3)%
C^{\tau}_{1}+(4S-3)(4T-3)C^{sT}_{1}\big{]}$$
$$\displaystyle\leavevmode\nobreak\ \times\leavevmode\nobreak\ \frac{1}{16}\big{%
[}(2S+1)(2T+1)\rho_{0}\tau_{0}+(2S-1)(2T+1)\mathbf{s}_{0}\cdot\mathbf{T}_{0}%
\leavevmode\nobreak\ +\leavevmode\nobreak\ (2S+1)(2T-1){\vec{\rho}}_{1}{\vec{%
\tau}}_{1}+(2S-1)(2T-1){\vec{\mathbf{s}}}_{1}\cdot{\vec{\mathbf{T}}}_{1}\big{]}.$$
Appendix C Particle-hole residual interaction
With the definition of densities given in Eqs. (33b)-
(33e) the particle-hole residual interaction is obtained
through
$$\displaystyle\langle\mathbf{x}^{\prime}_{1}\sigma^{\prime}_{1}q^{\prime}_{1},%
\mathbf{x}^{\prime}_{2}\sigma^{\prime}_{2}q^{\prime}_{2}|V_{\mathrm{p-h}}|%
\mathbf{x}_{1}\sigma_{1}q_{1},\mathbf{x}_{2}\sigma_{2}q_{2}\rangle$$
$$\displaystyle\leavevmode\nobreak\ =\leavevmode\nobreak\ \frac{\delta^{2}{%
\mathcal{E}}}{\delta\rho(\mathbf{x}_{1}\sigma_{1}q_{1},\mathbf{x}^{\prime}_{1}%
\sigma^{\prime}_{1}q^{\prime}_{1})\leavevmode\nobreak\ \delta\rho(\mathbf{x}_{%
2}\sigma_{2}q_{2},\mathbf{x}^{\prime}_{2}\sigma^{\prime}_{2}q^{\prime}_{2})},$$
(40)
which, for the central, spin-scalar, isoscalar part of the functional,
i. e.
$$\displaystyle{\mathcal{H}}^{\mathrm{sv}}$$
$$\displaystyle=$$
$$\displaystyle C^{\rho}_{0}(\rho_{0})\leavevmode\nobreak\ \rho_{0}^{2}+C^{%
\Delta\rho}_{0}\leavevmode\nobreak\ \rho_{0}\Delta\leavevmode\nobreak\ \rho_{0%
}+C^{\tau}_{0}\leavevmode\nobreak\ (\rho_{0}\leavevmode\nobreak\ \tau_{0}-%
\mathbf{j}_{0}^{2}),$$
(the generalization to the full central part, omitted here for the sake of
brevity, being straightforward), reads
$$\displaystyle V_{\mathrm{p-h}}$$
$$\displaystyle=$$
$$\displaystyle 2C^{\rho,0}_{0}+\sum_{i=1}^{2}C^{\rho,i}_{0}\left(\frac{i}{3}+2%
\right)\left(\frac{i}{3}+1\right)\rho_{0}^{i/3}$$
(42)
$$\displaystyle+C^{\tau}_{0}\Big{(}\boldsymbol{\nabla}^{\prime}_{1}\cdot%
\boldsymbol{\nabla}_{1}+\boldsymbol{\nabla}^{\prime}_{2}\cdot\boldsymbol{%
\nabla}_{2}$$
$$\displaystyle\leavevmode\nobreak\ +\frac{1}{2}(\boldsymbol{\nabla}^{\prime}_{1%
}-\boldsymbol{\nabla}_{1})\cdot(\boldsymbol{\nabla}^{\prime}_{2}-\boldsymbol{%
\nabla}_{2})\Big{)}$$
$$\displaystyle+C^{\Delta\rho}_{0}\Big{(}(\boldsymbol{\nabla}^{\prime 2}_{1}+2%
\boldsymbol{\nabla}^{\prime}_{1}\cdot\boldsymbol{\nabla}_{1}+\boldsymbol{%
\nabla}_{1}^{2})$$
$$\displaystyle\leavevmode\nobreak\ +(\boldsymbol{\nabla}^{\prime 2}_{2}+2%
\boldsymbol{\nabla}^{\prime}_{2}\cdot\boldsymbol{\nabla}_{2}+\boldsymbol{%
\nabla}_{2}^{2})\Big{)}.$$
When computing the momentum-space matrix element, Eq. (14), one
makes the substitutions $\boldsymbol{\nabla}_{1}=i(\mathbf{q}_{1}+\mathbf{q})$,
$\boldsymbol{\nabla}_{2}=i\mathbf{q}_{2}$, $\boldsymbol{\nabla}^{\prime}_{1}=-i\mathbf{q}_{1}$ and
$\boldsymbol{\nabla}^{\prime}_{2}=-i(\mathbf{q}_{2}+\mathbf{q})$ (with $\hbar=1$),
which yields expressions (17a)-(18d).
References
Bender et al. (2003)
M. Bender,
P.-H. Heenen,
and P.-G.
Reinhard, Rev. Mod. Phys.
75, 121 (2003).
Chabanat et al. (1997)
E. Chabanat,
P. Bonche,
P. Haensel,
J. Meyer, and
R. Schaeffer,
Nucl. Phys. A627,
710 (1997).
Chabanat
et al. (1998a)
E. Chabanat,
P. Bonche,
P. Haensel,
J. Meyer, and
R. Schaeffer,
Nucl. Phys. A635,
231 (1998a).
Chabanat
et al. (1998b)
E. Chabanat,
P. Bonche,
P. Haensel,
J. Meyer, and
R. Schaeffer,
Nucl. Phys. A643,
441 (1998b).
Lattimer and Prakash (2004)
J. M. Lattimer and
M. Prakash,
Science 304,
536 (2004).
Chen et al. (2005)
L.-W. Chen,
C. M. Ko, and
B.-A. Li,
Phys. Rev. Lett. 94,
032701 (2005).
Typel and Brown (2001)
S. Typel and
B. A. Brown,
Phys. Rev. C 64,
027302 (2001).
Li (2004)
B.-A. Li,
Phys. Rev. C 69,
064602 (2004).
Li et al. (2004)
B.-A. Li,
C. B. Das,
S. D. Gupta, and
C. Gale,
Phys. Rev. C 69,
011603 (2004).
Bethe (1990)
H. Bethe, Rev.
Mod. Phys. 62, 801
(1990).
Farine et al. (2001)
M. Farine,
J. M. Pearson,
and F. Tondeur,
Nucl. Phys. A696,
396 (2001).
Bombaci and Lombardo (1991)
I. Bombaci and
U. Lombardo,
Phys. Rev. C 44,
1892 (1991).
Kubis and Kutschera (1997)
S. Kubis and
M. Kutschera,
Phys. Lett. B399,
191 (1997).
Zuo et al. (1999)
W. Zuo,
I. Bombaci, and
U. Lombardo,
Phys. Rev. C 60,
024605 (1999).
Greco et al. (2001)
V. Greco,
M. Colonna,
M. D. Toro,
G. Fabbri, and
F. Matera,
Phys. Rev. C 64,
045203 (2001).
Hofmann et al. (2001)
F. Hofmann,
C. Keil, and
H. Lenske,
Phys. Rev. C 64,
034314 (2001).
Liu et al. (2002)
B. Liu,
V. Greco,
V. Baran,
M. Colonna, and
M. D. Toro,
Phys. Rev. C 65,
045201 (2002).
Rizzo et al. (2004)
J. Rizzo,
M. Colonna,
M. D. Toro, and
V. Greco,
Nucl. Phys. A732,
202 (2004).
Ma et al. (2004)
Z.-Y. Ma,
J. Rong,
B.-Q. Chen,
Z.-Y. Zhu, and
H.-Q. Song,
Phys. Lett. B604,
170 (2004).
van Dalen et al. (2005)
E. van Dalen,
C. Fuchs, and
A. Faessler,
Phys. Rev. Lett. 95,
022302 (2005).
Satula et al. (2006)
W. Satula,
R. A. Wyss, and
M. Rafalski,
Phys. Rev. C 74,
011301 (2006).
Bender et al. (2002)
M. Bender,
J. Dobaczewski,
J. Engel, and
W. Nazarewicz,
Phys. Rev. C 65,
054322 (2002).
Jeukenne et al. (1976)
J. P. Jeukenne,
A. Lejeune, and
C. Mahaux,
Phys. Rep. 25,
83 (1976).
Baldo (1999)
M. Baldo,
Nuclear methods and the nuclear equation of state,
Int. Rev. Nucl. Phys., Vol 9 (World Scientific,
Singapore, 1999).
Kohn and Sham (1964)
W. Kohn and
L. J. Sham,
Phys. Rev. 137,
A1697 (1964).
Blaizot and Gogny (1977)
J. P. Blaizot and
D. Gogny,
Nucl. Phys. A284,
429 (1977).
Severyukhin et al. (2002)
A. P. Severyukhin,
C. Stoyanov,
V. V. Voronov,
and Nguyen Van Giai,
Phys. Rev. C 66,
034304 (2002).
Meyer et al. (1995)
J. Meyer,
P. Bonche,
M. S. Weiss,
J. Dobaczewski,
H. Flocard, and
P.-H. Heenen,
Nucl. Phys. A588,
597 (1995).
Bernard and Giai (1980)
V. Bernard and
Nguyen Van Giai,
Nucl. Phys. A348,
75 (1980).
Goriely et al. (2003)
S. Goriely,
M. Samyn,
M. Bender, and
J. M. Pearson,
Phys. Rev. C 68,
054325 (2003).
Liu and Giai (1976)
K. F. Liu and
Nguyen Van Giai,
Phys. Lett. B65,
23 (1976).
Charity et al. (2006)
R. J. Charity,
L. G. Sobotka,
and W. H.
Dickhoff (2006), eprint nucl-ex/0605026.
Lane (1962)
A. M. Lane,
Nucl. Phys. 35,
676 (1962).
Sammarruca et al. (2005)
F. Sammarruca,
W. Barredo, and
P. Krastev,
Phys. Rev. C 71,
064306 (2005).
M. Jaminon (1989)
C. M. M. Jaminon,
Phys. Rev. C 40,
354 (1989).
Bender et al. (1999)
M. Bender,
K. Rutz,
P.-G. Reinhard,
J. A. Maruhn,
and W. Greiner,
Phys. Rev. C 60,
034304 (1999).
Kruppa et al. (2000)
A. T. Kruppa,
M. Bender,
W. Nazarewicz,
P.-G. Reinhard,
T. Vertse, and
S. Cwiok,
Phys. Rev. C 61,
034313 (2000).
Bender et al. (2001)
M. Bender,
W. Nazarewicz,
and P.-G.
Reinhard, Phys. Lett.
B515, 42 (2001).
Berger et al. (2001)
J.-F. Berger,
L. Bitaud,
J. Dechargé,
M. Girod, and
K. Dietrich,
Nucl. Phys. A685 (1-4),
1c (2001).
Paar et al. (2005)
N. Paar,
T. Niks̆ić,
D. Vretenar, and
P. Ring,
Phys. Lett. B606,
288 (2005).
Duguet et al. (2002a)
T. Duguet,
P. Bonche,
P.-H. Heenen,
and J. Meyer,
Phys. Rev. C 65,
014310 (2002a).
Rigollet et al. (1999)
C. Rigollet,
P. Bonche,
H. Flocard, and
P.-H. Heenen,
Phys. Rev. C 59,
3120 (1999).
Cochet et al. (2004)
B. Cochet,
K. Bennaceur,
P. Bonche,
T. Duguet, and
J. Meyer,
Nucl. Phys. A731,
34 (2004).
Cao et al. (2006)
L. G. Cao,
U. Lombardo,
C. W. Shen, and
Nguyen Van Giai,
Phys. Rev. C 73,
014313 (2006).
Dobaczewski et al. (1984)
J. Dobaczewski,
H. Flocard, and
J. Treiner,
Nucl. Phys. A422,
103 (1984).
Bohigas et al. (1979)
O. Bohigas,
A. M. Lane, and
J. Martorell,
Phys. Rep. 51,
267 (1979).
F.Bertsch et al. (2005)
G. F.Bertsch,
B. Sabbey, and
M. Uusnakki,
Phys. Rev. C 71,
054311 (2005).
Beiner et al. (1975)
M. Beiner,
H. Flocard,
Nguyen Van Giai, and
P. Quentin,
Nucl. Phys. A238,
29 (1975).
Bartel et al. (1982)
J. Bartel,
P. Quentin,
M. Brack,
C. Guet, and
H.-B. Håkansson,
Nucl. Phys. A386,
79 (1982).
Chappert et al. (2006)
F. Chappert,
M. Girod, and
J.-F. Berger
(2006), private communication.
Akmal et al. (1998)
A. Akmal,
V. R. Pandharipande,
and D. G.
Ravenhall, Phys. Rev. C
58, 1804 (1998).
Bulgac and Yu (2002)
A. Bulgac and
Y. Yu, Phys.
Rev. Lett. 88, 042504
(2002).
Duguet et al. (2002b)
T. Duguet,
P. Bonche,
P.-H. Heenen,
and J. Meyer,
Phys. Rev. C 65,
014311 (2002b).
Hosmer et al. (2005)
P. T. Hosmer,
H. Schatz,
A. Aprahamian,
O. Arndt,
R. R. C. Clement,
A. Estrade,
K.-L. Kratz,
S. N. Liddick,
P. F. Mantica,
W. F. Mueller,
et al., Phys. Rev. Lett.
94, 112501
(2005).
sp2 (2006)
Physics case of spiral 2 (2006),
eprint http://www.ganil.fr/.
Dobaczewski (2006)
J. Dobaczewski
(2006), eprint nucl-th/0604043.
Audi et al. (2003)
G. Audi,
A. H. Wapstra,
and C. Thibault,
Nucl. Phys. A739,
337 (2003).
Duguet and Bennaceur (2006)
T. Duguet and
K. Bennaceur
(2006), in preparation.
Colò et al. (1995)
G. Colò,
Nguyen Van Giai, and
H. Sagawa,
Phys. Lett. B363,
5 (1995).
Ritman et al. (1993)
J. Ritman,
F.-D. Berg,
W. Kühn,
V. Metag,
R. Novotny,
M. Notheisen,
P. Paul,
M. Pfeiffer,
O. Schwalb,
H. Löhner,
et al., Phys. Rev. Lett.
70, 533 (1993).
Erell et al. (1986)
A. Erell,
J. Alster,
J. Lichtenstadt,
M. A. Moinester,
J. D. Bowman,
M. D. Cooper,
F. Irom,
H. S. Matis,
E. Piasetzky,
and
U. Sennhauser,
Phys. Rev. C 34,
1822 (1986).
Baldo (2006)
M. Baldo (2006),
private communication.
Wiringa et al. (1995)
R. B. Wiringa,
V. G. J. Stocks,
and
R. Schiavilla,
Phys. Rev. C 51,
38 (1995).
Grangé et al. (1989)
P. Grangé,
A. Lejeune,
M. Martzolff,
and J.-F.
Mathiot, Phys. Rev. C
40, 1040 (1989).
Lejeune et al. (2000)
A. Lejeune,
U. Lombardo, and
W. Zuo,
Phys. Lett. B477,
45 (2000).
Farine and Tondeur (1997)
M. Farine and
J. M. P. F. Tondeur,
Nucl. Phys. A615,
135 (1997).
Fetter and Walecka (1971)
A. Fetter and
J. D. Walecka,
Quantum Theory of Many-Particle Systems
(McGraw-Hill, New-York,
1971).
García-Recio
et al. (1992)
C. García-Recio,
J. Navarro,
Nguyen Van Giai, and
N. N. Salcedo,
Ann. of Phys. 214,
293 (1992).
Margueron et al. (2006)
J. Margueron,
J. Navarro, and
Nguyen Van Giai
(2006), unpublished,
eprint nucl-th/0604019.
Midgal (1967)
A. B. Midgal,
Theory of Finite Fermi Systems and Applications to
Atomic Nuclei (Wiley, New York,
1967).
Terasaki and Engel (2006)
J. Terasaki and
J. Engel
(2006), eprint nucl-th/0608007.
Terasaki et al. (2005)
J. Terasaki,
J. Engel,
M. Bender,
J. Dobaczewski,
W. Nazarewicz,
and M. Stoitsov,
Phys. Rev. C 71
(2005). |
Market structure explained by pairwise interactions
Thomas Bury
tbury@ulb.ac.be
service OPERA (CP194/5), Université libre de Bruxelles, Avenue F.D. Roosevelt 50, 1050 Brussels, Belgium
(December 4, 2020)
Abstract
Financial markets are a typical example of complex systems where interactions between constituents lead to many remarkable features. Here we give empirical evidence, by making as few assumptions as possible, that the market microstructure capturing almost all of the available information in the data of stock markets does not involve higher order than pairwise interactions. We give an economic interpretation of this pairwise model. We show that it accurately recovers the empirical correlation coefficients thus the collective behaviors are quantitatively described by models that capture the observed pairwise correlations but no higher-order interactions. Furthermore, we show that an order-disorder transition occurs as predicted by the pairwise model. Last, we make the link with the graph-theoretic description of stock markets recovering the non-random and scale-free topology, shrinking length during crashes and meaningful clustering features as expected.
pacs:
89.65.Gh,
89.75.Fb,
64.60.Cn,
64.60.De
I Introduction
Complex systems are particularly interesting because they exhibit very sophisticated behaviors caused by, a priori, simple rules. Indeed, magnetic materials and neural networks, for instance, have some striking features such as phase transitions, memory, complicated equilibria structures and clustering. It is remarkable that these properties are caused by such simple interactions as pairwise ones.
We believe that the markets are also driven by such simple rules and that the higher-order interactions encountered in financial systems are the pairwise ones. Typical characteristics of a complex system are numerous entities and interaction rules (with a degree of non-linearity), all leading to the emergence of collective behaviors. Those behaviors in general depend more on the interactions (e.g their scaling and their order) and their effects than on the intrinsic nature of the elementary constitutive entities taken individually. The market can be viewed as such a system. The entities can be stocks or traders interacting through non-obvious rules. We note that we should interpret interaction at the larger sense of mutual or reciprocal influence.
What one knows is that the markets exhibit features such as synchronization Dal , structural reorganization OnnelaPRE ; PeronChaos , power laws stanley-gabaix1 ; stanley-gabaix2 , hierarchical and non-randomness Petra .
What one does not know is the true market dynamics. Even if trading rules are known, microscopic equations of motion are not known. This is a fundamental difference between finance and physics (or neuroscience).
A natural approach, given the above considerations, is a statistical modeling collecting and using at best the available amount of information and allowing (in a certain sense) the emergence of critical properties. This is exactly the purpose of the maximum entropy modeling in complex systems theory.
Indeed the maximum entropy principle (MEP) allows the selection of the less restricting model on the basis of incomplete information cover . We choose this data-based approach to avoid the use of any particular microscopic schemes (e.g. trader-agent-based rules, a priori unknown) which are difficult to assess experimentally or to avoid any analogy (even if some of such models are valuable Rosenow ). The reason is that, even if one does not know the underlying microscopic processes, the macroscopic collective behaviors can still be described by an effective model.
One has long experience of this powerful approach in the description of phase transitions and magnetic materials ref8 . More recently, it has led to valuable results about the description of real neural networks ref13 . Moreover, this approach also has counterparts in economics. Indeed, in addition of the statistical meaning of the entropy, one can interpret it as a measure of the economic activity Aoki and it is linked to the central concept of utility of many interacting economic entities Brock ; mas .
An important outcome of such a modeling is a convenient simplified version of the real interaction structure that is still consistent with the data. In the following, we derive the model in this point of view and we study the structural properties of the resulting complex network. The critical properties will be investigated in another work.
The paper is organized as follow. In section II, we present the model, its economic interpretation and the link between the interaction matrix and the moments. In section III, we give evidence that the information embedded in the data is mostly explained by the pairwise but no higher-order interactions. In section IV, we show an order-disorder transition through actual data. In section V, we highlight the properties of the interaction matrix and its link to the crises. Finally, in section VI we explain the link with the graph-theoretic approach and the topological evolution of the market network.
II The model
II.1 Model derivation
The aim is to set up a statistical model describing the market state. This requires a way to infer the probability distribution in order to get the observables (here, the associated moments). The model will also allow the study of the market structure. All these quantities will be defined below.
We consider a set of $N$ market indices or $N$ stocks with binary states $s_{i}$ ($s_{i}=\pm 1$ for all $i=1,\cdots,N$). A system configuration will be described by a vector $\textbf{s}=(s_{1},\cdots,s_{N})$. The binary variable will be equal to 1 if the associated index is bullish and equal to $-1$ if not. A configuration $(s_{1},\cdots,s_{N})$ is a binary version of the index returns.
One knows that this approximation is already useful in the description of neural populations ref13 and that neural networks are similar to financial networks Petra . We may think that it will also be the case in finance; this will be justified a posteriori as the model gives consistent results.
We seek to establish a less structured model explaining only the measured index mean orientation $q_{i}=\langle s_{i}\rangle$ and instantaneous pairwise correlations $q_{kl}=\langle s_{k}s_{l}\rangle$. The brackets $\langle\cdot\rangle$ denote the average with respect to the unknown distribution $p(\textbf{s})$. As the entropy of a distribution measures the randomness or the lack of interaction among the binary variables, a way to infer such probability distribution knowing the mean orientations and the correlations is the maximum entropy principle. Jaynes showed how to derive the probability distribution using the maximum entropy principle ref12 ; for supplementary information see cover . It consists in the following constrained maximization:
$$\displaystyle\max$$
$$\displaystyle S(\textbf{s})=-\sum_{\{\textbf{s}\}}p(\textbf{s})\,\log p(%
\textbf{s})$$
$$\displaystyle\mathrm{s.t}$$
$$\displaystyle\sum_{\{\textbf{s}\}}p(\textbf{s})=1,\quad\sum_{\{\textbf{s}\}}p(%
\textbf{s})s_{i}=q_{i},\quad\sum_{\{\textbf{s}\}}p(\textbf{s})s_{i}s_{j}=q_{ij}$$
The resulting two-agents distribution $p_{2}(\textbf{s})$ is the following
$$p_{2}(\textbf{s})=\mathcal{Z}^{-1}\exp\left(\frac{1}{2}\sum_{i,j}^{N}J_{ij}s_{%
i}s_{j}+\sum_{i=1}^{N}h_{i}s_{i}\right)\equiv\frac{e^{-\mathcal{H}(\textbf{s})%
}}{\mathcal{Z}}$$
(2)
where $J_{ij}$ and $h_{i}$ are Lagrange multipliers and $\mathcal{Z}$ a normalizing constant (the partition function). They can be expressed in terms of partial derivatives of the entropy as
$$\frac{\partial S(\textbf{s})}{\partial q_{i}}=-h_{i}\qquad\frac{\partial S(%
\textbf{s})}{\partial q_{ij}}=-J_{ij}$$
(3)
Thus preferences are conjugated to mean orientations and interaction strengths to pairwise correlations. Including higher-order correlations in constraints in (II.1) could bring more information and thus decrease the maximum entropy. We will show below that this will not be the case.
The Gibbs distribution (2) is similar to the one given by Brock and Durlauf in the discrete choice problem Brock and to the one in stochastic models in macroeconomics Aoki , and also to the Ising model used in description of magnetic materials and neural networks ref8 ; ref13 . It is also a special case of Markov random fields ref15 . It is to be noted that the Gibbs distribution and Shannon entropy naturally arise from the stochastic modeling in economics; this is discussed in Aoki .
We obtain the parameters $\{J_{ij},h_{i}\}$ by performing explicitly the maximization (II.1) so that the theoretical moments $\langle s_{i}\rangle$ and $\langle s_{i}s_{j}\rangle$ match the measured ones $q_{i}$ and $q_{ij}$. We note that this requires the computation of $2^{N}$ terms. If this number is large, the computation will take a while and we can benefit from one of the methods described in ref4 .
Last, we show how the cumulants are obtained from this model and their relation to interaction strengths. As the statistical model (2) is expressed as a Gibbs distribution, we have the relations
$$\langle s_{i_{1}}\ldots s_{i_{N}}\rangle_{\mathrm{c}}=\partial^{N}\ln\mathcal{%
Z}/\partial h_{i_{1}}\ldots\partial h_{i_{N}}$$
(4)
where $\langle\cdot\rangle_{\mathrm{c}}$ is the cumulant average Kubo ; it gives the relation between $\mathbf{J}$ and the correlation functions. If the partition function $\mathcal{Z}$ cannot be explicitly computed, we can use the Plefka series Plef or a variational cumulant expansion Barb .
Hereafter, we will show that the covariances are consistently deduced from this statistical model and thus that they are a function of the interaction strengths.
II.2 Interpretation
We interpret the objective function $\mathcal{H}(\textbf{s})$ defined by the MEP in the distribution (2) as follows.
The pairwise interactions between economic agents are modeled by interaction strengths $J_{ij}$ which describe how $i$ and $j$ influence each other. Here by interaction, we mean a measure of mutual influence or a measure of share comovement. In this framework, our intention is not to give a description of these interactions but to study their effects. Actually, the causes underlying the interaction process seem to be unnecessary in the description of emergent macroscopic behaviors. Indeed the complicated interactions between magnetic moments or between neurons are efficiently simplified in their maximum entropy description but one still recovers the main macroscopic features observed in these systems. In this description, the crucial features are the scaling (dependence on or independence of the system size) of interaction strengths and the order of interactions. The matrix J is set to be symmetric in this first approach. There is disagreement or conflict between entities when the weighted product of their orientations $J_{ij}s_{i}s_{j}$ is negative. If two shares are supposed to move together ($J_{ij}>0$), a conflicting situation is the one where they do not have the same orientation (bearish or bullish).
We include the idiosyncratic preferences of the economic agents, here the willingness to be bullish or not. These Lagrange multipliers $h_{i}$ can also be interpreted as the external influences on entities $i$ induced by the macroeconomic background. By example a company can prosper and make benefits during a crisis period and the associated stock can still fall simultaneously because the investors are negatively influenced by the economic background. It results that the stock will have a propensity to fall. We denote the external influence by $h_{i}$. If $i$’s orientation satisfies its preference, $h_{i}s_{i}$ is positive.
The total conflict of the system is thus given by
$$\mathcal{H}(\textbf{s})=-\frac{1}{2}\sum_{i=1}^{N}\sum_{j=1}^{N}J_{ij}\,s_{i}s%
_{j}-\sum_{i=1}^{N}h_{i}s_{i}$$
(5)
So, we interpret $\mathcal{H}(\textbf{s})$ as the opposite of the so-called utility function $\mathcal{U}(\textbf{s})=-\mathcal{H}(\textbf{s})$ with pairwise interacting and idiosyncratic parts. Consequently the interaction strengths can be viewed as the incentive complementarities Brock ; mas . Indeed we have $\partial^{2}\mathcal{U}/\partial s_{i}\partial s_{j}=J_{ij}$.The larger $J_{ij}s_{i}s_{j}$ is, the stronger the strategic interaction between $i$ and $j$ is.
We emphasize that this Ising like model is forced upon us as the statistically consistent model with the measured orientations and correlations. It is not an analogy based on specific hypotheses about the market dynamics.
III Consistency of the pairwise modeling
One of the most exciting features of the model is the emergence of collective behaviors even if the interactions are weak. If the model is able to explain the recorded data, the system is therefore dominated by pairwise correlations.
The aim is to provide quantitative empirical evidence that the pairwise modeling is a consistent paradigm to explain the financial data and exhibited behaviors in the market.
In the following, we apply the pairwise model to a set of six major market indices (AEX, Bel-20, CAC 40, Xetra Dax, Eurostoxx 50, FTSE 100). We selected only European indices because some financial issues are specific to Europe and we consider indices because they are the driving force of the respective stock markets Shap , they will reflect the main properties of the subjacent stock set. We will say that they are up or bullish if the closing price is higher than the opening price and they are down or bearish if not. These will constitue our binary states. We observe 2253 configurations from 6/06/2002 to 14/06/2011 ref3 . We take a nine year long time series including two large crises. The daily sampling is enough since we want to study large crises, and the two principal peaks of the Fourier transform are centered on frequencies $f_{1}=6\times 10^{-4}\,\mathrm{d}^{-1}$ and $f_{2}=1.2\times 10^{-3}\,\mathrm{d}^{-1}$; the unit day stands for trading day. The first frequency $f_{1}$ is the crisis occurrence frequency in our time window, the corresponding period is $T_{1}=1.7\times 10^{3}\,\mathrm{d}$ . Later, we will also analyze the stocks composing the Dow Jones and the S$\&$P100 indices, and another set of 116 stocks.
First of all, we give the magnitude order of the interaction strengths and of the empirical pairwise correlations in Fig-1.
The $J_{ij}$ are all positive; we can therefore use net mean orientation (net magnetization) as an order parameter. The mean value of $h_{i}$ is about $0.0113$.
As mentioned above, higher-order interactions can be involved in the interaction structure. In order to show that pairwise correlations are prevailing, we compute the Kullback-Leibler (KL) divergence, $D_{\scriptstyle{\mathbf{KL}}}(P_{2}\|P_{\scriptstyle{\mathbf{data}}})$ between the two-agents maximum entropy (ME) distribution $P_{2}$ and the empirical one $P_{\scriptstyle{\mathbf{data}}}$. The KL divergence is equal to $2.27\times 10^{-2}$ for the ME distribution inferred from 2253 observations. It must be compared to $D_{\scriptstyle{\mathbf{KL}}}(P_{1}\|P_{\scriptstyle{\mathbf{data}}})=1.4801$ for the independent agents model $P_{1}$. The closer to zero this quantity is, the closer $P_{2}$ to $P_{\scriptstyle{\mathbf{data}}}$ is. Specifically, a consistent way to test if the pairwise correlation model satisfactorily explains data statistics is to evaluate the ratio between $S(P_{1})-S(P_{2})$ and the Kullback-Leibler discrepancy $I_{N}\equiv D_{\mathrm{KL}}\left(P_{N}||P_{1}\right)$, where $S(P_{2})$ is the entropy of the pairwise model. If this ratio is close to 1, the pairwise correlations explain most of available information. Indeed the multi-information $I_{N}=S(P_{1})-S(P_{N})$ measures the total amount of correlations in the system ref16 .
In this application, we obtain $I_{2}/I_{N}=98.5\%$. The pairwise correlations model is effective since it explains almost all the available information; only $1.5\%$ of information is due to higher-order interactions.
As a further test of the pairwise model consistency, we show below that this statistical model is able to recover the observed empirical moments. We compare the average index orientations $q_{i}=T^{-1}\sum_{t=1}^{T}s_{i,t}$ obtained by simulation to the real ones. We simulated the process by doing $1\times 10^{5}$ equilibration Monte Carlo time steps (MCS) and we take the average on the next $2\times 10^{7}$ MCS in order to reduce the variance of the estimator. The flipping attempts are simulated by the Glauber dynamics. Namely, we take an entity $i$ chosen randomly and the attempt to flip the associated binary variable $s_{i}$ is performed with a rate depending on the exponential weight, the other orientations remaining fixed ref2 . We take the time average for each index from the data and we compare it to the value obtained with the simulation; they are illustrated in Fig-2.
The root mean squared error (RMSE) is equal to $7\times 10^{-4}$, which represents $1.5\%$ of the root mean squared (RMS) value of the six arithmetic means (equal to $4.90\times 10^{-2}$). We recover quantitatively the average orientation of the six indices on the observation period. Moreover, since we obtained the probability distribution, we can compare the correlation coefficients resulting from the sampling of the proposed probability distribution to the empirical ones. We sample the probability law $p_{2}(\textbf{s},\mathbf{J}^{\mathrm{\scriptstyle{ME}}},\mathbf{h}^{\mathrm{%
\scriptstyle{ME}}})$ by a Monte Carlo Markov chain (MCMC). We take $1.2\times 10^{6}$ equilibration steps and $1.2\times 10^{4}$ independent sampling steps between each sample. Fig-3 illustrates the recovered correlation coefficients with the maximum entropy estimation versus the empirical ones. The results for only 130 observations (chosen arbitrarily corresponding to half a year) are conclusive. Indeed the RMSE represents $8.3\%$ of the RMS value and the correlation coefficient of the empirical and simulated values is equal to $0.963$. Including more observations (2258 trading days) allows us to reduce the dispersion in the results (correlation coefficient of the empirical and simulated values equal to $0.997$; the RMSE represents $1.8\%$ of the RMS value). We note that it is effective even with few data.
We perform the same work for the Dow Jones and the S$\&$P100 indices (2500 configurations observed from 10/10/2001 to 02/08/2011). We also consider 116 stocks from the New York Stock Exchange available on the Onnela’s website (http://jponnela.com/) extending from the beginning of 1982 to the end of 2000 (4800 trading days). For these larger stock sets, the exact entropy maximization (II.1) is not computationally tractable. There are several approximate inversion methods to estimate the parameters. The mean field methods (naïve, TAP and Tanaka’s inversion see ref4 ; Sc ; refTanaka ) are the faster ones and they are accurate if the interaction strengths are weak (the weakness will be investigated in a further work). These methods will be used in the investigation of the structure evolution due to their reasonable accuracy and quickness. Two other valuable inference methods are minimum probability flow (MPF) Sohl and regularized pseudo-likelihood maximization (rPLM) Aurell . In our application the rPLM method performs best.
The results for the first and second recovered moments ($2\times 10^{6}$ equilibration MCS, values estimated on $2\times 10^{7}$ samples recorded each $N$ MCS) are illustrated in Fig-4 and Fig-5.
The correlation coefficient between the recovered and empirical values is respectively $0.998$, $0.996$ and $0.997$ for the net orientations illustrated in Fig-4 and $0.989$, $0.964$, $0.997$ for the covariances illustrated in Fig-5 which shows the strong linear statistical relation between the empirical and the recovered values.
The relative deviation between the RMSE and the RMS values is respectively $2\%$, $7\%$ and $6\%$ for the net orientations and $9\%$, $17\%$, $8\%$ for the covariances.
We have seen that, in addition of the multi-information criterion, the net orientations and the covariances are recovered from this model even with few data. We conclude that the proposed pairwise interaction structure is a trustful one; this means that interactions are believed to be pairwise and symmetric ones and that they cause correlations.
