Neutron–neutron scattering length from the reaction γd → π+nn
employing chiral perturbation theory
V. Lenskya, V. Barub, E. Epelbauma,c, C. Hanharta, J. Haidenbauera,
A. Kudryavtsevb, Ulf-G. Meißnerc,a
a Institut für Kernphysik, Forschungszentrum Jülich GmbH,
D–52425 Jülich, Germany
b Institute of Theoretical and Experimental Physics,
117259, B. Cheremushkinskaya 25, Moscow, Russia
c Helmholtz-Institut für Strahlen- und Kernphysik (Theorie),
Universität Bonn, Nußallee 14-16, D–53115 Bonn, Germany
November 4, 2018
Abstract
We discuss the possibility to extract the neutron-neutron scattering length ann from experi-
mental spectra on the reaction γd → π+nn. The transition operator is calculated to high accuracy
from chiral perturbation theory. We argue that for properly chosen kinematics, the theoretical
uncertainty of the method can be as low as 0.1 fm.
FZJ-IKP-TH-2007-13, HISKP-TH-07/12
1 Introduction
A precise knowledge of the neutron-neutron scattering length ann is, e.g., important for an understand-
ing of the effects of charge symmetry breaking in nucleon–nucleon forces [1]. The scattering length
ann characterizes scattering at low energies. It is related to the on–shell
1S0 scattering amplitude f
fon(pr) =
pr cot δ(pr)− ipr
−a−1nn + 12 rnnp2r +O(p4r)− i pr
, (1)
where pr is the relative momentum between the two neutrons, δ(pr) the scattering phase shift in the
1S0 partial wave and rnn is the effective range. At low energies the terms of order p
r can be neglected
to very high accuracy. Obviously, a direct determination of ann in a scattering experiment is extremely
difficult due to the absence of a free neutron target. For this reason, the value for ann is to be obtained
from analyses of reactions where there are three particles in the final state, e.g. π−d → γnn [2, 3, 4]
or nd → pnn [5, 6, 7]. There is some spread in the results for ann obtained by the various groups. In
particular, two independent analyses of the reaction nd → pnn give significantly different values for
ann, namely ann = −16.1± 0.4 fm [6] and ann = −18.7± 0.6 fm [7], whereas the latest value obtained
from the reaction π−d → γnn is ann = −18.5 ± 0.3 fm [4]. At the same time, for the proton-proton
scattering length, which is directly accessible, a very recent analysis reports app = −17.3 ± 0.4 fm
[8] after correcting for electromagnetic effects. This means that even the sign of ∆a = app − ann is
http://arxiv.org/abs/0704.0443v1
not fixed.1 It should be mentioned, however, that state of the art calculations for the binding energy
difference of tritium and 3He suggest that ∆a > 0 [9, 10].
In the present work we discuss the possibility to determine ann from differential cross sections in
the reaction γd → π+nn. Specifically, we show that one can extract the value of ann reliably by fitting
the shape of a properly chosen momentum spectrum. In this case the main source of inaccuracies,
caused by uncertainties in the single–nucleon photoproduction multipole E0+, is largely suppressed.
Furthermore there is a suppression of the quasi-free pion production at specific angles. We show that
at these angular configurations the extraction of ann can be done with minimal theoretical uncertainty.
Our investigation is based on the recent work of Ref. [11] in which the transition operator for
the reaction γd → π+nn was calculated up to order χ5/2 in chiral perturbation theory (ChPT) with
χ = mπ/MN ≃ 1/7, where mπ (MN ) is the pion (nucleon) mass. Half-integer powers of χ in the
expansion arise from the unitarity (two– and three–body) cuts (see also [12]). The results of Ref. [11]
for the total cross section are in very good agreement with the experimental data. The only input
parameter that entered the calculation was the leading single–nucleon photoproduction multipole E0+,
which was fixed from a fourth-order one-loop calculation of Bernard et al. [13]. The uncertainty in
E0+ is the main theoretical error in the calculation presented in Ref. [11]. Besides this transition
operator, in the present study we use nucleon–nucleon (NN) wave functions constructed likewise
in the framework of ChPT, namely those of the NNLO interaction of Ref. [14]. This allows us to
estimate the theoretical uncertainty which arises from variations in the wave functions. In fact, as
soon as we include consistently all terms up to order χ5/2, we expect the ambiguities due to different
wave functions not to be larger than a χ3 correction, for only at this order the leading counter term
which absorbs these effects enters. This expectation is indeed quantitatively confirmed in the concrete
calculations.
Since we work within chiral perturbation theory we can estimate the effect of higher orders in
terms of established expansion parameters together with the standard assumption that additional
short ranged operators, that enter at higher orders, behave in accordance with the power counting
(the so-called naturalness assumption). This method was also applied in Refs. [15, 16], where the
reaction π−d → γnn was investigated as a tool to extract ann. However, to know the effect of higher
orders for sure, one has to calculate them. Therefore, to derive a reliable uncertainty estimate for the
extraction of ann from the γd reaction, we use our leading order calculation as baseline result and
estimate the theoretical uncertainty from the effects of the higher orders that we calculated completely.
Based on this, we find a theoretical uncertainty δann . 0.1 fm. We therefore argue that the reaction
γd → π+nn appears to be a good tool for the extraction of ann.
To end this section, we remark that in Ref. [17] a method was proposed to extract scattering
lengths from γd induced meson production. However, this approach should not be used here, since
the momentum transfer is not sufficiently large to use this method and with our explicit calculation
of the transition operator we can reach a significantly higher accuracy.
2 ChPT calculation for γd → π+nn
The diagrams that contribute to the reaction γd → π+nn are shown on Fig. 1. The kinematical
variables are defined in Fig. 2.
Before going into the details some comments are necessary regarding the relevant scales of the
problem. In the near threshold regime of interest here (excess energies of at most 20 MeV above the
1Note, that, in contrast to app, ann is not corrected for electromagnetic effects. However, since those are only of the
order of 0.3 fm [1] they are not relevant for the sign of ∆a. But they ought to be taken into account for determining
charge symmetry breaking effects quantitatively.
χ , χ , χ χ , χ0 1 2 2 5/2
(d2)(b2)
(b1) (c1) (d1)
Figure 1: Diagrams for γd → π+nn. Shown are one–body terms (diagram (a) and (b) ), as well as the
corresponding rescattering contribution (c)—all without and with final state interaction. Diagrams
(d) shows the class of diagrams with intermediate NN interaction. Solid, wavy, and dashed lines
denote nucleons, photons and pions, in order. Filled squares and ellipses stand for the various vertices
(see Ref. [11] for the details), the hatched area shows the deuteron wave function and the filled circle
denotes the nn scattering amplitude. Crossed terms (where the external lines are interchanged) are
not shown explicitly.
pion production threshold) the outgoing pion momenta are small compared even to the pion mass.
Thus, in addition to the conventional expansion parameters of ChPT mπ/Λχ and qγ/Λχ, where Λχ
denotes the chiral symmetry breaking scale of order of (and often identified with) the nucleon mass,
and qγ denotes the photon momentum in the center–of–mass system which is of order of the pion
mass, we can also regard kπ/mπ as small, where kπ denotes the momentum of the outgoing pion. In
what follows we will perform an expansion in two parameters, namely
χm = mπ/MN and χQ = kπ/mπ .
Obviously, the value of the second parameter depends on the excess energy Q. The energy regime
of interest to us corresponds to excess energies up to 20 MeV. The maximum value of χQ, χ
2Q/mπ, at the highest energy considered is thus about 1/2. Since this is numerically close to
we use the following assignment for the expansion parameter:
χ ∼ χm ∼ χ2Q . (2)
The tree level γp → π+n vertex, as it appears in diagrams (a1) and (a2) in Fig 1 (the vertex is labeled
as filled square), contributes at leading order (order χ0), and orders χ1 and χ2, depending on the
one–body operator used. Note that the loop diagrams with πN rescattering (see diagrams (b), (c)
and (d) in Fig. 1) contribute at order χ2m as well as at χ
mχQ, χ
m and at χ
Q. The origin of the
non–integer power of χ are the two–body (πN) and three–body (πNN) singularities. Thus, all terms
up to χ5/2 are explicitly taken into account in our calculation of the transition operator.
As already emphasized, we employ wave functions evaluated in the same framework in order to
have a fully consistent calculation. In our work, we use the N2LO wave functions corresponding to
the chiral NN forces introduced in Ref. [18] and based on the spectral function regularization (SFR)
scheme [19]. At this order, the NN force receives contributions from one-pion exchange, two-pion
p−q 2
Figure 2: Kinematical variables for γd → π+nn. The relative neutron–neutron momentum is defined
as ~pr =
(~p1 − ~p2).
exchange at the subleading order as well as from all possible short-range contact interactions with up
to two derivatives. In addition, the dominant isospin-breaking correction due to the charged-to-neutral
pion mass difference in the one-pion exchange potential together with the two leading isospin-breaking
S-wave contact interactions were taken into account [18]. The two corresponding low-energy constants
were adjusted to reproduce the scattering lengths ann and app. The SFR cutoff Λ̃ is varied in the range
500 . . . 700 MeV. It was argued in Ref. [19] that such a choice for Λ̃ provides a natural separation of
the long- and short-range parts of the nuclear force and allows to improve the convergence of the chiral
expansion [19]. The cutoff Λ in the Lippmann-Schwinger equation is varied in the range 450 . . . 600
MeV. For an extensive discussion on the choice of Λ and Λ̃ the reader is referred to [14, 18].
