$\langle\phi\rangle \neq 0$, breaking electroweak symmetry down to its electromagnetic subgroup, $SU(2)_W \times U(1)_Y \rightarrow U(1)_{e.m.}$. We do not consider fermions as they do not play any qualitatively important role in the phenomena that we are interested in.

The particle content of the model is simple. In the $T=0$ broken phase, the Higgs field acquires the mass $m_H = \sqrt{2\lambda}v$. The model also possesses the massless photon, $A_\mu = W_\mu^3 \sin\theta_W + Y_\mu \cos\theta_W$, and three massive gauge bosons, which include the electrically (off-diagonal) charged W bosons $W_\mu^\pm = W_\mu^1 \pm iW_\mu^2$, and the neutral (diagonal) Z boson, $Z_\mu = W_\mu^3 \cos\theta_W - Y_\mu \sin\theta_W$, with the masses $m_W = gv/2$ and $m_Z = m_W/\cos\theta_W$, respectively.

The electroweak model (16.6) in the background of the magnetic field has been extensively studied in Ref. [509] using first-principle numerical simulations. In simulations, we used the hypermagnetic field $B_Y$ associated with the hypergauge field $Y_\mu$ which, in the broken phase, corresponds to the usual magnetic field $B$ via the relation $g'B_Y = eB$ and plays an independent role in the symmetry-restored phase, in which the magnetic field cannot be defined. The results of Ref. [509] reveal the existence of three phases separated by two (pseudo)critical magnetic fields:

$$eB_{c1} = 0.68(5)m_W^2, \quad eB_{c2} = 0.99(2)m_H^2. \qquad (16.7)$$

The first critical field $B_{c1} \approx 0.7B_W^{EW}$ appears to be about 30% weaker than the value (16.3) predicted by the classical theoretical analysis that does not take into account quantum fluctuations. However, the second critical field $B_{c2} \approx B_H^{EW}$ agrees precisely with the theoretical value (16.4) within the accuracy of 1%.

The existence of these phases can be readily seen in Fig. 16.1(a), where a normalized expectation value of the Higgs condensate is shown against the magnetic field. One can distinguish (i) the low-field phase at $B < B_{c1}$, where the Higgs expectation value is independent of the strength of the magnetic field; (ii) the intermediate phase at $B_{c1} < B < B_{c2}$, where the field gradually diminishes; and (iii) the symmetry restored phase at $B > B_{c2}$. Thus, the phase structure of the electroweak model in a strong magnetic field is indeed nontrivial, and it agrees rather well with the predictions of the theory. However, what is the nature of the intermediate phase and the bordering phase transitions?

<figure><img src="image_61.png" /><figcaption>Figure 16.1: (a) A normalized value of Higgs condensate squared $\langle\phi^2\rangle_n$. The insets represent the density plots of the $Z_{12}$ fluxes in the cross-sections normal to the magnetic field axis obtained in numerical simulations at the values of the magnetic field shown by the arrows. (b) An illustration of a hexagonal crystalline order of the vortices predicted by the theory (top) and the result obtained in the numerical simulations (bottom); (c) A typical 3d configuration in the superconducting phase ($B_{c1} < B < B_{c2}$) in the magnetic field background $eB = 1.1m_W^2$. The blue and red (green) surfaces show equipotential surfaces of the W (Higgs) condensate in complementary regions. Figure adapted from Ref. [509].</figcaption></figure>

Theoretically, the intermediate phase should be described by a classical inhomogeneous W-condensate solution. It can best be imagined as a crystal made of parallel vortex-like structures that share a geometric similarity with the lattice
of Abrikosov vortices of a conventional type-II superconductor. For realistically heavy Higgs masses, $m_H > m_Z$, the vortices in the W condensate arrange themselves into a hexagonal lattice [495, 515, 516]. It is worth also noticing that the similarity between the magnetized vortex phase of the electroweak vacuum and a type-II superconductor is geometric rather than physical because the properties of the two systems differ significantly. For example, the W vortices expel the magnetic field from their cores [496, 517] while the Abrikosov vortices carry the magnetic field lines inside their cores [518]. In addition, the vacuum in this exotic state exhibits superconductivity (associated with the condensation of the W bosons [500]) and superfluidity (supported by the condensation of the Z bosons [501]) only along the direction of the magnetic field, being an insulator in the transverse plane [500, 519]. On the contrary, an ordinary superconductor supports dissipationless transport in all spatial directions.

The numerical results on the inhomogeneities of the vacuum phases are presented in the insets of Fig. 16.1(a) corresponding to different values of the background magnetic field. The snapshots of the cross-sections of typical lattice configurations do indeed reveal the presence of flux-like vortices in the intermediate phase. Contrary to a crystalline hexagonal structure suggested by the theory, the vortices form an amorphous medium, which can be characterized as a disordered solid or a liquid but not a crystal. We attribute the difference between the numerical results and the theoretical predictions on the structure of the inhomogeneous state to the quantum fluctuations that were not taken into account by the semiclassical estimations. A qualitative comparison between the vortex arrangement in the cross-section 2d plane proposed by the theory and found in the simulations is given in Fig. 16.1(b), while the 3d visualization of a typical field configuration found in the simulations is presented in Fig. 16.1(c). Our simulations also reveal that at a higher magnetic field, the solid state of the vacuum melts down (at zero temperature) and disappears: the vacuum enters the homogeneous symmetry-restored state at $B > B_{c2}$. This observation agrees with earlier theoretical estimations [520, 521].

What is also interesting—and this property should definitely be crucial in an analysis of the corresponding phase in QCD—is that both numerically observed transitions are smooth crossovers. As it is demonstrated in Ref. [509], the susceptibility of the Higgs field does not possess a local maximum across either $B_{c1}$ or $B_{c2}$ as it could be expected in a case of a phase transition of first or second order, or in a case of a strong crossover. Therefore, these transitions are not marked with thermodynamic singularities. As a consequence, no instability may occur.

## 16.3. Magnetic-field induced vortex phase in QCD

We expect that a similar scenario can also be realized in QCD, where the role of the W bosons is played by the charged $\rho$ mesons. The numerical simulations of the electroweak model reveal the absence of any tachyonic instability at the critical transition field, while the vortex phase does exist at the intermediate values of the magnetic field. Likewise, the tachyonic instability has not been found in QCD as well. However, do vortex-like structures appear in QCD in the background magnetic field? This question has been addressed in Ref. [522] in the quenched approximation in the simplest, two-color version of the theory. In this section, we highlight certain findings of that paper.

The superconducting ground state should be characterized by the presence of the mixed quark-antiquark condensates, composed of u and d quarks, $\langle \bar{u}\gamma_1 d \rangle = \rho$ and $\langle \bar{u}\gamma_2 d \rangle = -i\rho$, that carry the quantum numbers of the electrically charged $\rho$ mesons. Effective models predict that these condensates should have an inhomogeneous vortex-like structure in the plane normal to the axis of the magnetic field. This phenomenon can be interpreted as the $\rho$-meson condensation.

To probe the emergence of these inhomogeneous states, Ref. [522] employed lattice Monte-Carlo simulations of $SU(2)$ Yang-Mills lattice gauge theory. The quark fields were introduced by the overlap lattice Dirac operator $\mathcal{D}$ with exact chiral
symmetry [523]. The background magnetic field has been introduced via the twisted spatial boundary conditions [524]. The quarks, treated in the quenched approximation, were subjected to the background magnetic field in the range $eB = (0...2.14) \text{ GeV}^2$, which includes the critical field (16.2).

Due to the oscillating phase of the complex $\rho$-meson condensate and the motion of vortices, the calculation of the local value of this condensate appears to be a challenging task. Therefore, the inhomogeneous properties of the vacuum were accessed via the following $\rho$-meson correlator:

$$ \phi(x) \equiv \phi(x; A, B) = \langle \rho^{\dagger}(0)\rho(x) \rangle_{A,B} \equiv \text{Tr} \left( \frac{1}{\mathcal{D}_u(A, B) + m} \gamma_{\mu} \frac{1}{\mathcal{D}_d(A, B) + m} \gamma_{\nu} \right), \quad (16.8) $$

computed in the background of both the non-Abelian gauge field A and the Abelian magnetic field B. The point $x=0$ has been taken at the origin of the lattice.

The potential inhomogeneous properties of the $\rho$-meson condensation can be probed in the spirit of Ref. [525], where the existence of a confining chromoelectric flux tube has been revealed in lattice Yang-Mills theory. The flux tube has been introduced with the help of a rectangular Wilson loop $W$ as a source of heavy quark-antiquark pair and then studied the local energy density operator $\mathcal{E} = \mathcal{E}(x)$ as a probe of the flux tube formed due to the presence of the pair.