IV Order-disorder transition
As the previous pairwise model describes market indices quantitatively, we expect to observe an order-disorder transition in this system; we give below some empirical evidence that these transitions actually appear.
As the interaction strengths are all positive, the system is ordered if the net orientation distribution has two modes near the extreme values $-1$ and $1$ and disordered if the distribution has a unique mode. Indeed in an ordered situation, each index tends to have the same orientation as the others. Furthermore, in the absence of external influences, both extreme values are equivalent (as a consequence of the symmetry under sign exchange), and the distribution is thus bimodal. One of the extreme values can be favored following the values taken by the external influences $h_{i}$. It will be a first clue that the system is reorganized if the distribution changes in such a way (having two modes and then a unique one, and reciprocally).
We compute the system net orientation $q(\tau,\Delta t)=(\Delta t\,N)^{-1}\sum_{i}\sum_{t=\tau}^{\tau+\Delta t}s_{i,t}$ on successive periods $\Delta t$ of 25 trading days (without overlapping), and we show that the net orientation probability distribution can be bimodal or not on successive time windows. The resulting empirical distributions for observations from 5 November 2010 to 30 March 2011 are illustrated in Fig-6.
In Fig-6 we see that the empirical probability distribution has initially two modes at extreme orientation values then has no clear mode, and finally again has two modes. During this period, initially the indices move in an organized way then in a disorganized fashion, and finally the Fukushima nuclear accident caused a large global market fall followed by a large recovery. During this event, the indices were in comovement. So the system is initially ordered then disordered for two periods and then again ordered.
Another way to characterize financial irregularities is to study the entropy $S(\textbf{s})$ on a sliding window (here, 300 trading days shifted by 1 day). We compute the mean-field approximation of the entropy on those time windows (much faster than the exact computation). The mean-field entropy binder is
$$S_{\mathbf{MF}}(\textbf{s})=-\sum_{i=1}^{N}\frac{1+q_{i}}{2}\ln(\frac{1+q_{i}}%
{2})+\frac{1-q_{i}}{2}\ln(\frac{1-q_{i}}{2})$$
(6)
The entropy is maximal when the average orientations, computed on the corresponding time window, are equal to zero and is minimal when the indices have the same orientation. During a disordered period, the entropy should be large and during a synchronized (ordered) period the entropy should be low. We should thus observe entropy minima simultaneously to orientation extrema (bubbles or crashes). We check in the results illustrated in Fig-7 that orientation extrema and entropy minima are related to the periods of synchronization described in Dal .
We observe large falls of the entropy when the net orientation is much larger than its mean (the mean is set to zero in Fig-7). The shaded portions show the orientation extrema and entropy minima on this time window. They correspond (chronologically) to the end of the growth period and the end of the collapse. Furthermore the correlation coefficient of the net-orientation and the financial time series is equal to $0.82$ showing a high degree of linear statistical dependency. We conclude that the entropy minima are thus related to financial irregularities (large upward or downward movements).
This is an empirical evidence that order-disorder transitions occur in markets. This interpretation is supported by the recent results obtained in Dal , where the authors showed that market irregularities present a high degree of synchronization, meaning an ordered state.
The economic consequence is that the whole market is correlated when such transitions occur. It also means the absence of a characteristic scale for the fluctuations and the emergence of power-laws.
In appendix, we illustrate in Fig-14 a larger version of Fig-7.
V Dynamics of interactions
Linked to the above, such a transition occurs if the stochasticity changes or the interaction strengths change. A possible interpretation of time-varying interaction strengths is that some learning or adaptive process takes place through time. This means that the market adjusts the interactions between its entities in some adaptive processes so the $\{J_{ij},h_{i}\}$ are time dependent. The reason is that the background, namely worldwide economic conditions, changes through time and goes through economic fluctuations with contractions (recessions) and expansions (growths). As the correlations are explained by the pairwise interactions, it also means that the correlations to be do not necessarily match past correlations.
Following this interpretation, we expect that the temporal behaviors of the interaction strengths and external influences are related to market evolution. This is indeed true, as we will see below. First of all, we study the preference evolution of the six previous indices (reflecting the current state of the European economy) and its link to the crises.
We use a sliding temporal window of width $T=200$ trading days shifted by a constant amount of $\Delta t=2$ trading days. We show that the aggregate preference $h=\sum_{i}h_{i}$ is negative during a crisis (or during a significative contraction) as illustrated in Fig-8.
The first negative incursion corresponds to the 2002-2003 crisis and the second one to the 2008-2009 crisis ref3 .
As expected the external influences are decreasing when the market is subject to a crisis.
More interestingly, we will study the spectrum of the interaction matrix. Indeed the spectrum evolution will be related to the market evolution. The spectrum of the interaction matrix of a stock set has an interesting feature; we will show it for the Dow Jones index. We collected data for the Dow Jones index from the 10 October 2001 to 1 August 2011 ref3 , and we extract the interaction strengths using the third-order approximation described in refTanaka .
The trace of the interaction matrix, the sum of its eigenvalues, has the following interesting property. It decreases during a crisis; specifically, the trace minus its temporal average becomes negative if there is a substantial fall of the index, this feature is illustrated in Fig-9.
The trace of the exact interaction matrix should be zero (without self-interactions) but, with the Tanaka’s diagonal trick, the diagonal entries are related to the second-order term and to a part of the third-order of the Plefka series refTanaka ; Plef . The second-order term of the Plefka series is negative, the sign of the third-order term depends on the product of the interaction strengths. The temporal variation of the trace reflects the temporal variation of these second and third order terms. These terms are particularly important near a transition. This explains why the trace of the interaction matrix is smaller than its mean value during a crisis. Indeed during a crisis all the stocks act in similar way: they fall down. They thus have similar mean orientation (down) and the resulting system state is an ordered one. Before the crisis, during a common market growth or steady state, the price of some stocks rises (on average) and some others fall leading to a dispersion of the mean orientations. This is indirect evidence of a transition from one regime to another and of coordination. It is consistent with the results obtained above and in Dal ; Jr . In appendix, we illustrate in Fig-13 a larger version of the Fig-9.
VI Link to the graph-theoretic approach
Hereafter, we make the link with the previous spectrum feature and the observation that the length of the minimum spanning tree (MST) based on the Sornette-Mantegna distance decreases during a crash Mant ; Onnela , meaning that stocks are highly correlated during these events (as they should be in an order-disorder transition). We will see that we recover this feature with the pairwise model with a distance based on interaction strengths in place of correlation coefficients. Indeed the interaction matrix can be thought of as the weight matrix of an undirected complete graph. Using a modified version of the method proposed in Ding and computing the minimum spanning tree length $L(t)$ (the sum of the edges weights of the MST), we also observe that this length decreases during a crash, as expected; the results for the Dow Jones index are illustrated in Fig-10.
Moreover, it also allows cluster identification. Indeed, it is known that the asset tree based on the Sornette-Mantegna distance allows regrouping some stocks in clusters following their economic sectors Onnela . As the correlations are caused by the interactions, it is not surprising that the MST of the network defined by the interaction matrix also allows cluster identification. This approach has the advantage of not being limited to linear or monotonic statistical dependencies. The clustering feature is illustrated in Fig-11.
We note that General Electric (GE) is not the most connected node but it is a cental one in the sense that it appears in three different clusters, as such it is still considered as the root of the MST and defines the generational direction. This approach provides a different classification as given in Onnela or given by the Forbes for instance. Indeed, Forbes classification is given by sector then by industry. Disney and Walmart are classified in the same sector, services; this category is too vague to be an useful tag. Similarly, General Electric is tagged by Forbes as industrial goods and then as diversified machinery but this company also provides financial services, aircraft engines, TV channel broadcasting, etc. It is then clear that this company should be classified with more than one tag, as does the proposed method. In this point of view, the internal structure of each company seems to be the crucial information to identify stock clusters.
We can also study the topological structure of the remaining asset tree during a crash and a growth period. We will see that, as expected, the degree distribution follows a power law. We consider the stocks of the S&P100 index on two intervals, from 1/10/2007 to 01/02/2009 (360 trading day crisis period) and from 1/02/2005 to 1/07/2007 (600 trading day growth period). The occurring frequencies of the vertex degrees are illustrated in Fig-12.
Thid reveals that the degree distribution is a power law, $f(n)\sim n^{-\alpha}$, and the value of the exponent is similar for the both periods. For the growth period, we obtain $\hat{\alpha}=1.64\pm 0.17$ and during a crash $\hat{\alpha}=1.58\pm 0.12$. They can be included in the confidence interval of each other, so they are very similar.
The maximum degree is $n=8$ in the both periods. They are 58 vertices of degree $n=1$ during the crash. This value is slightly larger (about $10\%$) than the one corresponding to the growth period, 52 vertices of degree $n=1$. This explains the difference between both exponents. The asset tree topology is thus slightly different during a crash. The main change is the variation of the interaction strengths (the graph weights) rather than the variation of the vertex degrees. In both regimes, the asset trees are thus scale-free networks. This implies that the edges are not drawn at random and the asset trees exhibit small-worldness, as observed with another method in Petra . Furthermore, the low value of this exponent implies that hubs (high-degree vertices) represent a significant part of the total number of vertices. The market is thus sensitive to the failure of a hub (a highly connected company) whereas the failure of a leaf (terminal node) will only slightly affect the market. By example the hypothetic failure of the American Express Company (AXP) would leave a fragmented market whereas the bankruptcy of Kraft Food Inc. (KFT) would not change the topology of the asset tree significantly; see Fig-11. This could help in selecting the companies one has to save from an eventual bankruptcy in order to minimize the impact of such an event. This could also help to select which companies one has to monitor to prevent a hypothetical dramatic system failure.
VII Conclusion
We have seen that, without making assumptions on the market dynamics, the maximum entropy principle provides a rigorous pairwise model which is able to describe the data and the observed collective behaviors quantitatively. We showed that including higher-order interactions does not explain more than using the pairwise model, and thus that the collective phenomena emerge from simple pairwise interactions. To confirm this result, we showed that this statistical model is able to recover the empirical moments computed from the data, especially the mean orientations and the correlations.
The success of the pairwise model implies that markets exhibit some properties observed in magnetic materials and in neural networks. Indeed, we showed that an order-disorder transition occurs in such a system, as described by a pairwise model equivalent to the Ising model. Furthermore, we showed that the interaction strengths are time dependent meaning that an adaptive process occurs and that they are the starting point of the graph-theoretic approach of the market.
In this view the system is more than the sum of its parts, is ruled by its entities pairs, exhibits collective behaviors and is quantitatively described by a pairwise model. It is surprising that such sophisticated collective behaviors, emergent structures and underlying complex trading rules are captured by a simple (a priori) scheme of interdependence involving only pairwise but no higher-order interactions.
Acknowledgments
I would like to thank B. De Rock and P. Emplit for their helpful comments and discussions. This work was undertaken with financial support from the Solvay Brussels School of Economics and Management.
References
(1)
Dal’Maso Peron, T. & Rodrigues, F.
Collective behavior in financial markets.
EPL (Europhysics Letters)
96, 48004 (2011).
(2)
Onnela, J.-P., Chakraborti, A.,
Kaski, K., Kertész, J. &
Kanto, A.
Dynamics of market correlations: Taxonomy and
portfolio analysis.
Phys. Rev. E 68,
056110 (2003).
URL http://link.aps.org/doi/10.1103/PhysRevE.68.056110.
(3)
Peron, T. K. D., da Fontoura Costa, L. &
Rodrigues, F. A.
The structure and resilience of financial market
networks.
Chaos: An Interdisciplinary Journal of
Nonlinear Science 22, 013117
(2012).
URL http://link.aip.org/link/?CHA/22/013117/1.
(4)
Stanley, H., Gabaix, X.,
Gopikrishnan, P. & Plerou, V.
Economic fluctuations and statistical physics:
Quantifying extremely rare and less rare events in finance.
Physica A: Statistical Mechanics and its
Applications 382, 286–301
(2007).
(5)
Stanley, H., Plerou, V. &
Gabaix, X.
A statistical physics view of financial fluctuations:
Evidence for scaling and universality.
Physica A: Statistical Mechanics and its
Applications 387, 3967–3981
(2008).
(6)
Vértes, P., Nicol, R.,
Chapman, S., Watkins, N. &
Bullmore, E.
Frontiers: Topological isomorphisms of human brain
and financial market networks.
Frontiers in Systems Neuroscience
(2011).
(7)
Cover, T., Thomas, J.,
Wiley, J. et al.
Elements of information theory,
vol. 6 (Wiley Online Library,
1991).
(8)
Rosenow, B., Gopikrishnan, P.,
Plerou, V. & Stanley, H.
Random magnets and correlations of stock price
fluctuations.
Physica A 314,
762–767 (2002).
(9)
Fischer, K. H. & Hertz, J. A.
Spin Glasses (Cambridge
University Press, Cambridge, 1991).
(10)
Schneidman, E., Berry, M. J.,
Segev, R. & Bialek, W.
Weak pairwise correlations imply strongly correlated
network states in a neural population.
Nature 440,
1007–1012 (2006).
URL http://dx.doi.org/10.1038/nature04701.
(11)
Aoki, M.
New approaches to macroeconomic modeling:
evolutionary stochastic dynamics, multiple equilibria, and externalities as
field effects (Cambridge Univ Pr,
1998).
(12)
Brock, W. A. & Durlauf, S. N.
Discrete choice with social interactions.
Rev. Econ. Stud.
68, 235–260
(2001).
URL http://dx.doi.org/10.2307/2695928.
(13)
Mas-Colell, A., Whinston, M.,
Green, J. & de Cičncies
Econňmiques i Empresarials, U. P. F. F.
Microeconomic theory,
vol. 1 (Oxford university press New
York, 1995).
(14)
Jaynes, E. T.
Information theory and statistical mechanics.
Phys. Rev. 106,
620–630 (1957).
URL http://dx.doi.org/10.1103/PhysRev.106.620.
(15)
Kindermann, R.
Markov Random Fields and Their Applications
(Contemporary Mathematics ; V. 1) (Amer Mathematical
Society, 1980).
URL http://www.worldcat.org/isbn/0821850016.
(16)
Roudi, Y., Tyrcha, J. &
Hertz, J.
Ising model for neural data: Model quality and
approximate methods for extracting functional connectivity.
Phys. Rev. E 79,
051915+ (2009).
URL http://dx.doi.org/10.1103/PhysRevE.79.051915.
(17)
Kubo, R.
Generalized cumulant expansion method.
Journal of the Physical Society of Japan
17, 1100 (1962).
(18)
Plefka, T.
Convergence condition of the tap equation for the
infinite-ranged ising spin glass model.
Journal of Physics A: Mathematical and
General 15, 1971
(1982).
URL http://stacks.iop.org/0305-4470/15/i=6/a=035.
(19)
Barber, D. & van de Laar, P.
Variational cumulant expansions for intractable
distributions.
Journal of Artificial Intelligence Research
10, 435–455
(1999).
(20)
Shapira, Y., Kenett, D. Y. &
Ben-Jacob, E.
The index cohesive effect on stock market
correlations.
The European Physical Journal B - Condensed
Matter and Complex Systems (2009).
URL http://dx.doi.org/10.1140/epjb/e2009-00384-y.
(21)
Yahoo (2011).
Http://finance.yahoo.com/.
(22)
Schneidman, E., Still, S.,
Berry, M. J. & Bialek, W.
Network information and connected correlations.
Phys. Rev. Lett.
91, 238701+
(2003).
URL http://dx.doi.org/10.1103/PhysRevLett.91.238701.
(23)
Glauber, R. J.
Time-Dependent statistics of the ising model.
J. Math. Phys.
4, 294–307
(1963).
URL http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=JMAPAQ000004000002000294000001&idtype=cvips&gifs=yes.
(24)
Schaub, M. & Schultz, S.
The ising decoder: reading out the activity of large
neural ensembles.
Journal of Computational Neuroscience
1–18 (2010).
(25)
Tanaka, T.
Mean-field theory of boltzmann machine learning.
Phys. Rev. E 58,
2302–2310 (1998).
URL http://link.aps.org/doi/10.1103/PhysRevE.58.2302.
(26)
Sohl-Dickstein, J., Battaglino, P. B. &
DeWeese, M. R.
New method for parameter estimation in probabilistic
models: Minimum probability flow.
Phys. Rev. Lett.
107, 220601
(2011).
URL http://link.aps.org/doi/10.1103/PhysRevLett.107.220601.
(27)
Aurell, E. & Ekeberg, M.
Inverse ising inference using all the data.
Phys. Rev. Lett.
108, 090201
(2012).
URL http://link.aps.org/doi/10.1103/PhysRevLett.108.090201.
(28)
Binder, K. & Young, A.
Spin glasses: Experimental facts, theoretical
concepts, and open questions.
Reviews of Modern physics
58, 801 (1986).
(29)
Sandoval, L. & Franca, I.
Correlation of financial markets in times of crisis.
Physica A Statistical Mechanics and its
Applications 391, 187–208
(2012).
(30)
Mantegna, R.
Hierarchical structure in financial markets.
The European Physical Journal B-Condensed
Matter and Complex Systems 11,
193–197 (1999).
(31)
Onnela, J., Chakraborti, A.,
Kaski, K. & Kertesz, J.
Dynamic asset trees and black monday.
Physica A: Statistical Mechanics and its
Applications 324, 247–252
(2003).
(32)
Ding, C., He, X., Xiong,
H. & Peng, H.
Transitive closure and metric inequality of weighted
graphs: detecting protein interaction modules using cliques.
International journal of data mining and
bioinformatics 1, 162–177
(2006). |
Superluminal self-interacting neutrino
Ernst Trojan
Moscow Institute of Physics and Technology
PO Box 3, Moscow, 125080, Russia
Abstract
The effect of nonlinear self-interaction can be associated with superluminal
velocity of neutrino. The power energy spectrum $E=p+Cp^{a}$ is derived from
the nonlinear Dirac equation when interaction term $V=\lambda(\bar{\psi}\gamma_{\mu}\psi\bar{\psi}\gamma^{\mu}\psi)^{a}$ is added to the Lagrangian
of a free spin-1/2 particle. The superluminal velocity recorded by the OPERA
and MINOS collaborations is achieved when the coupling constants are taken
in the range $a=0.4\div 1.18$ and $\lambda=-\left(0.5\div 1.6\right)\times 10^{-4}$. The self-interaction Lagrangian $V=\lambda\bar{\psi}\gamma_{\mu}\psi\bar{\psi}\gamma^{\mu}\psi$ with the coupling constant $\lambda=-\left(0.7\div 0.9\right)\times 10^{-4}$ yields the same result. Scalar
interaction $V=\lambda(\bar{\psi}\psi)^{b}$ and scalar-vector interaction $\lambda\left(\psi^{\dagger}\psi\right)^{b+1}/\left(\bar{\psi}\psi\right)^{b}$ cannot be responsible for the observed superluminal neutrino.
1 Introduction
Neutrino was believed to be a massless spin-1/2 fermion with energy
$$E=pc$$
(1)
and group velocity $v=c$ equal to the speed of light $c=1$ (in relativistic
units). The modern theory expects, however, that neutrino has finite mass
[1]
$$m=m_{\nu}<0.28\,\mathrm{eV}$$
(2)
that implies deviation from the energy spectrum (1) and velocity
$$v=\frac{dE}{dp}\neq 1$$
(3)
Recent experiments of the OPERA Collaboration [2] revealed
superluminal motion of neutrino with energy $E=17\,\mathrm{GeV}$ at the
average velocity
$$v=1+\mathbf{2.37}\,\mathbf{\times}10^{-5}$$
(4)
while the the MINOS Collaboration detected velocity
$$v=1+\mathbf{5.1\times}10^{-5}$$
(5)
for the low energy neutrino with energy spectrum peaked at approximately $E=3\,\mathrm{GeV}$. Superluminal neutrino was also observed in supernova
explosion SN1987a [4].
This fact is a serious puzzle to the researchers. Superluminal velocity (4) cannot belong to a free massive particle with energy spectrum $E=\sqrt{p^{2}+m^{2}}$ whose velocity is always subluminal because
$$v-1=-\frac{1}{2}\frac{m^{2}}{p^{2}}\simeq-\frac{1}{2}\frac{m^{2}}{E^{2}}<0$$
(6)
The tachyonic energy spectrum $E=\sqrt{p^{2}-m^{2}}$ results in velocity above
the speed of light
$$v-1=\frac{1}{2}\frac{m^{2}}{p^{2}}\simeq\frac{1}{2}\frac{m^{2}}{E^{2}}>0$$
(7)
that does not exceed $3\times 10^{-22}$ even at maximum possible neutrino
mass $m$ (2) and $E=17\,\mathrm{GeV}$. Nevertheless, in the frames of
the accuracy of measurements, the energy spectrum of superluminal neutrino
of OPERA [2] and MINOS [3] can be fitted to a power law [5, 6, 7]
$$E=p+Cp^{a}$$
(8)
where the coefficients must be taken in the range [8]
$$a=0.40\div 1.18\qquad C=4.15\times 10^{-4}\div 1.5\times 10^{-5}$$
(9)
and $C=\left(2.2\,\div 3.03\right){\times}10^{-5}$ if we choose $a\equiv 1$. Indeed, neutrino is not a free particle, but there are several
interesting hypotheses to explain its superluminal motion [9].
In the present paper we consider massive neutrino whose Lagrangian
$$L=\bar{\psi}\left(i\gamma^{\mu}\partial_{\mu}-m\right)\psi+V\left(\bar{\psi},%
\psi\right)$$
(10)
includes nonlinear self-interaction term $V\left(\bar{\psi},\psi\right)$.
There is no additional interaction with external fields and the medium,
while the superluminal velocity is hidden in the very nature of neutrino. We
need to find the energy spectrum of nonlinear Dirac equation
$$\left(i\gamma^{\mu}\partial_{\mu}-m\right)\psi+\frac{\partial V}{\partial\bar{%
\psi}}=0$$
(11)
and check whether the Lagrangian (10) is adjustable to reproduce the
superluminal velocity (4)-(5) detected by the OPERA [2] and MINOS [3] collaborations.
2 Effect of self-interaction
Consider the Lagrangian (10) with a simple self-interaction term
$$V\left(\bar{\psi},\psi\right)=\frac{\lambda}{b+1}\left(\psi^{\dagger}\psi%
\right)^{b+1}\qquad\psi^{\dagger}=\bar{\psi}\gamma^{0}$$
(12)
The relevant Dirac equation (11) is written
$$\left(i\gamma^{\mu}\partial_{\mu}-m+F\right)\psi=0$$
(13)
where
$$F=\gamma^{0}\omega$$
(14)
and
$$\omega=\lambda\left(\psi^{\dagger}\psi\right)^{b}$$
(15)
The Dirac equation (13) at $\omega=0$ has well-known solution in the
form of plane wave
$$\psi_{0}=\left(\begin{array}[]{c}\phi_{0}\\
\chi_{0}\end{array}\right)\exp\left(-iE_{0}t\right)\qquad\chi_{0}=\frac{\vec{%
\sigma}\cdot\vec{p}}{E_{0}+m}\phi_{0}\qquad E_{0}=\sqrt{m^{2}+p^{2}}$$
(16)
Substituting stationary wave function
$$\psi=\varphi\left(\vec{r}\right)\exp\left(-iEt\right)$$
(17)
in (13), we have equation
$$\left(-i\vec{\alpha}\cdot\nabla+\beta m\right)\varphi=\left(E+\omega\right)\varphi$$
(18)
where
$$\vec{\alpha}=\beta\vec{\gamma}\qquad\beta=\gamma^{0}=-\gamma_{0}$$
(19)
Substituting a plane-wave bispinor
$$\varphi=\left(\begin{array}[]{c}\phi\\
\chi\end{array}\right)\exp\left(i\vec{p}\cdot\vec{r}\right)$$
(20)
in (18), we obtain a linear system of equations
$$\begin{array}[]{c}\vec{\sigma}\cdot\vec{p}\chi=\left(E+\omega-m\right)\phi\\
\vec{\sigma}\cdot\vec{p}\phi=\left(E+\omega+m\right)\chi\end{array}$$
(21)
that has solution if and only if
$$E=\sqrt{m^{2}+p^{2}}-\omega$$
(22)
where $p=\left|\vec{p}\right|$.
We can estimate the energy spectrum (22) in the frames of mean-field
approximation if we neglect correlations between the field operators in (15) and apply effective interaction
$$\omega\simeq\omega_{*}=\lambda n^{b}$$
(23)
where
$$n=\left\langle\psi^{\dagger}\psi\right\rangle$$
(24)
is the particle number density. The latter can be adjusted as $n=1/V$ for 1
particle in volume $V$. For a many-particle system the quantity (24) is
determined according to formula
$$n=\frac{2}{\left(2\pi\right)^{q}}\int\limits_{0}^{\infty}f_{k}\,d^{q}k$$
(25)
where $f_{p}$ is the distribution function in $q$-dimensional momentum space.
The latter is the Fermi-Dirac distribution function if it is an ideal gas in
equilibrium, however, the OPERA [2] and MINOS [3] collaborations
considered a neutrino beam with the energy peaked at $k=p$, and its
distribution function corresponds to a delta-function
$$f_{k}=\delta\left(1-\frac{k}{p}\right)$$
(26)
in 1-dimensional momentum space, so that the particle number density (25) is estimated so
$$n=\frac{p}{\pi}$$
(27)
and, according to (22) and (23), the energy spectrum is
$$E=\sqrt{p^{2}+m^{2}}-\lambda n^{b}=\sqrt{p^{2}+m^{2}}-\frac{\lambda}{\pi^{b}}%
\,p^{b}$$
(28)
3 Vector self-interaction
Consider more general form of vector self-interaction [10]
$$V=\frac{\lambda}{b+1}\left(\bar{\psi}\gamma_{\nu}\psi\bar{\psi}\gamma^{\nu}%
\psi\right)^{b+1}$$
(29)
corresponding to the Dirac equation
$$\left(i\gamma^{\mu}\partial_{\mu}-m+F\right)\psi=0$$
(30)
where
$$F=\lambda\left(\bar{\psi}\gamma_{\nu}\psi\bar{\psi}\gamma^{\nu}\psi\right)^{b}%
\gamma_{\mu}\left(\bar{\psi}\gamma^{\mu}\psi\right)$$
(31)
At $b=1$ it is no more than the Heisenberg model of self-interaction [11]
$$F=\lambda\gamma_{\mu}\left(\bar{\psi}\gamma^{\mu}\psi\right)$$
(32)
When we consider 1-dimensional neutrino beam, we take into account that
gamma-matrices in (1+1)-dimensional representation [10]
$$\gamma^{0}=\left(\begin{array}[]{cc}1&0\\
0&-1\end{array}\right)\qquad\gamma^{x}=\left(\begin{array}[]{cc}0&i\\
i&0\end{array}\right)$$
(33)
satisfy standard commutation relations
$$\left\{\gamma_{\mu},\gamma_{\nu}\right\}=2\eta_{\mu\nu}\qquad\eta_{\mu\nu}=%
\left(\begin{array}[]{cc}1&0\\
0&-1\end{array}\right)$$
(34)
Substituting stationary plane wave solution
$$\psi=\left(\begin{array}[]{c}u\\
v\end{array}\right)\exp\left(ip_{x}x-iEt\right)\qquad\vec{p}=\left(p_{x},0,0\right)$$
(35)
in the Dirac equation (45), we have [10]
$$\begin{array}[]{c}ip_{x}v=\left(E+\omega-m\right)u\\
-ip_{x}u=\left(E+\omega+m\right)v\end{array}$$
(36)
instead of (21), while
$$\omega=\lambda\left(\left|u\right|^{2}+\left|v\right|^{2}\right)^{b}=\lambda%
\left(\psi^{\dagger}\psi\right)^{b}$$
(37)
formally coincides with (15). The linear system (36) has solution
if and only if condition (22) is satisfied. Again, applying the
mean-field approximation
$$\omega\simeq\omega_{*}=\lambda\left\langle\psi^{\dagger}\psi\right\rangle^{b}=%
\lambda n^{b}$$
(38)
we obtain the same formula for the energy spectrum (28) of a
1-dimensional neutrino beam.
The group velocity is immediately calculated
$$v=\frac{p}{\sqrt{p^{2}+m^{2}}}-\frac{\lambda b}{\pi^{b}}\,p^{b-1}$$
(39)
that tends to
$$v\simeq 1-\frac{\lambda b}{\pi^{b}}p^{b-1}-\frac{m^{2}}{2p^{2}}$$
(40)
in the ultra-relativistic limit $E\simeq p\gg m$. The latter term in (40) is negligible even at the upper bound of neutrino mass (2),
and superluminal velocity (4)-(5) can be explained by the
second term if $\lambda b<0$. Indeed, the power energy spectrum (8) is
compatible with (40) if
$$b=a=0.4\div 1.18$$
(41)
$$\lambda=-\left(0.5\div 1.6\right)\times 10^{-4}$$
(42)
Particularly, the coupling constant may vary in the range
$$\lambda=-\left(0.7\div 0.9\right)\times 10^{-4}$$
(43)
when $a=b=1$ that corresponds to the Heisenberg model (32).
Therefore, the origin of superluminal velocity of neutrino [2, 3] can
be associated with vector self-interaction (29) if the coupling
constants are properly identified (41)-(42).
4 Scalar self-interaction
Consider the Lagrangian (10) with scalar self-interaction
$$V=\frac{\lambda}{b+1}\left(\bar{\psi}\psi\right)^{b+1}$$
(44)
The Dirac equation (11) has the form
$$\left(i\gamma^{\mu}\partial_{\mu}-m+W\right)\psi=0$$
(45)
where
$$W=\lambda\left(\bar{\psi}\psi\right)^{b}$$
(46)
Substituting stationary wave function $\psi=\varphi\left(\vec{r}\right)\exp\left(-iEt\right)$ and the effective mass
$$m_{*}=m-W$$
(47)
in (45), we obtain equation
$$\left(-i\vec{\alpha}\cdot\nabla+\beta m_{*}\right)\varphi=E\varphi$$
(48)
that has plane-wave solution
$$\varphi=\left(\begin{array}[]{c}\phi\\
\chi\end{array}\right)\exp\left(i\vec{p}\cdot\vec{r}\right)\qquad\chi=\frac{%
\vec{\sigma}\cdot\vec{p}}{E+m_{*}}\phi$$
(49)
for a free particle with the energy spectrum
$$E=\sqrt{\vec{p}^{2}+m_{*}^{2}}$$
(50)
and relevant group velocity
$$v=\frac{1}{\sqrt{p^{2}+m_{*}^{2}}}\left(p+m_{*}\frac{dm_{*}}{dp}\right)$$
(51)
Solution (49) also implies
$$\bar{\psi}\psi=\left\|\phi\right\|^{2}-\left\|\chi\right\|^{2}=\frac{m_{*}}{E}%
\psi^{\dagger}\psi$$
(52)
where
$$\psi^{\dagger}\psi=\left\|\phi\right\|^{2}+\left\|\chi\right\|^{2}$$
(53)
Applying the mean-field approximation, we neglect correlations in (46)
and use the effective interaction
$$W\simeq W_{*}=\lambda n_{s}^{b}$$
(54)
where
$$n_{s}=\left\langle\bar{\psi}\psi\right\rangle=\frac{m_{*}}{E}n$$
(55)
is the scalar density and $n$ is the particle number density (24). For
a many-particle system it is determined by formula
$$n_{s}=\frac{2}{\left(2\pi\right)^{q}}\int\limits_{0}^{\infty}\frac{m_{*}\left(%
k\right)}{E\left(k\right)}f_{k}\,d^{q}k$$
(56)
The distribution function $f_{k}$ of a neutrino beam with the energy peaked at
$k=p$ is presented by a delta-function (26) in 1-dimensional momentum
space, so that the scalar density (55) is estimated
$$n_{s}=\frac{p}{\pi}\frac{m_{*}\left(p\right)}{E\left(p\right)}$$
(57)
Substituting (54) and (57) in (47), we obtain
self-consistent equation for the effective mass
$$m_{*}\left(p\right)=m-\lambda n_{s}^{b}=m-\lambda\left[\frac{p}{\pi}\frac{m_{*%
}\left(p\right)}{\sqrt{\left(p^{2}+m_{*}^{2}\right)}}\right]^{b}$$
(58)
Consider the ultra–relativistic limit $E\simeq p\gg m_{*}$. The energy (50) is reduced to
$$E=p+\frac{m_{*}^{2}}{2p}=p+\frac{m^{2}}{2p}-\frac{mW_{*}}{p}+\frac{W_{*}^{2}}{%
2p}$$
(59)
while the effective interaction (54) is reduced to
$$W_{*}\simeq\frac{\lambda}{\pi^{b}}m_{*}^{b}$$
(60)
and velocity (51) is estimated
$$v\simeq 1-\frac{m_{*}^{2}}{2p^{2}}$$
(61)
Although the deviation from the speed of light may be visible at large $W_{*}$, the velocity (61) is subluminal at any choice of constants $\lambda$ and $b$. Therefore, the scalar self-interaction cannot be
responsible for the superluminal neutrino velocity.