3 Differential cross sections: relevant features
In this section we outline the features of the differential cross section for unpolarized particles that are
important for our considerations. For later convenience let us consider the function F proportional to
the square of the matrix element as well as the five–fold differential cross section
F (pr, θr, φr, θπ, φπ) = C pr kπ(pr)|M(pr, θr, φr, θπ, φπ)|2 ∝
d5σ(pr, θr, φr, θπ, φπ)
dΩ~prdΩ~kπ
, (3)
where ~pr (~kπ) stands for the relative momentum of the two final neutrons (momentum of the final
pion) in the center–of–mass frame, θr, φr (θπ, φπ) for the corresponding polar and azimuthal angles,
respectively, and |M|2 for the squared and averaged amplitude. In Eq. (3) C is an irrelevant dimen-
sionful constant. In what follows we will consider only shapes of cross sections and therefore the value
of C is not important for our considerations. The value of kπ at given pr and excess energy Q is fixed
by energy conservation:
, (4)
hence we write kπ(pr) in Eq. (3).
In the following we choose the momentum ~qγ of the initial photon to be along the z–axis. Then
the cross sections at a certain excess energy Q depend on four variables, namely the magnitude of
the relative momentum of the two final neutrons pr, the polar angles of the vectors ~pr and ~kπ, and
the difference between the azimuthal angles of those two momenta. Unpolarized cross sections are
invariant under rotations around the beam axis, which makes the dependence on the missing angle
trivial.
Typical differential cross sections F are shown in Fig. 3 as a function of pr at some fixed set of
angles {φr, θπ, φπ} and Q = 5 MeV for two different values of θr. One can see from this figure that
for the differential cross section F of Eq. (3) there are two characteristic regions:
1. The region of quasi-free production (QF) at large pr, which corresponds to the dominance of those
diagrams of Fig. 1 that do not contain the NN interaction in the final or intermediate states.
In the Appendix we give explicit expressions for the diagram a1 – the most significant diagram
of this type. At large pr the pion momentum kπ is small (see Eq. (4)) and the arguments of the
deuteron wave function in Eqs. (A.1) and (A.2) may become small for particular combinations
of ± ~pr and ~qγ/2. This feature gives rise to a peak in the differential cross section at large pr.
2. The region with prominence of the strong nn final–state interaction (FSI) at small pr (in fact,
we would have the strongest final state interaction at zero relative momentum, however the cross
section goes to zero at pr = 0 due to the phase space, therefore we see a peak shape).
One can see from Fig. 3 that the FSI peak depends on the value of θr only marginally, whereas the
quasi-free peak shows significant dependence on this angle. In particular, the quasi-free production
is largely suppressed at θr = 90
◦ — at this angle the arguments of the wave functions in both terms
in the r.h.s. of Eqs. (A.1) and (A.2) are large. It can also be seen from Fig. 3 (right panel) that
the effect of higher orders is more important for the quasi-free production amplitude — the influence
of higher-order effects on the FSI production is quite small. Another interesting observation is that
the contributions of higher orders change the relative height of the two peaks – the FSI peak goes up
whereas the QF peak goes down when we proceed from the LO calculation to the order χ5/2. In order
to suppress the distortions of the spectrum due to higher orders in the chiral expansion, which is the
condition for an extraction of ann with small theoretical uncertainty, configurations should be chosen
where θr = 90
We now briefly discuss the dependence of the cross section on the remaining angles θπ, φπ (we always
may choose φr to be zero). The dependence on θπ is illustrated in Fig. 4. One can see from this figure
that the dependence on θπ is significant for both the quasi–free as well as the FSI peak. This can be
easily understood from the explicit expressions for the matrix elements given in the Appendix keeping
in mind that already at Q = 5 MeV the maximal value of kπ is about mπ/3 while qγ ≈ mπ. Thus,
the momentum transfer to the nucleon pair, |~qγ −~kπ|, varies in the range 2mπ/3 to 4mπ/3 depending
on θπ. Since the S-wave deuteron wave function is large only for very small arguments, the influence
of the direction of ~kπ is significant. In addition, from Fig. 4 it follows that a variation of θπ not only
changes the magnitude but also the shape of the cross section, even in the FSI region. This has to be
taken into account in the analysis of any experiment.
In contrast to the polar angles, the dependence of F on φπ is negligible for all configurations (there
is no dependence at all for θr = 0
◦ and at θr = 90
◦, only the anyway small QF contribution changes
by just 5 %).
4 Extraction of ann and estimate of the theoretical uncertainty
In this section we discuss how to extract the scattering length from future data on γd → π+nn as well
as the resulting theoretical uncertainty. Our focus is especially the latter point. As in the previous
section we will only discuss results at excess energy Q = 5 MeV. However, the analysis can be repeated
analogously at any excess energy within the range of applicability of the formalism, i.e. Q ≤ 20 MeV.
We are interested in extracting the value of ann, which, in turn, is a low-energy characteristic of
neutron-neutron scattering and manifests itself in the momentum dependence of the cross section at
small values of the momentum pr. The influence of the value of ann on the cross section is illustrated in
0 10 20 30 40 50 60 70
 [MeV]
0 10 20 30 40 50 60 70
 [MeV]
.] FSI
QF 0 degrees
QF 90 degrees
Figure 3: Left panel: Differential cross section. The solid line corresponds to the configuration
when the quasi–free peak is suppressed (θr = 90
◦), whereas the dashed line corresponds to one of
the configurations when the quasi–free production amplitude is maximal (θr = 0
◦). The values of
the remaining angles are θπ = 135
◦, φr = φπ = 0
◦; they are the same for both curves. Right panel:
Differential cross section — relative strength of QF and FSI peaks. Here the dashed curves correspond
to the calculation at LO, the solid ones to the calculation at χ5/2. Curves denoted by ”FSI” (”QF”)
are obtained by retaining only those diagrams of Fig. 1 that contain (do not contain) the final or
the intermediate nucleon–nucleon interaction. The labels ”0 degrees” and ”90 degrees” denote the
corresponding values of θr for the ”QF” curves whereas the ”FSI” curves are almost insensitive to this
angle. The values or the remaining angles are as on the left panel of this figure. The overall scale is
arbitrary in both panels but the relative normalization is the same for all curves.
Fig. 5, where the cross sections are shown for three different values of ann, namely −18, −19, −20 fm.
For each value there are two curves, the dashed one corresponds to θr = 0
◦, and the solid one to
θr = 90
◦. One can see from Fig. 5 that the influence of different values of ann is significant in the FSI
peak and marginal in the quasi–free peak, as one would have expected.
In the previous section we have shown (see right panel in Fig. 3) that the relative height of the quasi-
free and the FSI peak changes if the effects of higher orders are included in the cross sections. Therefore
those angular configurations are to be preferred, where the quasi-free production is suppressed.
The central point of this study is to demonstrate that there is a large sensitivity of the momentum
spectra to the scattering length and that this scattering length can be extracted with a small and
controlled theoretical uncertainty. As outlined in the Introduction, we can estimate this uncertainty
reliably, because the effect of the higher orders up to χ5/2 are calculated completely. In order to
demonstrate the effect of those higher orders on the shape of the momentum distribution, in Fig. 6
we show as the light band the spread in the results for the calculation from LO to χ5/2. The results
also include higher partial waves for the pion as well as the final nn system.
There is some sensitivity to the behavior of the deuteron wave function at short distances. For the
reaction π−d → γnn this sensitivity was identified as the largest effect at N3LO in Ref. [15] 2. Guided
by that observation we include in the uncertainty estimate also the spread in the results due to the
use of different wave functions. In order to remove the effect of the change in normalization when,
e.g., changing the chiral order, all curves are normalized at pnorm = 30 MeV in Fig. 6. In the same
2Within the framework of ChPT with a consistent power counting scheme, the quantitative impact of the wave-
function dependence is governed by the order at which a counter term appears that can absorb this model dependence.
The corresponding counter term for the γd as well as the πd reaction arises at N3LO.
0 10 20 30 40 50 60 70
 [MeV]
0 10 20 30 40 50 60 70
 [MeV]
Figure 4: Dependence of the differential cross section on θπ. The left panel corresponds to the
suppressed quasi–free amplitude (θr = 90
◦), the right panel to the maximal quasi–free amplitude
(θr = 0
◦). Solid, dashed, dotted, and dash-dotted lines correspond to θπ = 0
◦, 45◦, 90◦, 135◦
respectively. The values of the remaining angles are φr = φπ = 0
◦. The overall scale is arbitrary in
both panels but the relative normalization is the same for all curves.
Figure (with the same normalization) we also show the change in the shape that comes from different
values of the scattering length: the dark band is generated by a variation of ann by ±1 fm around
the central value of −18.9 fm. Clearly, the theoretical uncertainty is negligibly small compared to the
signal of interest.
One way to quantify the theoretical uncertainty is through the use of the function S, defined as
S(ann,Φ) =
F (pr|a(0)nn ,Φ(0))−N(ann,Φ) F (pr|ann,Φ)
w(pr) , (5)
where pmax =
MNQ is the maximum value of pr, F (pr|ann,Φ) is proportional to the five-fold
differential cross section as defined in Eq. (3). In the latter we refrained from showing the angular
dependence in favor of the parametric dependence of the cross section on the nn scattering length ann
as well as the multi–index Φ, which symbolizes the dependence of the cross section on the chosen chiral
order and the wave functions used, as outlined above. The weight function w(pr) was introduced to
allow us to suppress particular regions of momenta in the analysis — the role of w(pr) will be discussed
in detail below. For simplicity we may assume that S is dimensionless; all dimensions can be absorbed
into the constant C defined in Eq. (3).