Working in analogy to Ref. [525], one can construct the normalized scalar energy of the $\rho$-meson field $\mathcal{E}(x)$, the normalized electric (super)current density $j_{\mu}(x)$ generated by the $\rho$-meson field, and the local vortex density $v(x)$, respectively:

$$ \mathcal{E}(x) = \frac{|D_{\mu}\phi(x)|^2}{|\phi(x)|^2}, \quad D_{\mu} = \partial_{\mu} - ieA_{\mu}, \quad (16.9) $$

$$ j_{\mu}(x) = \frac{\phi^{*}(x) \overleftrightarrow{D}_{\mu} \phi(x) - \phi^{*}(x) \overleftarrow{D}_{\mu} \phi(x)}{2i|\phi(x)|^2}, \quad (16.10) $$

$$ v(x) = \text{sing arg } \phi(x) \equiv \frac{\epsilon^{ab}}{2\pi} \frac{\partial}{\partial x_a} \frac{\partial}{\partial x_b} \arg \phi(x), \quad a, b = 1, 2. \quad (16.11) $$

Our experience with the electroweak model tells us that the ideal semiclassical hexagonal lattice of $\rho$ vortices, shown in Fig. 16.2(a), should get perturbed by quantum fluctuations, forming a disordered solid or a fluid. The fluctuations impact vortex patterns in the cross-sections. For example, a transverse $(x, y)$ cross-section of the volume may contain some linear segments of the vortices instead of a set of localized points in an ideal vortex lattice. In addition, the linear vortex instructions may disappear from one longitudinal $(x, z)$ plane as the vortices may propagate to another slice. The orientations of the cross-sections are illustrated in the first plots of the panels in Figs. 16.2(b) and 16.2(c), respectively.

Within the core of a physical $\rho$-vortex, the energy density is elevated relative to the surrounding regions. Consequently, if physical $\rho$-vortices emerge within a sufficiently strong magnetic field, one would expect to observe localized point-like energy density concentrations in the transverse $(x, y)$ plane, as indeed seen in Fig. 16.2(b). In the picture, the magnetic field is oriented along the $z$-axis. We can also expect the formation of long linear structures in the longitudinal $(x, z)$ plane, which is also the case according to Fig. 16.2(c). In these figures, typical examples of the distribution of the energy density in the transverse $(x, y)$ and longitudinal $(x, z)$ planes are shown for weak ($eB = 0.356 \text{ GeV}^2$), moderate ($eB = 1.07 \text{ GeV}^2$) and high ($eB = 2.14 \text{ GeV}^2$) magnetic fields.

In accordance with our qualitative expectations, at a low magnetic field, the vortex lattice is not formed. At the moderate magnetic field, the appearance of a coherent vortex structure is seen. However, the vortices are not rigidly ordered in the transverse plane, and they are not quite parallel to the magnetic field. At a higher magnetic field, the physical picture shown in Fig. 16.2 becomes visually consistent with the presence of a melted lattice (liquid) of the $\rho$ vortices. This picture of the vortex formation in QCD [522] strongly resembles the formation of the superconducting W vortices in electroweak model [509].
<figure><img src="image_62.png" /><figcaption>**Figure 16.2:** (a) An illustration of the classical vortex solution in the superconducting QCD vacuum. The vortices arrange themselves parallel to the background magnetic field in a hexagonal structure; (b) the longitudinal and (c) transverse cross-sections of typical gauge-field configurations in quenched SU(2) QCD as revealed by the energy correlator (16.9), preceded by an illustration of the relevant cross-section for a partially disordered, realistic configuration. Figure adapted from Ref. [522].</figcaption></figure>

One can also numerically observe [522] that the lumps in the energy density (16.9) are surrounded by circulating electric currents (16.10) while the position of the energy lumps correspond to the locations of the vortex cores as revealed by the vorticity density (16.11). These expected features correspond to the properties of the $\rho$ vortices obtained from the analytical solutions in Ref. [500]. Thus, the results of quenched two-color QCD strongly suggest that similar superconducting vortex patterns may also emerge in realistic QCD with dynamical quarks.

## 16.4. Conclusions

First-principle numerical simulations of the electroweak model in a strong magnetic field reveal the presence of an inhomogeneous phase made of vortices parallel to the magnetic field [509]. Contrary to various theoretical estimations, the transition to this phase proceeds via a very smooth crossover, which does not show any signature of a tachyonic instability, claimed to be responsible for the emergence of this new phase. Still, the inhomogeneous phase does exist, and most of its features qualitatively coincide with the predictions coming from the semiclassical analysis [495, 496, 497, 498, 499]. A notable difference between the theory and simulations appears to be in the geometrical structure of this phase, which, contrary to the hexagonal crystalline order, resides in an exotic disordered solid/liquid state. The observed condensation of the W and Z fields implies that the vacuum in this phase should also be an unusual, strongly anisotropic superconductor and, simultaneously, an anisotropic superfluid [500, 501]. Such anisotropic superconducting properties also imply that the vacuum becomes a hyperbolic metamaterial that behaves as a diffractionless 'perfect lens.' [526, 527, 528].

The results obtained in the electroweak model—where the existence of the superconducting phase can be revealed analytically at a classical level—suggest that a similar situation can occur in QCD: the superconducting inhomogeneous phase may be reached in strong magnetic fields in the absence of a tachyonic instability and with a perfectly analytical behavior of masses of the $\rho$ mesons. In the absence of the Higgs field in QCD, which has a global expectation value in
the vortex phase in the electroweak model, the existence of the inhomogeneous phase of QCD is difficult to reveal. The observables that could show the existence of the vortex structure should perhaps be related to local correlators of the $\rho$-meson fields such as energy (16.9), electric current (16.10), and vorticity (16.11) that were shown to exhibit robust vortex-like features in quenched QCD in strong magnetic fields [522].

# 17. Outlook

The chiral magnetic effect in QCD is by now understood as an inherently out-of-equilibrium phenomenon. The main challenge for the theory remains to find reliable ways to estimate the size of the effect including interactions. Comparison to heavy-ion experiments can be carried out through hydrodynamic simulations, which need the CME conductivity as an input. Via linear response theory, out-of-equilibrium transport coefficients can be determined using lattice Monte-Carlo simulations of equilibrium QCD. Nevertheless, purely non-equilibrium properties are hidden in the spectral function, in contrast to the Euclidean correlators directly available on the lattice. As a desirable extension of current lattice simulations, an analytic continuation (spectral reconstruction) must be performed to obtain the out-of-equilibrium CME conductivity. The calculation of this conductivity from first-principle lattice QCD simulations could, in combination with other effective theory approaches to strongly interacting matter phenomenology, provide valuable information towards the goal of detecting the CME in heavy-ion collision experiments.

Lattice simulations turned out to be very efficient in studying equilibrium properties of QCD, such as the QCD phase diagram or its equation of state. The introduction of an external magnetic field, from one side, leads to plenty of new phenomena, including inverse magnetic catalysis, possible critical endpoints, enhancement of fluctuations and others. From another side, it makes the parameter space of the system very large and complicates lattice studies, especially if one takes the continuum limit. Despite recent progress in understanding QCD bulk properties in the temperature-magnetic field-baryon density space, there are still multiple open issues, which could be studied further. These include, e.g., the existence of the critical endpoint at large magnetic fields and its connection to the chiral critical endpoint in QCD in the $\mu-T$ plane, or the existence of new non-trivial phases. A more detailed study of QCD properties in a larger parameter space, including strangeness or electric chemical potential, as well as the tuning of the parameters to the conditions relevant for the heavy ion collision experiments would be beneficial. One could also possibly extend existing studies and results to various other parameters, including the chiral chemical potential $\mu_5$, electric field $eE$, non-homogeneous fields and others.

Novel chiral hydrodynamic effects arising in extreme magnetic fields, see Sec. 13, have the potential to heavily impact various disciplines. Developing chiral hydrodynamics as an effective field theory has lead to the conclusion that in strong magnetic fields, a large number of novel transport coefficients and susceptibilities can arise, see also Fig. 13.1. The relevance of these novel transport effects for QCD plasma, neutron star mergers, and astrophysical plasma has to be estimated. If they turn out to be relevant, they should be included in hydrodynamic codes used for the analysis of heavy-ion collision data or for the simulation of neutron star evolution, especially for magnetars. Such effects can, for example, account for the observed neutron star kicks. A subset of the novel transport effects are susceptibilities obeying time-independent Kubo relations, and can thus be computed in equilibrium from lattice QCD. For the example of the *magnetic vorticity susceptibility*, $M_2$, Eq. (13.4) and Fig. 13.1 illustrate what would need to be calculated for such susceptibilities: In a homogeneous magnetic field pointing in the $z$-direction, one needs a $y$-dependent magnetic field in the $x$-direction in order to have non-vanishing *magnetic vorticity*. In this state, the $\langle T^{xz}(+k)T^{yz}(-k) \rangle$ correlator needs to be calculated. No
time-dependence is needed here, only spatial inhomogeneity, making $M_2$ and its peers ($M_1$, $M_3$, $M_4$, and $M_5$) a well
defined target for future lattice computations. A complete extension of the chiral hydrodynamics framework from Sec. 13
into chiral magnetohydrodynamics, including dynamical magnetic fields obeying Maxwell's equations is an open problem.
In addition, the framework should be extended to include axial $U(1)$ and vector $U(1)$ symmetries simultaneously. Taking
chiral transport far away from equilibrium, holographic models show that the time evolution of the CME currents very
strongly depends on the choice of initial conditions and parameter choices. A fully dynamical study including dynamical
magnetic field and dynamically generated axial charges is a future goal for holographic models. In general, holographic
models including magnetic fields, serve as discovery tools for new physics and as testing grounds for novel hydrodynamic
effects. They can even be pushed to investigate plasma dynamics far from equilibrium, as illustrated in the CME case
study.