5 Scalar-vector self-interaction
Now consider a mixed variant of scalar-vector self-interaction in the form
[12, 13]
$$V=\lambda\frac{\left(\psi^{\dagger}\psi\right)^{b+1}}{\left(\bar{\psi}\psi%
\right)^{b}}$$
(62)
that corresponds to the Dirac equation
$$\left(i\gamma^{\mu}\partial_{\mu}-m+W+\gamma^{0}\omega\right)\psi=0$$
(63)
where
$$\omega=\lambda\left(b+1\right)\frac{\left(\psi^{\dagger}\psi\right)^{b}}{\left%
(\bar{\psi}\psi\right)^{b}}\qquad W=-\lambda b\frac{\left(\psi^{\dagger}\psi%
\right)^{b+1}}{\left(\bar{\psi}\psi\right)^{b+1}}$$
(64)
Substituting a stationary plane wave solution
$$\varphi=\left(\begin{array}[]{c}\phi\\
\chi\end{array}\right)\exp\left(i\vec{p}\cdot\vec{r}-iEt\right)$$
(65)
and the effective mass (47) in (63), we have a linear system of
equations for the spinors
$$\begin{array}[]{c}\vec{\sigma}\cdot\vec{p}\chi=\left(E+\omega-m_{*}\right)\phi%
\\
\vec{\sigma}\cdot\vec{p}\phi=\left(E+\omega+m_{*}\right)\chi\end{array}$$
(66)
that has solution
$$\chi=\frac{\vec{\sigma}\cdot\vec{p}}{E+\omega+m_{*}}\phi$$
(67)
if and only if the particles has the energy spectrum
$$E=\sqrt{m_{*}^{2}+p^{2}}-\omega$$
(68)
where the effective mass is
$$m_{*}=m-W$$
(69)
According to (65) and (67) we have
$$\frac{\psi^{\dagger}\psi}{\bar{\psi}\psi}=\frac{\left\|\phi\right\|^{2}+\left%
\|\chi\right\|^{2}}{\left\|\phi\right\|^{2}-\left\|\chi\right\|^{2}}=\frac{%
\sqrt{m_{*}^{2}+p^{2}}}{m_{*}}$$
(70)
and the effective mass (69) is determined by self-consistent equation
$$m_{*}=m-\lambda b\frac{\sqrt{\left(m_{*}^{2}+p^{2}\right)^{b+1}}}{m_{*}^{b+1}}$$
(71)
that is reduced to
$$m_{*}=m-\lambda b\frac{p^{b+1}}{m_{*}^{b+1}}$$
(72)
in the ultra-relativistic limit $p\gg m_{*}$, so that
$$\left[\frac{b+2}{b+1}m_{*}-m\right]m_{*}^{\prime}=\left(m_{*}-m\right)\frac{m_%
{*}}{p}$$
(73)
The energy spectrum (68) is, then, reduced to
$$E\simeq p+\frac{m_{*}^{2}}{2p}-\lambda\left(b+1\right)\frac{p^{b}}{m_{*}^{b}}$$
(74)
At large coupling constant $\lambda$, when
$$m_{*}\gg m$$
(75)
equation (73) implies
$$m_{*}^{\prime}=\frac{b+1}{b+2}\frac{m_{*}}{p}$$
(76)
and, according to equation (74), the neutrino velocity is subluminal
$$v\simeq 1-\frac{m_{*}^{2}}{2p^{2}}$$
(77)
When
$$m_{*}\rightarrow m$$
(78)
equation (73) implies $m_{*}^{\prime}\rightarrow 0$ and, according
to equation (74), the neutrino velocity is also subluminal
$$v\simeq 1-\frac{m^{2}}{2p^{2}}$$
(79)
When
$$m_{*}\ll m$$
(80)
equation (73) implies
$$m_{*}^{\prime}\simeq\frac{m_{*}}{p}$$
(81)
and, according to equation (74), the neutrino velocity is
superluminal
$$v-1\simeq\frac{m_{*}^{2}}{2p^{2}}\ll\frac{m^{2}}{2p^{2}}$$
(82)
but it negligible with respect to estimation (7) for a free massive
neutrino and, hence, cannot achieve velocity (4)-(5) that
was observed in experiments [2, 3].
6 Conclusion
The model of neutrino with self-interaction can explain superluminal
velocity (4)-(5) recorded by the OPERA and MINOS collaborations [2, 3]. When the Lagrangian includes vector
self-interaction in the form of (12) or (29), the energy spectrum
of a 1-dimensional neutrino beam (28) includes additional term (37), and the neutrino can move faster than light. Its energy spectrum is
compatible with the power energy spectrum (8), and the neutrino
velocity is associated with the observed data when the coupling constants
satisfy constraints (41) and (42). The Heisenberg model of
self-interacting spin-1/2 particle (32) can also give satisfactory
result under appropriate choice of the coupling constant (42)
Neutrino with scalar self-interaction (44) is always
subluminal (61). Scalar-vector self-interaction (62) can result
in the group velocity (82) above the speed of light but it is much
smaller than the observed values (4)-(5).
Although the only model of vector self-interaction (29) is suitable
for explanation of superluminal neutrino, it looks very promising because
this effect is universal and does not depend on the medium or external
fields. It should be emphasized that the neutrino mass is not important
here, and the observed superluminal velocity can be produced by a massless
particle. We have applied the mean-field approximation (38) for
obtaining the effective energy spectrum (22) of a plane-wave solution (35). It is enough for estimation of the neutrino velocity. Exact
analysis of equation (36) may concern the problem of neutrino flavor
oscillations that is the subject for further research.
The author is grateful to Erwin Schmidt for discussions.
References
[1]
S. A. Thomas, F. B. Abdalla, and O. Lahav, Phys. Rev. Lett.
105, 031301 (2010). arXiv:0911.5291v2 [astro-ph.CO]
[2]
T. Adam et al. [OPERA Collaboration], Measurement of the
neutrino velocity with the OPERA detector in the CNGS beam, arXiv:1109.4897 [hep-ex].
[3]
P. Adamson et al. [MINOS Collaboration], Phys. Rev. D 76 (2007) 072005. arXiv:0706.0437 [hep-ex]
[4]
K. Hirata et al [Kamiokande-II Collaboration], Phys. Rev.
Lett. 58 (1987) 1490.
[5]
G. Cacciapaglia, A. Deandrea, and L. Panizzi, JHEP 11, 137, (2011). arXiv:1109.4980 [hep-ph]
[6]
N. D. Hari Dass, OPERA, SN1987a and energy dependence
of superluminal neutrino velocity, arXiv:1110.0351 [hep-ph]
[7]
G. Amelino-Camelia, G. Gubitosi, N. Loret, F. Mercati, G.
Rosati, and P. Lipari, OPERA-reassessing data on the energy
dependence of the speed of neutrinos, arXiv:1109.5172 [hep-ph]
[8]
E. Trojan, Superluminal neutrino energy spectrum of
OPERA and MINOS, arXiv:1112.2689 [hep-ph]
[9]
G. Dvali and A. Vikman, Price for Environmental
Neutrino-Superluminality, arXiv:1109.5685 [hep-ph]. L. Iorio,
Environmental fifth-force hypothesis for the OPERA superluminal
neutrino phenomenology: constraints from orbital motions around the Earth,
arXiv:1109.6249 [gr-qc]. A. Kehagias, Relativistic
Superluminal Neutrinos, arXiv:1109.6312 [hep-ph]. F.R. Klinkhamer,
Superluminal muon-neutrino velocity from a Fermi-point-splitting
model of Lorentz violation, arXiv:1109.5671 [hep-ph]. R. A
Konoplya, Superluminal neutrinos and the tachyon’s stability in the
rotating Universe, arXiv:1109.6215 [hep-th]. M. Matone, Superluminal Neutrinos and a Curious Phenomenon in the Relativistic Quantum
Hamilton-Jacobi Equation, arXiv:1109.6631 [hep-ph]. J. W. Moffat,
Bimetric Relativity and the Opera Neutrino Experiment, arXiv:1110.1330v3 [hep-ph]. A. Nicolaidis, Neutrino Shortcuts in
Spacetime, arXiv:1109.6354 [hep-ph]. E. N. Saridakis, Superluminal neutrinos in Horava-Lifshitz gravity, arXiv:1110.0697
[gr-qc]. F. Tamburini and M. Laveder, Apparent Lorentz violation
with superluminal Majorana neutrinos at OPERA? arXiv:1109.5445
[hep-ph]. P. Wang, H. Wu and H. Yang, Superluminal neutrinos and
domain walls, arXiv:1109.6930 [hep-ph]. E. Ciuffoli, J. Evslin, J.
Liu, and X. Zhang, OPERA and a Neutrino Dark Energy Model, arXiv:1109.6641 [hep-ph]. B. Alles, Relativity accommodates
superluminal mean velocities, arXiv:1111.0805 [hep-ph]. S. Nojiri
and S. D. Odintsov, Could the dynamical Lorentz symmetry breaking induce the
superluminal neutrinos? arXiv:1110.0889 [hep-ph]. M. Li and T.
Wang, Mass-dependent Lorentz Violation and Neutrino Velocity,
arXiv:1109.5924 [hep-ph].
[10]
F. Cooper, A. Khare, B. Mihaila, and A. Saxena, Solitary waves in the Nonlinear Dirac Equation with arbitrary nonlinearity,
arXiv:1007.3194
[11]
W. Heisenberg, Rev. Mod. Phys. 29, 269 (1957).
[12]
W. K. Ng, R. R. Parwani, SIGMA 5, 023 (2009). arXiv:0707.1553 [hep-th]
[13]
W. K. Ng, R. R. Parwani, Mod. Phys. Lett. A 25, 793
(2010). arXiv:0805.3015 [hep-ph] |
DC spin generation by junctions with AC driven spin-orbit interaction
M. Jonson
Department of Physics, University of Gothenburg, SE-41296 Göteborg, Sweden
R. I. Shekhter
Department of Physics, University of Gothenburg, SE-41296 Göteborg, Sweden
O. Entin-Wohlman
Raymond and Beverly Sackler School of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978, Israel
Physics Department, Ben Gurion University, Beer Sheva 84105, Israel
A. Aharony
Raymond and Beverly Sackler School of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978, Israel
Physics Department, Ben Gurion University, Beer Sheva 84105, Israel
H. C. Park
Center for Theoretical Physics of Complex Systems, Institute for Basic Science (IBS), Daejeon 34051, Republic of Korea
D. Radić
Department of Physics, Faculty of Science, University of Zagreb, Bijenička 32, Zagreb 10000, Croatia
(December 2, 2020)
Abstract
An unbiased one-dimensional weak link between two terminals, subjected to the Rashba spin-orbit interaction caused by an AC electric field which rotates periodically in the plane perpendicular to the link,
is shown to inject spin-polarized electrons into the terminals. The injected spin-polarization has a DC component along the link and a rotating transverse component in the perpendicular plane. In the adiabatic, low rotation-frequency regime, these polarization components are proportional to the frequency.
The DC component of the polarization vanishes for a linearly-polarized electric field.
Introduction. – Spintronics takes advantage of the electronic spins in designing a variety of applications, including giant magnetoresistance sensing, quantum computing, and quantum-information processing wolf ; zutic ; Kim . A promising approach for the latter exploits mobile qubits, which carry the quantum information via the spin polarization of moving electrons. The spins of mobile electrons can be manipulated by the spin-orbit interaction (SOI), which causes the spin of an electron moving through a spin-orbit active material (e.g., semiconductor heterostructures Kohda ) to rotate around an effective magnetic field winkler ; manchon . In the particular case of the Rashba SOI rashba , the magnitude and direction of this field can be tuned by gate voltages Nitta ; Sato ; Beukman ; comDres .
The Rashba SOI is mostly significant at surfaces and interfaces because of strong internal uncompensated atomic electric fields perpendicular to the surface/interface. These occur since the (weaker) surface/interface potential breaks the symmetry of the atomic orbitals there, so that the corresponding strong atomic fields no longer cancel as they do in the bulk. An electric field induced by external gates can then modulate the resulting SOI to a certain extent by changing the degree of orbital asymmetry.
One aim of spintronics is to build logic devices Kim , which produce spin-polarized electrons, so that one can use their electronic spinors as qubits. In the simplest device, electrons move between two large electronic reservoirs, via a nano-scale quantum network. For this two-terminal case, time-reversal symmetry and unitarity of the Hamiltonian prevent any spin splitting of the transport between the reservoirs bardarson .
Since the time-independent SOI obeys time-reversal symmetry, it alone cannot generate spin splitting. Time-reversal symmetry can be broken by applying a magnetic field, either via a magnetic flux, which penetrates SOI-active loops of Aharonov-Bohm interferometers lyanda ; us ; Saarikoski , or by a Zeeman magnetic field
Shmakov ; Nagasawa . Alternatives utilize ferromagnetic terminals datta ; sarkar .
Here we explore yet another means to break time-reversal symmetry, exploiting time-dependent Hamiltonians. Several papers proposed the generation of spin-splitting by quantum spin-pumping, in which different terms in the system’s Hamiltonian vary adiabatically periodically with time. Some of these require DC or AC magnetic fields marcus ; wang ; janine . Here we concentrate on all-electrical devices, which pump polarized electrons. One such device used an out-of-phase oscillation of the heights of the barriers representing the contacts between a planar quantum dot and the two leads to yield a spin current with polarization perpendicular to the plane brouwer . Alternatively, polarized spins were created by periodic variations of one barrier height and of the strength of a uniaxial-SOI (induced by an electric field perpendicular to the quantum dot’s plane) governale . In a third example, a one-dimensional wire was split into two regions, with two differently-oriented SOI-generating electric fields which oscillate periodically with time avishai .
In these examples, the two gate voltages act at different locations of the system, and the calculation yields only the average spin current, integrated over a period of the oscillation.
Below we consider
the possibility to activate spin splitting via weak links (also called ‘junctions’) by breaking time-reversal symmetry with an AC Rashba SOI created by an electric field that rotates slowly with frequency $\Omega$
perpendicularly to the (one-dimensional) weak link.
A rotating field can result from two external fields along perpendicular directions, which are normal to a thin cylindrical wire. When the two fields oscillate periodically with time, with a phase difference of $\pi/2$, the resultant vector rotates around the wire. Such fields can be produced by gate voltages $V_{y}(t)$ and $V_{z}(t)$, applied to electrodes as in
Fig. 1(a) com2 .
They can also be generated by rotating
a bent wire periodically under a uniform electric field extension , or from a circularly-polarized electromagnetic field.
Even in the absence of a voltage bias we find that a time-independent DC flow, towards both terminals, of electrons whose spins are polarized parallel to the junction’s direction, is created in the junction [as indicated by arrows in the weak link shown in Fig. 1]. In addition, the time-dependence of the SOI in the weak link gives rise to transverse components of the polarization, which rotate in the plane perpendicular to the junction in parallel to the effective SOI magnetic field. These transverse components vanish upon averaging over a period, and thus would not appear in the ‘standard’ spin-pumping approach. We analyze in detail the case of a circularly-polarized electric field, and then extend the discussion to allow for an elliptic variation of the field, all the way to the limit of a longitudinal uniaxial oscillation, where the
DC spin polarization is found to vanish.
Model and currents. –
Electronic transport
through a spin-orbit-active weak link
can be analysed within the framework of an effective tunneling Hamiltonian,
$$\displaystyle{\cal H}_{\rm tun}(t)=\sum_{k,p}\sum_{\sigma,\sigma^{\prime}}\Big%
{(}[W_{LR}(t)]_{\sigma\sigma^{\prime}}c^{\dagger}_{k\sigma}c_{p\sigma^{\prime}%
}+{\rm H.c.}\Big{)}\ ,$$
(1)
where
$c^{\dagger}_{k\sigma}$ ($c_{k\sigma}$)
creates (annihilates)
an electron of wave vector $k$ and spin index $\sigma$ in the left electrode com1 ; the wave vectors on the right are denoted by $p$.
The tunneling amplitude $W_{LR}(t)$ from right to left is a (2$\times$2) matrix in spin space, independent of the wave vectors, i.e., it is approximated by its value at the Fermi energy.
For the Rashba SOI, this tunneling amplitude has the form
$$\displaystyle W_{LR}(t)=W_{0}\exp[i{\mbox{\boldmath{$\varphi$}}}_{\rm AC}(t)]\ ,$$
(2)
where $W_{0}$ sets the magnitude of the tunnel coupling (in units of energy) and the Aharonov-Casher Aharonov-Casher ; ora
phase operator is
$$\displaystyle{\mbox{\boldmath{$\varphi$}}}_{\rm AC}(t)=k_{\rm so}d[\hat{\bf x}%
\times\hat{\bf n}(t)]\cdot{\mbox{\boldmath{$\sigma$}}}\ .$$
(3)
Here,
${\mbox{\boldmath{$\sigma$}}}=(\sigma_{x},\sigma_{y},\sigma_{z})$ is the vector of the Pauli matrices,
the unit vector $\hat{\bf x}$ is along the weak link whose length is $d$,
and $k_{\rm so}$ is the strength of the SOI (in units of inverse length), resulting from an electric field directed along $\hat{\bf n}(t)$; its explicit form is specified below.
The electronic transfer through the SOI-active weak link can be considered as an instantaneous process
at small enough frequencies $\Omega$ such that
$\Omega\ll\tau^{-1}$, where $\tau$ is the dwell time in the weak link before the electron escapes to an adjacent reservoir
(to be estimated below). In this adiabatic framework
the time-dependent tunneling through the weak link is given by Eqs. (2) and (3).
The entire tunnel junction is modelled by the Hamiltonian
$$\displaystyle{\cal H}={\cal H}_{\rm leads}+{\cal H}_{\rm tun}(t)\ ,$$
(4)
where the first term describes the non-polarized (free-electron) leads
$$\displaystyle{\cal H}_{\rm leads}=\sum_{k,\sigma}\epsilon_{k}c^{\dagger}_{k%
\sigma}c_{k\sigma}+\sum_{p,\sigma}\epsilon_{p}c^{\dagger}_{p\sigma}c_{p\sigma}\ ,$$
(5)
and $\epsilon_{k(p)}$ is the single-electron energy in the left (right) lead.
The particle current into the left electrode, $I_{L}(t)$, is the rate of change of the particle occupation there,
$$\displaystyle I_{L}(t)=\frac{d}{dt}\sum_{\sigma}\sum_{k}\langle c^{\dagger}_{k%
\sigma}(t)c_{k\sigma}(t)\rangle\equiv\sum_{\sigma}[R_{L}(t)]_{\sigma\sigma}\ ,$$
(6)
while the rate of change of the total spin
in the left terminal, and hence the spin current into that terminal, is $(\hbar/2)\dot{\bf M}_{L}$, with
$$\displaystyle\dot{\bf M}_{L}=\frac{d}{dt}\sum_{\sigma,\sigma^{\prime}}\sum_{k}%
\langle c^{\dagger}_{k\sigma}(t){\mbox{\boldmath{$\sigma$}}}_{\sigma\sigma^{%
\prime}}c_{k\sigma^{\prime}}(t)\rangle\equiv\sum_{\sigma,\sigma^{\prime}}[R(t)%
]_{\sigma\sigma^{\prime}}{\mbox{\boldmath{$\sigma$}}}_{\sigma\sigma^{\prime}}$$
(7)
and angular brackets indicate quantum averaging.
It follows that the rate of change of magnetization in the left lead is $(g\mu_{B}/2)\dot{\bf M}_{L}$.
Perturbation theory to second order in the tunneling amplitude gives ($\hbar=1$, $\eta\rightarrow 0^{+}$) supp
$$\displaystyle[R_{L}]_{\sigma\sigma^{\prime}}(t)=\sum_{k,p}\Big{[}f_{R}(%
\epsilon_{p})-f_{L}(\epsilon_{k})\Big{]}\int_{-\infty}^{t}dt_{1}e^{\eta t_{1}}$$
$$\displaystyle\times\Big{(}e^{i(\epsilon_{k}-\epsilon_{p})(t-t_{1})}[W_{LR}(t)W%
^{\dagger}_{LR}(t_{1})]_{\sigma^{\prime}\sigma}+{\rm H.c.}\Big{)}\ .$$
(8)
Here,
$f_{L(R)}(\epsilon_{k(p)})=\big{(}\exp[(\epsilon_{k(p)}-\mu_{L(R)})/(k_{B}T)]+1%
\big{)}^{-1}$ is the equilibrium Fermi function in the left (right) lead, whose chemical potential is $\mu_{L(R)}$.
Circularly rotating field. – For a circularly-polarized electric field
${\bf n}(t)=\cos(\Omega t)\hat{\bf z}-\sin(\Omega t)\hat{\bf y}$, and thus the tunneling amplitude is
$$\displaystyle W_{LR}(t)=W_{0}[\cos(k_{\rm so}d)+i\sin(k_{\rm so}d){\mbox{%
\boldmath{$\sigma$}}}\cdot\hat{\bf v}(t)]\ ,$$
(9)
where
$$\displaystyle\hat{\bf v}(t)=[0,\cos(\Omega t),\sin(\Omega t)]$$
(10)
lies in $y-z$ plane as well.
Consequently,
$$\displaystyle W_{LR}(t)W^{\dagger}_{LR}(t_{1})/|W_{0}|^{2}=\cos^{2}(k_{\rm so}%
d)+\sin^{2}(k_{\rm so}d)\hat{\bf v}(t)\cdot\hat{\bf v}(t_{1})$$
$$\displaystyle+i{\mbox{\boldmath{$\sigma$}}}\cdot\Big{(}\cos(k_{\rm so}d)\sin(k%
_{\rm so}d)[\hat{\bf v}(t)-\hat{\bf v}(t_{1})]+\sin^{2}(k_{\rm so}d)\hat{\bf v%
}(t)\times\hat{\bf v}(t_{1})\Big{)}\ .$$
(11)
Inserting Eq. (11) into Eq. (6) (details are given in Ref. supp, ) one finds that
there is no effect of the SOI on the particle current if the densities of states in the terminals are assumed to be energy independent mjcomment ,
${\cal N}_{L(R)}(\epsilon)={\cal N}_{L(R)}(\epsilon_{\rm F})\equiv{\cal N}_{L(R)}$ (wide-band approximation). Therefore, if $\mu_{R}-\mu_{L}=eV$, one finds in the limit of low bias-voltage $V$ and low temperature the expected result
$$\displaystyle I_{L}=GV/e;\quad G=4\pi^{2}|W_{0}|^{2}{\cal N}_{L}{\cal N}_{R}G_%
{0}\ ,$$
(12)
where $G_{0}=e^{2}/(\pi\hbar)$ is the quantum of conductance.
The current into the right terminal,
$I_{R}$, is supp $I_{R}=-I_{L}$,
demonstrating that particle number is conserved in the junction.
For the spin currents, however, there is no such conservation law, and in fact spin-flip transitions generated by the
SOI in the weak link may result in the accumulation of spin polarization.
Indeed, as seen in Eqs. (13) and (15) below for the spin polarization in the left lead, interchanging $L$ with $R$ in each of them to obtain the spin polarization in the right one leaves them intact, $\dot{\bf M}_{L}(t)=\dot{\bf M}_{R}(t)$;
the total spin is not conserved, and the junction injects the same amount of spin polarization into the two leads, even in the absence of any bias voltage.
Inserting Eq. (11) into Eq. (7)
(see Ref. supp, ) yields that the $x-$component of the spin-polarization flow is
proportional to
$$\displaystyle\dot{M}_{L,x}=(G/G_{0}){\cal F}(\Omega)\sin^{2}(k_{\rm so}d)\ ,$$
(13)
where
$$\displaystyle{\cal F}(\Omega)$$
$$\displaystyle=\int\frac{d\omega d\omega^{\prime}}{2\pi}[f_{L}(\omega)-f_{R}(%
\omega^{\prime})]$$
$$\displaystyle\times[\delta(\omega-\omega^{\prime}+\Omega)-\delta(\omega-\omega%
^{\prime}-\Omega)]\ .$$
(14)
Interestingly, the $x-$component of the injected spin polarization is time-independent; the AC electric field yields a DC polarization in the leads, parallel to the junction.
The difference of the two delta-functions in Eq. (14), that implies spin current proportional to $\Omega$ in the adiabatic approximation,
indicates inelastic processes: electrons exchange photons of energy $\Omega$ with the electric field, and accordingly flip their spins. For instance, only absorption processes are allowed in an un-biased junction held at zero temperature; these
lead to pumping of the $x-$component of the spin polarization into the terminals.
The transverse components of the spin-polarization flow do oscillate with time, since supp
$$\displaystyle\dot{\bf M}^{\rm tr}_{L}(t)=\frac{G}{G_{0}}\frac{{\cal F}(\Omega)%
}{2}\sin(2k_{\rm so}d)[0,\sin(\Omega t),-\cos(\Omega t)]\ .$$
(15)
The sum of the two transverse spin components is directed along the vector $[0,\sin(\Omega t),-\cos(\Omega t)]$.
Integration over time yields a transverse spin polarization ${\bf M}^{\rm tr}_{L}(t)=\int^{t}\dot{\bf M}^{\rm tr}_{L}(t^{\prime})dt^{\prime}$, which is parallel to the effective magnetic field, i.e., to $\hat{\bf v}(t)$, Eq. (10).
Elliptically rotating field. –
The DC character of the flow of the longitudinal ($x-$) component of the spin polarization is our main result. It is a remarkable consequence of the AC electric field responsible for the SOI and crucially depends on the fact that this electric field is rotating in the plane perpendicular to the weak link.
To elucidate this point we allow for different amplitudes of the electric field components oscillating in the two transverse ($y-$ and $z-$) directions.
In that case the tunneling amplitude takes the form
$$\displaystyle W^{(n)}_{LR}(t)/W_{0}=\cos[U(t)k_{\rm so}d]+i\sin[U(t)k_{\rm so}%
d]{\mbox{\boldmath{$\sigma$}}}\cdot\hat{\bf u}(t)\ ,$$
(16)
where
$$\displaystyle\hat{\bf u}(t)={\bf U}(t)/U(t)\equiv[0,\cos(\Omega t),\gamma\sin(%
\Omega t)]/U(t)\ ,$$
(17)
and
$$\displaystyle U(t)=\sqrt{\cos^{2}(\Omega t)+\gamma^{2}\sin^{2}(\Omega t)}\ .$$
(18)
As seen, $\gamma$
measures the deviation away from the circular polarization: $\gamma=0$ corresponds to a linear-polarized electric field, while $\gamma=1$ restores the circularly-polarized field, Eq. (10).
Whereas a circularly-polarized electric field implies single-photon absorption and emission processes [as expressed by the delta-functions in Eq. (14)], the intricate time dependence [see Eq. (18)] of the non-circular polarization leads to an infinite Fourier series in powers of $\exp[in\Omega t]$, and consequently to an infinite series of delta-functions expressing multiple-photon processes of emission and absorption. However, upon retaining only terms up to second order in $(k_{\rm so}d)$ Eq. (16) becomes
$$\displaystyle W^{(n)}_{LR}(t)/W_{0}\approx 1-[U(t)(k_{\rm so}d)]^{2}/2+i(k_{%
\rm so}d)\,{\mbox{\boldmath{$\sigma$}}}\cdot{\bf U}(t)\ .$$
(19)
The spin part here, which determines $\dot{\bf M}_{L}(t)$, has the same form as in a similarly-expanded Eq. (9), except that the coefficient of $\sigma_{z}$ is multiplied by $\gamma$. Repeating the previous calculation, this reproduces Eqs. (13) and (15), except that the $x-$ and $z-$components of $\dot{\bf M}_{L}$ are now multiplied by $\gamma$. Therefore, in the longitudinal limit $\gamma=0$ and for a small SOI there is no DC spin current, while the transverse spin current oscillates only in the $\hat{\bf y}-$direction.
Discussion. – In reality, the flow of the spin-polarized electrons
injected from the junction into the adjacent parts of the terminals has a certain spatial dependence.
For one-dimensional leads (in the absence of the SOI and magnetic fields), we expect the extra charge and spin polarization in the terminals to follow a classical trajectory with the fermi velocity $v_{\rm F}$ in the leads, e.g.
$M_{L,x}(r,t)=M_{L,x}(0,t-r/v_{\rm F})$ at a distance $r$ from the edge of the junction, up to a certain length determined by the impurity scattering length in the terminal.
The periodic rotation of the transverse spin components will translate into a periodic rotation in space.
In higher dimensions, the ballistic electronic motion can be treated as in the theory of point-contact spectroscopy of metals, with the corresponding densities decaying as $(r_{0}/r)^{\xi}$,
where $r_{0}$ characterizes the cross-section of the junction and $\xi=2\ (\xi=1)$
in the ballistic (diffusive) transport regime 30 .
In the closed-circuit configuration sketched in Fig. 1(a) the magnetization injected into the leads can be measured, e.g., by a properly positioned SQUID, or by
a magnetic-resonance force microscope.
Alternatively, an open circuit such as the one sketched in Fig. 1(b), where magnetization is accumulated in two terminals, can be used. Here low-dimensional contacts connect the weak link to terminals whose linear dimension significantly exceeds the cross section of the contacts so that the terminals can be thought of as reservoirs where injected polarized spins spend a significant time — much longer than the spin relaxation time — before they are reflected back to the weak link.
An estimate for the amount of magnetization accumulated in one of the terminals during a time interval of the order of the spin relaxation time $\tau_{s}$ will then be $(g\mu_{B}/2)\dot{M}_{L,x}\tau_{s}$. Using Eqs. (13) and (14) without any bias voltage, $\mu_{L}=\mu_{R}$, we find that in the adiabatic limit
$$\displaystyle\dot{M}_{L,x}=(\Omega/\pi)(G/G_{0})\sin^{2}(k_{\rm so}d)\ .$$
(20)
Consider now an SOI-active weak link in the form of an InAs nanowire and adopt the value $k_{\rm so}=1/(100\,{\rm nm})$ measured by
Scherübl et al. nygard .
A wire length of $d=100$nm would then give $k_{\rm so}d=1$ and hence $\sin^{2}(k_{\rm so}d)\approx 1$.
For $\Omega=2\pi\times 20$ GHz and using the typical value $G\sim 0.5G_{0}$ for the normal conductance of InAs nanowires
nygard ; chuang ; dayeh
one then finds that $\dot{M}_{L,x}\approx 2\times 10^{10}$ s${}^{-1}$. Next consider $n$-type bulk GaAs terminals [see Fig. 1(b)] moderately doped to give a low electron density of $1\times 10^{16}\,$ cm${}^{-3}$, for which spin relaxation times as long as $\tau_{s}=100\,$ns have been measured at low temperatures kikkawa .
Using the measured value $g=-0.45$ for the $g$-factor of conduction electrons in bulk GaAs pidgeon , we then arrive at the conclusion that spins corresponding to a magnetization of about 500 bohr magnetons may accumulate in the terminals, which if they were cubes with side-lengths of 1$\,\mu$m would contain $\sim$10,000 electrons.
Finally, we need to verify that we are in the adiabatic limit, $\Omega\ll 1/\tau$, or equivalently that $\hbar\Omega\ll\Gamma$, where $\Gamma=\hbar/\tau$ is the level width in the wire. For this purpose we use the estimate $\Gamma=D\hbar v_{\rm F}/d$, where $D\sim G/G_{0}=0.5$ is the transparency of the barriers between the wire and the terminals
and $v_{\rm F}=\ell_{e}e/(m^{\ast}\mu_{e})$ is the fermi velocity of electrons in the wire, here related to the electron mobility $\mu_{e}$, the electron mean free path $\ell_{e}$, and the effective mass $m^{\ast}=0.023m$. Using the typical InAs-nanowire values $\mu_{e}=3000\,$ cm${}^{2}$/(Vs) and $\ell_{e}=55\,$nm (taken from Ref. dayeh, ) we find that $v_{\rm F}\approx 1\times 10^{8}\,$cm/s and hence $\Gamma\approx 3\,$meV. Since $\hbar\Omega\approx 0.1\,$meV for $\Omega=2\pi\times 20\,$GHz we are indeed in the adiabatic regime.
In conclusion we have shown that a rotating electric field, acting on a weak link between two non-magnetic metals, generates both a DC spin current and transverse spin components which rotate around the link, flowing into both metals. This is a novel simple device, with potential uses as a logic element in quantum data processing. Our estimates show that it can realistically be made with existing materials and technology.
We thank Jungho Suh and Chulki Kim for fruitful discussions.
This research was partially supported by the Israel Science Foundation (ISF), by the infrastructure program of Israel Ministry of Science and Technology under contract 3-11173, by the Pazy Foundation, by
the Croatian Science Foundation, project IP-2016-06-2289, and by the
QuantiXLie Centre of Excellence, a project cofinanced by the Croatian
Government and the European Union through the European Regional
Development Fund - the Competitiveness and Cohesion Operational Programme
(Grant KK.01.1.1.01.0004). MJ, RJS, OEW, AA, and DR acknowledge the hospitality of the PCS at IBS, Daejeon, Korea, where part of this work was done.
References
(1)
S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M. Daughton, S. von Molnár, M. L. Roukes,
A. Y. Chtchelkanova, and D. M. Treger,
Spintronics: a spin-based electronics vision for the future,
Science 294, 1488 (2001).
(2)
I. Žutić, J. Fabian, and S. Das Sarma,
Spintronics: fundamentals and applications,
Rev. Mod. Phys. 76, 323 (2004).
(3)
J. Kim, A. Paul, P. A. Crowell, S. J. Koester, S. S. Sapatnekar, J. Wang, and C. H. Kim,
Spin-Based Computing: Device Concepts, Current Status, and a Case Study on a High-Performance Microprocessor, Proceeding of the IEEE 103, 106 (2015).
(4)
M. Kohda and J. Nitta,
Enhancement of spin-orbit interaction and the effect of interface diffusion in quaternary InGaAsP/InGaAs heterostructures,
Phys. Rev. B 81, 115118 (2010).
(5)
R. Winkler,
Spin-Orbit Coupling Effects in Two-Dimensional Electron and Hole Systems,
(Springer-Verlag, Berlin, 2003).
(6)
A. Manchon, H. C. Koo, J. Nitta, S. M. Frolov, and R. A. Duine,
New Perspectives for Rashba Spin-Orbit Coupling,
Nat. Mater. 14, 871 (2015).
(7)
E. I. Rashba,
Properties of semiconductors with an extremum loop .1. Cyclotron and combinational resonance in a magnetic field perpendicular to the plane of the loop, Fiz. Tverd. Tela (Leningrad) 2, 1224 (1960) [Sov.
Phys. Solid State 2, 1109 (1960)]; Y. A. Bychkov and E. I. Rashba, Oscillatory effects and the magnetic susceptibility of carriers in inversion layers,
J. Phys. C 17, 6039 (1984).
(8)
J. Nitta, T. Akazaki, H. Takayanagi, and T. Enoki,
Gate Control of Spin-Orbit Interaction in an Inverted In${}_{0.53}$Ga${}_{0.47}$As/In${}_{0.52}$Al${}_{0.48}$As Heterostructure,
Phys. Rev. Lett. 78, 1335 (1997).
(9)
Y. Sato, T. Kita, S. Gozu, and S. Yamada,
Large spontaneous spin splitting in gate-controlled two-dimensional electron gases at normal In${}_{0.75}$Ga${}_{0.25}$As/In${}_{0.75}$Al${}_{0.25}$As
heterojunctions,
J. Appl. Phys. 89, 8017 (2001).
(10)
A. J. A. Beukman, F. K. de Vries, J. van Veen, R. Skolasinski, M. Wimmer, F. Qu, D. T. de Vries, B. M. Nguyen,
W. Yi, A. A. Kiselev, M. Sokolich, M. J. Manfra, F. Nichele,
C. M. Marcus, and L. P. Kouwenhoven,
Spin-orbit interaction in a dual gated InAs/GaSb quantum well,
Phys. Rev. B 96, 241401 (2017).