The value a
nn denotes the central value of the scattering length (−18.9 fm) for which we perform
the estimate of the theoretical uncertainty3 whereas Φ(0) corresponds to the baseline type of calcu-
lation, namely leading order with chiral wave functions as specified in the Appendix. The relative
normalization N(ann,Φ) is fixed by demanding that S gets minimized for any given pair of parameters
ann,Φ (∂S/∂N = 0). This gives
N(ann,Φ) =
dpr F (pr|a
nn ,Φ
(0))F (pr|ann,Φ)w(pr)
dpr F 2(pr|ann,Φ)w(pr)
. (6)
3Note that the theoretical uncertainty practically does not change when the central value of the scattering length
varies in the relevant interval ±1 fm.
0 10 20 30 40 50 60 70
 [MeV]
-20 fm
-19 fm
-18 fm
Figure 5: The effect of varying the value of ann on the differential cross section. The solid and dashed
lines correspond to the same angular configurations as in Fig. 3, left panel. The different values of
ann are shown on the figure. The overall scale is arbitrary but the relative normalization is the same
for all curves.
Obviously S is the continuum version of the standard χ2 sum, i.e. it characterizes the mean-square
deviation from the baseline cross section F (pr|a
nn ,Φ
(0)). In this way we determine the theoretical
uncertainty in full analogy to the standard method of data analysis.
In order to quantify the theoretical uncertainty we may define Φmax as that chiral order and choice
of wave function, where S(a(0)nn ,Φmax) gets maximal:
S(a(0)nn ,Φmax) = max
S(a(0)nn ,Φ)
. (7)
Therefore S(a(0)nn ,Φmax) provides an integral measure of the theoretical uncertainty of the differential
cross section. Demanding that the effect of a change in the scattering length by the amount ∆ann
matches that by the inclusion of higher orders etc., we can identify ∆ann as an uncertainty in the
scattering length. Expressed in terms of S, we may define ∆ann via
S(a(0)nn +∆ann,Φ(0)) = S(a(0)nn ,Φmax) . (8)
This relation is illustrated in Fig. 7. The dashed horizontal line corresponds to S(a(0)nn ,Φmax), where
we use w(pr) = 1. The dashed parabolic line shows the corresponding S(a
nn + ∆ann,Φ
(0)) as a
function of ∆ann. The calculation is performed for θr = 90
◦, and θπ = 0
◦. The crossing point of the
curves corresponds to ∆ann = 0.16 fm, which can be identified as the theoretical uncertainty for the
extraction of the scattering length.
In the previous section we showed that the signal region is located at momenta lower than 30 MeV.
On the other hand, the theoretical uncertainty of the differential cross section is largest for large values
of pr due to the onset of the quasi–free contribution. In view of these two facts it seems reasonable
to use such weight functions w(pr) that suppress the contribution of large momenta. For instance, we
may use w(pr) = Θ(p
cut−pr) for the weight function. If we choose, e.g., pcut = 30 MeV the theoretical
uncertainty of the extraction of the scattering length reduces to 0.07 fm, as is demonstrated by the
solid lines in Fig. 7. This figure nicely illustrates that the parabolic curve that represents the signal
changes only very little when a restriction to small values of pr is applied. At the same time this
procedure significantly reduces the value of the uncertainty S(a(0)nn ,Φmax).
0 10 20 30 40 50 60 70
 [MeV]
Figure 6: The light (white) band is the error band, and dark (blue) band correspond to ±1 fm shift
in the scattering length from the central value −18.9 fm.
The observation that the dependence of S(a(0)nn +∆ann,Φ(0)) on ∆ann is very well approximated by
a parabola allows for a more systematic study of the pcut dependence of the theoretical uncertainty.
We therefore define
α(pcut) =
S(a(0)nn +∆ann,Φ(0)|pcut)
(∆ann)2
, (9)
where the explicit pcut dependence is introduced into the function S through the weight function w as
explained above. The dashed and the solid parabola in Fig. 7 can then be written as α(pcut) (∆ann)
with α(pmax) = 41 fm
−2 and α(30 MeV) = 33 fm−2. In the left panel of Fig. 8 we show α(pcut) as
the solid line. In the same panel the dashed line represents the measure of the theoretical uncertainty
given by S(a(0)nn ,Φmax|pcut), multiplied by a factor of 40. This figure makes more quantitative the
statement made above: for very small values of pcut we cut into the signal region and therefore α
shows a very rapid variation. However, as soon as pcut is larger than 30 MeV it goes to a plateau (in
the figure indicated by the arrow). On the other hand, the theoretical uncertainty is monotonously
growing once pcut is larger than 30 MeV. From this figure we deduce that the ideal value for pcut
is between 25 and 40 MeV. This translates into a theoretical uncertainty between 0.05 and 0.1 fm,
as illustrated in the right panel of the same figure. The value of θπ also has some impact on the
theoretical uncertainty, however, in its whole parameter range the estimated uncertainty stays below
0.1 fm for pcut = 30 MeV.
Clearly, also the experimental data, once they exist, should be analyzed using a procedure analogous
to the one given above. This means that the scattering length is to be extracted from a χ2 fit of the
theoretical curves to the data. In this work we used the calculation at LO as baseline result and the
results at higher orders to estimate the theoretical uncertainty. Consequently, we propose to use the
momentum spectrum calculated at LO in the fitting procedure of the experiment. The corresponding
analytical expressions are given in the Appendix. The only parameter to be adjusted besides the
scattering length is the overall normalization. In this fitting procedure only those data points should
be included that are below a given pcut, in order to keep the theoretical uncertainty small.
-0.2 -0.1 0 0.1 0.2
 [fm]
Figure 7: The functions S(a(0)nn ,Φmax) and S(a
nn + ∆ann,Φ
(0)) are shown by the horizontal and
parabolic curves, respectively. The solid curves are obtained by adding the weight factor in Eq. (5)
that cuts all momenta above 30 MeV in distinction from the dashed ones. The calculation is performed
for the scattering length a
nn = −18.9 fm, θr = 90◦, and θπ = 0◦. The value of ∆ann corresponding
to the crossing point of the horizontal and parabolic curves determines the theoretical uncertainty of
the calculation.
5 Discussion and conclusions
In the previous section it was shown that for the angular configurations that suppress the quasi-free
production the inclusion of higher order effects (NLO, N2LO, and χ5/2) as well as the use of different
wave functions leads only to a minor change in the momentum dependence of the five-fold differential
cross sections.
Based on this observation we propose to use the momentum spectrum calculated at LO for the
extraction of the neutron–neutron scattering length from the data. This procedure has the advantage
that the corresponding matrix elements can be given in an analytic form (see Appendix) that could
be used directly in the Monte Carlo codes for the experiment analysis. In this way the non–trivial
dependence of the spectra on θπ, discussed above, can be easily controlled. The scattering length can
then be extracted by a two parameter fit to the data where, simultaneously to a variation in ann, the
normalization constant needs to be adjusted.
Note that the leading order calculation basically agrees to the expression given in Ref. [20] long
ago. However, a systematic and reliable study of the theoretical uncertainties of the extraction was
possible only within our full calculation up to order χ5/2 in ChPT. In this way we could show that
the reaction γd → π+nn is very well suited for a determination of the nn scattering length. The
theoretical uncertainty of order 0.1 fm for the extracted scattering length, estimated in this paper, is
of the same order as that claimed for π−d → γnn [16] and nd → pnn [4, 7].
We discussed in detail the theoretical uncertainty for a fixed excess energy of Q = 5 MeV only,
however, it should be clear that the procedure can be easily repeated for any energy within the range of
applicability of our approach (Q ≤ 20 MeV). For example, we checked that the theoretical uncertainty
stays below 0.1 fm also at Q = 10 MeV. Note that the number of events in the signal region scales
roughly with
Q, the phase space available for the pion. It remains to be seen which energy is the
best for the corresponding experiment.
We showed that for a proper choice of both kinematics and weight function w, the theoretical
0 10 20 30 40 50 60 70
 [MeV]
0 10 20 30 40 50 60 70
 [MeV]
Figure 8: Left panel: Comparison of the pcut dependence of functions S(a(0)nn ,Φmax|pcut) (dashed curve)
and α(pcut) (solid curve). The calculation is performed for the scattering length a
nn = −18.9 fm,
θr = 90
◦, and θπ = 0
◦. Right panel: The corresponding theoretical uncertainty ∆ann as a function of
pcut.
uncertainty for the extraction of the neutron–neutron scattering length from γd → π+nn can be as
low as 0.1 fm. It should be stressed, however, that this error was evaluated most conservatively – we
use our LO calculation as baseline result and estimate the theoretical uncertainty from the effects of
the higher orders that we calculated completely. This error can be significantly reduced by further
studies. For example, if we include in the uncertainty estimate only the spread in the results due to
the use of different wave functions, which is identified as the largest effect at N3LO for the reaction
π−d → γnn [15], the theoretical uncertainty of the extracted scattering length reduces by one order
of magnitude. This indicates that the theoretical uncertainty is indeed under control. However to
put this N3LO estimation on more solid ground a complete calculation should be performed to this
order. Most of the operators that are relevant at this order are the same as those of π−d → γnn, given
explicitly in Ref. [21]. One counter term enters, which can be fixed from other processes [16], e.g.,
from nd scattering [22], the reaction NN → NNπ [23], or from weak decays [16]. Once this is done
we may use our calculation to order χ5/2 as baseline result and estimate the theoretical uncertainty
from the then available N3LO calculation.
Although we have identified the angles θr = 90
◦ as the preferred kinematics, also other configura-
tions could be studied in order to control the systematics. However, then the spectra calculated at
χ5/2 should be used in the analysis.