Another facet of the aforementioned transport phenomena with great potential to be explored is their condensed
matter analogues. The table-top setups used in this kind of study are enormously simpler than high energy experiments
and have attracted attention of distinct communities, providing technological applications, insight on fundamental physics
and hopefully will provide some insight on their non-Abelian cousins. So far, a few groups have claimed to have observed
the CME in (3+1)D materials. However, it is still hard to state firmly that the samples really belong to the Weyl type.
ARPES (Angle-resolved Photoemission Spectroscopy) technique has given support on this issue but at this point it is very
useful to build analogues in materials whose charge carriers can be more easily identified as relativistic-like. Reminding
that the CME is one example of a plethora of transport phenomena, anomalous or not, that may take place in the
quark-gluon plasma, increasing our understanding and confidence on the nature of these materials and having other kinds
of structures that may harbor analogues will allow for the search of other transport mechanisms such as chiral separation
effect, chiral magnetic wave, etc.

Finally it is worth to remark that by constructing these type of analogues is a rich procedure to export benefits from
high energy to condensed matter, since several theoretical aspects of the theory of anomalies and transport in general
have been well developed and understood by particle physicists. On the other hand, it would be breakthrough if one
could formulate how to explore tabletop setups in order to have more concrete insight about their hard to access high
energy counterparts.

The relevance of the strong magnetic fields of the electroweak scale to the physics of the present-day Universe poses an
intriguing question. Such fields with the typical electroweak magnitudes of $10^{20}$ T may still exist in the atmospheres
of the extreme magnetically charged black holes. These magnetized Reissner-Nordström black holes should have zero
temperature, implying that they do not radiate Hawking radiation and are, potentially, stable. In-falling matter will,
however, heat the atmospheres of the black holes and lead to thermal fluctuations of the electroweak superconducting
vortices. Being positioned above the event horizon of such a black hole, the vortices should emit high-energy rays that, after
a gravitational well red-shift, will appear, for a distant observer, as radiation in a lower-energy electromagnetic spectrum.
If the population of such black holes is statistically significant, their radiation can potentially have an observable imprint
on the cosmic background.

Our idea poses the challenging question of finding properties of the superconducting electroweak phase subjected to
the strong gravitational field of a black hole. To this end, the lattice gauge theory methods adapted to curved space-times
can provide us with insight into the realistic structure of this vortex-populated phase and give us vital knowledge for
detecting magnetic black holes using astrophysical observations.
# Effective Models

## 18. Introduction

Strongly interacting matter is one of the many fascinating physical systems being intensively investigated in the last decade. Nuclear matter, quark matter and even cold atoms may be considered strongly interacting matter, though in different scales. Most of the problems in dealing with such systems arise from the (infinite) degrees of freedom and from the fact that the couplings are strong (non-perturbative). Without any extreme conditions, that may come from higher temperatures, higher densities, strong magnetic fields or any other environmental condition, strong interacting matter is already a complex system due to its many-body nature. This means that the description of such system has to undergo some approximation, even in ab initio frameworks. Many different approaches have been used and most of what we currently know about strongly interacting matter is the result of a great effort during decades in order to understand systems that are supported by strong interactions.

Many effective models have been developed to describe few-nucleon systems, nuclear matter, nuclei and QCD matter. Although they were designed to describe some specific properties, they provided a great knowledge on strong interactions that would be extremely difficult to acquire directly from the fundamental theory with first-principle calculations. In this Part, several effective approaches aimed to describe strongly interacting magnetized matter will be discussed.

## 19. Modified NJL models

Among the many effective models that have been developed to study strongly interacting matter under extreme conditions, there is one approach that rely on the NJL model [529] for a basic strong interaction, in general, two and three-flavor quark matter have been considered. The unknown details are represented by a thermo-magnetic coupling $G(eB, T)$ which encodes the temperature and magnetic field effects. The first attempts in this direction were made one decade ago and considered an analytical *ansatz* for the thermo-magnetic coupling [17, 18]. Later, in 2017, an improvement of this approach was developed by numerically building $G(eB, T)$ with the constrain that the average of $u$ and $d$ condensates obtained with lattice QCD are reproduced by the effective model for given values of $T$ and $eB$ [22]. The beauty of this approach is that it reproduces the inverse magnetic catalysis observed in lattice QCD calculations in a very simple way, a feature that the standard NJL model fails do describe.

In Fig. 19.1, we show how the average $u$ and $d$ quark condensates obtained from lattice QCD simulations can be reproduced in the NJL model by introducing a thermo-magnetic coupling, $G(eB, T)$, which is constructed by a fit to the lattice QCD results. Note that the standard NJL model gives a very poor description of the *ab-initio* results.

With the thermo-magnetic coupling, the inverse magnetic catalysis is obtained with the thermo-magnetic coupling, as can be observed in the thermal susceptibilities displayed in Fig. 19.2 and in the pseudo-critical temperature for the chiral transition shown in Fig. 19.3.

Another interesting result is obtained, at zero temperature, by using a $B$-dependent coupling which is also built by performing a fit procedure with lattice QCD results [108]. This can be observed in Fig. 19.4 where we display the condensates and the pion mass as functions of the magnetic field at $T = 0$. The $B$-dependent coupling is constructed
<figure><img src="image_63.png" /><figcaption>**Figure 19.1:** Average *u* and *d* quark condensates as a function of the temperature for some values of the magnetic field, for fixed coupling *G* (left panel) and thermo-magnetic coupling *G*(*eB*, *T*) (right panel).</figcaption></figure>

<figure><img src="image_64.png" /><figcaption>**Figure 19.2:** Thermal susceptibilities as a function of the temperature for some values of the magnetic field, for fixed coupling *G* (left panel) and thermo-magnetic coupling *G*(*eB*, *T*) (right panel).</figcaption></figure>

<figure><img src="image_65.png" /><figcaption>**Figure 19.3:** Pseudo-critical temperature as a function of the magnetic field in three strategies, for a fixed coupling *G* (left panel) and for the thermo-magnetic coupling *G*(*eB*, *T*) (right panel).</figcaption></figure>

to reproduce the average condensates, and the pion mass is a prediction based on the new coupling. In ref.[108], we emphasize the importance of a proper regularization for the excellent agreement between the pion pole mass calculation and LQCD results. This issue is a key point for any calculation using effective models under extreme electromagnetic fields, although, many authors still do not use correct regularization techniques causing, in general, unphysical results for the observables. A common signal of improper regularizations is unphysical oscillations that are artifacts of a wrong calculation [131].
The effects produced by the thermo-magnetic coupling also affects the magnetization of the hadronic matter [199]. A numerical evaluation is summarized in Figs. 19.5 and 19.6, where the pseudo-critical temperature and the renormalized magnetization, obtained in our modified NJL model, are compared to lattice QCD calculations. In another study [530], we computed the magnetization in the case of zero temperature and finite density. Figure 19.7 displays the magnetization of cold hadronic matter for barionic densities between zero and $1 \text{ fm}^{-3}$, considering two values of the magnetic field. In the presence of strong magnetic fields, the rotational symmetry breaking due to the magnetic field, makes the parallel and perpendicular pressure related to the direction of the external field to be different, and the size of this difference is determined by the magnetization. Our results in [530] show that for typical magnetic fields found in magnetars, i.e., a special class of neutron stars with strong surface magnetic fields, the parallel and perpendicular pressures are almost identical.

As a last sample of a recent application of the NJL model, we show the calculation of the neutral $\rho$ meson pole mass in a full random phase approximation, considering vector interactions [123]. In Fig. 19.8, we show the $\rho^0$ mass as a function of the magnetic field background for three different values of the vector interaction.