(11)
Other types of SOI, e.g. the Dresselhaus SOI [G. Dresselhaus, Spin-Orbit Coupling Effects in Zinc Blende Structures, Phys. Rev. 100, 580 (1955)], or those resulting from strains in the wire [M. S. Rudner and E. I. Rashba, Spin relaxation due to deflection coupling in nanotube quantum dots, Phys. Rev. B 81, 125426 (2010); K. Flensberg and C. M. Marcus, Bends in nanotubes allow electric spin control and coupling, Phys. Rev. B 81, 195418 (2010);
G. A. Steele, F. Pei, E. A. Laird, J. M. Jol, H. B. Meerwaldt, and L. P. Kouwenhoven, Large spin-orbit coupling in carbon nanotubes,
Nat. Commun. 4, 1573 (2013)], yield similar effects, with other directions of the effective SOI magnetic field around which the spins rotate.
(12)
J. H. Bardarson,
A proof of the Kramers degeneracy of transmission eigenvalues from antisymmetry of the scattering matrix,
J. Phys. A: Math. Theor. 41, 405203 (2008).
(13)
A. G. Aronov and Y. B. Lyanda-Geller,
Spin-Orbit Berry Phase in Conducting Rings,
Phys. Rev. Lett. 70, 343 (1993).
(14)
A. Aharony, Y. Tokura, G. Z. Cohen, O. Entin-Wohlman, and S. Katsumoto,
Filtering and analyzing mobile qubit information via Rashba-Dresselhaus-Aharonov-Bohm interferometers,
Phys. Rev. B 84, 035323 (2011) and references therein.
(15)
H. Saarikoski, A. A. Reynoso, J. P. Baltanás, D. Frustaglia, and J. Nitta,
Spin interferometry in anisotropic spin-orbit fields,
Phys. Rev, B 97, 125423 (2018);
F. Nagasawa, A. A. Reynoso, J. P. Baltanás, D. Frustaglia, H. Saarikoski, and J. Nitta,
Gate-controlled anisotropy in Aharonov-Casher spin interference: Signatures of Dresselhaus spin-orbit inversion and spin phases,
Phys. Rev, B 98, 245301 (2018).
(16)
P. M. Shmakov, A. P. Dmitriev, and V. Yu. Kachorovskii, High-temperature Aharonov-Bohm-Casher interferometer, Phys. Rev. B
85, 075422 (2012); Aharonov-Bohm conductance of a disordered single-channel quantum ring, Phys. Rev. B 87, 235417 (2013).
(17)
F. Nagasawa, J. Takagi, Y. Kunihashi, M. Kohda, and J. Nitta,
Experimental Demonstration of Spin Geometric Phase: Radius Dependence of Time-Reversal Aharonov-Casher Oscillations,
Phys. Rev. Lett. 108, 086801 (2012).
(18)
S. Datta and B. Das,
Electronic analog of the electro-optic modulator,
Appl. Phys. Lett. 56, 665 (1990).
(19)
A. Aharony, O. Entin-Wohlman, K. Sarkar, R. I. Shekhter, and M. Jonson, Effects of different lead magnetizations on the Datta-Das spin field-effect transistor, arXiv:1901.07554 and references therein.
(20)
S. K. Watson, R. M. Potok, C. M. Marcus, and V. Umansky, Experimental Realization of a Quantum Spin Pump, Phys. Rev. Lett. 91, 258301 (2003).
(21)
B. Wang, J. Wang, and H. Guo, Quantum spin field effect transistor, Phys. Rev. B 67, 092408 (2003).
(22)
J. Splettstoesser, M. Governale, and J. König, Adiabatic charge and spin pumping through quantum dots with ferromagnetic leads, Phys. Rev. B 77, 195320 (2008).
(23)
P. Sharma and P. W. Brouwer, Mesoscopic Effects in Adiabatic Spin Pumping, Phys. Rev. Lett. 91, 166801 (2003).
(24)
M. Governale, F. Taddei, and R. Fazio, Pumping spin with electrical fields, Phys. Rev. B 68, 155324 (2003).
(25)
Y. Avishai, D. Cohen, and N. Nagaosa, Purely Electric Spin Pumping in One Dimension, Phys. Rev. Lett. 104, 196601 (2010).
(26)
Alternatively, the fields can be generated by a single back gate and two other gates placed very close to the sides of the wire as in Ref. nygard, .
(27)
Z. Scherübl, G. Fülöp, M. H. Madsen, J. Nygard, and S. Csonka, Electrical tuning of Rashba spin-orbit interaction in multi-gated InAs nanowires, Phys. Rev. B 94, 035444 (2016)].
(28)
M. Jonson, R. I. Shekhter, O. Entin-Wohlman, A. Aharony, H. C. Park, and D. Radić,
Mechanically driven spin-orbit-active weak links,
Low Temp. Phys. 44 (12) 1228 (2018) [Fiz. Niz. Temp. 44 (12) 1577 (2018)].
(29)
The final results are expressed as traces over the $\sigma$’s, and therefore they do not depend on the quantization axes which define the indices $\sigma,~{}\sigma^{\prime}$.
(30)
Y. Aharonov and A. Casher, Topological Quantum Effects for Neutral Particles, Phys. Rev. Lett. 53, 319 (1984).
(31)
Y. Meir, Y. Gefen, and O. Entin-Wohlman, Universal effects of spin-orbit scattering in mesoscopic systems, Phys. Rev. Lett. 63, 798 (1989); Y. Oreg and O. Entin-Wohlman, Transmissions through low-dimensional mesoscopic systems subject to spin-orbit scattering,
Phys. Rev. B 46, 2393 (1992).
(32)
See Supplemental Material at [URL will be inserted by publisher] for further details.
(33)
If the densities of states in the terminals are energy dependent (and equal),
${\cal N}_{L}(\epsilon)={\cal N}_{R}(\epsilon)={\cal N}(\epsilon)$, $G$ in Eq. (12)
should be multiplied by
$\{1+\sin^{2}(k_{\rm so}d)[{\cal N}(\epsilon_{\rm F}+\Omega)+{\cal N}(\epsilon_%
{\rm F}-\Omega)-2{\cal N}(\epsilon_{\rm F})]/(2{\cal N}(\epsilon_{\rm F}))\}$. Hence,
there is a small SOI effect on the particle current of order $(\Omega/\epsilon_{\rm F})^{2}$.
(34)
I. O. Kulik, R. I. Shekhter, and A. G. Shkorbatov, Point contact spectroscopy of electron-phonon coupling in metals with a small electron mean free path, Zh. Eksp. Teor.
Fiz. 81, 2126 (1981) [Sov. Phys. JETP 54, 1130 (1981)]:
Eqs. (2.11), (2.12), and (3.3-3.5);
R. I. Shekhter and I. O. Kulik, Spectroscopy of phonons in
heterocontacts, Fiz. Nizk. Temp. 9, 46 (1983) [Sov. J. Low Temp.
Phys. 9, 22 (1983)]: the Introduction and Sec. 1.
(35)
S. Chuang, Q. Gao, R. Kapadia, A. C. Ford, J. Guo, and A. Javey, Ballistic InAs nanowire transistors,
Nano Lett. 13, 555 (2012).
(36)
S. A. Dayeh, Electron transport in indium arsenide nanowires,
Semicond. Sci. Technol. 25, 024004 (2010); Section 2.
(37)
J. M. Kikawa and D. D. Awschalom, Resonant spin amplification in $n$-type GaAs,
Phys. Rev. Lett. 80, 4313 (1998).
(38)
W. Zawadzki, P. Pfeffer, R. Bratschitsch, Z. Chen, S. T. Cundiff, B. N. Murdin, and C. R. Pidgeon,
Temperature dependence of the electron spin g factor in GaAs,
Phys. Rev. B 78, 245203 (2008). |
Quantum hydrodynamic modelling of edge modes in chiral Berry plasmons
Ya Zhang${}^{1}$, Feng Zhai${}^{2}$, Bin Guo${}^{1}$, Wei Jiang${}^{3}$
${}^{1}$ Department of Physics, Wuhan University of Technology, Wuhan 430070, China
${}^{2}$ Department of Physics, Zhejiang Normal University, Zhejiang 321004, China
${}^{3}$ School of Physics, Huazhong University of Science and Technology, Wuhan 430074, China
binguo@whut.edu.cn
weijiang@hust.edu.cn
Abstract
A quantum hydrodynamic model is used to study the edge modes of chiral Berry plasmons propagating along a boundary of a two-dimensional electron gas confined to a half-plane. The transcendental equation of the dispersion relation is solved nonlinearly and semi-analytically. Without quantum effect, for small transverse wave vector $q$, the solutions agree with the linear approximate results, while for large transverse wave vector $q$, the nonlinear solution gives rise to a new edge mode. With quantum effect several new edge modes emerges due to the strong coupling of the chiral Berry and quantum effect terms. Indeed the $\omega_{\pm edge}$ indicate different limits, respectively, for $q\rightarrow 0^{-}$ and $q\rightarrow 0^{+}$.
As a result, quantum effect enhances the chirality significantly in a broad range of $q$. Both counterpropagating edge modes becomes more confined to the edge with considering the quantum effect. In addition with increasing Berry flux $F$, new localized edge modes are found in both cases without and with quantum effect.
I Introduction
The anomalous Hall effect first discovered by Hall in 1881, occurs due to broken time-reversal symmetry in metallic
ferromagnets with a strong spin-orbit couplingNagaosa et al. (2010). Time-reversal
symmetry can be broken by the inclusion of a proper weak magnetic fieldZhai and Xu (2005).
At early, the ”anomalous behavior” was reported by Karplus and LuttingerKarplus and Luttinger (1954), which naturally arose in the first microscopic theory of the anomalous Hall effect.
The Berry curvature concept was adopted theoretically to relate between the anomalous Hall effect and the topological nature of the Hall currents. Due to the topological Berry phase the intrinsic anomalous Hall effect can be expressed as an integral over the Fermi surfaceHaldane (2004).
In ferromagnets electrons have an ”anomalous velocity” perpendicular to the electric field due to their Berry curvature. Culcer et al.Culcer et al. (2003) investigated the anomalous Hall effect in paramagnetic two-dimensional
systems with a external magnetic field. Wang et al.Wang et al. (2014) have investigated the quantum anomalous
Hall effect in conventional diluted magnetic semiconductors with magnetically doped InAs/GaSb
quantum wells based on the Kane model.
The necessary background and the basic concept of the Berry curvature were summarized in the review paper of Xiao et al. Xiao et al. (2010).
All of the studies mentioned above were performed magnetically. However, the Berry phase can
manifest in various magnetic and nonmagnetic materials. Without magnetic field, the Berry phase manifest in gapped Dirac materials, where the time reversal asymmetry was realized by a nonequilibrium
valley polarizationMak et al. (2014); Lensky et al. (2015). Gapped Dirac materials with several valleys, such as
graphene and transition metal dichalcogenide monolayers were investigated by Xiao et al.Xiao et al. (2007, 2012).
Kumar et al.Kumar et al. (2016) observed chirality in bulk and edge plasmons
without an external magnetic field, where the valley imbalance leaded to a net Berry curvature (Berry flux).
The Berry flux can cause chiral plasmons
in the absence of a magnetic fieldKumar et al. (2016); Song and Rudner (2016). Song et al. confirmed counterpropagating charge density waves in chiral plasmons, where the splitting dispersion occurred
for transverse edge modes in opposite directions. This modes were regarded as chiral Berry plasmons (CBPs)Song and Rudner (2016).
However, quantum effects may result in other interesting phenomena in this CBPs.
In this letter we use a quantum hydrodynamic method coupled with the anomalous velocity to examine the quantum effect on the splitting dispersive relation and the transverse mode confinement. Haas et al. first introduced the quantum hydrodynamic theory by solving the nonlinear Schrördinger-Poisson or the Wigner-Poisson kinetic modelsHaas et al. (2000); Manfredi and Haas (2001). The quantum effects always exist in a metallic electron gasHaas (2011).
The quantum effect establishes new chiral edge modes, and enhances the splitting as well as the transverse confinement significantly in a wide range of transverse mode.
Gauss units will be adopted throughout the paper
except specific definitions.
II Model and formalism
We consider CBPs along the edges of a two-dimensional electron gas (2DEG) with considering quantum effect without magnetic field.
The excitations of the 2DEG
are described by the quantum hydrodynamic (QHD) modelHaas et al. (2000); Manfredi and Haas (2001); Marklund and Brodin (2007); Brodin et al. (2008) with
the continuity equation,
$$\frac{\partial n_{e}}{\partial t}+\nabla\cdot(n_{e}\mathbf{V})=0,$$
(1)
the momentum-balance equation
$$\frac{\partial\mathbf{u}_{e}}{\partial t}+(\mathbf{u}_{e}\cdot\nabla)\mathbf{u%
}_{e}=\frac{e}{m_{e}}\nabla\phi-\frac{\pi\hbar^{2}}{m_{e}^{2}}\nabla n_{e}-%
\frac{\hbar^{2}}{2m_{e}}\nabla(\frac{1}{\sqrt{n_{e}}}\nabla^{2}\sqrt{n_{e}}),$$
(2)
and the integral-form of the electric potential
$$\phi(\mathbf{r},t)=\int d^{2}\mathbf{r}^{\prime}W(\mathbf{r}-\mathbf{r}^{%
\prime})(n_{e}(\mathbf{r^{\prime}},t)-n_{0}).$$
(3)
Here $m$ is the effective electron mass, $e$ is the elementary charge, $\hbar$ is
the Planck constant, $W(\mathbf{r}-\mathbf{r}^{\prime})$ is the Coulomb interaction,
$n_{e}$ is the density, and $\mathbf{u}_{e}$ is the homogenous fluid velocity.
In particular,
$$\mathbf{V}=\mathbf{u}_{e}+\mathbf{V}_{a},\mathbf{V}_{a}=\frac{eF}{\hbar}[(%
\nabla\phi)\times\mathbf{e}_{z}],$$
(4)
with the chiral velocity $\mathbf{V}_{a}$ which is a self-induced anomalous
velocity componentXiao et al. (2010), and $F$ is the (dimensionless) Berry fluxSong and Rudner (2016).
Note that, in Eq. (2) at the right hand side, the last two terms are regarded as quantum effectsHaas et al. (2000); Manfredi and Haas (2001); Marklund and Brodin (2007); Brodin et al. (2008), namely, the second term is the quantum statistical effect (second order quantum correction) which is the force due to the internal interactions in the electron species, the third term is the quantum diffraction effect (fourth order quantum correction) coming from the quantum pressure (so called Bohm potential).
For definiteness, only the quantum statistical effect (second order) is considered in this semi-analytical study, since the inclusion of the quantum pressure term (fourth order) can not yield a analytical solution.
The above nonlinear equations can be linearized under a weak perturbation of the plasmons, where $n_{e}(\mathbf{r},t)$ is represented by the first-order
perturbed value $n_{e}=n_{0}+n_{e1}$ with $n_{e1}\ll n_{0}$Fetter (1985). $n_{0}$ is the initial equilibrium density.
Thus the linearized QHD theory gives rise to
$$\frac{\partial n_{e1}}{\partial t}+n_{0}\nabla\cdot\mathbf{V}=0,$$
(5)
$$\frac{\partial\mathbf{u}_{e}}{\partial t}=\frac{e}{m}\mathbf{\nabla}\phi-\frac%
{\pi\hbar^{2}}{m^{2}}\nabla n_{e1},$$
(6)
and
$$\phi(\mathbf{r},t)=\int d^{2}\mathbf{r}^{\prime}W(\mathbf{r}-\mathbf{r}^{%
\prime})n_{e1}(\mathbf{r^{\prime}},t).$$
(7)
A semi-infinite ($x\geq 0$) 2DEG in half $x-y$ plane is considered, where
$n_{e}(\mathbf{r},t)$ and $\mathbf{V}(\mathbf{r},t)$ take finite values for $x\geq 0$, while $n_{e}(\mathbf{r},t)=0$ and $\mathbf{V}(\mathbf{r},t)=0$ for $x<0$, as shown in Fig. 1. Thus an edge plasmon is formed along the boundary $x=0$, propagating as a plane wave along $y$ axis with a frequency $\omega$ and a wave number $q$. The potential, perturbed density and velocities can be written as
$$\begin{array}[]{lcl}\phi(\mathbf{r},t)=\phi_{q}(x)\exp(i\omega t-iqy)\\
n_{e1}(\mathbf{r},t)=n_{e1q}(x)\exp(i\omega t-iqy)\\
u_{e}(\mathbf{r},t)=u_{eq}(x)\exp(i\omega t-iqy)\\
\mathbf{V}(\mathbf{r},t)=\mathbf{V}_{q}(x)\exp(i\omega t-iqy)\end{array}\text{.}$$
(8)
Thus, Eq. (5) can be rewritten as $n_{e1}(\mathbf{r})=-n_{0}\mathbf{\nabla}\cdot\mathbf{V}(\mathbf{r},t)/i\omega$, and by using the step-like property of the velocity $\mathbf{V}(\mathbf{r},t)$ [$\mathbf{V}(\mathbf{r},t)\neq 0$ for $x\geq 0$ and $\mathbf{V}(\mathbf{r},t)=0$ for $x<0$], a jump condition for the $x$ derivative of the potential $\phi_{q}(x)$ can be obtained based on Eq. (7)Song and Rudner (2016),
$$\frac{\partial\phi_{q}}{\partial x}\mid_{x=0^{+}}-\frac{\partial\phi_{q}}{%
\partial x}\mid_{x=0^{-}}=\frac{1}{i\omega}(\frac{\partial W_{q}}{\partial x}%
\mid_{x=0^{-}}-\frac{\partial W_{q}}{\partial x}\mid_{x=0^{+}})n_{0}\mathbf{V}%
_{x}\mid_{x=0^{+}}.$$
(9)
Here $\mathbf{V}_{x}\mid_{x=0^{+}}=\mathbf{u}_{ex}\mid_{x=0^{+}}+\mathbf{V}_{ax}\mid%
_{x=0^{+}}$, and $W_{q}=-e\int dkexp(ikx)/\sqrt{q^{2}+k^{2}}$ by assuming a wave number $k$ along $x$ axis.
According to lituratureSong and Rudner (2016); Fetter (1985), $W_{q}$ can be simplified as
$$W_{q}(x)\approx-4\pi e\int\frac{dk}{2\pi}\frac{|q|}{k^{2}+2q^{2}}\exp(ikx)=-%
\frac{4\pi e}{2\sqrt{2}}\exp(-|q|x),x\geq 0.$$
(10)
for small $k/q$.
Again by substituting the simplified $W_{q}(x)$ into the integral potential
$$\phi_{q}(x)=\int dx^{\prime}W_{q}(x-x^{\prime})n_{e1q}(x^{\prime}),$$
(11)
and using the convolution theorem, we can obtain a differential
equation of $\phi_{q}(x)$ satisfying,
$$\begin{array}[]{lcl}(x^{2}-2q^{2})\phi_{q}(x)=0,x<0\\
(x^{2}-2q^{2})\phi_{q}(x)=4\pi e|q|n_{e1q}(x),x\geq 0\end{array}\text{.}$$
(12)
Thus $\phi_{q}(x)$ has the solutions of
$$\begin{array}[]{lcl}\phi_{q}(x)=\phi_{1}\exp(\kappa_{1}x),x<0\\
\phi_{q}(x)=\phi_{2}\exp(-\kappa_{2}x),x\geq 0\end{array}\text{.}$$
(13)
Here $\kappa_{1}=\sqrt{2}|q|$,
and $\kappa_{2}$ is obtained as follows.
Applying $\partial/\partial t$ to the continuity relation in Eq. (5), and
substituting $\frac{\partial\mathbf{V}}{\partial t}=\frac{\partial\mathbf{u_{e}}}{\partial t%
}+\frac{\partial\mathbf{V_{a}}}{\partial t}$ to Eq. (5), yields
$$\frac{\partial^{2}n_{e1}}{\partial t^{2}}=-\frac{e}{m}n_{0}\nabla^{2}\phi+%
\frac{\pi\hbar^{2}n_{0}}{m^{2}}\nabla^{2}n_{e1}.$$
(14)
By using the plane wave form of $n_{e1}(\mathbf{r},t)$ and $\phi(\mathbf{r},t)$, and the relation $n_{e1q}(x)=(x^{2}-2q^{2})\phi_{q}(x)/(4\pi e|q|)$ based on Eq. (12), $n_{e1}(\mathbf{r},t)$ is given by
$$n_{e1}(\mathbf{r},t)=\frac{\phi_{2}}{4\pi e|q|}(\kappa_{2}^{2}-2q^{2})e^{-%
\kappa_{2}x}e^{i\omega t-iqy},$$
(15)
for $x\geq 0$.
Therefore, Eq. (14) is rewritten as
$$1=\frac{2\omega_{P}^{2}}{\omega^{2}}\frac{\kappa_{2}^{2}-q^{2}}{\kappa_{2}^{2}%
-2q^{2}}-A_{Q}\frac{\kappa_{2}^{2}-q^{2}}{\omega^{2}},$$
(16)
where $\omega_{P}=\sqrt{2\pi n_{0}e^{2}|q|/m}$ according to literatureFetter (1985) and assuming $q\ll k_{F}=\sqrt{2\pi n_{0}}$. $k_{F}$ is the Fermi wave number. In particular, $A_{Q}=\pi\hbar^{2}n_{0}/m^{2}$ is the coefficient of quantum term, a dimensionless parameter, characterizing the strength of the quantum effect.
Equation (16) leads to a nonlinear equation of $\kappa_{2}$. $\kappa_{2}^{2}$ can be explicitly expressed as
$$\kappa_{2}^{2}=\frac{3A_{Q}q^{2}-\omega^{2}+2\omega_{P}^{2}\mp\sqrt{A_{Q}^{2}q%
^{4}+2A_{Q}q^{2}\omega^{2}+\omega^{4}+4A_{Q}q^{2}\omega_{P}^{2}-4\omega^{2}%
\omega_{P}^{2}+4\omega_{P}^{4}}}{2A_{Q}}.$$
(17)
At $q=0$, $\kappa_{2}^{2}$ has two solutions: $\kappa_{2}^{2}=0$ and $\kappa_{2}^{2}=-2\omega^{2}$. We only take care of positive $\kappa_{2}^{2}$, since a mode for negative $\kappa_{2}^{2}$ joins the bulk. Thus the potential decays exponentially away from the boundary $x=0$, leading to non-vanishing edge modes.
In what follows in order to give the dispersion relation, we first express
$${V}_{x}|_{x=0^{+}}={u}_{ex}|_{x=0^{+}}+{V}_{ax}|_{x=0^{+}}.$$
(18)
Here
$$\displaystyle{u}_{ex}|_{x=0^{+}}=\frac{1}{i\omega}[-\frac{e}{m}\kappa_{2}\phi_%
{2}+\frac{\pi\hbar^{2}}{m^{2}}\frac{1}{4\pi e|q|}\kappa_{2}(\kappa_{2}^{2}-2q^%
{2})\phi_{2}],$$
(19)
based on the Fourier transform of the momentum-balance equation and Eq. (15),
and
$${V}_{ax}|_{x=0^{+}}=\frac{-iqeF}{\hbar}\phi_{2}.$$
(20)
The dispersion relation is determined from the continuity condition of $\phi$ and the jump boundary condition given by Eq. (9),
$$\sqrt{2}|q|+\kappa_{2}=\frac{2\omega_{p}^{2}}{\omega^{2}}\kappa_{2}-A_{F}\frac%
{q^{2}sgn(q)}{\omega}-A_{Q}\frac{\kappa_{2}(\kappa_{2}^{2}-2q^{2})}{\omega^{2}},$$
(21)
indicating a nonlinear equation of $\kappa_{2}$ and $\omega$. Here $A_{F}=4\pi e^{2}F/\hbar$ is the coefficient of chiral Berry term.
At $q=0$ the above dispersion relation gives rise to
$\kappa_{2}=-A_{Q}\kappa_{2}^{3}/\omega^{2}$, indicating a diverging mode or $\kappa_{2}=0$, which will lead to a jump for frequency of edge modes at $q=0$ .
We can couple two nonlinear equations (16) and (21) to semi-analytically solve $\kappa_{2}$ and $\omega$ for a given $q$. The solutions obey two coupled polynomial functions of fifth and forth orders,respectively.
It is instructive to discuss the properties of equations (16) and (21) near $q=0$.
Indeed the coupling of this two equation yields the solution
$$\displaystyle\omega=$$
$$\displaystyle-(2.8A_{Q}q\kappa_{2}^{4})/A_{F}-(2.8A_{Q}q\kappa_{2}^{3})/A_{F}+%
(0.5(-1.4A_{F}^{2}q^{2}+9.9A_{Q}q^{2}+5.7\omega_{P}^{2})\kappa_{2}^{2})/(A_{F}q)$$
$$\displaystyle+(2.9(A_{Q}q^{2}+\omega_{P}^{2})\kappa_{2})/(A_{F}q)-(-0.4A_{F}^{%
2}q^{2}+2.1A_{Q}q^{2}+2.1\omega_{P}^{2})/(A_{F}q).$$
(22)
Here, $\omega$ represents, namely, $\omega_{+edge}$ for negative $q$ and $\omega_{-edge}$ for positive $q$.
Thus we can obtain the limits of $\omega$ and $\kappa_{2}$, respectively for $q\rightarrow 0^{-}$ and $q\rightarrow 0^{+}$
$$\begin{array}[]{lcl}\kappa_{2}|q\rightarrow 0^{-}=O(q^{4})\\
\omega|q\rightarrow 0^{-}=\frac{21\omega_{p}^{2}}{10A_{F}|q|}\\
\kappa_{2}|q\rightarrow 0^{+}=\frac{\sqrt{2}}{2}q\\
\omega|q\rightarrow 0^{+}=\frac{\sqrt{2}}{8}A_{F}q\end{array}\text{.}$$
(23)
The following results
take $n_{0}=6\times 10^{10}$ cm${}^{-2}$, $m=0.03m_{e}$ according to literatureYan et al. (2012); Song and Rudner (2016). $m_{e}$ is free electron mass.
For definiteness, we take the unit of $q$ as $k_{F}$ for large $q$, while as cm${}^{-1}$ for small $q$, and the unit of plasmon frequency $\omega$ as $E_{F}/\hbar$ with the Fermi energy $E_{F}=\hbar^{2}k_{F}^{2}/2m$.
III results and discussion
Figures 2(a-b) show the dispersion relations of CBPs with Berry flux $F=1$, comparing the edge modes without (circle) and with quantum effect (triangle).
The $\omega_{+edge}$ mode without quantum effect diverges between $-0.0028<q/k_{F}<-0.0013$, and both $\omega_{\pm edge}$ modes with quantum effect diverge between $0.0024<|q|/k_{F}\leq 0.0033$ and $q=0$.
Indeed, with quantum effect, substituting these $q$ values into
Eqs.(16) and (21) gives negative decay constant $\kappa_{2}$ and edge frequencies $\omega_{\pm edge}$, inducing the diverging modes.
Without quantum effect, in order to validate our model, we first compare our results with that of literatureSong and Rudner (2016), for small $q$, where the splitting trends of the edge modes agree well with the linear approximate results (the first order of $q$ modes) in Ref.Song and Rudner (2016) as shown in Fig. 2(a) (square and circle).
For large amplitude and negative sign $q$, a new edge mode emerges for $|q|/k_{F}>0.0028$ (circle) due to the dominant feature of the quadratic ($\propto q^{2}$) terms, which was not present in the linear approximation limit.
With quantum effect,
$\omega_{+edge}$ takes its maximum value near $q=0^{-}$, and shrinks with increasing $|q|$ for $|q|/k_{F}\leq 0.0024$, since the dispersion relation with quantum effect has a vanishing solution at $q=0$ (see the description below Eq. (21).
The situation is reversed for $\omega_{-edge}$ mode, where $\omega_{-edge}$ takes its minimum values near $q=0^{+}$, and increases quickly with increasing $|q|$ for large $q$.
Figure 2(b) displays
a larger deviation of $\omega_{\pm edge}$ from $\omega_{P}$ with quantum effect compared to that without quantum effect, near $q=0$.
Indeed the $\omega_{\pm edge}$ indicate different limits (see Eq. (23)), respectively for $q\rightarrow 0^{-}$ and $q\rightarrow 0^{+}$.
The splittings have crucial applications in subwavelength optical
nonreciprocitySounas et al. (2013) in the THz range.
The corresponding transverse confinement of this edge modes are displayed in Fig. 3 for negative $q$ (a) and positive $q$ (b), with the same condition in Fig. 2. Without quantum effect, Fig. 3(a) again presents the divergence of $\omega_{+edge}$ for $-0.0028<q/k_{F}<-0.0013$ corresponding to Fig. 2(a) (circle).
With quantum effect, both $\omega_{\pm edge}$ modes are more localized in a wider range of $q$, ascribed to the coupling of the quantum effect and the chiral velocity (see Eq.(22)). This largely enhanced transverse localization is preferable for the realization of subwavelength optical non-reciprocal devises in THz without any magnetic field. Such possibility is immune from the current requirements of large magnetic-based devices for realizing nonreciprocity in THz, enhancing the industrial potential in microwave and photonic components.
Without quantum effect, the splitting between $\omega_{\pm edge}$ has rose linearly, respectively, with increasing $q$ and $F$, due to the relation $\omega\propto Fq$ and $\mathbf{V}_{a}=\frac{eF}{\hbar}[(\nabla\phi)\times\mathbf{e}_{z}]$ in Ref.Song and Rudner (2016). However, with quantum effect, the splitting between $\omega_{\pm edge}$ presents a nonmonotonous relation to $q$ and $F$, due to the diverging solutions at some particular $q$ and $F$. Indeed the diverging solutions usually occurs for a higher order polynomial function. The $q$-related modes are shown in above Figs.2-3.
The comparison of 4(a) the edge modes $\omega_{\pm edge}$ as well as 4(b) the corresponding transverse confinement lengths, with and without quantum effect as a function of $F$, are shown in figure 4, for a fixed $q/k_{F}=0.0098$.
Without quantum effect,
in Fig. 4(a), the splitting modes (square and sphere) agree well with those in literatureSong and Rudner (2016), but a new branch of $\omega_{+edge}$ mode emerging due to the quadratic $q^{2}$ terms for $F\geq 0.55$. This new mode was easily omitted in linear $q$ approximation. As a result, this new mode introduces a new branch of confinement length for $F\geq 0.55$ as clearly seen from Fig. 4(b) (square). Whereas, the confinement lengths of the $\omega_{-edge}$ mode and the up-growing branch of the $\omega_{+edge}$ have good agreement with the curves in Ref.Song and Rudner (2016). This up-growing branch diverges when $F\geq 0.35$, then a new branch of $\omega_{+edge}$ mode occurs when $F\geq 0.55$ which is more compressed to the edge than its counterpropagating $\omega_{-edge}$ mode.
With quantum effect, the $\omega_{+edge}$ vanishes for $F<0.55$. This is because small $F$ leads negative decay constants and vanishing frequencies. Again the oscillations in $\omega_{-edge}$ mode can be ascribed that the dispersion relation is determined by polynomial functions of high order decay constant $\kappa_{2}$, where the coefficients changes non-monotonically with increasing $F$ as shown in Eq.(22). With increasing $F$ the $\omega_{+edge}$ mode introduces a cut-off confinement of 0.4 (up-triangle), and this mode is mostly confined to the edge in both cases of with and without quantum effect. Such a confinement will have crucial importance for realizing nonreciprocal devices without magnetic field.
Last but not least, we briefly discuss the property of the CBPs with quantum effect, compared to the magnetoplasmons.
For magnetoplasmons, the authors reported that the quantum effect whose strength was defined as $\propto sq$ in Ref.Fetter (1985), had almost no effect on the edge modes in the limit $sq<<\omega_{P}$. This means the edge magnetographs are nearly independent of quantum effect.
In contrast, for the chiral Berry plasmons, the quantum effect give rise to several new edge modes (see $\omega_{\pm edge}$ with quantum effect in above figures) even for this limit $sq<<\omega_{P}$ (defined as $A_{Q}q^{2}<<\omega_{P}^{2}$ in the present work). Indeed, the quantum effect clearly enhances the chirality as well as the transverse confinement, indicating a strongly coupling of chirality and quantum effect (see Eq.(22)).
The chirality in CBPs emerges from the coupling action of the self-induced electric field $-\nabla\phi$ due to the collective density oscillation and the anomalous velocity $\mathbf{V}_{a}=\frac{eF}{\hbar}[(\nabla\phi)\times\mathbf{e}_{z}]$ whose strength depends on $-\nabla\phi$ and the Berry flux $F$. The anomalous velocity can be realized by the Berry curvature, which is an intrinsic quantum-mechanical property of a perfect crystal because it only depends on the Bloch wave function.
Indeed, the anomalous velocity can be expressed by the Berry curvature $\Omega_{n}(\mathbf{k})$ in a given Bloch state $u_{n}(\mathbf{k})$ in $n^{th}$ bandXiao et al. (2010)
$$\mathbf{V}_{a}=\frac{e}{\hbar}\nabla\phi\times\Omega_{n}(\mathbf{k}),\Omega_{n%
}(\mathbf{k})=i<\nabla_{\mathbf{k}}u_{n}(\mathbf{k})|\times|\nabla_{\mathbf{k}%
}u_{n}(\mathbf{k})>.$$
(24)
In summary, we use a quantum hydrodynamic theory to study the chiral Berry plasmons in a semi-infinite 2DEG, where the chiral Berry term and quantum term are considered simultaneously. This semi-infinite 2DEG can be realized easily in experiment in 2D gapped Dirac
materials, such as graphene and transition metal dichalcogenide monolayersXiao et al. (2007, 2012).
The transverse decay constant and the edge plasmon frequency are obtained by nonlinearly and semi-analytically solving two coupled transcendental equations. First our model is validated by comparing our results with that based on the linear approximate method without quantum effect for small $q$.