Acknowledgments
We thank A. Bernstein for useful discussions and interest in this work. We also thank D. R. Phillips
and A. G̊ardestig for helpful discussions. This research is part of the EU Integrated Infrastructure
Initiative Hadron Physics Project under contract number RII3-CT-2004-506078, and was supported
also by the DFG-RFBR grant no. 05-02-04012 (436 RUS 113/820/0-1(R)) and the DFG SFB/TR 16
”Subnuclear Structure of Matter”. A. K. and V. B. acknowledge the support of the Federal Program
of the Russian Ministry of Industry, Science, and Technology No 02.434.11.7091. E. E. acknowledges
the support of the Helmholtz Association (contract no. VH-NG-222).
A Leading amplitudes
In this appendix we give explicit expressions for the amplitudes that appear at leading order in the
calculation for γd → π+nn. As outlined in the main text these expressions can be used directly in
the analysis of the data, once available. In addition, they should also proof useful for the design of
the experiment. Note, as outlined in the text, only near θr = 90
◦ the leading order calculation gives
a sufficiently accurate representation of the spectra. At all other angles one should use the complete
calculation.
At leading order only diagrams a1 and a2 of Fig. 1 contribute. Since only the momentum dependence
of the amplitudes is relevant for the experimental analysis we drop an overall factor compared to
Ref. [11]. The corresponding amplitudes read:
M sa1 =
u(~pr − ~kπ/2 + ~qγ/2) + u(−~pr − ~kπ/2 + ~qγ/2)
(A.1)
M ta1 =
u(~pr − ~kπ/2 + ~qγ/2)− u(−~pr − ~kπ/2 + ~qγ/2)
(A.2)
Ma2 = 8π
fon(pr)
g(pr)
(2π)3
u(~p− ~kπ/2 + ~qγ/2) g(p)
p2 − p2r − i0
fon(pr)
iqπγ g(pr)
p2r + β
αi − ipr + iqπγ
αi − ipr − iqπγ
αi + βj − iqπγ
αi + βj + iqπγ
(A.3)
where u(~p) denotes the S–wave part of the deuteron wave function in momentum space. We checked
by explicit calculations that the inclusion of the deuteron D-wave changes only the absolute scale of
the differential cross sections but not its momentum dependence. Thus, the D-wave contribution is
not taken into account in the parameterization. The quantity qπγ is defined as qπγ = |~kπ − ~qγ|/2. The
labels s and t stand for spin singlet and triplet final two-nucleon states, respectively — we do not
write out the corresponding spin structures. We take into account only the 1S0 partial wave in the
final state interaction. For a discussion of the effect of nn P–waves see Ref. [11].
To derive the expression forMa2 we used the fact that the neutron–neutron scattering amplitude can
be represented to high accuracy in separable form [11, 24]. The neutron–neutron scattering amplitude,
f(p, k;E), can be written in half off–shell kinematics as
f(p, k; k2/MN ) =
2π2MNg(p)g(k)
g2(q)
q2−k2−i0
= fon(k)
, (A.4)
where the corresponding on–shell amplitude fon(k) can then be expressed in terms of the scattering
phase–shifts through
fon(k) = f(k, k; k2/MN ) =
k cot δ(k) − ik
For small momenta one can use the effective range expansion for k cot δ = −1/ann + rnnk2/2+O(k4),
in agreement with Eq. (1). Here ann is the parameter to be fitted to the data and rnn = 2.76 fm. We
checked that changing the value of rnn within the bounds allowed (±0.1 fm [1]) leads to negligible
effects on the extraction of the scattering length. In this way we expressed the matrix element explicitly
in terms of the scattering length. We checked that the ratio g(p)/g(k) in Eq. (A.4) does not change
when we vary the scattering length within acceptable range bounds.
In order to evaluate the convolution of the deuteron wave function with the nn final state interaction
analytically, we needed to employ the following parameterizations for the 1S0 nn form factor g(p) (see
Eq. (A.4)) and the S-wave deuteron wave function
g(p) =
p2 + β2i
; u(p) =
p2 + α2i
; (A.5)
1S0 form factor S-wave deuteron w.f.
βi [MeV] Di [MeV] αi [MeV] Ci [MeV
1 164.53278 31.101228 45.334919 43.543212
2 246.85751 -1310.3056 242.66091 -35.643003
3 329.18224 9455.9603 439.98691 419.25214
4 411.50697 -9666.0268 637.31291 -1833.4708
5 493.83170 -55571.615 834.63891 -3710.8173
6 576.15643 64600.071 1031.9649 24903.150
7 658.48116 149128.85 1229.2909 -31673.576
8 740.80589 -84844.967 1426.6169 26476.636
9 823.13062 -295594.17 1623.9429 -118733.48
10 905.45536 -30332.710 1821.2689 259759.15
11 987.78009 560829.89 2018.5949 -223816.07
12 1070.1048 -307006.25 2215.9209 −
i=1Ci
Table 1: Parameters of the 1S0 form factor and the S-wave deuteron wave function for the separable
representation of the N2LO chiral NN potential.
where the parameters corresponding to the ChPT calculation at N2LOwith cut offs {Λ, Λ̃}= {550 MeV,
600 MeV} (see Ref [18] for details) are listed in Table 1. Note that the coefficients in the parameter-
ization of the wave function have to fulfill the relation
Ci = 0 in order to ensure the regularity of
the deuteron wave function at the origin in coordinate space [25].
The squared and averaged amplitude to be used in the expression for the differential cross section,
defined in Eq. (3) is
|M(pr, θr, φr, θπ, φπ)fit|2 = |M sa1 +Ma2|
∣M ta1
. (A.6)
In a fit to data two parameters are to be adjusted, namely the overall normalization C of Eq. (3)
and the object of desire, ann.
References
[1] G. Miller, B. Nefkens and I. Šlaus. Phys. Rep. 194, 1 (1990).
[2] B. Gabioud et al., Nucl. Phys. A 420, 496 (1984).
[3] O. Schori et al., Phys. Rev. C 35, 2252 (1987).
[4] C. R. Howell et al., Phys. Lett. B 444, 252 (1998).
[5] D. E. Gonzáles Trotter et al., Phys. Rev. Lett. 83, 3788 (1999).
[6] V. Huhn et al., Phys. Rev. Lett. 85, 1190 (2000); Phys. Rev. C 63, 014003 (2000).
[7] D. E. Gonzáles Trotter et al., Phys. Rev. C 73, 034001 (2006).
[8] R. B. Wiringa, V. G. J. Stoks and R. Schiavilla, Phys. Rev. C 51, 38 (1995) [arXiv:
nucl-th/9408016].
http://arxiv.org/abs/nucl-th/9408016
[9] R. Machleidt and H. Müther, Phys. Rev. C 63, 034005 (2001) [arXiv:nucl-th/0011057].
[10] A. Nogga et al., Phys. Rev. C 67, 034004 (2003) [arXiv:nucl-th/0202037].
[11] V. Lensky, V. Baru, J. Haidenbauer, C. Hanhart, A. Kudryavtsev and U.-G. Meißner,
Eur. Phys. J. A 26, 107 (2005) [arXiv: nucl-th/0505039].
[12] V. Baru, C. Hanhart, A. E. Kudryavtsev and U.-G. Meißner, Phys. Lett. B 589, 118 (2004)
[arXiv: nucl-th/0402027].
[13] V. Bernard, N. Kaiser and U.-G. Meißner, Phys. Lett. B 383, 116 (1996) [arXiv:hep-ph/9603278].
[14] E. Epelbaum, Prog. Part. Nucl. Phys. 57, 654 (2006) [arXiv: nucl-th/0505032].
[15] A. G̊ardestig and D. R. Phillips, Phys. Rev. C 73, 014002 (2006) [arXiv: nucl-th/0501049].
[16] A. G̊ardestig and D. R. Phillips, Phys. Rev. Lett. 96 (2006) 232301 [arXiv:nucl-th/0603045].
[17] A. Gasparyan, J. Haidenbauer, C. Hanhart and K. Miyagawa, arXiv:nucl-th/0701090;
Eur. Phys. J. A in print.
[18] E. Epelbaum, W. Glöckle and U.-G. Meißner, Nucl. Phys. A 747, 362 (2005) [arXiv:
nucl-th/0405048]
[19] E. Epelbaum, W. Glöckle and U.-G. Meißner, Eur. Phys. J. A 19, 125 (2004) 125
[arXiv:nucl-th/0304037].
[20] J. M. Laget, Phys. Rep. 69 (1981) 1.
[21] A. G̊ardestig, Phys. Rev. C 74, 017001 (2006) [arXiv:nucl-th/0604035].
[22] E. Epelbaum, A. Nogga, W. Glöckle, H. Kamada, U.-G. Meißner and H. Witala, Phys. Rev. C
66, 064001 (2002) [arXiv:nucl-th/0208023].
[23] C. Hanhart, U. van Kolck and G. A. Miller, Phys. Rev. Lett. 85 (2000) 2905
[arXiv:nucl-th/0004033].
[24] J. Haidenbauer and W. Plessas, Phys. Rev. C 30, 1822 (1984).
[25] M. Lacombe, B. Loiseau, R. Vinh Mau, J. Cote, P. Pires and R. de Tourreil, Phys. Lett. B 101,
139 (1981).
http://arxiv.org/abs/nucl-th/0011057
http://arxiv.org/abs/nucl-th/0202037
http://arxiv.org/abs/nucl-th/0505039
http://arxiv.org/abs/nucl-th/0402027
http://arxiv.org/abs/hep-ph/9603278
http://arxiv.org/abs/nucl-th/0505032
http://arxiv.org/abs/nucl-th/0501049
http://arxiv.org/abs/nucl-th/0603045
http://arxiv.org/abs/nucl-th/0701090
http://arxiv.org/abs/nucl-th/0405048
http://arxiv.org/abs/nucl-th/0304037
http://arxiv.org/abs/nucl-th/0604035
http://arxiv.org/abs/nucl-th/0208023
http://arxiv.org/abs/nucl-th/0004033
	Introduction
	ChPT calculation for d+nn
	Differential cross sections: relevant features
	Extraction of ann and estimate of the theoretical uncertainty
	Discussion and conclusions
	Leading amplitudes

  We discuss the possibility to extract the neutron-neutron scattering length
a_{nn} from experimental spectra on the reaction gamma d --> pi^+ nn. The
transition operator is calculated to high accuracy from chiral perturbation
theory. We argue that for properly chosen kinematics, the theoretical
uncertainty of the method can be as low as 0.1 fm.