<figure><img src="image_66.png" /><figcaption>**Figure 19.4:** Left panel: average $u$ and $d$ quark fields as a function of the magnetic field, for fixed coupling $G$ (dashed red line) and thermo-magnetic coupling $G(eB, T)$ (solid blue line), compared to lattice QCD (black points). Right panel: Pion mass as a function of the magnetic field, for fixed coupling $G$ (dashed red line and dotted black line) and thermo-magnetic coupling $G(eB, T)$ (solid blue line), compared to lattice QCD (black points).</figcaption></figure>

<figure><img src="image_67.png" /><figcaption>**Figure 19.5:** Left panel: pseudo-critical temperature as a function of the magnetic field with the thermo-magnetic coupling $G(eB, T)$. Right panel: renormalized magnetization at zero temperature, as a function of the magnetic field, in different calculations.</figcaption></figure>
<figure><img src="image_68.png" /><figcaption>**Figure 19.6:** Renormalized magnetization as a function of the magnetic field, compared to lattice QCD simulations, below the chiral transition (left panel) and above the chiral transition (right panel).</figcaption></figure>

<figure><img src="image_69.png" /><figcaption>**Figure 19.7:** Magnetization as a function of the barionic density for two values of the magnetic field.</figcaption></figure>
<figure><img src="image_70.png" /><figcaption>**Figure 19.8:** Neutral ρ meson mass as a function of the magnetic field for different values of the vector interaction, compared to the effective quark mass times two.</figcaption></figure>
# 20. Background magnetic field helps fixing model parameters

## 20.1. Motivation

In this section of the review, we will briefly focus on how the known properties of a QCD medium in the presence of an external magnetic field such as magnetic catalysis (MC) and inverse magnetic catalysis (IMC) can be utilised to constrain the parameter space of the model. In turn, the constraints can be tested by predicting some observables. It is also interesting to observe how these medium properties can be used to understand better the working principles of such effective models.

After being first observed in lattice QCD, the IMC effect has been observed in various effective models, including the NJL model (local version), the LSM, and the QM model by tweaking the models, particularly by introducing a magnetic field and a temperature-dependent coupling constant [17, 18, 22]. This obviously requires the addition of further parameters to the models. On the other hand, the non-local NJL model can capture the IMC effect through its very construction and does not require the introduction of any additional parameters.

We realise that the running of the coupling constant is the crucial property that is responsible for producing the IMC effect. In the non-local NJL model, the coupling constant becomes momentum-dependent through the form factor. On the other hand, in local versions, the coupling constant becomes a running one only after it is made dependent on temperature and magnetic field.

This particular feature of the non-local NJL model gives us an opportunity to check whether such a model is capable of reproducing other known observables predicted by lattice QCD, even without introducing any new external parameters. In the process, it hands us the scope to constrain the model parameters which are otherwise completely unknown or not so precisely known.

One such parameter is the strength of the 't Hooft determinant term in a 2-flavour NJL model [531]. Its value is governed by a dimensionless parameter, c. There exists literature that provides us with some possible ranges of it [532, 533]. However, no efforts had been made to directly determine its value until recently, when a 2-flavour non-local NJL model in the presence of a magnetic field was utilized to fit its value [534].

Once its value is estimated, it can be further tested by calculating observables such as topological susceptibility ($\chi_t$) at zero magnetic field and predicting its ($\chi_t$) values for non-zero magnetic field [534]. This value of c is further utilised in a study that investigates the intertwined effects of the 't Hooft determinant term and a difference in light quark masses on the QCD corrections to the pion mass difference [535].

For this part of the review, we will first provide the required formalism in short and then briefly describe the process of fitting the parameter c while describing some associated results. We, then, present the results obtained with that fitted c value for different observables

## 20.2. Determining the strength of the 't Hooft determinant term

The version of the Lagrangian that we utilise here is a 2-flavour non-local NJL model written in the Euclidean space as,

$$ \mathcal{L}_{\text{NJL}} = \bar{\psi}(-i\not\partial + \hat{m})\psi - G_1\{j_a(x)j_a(x) + \tilde{j}_a(x)\tilde{j}_a(x)\} - G_2\{j_a(x)\tilde{j}_a(x) - \tilde{j}_a(x)j_a(x)\}, \quad (20.1) $$

where the currents are given by

$$ j_a(x)/\tilde{j}_a(x) = \int d^4z \mathcal{H}(z)\bar{\psi}\left(x + \frac{z}{2}\right)\Gamma_a/\tilde{\Gamma}_a\psi\left(x - \frac{z}{2}\right). \quad (20.2) $$
The Γ's are defined as $\Gamma = (\Gamma_0, \Gamma_j) = (\mathbb{I}, i\gamma_5\tau^j)$ and $\tilde{\Gamma} = (\tilde{\Gamma}_0, \tilde{\Gamma}_j) = (i\gamma_5, \tau^j)$ with $\tau = (\tau^1, \tau^2, \tau^3)$ representing the Pauli matrices. The current quark mass matrix, $\hat{m} = m \times \mathbb{I}$, with $m_u = m_d = m$. $\mathcal{H}(z)$ is the form factor, which if taken to be a $\delta$-function reduces the Lagrangian in Eq. 20.1 into a local one. It is important to note that the second part of the interaction term breaks the $U(1)_A$ term. Thus, putting $G_2 = 0$ makes the Lagrangian symmetric under $SU(2)_V \times SU(2)_A \times U(1)_V \times U(1)_A$ in the chiral limit.

In the absence of isospin symmetry breaking (equal non-zero current quark mass does not break it), the symmetry allows only $\langle \bar{\psi}\psi \rangle$ condensate that depends on the $G_1 + G_2$ combination. On the other hand, if there is an isospin symmetry-breaking agent such as isospin chemical potential ($\mu_I$) and/or magnetic field (eB) there is an explicit breaking of $SU(2)_V$ symmetry. In that circumstance, one can have another condensate such as $\langle \bar{\psi}\tau_3\psi \rangle$ that couples to the fermions with coupling strength $G_1 - G_2$. This motivates to parameterise the coupling constants $G_1 = (1-c)G_0/2$ and $G_2 = cG_0/2$ in terms of a single coupling constant, $G_0$ and the parameter, c. Its value (c) determines the strength of the 't Hooft determinant term, and c = 1/2 reduces the Lagrangian in Eq. 20.1 to that of the usual NJL model.

It is well-known how to obtain the free energy starting from the mean-field version of the Lagrangian given in Eq. 20.1. Eventually, to perform numerical calculation a few more things need to be known. One of them is the form of $\mathcal{H}(z)$, the form factor. The Fourier transform of which is denoted as $g(p)$ and can only be a function of $p^2$ due to the Lorentz invariance: $g(p) = e^{-p^2/\Lambda^2}$. The other important things are the parameters in the model such as $m$, $G_0$ and $\Lambda$. It should be mentioned that $\Lambda$ does not serve as a cut-off in the same way as in the usual local model; instead, it sets the scale of the theory.

These parameters are fixed by fitting known QCD observables such as pion mass ($m_\pi$), pion decay constant ($F_\pi$) and the chiral condensate ($\langle \bar{\psi}\psi \rangle$). Occasionally, these observables are chosen from different studies. However, in the pursuit of determining the c parameter [534] these are taken from a single lattice QCD study [536] to maintain consistency. One needs to be careful while fitting such observables obtained in lattice QCD calculation, for which generally the scale is set at 2 GeV. On the other hand, the scale of the effective model is roughly 1 GeV. In order to express the lattice QCD results at a desired scale of 1 GeV one can utilise the perturbative renormalisation group (RG) running [537].

After utilising perturbative RG running the condensate value becomes $\langle \bar{\psi}\psi \rangle^{1/3}|_{\mu=1 \text{ GeV}} = 224.8(3.7) \text{ MeV}$. On the other hand, $F_\pi$ being a scale-independent quantity remains the same, $F_\pi = 87.3(5.6) \text{ MeV}$. The pion mass is taken to be 135 MeV. To explore the whole observable space while fitting the model parameters one needs to include the errors in both in $\langle \bar{\psi}\psi \rangle$ and $F_\pi$. This makes a total of nine different sets of observables to fit. Out of which, considering the four corner sets along with the central one will be enough to understand all possible physical scenarios arising from different combinations.

Out of those five observable sets, the most important three are quoted here along with the fitted parameter values in Table 20.1. The rest of the sets can be found in Ref. [534]. The letters C, H and L stand for the central, highest and lowest values of the observables, respectively. For each parameter set, the first letter corresponds to the value of the condensate and the second to that of the pion decay constant. With the fitted parameters, one can now proceed with the calculation. So far the whole discussion is in a vacuum. To introduce the magnetic field the non-local currents in Eq. 20.1 need to be modified as

$$j_a(x)/\tilde{j}_a(x) = \int d^4 z \mathcal{H}(z) \bar{\psi} \left(x + \frac{z}{2}\right) W^\dagger \left(x + \frac{z}{2}, x\right) \Gamma_a / \tilde{\Gamma}_a W \left(x, x - \frac{z}{2}\right) \psi \left(x - \frac{z}{2}\right), \quad (20.3)$$

where, $W(s, t) = P \exp [-i\hat{Q} \int_s^t dr_\mu A_\mu(r)]$ with r running over arbitrary path and connecting s and t. $A_\mu(x)$ represents the background magnetic field and $\hat{Q} = \text{diag}(q_u, q_d)$ with $q_u$ and $q_d$ being the electric charge of u and d quark, respectively.
It is to be noted here that the very definition of the currents in Eq. 20.3 helps produce the IMC effect in a non-local NJL model. To get to the non-zero temperature Matsubara formalism is used, which connects the Euclidean time component to the temperature.