In our calculation, without quantum effect, a new chiral edge mode emerges due to the contribution of quadratic terms ($\propto q^{2}$). With quantum effect our results predict that the chiral edge modes as well as the transverse confinement in a bounded 2DEG can be tuned. In addition quantum effect introduces new intriguing phenomena that several new chiral edge modes are found, indicating a
strong coupling of the chiral Berry and quantum terms. Indeed the $\omega_{\pm edge}$ indicate different limits, respectively, for $q\rightarrow 0^{-}$ and $q\rightarrow 0^{+}$, in contrast to the case without quantum effect and $\omega_{P}$.
The discovered new chiral edge modes will open up new
paradigms in condensed matter and plasma-optics physicsSounas et al. (2013); Barnes et al. (2003); Yao et al. (2008) to study bounded 2DEGs, like graphene and traditional quantum dots.
References
Nagaosa et al. (2010)
N. Nagaosa,
J. Sinova,
S. Onoda,
A. MacDonald,
and N. Ong,
Reviews of modern physics 82,
1539 (2010).
Zhai and Xu (2005)
F. Zhai and
H. Xu,
Physical review letters 94,
246601 (2005).
Karplus and Luttinger (1954)
R. Karplus and
J. Luttinger,
Physical Review 95,
1154 (1954).
Haldane (2004)
F. Haldane,
Physical review letters 93,
206602 (2004).
Culcer et al. (2003)
D. Culcer,
A. MacDonald,
and Q. Niu,
Physical Review B 68,
045327 (2003).
Wang et al. (2014)
Q.-Z. Wang,
X. Liu,
H.-J. Zhang,
N. Samarth,
S.-C. Zhang, and
C.-X. Liu,
Physical review letters 113,
147201 (2014).
Xiao et al. (2010)
D. Xiao,
M.-C. Chang, and
Q. Niu,
Reviews of modern physics 82,
1959 (2010).
Mak et al. (2014)
K. F. Mak,
K. L. McGill,
J. Park, and
P. L. McEuen,
Science 344,
1489 (2014).
Lensky et al. (2015)
Y. D. Lensky,
J. C. Song,
P. Samutpraphoot,
and L. S.
Levitov, Physical review letters
114, 256601
(2015).
Xiao et al. (2007)
D. Xiao,
W. Yao, and
Q. Niu,
Physical Review Letters 99,
236809 (2007).
Xiao et al. (2012)
D. Xiao,
G.-B. Liu,
W. Feng,
X. Xu, and
W. Yao,
Physical Review Letters 108,
196802 (2012).
Kumar et al. (2016)
A. Kumar,
A. Nemilentsau,
K. H. Fung,
G. Hanson,
N. X. Fang, and
T. Low,
Physical Review B 93,
041413 (2016).
Song and Rudner (2016)
J. C. Song and
M. S. Rudner,
Proceedings of the National Academy of Sciences p.
201519086 (2016).
Haas et al. (2000)
F. Haas,
G. Manfredi, and
M. Feix,
Physical Review E 62,
2763 (2000).
Manfredi and Haas (2001)
G. Manfredi and
F. Haas,
Physical Review B 64,
075316 (2001).
Haas (2011)
F. Haas,
Introduction (Springer,
2011).
Marklund and Brodin (2007)
M. Marklund and
G. Brodin,
Physical review letters 98,
025001 (2007).
Brodin et al. (2008)
G. Brodin,
M. Marklund, and
G. Manfredi,
Physical Review Letters 100,
175001 (2008).
Fetter (1985)
A. L. Fetter,
Physical Review B 32,
7676 (1985).
Yan et al. (2012)
H. Yan,
Z. Li,
X. Li,
W. Zhu,
P. Avouris, and
F. Xia, Nano
letters 12, 3766
(2012).
Sounas et al. (2013)
D. L. Sounas,
C. Caloz, and
A. Alu,
Nature communications 4
(2013).
Barnes et al. (2003)
W. L. Barnes,
A. Dereux, and
T. W. Ebbesen,
Nature 424,
824 (2003).
Yao et al. (2008)
W. Yao,
D. Xiao, and
Q. Niu,
Physical Review B 77,
235406 (2008).
Figures and figure Captions |
Robust Beamforming Design for IRS-Aided URLLC in D2D Networks
Jing Cheng, , Chao Shen, , Zheng Chen, , Nikolaos Pappas
Manuscript received January 18, 2022; revised May 6, 2022 and June 4, 2022; accepted July 5, 2022. Date of publication xxx xx, 2022; date of current version xxx xx, 2022. This work was supported in part by the National Key R&D Program of China under Grant 2021YFB2900301; in part by the State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University under Contract RCS2021ZP002; in part by the NSFC, China under Grants 61871027, 62031008, and U1834210; and in part by the China Scholarship Council (CSC) under Grant 202007090174. The work of N. Pappas was supported by the Swedish Research Council (VR), ELLIIT, and CENIIT. A part of this work has been presented at the 25th International ITG Workshop on Smart Antennas (WSA), 2021 [1]. The editor coordinating the review of this article and approving it for publication was Behrooz Makki. (Corresponding author: Chao Shen.)Jing Cheng is with the State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University, Beijing 100044, China (e-mail: chengjing@bjtu.edu.cn).Chao Shen is with the State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University, Beijing 100044, China, and also with the Shenzhen Research Institute of Big Data, Shenzhen, China (email: chaoshen@sribd.cn).Zheng Chen is with the Department of Electrical Engineering, Linköping University, 58183 Linköping, Sweden (e-mail: zheng.chen@liu.se).Nikolaos Pappas is with the Department of Computer and Information Science, Linköping University, 58183 Linköping, Sweden (e-mail: nikolaos.pappas@liu.se).
Abstract
Intelligent reflecting surface (IRS) and device-to-device (D2D) communication are two promising technologies for improving transmission reliability between transceivers in communication systems. In this paper, we consider the design of reliable communication between the access point (AP) and actuators for a downlink multiuser multiple-input single-output (MISO) system in the industrial IoT (IIoT) scenario. We propose a two-stage protocol combining IRS with D2D communication so that all actuators can successfully receive the message from AP within a given delay. The superiority of the protocol is that the communication reliability between AP and actuators is doubly augmented by the IRS-aided first-stage transmission and the second-stage D2D transmission. A joint optimization problem of active and passive beamforming is formulated, which aims to maximize the number of actuators with successful decoding. We study the joint beamforming problem for cases where the channel state information (CSI) is perfect and imperfect. For each case, we develop efficient algorithms that include convergence and complexity analysis. Simulation results demonstrate the necessity and role of IRS with a well-optimized reflection matrix, and the D2D network in promoting reliable communication. Moreover, the proposed protocol can enable reliable communication even in the presence of stringent latency requirements and CSI estimation errors.
Index Terms:
D2D communication, industrial Internet of things (IIoT), intelligent reflecting surface (IRS), robust beamforming, URLLC
I Introduction
With the development of the Internet of things (IoT) and the fifth-generation (5G) and beyond wireless networks, the communication paradigm shifts from human-to-human (H2H) communication to machine-to-machine (M2M) communication. One of the IoT use cases is critical industrial IoT (IIoT), which is envisioned to support mission-critical applications such as intelligent transportation systems, remote healthcare and smart manufacturing [2]. The implementation of critical IIoT requires establishing ultra-reliable and low-latency communication (URLLC) among IIoT devices. To meet the demanding low-latency with the order of milliseconds required for critical IIoT applications, short packets are usually transmitted. However, this will inevitably cause a loss of coding gain. That is, low latency is achieved at the expense of reliability. To enable ultra-reliable communication between the transmitter and the receiver, a retransmission mechanism [3] utilizing temporal diversity is proposed. However, URLLC packets scheduled in mini-slots are subject to quasi-static fading channels, which can degrade the retransmission performance. Moreover, temporal diversity cannot guarantee the reliability requirements of URLLC packet transmission if the channel exhibits deep fading over a long period of time. Therefore, it is crucial to investigate new technologies or develop new communication protocols to ensure ultra-reliable communication between IIoT devices within a certain millisecond delay.
I-A Related Work
Intelligent reflecting surface (IRS), a metasurface equipped with massive reflecting elements [4], is a promising technology for enabling URLLC [1]. It is able to reconfigure the wireless environment and turn the random wireless channels into partially deterministic ones by beamforming design [5, 6, 7]. As a result, the received signal-to-noise ratio (SNR) can be significantly improved. Even though the direct link between transceivers is hindered, IRS can create a virtual line-of-sight (LoS) link to bypass obstacles between transceivers via smart reflection [8]. Thus, the integration of IRS into the communication system helps to enhance reliability, reduce packet retransmission, and minimize the delay. Consequently, IRS can be a potential and cost-effective solution to realize URLLC. In [9], the authors presented the performance analysis of the average achievable rate and error probability over an IRS-aided URLLC transmission with/without phase noise. Considering non-linear energy harvesting, the end-to-end performance of the IRS-assisted wireless system was analyzed in [10] for industrial URLLC applications, and the approximate closed-form expression of block error rate was derived. Authors in [11] studied an IRS-assisted downlink multiuser URLLC system and jointly optimized the user grouping and the blocklength allocation at the base station (BS), as well as the reflective beamforming at the IRS for latency minimization.
Device-to-device (D2D) communication is another potential technology to achieve URLLC. In most mission-critical applications, devices (e.g. sensors, machines, actuators, robots) are in close proximity to each other. Thus, the channel between the devices is much more reliable than that between the access point (AP) and the device, thereby rendering the D2D network promising to reduce resource consumption, lower communication latency, and improve reliability [12, 13]. Recently, there are studies on the design of D2D-based URLLC systems. A probability-based D2D activation and power allocation scheme was proposed in [14] to deal with the extremely high quality-of-service (QoS) requirements in URLLC for real-time wireless control systems, where each sensor autonomously decides whether to participate in the control process without interactive communications. The authors in [15] developed a D2D-based two-phase transmission protocol for URLLC, where each group’s messages are combined and the BS multicasts them to the leaders in groups in the first phase, while leaders help to relay messages to other users in their groups in the second phase. In [16], authors investigated the contention-based radio resource management for URLLC-oriented D2D communications.
Some research works further combined the technologies of IRS and D2D communication and studied the IRS-assisted D2D network design. For instance, the authors in [17] considered deploying IRS in the integrated data and energy network coexisting with D2D communication to maximize the minimum throughput of the information-demanded users. In [18], a resource allocation design for the IRS-aided joint processing coordinated multipoint (JP-CoMP) system with underlaying D2D network was investigated. The authors in [19] studied an IRS-aided D2D communication system over Rician fading channels with the consideration of practical hardware impairments at both the terminals and IRSs. However, the optimization of most of these works is based on Shannon capacity with assumptions of infinite blocklength and zero error probability. If we directly apply the results and conclusion of these works to URLLC-oriented applications, we may get the underestimated delay performance and overestimated reliability performance [20]. This necessitates the IRS-assisted D2D network design for URLLC under the finite blocklength (FBL) regime. Towards this end, one practical factor needed to be considered is the channel state information (CSI) estimation. On the one hand, IRS can only passively reflect signals and is not able to transmit or receive pilot symbols. On the other hand, the transmission time interval (TTI) of URLLC systems is very short, so that the time for channel training is highly restricted. Consequently, the perfect knowledge of CSI may not be available in practice. This entails a robust IRS-assisted D2D network design for URLLC under the imperfect CSI scenario.
I-B Contributions
In this paper, we propose a two-stage protocol to enable reliable communication between the AP and actuators in the IIoT scenario assisted by IRS and D2D networks under the scenarios of perfect and imperfect CSI. This is achieved by jointly optimizing the active beamforming at the AP and the reflective beamforming at the IRS to maximize the number of actuators with successful decoding. The main contributions of this paper are summarized as follows.
•
A communication protocol for dual augmented reliability by combining IRS and D2D networks in a specified latency requirement is proposed. The reliable communication design is investigated under the scenarios of perfect and imperfect CSI. In this way, URLLC in IIoT scenario can be enabled.
•
With perfect CSI, we propose two efficient algorithms with guaranteed convergence and polynomial time complexity. One is the AltMin algorithm, dealing with unit-modulus constraints by relaxation first and then projection into a feasible region. The other one is a penalty-based successive convex approximation (SCA) algorithm which decomposes the product of active and passive beamformers and avoids using the alternating optimization method. For the imperfect CSI case, a semidefinite relaxation (SDR)-based block coordinate descent (BCD) algorithm is proposed for robust design. The proposed algorithms show better performance than other baseline schemes.
•
The advantages of IRS with well-optimized phase shifts and the D2D network to enhance reliable communication are verified by simulations compared to other baseline schemes. Due to the doubly improved reliability from the combined usage of IRS and D2D network, as well as the multiuser diversity, the proposed two-stage protocol can ensure reliable communication between AP and actuators even under stringent delay requirements and it is shown to be robust to the uncertainties of CSI.
The rest of the paper is organized as follows. In Section II, we present the system model, the two-stage communication protocol, and the problem formulation. The reliable communication design under the perfect and imperfect CSI scenarios are given in Section III and Section IV, respectively. Simulation results are presented in Section V and Section VI concludes the paper.
Notations: $\mathbf{I}_{n}$ refers to an $n\times n$ identity matrix. $\mathbf{x}_{i},\mathbf{X}_{ij}$ stand for the $i$-th element of a vector $\mathbf{x}$ and the $(i,j)$-th element of a matrix $\mathbf{X}$, respectively. $\mathrm{diag}(\mathbf{x})$ denotes a diagonal matrix whose diagonal elements are extracted from a vector $\mathbf{x}$.
II System Model
We consider a downlink multiple-input single-output (MISO)-URLLC system between AP and $K$ actuators in the IIoT scenario, where AP is equipped with $N_{t}$ antennas and all actuators indexed by $k=\{1,\cdots,K\}$ are equipped with a single antenna. As shown in Fig. 1, inspired by a D2D-based two-phase transmission protocol [15], we leverage an IRS with $M$ reflecting elements and the D2D network to doubly enhance the transmission reliability. Thus, all actuators can successfully receive critical messages in the form of short packets from AP within a delay of $\tau$ seconds. The symbols used throughout the paper and their definitions are listed in Table I.
The channels from the AP to the actuator $k$, from the AP to the IRS, and from the IRS to the actuator $k$ are denoted by $\mathbf{h}_{d,k}\in\mathbb{C}^{N_{t}},\mathbf{H}_{dr}\in\mathbb{C}^{M\times N_{t}},\mathbf{h}_{r,k}\in\mathbb{C}^{M}$, respectively. We denote the channel from the actuator $j$ to actuator $k$ by $g_{jk}\in\mathbb{C}$. The coherence time and bandwidth of all channels are assumed to be the same for tractability, denoted as $T$ seconds and $B$ Hz, respectively. Given that the positions of all actuators are fixed and the target latency requirement $\tau$ is much smaller than the channel coherence time $T$, we adopt a quasi-static block-fading channel model, i.e., the channel gains remain constant within one coherence block and vary independently across blocks.
The transmission from AP to the actuators follows a two-stage IRS-aided D2D communication protocol, as shown in Fig. 2. The description of each stage is given below.
II-A IRS-aided first-stage transmission
In the first stage with a duration of $\tau_{1}$ seconds, the transmitted signal at the AP is $\mathbf{x}^{(\text{I})}=\mathbf{w}s$, where $\mathbf{w}\in\mathbb{C}^{N_{t}}$ is the beamforming vector for broadcasting at the AP, and $s$ denotes the combined symbol with unit power intended for all actuators. The received signal at the actuator $k$ consists of a direct signal from AP and a reflected signal from IRS, which can be given by $y_{k}^{(\text{I})}=\left(\mathbf{h}_{d,k}^{H}+\mathbf{h}_{r,k}^{H}\boldsymbol{\Phi}\mathbf{H}_{dr}\right)\mathbf{x}^{(\text{I})}+z_{k}^{(\text{I})}$, where $\boldsymbol{\Phi}=\mathrm{diag}(\phi_{1},\cdots,\phi_{M})$ is a reflection matrix of IRS with $\phi_{m}=e^{j\theta_{m}},\forall m$ on it diagonal, $\theta_{m}\in[0,2\pi]$ is the phase shift of the $m$-th reflecting element, and $z_{k}^{(\text{I})}\sim\mathcal{CN}(0,\sigma_{k}^{2})$ is the additive white Gaussian noise (AWGN).
Denote the cascaded AP-IRS-actuator channel and the vector containing diagonal elements of matrix $\boldsymbol{\Phi}$ by $\mathbf{G}_{k}=\mathrm{diag}\left(\mathbf{h}_{r,k}^{H}\right)\mathbf{H}_{dr}$ and $\widetilde{\mathbf{v}}=[\phi_{1},\cdots,\phi_{M}]^{H}\in\mathbb{C}^{M}$, respectively. The SNR of the $k$-th actuator in the first stage can be represented as
$$\gamma_{k}^{(\text{I})}=\frac{\begin{vmatrix}\left(\mathbf{h}_{d,k}^{H}+\widetilde{\mathbf{v}}^{H}\mathbf{G}_{k}\right)\mathbf{w}\end{vmatrix}^{2}}{\sigma_{k}^{2}}.$$
(1)
Based on the normal approximation of the maximum achievable rate with finite blocklength codes [21], we characterize the maximum achievable rate of the actuator $k$ over $L_{1}=\tau_{1}B$ channel uses in the first stage by
$$\frac{D}{L_{1}}=\log_{2}\left(1+\gamma_{k}^{(\text{I})}\right)-\sqrt{\frac{V_{k}^{(\text{I})}}{L_{1}}}Q^{-1}\left(\varepsilon_{k}^{(\text{I})}\right),$$
(2)
where $D$ is the number of data bits, $V_{k}^{(\text{I})}=(\log_{2}e)^{2}\left(1-\left(1+\gamma_{k}^{(\text{I})}\right)^{-2}\right)$ is the channel dispersion, $\varepsilon_{k}^{(\text{I})}$ is the decoding error probability, $Q^{-1}(\cdot)$ is the inverse of Q-function $Q(x)=\frac{1}{\sqrt{2\pi}}\int_{x}^{\infty}e^{-t^{2}/2}dt$. Note that equation (2) characterizes the relationship between the achievable rate, transmission time (channel uses) and decoding error probability. From (2), the decoding error probability can be represented as
$$\varepsilon_{k}^{(\text{I})}\!=\!Q\left(\frac{\sqrt{L_{1}}\log_{2}\left(1\!+\!\gamma_{k}^{(\text{I})}\right)\!-\!D/\sqrt{L_{1}}}{\sqrt{V_{k}^{(\text{I})}}}\right)\!\triangleq\!Q\left(f_{1}\left(\gamma_{k}^{(\text{I})}\right)\right).\\
$$
(3)
Note that $f_{1}\left(\gamma_{k}^{(\text{I})}\right)$ is a monotonically increasing function of $\gamma_{k}^{(\text{I})}$ [22, Lemma 1] and Q-function is a monotonically decreasing function. So it can be obtained that $\varepsilon_{k}^{(\text{I})}$ decreases with $\gamma_{k}^{(\text{I})}$. Denote the maximum allowed packet error probability (PEP) by $\varepsilon_{\text{th}}$, then we can have the minimum SNR threshold $\gamma_{\text{th}}^{(\text{I})}$ required for successful decoding in the first stage corresponding to the maximum PEP $\varepsilon_{\text{th}}$.
If the SNR of actuator $k$ can reach the threshold $\gamma_{\text{th}}^{(\text{I})}$, we regard that the $k$-th actuator can successfully decode the message in the first stage, otherwise, the packet reception of actuator $k$ ends with failure in the first stage. Thus, for any actuator $k$, we define an indicator $a_{k}^{(\text{I})}$ as follows
$$a_{k}^{(\text{I})}=\left\{\begin{array}[]{ll}1,&\mathrm{if}~{}\gamma_{k}^{(\text{I})}\geq\gamma_{\text{th}}^{(\text{I})},\\
0,&\mathrm{otherwise}.\end{array}\right.$$
(4)
II-B Second-stage D2D transmission
In the second stage with a duration of $\tau_{2}=\tau-\tau_{1}$ seconds, the actuators with successful decoding in the first stage relay messages over a D2D network to the rest of actuators. The transmit signal of the actuator $j$ in the second stage is $x_{j}^{(\text{II})}=a_{j}^{(\text{I})}\sqrt{P}s$, where $P$ is the common transmit power for all actuators. Here, we don’t consider power control due to the following two factors: 1) the global CSI of the D2D network is not available at the AP; 2) the actuators that act as relays transmit the signal at the maximum power, which can improve the received SNR and is more aligned with the requirements of the URLLC scenario.
If actuator $k$ cannot decode the message successfully in the first stage, it will receive relayed signals in the second stage from actuators with successful reception. Thus, the received signal of actuator $k$ in the second stage can be expressed as $y_{k}^{(\text{II})}=\sum\limits_{j\neq k}g_{jk}x_{j}^{(\text{II})}+z_{k}^{(\text{II})}$, where $z_{k}^{(\text{II})}\sim\mathcal{CN}(0,\sigma_{k}^{2})$ is the AWGN. At the end of this stage, the actuator $k$ jointly decodes the signals received in two stages via maximum-ratio combining (MRC). The received SNR of actuator $k$ in the second stage is given by
$$\gamma_{k}^{(\text{II})}=\gamma_{k}^{(\text{I})}+\frac{P\begin{vmatrix}\sum\limits_{j\neq k}a_{j}^{(\text{I})}g_{jk}\end{vmatrix}^{2}}{\sigma_{k}^{2}}.$$
(5)
Then, the maximum achievable rate of actuator $k$ over $L_{2}=\tau_{2}B$ channel uses in the second stage can be characterized by
$$\frac{D}{L_{2}}=\log_{2}\left(1+\gamma_{k}^{(\text{II})}\right)-\sqrt{\frac{V_{k}^{(\text{II})}}{L_{2}}}Q^{-1}\left(\varepsilon_{k}^{(\text{II})}\right),$$
(6)
where $\varepsilon_{k}^{(\text{II})}$ is the decoding error probability. From equation (6), $\varepsilon_{k}^{(\text{II})}$ can be expressed as
$$\!\varepsilon_{k}^{(\text{II})}\!=\!Q\left(\!\frac{\sqrt{L_{2}}\log_{2}\left(1\!+\!\gamma_{k}^{(\text{II})}\right)\!-\!D/\sqrt{L_{2}}}{\sqrt{V_{k}^{(\text{II})}}}\right)\!\triangleq\!Q\left(f_{2}\left(\gamma_{k}^{(\text{II})}\right)\right).\\
$$
(7)
Similar to the conclusion derived from (3), we know that $\varepsilon_{k}^{(\text{II})}$ decreases with $\gamma_{k}^{(\text{II})}$. Then, we can obtain the minimum SNR threshold $\gamma_{\text{th}}^{(\text{II})}$ for successful decoding in the second stage corresponding to the maximum PEP $\varepsilon_{\text{th}}$. To indicate whether actuator $k$ failed in the first stage can successfully decode the message in the second stage, we denote an indicator by
$$a_{k}^{(\text{II})}=\left\{\begin{array}[]{ll}1,&\mathrm{if}~{}a_{k}^{(\text{I})}=0,\gamma_{k}^{(\text{II})}\geq\gamma_{\text{th}}^{(\text{II})},\\
0,&\mathrm{if}~{}a_{k}^{(\text{I})}=0,\gamma_{k}^{(\text{II})}<\gamma_{\text{th}}^{(\text{II})}.\end{array}\right.$$
(8)
II-C Problem Formulation
In this paper, we aim to develop a reliable communication design between an AP and multiple actuators within a given delay requirement. Based on the two-stage communication protocol, our goal is to maximize the number of actuators with successful decoding by jointly designing the active beamforming at the AP and the reflection matrix at the IRS, i.e.,
$$\displaystyle\mathrm{P1}:\max_{\mathbf{w},\widetilde{\mathbf{v}}}\quad$$
$$\displaystyle\sum_{k=1}^{K}\left(a_{k}^{(\text{I})}+a_{k}^{(\text{II})}\right)$$
(9a)
$$\displaystyle\mathrm{s.t.}\quad$$
$$\displaystyle\|\mathbf{w}\|^{2}\leq P_{\text{max}},$$
(9b)
$$\displaystyle|\widetilde{\mathbf{v}}_{m}|=1,\forall m,$$
(9c)
where constraints (9b) and (9c) denote the maximum transmit power constraint at AP and the unit-modulus constraint of each reflecting element.
We assume that at least one actuator can successfully decode the message in the first stage. Note that the higher the number of actuators that can successfully decode the signal in the first stage (i.e., the higher the number of actuators that act as relays to relay messages in the second stage), the higher the probability of successful reception that the remaining actuators have via D2D network in the second stage [15]. This is mainly due to the multiuser diversity and the strong D2D link between actuators in close proximity. Motivated by this, we transform problem $\mathrm{P1}$ into an alternative form that aims to maximize the number of actuators with successful decoding in the first stage. As a result, fewer remaining actuators are required to reach the SNR threshold $\gamma_{\text{th}}^{(\text{II})}$ for reliable reception in the second stage. Therefore, problem $\mathrm{P1}$ can be reformulated as
$$\displaystyle\mathrm{P2}:\max_{\mathbf{w},\widetilde{\mathbf{v}}}\quad$$
$$\displaystyle\sum_{k=1}^{K}a_{k}^{(\text{I})}$$
(10a)
$$\displaystyle\mathrm{s.t.}\quad~{}$$
$$\displaystyle\sum_{k=1}^{K}a_{k}^{(\text{I})}\geq 1,\forall k,$$
(10b)
$$\displaystyle\|\mathbf{w}\|^{2}\leq P_{\text{max}},$$
(10c)
$$\displaystyle|\widetilde{\mathbf{v}}_{m}|=1,\forall m.$$
(10d)
We can obtain the active beamformer $\mathbf{w}$ at the AP, phase shifts $\widetilde{\mathbf{v}}$ and indicator $a_{k}^{(\text{I})},\forall k$ by solving problem $\mathrm{P2}$. With those solutions, we can calculate $\gamma_{k}^{(\text{I})}$ and $\gamma_{k}^{(\text{II})}$ based on equations (1) and (5). Then, we can obtain $a_{k}^{(\text{II})}$ based on equation (8). It is necessary to remark that if $\sum_{k=1}^{K}\begin{pmatrix}a_{k}^{(\text{I})}+a_{k}^{(\text{II})}\end{pmatrix}=K$, then all actuators can successfully receive the critical signals from AP after the two-stage transmission.
III Active and Reflective Beamforming with Perfect CSI
In this section, to achieve a reliable communication design with perfect CSI, we propose two effective algorithms with guaranteed convergence, namely the AltMin algorithm and the penalty-based SCA algorithm.
III-A Problem Transformation
In this case, the CSI of direct AP-actuator channels $\mathbf{h}_{d,k},\forall k$ and cascaded AP-IRS-actuator channels $\mathbf{G}_{k},\forall k$ are assumed to be perfectly known at the AP, which can be obtained by channel estimation methods in [23].
In problem $\mathrm{P2}$, $a_{k}^{(\text{I})}$ given by equation (4) is a discrete function over beamformers $\mathbf{w}$ and $\widetilde{\mathbf{v}}$. This discrete nature complicates the solution and makes it difficult to apply the standard optimization techniques. In view of this, we introduce a set of auxiliary variables $\mathbf{q}=[q_{1},\cdots,q_{k}]^{T}\succeq\mathbf{0}$ to characterize the SNR gap between the achievable SNR $\gamma_{k}^{(\text{I})}$ and the SNR threshold $\gamma_{\text{th}}^{(\text{I})}$. In particular, $q_{k}=0$ means the SNR condition $\gamma_{k}^{(\text{I})}\geq\gamma_{\text{th}}^{(\text{I})}$ is satisfied and $a_{k}^{(\text{I})}=1$. Otherwise, $a_{k}^{(\text{I})}=0$ for nonzero $q_{k}$. That is, the number of zero elements in vector $\mathbf{q}$ denotes the number of actuators with successful decoding. Therefore, maximizing $\sum_{k=1}^{K}a_{k}^{(\text{I})}$ (the number of actuators with successful decoding in the first stage) is equivalent to minimizing $\|\mathbf{q}\|_{0}$ (the number of nonzero elements in vector $\mathbf{q}$). In this way, we transform $\mathrm{P2}$ into an equivalent but a more tractable and continuous form, which can be formulated as
$$\displaystyle\mathrm{P3}:\min_{\mathbf{q,w},\widetilde{\mathbf{v}}}\quad$$
$$\displaystyle\|\mathbf{q}\|_{0}$$
(11a)
$$\displaystyle\mathrm{s.t.}~{}~{}~{}~{}$$
$$\displaystyle\gamma_{k}^{(\text{I})}+q_{k}\geq\gamma_{\text{th}}^{(\text{I})},\forall k,$$
(11b)
$$\displaystyle q_{k}\geq 0,\forall k,$$
(11c)
$$\displaystyle\|\mathbf{q}\|_{0}\leq K-1,$$
(11d)
$$\displaystyle\|\mathbf{w}\|^{2}\leq P_{\text{max}},$$
(11e)
$$\displaystyle|\widetilde{\mathbf{v}}_{m}|=1,\forall m,$$
(11f)
where (11d), equivalent to (10b), guarantees that at least one actuator can successfully decode the message in the first stage.
The challenges in solving $\mathrm{P3}$ mainly include the $\ell_{0}$-norm in the objective function and constraint (11d), the strictly nonconvex constraint (11b) due to the coupling of $\mathbf{w}$ and $\widetilde{\mathbf{v}}$ in the form of a product, and the unit-modulus constraint (11f). To overcome these challenges, we propose two effective algorithms below.
III-B Proposed AltMin Algorithm
In this algorithm, we apply an alternating optimization technique to solve this minimization problem. Hence, we call this algorithm AltMin. Specifically, we alternately optimize $\mathbf{q,w}$ with given $\widetilde{\mathbf{v}}$ and $\mathbf{q},\widetilde{\mathbf{v}}$ with given $\mathbf{w}$. In the following, we elaborate the design of the AltMin algorithm.
With given $\widetilde{\mathbf{v}}$, let $\mathbf{b}_{k}=\mathbf{h}_{d,k}+\mathbf{G}_{k}^{H}\widetilde{\mathbf{v}}$, then constraint (11b) can be recast as
$$|\mathbf{b}_{k}^{H}\mathbf{w}|^{2}\geq\sigma_{k}^{2}\left(\gamma_{\text{th}}^{(\text{I})}-q_{k}\right),\forall k.$$
(12)
Further, based on the first-order Taylor approximation, $|\mathbf{b}_{k}^{H}\mathbf{w}|^{2}$ is lower bounded by
$$\mathcal{F}_{k}(\mathbf{w})=2\mathrm{Re}\begin{Bmatrix}\left(\mathbf{w}^{(i)}\right)^{H}\mathbf{b}_{k}\mathbf{b}_{k}^{H}\mathbf{w}\end{Bmatrix}-|\mathbf{b}_{k}^{H}\mathbf{w}^{(i)}|^{2},$$
(13)
where $\mathbf{w}^{(i)}$ is the solution obtained in the $i$-th iteration. In this way, (11b) can be approximated by a convex constraint, i.e.,
$$\mathcal{F}_{k}(\mathbf{w})\geq\sigma_{k}^{2}\left(\gamma_{\text{th}}^{(\text{I})}-q_{k}\right),\forall k.$$
(14)
Next, we apply the reweighted $\ell_{1}$ technique [24] to deal with the $\ell_{0}$-norm. The principle is to use the weighted $\ell_{1}$-norm to approximate the $\ell_{0}$-norm and then update the weights. To be more specific, $\|\mathbf{q}\|_{0}$ can be approximated by $\sum_{k=1}^{K}\omega_{k}q_{k}$, where $\omega_{k},\forall k$ are positive weights. Here, weights are updated by $\omega_{k}^{(i+1)}=1/\left(q_{k}^{(i)}+\nu\right)$, where $q_{k}^{(i)}$ is the solution obtained in the $i$-th iteration and $\nu$ is a small number. Note that the weights can be interpreted as penalties, i.e., large weights encourage zero entry $q_{k}$ while small weights discourage nonzero $q_{k}$.
Based on the above transformations, the optimization problem of $\mathbf{q,w}$ with given $\widetilde{\mathbf{v}}$ can be given by
$$\displaystyle\!\!\!\!\mathrm{AltMin\!-\!P4\!-\!1}:\min_{\mathbf{q,w}}\quad$$
$$\displaystyle\sum_{k=1}^{K}\omega_{k}q_{k}$$
(15a)
$$\displaystyle\mathrm{s.t.}~{}~{}~{}$$
$$\displaystyle\mathcal{F}_{k}(\mathbf{w})\geq\sigma_{k}^{2}\left(\gamma_{\text{th}}^{(\text{I})}-q_{k}\right),\forall k,$$
(15b)
$$\displaystyle q_{k}\geq 0,\forall k,$$
(15c)
$$\displaystyle\sum_{k=1}^{K}\omega_{k}q_{k}\leq K-1,$$
(15d)
$$\displaystyle\|\mathbf{w}\|^{2}\leq P_{\text{max}}.$$
(15e)
It is a convex problem, which can be solved by existing convex program solvers like CVX [25].