Introduction
A precise knowledge of the neutron-neutron scattering length ann is, e.g., important for an understand-
ing of the effects of charge symmetry breaking in nucleon–nucleon forces [1]. The scattering length
ann characterizes scattering at low energies. It is related to the on–shell
1S0 scattering amplitude f
fon(pr) =
pr cot δ(pr)− ipr
−a−1nn + 12 rnnp2r +O(p4r)− i pr
, (1)
where pr is the relative momentum between the two neutrons, δ(pr) the scattering phase shift in the
1S0 partial wave and rnn is the effective range. At low energies the terms of order p
r can be neglected
to very high accuracy. Obviously, a direct determination of ann in a scattering experiment is extremely
difficult due to the absence of a free neutron target. For this reason, the value for ann is to be obtained
from analyses of reactions where there are three particles in the final state, e.g. π−d → γnn [2, 3, 4]
or nd → pnn [5, 6, 7]. There is some spread in the results for ann obtained by the various groups. In
particular, two independent analyses of the reaction nd → pnn give significantly different values for
ann, namely ann = −16.1± 0.4 fm [6] and ann = −18.7± 0.6 fm [7], whereas the latest value obtained
from the reaction π−d → γnn is ann = −18.5 ± 0.3 fm [4]. At the same time, for the proton-proton
scattering length, which is directly accessible, a very recent analysis reports app = −17.3 ± 0.4 fm
[8] after correcting for electromagnetic effects. This means that even the sign of ∆a = app − ann is
http://arxiv.org/abs/0704.0443v1
not fixed.1 It should be mentioned, however, that state of the art calculations for the binding energy
difference of tritium and 3He suggest that ∆a > 0 [9, 10].
In the present work we discuss the possibility to determine ann from differential cross sections in
the reaction γd → π+nn. Specifically, we show that one can extract the value of ann reliably by fitting
the shape of a properly chosen momentum spectrum. In this case the main source of inaccuracies,
caused by uncertainties in the single–nucleon photoproduction multipole E0+, is largely suppressed.
Furthermore there is a suppression of the quasi-free pion production at specific angles. We show that
at these angular configurations the extraction of ann can be done with minimal theoretical uncertainty.
Our investigation is based on the recent work of Ref. [11] in which the transition operator for
the reaction γd → π+nn was calculated up to order χ5/2 in chiral perturbation theory (ChPT) with
χ = mπ/MN ≃ 1/7, where mπ (MN ) is the pion (nucleon) mass. Half-integer powers of χ in the
expansion arise from the unitarity (two– and three–body) cuts (see also [12]). The results of Ref. [11]
for the total cross section are in very good agreement with the experimental data. The only input
parameter that entered the calculation was the leading single–nucleon photoproduction multipole E0+,
which was fixed from a fourth-order one-loop calculation of Bernard et al. [13]. The uncertainty in
E0+ is the main theoretical error in the calculation presented in Ref. [11]. Besides this transition
operator, in the present study we use nucleon–nucleon (NN) wave functions constructed likewise
in the framework of ChPT, namely those of the NNLO interaction of Ref. [14]. This allows us to
estimate the theoretical uncertainty which arises from variations in the wave functions. In fact, as
soon as we include consistently all terms up to order χ5/2, we expect the ambiguities due to different
wave functions not to be larger than a χ3 correction, for only at this order the leading counter term
which absorbs these effects enters. This expectation is indeed quantitatively confirmed in the concrete
calculations.
Since we work within chiral perturbation theory we can estimate the effect of higher orders in
terms of established expansion parameters together with the standard assumption that additional
short ranged operators, that enter at higher orders, behave in accordance with the power counting
(the so-called naturalness assumption). This method was also applied in Refs. [15, 16], where the
reaction π−d → γnn was investigated as a tool to extract ann. However, to know the effect of higher
orders for sure, one has to calculate them. Therefore, to derive a reliable uncertainty estimate for the
extraction of ann from the γd reaction, we use our leading order calculation as baseline result and
estimate the theoretical uncertainty from the effects of the higher orders that we calculated completely.
Based on this, we find a theoretical uncertainty δann . 0.1 fm. We therefore argue that the reaction
γd → π+nn appears to be a good tool for the extraction of ann.
To end this section, we remark that in Ref. [17] a method was proposed to extract scattering
lengths from γd induced meson production. However, this approach should not be used here, since
the momentum transfer is not sufficiently large to use this method and with our explicit calculation
of the transition operator we can reach a significantly higher accuracy.
2 ChPT calculation for γd → π+nn
The diagrams that contribute to the reaction γd → π+nn are shown on Fig. 1. The kinematical
variables are defined in Fig. 2.
Before going into the details some comments are necessary regarding the relevant scales of the
problem. In the near threshold regime of interest here (excess energies of at most 20 MeV above the
1Note, that, in contrast to app, ann is not corrected for electromagnetic effects. However, since those are only of the
order of 0.3 fm [1] they are not relevant for the sign of ∆a. But they ought to be taken into account for determining
charge symmetry breaking effects quantitatively.
χ , χ , χ χ , χ0 1 2 2 5/2
(d2)(b2)
(b1) (c1) (d1)
Figure 1: Diagrams for γd → π+nn. Shown are one–body terms (diagram (a) and (b) ), as well as the
corresponding rescattering contribution (c)—all without and with final state interaction. Diagrams
(d) shows the class of diagrams with intermediate NN interaction. Solid, wavy, and dashed lines
denote nucleons, photons and pions, in order. Filled squares and ellipses stand for the various vertices
(see Ref. [11] for the details), the hatched area shows the deuteron wave function and the filled circle
denotes the nn scattering amplitude. Crossed terms (where the external lines are interchanged) are
not shown explicitly.
pion production threshold) the outgoing pion momenta are small compared even to the pion mass.
Thus, in addition to the conventional expansion parameters of ChPT mπ/Λχ and qγ/Λχ, where Λχ
denotes the chiral symmetry breaking scale of order of (and often identified with) the nucleon mass,
and qγ denotes the photon momentum in the center–of–mass system which is of order of the pion
mass, we can also regard kπ/mπ as small, where kπ denotes the momentum of the outgoing pion. In
what follows we will perform an expansion in two parameters, namely
χm = mπ/MN and χQ = kπ/mπ .
Obviously, the value of the second parameter depends on the excess energy Q. The energy regime
of interest to us corresponds to excess energies up to 20 MeV. The maximum value of χQ, χ
2Q/mπ, at the highest energy considered is thus about 1/2. Since this is numerically close to
we use the following assignment for the expansion parameter:
χ ∼ χm ∼ χ2Q . (2)
The tree level γp → π+n vertex, as it appears in diagrams (a1) and (a2) in Fig 1 (the vertex is labeled
as filled square), contributes at leading order (order χ0), and orders χ1 and χ2, depending on the
one–body operator used. Note that the loop diagrams with πN rescattering (see diagrams (b), (c)
and (d) in Fig. 1) contribute at order χ2m as well as at χ
mχQ, χ
m and at χ
Q. The origin of the
non–integer power of χ are the two–body (πN) and three–body (πNN) singularities. Thus, all terms
up to χ5/2 are explicitly taken into account in our calculation of the transition operator.
As already emphasized, we employ wave functions evaluated in the same framework in order to
have a fully consistent calculation. In our work, we use the N2LO wave functions corresponding to
the chiral NN forces introduced in Ref. [18] and based on the spectral function regularization (SFR)
scheme [19]. At this order, the NN force receives contributions from one-pion exchange, two-pion
p−q 2
Figure 2: Kinematical variables for γd → π+nn. The relative neutron–neutron momentum is defined
as ~pr =
(~p1 − ~p2).
exchange at the subleading order as well as from all possible short-range contact interactions with up
to two derivatives. In addition, the dominant isospin-breaking correction due to the charged-to-neutral
pion mass difference in the one-pion exchange potential together with the two leading isospin-breaking
S-wave contact interactions were taken into account [18]. The two corresponding low-energy constants
were adjusted to reproduce the scattering lengths ann and app. The SFR cutoff Λ̃ is varied in the range
500 . . . 700 MeV. It was argued in Ref. [19] that such a choice for Λ̃ provides a natural separation of
the long- and short-range parts of the nuclear force and allows to improve the convergence of the chiral
expansion [19]. The cutoff Λ in the Lippmann-Schwinger equation is varied in the range 450 . . . 600
MeV. For an extensive discussion on the choice of Λ and Λ̃ the reader is referred to [14, 18].
3 Differential cross sections: relevant features
In this section we outline the features of the differential cross section for unpolarized particles that are
important for our considerations. For later convenience let us consider the function F proportional to
the square of the matrix element as well as the five–fold differential cross section
F (pr, θr, φr, θπ, φπ) = C pr kπ(pr)|M(pr, θr, φr, θπ, φπ)|2 ∝
d5σ(pr, θr, φr, θπ, φπ)
dΩ~prdΩ~kπ
, (3)
where ~pr (~kπ) stands for the relative momentum of the two final neutrons (momentum of the final
pion) in the center–of–mass frame, θr, φr (θπ, φπ) for the corresponding polar and azimuthal angles,
respectively, and |M|2 for the squared and averaged amplitude. In Eq. (3) C is an irrelevant dimen-
sionful constant. In what follows we will consider only shapes of cross sections and therefore the value
of C is not important for our considerations. The value of kπ at given pr and excess energy Q is fixed
by energy conservation:
, (4)
hence we write kπ(pr) in Eq. (3).