<table><caption>**Table 20.1:** Central and two other sets along the fitted parameters of the model given on the right of the double bars.</caption><thead><tr><th></th><th>⟨ψ̄<sub>f</sub>ψ<sub>f</sub>⟩<sup>1/3</sup>(MeV)</th><th>m<sub>π</sub>(MeV)</th><th>F<sub>π</sub>(MeV)</th><th>F<sub>π,0</sub>(MeV)</th><th>m(MeV)</th><th>G<sub>0</sub>(GeV<sup>-2</sup>)</th><th>Λ(MeV)</th></tr></thead><tbody><tr><td>Parameter Set CC</td><td>224.8</td><td>135</td><td>87.3</td><td>84.25</td><td>5.87</td><td>43.34</td><td>697.22</td></tr><tr><td>Parameter Set HH</td><td>228.6</td><td>135</td><td>92.9</td><td>90.63</td><td>6.31</td><td>57.15</td><td>660.46</td></tr><tr><td>Parameter Set LH</td><td>221.1</td><td>135</td><td>92.9</td><td>91.00</td><td>6.94</td><td>80.26</td><td>605.05</td></tr></tbody></table>

In Fig. 20.1, the QCD phase diagram (PD) in the $T-eB$ plane is shown. The band represents the lattice QCD data [13]. The plot displays five data sets [534] including the three given in Table 20.1 that reproduces the decreasing behaviour of the crossover temperature ($T_{CO}$). The IMC effect is also captured for these three sets. It is to be mentioned that although decreasing $T_{CO}$ and the IMC effect accompany each other, the simultaneous occurrence of the two is not guaranteed [538]. The $T_{CO}$ is calculated from the inflection point of the condensate average for each value of $eB$ following the lattice QCD given definition of the condensate [14] as.

$$ \Sigma_{B,T}^{f} = \frac{2m}{\mathcal{N}^4} \left[ \langle \bar{\psi}_f \psi_f \rangle_{B,T}^{\text{reg}} - \langle \bar{\psi}_f \psi_f \rangle_{0,0}^{\text{reg}} \right] + 1, \qquad (20.4) $$

where $\mathcal{N} = (m_\pi F_{\pi,0})^{1/2}$ with $F_{\pi,0}$ being the pion decay constant in the chiral limit. From the three sets which capture

<figure><img src="image_71.png" /><figcaption>**Figure 20.1:** QCD phase diagram in the $T-eB$ plane for different parameter sets and also in lattice QCD.</figcaption></figure>

the decreasing behaviour of $T_{CO}$, the sets LH and HH do it better and are comparable. Thus, these are the two parameter sets for which the fitted $c$-values are further considered.

<figure><img src="image_72.png" /><figcaption>**Figure 20.2:** The condensate difference fitted in $c$.</figcaption></figure>

<table><caption>**Table 20.2:** $\chi^2$ fitting of condensate difference in $c$.</caption><thead><tr><th></th><th>c</th><th>χ<sup>2</sup> per DoF</th></tr></thead><tbody><tr><td>Parameter Set HH</td><td>0.044 ± 0.079</td><td>0.149</td></tr><tr><td>Parameter Set LH</td><td>0.149 ± 0.103</td><td>0.634</td></tr></tbody></table>

In Fig. 20.2, a specimen of the fitting of the light quark condensate difference in $c$ is shown for the set CC. This is
done for zero temperature and the result for $c = 1/2$ is also shown. Similar figures can be obtained for other parameter sets. Two such fitted values for HH and LH sets are given in Table 20.2.

To narrow down further on the best possible parameter set and decide on the $c$-value the $\eta^*$ phenomenology is evoked, $\eta^*$ is nothing but the mass of the fluctuations in the isoscalar pseudoscalar channel [539]. However, it cannot be directly used to fit for $c$ as there is no physical particle representing the $\eta^*$. But physical consideration can help constrain $c$. Considering it to be an admixture of $\eta$ and $\eta'$, the mass of $\eta^*$ to be a few times $\pi^0$. Imposing the physically motivated constrain that $M_{\eta^*} > 400$ MeV it is possible to find a lower bound on $c$ [534]. It is found to be $c > 0.12$. With this constraint, the only viable parameter set is LH and the corresponding fitted value of $c = 0.149_{-0.029}^{+0.103}$.

<figure><img src="image_73.png" /><figcaption>Figure 20.3: The condensate difference for the fitted value of $c$ in the LH parameter set.</figcaption></figure>

With the fitted value of $c$ at zero $T$, the lattice-defined condensate difference at non-zero $T$ for different values of $eB$ is obtained as a prediction of the model. This is shown in Fig. 20.3 along with the lattice QCD result. The gray band in the figure gets excluded with a consideration of $M_{\eta^*} > 400$ MeV resulting in a better match with the lattice QCD result.

## 20.3. Topological susceptibility

The CP-violating topological term that can be added to the QCD Lagrangian is written as [540],

$$ \Delta\mathcal{L} = \epsilon^{\mu\nu\sigma\lambda} \frac{\theta}{64\pi^2} \mathcal{F}_{\mu\nu}^a \mathcal{F}_{\sigma\lambda}^a, \qquad (20.5) $$

where $\theta$ can be related to the axion field ($a$) as $\theta = a/f_a$, with $f_a$ being axion decay constant. This term can be removed by making a suitable $U(1)_A$ transformation on the fermionic fields. Under such a transformation the first interaction term in Eq. 20.1 remains invariant but the second one picks up phase as

$$ G_2 \left[ \cos\theta \{j_a(x)j_a(x) - \tilde{j}_a(x)\tilde{j}_a(x)\} + 2\sin\theta \{j_0(x)\tilde{j}_0(x) - j_i(x)\tilde{j}_i(x)\} \right], \qquad (20.6) $$

with $i$ running from 1 to 3. With this modification the free energy ($\Omega$) is calculated and from there the topological susceptibility ($\chi_t$),

$$ \chi_t = \left. \frac{d^2\Omega}{d\theta^2} \right|_{\theta=0}. \qquad (20.7) $$

In the left panel of Fig 20.4, we show the sensitivity of $\chi_t$ to $c$ for zero $eB$. We have both the fitted $c$-value and $c = 1/2$, the standard NJL model results. For both values of $c$, $\chi_t$ below $T_{CO}$ is the same and is within the lattice result given by the red band [541]. With increasing temperature, they become different and fall at a faster rate than the lattice results [541, 542]. However, at non-zero $eB$ (in the right panel), the model predicts an enhancement in $\chi_t$ below $T_{CO}$, which could be tested in future lattice QCD simulations.
<figure><img src="image_74.png" /><figcaption>**Figure 20.4:** $\chi_t$ for zero $eB$ (left panel) and non-zero $eB$ (right panel). The red and the gray bands are lattice results.</figcaption></figure>

<figure><img src="image_75.png" /><figcaption>**Figure 20.5:** Contour plot of $(\Delta M_{\pi})$ in $\Delta m$ and $c$. The bands are from $\chi$PT calculation (please see text).</figcaption></figure>

## 20.4. QCD corrections to the pion mass difference

We know that the charged and neutral pion mass difference ($\Delta M_{\pi}$) can arise because of both the QED and QCD contributions. The current quark mass difference ($\Delta m$) is responsible for the QCD corrections. In a 2-flavour non-local NJL model (Eq. 20.1) this QCD contribution has been calculated [535] and it is found that 't Hooft determinant parameter, $c$ plays an important role when intertwined with $\Delta m$. This is shown using a contour plot in Fig. 20.5, which has been drawn for different $\Delta M_{\pi}$-values. The magenta and grey bands are from $\chi$PT calculation [261] and [543], respectively.

In the left panel of Fig. 20.6, $\Delta M_{\pi}$ is plotted as a function of $\Delta m$ with the fitted value of $c = 0.149_{-0.029}^{+0.103}$. Using that fitted $c$-value and different $\Delta M_{\pi}$ values from Refs. [261, 543], one can determine the allowed range of $\Delta m$ in the model as

<figure><img src="image_76.png" /><figcaption>**Figure 20.6:** Left panel: $\Delta M_{\pi}$ as a function of $\Delta m$ for the fitted $c$ value from Ref. [534]. The bands are obtained using $\chi$PT calculations. Right panel: the $\Delta m$ dependence of the low energy constant $l_7$. For details, please check the main text.</figcaption></figure>
$$ \Delta m = 2.59 \binom{+0.22}{-0.24} \binom{+1.26}{-0.35} \text{ MeV for } \Delta M_{\pi} \text{ from Ref. [261], and} \qquad (20.8) $$

$$ \Delta m = 3.55 \binom{+0.97}{-1.37} \binom{+1.73}{-0.48} \text{ MeV for } \Delta M_{\pi} \text{ from Ref. [543].} $$

In turn, these $\Delta m$-values enable us to extract the chiral perturbation theory low-energy constant, $l_7$, given in the right panel of Fig. 20.6. It is also observed that the value of $l_7$ increases with increasing $c$. The estimated $l_7$ value using the 2-flavour non-local NJL model is $l_7 = 17.2_{-9.4}^{+5.8} \times 10^{-3}$. Such an estimation is within the range provided by other studies [260, 544, 545].
# 21. van der Waals-type models