With given $\mathbf{w}$, constraint (11b) can be rewritten as
$$|c_{k}+\widetilde{\mathbf{v}}^{H}\mathbf{d}_{k}|^{2}\geq\sigma_{k}^{2}\left(\gamma_{\text{th}}^{(\text{I})}-q_{k}\right),\forall k,$$
(16)
where $c_{k}=\mathbf{h}_{d,k}^{H}\mathbf{w},\mathbf{d}_{k}=\mathbf{G}_{k}\mathbf{w}$. By applying the first-order Taylor approximation, $|c_{k}+\widetilde{\mathbf{v}}^{H}\mathbf{d}_{k}|^{2}$ is lower bounded by
$$\displaystyle\mathcal{G}_{k}(\widetilde{\mathbf{v}})=$$
$$\displaystyle 2\mathrm{Re}\begin{Bmatrix}\left(c_{k}\mathbf{d}_{k}^{H}+\left(\widetilde{\mathbf{v}}^{(i)}\right)^{H}\mathbf{d}_{k}\mathbf{d}_{k}^{H}\right)\left(\widetilde{\mathbf{v}}-\widetilde{\mathbf{v}}^{(i)}\right)\end{Bmatrix}$$
(18)
$$\displaystyle+\begin{vmatrix}c_{k}+\left(\widetilde{\mathbf{v}}^{(i)}\right)^{H}\mathbf{d}_{k}\end{vmatrix}^{2},$$
(20)
where $\widetilde{\mathbf{v}}^{(i)}$ is the solution obtained in the $i$-th iteration. Thus, we can convert constraint (11b) into a convex one
$$\mathcal{G}_{k}(\widetilde{\mathbf{v}})\geq\sigma_{k}^{2}\left(\gamma_{\text{th}}^{(\text{I})}-q_{k}\right),\forall k.$$
(21)
The same technique is applied to deal with $\|\mathbf{q}\|_{0}$. The remaining challenge is the nonconvex unit-modulus constraint (11f). To make it more tractable, we first relax it as
$$|\widetilde{\mathbf{v}}_{m}|\leq 1,\forall m.$$
(22)
Accordingly, we have the following convex optimization problem
$$\displaystyle\mathrm{AltMin\!-\!P4\!-\!2}:\min_{\mathbf{q,\widetilde{\mathbf{v}}}}\quad$$
$$\displaystyle\sum_{k=1}^{K}\omega_{k}q_{k}$$
(23a)
$$\displaystyle\mathrm{s.t.}~{}~{}~{}$$
$$\displaystyle\mathcal{G}_{k}(\widetilde{\mathbf{v}})\geq\sigma_{k}^{2}\left(\gamma_{\text{th}}^{(\text{I})}-q_{k}\right),\forall k,$$
(23b)
$$\displaystyle q_{k}\geq 0,\forall k,$$
(23c)
$$\displaystyle\sum_{k=1}^{K}\omega_{k}q_{k}\leq K-1,$$
(23d)
$$\displaystyle|\widetilde{\mathbf{v}}_{m}|\leq 1,\forall m.$$
(23e)
Note that the solution obtained by solving $\mathrm{AltMin\!-\!P4\!-\!2}$ may not satisfy the unit-modulus constraint. One way to construct feasible phase shifts is given by
$$\widetilde{\mathbf{v}}^{*}_{m}=\widetilde{\mathbf{v}}_{m}/|\widetilde{\mathbf{v}}_{m}|,\forall m.$$
(24)
With given $\mathbf{w}$ and the obtained solution $\widetilde{\mathbf{v}}^{*}$, the remaining $q_{k}$ can be obtained by the closed-form expression $q_{k}=\gamma_{\text{th}}^{(\text{I})}-\gamma_{k}^{(\text{I})}$.
The overall AltMin algorithm is summarized in Algorithm 1. This algorithm alternately optimizes two blocks defined by problems $\mathrm{AltMin\!-\!P4\!-\!1}$ and $\mathrm{AltMin\!-\!P4\!-\!2}$, respectively. The objective function denoted by $\mathcal{A}=\sum_{k=1}^{K}\omega_{k}q_{k}$ follows that $\mathcal{A}(\mathbf{w}^{(i)},\widetilde{\mathbf{v}}^{(i)})\geq\mathcal{A}(\mathbf{w}^{(i+1)},\widetilde{\mathbf{v}}^{(i)})\geq\mathcal{A}(\mathbf{w}^{(i+1)},\widetilde{\mathbf{v}}^{(i+1)})$, where $\mathbf{w}^{(i+1)},\widetilde{\mathbf{v}}^{(i+1)}$ are the optimal solutions obtained
for both blocks. The inequalities hold since $\gamma_{k}^{(\text{I})}$ can be maximized with optimal $\mathbf{w}^{(i+1)},\widetilde{\mathbf{v}}^{(i+1)}$, thereby reducing $q_{k}$ and lowering the objective function. Hence, the algorithm is guaranteed to converge to a stationary point of problem $\mathrm{P3}$ in polynomial time [26].
In each iteration, the complexity of solving problem $\mathrm{AltMin\!-\!P4\!-\!1}$ is $\mathcal{O}((K+N_{t})^{3}(2K+2))$ [27], where $K+N_{t}$ is the number of optimization variables and $2K+2$ is the number of affine and convex constraints. Similarly, the complexity of solving problem $\mathrm{AltMin\!-\!P4\!-\!2}$ is $\mathcal{O}((K+M)^{3}(2K+M+1))$. Thus, the computational complexity order per iteration is $\mathcal{O}(2K(K+N_{t})^{3}+(2K+M)(K+M)^{3})$.
III-C Proposed Penalty-based SCA Algorithm
In this algorithm, we first handle the coupling problem. Let $\mathbf{R}_{k}=\left[\mathbf{G}_{k}^{H}\quad\mathbf{h}_{d,k}\right]\in\mathbb{C}^{N_{t}\times(M+1)}$, and $\mathbf{v}=[\widetilde{\mathbf{v}}^{T}\quad 1]^{T}$, then we can equivalently rewrite constraint (11b) as
$$\mathrm{Tr}(\mathbf{W}\mathbf{R}_{k}\mathbf{V}\mathbf{R}_{k}^{H})\geq\sigma_{k}^{2}\left(\gamma_{\text{th}}^{(\text{I})}-q_{k}\right),\forall k,$$
(25)
where $\mathbf{W}=\mathbf{w}\mathbf{w}^{H},\mathbf{V}=\mathbf{v}\mathbf{v}^{H}$. Note that $\mathbf{W}$ and $\mathbf{V}$ remain coupled in the form of a product in constraint (25). However, we can leverage the following equality to decompose the product, which is given by
$$\langle\mathbf{A,B}\rangle_{F}=\frac{1}{2}\left(\|\mathbf{A}+\mathbf{B}\|_{F}^{2}-\|\mathbf{A}\|_{F}^{2}-\|\mathbf{B}\|_{F}^{2}\right),$$
(26)
where $\langle\mathbf{A,B}\rangle_{F}=\mathrm{Tr}(A^{H}B)$ is the Frobenius inner product. Therefore, constraint (25) can be equivalently written as
$$\displaystyle\frac{1}{2}\left(\|\mathbf{W}+\mathbf{R}_{k}\mathbf{V}\mathbf{R}_{k}^{H}\|_{F}^{2}-\|\mathbf{W}\|_{F}^{2}-\|\mathbf{R}_{k}\mathbf{V}\mathbf{R}_{k}^{H}\|_{F}^{2}\right)$$
$$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\geq\sigma_{k}^{2}\left(\gamma_{\text{th}}^{(\text{I})}-q_{k}\right),\forall k.$$
(27)
Note that the left-hand-side (LHS) term in constraint (III-C) is in the form of difference-of-convex function. We apply the first-order Taylor approximation to the convex function $\|\mathbf{W}+\mathbf{R}_{k}\mathbf{V}\mathbf{R}_{k}^{H}\|_{F}^{2}$, thus the LHS term is lower bounded by (29) as shown at the top of the next page,
where $\mathbf{W}^{(i)},\mathbf{V}^{(i)}$ are the solutions obtained in the $i$-th iteration. In this way, we can transform constraint (III-C) into a convex one
$$\mathcal{H}_{k}(\mathbf{W},\mathbf{V})\geq\sigma_{k}^{2}\left(\gamma_{\text{th}}^{(\text{I})}-q_{k}\right),\forall k.$$
(36)
For the $\|\mathbf{q}\|_{0}$, we use reweighted $\ell_{1}$-norm for approximation. Through above transformations, problem $\mathrm{P3}$ can be reformulated as
$$\displaystyle\!\!\!\!\mathrm{SCA\!-\!P4}:\min_{\mathbf{q,W,V}}\quad$$
$$\displaystyle\sum_{k=1}^{K}\omega_{k}q_{k}$$
(37a)
$$\displaystyle\mathrm{s.t.}\quad~{}$$
$$\displaystyle\mathcal{H}_{k}(\mathbf{W},\mathbf{V})\geq\sigma_{k}^{2}\left(\gamma_{\text{th}}^{(\text{I})}-q_{k}\right),\forall k,$$
(37b)
$$\displaystyle q_{k}\geq 0,\forall k,$$
(37c)
$$\displaystyle\sum_{k=1}^{K}\omega_{k}q_{k}\leq K-1,$$
(37d)
$$\displaystyle\mathrm{Tr}(\mathbf{W})\leq P_{\text{max}},$$
(37e)
$$\displaystyle\mathbf{V}_{mm}=1,m=1,\cdots,M+1,$$
(37f)
$$\displaystyle\mathbf{W}\succeq\mathbf{0},\mathbf{V}\succeq\mathbf{0},$$
(37g)
$$\displaystyle\mathrm{rank}(\mathbf{W})=1,\mathrm{rank}(\mathbf{V})=1,$$
(37h)
where constraint (37f) is introduced to guarantee the unit-modulus of each IRS element.
Our main focus now is to deal with the rank-one constraints in (37h). To this end, we apply the following lemma to represent them equivalently.
Lemma 1.
For any $\mathbf{A}\in\mathbb{H}^{n}$, the constraint that $\mathbf{A}$ is rank-one is equivalent to
$$\|\mathbf{A}\|_{*}-\|\mathbf{A}\|_{2}\leq 0.$$
(38)
Proof.
For any $\mathbf{A}\in\mathbb{H}^{n}$, the inequality $\|\mathbf{A}\|_{*}=\sum_{i}\sigma_{i}\geq\|\mathbf{A}\|_{2}=\max_{i}\{\sigma_{i}\}$ holds, where $\sigma_{i}$ is the $i$-th singular value of matrix $\mathbf{A}$. And the equality holds if and only if $\mathbf{A}$ is rank-one. Thus, the implicit constraint of Hermitian matrix $\mathbf{A}$, i.e., $\|\mathbf{A}\|_{*}-\|\mathbf{A}\|_{2}\geq 0$ functions simultaneously with the constraint (38), which requires that $\|\mathbf{A}\|_{*}-\|\mathbf{A}\|_{2}=0$, i.e., $\mathbf{A}$ is a rank-one matrix.
∎
According to Lemma 1, rank-one constraints of $\mathbf{W,V}$ can be equivalently reformulated as
$$\|\mathbf{W}\|_{*}-\|\mathbf{W}\|_{2}\leq 0,~{}\|\mathbf{V}\|_{*}-\|\mathbf{V}\|_{2}\leq 0.$$
(39)
Then, we adopt the penalty-based method by moving constraint (39) into the objective function, thereby resulting in the following optimization problem
$$\displaystyle\!\!\!\!\mathrm{SCA\!-\!P5}:\min_{\mathbf{q,W,V}}\quad$$
$$\displaystyle\sum_{k=1}^{K}\omega_{k}q_{k}\!\!+\!\!\frac{1}{\rho}(\|\mathbf{W}\|_{*}\!\!-\!\!\|\mathbf{W}\|_{2}\!\!+\!\!\|\mathbf{V}\|_{*}\!\!-\!\!\|\mathbf{V}\|_{2})$$
(40)
$$\displaystyle\mathrm{s.t.}\quad~{}$$
$$\displaystyle\eqref{SCAP4C1}-\eqref{PSD},$$
where $\rho>0$ is a penalty factor penalizing the violation of constraint (39). The following proposition demonstrates the equivalence of problems $\mathrm{SCA\!-\!P4}$ and $\mathrm{SCA\!-\!P5}$. By solving problem $\mathrm{SCA\!-\!P5}$, we can recover $\mathbf{w},\mathbf{v}$ from rank-one solutions $\mathbf{W},\mathbf{V}$ by eigenvalue decomposition.
Proposition 1.
Let $\mathbf{W}_{s},\mathbf{V}_{s}$ be the optimal solutions of problem $\mathrm{SCA\!-\!P5}$ with penalty factor
$\rho_{s}$. For sufficiently small $\rho_{s}\rightarrow 0$, any limit points $\overline{\mathbf{W}},\overline{\mathbf{V}}$ of the sequence $\{\mathbf{W}_{s},\mathbf{V}_{s}\}$ are optimal solutions of problem $\mathrm{SCA\!-\!P4}$.
Proof.
Please refer to Appendix A.
Note that nuclear norm $\|\mathbf{W}\|_{*},\|\mathbf{V}\|_{*}$ and spectral norm $\|\mathbf{W}\|_{2},\|\mathbf{V}\|_{2}$ are convex functions [28]. Hence, the penalty terms are in a form of difference-of-convex function. To make it tractable, we exert the first-order Taylor series approximation on $\|\mathbf{W}\|_{2}$ and $\|\mathbf{V}\|_{2}$, thereby approximating the penalty terms as equations (33) and (36),
where $\boldsymbol{\lambda}_{\max}(\mathbf{W}),\boldsymbol{\lambda}_{\max}(\mathbf{V})$ denote the eigenvector corresponding to the largest eigenvalue of matrix $\mathbf{W,V}$, respectively, and $\mathbf{W}^{(i)},\mathbf{V}^{(i)}$ are the solutions obtained in the $i$-th iteration.
Therefore, the optimization problem that needs to be solved in the $i$-th iteration can be expressed as
$$\displaystyle\mathrm{SCA\!-\!P6}:\min_{\mathbf{q,W,V}}\quad$$
$$\displaystyle\sum_{k=1}^{K}\omega_{k}q_{k}+\mathcal{P}_{w}+\mathcal{P}_{v}$$
(37)
$$\displaystyle\mathrm{s.t.}\quad~{}$$
$$\displaystyle\eqref{SCAP4C1}-\eqref{PSD},$$
which can be efficiently solved by standard convex program solvers such as CVX [25].
According to the preceding analysis, we propose a penalty-based SCA algorithm to solve problem $\mathrm{SCA\!-\!P6}$, which is summarized in Algorithm 2. By iteratively solving a sequence of problem $\mathrm{SCA\!-\!P6}$ at feasible points, we can obtain a stationary-point solution of problem $\mathrm{P3}$ after convergence [26].
For problem $\mathrm{SCA\!-\!P6}$, there are in total $N_{t}^{2}+(M+1)^{2}+K$ optimization variables and $2K+M+5$ affine and convex constraints. Therefore, the computational complexity order per iteration is $\mathcal{O}((N_{t}^{2}+M^{2}+K)^{3}(2K+M))$ [27].
IV Worst-case Robust Beamforming
In this section, we proceed to investigate a robust design to achieve reliable communication in the worst-case. To resolve the robust beamforming problem, we propose a SDR-based BCD algorithm. Moreover, convergence and complexity analysis is given for the proposed algorithm.
IV-A Uncertainty Modeling
Although the CSI of the direct channel $\mathbf{h}_{d,k}$ from the AP to the actuator $k$ can be estimated by conventional methods, CSI acquisition of AP-IRS channel $\mathbf{H}_{dr}$ and IRS-actuator channel $\mathbf{h}_{r,k}$ is more challenging. The is because without the help of active RF chains, a passive IRS cannot transmit and receive signals for channel estimation. Instead of estimating these two individual channels $\mathbf{H}_{dr}$ and $\mathbf{h}_{r,k}$, it is more preferable to attain the estimated CSI of the cascaded AP-IRS-actuator channel $\mathbf{G}_{k}$. In addition, the difficulty of channel estimation increases in URLLC systems since the time for pilot training is highly limited due to short TTI. Thus, it is more practical to consider the case with imperfect CSI of both direct AP-actuator channels and cascaded AP-IRS-actuator channels. This is different from most existing IRS-related works that only consider the uncertainty of cascaded AP-IRS-actuator channels [29, 30].
Channel uncertainty modeling mainly falls into two categories: 1) stochastic CSI error, where the CSI error is assumed to follow the circularly symmetric complex Gaussian (CSCG) distribution; 2) bounded CSI error, where all CSI errors lie in a norm-bounded uncertainty region. From [31], we know that based on the worst-case bounding approach, we can convert the probabilistic constraint involving Gaussian CSI uncertainties into a deterministic norm-bounded form. Hence, in this paper, we adopt the norm-bounded CSI error model. Specifically, the CSI of the direct AP-actuator channel $\mathbf{h}_{d,k}$ and the cascaded AP-IRS-actuator channel $\mathbf{G}_{k}$ can be modeled as
$$\displaystyle\mathbf{h}_{d,k}=\hat{\mathbf{h}}_{d,k}+\Delta\mathbf{h}_{d,k},$$
$$\displaystyle\Delta\mathbf{h}_{d,k}\in\Omega_{\mathbf{h}_{d,k}}\triangleq\{\Delta\mathbf{h}_{d,k}:\|\Delta\mathbf{h}_{d,k}\|_{2}\leq\delta_{\mathbf{h}_{d,k}}\},\forall k,$$
$$\displaystyle\mathbf{G}_{k}=\widehat{\mathbf{G}}_{k}+\Delta\mathbf{G}_{k},$$
$$\displaystyle\Delta\mathbf{G}_{k}\in\Omega_{\mathbf{G}_{k}}\triangleq\{\Delta\mathbf{G}_{k}:\|\Delta\mathbf{G}_{k}\|_{F}\leq\delta_{\mathbf{G}_{k}}\},\forall k,$$
(38)
where $\hat{\mathbf{h}}_{d,k},\widehat{\mathbf{G}}_{k}$ are the estimates of the direct AP-actuator channel and the cascaded AP-IRS-actuator channel, respectively. The corresponding estimation errors $\Delta\mathbf{h}_{d,k},\Delta\mathbf{G}_{k}$, lying in continuous sets $\Omega_{\mathbf{h}_{d,k}},\Omega_{\mathbf{G}_{k}}$ are norm-bounded by $\delta_{\mathbf{h}_{d,k}},\delta_{\mathbf{G}_{k}}$, respectively. Meanwhile, $\delta_{\mathbf{h}_{d,k}}>0,\delta_{\mathbf{G}_{k}}>0$ represent the level of channel uncertainty, which can be chosen smaller for more accurate quantization and channel estimation.
IV-B Proposed SDR-based BCD Algorithm
Under the case with imperfect CSI, $\mathbf{R}_{k}=\left[\mathbf{G}_{k}^{H}\quad\mathbf{h}_{d,k}\right]$, the effective channel from AP to actuators, contains the uncertainty of direct AP-actuator channels and cascaded AP-IRS-actuator channels, which can be expressed as
$$\!\!\mathbf{R}_{k}\!\!=\!\!\left[\mathbf{G}_{k}^{H}\quad\mathbf{h}_{d,k}\right]\!\!=\!\![\widehat{\mathbf{G}}_{k}^{H}\quad\hat{\mathbf{h}}_{d,k}]\!+\![\Delta\mathbf{G}_{k}^{H}\quad\Delta\mathbf{h}_{d,k}]\!\!=\!\!\widehat{\mathbf{R}}_{k}\!+\!\Delta\mathbf{R}_{k},$$
(39)
where
$$\|\Delta\mathbf{R}_{k}\|_{F}\!=\!\sqrt{\|\Delta\mathbf{G}_{k}\|_{F}^{2}+\|\Delta\mathbf{h}_{d,k}\|_{2}^{2}}\!\leq\!\sqrt{\delta_{\mathbf{G}_{k}}^{2}+\delta_{\mathbf{h}_{d,k}}^{2}}\!\triangleq\!\delta_{k}.$$
(40)
The robust beamforming problem for imperfect CSI can be given by
$$\displaystyle\!\!\!\mathrm{ICSI\!-\!P4}:$$
$$\displaystyle\min_{\mathbf{q,W,V}}\quad$$
$$\displaystyle\|\mathbf{q}\|_{0}$$
(41a)
$$\displaystyle\mathrm{s.t.}\quad~{}$$
$$\displaystyle\mathrm{Tr}(\mathbf{W}\mathbf{R}_{k}\mathbf{V}\mathbf{R}_{k}^{H})\!\!\geq\!\!\sigma_{k}^{2}\left(\gamma_{\text{th}}^{(\text{I})}\!-\!q_{k}\right),\!\forall\|\Delta\mathbf{R}_{k}\|_{F}\!\!\leq\!\!\delta_{k},\!\forall k,$$
(41b)
$$\displaystyle q_{k}\geq 0,\forall k,$$
(41c)
$$\displaystyle\|\mathbf{q}\|_{0}\leq K-1,$$
(41d)
$$\displaystyle\mathrm{Tr}(\mathbf{W})\leq P_{\text{max}},$$
(41e)
$$\displaystyle\mathbf{V}_{mm}=1,m=1,\cdots,M+1,$$
(41f)
$$\displaystyle\mathbf{W}\succeq\mathbf{0},\mathbf{V}\succeq\mathbf{0},$$
(41g)
$$\displaystyle\mathrm{rank}(\mathbf{W})=1,\mathrm{rank}(\mathbf{V})=1.$$
(41h)
Constraint (41b) has the following meaning. If the SNR of actuator $k$ can reach the SNR threshold for all possible CSI errors, we consider that the $k$-th actuator can successfully decode the message in the first stage, i.e., $q_{k}=0$. Otherwise, the decoding of actuator $k$ fails and $q_{k}$ is a positive number. Furthermore, it is challenging to deal with the infinite number of nonconvex SNR constraints in (41b) due to the continuity of the channel uncertainty set. The difficulties that remain to be solved are the $\|\mathbf{q}\|_{0}$ and rank-one constraints.
First, we need to apply the S-procedure in Lemma 2 to convert the infinite number of SNR constraints in (41b) into a finite number of constraints. To this end, we recast (41b) as
$$\!\!\mathrm{vec}^{H}(\mathbf{R}_{k})(\mathbf{V}^{T}\!\otimes\!\mathbf{W})\mathrm{vec}(\mathbf{R}_{k})\!\!\geq\!\!\sigma_{k}^{2}\left(\gamma_{\text{th}}^{(\text{I})}\!-\!q_{k}\right),\!\forall\|\Delta\mathbf{R}_{k}\|_{F}\!\!\leq\!\!\delta_{k},$$
(42)
by utilizing the following matrix identities $\mathrm{Tr}(\mathbf{ABC})=\mathrm{Tr}(\mathbf{BCA})=\mathrm{Tr}(\mathbf{CAB}),~{}\mathrm{Tr}(\mathbf{A}^{H}\mathbf{B})=\mathrm{vec}^{H}(\mathbf{A})\mathrm{vec}(\mathbf{B})$, and $\mathrm{vec}(\mathbf{AXB})=(\mathbf{B}^{T}\otimes\mathbf{A})\mathrm{vec}(\mathbf{X})$.
Further, denote $\mathbf{r}_{k}=\mathrm{vec}(\mathbf{R}_{k}),\hat{\mathbf{r}}_{k}=\mathrm{vec}(\widehat{\mathbf{R}}_{k}),\Delta\mathbf{r}_{k}=\mathrm{vec}(\Delta\mathbf{R}_{k})$ bounded by $\|\Delta\mathbf{r}_{k}\|_{2}\leq\delta_{k}$. Thus, constraint (41b) can be transformed into the following more tractable form as
$$\displaystyle\Delta\mathbf{r}_{k}^{H}(\mathbf{V}^{T}\otimes\!\mathbf{W})\Delta\mathbf{r}_{k}+2\mathrm{Re}\{\hat{\mathbf{r}}_{k}^{H}(\mathbf{V}^{T}\otimes\mathbf{W})\Delta\mathbf{r}_{k}\}$$
$$\displaystyle\!+\!\hat{\mathbf{r}}_{k}^{H}(\mathbf{V}^{T}\!\otimes\!\mathbf{W})\hat{\mathbf{r}}_{k}\!\geq\!\sigma_{k}^{2}\left(\gamma_{\text{th}}^{(\text{I})}\!-\!q_{k}\right),\forall\|\Delta\mathbf{r}_{k}\|_{2}\!\leq\!\delta_{k},\forall k.$$
(43)
Lemma 2.
(S-procedure for the complex case [28]): Let $\mathbf{F}_{1},\mathbf{F}_{2}\in\mathbb{H}^{n},\mathbf{s}_{1},\mathbf{s}_{2}\in\mathbb{C}^{n}$, $z_{1},z_{2}\in\mathbb{R}$. The following implication
$$\mathbf{x}^{H}\mathbf{F}_{1}\mathbf{x}\!+\!2\operatorname{Re}\!\left\{\mathbf{s}_{1}^{H}\mathbf{x}\right\}\!+\!z_{1}\!\leq\!0\!\Rightarrow\!\mathbf{x}^{H}\mathbf{F}_{2}\mathbf{x}\!+\!2\operatorname{Re}\!\left\{\mathbf{s}_{2}^{H}\mathbf{x}\right\}\!+\!z_{2}\!\leq\!0$$
holds true if and only if there exists a $\mu\geq 0$ such that
$$\left[\begin{array}[]{cc}\mathbf{F}_{2}&\mathbf{s}_{2}\\
\mathbf{s}_{2}^{H}&z_{2}\end{array}\right]\preceq\mu\left[\begin{array}[]{cc}\mathbf{F}_{1}&\mathbf{s}_{1}\\
\mathbf{s}_{1}^{H}&z_{1}\end{array}\right]$$
provided that there exists a point $\hat{\mathbf{x}}$ with $\hat{\mathbf{x}}^{H}\mathbf{F}_{1}\hat{\mathbf{x}}+2\operatorname{Re}\left\{\mathbf{s}_{1}^{H}\hat{\mathbf{x}}\right\}+z_{1}<0$.
According to Lemma 2, the infinite number of constraints in (IV-B) can be represented as the following $K$ matrix inequalities
$$\displaystyle\!\!\!\!\begin{bmatrix}\mu_{k}\mathbf{I}_{N_{t}(M+1)}\!\!+\!\!\mathbf{V}^{T}\!\!\otimes\!\!\mathbf{W}&(\!\mathbf{V}^{T}\!\!\otimes\!\!\mathbf{W})\hat{\mathbf{r}}_{k}\\
\hat{\mathbf{r}}_{k}^{H}(\mathbf{V}^{T}\!\!\otimes\!\!\mathbf{W}\!)&\hat{\mathbf{r}}_{k}^{H}(\!\mathbf{V}^{T}\!\!\otimes\!\!\mathbf{W}\!)\hat{\mathbf{r}}_{k}\!\!-\!\!\mu_{k}\delta_{k}^{2}\!\!-\!\!\sigma_{k}^{2}\left(\!\gamma_{\text{th}}^{(\text{I})}\!\!-\!\!q_{k}\!\right)\end{bmatrix}\!\!\succeq\!\!\mathbf{0},\!\forall k,$$
(46)
$$\displaystyle\Leftrightarrow\boldsymbol{\Xi}_{k}+\mathbf{E}_{k}^{H}(\mathbf{V}^{T}\otimes\mathbf{W})\mathbf{E}_{k}\succeq\mathbf{0},\forall k,$$
(47)
where $\mu_{k}\geq 0$, $\mathbf{E}_{k}=[\mathbf{I}_{N_{t}(M+1)}\quad\hat{\mathbf{r}}_{k}]$, and
$$\boldsymbol{\Xi}_{k}=\begin{bmatrix}\mu_{k}\mathbf{I}_{N_{t}(M+1)}&\mathbf{0}\\
\mathbf{0}&-\mu_{k}\delta_{k}^{2}-\sigma_{k}^{2}\left(\gamma_{\text{th}}^{(\text{I})}-q_{k}\right)\end{bmatrix}.$$
(48)
The reweighted $\ell_{1}$ method is applied to cope with $\|\mathbf{q}\|_{0}$. Note that the rank-one constraints in (41h) are nonconvex and difficult to solve. In view of this, we first drop the rank-one constraints to obtain a relaxed problem. Therefore, we have the following optimization problem
$$\displaystyle\!\!\!\mathrm{ICSI\!-\!P5}:\min_{\begin{subarray}{c}\mathbf{q,W,V}\\
\boldsymbol{\mu}\succeq\mathbf{0}\end{subarray}}\quad$$
$$\displaystyle\sum_{k=1}^{K}\omega_{k}q_{k}$$
(49c)
$$\displaystyle\mathrm{s.t.}\quad~{}$$
$$\displaystyle\boldsymbol{\Xi}_{k}+\mathbf{E}_{k}^{H}(\mathbf{V}^{T}\otimes\mathbf{W})\mathbf{E}_{k}\succeq\mathbf{0},\forall k,$$
(49d)
$$\displaystyle\sum_{k=1}^{K}\omega_{k}q_{k}\leq K-1,$$
(49e)
$$\displaystyle\eqref{ICSIP4C2},\eqref{ICSIP4C4}-\eqref{ICSIP4C6}.$$
It is essential to emphasize that $\mathbf{W},\mathbf{V}$ remain coupled in the form of the Kronecker product in the matrix inequality constraint (49d), which is different from the coupling problem of $\mathbf{W},\mathbf{V}$ in the real-valued inequality constraint (25) as mentioned in Section III-C. Therefore, we cannot leverage a similar technique for the product decomposition. In light of this, we use a BCD method to split problem $\mathrm{ICSI\!-\!P5}$ into two subproblems, i.e., we alternately optimize $\mathbf{q,W}$ with given $\mathbf{V}$ and $\mathbf{q,V}$ with given $\mathbf{W}$, which can be seen below. Both two subproblems are semidefinite program (SDP) that can be solved by the CVX tool [25]. The overall SDR-based BCD algorithm is summarized in Algorithm 3.
$$\displaystyle\mathrm{ICSI\!-\!P6\!-\!1}:$$
$$\displaystyle\min_{\mathbf{q,W},\boldsymbol{\mu}}\quad$$
$$\displaystyle\sum_{k=1}^{K}\omega_{k}q_{k}$$
(50a)
$$\displaystyle\mathbf{W}\succeq\mathbf{0},\boldsymbol{\mu}\succeq\mathbf{0},$$
(50b)
$$\displaystyle\eqref{ICSIP5C1},\eqref{ICSIP5C2},\eqref{ICSIP4C2},\eqref{ICSIP4C4}.$$
$$\displaystyle\mathrm{ICSI\!-\!P6\!-\!2}:$$
$$\displaystyle\min_{\mathbf{q,V},\boldsymbol{\mu}}\quad$$
$$\displaystyle\sum_{k=1}^{K}\omega_{k}q_{k}$$
(51a)
$$\displaystyle\mathbf{V}\succeq\mathbf{0},\boldsymbol{\mu}\succeq\mathbf{0},$$
(51b)
$$\displaystyle\eqref{ICSIP5C1},\eqref{ICSIP5C2},\eqref{ICSIP4C2},\eqref{ICSIP4C5}.$$
The objective function $\mathcal{A}=\sum_{k=1}^{K}\omega_{k}q_{k}$ follows that $\mathcal{A}(\mathbf{W}^{(i)},\mathbf{V}^{(i)})\geq\mathcal{A}(\mathbf{W}^{(i+1)},\mathbf{V}^{(i)})\geq\mathcal{A}(\mathbf{W}^{(i+1)},\mathbf{V}^{(i+1)})$, where $\mathbf{W}^{(i+1)},\mathbf{V}^{(i+1)}$ are the optimal solutions obtained for subproblems $\mathrm{ICSI\!-\!P6\!-\!1}$ and $\mathrm{ICSI\!-\!P6\!-\!2}$, respectively. The inequalities hold as $\gamma_{k}^{(\text{I})}$ can be maximized with optimal $\mathbf{W}^{(i+1)},\mathbf{V}^{(i+1)}$, thereby reducing $q_{k}$ and lowering the objective function. Therefore, the algorithm can be solved in polynomial time [26] with guaranteed convergence.
According to [27, Theorem 3.12], the complexity of an SDP problem with $m$ SDP constraints, where each constraint involves an $n\times n$ PSD matrix, is given by $\mathcal{O}\left(\sqrt{n}\log\frac{1}{\epsilon}\left(mn^{3}+m^{2}n^{2}+m^{3}\right)\right)$, where $\epsilon$ is the convergence tolerance. For SDP problems $\mathrm{ICSI\!-\!P6\!-\!1}$ and $\mathrm{ICSI\!-\!P6\!-\!2}$, we have $m=K+1,n=N_{\mathrm{t}}$ and $m=K+1,n=M+1$, respectively. Thus, the computational complexity of each iteration of the proposed BCD algorithm is $\mathcal{O}(\log\frac{1}{\epsilon}((\sqrt{N_{t}}+\sqrt{M})K^{3}+(N_{t}^{2.5}+M^{2.5})K^{2}+(N_{t}^{3.5}+M^{3.5})K))$.
When the SDR-based BCD algorithm converges, the obtained solutions $\mathbf{W,V}$ are not always rank-one. To obtain feasible rank-one solutions, we perform the following Gaussian randomization procedure [30]. First, we decompose $\mathbf{V}$ as $\mathbf{V}=\mathbf{U}_{v}\boldsymbol{\Sigma}_{v}\mathbf{U}_{v}^{H}$, where $\mathbf{U}_{v}=$ $\left[u_{v,1},\ldots,u_{v,M+1}\right]$ and $\boldsymbol{\Sigma}_{v}=\operatorname{diag}\left(\lambda_{v,1},\ldots,\lambda_{v,M+1}\right)$ are unitary matrices containing eigenvectors and a diagonal matrix with eigenvalues on its diagonal, respectively. Then, we generate a vector satisfying $\mathbf{v}=\mathbf{U}_{v}\boldsymbol{\Sigma}_{v}^{\frac{1}{2}}\mathbf{e}_{v}$, where $\mathbf{e}_{v}\sim\mathcal{C}\mathcal{N}\left(\mathbf{0},\mathbf{I}_{M+1}\right)$. Further, we normalize each element of $\mathbf{v}$ and obtain $\mathbf{v}^{*}$ as the optimal phase shift. Next, we decompose $\mathbf{W}$ as $\mathbf{W}=\mathbf{U}_{w}^{H}\boldsymbol{\Sigma}_{w}\mathbf{U}_{w}$ and obtain a sub-optimal solution as $\mathbf{w}=\mathbf{U}_{w}^{H}\boldsymbol{\Sigma}_{w}^{\frac{1}{2}}\mathbf{e}_{w}$, where $\mathbf{e}_{w}\sim\mathcal{C}\mathcal{N}\left(\mathbf{0},\mathbf{I}_{N_{t}}\right)$. For randomly generated vectors $\mathbf{e}_{w}$, the best $\mathbf{w}^{*}$ is selected if it satisfies the transmit power constraint (49d) and the SNR constraint (41e) while minimizing the objective function.