In the following we choose the momentum ~qγ of the initial photon to be along the z–axis. Then
the cross sections at a certain excess energy Q depend on four variables, namely the magnitude of
the relative momentum of the two final neutrons pr, the polar angles of the vectors ~pr and ~kπ, and
the difference between the azimuthal angles of those two momenta. Unpolarized cross sections are
invariant under rotations around the beam axis, which makes the dependence on the missing angle
trivial.
Typical differential cross sections F are shown in Fig. 3 as a function of pr at some fixed set of
angles {φr, θπ, φπ} and Q = 5 MeV for two different values of θr. One can see from this figure that
for the differential cross section F of Eq. (3) there are two characteristic regions:
1. The region of quasi-free production (QF) at large pr, which corresponds to the dominance of those
diagrams of Fig. 1 that do not contain the NN interaction in the final or intermediate states.
In the Appendix we give explicit expressions for the diagram a1 – the most significant diagram
of this type. At large pr the pion momentum kπ is small (see Eq. (4)) and the arguments of the
deuteron wave function in Eqs. (A.1) and (A.2) may become small for particular combinations
of ± ~pr and ~qγ/2. This feature gives rise to a peak in the differential cross section at large pr.
2. The region with prominence of the strong nn final–state interaction (FSI) at small pr (in fact,
we would have the strongest final state interaction at zero relative momentum, however the cross
section goes to zero at pr = 0 due to the phase space, therefore we see a peak shape).
One can see from Fig. 3 that the FSI peak depends on the value of θr only marginally, whereas the
quasi-free peak shows significant dependence on this angle. In particular, the quasi-free production
is largely suppressed at θr = 90
◦ — at this angle the arguments of the wave functions in both terms
in the r.h.s. of Eqs. (A.1) and (A.2) are large. It can also be seen from Fig. 3 (right panel) that
the effect of higher orders is more important for the quasi-free production amplitude — the influence
of higher-order effects on the FSI production is quite small. Another interesting observation is that
the contributions of higher orders change the relative height of the two peaks – the FSI peak goes up
whereas the QF peak goes down when we proceed from the LO calculation to the order χ5/2. In order
to suppress the distortions of the spectrum due to higher orders in the chiral expansion, which is the
condition for an extraction of ann with small theoretical uncertainty, configurations should be chosen
where θr = 90
We now briefly discuss the dependence of the cross section on the remaining angles θπ, φπ (we always
may choose φr to be zero). The dependence on θπ is illustrated in Fig. 4. One can see from this figure
that the dependence on θπ is significant for both the quasi–free as well as the FSI peak. This can be
easily understood from the explicit expressions for the matrix elements given in the Appendix keeping
in mind that already at Q = 5 MeV the maximal value of kπ is about mπ/3 while qγ ≈ mπ. Thus,
the momentum transfer to the nucleon pair, |~qγ −~kπ|, varies in the range 2mπ/3 to 4mπ/3 depending
on θπ. Since the S-wave deuteron wave function is large only for very small arguments, the influence
of the direction of ~kπ is significant. In addition, from Fig. 4 it follows that a variation of θπ not only
changes the magnitude but also the shape of the cross section, even in the FSI region. This has to be
taken into account in the analysis of any experiment.
In contrast to the polar angles, the dependence of F on φπ is negligible for all configurations (there
is no dependence at all for θr = 0
◦ and at θr = 90
◦, only the anyway small QF contribution changes
by just 5 %).
4 Extraction of ann and estimate of the theoretical uncertainty
In this section we discuss how to extract the scattering length from future data on γd → π+nn as well
as the resulting theoretical uncertainty. Our focus is especially the latter point. As in the previous
section we will only discuss results at excess energy Q = 5 MeV. However, the analysis can be repeated
analogously at any excess energy within the range of applicability of the formalism, i.e. Q ≤ 20 MeV.
We are interested in extracting the value of ann, which, in turn, is a low-energy characteristic of
neutron-neutron scattering and manifests itself in the momentum dependence of the cross section at
small values of the momentum pr. The influence of the value of ann on the cross section is illustrated in
0 10 20 30 40 50 60 70
 [MeV]
0 10 20 30 40 50 60 70
 [MeV]
.] FSI
QF 0 degrees
QF 90 degrees
Figure 3: Left panel: Differential cross section. The solid line corresponds to the configuration
when the quasi–free peak is suppressed (θr = 90
◦), whereas the dashed line corresponds to one of
the configurations when the quasi–free production amplitude is maximal (θr = 0
◦). The values of
the remaining angles are θπ = 135
◦, φr = φπ = 0
◦; they are the same for both curves. Right panel:
Differential cross section — relative strength of QF and FSI peaks. Here the dashed curves correspond
to the calculation at LO, the solid ones to the calculation at χ5/2. Curves denoted by ”FSI” (”QF”)
are obtained by retaining only those diagrams of Fig. 1 that contain (do not contain) the final or
the intermediate nucleon–nucleon interaction. The labels ”0 degrees” and ”90 degrees” denote the
corresponding values of θr for the ”QF” curves whereas the ”FSI” curves are almost insensitive to this
angle. The values or the remaining angles are as on the left panel of this figure. The overall scale is
arbitrary in both panels but the relative normalization is the same for all curves.
Fig. 5, where the cross sections are shown for three different values of ann, namely −18, −19, −20 fm.
For each value there are two curves, the dashed one corresponds to θr = 0
◦, and the solid one to
θr = 90
◦. One can see from Fig. 5 that the influence of different values of ann is significant in the FSI
peak and marginal in the quasi–free peak, as one would have expected.
In the previous section we have shown (see right panel in Fig. 3) that the relative height of the quasi-
free and the FSI peak changes if the effects of higher orders are included in the cross sections. Therefore
those angular configurations are to be preferred, where the quasi-free production is suppressed.
The central point of this study is to demonstrate that there is a large sensitivity of the momentum
spectra to the scattering length and that this scattering length can be extracted with a small and
controlled theoretical uncertainty. As outlined in the Introduction, we can estimate this uncertainty
reliably, because the effect of the higher orders up to χ5/2 are calculated completely. In order to
demonstrate the effect of those higher orders on the shape of the momentum distribution, in Fig. 6
we show as the light band the spread in the results for the calculation from LO to χ5/2. The results
also include higher partial waves for the pion as well as the final nn system.
There is some sensitivity to the behavior of the deuteron wave function at short distances. For the
reaction π−d → γnn this sensitivity was identified as the largest effect at N3LO in Ref. [15] 2. Guided
by that observation we include in the uncertainty estimate also the spread in the results due to the
use of different wave functions. In order to remove the effect of the change in normalization when,
e.g., changing the chiral order, all curves are normalized at pnorm = 30 MeV in Fig. 6. In the same
2Within the framework of ChPT with a consistent power counting scheme, the quantitative impact of the wave-
function dependence is governed by the order at which a counter term appears that can absorb this model dependence.
The corresponding counter term for the γd as well as the πd reaction arises at N3LO.
0 10 20 30 40 50 60 70
 [MeV]
0 10 20 30 40 50 60 70
 [MeV]
Figure 4: Dependence of the differential cross section on θπ. The left panel corresponds to the
suppressed quasi–free amplitude (θr = 90
◦), the right panel to the maximal quasi–free amplitude
(θr = 0
◦). Solid, dashed, dotted, and dash-dotted lines correspond to θπ = 0
◦, 45◦, 90◦, 135◦
respectively. The values of the remaining angles are φr = φπ = 0
◦. The overall scale is arbitrary in
both panels but the relative normalization is the same for all curves.
Figure (with the same normalization) we also show the change in the shape that comes from different
values of the scattering length: the dark band is generated by a variation of ann by ±1 fm around
the central value of −18.9 fm. Clearly, the theoretical uncertainty is negligibly small compared to the
signal of interest.
One way to quantify the theoretical uncertainty is through the use of the function S, defined as
S(ann,Φ) =
F (pr|a(0)nn ,Φ(0))−N(ann,Φ) F (pr|ann,Φ)
w(pr) , (5)
where pmax =
MNQ is the maximum value of pr, F (pr|ann,Φ) is proportional to the five-fold
differential cross section as defined in Eq. (3). In the latter we refrained from showing the angular
dependence in favor of the parametric dependence of the cross section on the nn scattering length ann
as well as the multi–index Φ, which symbolizes the dependence of the cross section on the chosen chiral
order and the wave functions used, as outlined above. The weight function w(pr) was introduced to
allow us to suppress particular regions of momenta in the analysis — the role of w(pr) will be discussed
in detail below. For simplicity we may assume that S is dimensionless; all dimensions can be absorbed
into the constant C defined in Eq. (3).
The value a
nn denotes the central value of the scattering length (−18.9 fm) for which we perform
the estimate of the theoretical uncertainty3 whereas Φ(0) corresponds to the baseline type of calcu-
lation, namely leading order with chiral wave functions as specified in the Appendix. The relative
normalization N(ann,Φ) is fixed by demanding that S gets minimized for any given pair of parameters
ann,Φ (∂S/∂N = 0). This gives
N(ann,Φ) =
dpr F (pr|a
nn ,Φ
(0))F (pr|ann,Φ)w(pr)
dpr F 2(pr|ann,Φ)w(pr)
. (6)
3Note that the theoretical uncertainty practically does not change when the central value of the scattering length
varies in the relevant interval ±1 fm.