## 21.1. Overview

Another class of effective models for description of strongly interacting matter, more specifically protons and neutrons as degrees of freedom at different regimes of temperature and density, is based on classical extension of ideal gases thermodynamics, namely, the van der Waals (vdW) model [546, 547], i.e., the nucleon is considered as a finite size hard-core object (not a particle anymore). In the context of the nuclear matter system, the exclude volume (EV) effect is responsible for the repulsive part of the nuclear interaction, and the intermediate range inter-particle attraction is represented by the pressure reduction. However, this phenomenology is not enough to completely describe nucleonic systems for which quantum statistical effects are clearly non-negligible (the classical Maxwell-Boltzmann fluid is not suitable), and even relativistic effects can also be important in certain domains. A complete treatment in which these effects are also taken into account was performed for the first time in Ref. [548]. The energy density at $T = 0$ for symmetric nuclear matter (SNM) in this approach reads

$$ \mathcal{E}(\rho) = \frac{2(1 - b\rho)}{\pi^2} \int_0^{k_F^*} dk \, k^2 (k^2 + M^2)^{1/2} - a\rho^2, \qquad (21.1) $$

where $M$ is the nucleon rest mass, $\rho$ is the total density, $k_F^* = (3\pi^2\rho^*/2)^{1/3}$, $\rho^* = \rho/(1-b\rho)$, and $b$ is the EV parameter. The two free parameters ($a$, and $b$) are found by imposing to the model a bound state at the saturation density $\rho_0$ ($\approx 0.15 \text{ fm}^{-3}$) with the correct value for the binding energy ($\approx -16 \text{ MeV}$ [549]). A similar implementation is addressed in Ref. [550]. The extension of this model to finite temperature regime allows the study of first-order liquid-gas phase transition in nuclear matter at moderate temperatures, and the thermodynamical analyses at sufficiently high temperatures. In this case, the Fermi-Dirac distribution is not a step function as in the zero temperature regime. As an example of $T > 0$ calculations, we cite the critical temperature found in Ref. [548], $T_c \approx 19.7 \text{ MeV}$, close to some experimental estimates for this quantity [551, 552, 553]. Furthermore, attractive and repulsive nucleon-nucleon interactions of the nucleonic vdW model were included in the hadron resonance gas model in Ref. [554]. The second-order susceptibilities of some conserved charges calculated from this approach are shown to be compatible with lattice data for a large range of $T$. An improvement to the aforementioned description was proposed in Ref. [555], where the generalized version of the model takes into account possible density dependence of the constants $a$ and $b$. The new energy density expression is constructed by making $b \to \mathcal{B}(\rho)$ and $a \to \mathcal{A}(\rho)$ in Eq. (21.1), in the notation of Refs. [556, 557]. A proper choice for the function $\mathcal{A}(\rho)$ selects the type of real gas model to be used in the description of SNM. Redlich-Kwong-Soave [558, 559], Peng-Robinson [560], and Clausius equations of state are possible options, see Ref. [555]. The function $\mathcal{B}(\rho)$ accounts for the method used to implement the excluded volume to the system, such as the one based on the Carnahan-Starling (CS) model [560]. The traditional EV procedure is recovered by taking $\mathcal{B}(\rho) = b$. This particular choice, along with $\mathcal{A}(\rho) = a$, identifies the previous nucleonic vdW model.

A direct challenge for these models is to make them more capable of describing as many strongly interacting matter properties as possible. In this direction, we address the reader to Ref. [561] for a generalized multicomponent system vdW model, useful to treat asymmetric nuclear matter. In a different way, the authors of Refs. [556, 557] proposed a new form for the $\mathcal{A}(\rho)$ function, namely,

$$ \mathcal{A}(\rho) = \frac{a}{(1 + b\rho)^n}, \qquad (21.2) $$

and a new term in the energy density, proportional to the squared difference between protons and neutrons densities inspired by the popular relativistic mean-field (RMF) models [562, 563, 564]. It mimics the exchange of the $\rho$ meson
and, consequently, generates the inequality of the numbers of protons and neutrons in the system. The four parameters of this model, named as density-dependent vdW (DD-vdW) model, are determined in order to reproduce the values of saturation density, binding energy, incompressibility, and symmetry energy (the last three quantities evaluated at $\rho = \rho_0$ and $y_p = 0.5$, with $y_p$ being the proton fraction of the system).

## 21.2. Causality

A typical drawback of EV models is the broken causality at high-density regimes for nuclear matter at $T = 0$ generating, as a consequence, superluminal equations of state. The Lorentz contraction of the hard-sphere nucleons should be taken into account in order to correct this issue, and then, it could be interpreted as a nucleon with the effective volume being a decreasing density-dependent function. An attempt to use the CS mechanism for this aim is still limited since the range of densities in which the model is causal is still not enough for studying, for instance, compact stars (a common dense matter system in which different hadronic models can be probed). However, the use of Eq. (21.2) in addition to the CS procedure in the DD-vdW model is shown to extend the region of causality, as depicted in Fig. 21.1a. Another way of

<figure><img src="image_77.png" /><figcaption>**Figure 21.1:** Squared sound velocity as a function of the ratio $r = \rho/\rho_0$ for (a) DD-vdW model, (b) CCS model with (full lines) and without (dashed lines) SRC included, for incompressibility values of $K_0 = 220$ MeV and $K_0 = 260$ MeV. Results for SNM at $T = 0$ in both panels.</figcaption></figure>

circumventing this problem is to implement the Lorentz contraction in the rest frame of the medium in a proper way, as done in Ref. [565] (effect presented in two finite size nucleons). One more option to overcome the difficulty in dense matter, with a nice application in neutron stars, was proposed in Ref. [566] with the use of the induced surface tension, generated by the repulsion between the nucleons at the surface and the outer nucleons. The increase of the causality range, in comparison with the simplest version of the vdW model, allows the generation of equations of state capable of reproducing massive stars, as well as configurations compatible with SNM constraints, such as the flow constraint [567].

## 21.3. Short-range correlations

One more very important feature observed in nuclear systems is the phenomenon of short-range correlations (SRC) [568, 569, 570, 571, 572], recently included into vdW-like models in Ref. [573]. SRC was probed in some experiments, such as those performed at the Thomas Jefferson National Accelerator Facility [572], where it was observed that collisions of very energetic incident particles in target nuclei induce the removal of correlated pairs of nucleons, mostly neutron-proton pairs, with large relative momentum due to the tensor force in the neutron-proton isosinglet channel. As a consequence, the single-nucleon momentum distribution ($T = 0$ case) is modified in comparison with the usual step function. The
asymmetric matter version of this quantity can be written as [574]

$$n_{p,n}(k) = \begin{cases} \Delta_{p,n}, & 0 < k < k_F^{p,n}, \\ C_{p,n} \frac{(k_F^{p,n})^4}{k^4}, & k_F^{p,n} < k < \phi_{p,n} k_F^{p,n}, \end{cases} \quad (21.3)$$

with $\Delta_{p,n} = 1 - 3C_{p,n}(1 - 1/\phi_{p,n})$, $C_p = C_0[1 - C_1(1 - 2y_p)]$, $C_n = C_0[1 + C_1(1 - 2y_p)]$, $\phi_p = \phi_0[1 - \phi_1(1 - 2y_p)]$ and $\phi_n = \phi_0[1 + \phi_1(1 - 2y_p)]$. Here the normalization condition was used, i.e., the momentum integral of $n_{p,n}(k)$ is imposed to be equal to $\rho_{p,n} = (k_F^{p,n})^3/3\pi^2$, with $k_F^{p,n}$ being the Fermi momentum of protons/neutrons. The constants $C_0, C_1, \phi_0$, and $\phi_1$ are determined from SRC experiments, as pointed out in Ref. [575]. It is worth noticing the new structure in $n_{p,n}(k)$ induced by SRC: the depletion under the nucleon Fermi momentum, and the so-called high momentum tail (HMT) above $k_F^{p,n}$. The shape of the HMT, proportional to $1/k^4$, is a property also observed in other quantum systems, such as two-component cold fermionic atoms [575]. Data from nucleon knockout reactions induced by high-energy particles can be used to determine the HMT fraction in SNM and in pure neutron matter, around 28% and 1.5% respectively [568, 576, 577].