V Simulation Results
In this section, we provide simulation results to evaluate the performance of the proposed two-stage IRS-aided D2D communication protocol in terms of reliable communication capability between AP and actuators under the perfect and imperfect CSI scenarios.
V-A Simulation Setup
The AP and IRS are located at (0,0,15) meter (m) and (50,50,15) m, respectively, and all actuators are randomly and uniformly distributed on a circle centered at (50,50,0) m with a radius of 20 m. The path loss is modeled as $\mathrm{PL}=C_{0}d^{-\alpha}$ [32], where $C_{0}$, set as -30 dB, is the path loss at the reference distance 1 m, $d$ is the link distance, and $\alpha$ is the path loss exponent. The small-scale fading of AP-IRS and IRS-actuator channels are assumed to be Ricean fading. The direct link between AP and actuators and the D2D link are modeled as Rayleigh fading. Assume the system operates in a mini-slot fashion, i.e., $\tau_{1}=\tau_{2}$. The carrier frequency is set as 2.4 GHz. Considering the imperfect CSI effects, an uncertainty ratio, denoted by $\kappa$, is defined as $\kappa=\frac{\delta_{k}}{\|\mathbf{R}_{k}\|_{F}}$, where $\kappa\in[0,1]$ [33]. Unless otherwise specified, other parameters are listed in Table II.
To evaluate the proposed protocol in terms of reliability, we define a performance metric called Probability of Reliable Communication (PRC) as the ratio of the number of experiments where all actuators can successfully decode the message over total experiments. For comparison, we consider the following schemes: 1) Upper bound: we relax the rank-one constraints, i.e., $\mathrm{rank}(\mathbf{W})=1,\mathrm{rank}(\mathbf{V})=1$ and then solve problem $\mathrm{P3}$ until convergence, which serves as a performance upper bound; 2) Proposed AltMin algorithm: the approach is proposed in Section III-B for the perfect CSI scenario; 3) Proposed penalty-based SCA algorithm: the method is proposed in Section III-C for the perfect CSI case; 4) Baseline 1: IRS is deployed for packet transmission, but we adopt random phase shifts; 5) Baseline 2: IRS is not introduced into the system; 6) Proposed SDR-based BCD algorithm: the approach is proposed in Section IV for the imperfect CSI scenario; 7) Penalty-based BCD algorithm: the penalty-based method is used to handle the rank-one constraints for the imperfect CSI case.
V-B Convergence of the Proposed Algorithms
We plot the convergence performance of the proposed AltMin algorithm and the proposed penalty-based SCA algorithm for $K=20,N_{t}=4,M=20,D=500$ bits, $\tau=0.5$ ms. As it can be seen from Fig. 3, both algorithms converge monotonically, and the proposed penalty-based SCA algorithm requires significantly more iterations to converge than the proposed AltMin algorithm. In particular, the proposed AltMin algorithm converges after about 5 iterations on average, while the penalty-based SCA algorithm needs another 30 iterations on average to converge. This is because the number of optimization variables and
constraints in the penalty-based SCA algorithm is much higher than those in the AltMin algorithm, which was discussed in more detail in Section III-B and Section III-C.
V-C Impact of Packet Size
Fig. 4 shows the PRC performance versus the packet size $D$ for different schemes when $K=20,N_{t}=2,M=10$, $\tau=1$ ms. It can be seen that the proposed AltMin algorithm and the penalty-based SCA algorithm can achieve reliable communication within 700 data bits. Moreover, AP and actuators in both proposed algorithms can also communicate reliably at 800 bits with $99\%$ probability. Particularly, the PRC performance of the two proposed algorithms is close to the upper bound scheme, which demonstrates the effectiveness of the proposed algorithms. In contrast, with the random-phase IRS, reliable communication between AP and actuators can be achieved within 500 bits in the baseline 1 scheme. The PRC performance degrades significantly compared to the proposed algorithms with optimal reflective beamforming. This shows the importance of optimizing the reflection matrix, which can greatly improve the quality of the received signal by intelligent reflection. In the baseline 2 scheme, without the support of IRS, the actuators can only communicate reliably within 300 bits, which has worse PRC performance. This is because the received SNR in the first stage is significantly reduced without the help of IRS. As a result, the number of actuators with successful decoding in the first stage decreases significantly, i.e., the number of actuators to relay messages in the second stage decreases. Consequently, the number of actuators that can successfully decode the message in the second stage also decreases. From the comparison of the proposed algorithms with baseline 1 and 2, we know that the integration of IRS with optimized phase shifts into the system play an important role in improving reliability, which confirms the necessity of the IRS in the two-stage protocol.
V-D Necessity of IRS
To clarify the necessity of IRS in a two-stage communication protocol, we investigate the effect of distance variation between the AP and actuators on the PRC performance by comparing the proposed algorithms and the No-IRS scheme for $K=10,N_{t}=2,D=900$ bits, $\tau=1$ ms. We set the location of actuators mentioned above as reference points, i.e., all actuators are randomly and uniformly distributed on a circle centered at (50,50,0) m with a radius of 20 m. Then in order to represent the distance variation between the AP and actuators, we increase the horizontal and vertical coordinates of the actuators at the same time. From Fig. 5, we can observe that when the distance between the AP and actuators is large, the actuators cannot communicate reliably with the AP under the No-IRS scheme. This is because of the poor channel quality of the direct link between the AP and the actuators. After the deployment of IRS into the communication protocol, by increasing the number of IRS reflecting elements, actuators can successfully receive the messages sent by the AP even when they are far from each other.
V-E Impact of Second-stage D2D Transmission
We now examine the role of the second-stage D2D transmission. For this purpose, we plot the average number of actuators with successful decoding in each stage and the total number in two stages under the proposed AltMin algorithm and the baseline 2 scheme for different packet size D. From Fig. 6, it can be seen that the average number of actuators with successful decoding in the first stage decreases as the packet size $D$ increases. Accordingly, the remaining actuators with decoding failures must rely on the second-stage D2D transmission to achieve reliable communication. In particular, the D2D network begins to function at $D>200$
bits in the proposed AltMin algorithm and at $D>100$ bits in the baseline 2 (i.e., No-IRS). Moreover, when the size of the data packets is larger, the second-stage D2D transmission plays a greater role. In addition, we find that when the data packet becomes larger (taking 800 bits as an example), about $30\%$ of the actuators in the proposed algorithm rely on the D2D network to achieve reliable communication, while in the baseline 1 scheme, the number of users who need to rely on D2D network to realize reliable communication is up to about $70\%$. This shows that the IRS-assisted first-stage transmission can effectively reduce the communication load of the second-stage D2D network.
V-F Impact of Delay
In Fig. 7, we study the effects of delay on the PRC performance for different schemes when $K=20,N_{t}=4,M=20,D=500$ bits. It can be observed that the baseline 1 and 2 schemes are sensitive to delay and their PRC performance is poor, especially for strict latency requirements. The proposed algorithms, on the other hand, can ensure reliable communication even for small delay (e.g. $\tau=0.5$ ms). This is due to the doubly improved reliability of the proposed two-stage protocol through the combined use of IRS and the D2D network. Thus, the proposed two-stage IRS-aided D2D communication protocol is crucial and better copes with URLLC-oriented applications.
V-G Impact of Number of Reflecting Elements
Fig. 8 depicts the PRC performance versus the number of reflecting elements $M$ for different $K$ and schemes when $N_{t}=2,D=900$ bits, $\tau=1$ ms. It can be seen that the proposed algorithms perform almost the same and their PRC performance approaches the upper bound. When the number of reflecting elements $M$ increases, the PRC performance improves and reliable communication of all actuators can be guaranteed. Although IRS is theoretically passive, i.e., it does not actively send and receive signals, it also requires a power supply to maintain the operation of each reflecting element and the intelligent controller. So in this case, 20 reflecting elements are sufficient for all actuators to successfully decode the 900-bit packet within 1 ms, which can provide some insight into the practical application. Furthermore, compared to baseline 1, the proposed algorithms have a large performance gap, which becomes smaller as the number of reflecting elements increases. This is because more reflecting elements can provide higher spatial degree of freedom (DoF), significantly improving the received signal. In particular, the PRC performance for $K=20$ is better than that for $K=10$, which benefits from the multiuser diversity and the second-stage D2D transmission between actuators with proximity to each other. At the same time, it is worth noting that the message that AP sends in the first stage and the successful-decoding actuators relay in the second stage is the same combined message, so there is no interference between the actuators. More specifically, the number of
actuators that can successfully decode the signal in the first stage is higher for $K=20$ than for $K=10$, i.e., the number of actuators that act as relays to relay messages in the second stage becomes larger, increasing the probability of successful reception of the remaining actuators.
V-H Impact of Number of Antennas
Fig. 9 shows the effects of the number of antennas $N_{t}$ on the PRC performance for different $K$ and schemes when $M=10,D=900$ bits, $\tau=1$ ms. As it can be seen from Fig. 9, the probability of reliable communication between AP and actuators improves as the number of antennas increases since more antennas provide a higher diversity gain. The PRC performance of the proposed AltMin algorithm is slightly better than that of the penalty-based SCA algorithm, and the performance gap between the proposed algorithms and the upper bound narrows as $N_{t}$ increases. The reason why the PRC performance for $K=20$ is better than that for $K=10$ was discussed in Section V-G. In addition, when we compare the proposed algorithms with baseline 1, we also find that as the number of antennas increases, the received signal of the direct link becomes stronger and the influence of the reflection matrix optimization becomes weaker. That is, reliable communication via the random phase IRS can be guaranteed if the number of antennas is sufficiently large, but it is achieved at the expense of the high cost of maintaining the antenna arrays. This demonstrates the necessity of using IRS with well-optimized phase shifts.
V-I Impact of CSI Uncertainty
In Fig. 10, we investigate the impact of CSI uncertainty on the PRC performance for different schemes when $K=10,N_{t}=4,M=10,D=400$ bits, $\tau=1$ ms. It can be seen that the PRC performance deteriorates when the accuracy of the CSI estimate decreases. This is because the difficulty of performing accurate active beamforming at AP and reflective beamforming at IRS increases with the poorer quality of the CSI estimate, resulting in poorer PRC performance. The proposed SDR-based BCD algorithm outperforms the penalty-based BCD algorithm and baseline 1 (i.e., IRS with random phase shifts) over the entire range of the considered CSI uncertainty ratio, which shows that the proposed algorithm can fully and efficiently exploit the spatial DoF to improve the reliability even when the CSI uncertainty exists.The proposed algorithm can guarantee reliable communication with a probability of $98\%$, even in the presence of large CSI estimation errors. This confirms the robustness of the proposed algorithm to CSI uncertainty.
VI Conclusion
In this work, we exploited the potential of IRS and D2D communication to enable URLLC between an AP and multiple actuators in the IIoT scenario. We proposed a two-stage protocol where MRC is adopted for joint decoding the superimposed signal (direct signal from AP and reflected signal from IRS) in the first stage and the signal from D2D links in the second stage. The optimization problem was formulated to maximize the number of actuators with successful reception by jointly optimizing the active beamforming at AP and the phase shifts at IRS. We studied the joint beamforming problem under the scenarios of perfect and imperfect CSI of both direct AP-actuator channels and cascaded AP-IRS-actuator channels, where efficient algorithms with complexity and convergence analysis were proposed for each case. Simulation results confirmed the role and importance of IRS and D2D communication in enhancing reliability compared to other baseline schemes. Thanks to the doubly enhanced reliability via IRS and D2D network, the proposed two-stage protocol can achieve reliable communication even under stringent latency requirements and even in the presence of CSI uncertainties.
Appendix A Proof of Proposition 1
For problem $\mathrm{SCA\!-\!P4}$, we denote the objective function by $f(\mathbf{W,V})$ and let $\mathbf{W}^{\star},\mathbf{V}^{\star}$ be its optimal solutions. Hence, we have $f\left(\mathbf{W}^{\star},\mathbf{V}^{\star}\right)\leq f(\mathbf{W,V})$. Denote the objective function of problem $\mathrm{SCA\!-\!P5}$ by $g(\mathbf{W,V};\rho)$ and its optimal solutions by $\mathbf{W}_{s},\mathbf{V}_{s}$ for penalty factor $\rho_{s}$. Thus, we have $g\left(\mathbf{W}_{s},\mathbf{V}_{s};\rho_{s}\right)\leq g\left(\mathbf{W}^{\star},\mathbf{V}^{\star};\rho_{s}\right)$, i.e.,
$$\displaystyle f\left(\mathbf{W}_{s},\mathbf{V}_{s}\right)+\frac{1}{\rho_{s}}\left(\left\|\mathbf{W}_{s}\right\|_{*}-\left\|\mathbf{W}_{s}\right\|_{2}+\left\|\mathbf{V}_{s}\right\|_{*}-\left\|\mathbf{V}_{s}\right\|_{2}\right)$$
$$\displaystyle\leq f\left(\mathbf{W}^{\star},\mathbf{V}^{\star}\right)\!+\!\frac{1}{\rho_{s}}\left(\left\|\mathbf{W}^{\star}\right\|_{*}\!-\!\left\|\mathbf{W}^{\star}\right\|_{2}\!+\!\left\|\mathbf{V}^{\star}\right\|_{*}\!-\!\left\|\mathbf{V}^{\star}\right\|_{2}\right)$$
$$\displaystyle\overset{(a)}{=}f\left(\mathbf{W}^{\star},\mathbf{V}^{\star}\right),$$
(52)
where $(a)$ holds as $\mathbf{W}^{\star},\mathbf{V}^{\star}$ are the optimal solutions of problem $\mathrm{SCA\!-\!P4}$, which satisfy the rank-one constraints, i.e., $\left\|\mathbf{W}^{\star}\right\|_{*}-\left\|\mathbf{W}^{\star}\right\|_{2}=0,\left\|\mathbf{V}^{\star}\right\|_{*}-\left\|\mathbf{V}^{\star}\right\|_{2}$. From (A), we have
$$\displaystyle\left\|\mathbf{W}_{s}\right\|_{*}-\left\|\mathbf{W}_{s}\right\|_{2}+\left\|\mathbf{V}_{s}\right\|_{*}-\left\|\mathbf{V}_{s}\right\|_{2}$$
$$\displaystyle\leq\rho_{s}\left[f\left(\mathbf{W}^{\star},\mathbf{V}^{\star}\right)-f\left(\mathbf{W}_{s}\mathbf{V}_{s}\right)\right].$$
(53)
Denote $\overline{\mathbf{W}},\overline{\mathbf{V}}$ by limit points of sequence $\left\{\mathbf{W}_{s},\mathbf{V}_{s}\right\}$ and there is an infinite subsequence $\mathcal{S}$ such that $\lim\limits_{s\in\mathcal{S}}\mathbf{W}_{s}=\overline{\mathbf{W}},\lim\limits_{s\in\mathcal{S}}\mathbf{V}_{s}=\overline{\mathbf{V}}$. By taking the limit as $s\rightarrow\infty$ for $s\in\mathcal{S}$ on both sides of (A), we can obtain that
$$\displaystyle\lim_{s\in\mathcal{S}}\left(\left\|\mathbf{W}_{s}\right\|_{*}-\left\|\mathbf{W}_{s}\right\|_{2}+\left\|\mathbf{V}_{s}\right\|_{*}-\left\|\mathbf{V}_{s}\right\|_{2}\right)$$
$$\displaystyle\overset{(b)}{=}\|\overline{\mathbf{W}}\|_{*}-\|\overline{\mathbf{W}}\|_{2}+\|\overline{\mathbf{V}}\|_{*}-\|\overline{\mathbf{V}}\|_{2}$$
$$\displaystyle\leq\lim_{s\in\mathcal{S}}\rho_{s}\left[f\left(\mathbf{W}^{\star},\mathbf{V}^{\star}\right)-f\left(\mathbf{W}_{s},\mathbf{V}_{s}\right)\right]\overset{\rho_{s}\rightarrow 0}{=}0,$$
where $(b)$ holds owing to the continuity of the function $\|\mathbf{X}\|_{*}-\|\mathbf{X}\|_{2}$. Combing (A) with $\|\overline{\mathbf{W}}\|_{*}-\|\overline{\mathbf{W}}\|_{2}\geq 0,\|\overline{\mathbf{V}}\|_{*}-\|\overline{\mathbf{V}}\|_{2}\geq 0$, we have $\|\overline{\mathbf{W}}\|_{*}-\|\overline{\mathbf{W}}\|_{2}=0,\|\overline{\mathbf{V}}\|_{*}-\|\overline{\mathbf{V}}\|_{2}=0$, so $\overline{\mathbf{W}},\overline{\mathbf{V}}$ are feasible for problem $\mathrm{SCA\!-\!P4}$. Moreover, by taking the limit as $s\rightarrow\infty$ for $s\in\mathcal{S}$ in (A), we have
$$\displaystyle\!\!f(\overline{\mathbf{W}},\overline{\mathbf{V}})\!\!\overset{(c)}{\leq}\!\!f(\overline{\mathbf{W}},\overline{\mathbf{V}})\!\!+\!\!\lim\limits_{s\in\mathcal{S}}\frac{1}{\!\rho_{s}}\!\left(\left\|\mathbf{W}_{s}\right\|_{*}\!\!-\!\!\left\|\mathbf{W}_{s}\right\|_{2}\!\!+\!\!\left\|\mathbf{V}_{s}\right\|_{*}\!\!-\!\!\left\|\mathbf{V}_{s}\right\|_{2}\right)$$
$$\displaystyle\leq f\left(\mathbf{W}^{\star},\mathbf{V}^{\star}\right),$$
(54)
where $(c)$ holds as the penalty factor $\rho_{s}$ and $\left\|\mathbf{W}_{s}\right\|_{*}-\left\|\mathbf{W}_{s}\right\|_{2}$, $\left\|\mathbf{V}_{s}\right\|_{*}-\left\|\mathbf{V}_{s}\right\|_{2}$
are nonnegative. Note that $\overline{\mathbf{W}},\overline{\mathbf{V}}$ are feasible for problem $\mathrm{SCA\!-\!P4}$ and its objective function $f(\overline{\mathbf{W}},\overline{\mathbf{V}})$ is no larger than $f\left(\mathbf{W}^{\star},\mathbf{V}^{\star}\right)$ obtained from optimal solutions $\mathbf{W}^{\star},\mathbf{V}^{\star}$ of problem $\mathrm{SCA\!-\!P4}$. Thus, $\overline{\mathbf{W}},\overline{\mathbf{V}}$ are also optimal to problem $\mathrm{SCA\!-\!P4}$. This completes the proof.
References
[1]
J. Cheng, C. Shen, Z. Chen, and N. Pappas, “RIS-aided D2D communication
design for URLLC packet delivery,” in the 25th International ITG
Workshop on Smart Antennas (WSA 2021), Nov. 2021, pp. 1–6.
[2]
G. J. Sutton, J. Zeng et al., “Enabling technologies for ultra-reliable
and low latency communications: From PHY and MAC layer perspectives,”
IEEE Commun. Surv. Tutor., vol. 21, no. 3, pp. 2488–2524, Feb. 2019.
[3]
R. Kotaba, C. N. Manchón, T. Balercia et al., “How URLLC can benefit
from NOMA-based retransmissions,” IEEE Trans. Wireless Commun.,
vol. 20, no. 3, pp. 1684–1699, Mar. 2021.
[4]
S. N. Sur and R. Bera, “Intelligent reflecting surface assisted MIMO
communication system: A review,” Phys. Commun., vol. 47, p. 101386,
Aug. 2021.
[5]
X. Zhai, G. Han, Y. Cai, and L. Hanzo, “Beamforming design based on two-stage
stochastic optimization for RIS-assisted over-the-air computation
systems,” IEEE Internet Things J., pp. 1–14, Aug. 2021.
[6]
Y. Guo, Z. Qin, Y. Liu, and N. Al-Dhahir, “Intelligent reflecting surface
aided multiple access over fading channels,” IEEE Trans. Commun.,
vol. 69, no. 3, pp. 2015–2027, Mar. 2021.
[7]
Y. Cai, M.-M. Zhao, K. Xu et al., “Intelligent reflecting surface aided
full-duplex communication: Passive beamforming and deployment design,”
IEEE Trans. Wireless Commun., vol. 21, no. 1, pp. 383–397, Jan. 2022.
[8]
S. Gong, X. Lu, D. T. Hoang et al., “Toward smart wireless
communications via intelligent reflecting surfaces: A contemporary survey,”
IEEE Commun. Surveys Tuts., vol. 22, no. 4, pp. 2283–2314, Jun. 2020.
[9]
R. Hashemi, S. Ali, N. Mahmood, and M. Latva-aho, “Average rate and error
probability analysis in short packet communications over RIS-aided URLLC
systems,” IEEE Trans. Veh. Technol., pp. 1–14, Aug. 2021.
[10]
S. Dhok, P. Raut, P. K. Sharma, K. Singh, and C.-P. Li, “Non-linear energy
harvesting in RIS-assisted URLLC networks for industry automation,”
IEEE Trans. Commun., pp. 1–14, Jul. 2021.
[11]
H. Xie, J. Xu, Y.-F. Liu, L. Liu, and D. W. K. Ng, “User grouping and
reflective beamforming for IRS-aided URLLC,” IEEE Wireless Commun.
Lett., pp. 1–5, Aug. 2021.
[12]
F. Jameel, Z. Hamid, F. Jabeen, S. Zeadally, and M. A. Javed, “A survey of
device-to-device communications: Research issues and challenges,” IEEE
Commun. Surveys Tuts., vol. 20, no. 3, pp. 2133–2168, Apr. 2018.
[13]
R. I. Ansari, C. Chrysostomou, S. A. Hassan et al., “5G D2D
networks: Techniques, challenges, and future prospects,” IEEE Syst.
J., vol. 12, no. 4, pp. 3970–3984, Dec. 2018.
[14]
B. Chang, L. Li, G. Zhao, Z. Chen, and M. A. Imran, “Autonomous D2D
transmission scheme in URLLC for real-time wireless control systems,”
IEEE Trans. Commun., vol. 69, no. 8, pp. 5546–5558, Apr. 2021.
[15]
L. Liu and W. Yu, “A D2D-based protocol for ultra-reliable wireless
communications for industrial automation,” IEEE Trans. Wireless
Commun., vol. 17, no. 8, pp. 5045–5058, Aug. 2018.
[16]
Y. Wu, D. Wu, L. Ao, L. Yang, and Q. Fu, “Contention-based radio resource
management for URLLC-oriented D2D communications,” IEEE Trans.
Veh. Technol., vol. 69, no. 9, pp. 9960–9971, Jun. 2020.
[17]
N. T. T. Van, H. T. Nguyen, N. C. Luong, N. M. Tien, D. Niyato, and D. I. Kim,
“Intelligence reflecting surface-aided integrated data and energy networking
coexisting D2D communications,” IEEE Trans. Wireless Commun., pp.
1–15, 2022.
[18]
W. Wang, L. Yang, A. Meng et al., “Resource allocation for IRS-aided
JP-CoMP downlink cellular networks with underlaying D2D
communications,” IEEE Trans. Wireless Commun., Dec. 2021.
[19]
Z. Peng, T. Li, C. Pan, H. Ren, and J. Wang, “RIS-aided D2D communications
relying on statistical CSI with imperfect hardware,” IEEE Commun.
Lett., Dec. 2021.
[20]
Y. Xu, C. Shen, T.-H. Chang et al., “Transmission energy minimization
for heterogeneous low-latency NOMA downlink,” IEEE Trans. Wireless
Commun., vol. 19, no. 2, pp. 1054–1069, Feb. 2020.
[21]
W. Yang, G. Durisi, T. Koch, and Y. Polyanskiy, “Quasi-static multiple-antenna
fading channels at finite blocklength,” IEEE Trans. Inf. Theory,
vol. 60, no. 7, pp. 4232–4265, Jul. 2014.
[22]
D. Malak, H. Huang, and J. G. Andrews, “Throughput maximization for
delay-sensitive random access communication,” IEEE Trans. Wireless
Commun., vol. 18, no. 1, pp. 709–723, Jan. 2019.
[23]
C. Hu, L. Dai, S. Han, and X. Wang, “Two-timescale channel estimation for
reconfigurable intelligent surface aided wireless communications,”
IEEE Trans. Commun., vol. 69, no. 11, pp. 7736–7747, Nov. 2021.
[24]
E. J. Candès, M. B. Wakin, and S. P. Boyd, “Enhancing sparsity by reweighted
$\ell_{1}$ minimization,” J Fourier Anal Appl, vol. 14, no. 5, pp.
877–905, 2007.
[25]
M. Grant and S. Boyd, “CVX: Matlab software for disciplined convex
programming, version 2.1,” 2014.
[26]
M. Razaviyayn, M. Hong, and Z. Q. Luo, “A unified convergence analysis of
block successive minimization methods for nonsmooth optimization,”
SIAM J. Optim., vol. 23, no. 2, pp. 1126–1153, Jun. 2013.
[27]
I. Pólik and T. Terlaky, Interior Point Methods for Nonlinear
Optimization. Berlin, Heidelberg:
Springer Berlin Heidelberg, 2010.
[28]
C.-Y. Chi, W.-C. Li, and C.-H. Lin, Convex Optimization for Signal
Processing and Communications: From Fundamentals to Applications. Boca Raton, FL, USA: CRC Press, Feb. 2017.
[29]
G. Zhou, C. Pan, H. Ren, K. Wang, and A. Nallanathan, “Outage constrained
transmission design for IRS-aided communications with imperfect cascaded
channels,” in Proc. IEEE Globecom, Dec. 2020, pp. 1–6.
[30]
J. Wang, Y.-C. Liang, S. Han, and Y. Pei, “Robust beamforming and phase shift
design for IRS-enhanced multi-user MISO downlink communication,” in
Proc. IEEE ICC, Jun. 2020, pp. 1–6.
[31]
S. Han, S. Xu, W.-X. Meng, and L. He, “Channel-correlation-enabled
transmission optimization for MISO wiretap channels,” IEEE Trans.
Wireless Commun., vol. 20, no. 2, pp. 858–870, Feb. 2021.
[32]
S. Mao, X. Chu, Q. Wu, L. Liu, and J. Feng, “Intelligent reflecting surface
enhanced D2D cooperative computing,” IEEE Wireless Commun. Lett.,
vol. 10, no. 7, pp. 1419–1423, Mar. 2021.
[33]
Q. Li and W.-K. Ma, “Optimal and robust transmit designs for MISO channel
secrecy by semidefinite programming,” IEEE Trans. Signal Process.,
vol. 59, no. 8, pp. 3799–3812, Apr. 2011. |
Quantum entropies of realistic states of a topological insulator
Nicolás Legnazzi
nicolaslegnazzi@gmail.com
Omar Osenda
osenda@famaf.unc.edu.ar
Facultad de Matemática, Astronomía, Física y Comptación,
Universidad Nacional de Córdoba, and Instituto de Física Enrique
Gaviola, CONICET, Av. Medina Allende s/n , Ciudad Universitaria,
CP:X5000HUA Córdoba, Argentina
Abstract
Nanowires of BiSe show topological states localized near the surface of
the material. The topological nature of these states can be analyzed using
well-known quantities. In this paper, we calculate the topological
entropy suggested by Kitaev and Preskill for these states together with a new
entropy based on a reduced density matrix that we propose as a measure
to distinguish topological one-electron states. Our results show that
the topological entropy is a constant independent of the parameters that
characterize a topological state as its angular momentum, longitudinal
wave vector, and radius of the nanowire. The new entropy is always larger
for topological states than for normal ones, allowing the identification
of the topological ones. We show how the reduced density matrices
associated with both entropies are constructed from the pure state
using positive maps and explicitly obtaining the Krauss operators.
††preprint: APS/123-QED
I Introduction
Topological states appear in different materials and
geometries [1]. Their appearance changes the conducting
properties of the material, giving place to many distinctive phenomena
like the quantum spin Hall effect [2, 3, 4],
topological superconductors [5, 6], etc. The
compound $\mbox{Bi}_{2}\mbox{Se}_{3}$ is a 3D topological
insulant [1, 7, 8, 9].
The calculation of the spectrum, spin currents, and density of
states (DOS) of BiSe cylinders or quantum dots is the subject of
numerous works [10, 11, 12, 13, 14].
Besides first principle calculations, the kp model is a preferred tool to
calculate the spectrum and eigenstates necessary to obtain the spin currents
and the DOS [7]. Once the eigenstates are available, it is possible
to characterize the topological states using quantum entropies to study
different physical regimes.
The cylindrical geometry is well-suited to calculating the Kitaev-Preskill
topological entropy [15]. This quantity should be a constant,
independent of all the quantum numbers that characterize a given topological
state. It can only depend on the topology of the problem. But what are the
values of this entropy for the normal in-band eigenstates that are
eigenfunctions of the kp model? In a situation without a boundary, the
topological entropy of a normal state should be null, but a cylindrical
nanowire necessarily has one. So, the topological entropy for a normal
eigenstate obtained using the kp method could be non-null, but its value
should depend on its quantum numbers.
Distinguishing topological from normal states can be done using different
quantum entropies and related quantities, such as the entanglement
spectrum [16, 17, 18, 19, 20]. To this
end, we propose a particular reduced-density matrix whose entropy distinguishes
the one-electron topological states of a $\mbox{Bi}_{2}\mbox{Se}_{3}$ cylinder
from the non-topological ones. Our study differs from those that employ
entanglement entropy to study many-electron wave functions. The entanglement
entropy detects the non-local character of topological states when applied, for instance, to quantum states that are good approximations of the many-electron ground-state wave function of the fractional Quantum Hall effect, that is, the Laughlin states [21, 16, 22].
For a given eigenstate of the kp Hamiltonian, which we calculate as a superposition of a basis set functions as is usual in the Rayleigh-Ritz variational method, the proposed reduced density matrix depends on the coefficients of the superposition and integrals of the basis set functions. The Rayleigh-Ritz variational method accurately provides the band structure near the gap between the conduction and valence bands in semiconductor nanostructures when applied to kp Hamiltonians. The application of the Rayleigh-Ritz variational method to kp Hamiltonians allowed the study of electronic and optical properties of core-shell nanowires [23, 24], edge states with and without an external magnetic field applied
to a quantum well
[25, 26, 27, 28], the transition between resonance and bound states in quantum dots embedded in nanowires [29], the entanglement entropy of edge states in quantum wells [30], spin currents in topological insulators [10], amongst other physical phenomena.
The paper is organized as follows. In Section 2, we present the kp Hamiltonian, whose eigenvalues give the band structure of a cylindrical nanowire made of $\mbox{Bi}_{2}\mbox{Se}_{3}$ and briefly describe how to obtain a numerical approximation to the spectrum and eigenvectors using the Rayleigh-Ritz method. Section 3 deals with the calculus of the topological entropy of an approximate variational eigenvector. The topological entropy depends on the von Neumann entropy of several reduced-density matrices. Obtaining each one of these matrices implies tracing out a real-space partition from a pure density one [15, 31].
We present in Section IV the mode-dependent reduced density matrix (RDM), $\rho_{MD}$, which contains information about a variational state through the coefficients of the variational expansion and spatial integrals of the basis set functions. We show how the von Neumann entropy and the entanglement spectrum [16] of the mode-dependent RDM allow us to distinguish between non-topological and topological states.
In Section V, we use quantum state processes [32, 33] as an alternative way to construct the mode-dependent RDM and some of the RDM necessary to calculate the topological entropy. Using the quantum process tomography algorithm described in Reference [34], the quantum process results in a sum of Kraus operators determined by a gradient-descent algorithm. The quantum process numerically calculated predicts an RDM, $\tilde{\rho}$, slightly different from the one intended, $\rho^{\prime}$. We compare both matrices, calculating their fidelity. Finally, in Section IV, we summarize and discuss our results.
II Model and Hamiltonian
We consider a cylindrical nanowire made of $Bi_{2}Se_{3}$, with a constant radius on the tens of nanometers and infinitely long in the axial direction.
To obtain the band structure and eigenstates, we employ the $\mathbf{k}\cdot\mathbf{p}$ Hamiltonian given by
$$H_{0}=\varepsilon(\mathbf{k})+\left(\begin{array}[]{c c c c}{{M(\bf{k})}}&{{B(k_{z})k_{z}}}&{{0}}&{{A\left(k_{\parallel}\right)k_{-}}}\\
{{B(k_{z})k_{z}}}&{{-M(\bf{k})}}&{{A(k_{\parallel})k_{-}}}&{{0}}\\
{{0}}&{{A\left(k_{\parallel}\right)k_{+}}}&{{M(\bf{k})}}&{{-B(k_{z})k_{z}}}\\
{{A\left(k_{\parallel}\right)k_{+}}}&{{0}}&{{-B(k_{z})k_{z}}}&{{-M(k_{z})k_{z}}}\end{array}\right),$$
(1)
where
•
$\varepsilon(\mathbf{k})=C_{0}+C_{1}k_{z}^{2}+C_{2}k_{\parallel}^{2}$,
•
$M(\mathbf{k})=M_{0}+M_{1}k_{z}^{2}+M_{2}k_{\parallel}^{2}$,
•
$B(k_{z})=B_{0}$,
•
$A(k_{\parallel})=A_{0}$.
The Hamiltonian in Equation 1 was introduced by Zhang [8] and collaborators to adjust the band structure found for different materials showing topological states. We consign the parameters that define the Hamiltonian in Equation 1 in Table 1.
We calculate approximate eigenvalues and eigenfunctions using the Rayleigh-Ritz variational method, which reduces the eigenvalue problem
$$Hf=ef,$$
(2)
where $H$is the $\mathbf{k}\cdot\mathbf{p}$ Hamiltonian, to an algebraic one. The Rayleigh-Ritz method is suitable for calculating the band structure of nano-structures near the gap between the conduction and valence bands and for states lying inside it, for instance, to find the topological states in three-dimensional topological insulators, to study the transition between localized and resonance states in quantum dots, to study properties of states in quantum wells in the Quantum Spin Hall Effect regime, etc. We consign the details about the implementation of the method to Appendix A.
III The topological entropy
The study of topological states has led to different entropic-like
quantities as, for instance, the real-space entropy, the topological
entropy of Kitaev-Preskill, and so on. Also, it is worth mentioning
the entanglement spectrum or, in the case of studies dealing with
topological states in spin chains, the Renyi entropies.