0 10 20 30 40 50 60 70
 [MeV]
-20 fm
-19 fm
-18 fm
Figure 5: The effect of varying the value of ann on the differential cross section. The solid and dashed
lines correspond to the same angular configurations as in Fig. 3, left panel. The different values of
ann are shown on the figure. The overall scale is arbitrary but the relative normalization is the same
for all curves.
Obviously S is the continuum version of the standard χ2 sum, i.e. it characterizes the mean-square
deviation from the baseline cross section F (pr|a
nn ,Φ
(0)). In this way we determine the theoretical
uncertainty in full analogy to the standard method of data analysis.
In order to quantify the theoretical uncertainty we may define Φmax as that chiral order and choice
of wave function, where S(a(0)nn ,Φmax) gets maximal:
S(a(0)nn ,Φmax) = max
S(a(0)nn ,Φ)
. (7)
Therefore S(a(0)nn ,Φmax) provides an integral measure of the theoretical uncertainty of the differential
cross section. Demanding that the effect of a change in the scattering length by the amount ∆ann
matches that by the inclusion of higher orders etc., we can identify ∆ann as an uncertainty in the
scattering length. Expressed in terms of S, we may define ∆ann via
S(a(0)nn +∆ann,Φ(0)) = S(a(0)nn ,Φmax) . (8)
This relation is illustrated in Fig. 7. The dashed horizontal line corresponds to S(a(0)nn ,Φmax), where
we use w(pr) = 1. The dashed parabolic line shows the corresponding S(a
nn + ∆ann,Φ
(0)) as a
function of ∆ann. The calculation is performed for θr = 90
◦, and θπ = 0
◦. The crossing point of the
curves corresponds to ∆ann = 0.16 fm, which can be identified as the theoretical uncertainty for the
extraction of the scattering length.
In the previous section we showed that the signal region is located at momenta lower than 30 MeV.
On the other hand, the theoretical uncertainty of the differential cross section is largest for large values
of pr due to the onset of the quasi–free contribution. In view of these two facts it seems reasonable
to use such weight functions w(pr) that suppress the contribution of large momenta. For instance, we
may use w(pr) = Θ(p
cut−pr) for the weight function. If we choose, e.g., pcut = 30 MeV the theoretical
uncertainty of the extraction of the scattering length reduces to 0.07 fm, as is demonstrated by the
solid lines in Fig. 7. This figure nicely illustrates that the parabolic curve that represents the signal
changes only very little when a restriction to small values of pr is applied. At the same time this
procedure significantly reduces the value of the uncertainty S(a(0)nn ,Φmax).
0 10 20 30 40 50 60 70
 [MeV]
Figure 6: The light (white) band is the error band, and dark (blue) band correspond to ±1 fm shift
in the scattering length from the central value −18.9 fm.
The observation that the dependence of S(a(0)nn +∆ann,Φ(0)) on ∆ann is very well approximated by
a parabola allows for a more systematic study of the pcut dependence of the theoretical uncertainty.
We therefore define
α(pcut) =
S(a(0)nn +∆ann,Φ(0)|pcut)
(∆ann)2
, (9)
where the explicit pcut dependence is introduced into the function S through the weight function w as
explained above. The dashed and the solid parabola in Fig. 7 can then be written as α(pcut) (∆ann)
with α(pmax) = 41 fm
−2 and α(30 MeV) = 33 fm−2. In the left panel of Fig. 8 we show α(pcut) as
the solid line. In the same panel the dashed line represents the measure of the theoretical uncertainty
given by S(a(0)nn ,Φmax|pcut), multiplied by a factor of 40. This figure makes more quantitative the
statement made above: for very small values of pcut we cut into the signal region and therefore α
shows a very rapid variation. However, as soon as pcut is larger than 30 MeV it goes to a plateau (in
the figure indicated by the arrow). On the other hand, the theoretical uncertainty is monotonously
growing once pcut is larger than 30 MeV. From this figure we deduce that the ideal value for pcut
is between 25 and 40 MeV. This translates into a theoretical uncertainty between 0.05 and 0.1 fm,
as illustrated in the right panel of the same figure. The value of θπ also has some impact on the
theoretical uncertainty, however, in its whole parameter range the estimated uncertainty stays below
0.1 fm for pcut = 30 MeV.
Clearly, also the experimental data, once they exist, should be analyzed using a procedure analogous
to the one given above. This means that the scattering length is to be extracted from a χ2 fit of the
theoretical curves to the data. In this work we used the calculation at LO as baseline result and the
results at higher orders to estimate the theoretical uncertainty. Consequently, we propose to use the
momentum spectrum calculated at LO in the fitting procedure of the experiment. The corresponding
analytical expressions are given in the Appendix. The only parameter to be adjusted besides the
scattering length is the overall normalization. In this fitting procedure only those data points should
be included that are below a given pcut, in order to keep the theoretical uncertainty small.
-0.2 -0.1 0 0.1 0.2
 [fm]
Figure 7: The functions S(a(0)nn ,Φmax) and S(a
nn + ∆ann,Φ
(0)) are shown by the horizontal and
parabolic curves, respectively. The solid curves are obtained by adding the weight factor in Eq. (5)
that cuts all momenta above 30 MeV in distinction from the dashed ones. The calculation is performed
for the scattering length a
nn = −18.9 fm, θr = 90◦, and θπ = 0◦. The value of ∆ann corresponding
to the crossing point of the horizontal and parabolic curves determines the theoretical uncertainty of
the calculation.
5 Discussion and conclusions
In the previous section it was shown that for the angular configurations that suppress the quasi-free
production the inclusion of higher order effects (NLO, N2LO, and χ5/2) as well as the use of different
wave functions leads only to a minor change in the momentum dependence of the five-fold differential
cross sections.
Based on this observation we propose to use the momentum spectrum calculated at LO for the
extraction of the neutron–neutron scattering length from the data. This procedure has the advantage
that the corresponding matrix elements can be given in an analytic form (see Appendix) that could
be used directly in the Monte Carlo codes for the experiment analysis. In this way the non–trivial
dependence of the spectra on θπ, discussed above, can be easily controlled. The scattering length can
then be extracted by a two parameter fit to the data where, simultaneously to a variation in ann, the
normalization constant needs to be adjusted.
Note that the leading order calculation basically agrees to the expression given in Ref. [20] long
ago. However, a systematic and reliable study of the theoretical uncertainties of the extraction was
possible only within our full calculation up to order χ5/2 in ChPT. In this way we could show that
the reaction γd → π+nn is very well suited for a determination of the nn scattering length. The
theoretical uncertainty of order 0.1 fm for the extracted scattering length, estimated in this paper, is
of the same order as that claimed for π−d → γnn [16] and nd → pnn [4, 7].
We discussed in detail the theoretical uncertainty for a fixed excess energy of Q = 5 MeV only,
however, it should be clear that the procedure can be easily repeated for any energy within the range of
applicability of our approach (Q ≤ 20 MeV). For example, we checked that the theoretical uncertainty
stays below 0.1 fm also at Q = 10 MeV. Note that the number of events in the signal region scales
roughly with
Q, the phase space available for the pion. It remains to be seen which energy is the
best for the corresponding experiment.
We showed that for a proper choice of both kinematics and weight function w, the theoretical
0 10 20 30 40 50 60 70
 [MeV]
0 10 20 30 40 50 60 70
 [MeV]
Figure 8: Left panel: Comparison of the pcut dependence of functions S(a(0)nn ,Φmax|pcut) (dashed curve)
and α(pcut) (solid curve). The calculation is performed for the scattering length a
nn = −18.9 fm,
θr = 90
◦, and θπ = 0
◦. Right panel: The corresponding theoretical uncertainty ∆ann as a function of
pcut.
uncertainty for the extraction of the neutron–neutron scattering length from γd → π+nn can be as
low as 0.1 fm. It should be stressed, however, that this error was evaluated most conservatively – we
use our LO calculation as baseline result and estimate the theoretical uncertainty from the effects of
the higher orders that we calculated completely. This error can be significantly reduced by further
studies. For example, if we include in the uncertainty estimate only the spread in the results due to
the use of different wave functions, which is identified as the largest effect at N3LO for the reaction
π−d → γnn [15], the theoretical uncertainty of the extracted scattering length reduces by one order
of magnitude. This indicates that the theoretical uncertainty is indeed under control. However to
put this N3LO estimation on more solid ground a complete calculation should be performed to this
order. Most of the operators that are relevant at this order are the same as those of π−d → γnn, given
explicitly in Ref. [21]. One counter term enters, which can be fixed from other processes [16], e.g.,
from nd scattering [22], the reaction NN → NNπ [23], or from weak decays [16]. Once this is done
we may use our calculation to order χ5/2 as baseline result and estimate the theoretical uncertainty
from the then available N3LO calculation.
Although we have identified the angles θr = 90
◦ as the preferred kinematics, also other configura-
tions could be studied in order to control the systematics. However, then the spectra calculated at
χ5/2 should be used in the analysis.
Acknowledgments
We thank A. Bernstein for useful discussions and interest in this work. We also thank D. R. Phillips
and A. G̊ardestig for helpful discussions. This research is part of the EU Integrated Infrastructure
Initiative Hadron Physics Project under contract number RII3-CT-2004-506078, and was supported
also by the DFG-RFBR grant no. 05-02-04012 (436 RUS 113/820/0-1(R)) and the DFG SFB/TR 16
”Subnuclear Structure of Matter”. A. K. and V. B. acknowledge the support of the Federal Program
of the Russian Ministry of Industry, Science, and Technology No 02.434.11.7091. E. E. acknowledges
the support of the Helmholtz Association (contract no. VH-NG-222).