<figure><img src="image_78.png" /><figcaption>**Figure 21.2:** CCS model for SNM at $T = 0$: (a) pressure as a function of ratio $r = \rho/\rho_0$ for $K_0 = 220$ MeV and $K_0 = 260$ MeV, model with SRC. Flow constraint extracted from Ref. [567]. (b) The slope of symmetry energy ($L_0$) as a function of symmetry energy ($J$), both quantities at the saturation density. Model for $K_0 = 220$ MeV with (full lines) and without (dashed lines) SRC included.</figcaption></figure>

An interesting application of SRC in dense fermionic system was analyzed in Ref. [578], where it is shown that one of the effects of this particular phenomenology is the capability of producing more massive neutron stars in comparison with the ones obtained from equations of state without SRC included. Furthermore, the average nucleon kinetic energy extracted from the RMF model used in Ref. [578] is shown to be compatible with electron-nucleus scattering data, feature not observed if SRC are not implemented in the model. The same is valid for the kinetic symmetry energy of the SRC induced model, i.e, this quantity is in agreement with results provided by microscopic many-body theories [579, 580, 581]. Later on, further studies with RMF-SRC models presented: (i) approximate analytical expressions for the nucleon chemical potentials [582], as well as the impact of SRC in the constraints established by the LIGO/Virgo Collaboration concerning the tidal deformabilities of compact stars, (ii) study on neutron star matter admixed with dark matter [583, 584, 585], and (iii) influence of SRC on the pasta phase [586]. As already mentioned, SRC were also applied to vdW-like models [573], more specifically, in parametrizations of the real gas Clausius model with CS method of excluded volume adopted and the density-dependent function given by $A(\rho) = a/(1 + c\rho)$, model named as CCS-SRC. In this case, it is verified that SRC move the break of causality to higher densities in comparison with the model without this phenomenology implemented, as displayed in Fig. 21.1b. At this dense matter region, it is also observed that CCS-SRC model completely satisfies the flow constraint [573], see Fig. 21.2a. It is also worth noticing the enhancement in the symmetry energy slope caused by SRC in the CCS-SRC model, as we can see in Fig. 21.2b. It is verified as well, that the values obtained for this quantity
are in agreement with the range of $L_0 = (106 \pm 37)$ MeV [587], compatible with the updated results provided by PREX-2 collaboration concerning the neutron skin thickness of $^{208}$Pb [588]. However, other studies are pointing out smaller ranges for $L_0$, see for instance Ref. [589], in which the range $L_0 = (54 \pm 8)$ MeV was obtained from theoretical uncertainties of the parity-violating asymmetry in $^{208}$Pb. Another different range, namely, $L_0 = (37 - 66)$ MeV, was found in Ref. [590] from ab initio calculations for the $^{208}$Pb nucleus. Analysis from the CREX Collaboration [591], performed in Ref. [592], determined the range of $L_0 = (-5 \pm 40)$ MeV. Combined studies of results from PREX-2 and CREX through Bayesian inference in Ref. [593], and through a covariance analysis in Ref. [594], found the ranges of found $L_0 = 15.3_{-41.5}^{+46.8}$ MeV and $L_0 = (82.32 \pm 22.93)$ MeV, respectively.

## 21.4. More improvements and new challenges

It is expected that at a high-density regime, the strongly interacting matter at zero temperature undergoes a hadron-quark phase transition. Such a phenomenon can be described by means of two different models representing the respective sectors of the system at this extreme condition. Nevertheless, this is not the unique procedure used for this description, see for instance, the concept of “quarkyonic matter”, presented in Ref. [595], and later on discussed in Ref. [596] in terms of a momentum space shell structure. In this particular phase of matter, deconfined quarks are surrounded by confined baryons, and a hadron-quark mixture is verified. The dynamical mechanism governing this phenomenon is the following: at a low-density regime, baryons degrees of freedom are responsible for the maximum momentum shell width (radius of the sphere in the momentum space). On the other hand, as density increases it is observed the emergence of a quark Fermi sea and, consequently, the shell width is decreased since at this stage quark matter is energetically more favorable than the baryonic one. An inner “quark” sphere appears, and the width is measured from the outer baryon surface to the inner quark one. Another picture of the hadron-quark mixture was studied in Ref. [597], where the opposite mechanism is proposed: confined baryons are surrounded by a shell of deconfined quarks, i.e., the quark surface is now the outer one, in contrast to the baryonic inner surface. The system described by this configuration is called “baryquark matter” [597]. Both, quarkyonic and baryquark matter properties were recently incorporated into vdW-like models in Ref. [598] with some interesting features observed, such as the value of the transition density (transition from normal matter to the quarkyonic/baryquark one) as being around $(1.5 - 2.0)\rho_0$; the need of introducing a regulator to avoid the singular behavior in the sound velocity at the onset of quark appearance; the baryquark matter being energetically favored; among other findings.

A natural improvement yet to come in this particular approach for describing strongly interacting matter at extreme conditions is the development of unified vdW-like models capable of simultaneously describing all aforementioned phenomenology shown in this section and in previous ones, as well as account for other current challenges such as (i) implementation of strangeness by means of inclusion of hyperons [599] with the aim of solving the so-called “hyperon-puzzle”; (ii) addition of magnetic field in the equations of state in order to enable the models to describe magnetars [600] as well as the magnetic field configuration presented in rapidly rotating pulsars; (iii) full agreement with the new observational astrophysical data coming from the advent of the multi-messenger astronomy, such as gravitational wave observations related to the binary neutron star merger (event called GW170817 [601]); and (iv) compatibility with observations provided by the Neutron Star Interior Composition Explorer (NICER) X-ray timing telescope, coupled to the International Space Station in 2017, concerning the neutron star radius [602, 603, 604, 605].
## 22. Supersymmetric Quantum Mechanics and the Dynamics of Dirac Matter Under External Electromagnetic Fields

Supersymmetry or SUSY is a theoretical framework in particle physics that proposes a symmetry between fermions and bosons in extensions to the Standard Model. The first mathematical construction dates back to the Wess-Zumino model [606]. Supersymmetry gained traction as it offered elegant solutions to several theoretical challenges (see, for example, Ref. [607]), such as the hierarchy problem, Higgs stabilization, grand unification of fundamental interactions and even provided a candidate for dark matter. Despite the lack of direct experimental confirmation of its existence to date, supersymmetry remains a pivotal and highly studied component of modern theoretical physics, promising profound insights into the fundamental structure of the universe.

Supersymmetry in quantum mechanics (SUSY-QM) is a powerful framework that offers significant advantages and insights in the study of quantum systems, making it a valuable tool in theoretical and mathematical physics [608]. One of the primary uses of SUSY-QM is its ability to provide exact solutions to certain quantum mechanical problems that are otherwise difficult to solve. By introducing a symmetry between bosonic and fermionic states, SUSY-QM allows for the construction of partner Hamiltonians whose spectra are related, simplifying the analysis of complex systems. SUSY-QM is instrumental in understanding the algebraic structures underlying quantum mechanics. It offers a natural context for exploring the properties of shape-invariant potentials, which are central to solving the Schrödinger equation for a variety of potentials [608]. This has practical applications in fields such as molecular physics, statistical mechanics, optics and condensed matter physics, just to mention some examples.

In discussing the main ideas of SUSY-QM, one often starts from two Schrödinger-like Hamiltonians $H^±$ which are intertwined by means of an operational relation

$$H^+L^- = L^-H^-. \qquad (22.1)$$

This is known as intertwining relation, where $L^- = d/dx + w(x)$ is the intertwining operator, and $w(x)$ the superpotential function. The corresponding potentials $V^±(x)$ can be written in terms of the superpotential as

$$V^{\pm}(x) - \epsilon = w^2(x) \pm \frac{dw(x)}{dx}, \qquad (22.2)$$

where $\epsilon$ is a constant called factorization energy. Thus, the supersymmetric partner Hamiltonians $H^±$ turn out to be factorized as

$$H^+ - \epsilon = L^-L^+, \quad H^- - \epsilon = L^+L^-, \qquad (22.3)$$

being $L^+ = (L^-)^† = -d/dx + w(x)$. All these elements satisfied the supersymmetric algebra, which is defined with the following commutation rules:

$$[H_{SS}, Q^{\pm}] = 0, \quad \{Q^{+}, Q^{-}\} = H_{SS}. \qquad (22.4)$$

Often, the supersymmetric Hamiltonian $H_{SS}$ and the superchargers are written as $2 \times 2$ matrices of the form

$$Q^+ = \begin{pmatrix} 0 & L^+ \\ 0 & 0 \end{pmatrix}, \quad Q^- = \begin{pmatrix} 0 & 0 \\ L^- & 0 \end{pmatrix}, \quad H_{SS} = \begin{pmatrix} H^- - \epsilon & 0 \\ 0 & H^+ - \epsilon \end{pmatrix}. \qquad (22.5)$$