Tracing out different subspaces of the whole Hilbert space leads to one
or other entropy. For instance, for a given multiparticle wavefunction $\psi$, the real-space entropy
$$S_{A}(\rho_{A})=-\mbox{Tr}(\rho_{A}\log{(\rho_{A})}),$$
(3)
is calculated from the reduced density operator defined by
$$\rho_{A}=\frac{1}{Z}\int_{\bar{A}}\,\psi^{\star}(\vec{x})\psi(\vec{x})d\vec{x}^{\prime},$$
(4)
where $\bar{A}$ is the spatial region outside region $A$, $Z$ is a normalization constant, and the integral involves a subset $\vec{x}^{\prime}$ of all the particle coordinates.
From the definition of the reduced density operator above, it is clear that the value of the real-space entropy could depend on the size of the spatial region. To avoid this problem, Kitaev and Preskill proposed to trace out over a set of spatial sectors and combine the corresponding real-space entropies to single out the topological entropy of the quantum state.
Following the argument in Reference [15], we consider three regions, $A$, $B$, and $C$, as shown in the cartoon in Figure 1. The triangular sectors are defined by
$$\displaystyle A:\varphi\in\left[-\pi/6,\pi/2\right]\quad\mbox{and}\quad\rho\leq R_{C},$$
(5)
$$\displaystyle B:\varphi\in\left[7\pi/6,11\pi/6\right]\quad\mbox{and}\quad\rho\leq R_{C},$$
(6)
$$\displaystyle C:\varphi\in\left[\pi/2,7\pi/6\right]\quad\mbox{and}\quad\rho\leq R_{C}.$$
(7)
Tracing out the spatial region outside $A$, $B$, $C$, or combinations of them, the topological entropy is given by
$$S_{t}=S_{A}+S_{B}+S_{C}-S_{AB}-S_{BC}-S_{AC}+S_{ABC}.$$
(8)
In our case, we calculate $\rho_{A}$ as follows
$$\rho_{A}=\int_{D}|\psi(\rho,\varphi,z)\rangle\langle\psi(\rho,\varphi,z)|\;\rho\,d\rho\,d\varphi,$$
(9)
where $|\psi\rangle$ is one of the eigenvectors obtained using the Rayleigh-Ritz method, and the integral is over the surface of the disk of radius $R$ minus the triangular sector $A$, see Figure 1. Proceeding in this way, $\rho_{A}$, $\rho_{B}$, and so on, are given by $4\times 4$ matrices. The corresponding von Neumann entropies are given by
$$S_{A}(\rho_{A})=-\sum_{i}\lambda_{i}\log(\lambda_{i}),$$
(10)
where the $\lambda_{i}$ are the eigenvalues of $\rho_{A}$.
The variational eigenvectors $|\psi\rangle$ are complex column vectors that depend on the three cylindrical coordinates and are labeled by two quantum numbers, $k_{z}$ and $L$,
$$|\psi_{L,k_{z}}\rangle=\sum_{n}^{N}\left(\!\!\!\begin{array}[]{c}{{b_{L,k_{z},n,\uparrow}A_{L,n}J_{L}\left(\alpha_{n}^{L}\rho/R\right)e^{iL\varphi}}}\\
{{c_{L,k_{z},n,\uparrow}A_{L,n}J_{L}\left(\alpha_{n}^{L}\rho/R\right)e^{iL\varphi}}}\\
{{b_{L,k_{z,n,\downarrow}}A_{L+1,n}J_{L+1,n}\left(\alpha_{n}^{L+1}\rho/R\right)e^{i(L+1)\varphi}}}\\
{{c_{L,k_{z,n,n,\downarrow}}A_{L+1,n}J_{L+1,n}\left(\alpha_{n}^{L+1}\rho/R\right)e^{i(L+1)\varphi}}}\end{array}\!\!\!\right)e^{ik_{z}.z},$$
(11)
where $c_{L,k_{z},n,\uparrow}$, $c_{L,k_{z},n,n,\downarrow}$, $b_{L,k_{z},n,\uparrow}$ and
$b_{L,k_{z},n,\downarrow}$ are the linear variational coefficients, $J_{L}$ is the Bessel function with index $L$, the $\alpha_{n}^{L}$ are its roots, and $A_{L,n}$ is a normalization constant. Note that the number of roots employed coincides with the number of basis set functions used.
Lou and collaborators [10] used the variational basis set in Equation (11) to calculate the band structure of $\mbox{Si}_{2}\mbox{Be}_{3}$ nanowires. They focused on nanowires with a diameter of 120 nanometers, and for these nanowires, they showed that there are two topological states for each pair $(k_{z},L)$. Of the two topological states, one has energy closer to the conduction band, while the other lies closer to the valence band. We will refer to the former state as the upper energy state and to the latter as the lower energy state, respectively. We will also focus on nanowires with a diameter of 120 nanometers for comparison purposes.
As is usual with states or eigenvalues calculated using the Rayleigh-Ritz
variational method, the larger the number of the basis set functions, the
greater the accuracy. If $\rho^{m}$ is an RDM obtained from a variational
eigenvector with $m$ basis set functions, we assess the accuracy of the
variational method by comparing two RDM with different values of $m$. The
fidelity between a succession of $\rho_{A}$ matrices obtained for one of the
two topological states with $L=0$, each matrix calculated with a different
number of basis set functions, is shown in Figure 2.
Note that, while each matrix on the succession is normalized, $\mbox{Tr}(\rho)=1$, the fidelity approaches the unity for larger basis sets. This convergence indicates that the successive matrices are more and more similar. Consequently, in what follows, we will present results obtained with the basis set with $N=40$. The convergence of the energy values of the topological states is even better. For given values of $k_{z}$ and $L$, the figures obtained using the two larger basis sets have a relative difference of less than $1\times 10^{-4}$.
The topological entropy in Equation (8) should be independent of the quantum numbers that label the topological quantum state. Figure 3a shows this property, where we plot the topological entropy of all the variational eigenstates obtained for different values of $k_{z}$ and $L$ as a function of the wave number $k_{z}$. The colored lines correspond to topological states with $L=0,1,2,3$. The topological entropy of the different topological states is constant as a function of $k_{z}$, except when its energy is close enough to a band, in which case the entropy value drops to zero.
In panel b) we show the behavior of the variational eigenvalues near the gap. The color code used to depict the energy of the topological states is the same one used in panel a) for the corresponding topological entropies. The black curves correspond to the valence and conduction bands and contain all the other eigenvalues obtained from the variational method. These are the eigenvalues of the normal non-topological states.
Comparing both panels, a) and b), it is clear that the topological entropy drops to zero when the energy of the topological state is close to the conduction or the valence bands.
The topological states reduced density matrices are difficult to calculate accurately close to $k_{z}=0$. Because of this, the topological entropy curves show a little slump near $k_{z}=0$. The difficulty increases for larger values of $L$ and, consequently, the deeper one corresponds to the larger $L$.
Finally, in Figure 3c, we show the topological entropy calculated for all the states whose eigenvalues we plotted in panel b). Again, the colored curves correspond to the topological states, while the black ones correspond to the normal non-topological ones. There are a few salient traits worth commenting on. The topological entropy of the non-topological states depends on $k_{z}$, and eventually, its value reaches up to $\log{2}$. Since we are dealing with a finite system, the length scale associated with the different eigenstates is finite and comparable with $R_{c}$, except for the topological states. So, changing the value of $R_{c}$ would change the topological entropy of a non-topological state but not the topological entropy of a topological one.
The topological entropy should also be independent of the length of the circular curve that defines the triangular sectors $A$, $B$, and $C$ or, equivalently, of the radius $R_{c}$. Figure 4 shows the topological entropy as a function of the wave number for three different values of $R_{c}$. As the Figure clearly shows, $S_{t}$ is a constant with very high precision.
IV Mode-dependent reduced density matrix, its entropy and entanglement spectrum
As the results in the previous Section show, the topological entropy is effectively a constant for the topological states up to the numerical precision. Nevertheless, this entropy is non-null for the non-topological ones. Its value depends on both quantum numbers $k_{z}$, $L$, and the radius $R_{c}$. Note that $\log(2)$ bounds from above the values that achieve $|S_{t}(\rho_{nt})|$, where $\rho_{nt}$ is the density matrix of a non-topological state. For some values of $k_{z}$, $L$ and $R_{c}$ $|S_{t}(\rho_{nt})|=log(2)$, which renders the Kitaev-Preskill entropy useless in this context to distinguish a topological state from normal ones.
In this Section, we propose a reduced-density matrix whose entropy for topological states is larger than the entropy of the normal ones. The reduced-density matrix depends on the variational coefficients and spatial integrals of the variational basis set functions.
Writing the variational eigenfunctions as
$$|\psi\rangle=\sum_{n,j}c(j,L,n)f_{j,L,n}(\rho,\varphi)e^{ik_{z}z}|j\rangle,$$
(12)
we define the mode-dependent reduced density matrix $\rho_{MD}$, with matrix elements
$$\displaystyle\left[\rho_{MD}\right]_{j,L,n;i,L^{\prime},m}=$$
$$\displaystyle\frac{1}{\mathcal{N}}c(j,L,n)(c(i,L^{\prime},m))^{*}\times$$
$$\displaystyle\int_{\Omega}f_{j,L,n}(\rho,\varphi)(f_{i,L^{\prime},m})^{*}(\rho,\varphi)\,\rho\,d\rho\,d\varphi,$$
(13)
where $\mathcal{N}$ is a normalization constant such that
$\mbox{Tr}(\rho_{MD})=1$, $\Omega$ is the ring outside $ABC$, and $\rho_{MD}$ is a $4N\times 4N$ matrix.
The mode-dependent RDM is closely related to the pure state given by
$$\rho_{p}=|\psi\rangle\langle\psi|,$$
(14)
which has matrix elements
$$\rho_{p}=c(j,l,n)(c(i,l^{\prime},m))^{*},$$
(15)
in the variational basis and is also a $4N\times 4N$ matrix.
In the next Section, we show that $\rho_{MD}$ is also given by
$$\rho_{MD}=\sum_{j}K_{j}\rho_{p}K^{\dagger}_{j},$$
(16)
i.e., $\rho_{MD}$ results from applying a quantum process to $\rho_{p}$, but before we want to analyze its spectrum and the behavior of the von Neumann entropy of $\rho_{MD}$ as a function of $k_{z}$.
Figure 5 shows a) the von Neumann entropy of the mode-dependent RDM, $(S\rho_{MD})$, and b) the spectrum of $\rho_{MD}$ as functions of $k_{z}$ and the eigenvalue number, respectively. The data in panel a) shows that the entropy of the topological states has larger values than the entropy of the normal ones, except when the energy of the topological states is too close to the bands. Moreover, up to the numerical accuracy, its value depends only on the quantum number $k_{z}$, but not depends on the angular momentum $L$.
On the other hand, Figure 5 b) shows the mode-dependent RDM spectrum obtained corresponding to all the variational eigenvectors calculated with $N=70$. Note that the vertical scale is logarithmic. The spectra of the topological states show a decaying compatible with $\lambda_{k}\sim e^{-\alpha k}$ over an ample range of values, where $\alpha$ is a constant. This behavior points to the topological character of quantum states in different physical systems. Figure 6 shows the behavior of the spectrum of the mode-dependent RDM calculated using three different basis set sizes.
The black, red, and blue curves shown in Figure 6 correspond to basis sets with $N=30$, $50$, and $70$ functions, respectively. The dashed black line is a guide to the eye. The number of eigenvalues that show an exponential decay grows with the number of basis functions used to obtain the RDM, although the increase is slow. This trait is another manifestation of the difficulties inherent to the obtention of the mode-dependent RDM or, more generally, how difficult it is to numerically calculate other quantities related to topological states beyond their spectrum. The entanglement spectrum, which is given by
$$\zeta_{k}=-\log(\lambda_{k}),$$
(17)
together with the exponential decay of $\lambda_{k}$ result in
$$\zeta_{k}=c+\alpha k.$$
(18)
The abrupt decay of the eigenvalues beyond the exponential regime, shown in Figure 6, marks when the numerical approximation becomes inaccurate, rendering all the remaining values indistinguishable from zero.
V Quantum process tomography
The results in the previous Section suggest different choices for the reduced density matrices if we wish to distinguish between topological and normal states using entropic-like quantities, at least when dealing with approximate ones obtained from phenomenological Hamiltonians.
For a given known quantum state, we could obtain the mode-dependent RDM using different basis sets, not only the particular one that we employ to implement the variational method. So, up to a point, some features of the entropy of the mode-dependent RDM should depend on the basis set chosen.
Instead of implementing our calculations in a different basis set, we prefer to show that the mode-dependent RDM is equivalent to many others by showing how to obtain it from the pure state in Equations 13 and 14 using quantum processes. Doing this has the twofold purpose of not dealing with the complicated calculations inherent to a change of functions basis set and showing that RDM obtained through operational processes leads to entropies that distinguish topological from normal states.
A quantum process takes a given RDM $\rho$ to another one, $\rho^{\prime}$ as follows
$$\rho^{\prime}=\mathcal{E}(\rho).$$
(19)
The quantum process $\mathcal{E}$ is a linear superoperator, and $\rho$ and $\rho^{\prime}$ as operators do not necessarily act on Hilbert spaces of the same dimension. The particular $\mathcal{E}$ that relates a given pair of $\rho$ and $\rho^{\prime}$ is determined using quantum process tomography, which usually is an expensive calculation.
There are numerous algorithms to calculate $\mathcal{E}$ [34, 35, 36, 37, 38, 39, 40, 41, 42, 43], which depend on the representation used for the process and the dimensionality of the involved RDM. We use the method proposed by Ahmed et al. [34], which assumes a Kraus representation for the quantum process
$$\mathcal{E}=\sum_{j}K_{j}\rho K_{j}^{\dagger},$$
(20)
and that the dimensionality of both Hilbert spaces, where $\rho$ and $\rho^{\prime}$ act, is a power of two. Ahmed et al. define a cost function that depends on the two RDM, the Kraus operators, $K_{j}$, and learn them using a gradient-descent method.
Starting with randomly chosen initial Kraus operators, the non-negative cost function $\mathcal{L}$ is minimized up to values near zero to learn the optimal Kraus operators. See Appendix B for details about the calculation. We studied two cases. For the first case, $\rho$ is the pure state in Equation (14), and $\rho^{\prime}=\rho_{ABC}$. For the second one, $\rho$ is again the pure state in Equation (14), and $\rho^{\prime}$ is the mode-dependent RDM.
The cost function is given by
$$\mathcal{L}(\mathbb{K})=\sum_{j}\left[d_{j}-\mbox{Tr}\left[\mathcal{M}_{j}\left(\sum_{l}K_{l}\rho_{p}K_{l}^{\dagger}\right)\right]\right]^{2}+\lambda||\mathbb{K}||_{1},$$
(21)
where
$$d_{j}=\mbox{Tr}(\mathcal{M}_{j}\rho^{\prime}),$$
(22)
$\mathcal{M}_{j}$ is a set of measurements, and $\mathbb{K}=\left[K_{1},K_{2},\ldots,K_{n_{k}}\right]$ is a $n_{k}M\times M^{\prime}$ matrix formed with all the $M\times M^{\prime}$ Kraus operators . The matrix norm is given by
$$||A||_{1}=max_{j}\sum_{i}|A_{ij}|,$$
(23)
and $\lambda\geq 0$ is the strength of the regularization imposed on $\mathcal{L}$. $\lambda$ is a hyperparameter of the minimization process and can be fixed or optimized. The Kraus operators must fulfill the condition
$$\sum_{l}K_{l}^{\dagger}K_{l}=\mathbb{I}.$$
(24)
As we said previously, we consider two cases $\rho^{\prime}=\rho_{ABC}$ and $\rho^{\prime}=\rho_{MD}$. In the following, we focus on the former case and will return to the latter near the end of the Section.
$\rho^{\prime}=\rho_{ABC}$ case
In Reference [34], Ahmed et al. discussed all the necessary details to implement the minimization of $\mathcal{L}$ and provided the code to reproduce their results. To obtain the results found in this Section, we adapted the code. For the case where $\rho^{\prime}=\rho_{ABC}$, the only adaptation needed arises from the different dimensions of the Hibert space where $\rho$ and $\rho^{\prime}$ act.
When the dimension of both Hilbert spaces is the same, it is reasonable to use random unitary square matrices as the initial random Kraus operators needed by the minimization algorithm. So,
$$\frac{1}{n_{k}}\sum U_{l}U_{l}^{\dagger}=1,$$
(25)
results in
$$K_{l}=\frac{1}{\sqrt{n_{k}}}U_{l}.$$
(26)
There are different numerical methods to construct square random unitary matrices.
For the case when $\rho^{\prime}=\rho_{ABC}$ is a $4\times 4$ matrix and $\rho_{p}$ is an $M\times M$matrix, with $M$ a multiple of four, the quantum process $\mathcal{E}$ requires $n_{k}$ $4\times M$ Kraus operators. Each Kraus operator is composed of $M/4$ blocks. We choose, as the initial Kraus operators, $4\times M$ matrices such that they have only one $4\times 4$ block different from zero, and this block is a random unitary matrix. If $K_{1}$ has only the first block different from zero, $K_{2}$ the second one, and so on, it is clear that the condition $Tr=1$ requires a renormalization of each random unitary matrix by a factor equal to $\frac{1}{\sqrt{n_{k}}}$.
The measurements $M_{j}$ are given in terms of the eigenvectors of all the operators of the form $\sigma^{\beta}\otimes\sigma^{\gamma}$, where $\beta=x,y,z$ and also $\gamma=x,y,z$ that is, all the operators that are the tensorial product of two Pauli matrices.
Figure 7 shows the typical behavior of the cost function as a function of the number of iterations performed by the gradient-descent minimization method. The algorithm that implements the gradient-descent minimization method has a tolerance parameter. Once the cost function value becomes smaller than the tolerance value, the algorithm does not further iterate. We used a tolerance value of $0.01$ for $n_{k}=64$. Imposing lower values for the tolerance does not necessarily improve the results obtained and, in some cases, leads to oscillations in the cost function behavior. Since the cost function value does not become zero, the predicted RDM will differ from the one employed as the target RDM, that is,
$\tilde{\rho}_{ABC}=\tilde{\mathcal{E}}(\rho_{p})\neq\rho_{ABC}$, where $\tilde{\mathcal{E}}$ is the quantum process found using a finite non-null tolerance. Other sources of errors that prevent the predicted RDM from becoming equal to the target RDM are the number of Kraus operators used, numerical precision, etc.
Figure 8 shows a comparison between the values obtained for the $S(\rho_{ABC})$ entropy and the entropy calculated with the RDM resulting from the quantum process $\tilde{\rho}_{ABC}$. In panel a), the entropy $S(\rho_{ABC})$ is shown using black solid dots, while for $S(\tilde{\rho}_{ABC})$, we use red solid dots.
It is easy to appreciate that the quantum process consistently results in RDM that gives larger entropy values than those corresponding to RDM calculated with the variational eigenvectors, as in Eq. 9. Panel b) shows the modulus of the difference between $S(\rho_{ABC})$ and $S(\tilde{\rho}_{ABC})$ using green points. The differences shown correspond to the data shown in panel a). The number next to each green dot is the fidelity $\mathcal{F}(\rho_{ABC},\tilde{\rho}_{ABC})$. Despite the excellent fidelity between both sets of reduced-density matrices, the calculated, $\rho_{ABC}$, and the predicted by the quantum process, $\tilde{\rho}_{ABC}$, the differences in their corresponding entropies can be as large as $0.047$ or a relative error of 7$\%$. Nevertheless, note that the predicted entropy is also a constant where the calculated entropy is a constant and that its value also drops to zero for the values of $k_{z}$ where the energy of the topological state becomes close enough to the band, see panel a).
$\rho^{\prime}=\rho_{MD}^{\prime}$ case
In this case, the quantum process takes $\rho_{p}$, which is $4M\times 4M$ matrix, to a predicted matrix $\rho_{MD}^{\prime}$, which is also a $4M\times 4M$ matrix. In our case, $M=64$ for the largest basis set size, resulting in $2^{8}\times 2^{8}$ matrices.
In Reference [34], the suggested number of measurements employed in Eq. 21 is $6^{n_{q}}$, where $n_{q}$ is the number of ”qubits” over which the quantum process acts. Allocating $6^{8}$ vectors, each one of $2^{8}$ components, as the method requires, becomes impractical. In our case, it is sufficient to consider only the measurements associated with the eigenvectors of the operator given by
$$\bigotimes_{i=1}^{8}\sigma_{x}^{i}.$$
(27)
The operator above has $2^{8}$ eigenvectors, a more manageable number than $6^{8}$. This adaptation is the only one needed to run the algorithm since the starting and predicted matrices have the same size, rendering all the Kraus operators square matrices.
The behavior of the cost function for this case is similar to the previous one see Figure 7. The number of steps necessary to reach the required tolerance (0.1) is higher, typically around seven hundred, and $n_{k}=30$.
Figure 9 shows in panel a) both entropies, $S(\rho_{MD})$ and $S(\tilde{\rho}_{MD})$, for several values of $k_{z}$, while panel b) shows the modulus of their difference and the fidelity between both matrices. Note that all the fidelities are better than $0.995$, and the modulus of the differences are all lower than $0.05$, which results in relative errors of less than $2\%$.
VI Discussion and conclusions
When dealing with one-particle wave functions or spinors, it is possible to trace out over only one of the coordinates and obtain a coordinate-dependent RDM, which leads to entropies (or information-like quantities) that detect transitions in quantum states of one-electron systems [44, 29].
In the case of a many-particle system, there are more possibilities since it is possible to trace over all the coordinates of a subset of particles (entanglement between particles) [22, 45, 21] or subsets of coordinates (real-space entanglement) [31, 46].
Our calculation of the topological entropy exploits the fact that the eigenvectors of the $\mathbf{k}\cdot\mathbf{p}$ Hamiltonian is a spinor with four components, $|\psi\rangle$. Using the spinor, we obtain a pure density matrix $|\psi\rangle\langle\psi|$, a coordinate-dependent $4\times 4$ matrix. Tracing out a spatial region, as in Equation 9, leads to a coordinate-independent $4\times 4$ reduced density matrix whose eigenvalues are easy to calculate.
Interestingly, for the topological states constructed as the spinors in Eq. 11, all the entropies that enter into the calculus of the topological entropy are constants independent of $k_{z}$ and $L$ except near $k_{z}=0$, that is, $S(\rho_{sp})=const.$, where $sp$ is any one of the triangular sectors $A,B$ or $C$, or a combination of them. Because all these entropies are constant, it is enough to look for a quantum process such that $\rho_{ABC}=\mathcal{E}(\rho_{p})$ to test the method.
A caveat about the ”constant” value of the topological entropy is in order since the statement is accurate as long as the radius of the triangular sectors $R_{c}$ is not too close to the radius of the nanowire, $R$, or becomes too small.
The results in Sections 3, 4, and V correspond to a nanowire 120 nanometers in diameter and a radius $R_{c}=15$ nanometers. We also ran numerous numerical tests for other diameters and values of $R_{c}$, and the results were qualitatively the same.
The entropy and entanglement spectrum obtained with the mode-dependent RDM are practical tools to distinguish topological from non-topological states, and we intend to test them in other contexts beyond the topological states in $\mbox{Bi}_{2}\mbox{Se}_{3}$ nanowires.
That the mode-dependent RDM could result from a quantum process makes us think that it is possible to define operatively an equivalent RDM, whose entropy allows us to distinguish between topological and non-topological states without resorting to a particular basis set of functions.
Acknowledgements.The authors acknowledge partial financial support from
CONICET (PIP 11220210100787CO). and SECYT-UNC.
Appendix A Variational Method
The simpler Rayleigh-Ritz variational method requires a set of appropriate basis functions that we denote as $f_{i}$. The expectation value of the Hamiltonian
$$\langle f|H|f\rangle,$$
(28)
is obtained using a test function given by
$$f=\sum_{i=1}^{N}c_{i}f_{i},$$
(29)
and the minimization is performed over the values of the $c_{i}$ coefficients. This procedure results in an algebraic problem
$$\left[H\right]\kappa_{j}=E^{v}_{j}\kappa_{j},$$
(30)
where $\left[H\right]$ is an $N\times N$ matrix whose entries are given by
$$\left[H\right]_{ij}=\left\langle f_{i}|H|f_{j}\right\rangle,$$
(31)
$\kappa_{j}$ is a vector that contains the $c$ coefficients, and the $E^{v}_{j}$ are the variational eigenvalues. $N$ is the number of functions in the basis set, and the algebraic problem has $N$ eigenvalues and their corresponding eigenvectors.
For dealing with a $\mathbf{k}\cdot\mathbf{p}$ Hamiltonian, like the one in Equation (1), the Rayleigh-Ritz method requires some adaptations. Instead of a test function, we need a test spinor to calculate the Hamiltonian expectation value. In particular, we employ the following,
$$|\psi_{L,k_{z}}\rangle=\sum_{n}\left(\!\!\!\begin{array}[]{c}{{b_{L,k_{z},n,\uparrow}A_{L,n}J_{L}\left(\alpha_{n}^{L}\rho/R\right)e^{iL\varphi}}}\\
{{c_{L,k_{z},n,\uparrow}A_{L,n}J_{L}\left(\alpha_{n}^{L}\rho/R\right)e^{iL\varphi}}}\\
{{b_{L,k_{z,n,\downarrow}}A_{L+1,n}J_{L+1,n}\left(\alpha_{n}^{L+1}\rho/R\right)e^{i(L+1)\varphi}}}\\
{{c_{L,k_{z,n,n,\downarrow}}A_{L+1,n}J_{L+1,n}\left(\alpha_{n}^{L+1}\rho/R\right)e^{i(L+1)\varphi}}}\end{array}\!\!\!\right)e^{ik_{z}.z},$$
(32)
where $J_{L}$ is the Bessel function with index $L$, $\alpha_{n}^{L}$ is its $n-th$ root, $L$ is an integer number, $R$ is the radius of the cylinder, $b_{L,k_{z,n,\uparrow}},c_{L,k_{z,n,\uparrow}},b_{L+1,k_{z,n,\downarrow}},c_{L+1,k_{z,n,\downarrow}}$ are the coefficients of the expansion (i.e., the linear variational parameters), and
$$A_{L,n}=\frac{1}{\sqrt{\pi}RJ_{L+1}(\alpha_{n}^{L})},$$
(33)
is a normalization constant.
With the provisos mentioned in the paragraph above, the resulting algebraic problem for the $\mathbf{k}\cdot\mathbf{p}$ Hamiltonian has $4N$ eigenvalues and their corresponding eigenvectors.
Finally, to calculate the matrix elements necessary for the Rayleigh-Ritz method, we write the operators $k_{\parallel}$ and $k_{\pm}$ as differential operators in cylindrical coordinates using that in Cartesian ones
$$\displaystyle k_{x}=-i\partial_{x}$$
(34)
$$\displaystyle k_{y}=-i\partial_{y}$$
(35)
$$\displaystyle k_{z}=k_{z}.$$
(36)
The change to cylindrical coordinates leads to
$$\displaystyle\partial_{x}=cos(\varphi)\partial_{\rho}-\frac{sin(\varphi)}{\rho}\partial_{\varphi},$$
(37)
$$\displaystyle\partial_{y}=sin(\varphi)\partial_{\rho}-\frac{cos(\varphi)}{\rho}\partial_{\varphi},$$
(38)
$$\displaystyle\partial^{2}_{x}=cos^{2}(\varphi)\partial^{2}_{\rho}+\frac{sin^{2}(\varphi)}{\rho}\partial_{\rho}+\frac{sin(2\varphi)}{\rho^{2}}\partial_{\varphi}+\frac{sin^{2}(\varphi)}{\rho^{2}}\partial^{2}_{\varphi}$$
(39)
$$\displaystyle\partial^{2}_{y}=sin^{2}(\varphi)\partial^{2}_{\rho}+\frac{cos^{2}(\varphi)}{\rho}\partial_{\rho}-\frac{sin(2\varphi)}{\rho^{2}}\partial_{\varphi}+\frac{cos^{2}(\varphi)}{\rho^{2}}\partial^{2}_{\varphi},$$
(40)
which results in
$$\displaystyle k_{\pm}=k_{x}\pm ik_{y}=e^{\pm i\varphi}\left(-i\left(\partial_{\rho}\pm\frac{i}{\rho}\partial_{\varphi}\right)\right),$$
(41)
$$\displaystyle k_{\parallel}=k_{x}^{2}+k_{y}^{2}=-\left(\partial^{2}_{\rho}+\frac{1}{\rho}\partial_{\rho}+\frac{1}{\rho^{2}}\partial^{2}_{\varphi}\right).$$
(42)
References
[1]
Xiao-Liang Qi and Shou-Cheng Zhang,
Rev. Mod. Phys. 83, 1057 (2011).
[2]
Kane C L and Mele E J, Phys. Rev. Lett. 95, 226801 (2005).
[3]
Bernevig B A and Zhang S-C, Phys. Rev. Lett. 96, 106802 (2006).
[4]
Shuichi Murakami, New J. Phys. 13, 105007 (2011).
[5]
Masatoshi Sato and Yoichi Ando, Rep. Prog. Phys. 80, 076501 (2017).
[6]
Kezilebieke, S., Huda, M.N., Vaňo, V. et al.. Nature 588, 424–428 (2020).
[7]
Chao-Xing Liu, Xiao-Liang Qi, HaiJun Zhang, Xi Dai, Zhong Fang, and Shou-Cheng Zhang
Phys. Rev. B 82, 045122 (2010)
[8]
Zhang, H., Liu, CX., Qi, XL. et al., Nature Phys 5, 438–442 (2009)
[9]
Liang Fu and C. L. Kane, Phys. Rev. B 76, 045302 (2007)
[10]
Wen-Kai Lou,
Fang Cheng,
Jun Li, Journal of Applied Physics 110, 093714 (2011).
[11]
M. Governale et al , New J. Phys. 22, 063042 (2020)
[12]
Iorio P, Perroni C A and Cataudella V, Eur. Phys. J. B 89, 97 (2016).
[13]
Linder J, Yokoyama T and Sudbø A, Phys. Rev. B 80, 205401 (2009)
[14]
L. Gioia, M. G. Christie, U. Zülicke, M. Governale, and A. J. Sneyd, Phys. Rev. B 100, 205417 (2019).
[15]
A. Kitaev y J. Preskill, Phys. Rev. Lett. 96, 110404 (2006).
[16]
Hui Li and F. D. M. Haldane, Phys. Rev. Lett. 101, 010504 (2008).
[17]
Lukasz Fidkowski, Phys. Rev. Lett. 104, 130502 (2010).
[18]
Xiao-Liang Qi, Hosho Katsura, and Andreas W. W. Ludwig
Phys. Rev. Lett. 108, 196402 (2012).
[19]
P. Calabrese and A. Lefevre
Phys. Rev. A 78, 032329 (2008).
[20]
Hong Yao and Xiao-Liang Qi
Phys. Rev. Lett. 105, 080501 (2010).
[21]
M. Haque, O. Zozulya, and K. Schoutens,
Phys. Rev. Lett. 98, 060401 (2007).
[22]
O. S. Zozulya, M. Haque, K. Schoutens, and E. H.
Rezayi, Phys. Rev. B 76, 125310 (2007).
[23]
V. V. Ravi Kishore, B. Partoens and F. M. Peeters, J. Phys.: Condens. Matter 24 135302 (2012).
[24]
V. V. Ravi Kishore, B. Partoens and F. M. Peeters, J. Phys.: Condens. Matter, 26, 095501 (2014).
[25]
S. S. Krishtopenko and F. Teppe, Phys. Rev. B 97, 165408 (2018).
[26]
S. S. Krishtopenko, W. Knap and F. Teppe, Scientific
Reports 6, 30755 (2016).
[27]
R. Skolasinski, D. I. Pikulin, J. Alicea, and
M. Wimmer, Phys. Rev. B 98, 201404(R) (2018).
[28]
Zewei Chen and Tai Kai Ng, Phys. Rev. B 99, 235157
(2019).
[29]
N. Giovenale and O. Osenda
Physica B 627, 413564 (2022)
[30]
N. Giovenale and O. Osenda
Physica E, 144, 115406 (2022)
[31]
A. Sterdyniak, A. Chandran, N. Regnault, B. A. Bernevig,
and Parsa Bonderson, Phys. Rev. B 85, 125308 (2012).
[32]
E. C. G. Sudarshan, P. M. Mathews, and Jayaseetha Rau, Phys. Rev. 121, 920 (1961).
[33]
Michael A Nielsen and Isaac L. Chuang, Quantum Computation and Quantum Information (10th ed.). Cambridge: Cambridge University Press. (2010).
[34]
S. Ahmed, F. Quijandría y A. F. Kockum, Phys. Rev. Lett. 130, 150402 (2023).
[35]
J. Fiurášek and Z. Hradil, Phys. Rev. A 63, 020101(R) (2001).
[36]
M. F. Sacchi, Phys. Rev. A 63, 054104
(2001).
[37]
A. Anis and A. I. Lvovsky, New J. Phys. 14,
105021 (2012).
[38]
K. Schultz, Phys. Rev. A 100, 062316 (2019).
[39]
G. C. Knee, E. Bolduc, J. Leach, and E. M. Gauger, Phys. Rev. A 98,
062336 (2018).
[40]
T. Surawy-Stepney, J. Kahn, R. Kueng, and M. Guta,
Quantum 6, 844 (2022).
[41]
C. H. Baldwin, A. Kalev, and I. H. Deutsch,
Phys. Rev. A 90, 012110 (2014).
[42]
Y. S. Teo, G. I. Struchalin, E. V. Kovlakov, D. Ahn, H.
Jeong, S. S. Straupe, S. P. Kulik, G. Leuchs, and L. L.
Sánchez-Soto, , Phys. Rev. A 101, 022334 (2020).
[43]
S. Xue, Y. Liu, Y. Wang, P. Zhu, C. Guo, and J. Wu,
Phys. Rev. A 105, 032427 (2022).
[44]
M. Garagiola and O. Osenda, Physica E 116, 113755 (2020)
[45]
O. Osenda, F. M Pont, A. Okopińska and P. Serra, J. Phys. A: Math. Theor. 48, 485301 (2015)
[46]
I. D. Rodríguez and G. Sierra, Phys. Rev. B 80, 153303 (2009) |
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