A Leading amplitudes
In this appendix we give explicit expressions for the amplitudes that appear at leading order in the
calculation for γd → π+nn. As outlined in the main text these expressions can be used directly in
the analysis of the data, once available. In addition, they should also proof useful for the design of
the experiment. Note, as outlined in the text, only near θr = 90
◦ the leading order calculation gives
a sufficiently accurate representation of the spectra. At all other angles one should use the complete
calculation.
At leading order only diagrams a1 and a2 of Fig. 1 contribute. Since only the momentum dependence
of the amplitudes is relevant for the experimental analysis we drop an overall factor compared to
Ref. [11]. The corresponding amplitudes read:
M sa1 =
u(~pr − ~kπ/2 + ~qγ/2) + u(−~pr − ~kπ/2 + ~qγ/2)
(A.1)
M ta1 =
u(~pr − ~kπ/2 + ~qγ/2)− u(−~pr − ~kπ/2 + ~qγ/2)
(A.2)
Ma2 = 8π
fon(pr)
g(pr)
(2π)3
u(~p− ~kπ/2 + ~qγ/2) g(p)
p2 − p2r − i0
fon(pr)
iqπγ g(pr)
p2r + β
αi − ipr + iqπγ
αi − ipr − iqπγ
αi + βj − iqπγ
αi + βj + iqπγ
(A.3)
where u(~p) denotes the S–wave part of the deuteron wave function in momentum space. We checked
by explicit calculations that the inclusion of the deuteron D-wave changes only the absolute scale of
the differential cross sections but not its momentum dependence. Thus, the D-wave contribution is
not taken into account in the parameterization. The quantity qπγ is defined as qπγ = |~kπ − ~qγ|/2. The
labels s and t stand for spin singlet and triplet final two-nucleon states, respectively — we do not
write out the corresponding spin structures. We take into account only the 1S0 partial wave in the
final state interaction. For a discussion of the effect of nn P–waves see Ref. [11].
To derive the expression forMa2 we used the fact that the neutron–neutron scattering amplitude can
be represented to high accuracy in separable form [11, 24]. The neutron–neutron scattering amplitude,
f(p, k;E), can be written in half off–shell kinematics as
f(p, k; k2/MN ) =
2π2MNg(p)g(k)
g2(q)
q2−k2−i0
= fon(k)
, (A.4)
where the corresponding on–shell amplitude fon(k) can then be expressed in terms of the scattering
phase–shifts through
fon(k) = f(k, k; k2/MN ) =
k cot δ(k) − ik
For small momenta one can use the effective range expansion for k cot δ = −1/ann + rnnk2/2+O(k4),
in agreement with Eq. (1). Here ann is the parameter to be fitted to the data and rnn = 2.76 fm. We
checked that changing the value of rnn within the bounds allowed (±0.1 fm [1]) leads to negligible
effects on the extraction of the scattering length. In this way we expressed the matrix element explicitly
in terms of the scattering length. We checked that the ratio g(p)/g(k) in Eq. (A.4) does not change
when we vary the scattering length within acceptable range bounds.
In order to evaluate the convolution of the deuteron wave function with the nn final state interaction
analytically, we needed to employ the following parameterizations for the 1S0 nn form factor g(p) (see
Eq. (A.4)) and the S-wave deuteron wave function
g(p) =
p2 + β2i
; u(p) =
p2 + α2i
; (A.5)
1S0 form factor S-wave deuteron w.f.
βi [MeV] Di [MeV] αi [MeV] Ci [MeV
1 164.53278 31.101228 45.334919 43.543212
2 246.85751 -1310.3056 242.66091 -35.643003
3 329.18224 9455.9603 439.98691 419.25214
4 411.50697 -9666.0268 637.31291 -1833.4708
5 493.83170 -55571.615 834.63891 -3710.8173
6 576.15643 64600.071 1031.9649 24903.150
7 658.48116 149128.85 1229.2909 -31673.576
8 740.80589 -84844.967 1426.6169 26476.636
9 823.13062 -295594.17 1623.9429 -118733.48
10 905.45536 -30332.710 1821.2689 259759.15
11 987.78009 560829.89 2018.5949 -223816.07
12 1070.1048 -307006.25 2215.9209 −
i=1Ci
Table 1: Parameters of the 1S0 form factor and the S-wave deuteron wave function for the separable
representation of the N2LO chiral NN potential.
where the parameters corresponding to the ChPT calculation at N2LOwith cut offs {Λ, Λ̃}= {550 MeV,
600 MeV} (see Ref [18] for details) are listed in Table 1. Note that the coefficients in the parameter-
ization of the wave function have to fulfill the relation
Ci = 0 in order to ensure the regularity of
the deuteron wave function at the origin in coordinate space [25].
The squared and averaged amplitude to be used in the expression for the differential cross section,
defined in Eq. (3) is
|M(pr, θr, φr, θπ, φπ)fit|2 = |M sa1 +Ma2|
∣M ta1
. (A.6)
In a fit to data two parameters are to be adjusted, namely the overall normalization C of Eq. (3)
and the object of desire, ann.
References
[1] G. Miller, B. Nefkens and I. Šlaus. Phys. Rep. 194, 1 (1990).
[2] B. Gabioud et al., Nucl. Phys. A 420, 496 (1984).
[3] O. Schori et al., Phys. Rev. C 35, 2252 (1987).
[4] C. R. Howell et al., Phys. Lett. B 444, 252 (1998).
[5] D. E. Gonzáles Trotter et al., Phys. Rev. Lett. 83, 3788 (1999).
[6] V. Huhn et al., Phys. Rev. Lett. 85, 1190 (2000); Phys. Rev. C 63, 014003 (2000).
[7] D. E. Gonzáles Trotter et al., Phys. Rev. C 73, 034001 (2006).
[8] R. B. Wiringa, V. G. J. Stoks and R. Schiavilla, Phys. Rev. C 51, 38 (1995) [arXiv:
nucl-th/9408016].
http://arxiv.org/abs/nucl-th/9408016
[9] R. Machleidt and H. Müther, Phys. Rev. C 63, 034005 (2001) [arXiv:nucl-th/0011057].
[10] A. Nogga et al., Phys. Rev. C 67, 034004 (2003) [arXiv:nucl-th/0202037].
[11] V. Lensky, V. Baru, J. Haidenbauer, C. Hanhart, A. Kudryavtsev and U.-G. Meißner,
Eur. Phys. J. A 26, 107 (2005) [arXiv: nucl-th/0505039].
[12] V. Baru, C. Hanhart, A. E. Kudryavtsev and U.-G. Meißner, Phys. Lett. B 589, 118 (2004)
[arXiv: nucl-th/0402027].
[13] V. Bernard, N. Kaiser and U.-G. Meißner, Phys. Lett. B 383, 116 (1996) [arXiv:hep-ph/9603278].
[14] E. Epelbaum, Prog. Part. Nucl. Phys. 57, 654 (2006) [arXiv: nucl-th/0505032].
[15] A. G̊ardestig and D. R. Phillips, Phys. Rev. C 73, 014002 (2006) [arXiv: nucl-th/0501049].
[16] A. G̊ardestig and D. R. Phillips, Phys. Rev. Lett. 96 (2006) 232301 [arXiv:nucl-th/0603045].
[17] A. Gasparyan, J. Haidenbauer, C. Hanhart and K. Miyagawa, arXiv:nucl-th/0701090;
Eur. Phys. J. A in print.
[18] E. Epelbaum, W. Glöckle and U.-G. Meißner, Nucl. Phys. A 747, 362 (2005) [arXiv:
nucl-th/0405048]
[19] E. Epelbaum, W. Glöckle and U.-G. Meißner, Eur. Phys. J. A 19, 125 (2004) 125
[arXiv:nucl-th/0304037].
[20] J. M. Laget, Phys. Rep. 69 (1981) 1.
[21] A. G̊ardestig, Phys. Rev. C 74, 017001 (2006) [arXiv:nucl-th/0604035].
[22] E. Epelbaum, A. Nogga, W. Glöckle, H. Kamada, U.-G. Meißner and H. Witala, Phys. Rev. C
66, 064001 (2002) [arXiv:nucl-th/0208023].
[23] C. Hanhart, U. van Kolck and G. A. Miller, Phys. Rev. Lett. 85 (2000) 2905
[arXiv:nucl-th/0004033].
[24] J. Haidenbauer and W. Plessas, Phys. Rev. C 30, 1822 (1984).
[25] M. Lacombe, B. Loiseau, R. Vinh Mau, J. Cote, P. Pires and R. de Tourreil, Phys. Lett. B 101,
139 (1981).
http://arxiv.org/abs/nucl-th/0011057
http://arxiv.org/abs/nucl-th/0202037
http://arxiv.org/abs/nucl-th/0505039
http://arxiv.org/abs/nucl-th/0402027
http://arxiv.org/abs/hep-ph/9603278
http://arxiv.org/abs/nucl-th/0505032
http://arxiv.org/abs/nucl-th/0501049
http://arxiv.org/abs/nucl-th/0603045
http://arxiv.org/abs/nucl-th/0701090
http://arxiv.org/abs/nucl-th/0405048
http://arxiv.org/abs/nucl-th/0304037
http://arxiv.org/abs/nucl-th/0604035
http://arxiv.org/abs/nucl-th/0208023
http://arxiv.org/abs/nucl-th/0004033
	Introduction
	ChPT calculation for d+nn
	Differential cross sections: relevant features
	Extraction of ann and estimate of the theoretical uncertainty
	Discussion and conclusions
	Leading amplitudes