The primary use of the supersymmetric transformations defined in Eq. (22.1) is to construct a supersymmetric partner Hamiltonian $H^+$ from an initial Hamiltonian $H^-$ whose eigenfunctions $\psi_n^-(x)$ and eigenvalues $E_n$ are known. This process
also involves determining the eigenfunctions $\psi_n^+(x)$ of $H^+$ from those of $H^-$. It is worth noting that a consequence of the supersymmetric algebra is that the supersymmetric partner Hamiltonians are either isospectral or their spectra differ only by one energy level (the factorization energy level), depending on the square-integrability conditions of the respective eigenfunctions. The supersymmetric algorithm to build a supersymmetric partner Hamiltonian is the following: Given a seed solution $u(x)$ satisfying $H^-u(x) = \epsilon u(x)$, the superpotential function can be obtained as

$$w(x) = -\frac{du(x)}{dx}, \qquad (22.6)$$

thus, using Eq. (22.2), we can calculate the supersymmetric partner potential $V^+(x)$. Moreover, from Eqs. (22.1) and (22.3), assuming the eigenfunctions $\psi_n^-(x)$ are normalized, we arrive at

$$\psi_n^{\pm}(x) = \frac{L^{\mp}\psi_n^{\mp}(x)}{\sqrt{E_n - \epsilon}}. \qquad (22.7)$$

Finally, the eigenfunction $\psi_\epsilon^+$ of $H^+$ corresponding to the factorization energy $\epsilon$ is given by

$$\psi_{\epsilon}^{+}(x) = \frac{1}{u(x)}, \qquad (22.8)$$

except for a normalization factor. This algorithm describes a first-order supersymmetric transformation since $L^\pm$ are first-order differential operators. However, the supersymmetric algorithm can be generalized if the intertwining operators are taken as second-order differential ones. In this process, two seed solutions and their respective factorization energies are necessary to construct the supersymmetric partner Hamiltonian but the intertwining relation in Eq. (22.1) remains. In general, we can carry out a k-th order supersymmetric transformation. For a more complete discussion on this issue, see Refs. [609, 610, 611, 612, 613] and references within.

General k-th order SUSY-QM is a hard nut to crack. Yet, second-order SUSY-QM has been proven to be a very useful tool. In a second-order SUSY transformation, the Hamiltonians $H^\pm$ are intertwined by a second-order operator $L_2^-$, which can be written as

$$L_2^- = \frac{d^2}{dx^2} + \eta(x) \frac{d}{dx} + \gamma(x), \qquad (22.9)$$

where $\eta(x)$ and $\gamma(x)$ are functions to be determined. The corresponding intertwining relation is

$$H^+ L_2^- = L_2^- H^- . \qquad (22.10)$$

The SUSY partner potentials $V^\pm(x)$ and the function $\gamma(x)$ can be written in terms of the function $\eta(x)$ as follows:

$$\begin{aligned} V^{+}(x) &= V^{-}(x) + 2\frac{d\eta(x)}{dx}, & \gamma(x) &= \frac{\eta^{2}(x)}{2} - \frac{d\eta(x)}{dx} - V^{-}(x) + \frac{\epsilon_{1} + \epsilon_{2}}{2}, \\ V^{-}(x) &= \frac{d^{2}\eta(x)}{dx^{2}} - \left(\frac{d\eta(x)}{dx}\right)^{2} - \frac{d\eta(x)}{dx} + \frac{\eta^{2}(x)}{4} + \left(\frac{\epsilon_{1} + \epsilon_{2}}{2}\right) + \left(\frac{\epsilon_{1} - \epsilon_{2}}{2\eta(x)}\right)^{2}. \end{aligned} \qquad (22.11)$$

This time, there are two factorization energies $\epsilon_1$ and $\epsilon_2$ associated with two seed solutions $u_1(x)$ and $u_2(x)$, which are solutions of the Hamiltonian $H^-$. In terms of the seed solutions, the function $\eta(x)$ is given by

$$\eta(x) = -\frac{dW(u_1(x), u_2(x))/dx}{W(u_1(x), u_2(x))}, \qquad (22.12)$$

being $W(f,g) = fg' - f'g$ the Wronskian. Furthermore, the eigenfunctions $\psi_n^\pm(x)$ for the Hamiltonians $H^\pm$ are related by the following equalities:

$$\psi_n^{\pm}(x) = \frac{L_2^{\mp} \psi_n^{\mp}(x)}{\sqrt{(E_n - \epsilon_1)(E_n - \epsilon_2)}}, \qquad (22.13)$$
while the functions $\psi_{\epsilon_j}^+$ turn out to be

$$ \psi_{\epsilon_1}^+ = \frac{u_2(x)}{W(u_1(x), u_2(x))}, \quad \psi_{\epsilon_2}^+ = \frac{u_1(x)}{W(u_1(x), u_2(x))}. \qquad (22.14) $$

Finally, the products of the intertwining operators $L_2^\pm$ are second-degree polynomials on $H^\pm$, i.e.,

$$ L_2^- L_2^+ = (H^+ - \epsilon_1)(H^+ - \epsilon_2), \quad L_2^+ L_2^- = (H^- - \epsilon_1)(H^- - \epsilon_2). \qquad (22.15) $$

First and second order SUSY-QM have been widely implemented in exploring the dynamics of non-relativistic charged particles in external static electromagnetic fields. This is valuable for exploring the dynamics of Dirac particles under similar circumstances, as we show below.

## 22.1. SUSY and the Dirac equation

The supersymmetric structure of the Dirac equation reveals a deep connection between SUSY and relativistic quantum mechanics. The Dirac equation, which describes the behavior of fermions such as leptons and quarks, can be reinterpreted within a supersymmetric framework, highlighting symmetries between different quantum states. The Dirac equation in natural units ($\hbar = c = 1$) is given by:

$$ (i\gamma^\mu \partial_\mu - m)\psi = 0, \qquad (22.16) $$

where $\gamma^\mu$ are the Dirac matrices, $\partial_\mu$ is the four-gradient, $m$ is the mass of the particle, and $\psi$ is the Dirac spinor. To reveal the SUSY structure in the Dirac equation, we consider the stationary form of the Dirac equation:

$$ H\psi = E\psi, \qquad (22.17) $$

with

$$ H = \alpha \cdot p + \beta m. \qquad (22.18) $$

Here, $\alpha_i = \gamma^0 \gamma^i$ and $\beta = \gamma^0$, with $p$ the momentum operator.

In a suitable representation, the Hamiltonian $H$ can be decomposed into a block form that resembles a pair of supersymmetric partner Hamiltonians:

$$ H = \begin{pmatrix} 0 & A \\ A^\dagger & 0 \end{pmatrix}, \qquad (22.19) $$

where $A$ and $A^\dagger$ are operators that act on the components of the spinor $\psi$. The supercharges $Q$ and $Q^\dagger$ can be defined as:

$$ Q = \begin{pmatrix} 0 & 0 \\ A & 0 \end{pmatrix}, \quad Q^\dagger = \begin{pmatrix} 0 & A^\dagger \\ 0 & 0 \end{pmatrix}. \qquad (22.20) $$

These supercharges satisfy the SUSY algebra:

$$ \{Q, Q^\dagger\} = H, \quad Q^2 = (Q^\dagger)^2 = 0. \qquad (22.21) $$

In this framework, the operators $A$ and $A^\dagger$ correspond to the creation and annihilation operators in the context of the Dirac equation. They map solutions of the Dirac equation to one another, revealing the underlying supersymmetric structure.

The supersymmetric structure shows that for every solution of the Dirac equation with positive energy, there is a corresponding solution with negative energy, forming a supersymmetric pair. This pairing reflects the symmetry between particles and antiparticles.
## 22.2. Bound states in external electromagnetic fields

The influence of external magnetic fields into the Dirac equation can be straightforwardly included through the minimal substitution replacement $\partial_{\mu} \rightarrow D_{\mu} = \partial_{\mu} + ieA_{\mu}$, where $A_{\mu}$ is the vector potential giving raise to the external fields. Bound states for Dirac fermions have been widely explored for static fields. In the case of magnetic fields, the uniform magnetic field configuration has lead to a lot of interest. For large magnetic fields compared to the rest mass of the Dirac particle, it is known that such configuration gives rise to the relativistic Landau levels, which are of the utmost importance in many branches of physics [614]. Going beyond this configuration, adding the influences of static electric fields, either parallel or perpendicular to the magnetic field, and the plane wave configuration are among the handful of examples that can be exactly solvable by standard technique [615].

SUSY-QM provides the means to analytically find the bound states in static magnetic fields with varying profiles, such as an exponentially decaying field, a Pöschl-Teller-like well, hyperbolic well, a trigonometric singular well, a singular field, a hyperbolic singular field [608, 616, 617]. Moreover, second-order SUSY-QM gives modifications of these profiles [618, 619]. Even, it is possible to obtain complex fields [620]. Many features of the SUSY structure of 2D Dirac fermions in uniform magnetic fields were discussed in [621].

The effect of combined electric and magnetic fields has been valuable to explore transport properties of Dirac matter. For instance, under the influence of the parallel electric and magnetic fields, a chiral magnetic current is induced in $ZrTe_5$ [622]. This current is directly proportional to the applied magnetic field strength and the chiral chemical potential difference induced by the electric field. A negative longitudinal magnetoresistance is observed, which is a signature of the chiral anomaly. This effect manifests as a decrease in electrical resistance with increasing magnetic field strength when the fields are aligned. This is the signature of the Chiral Magnetic Effect, which has been observed in several Dirac-Weyls semimetals as predicted first in the context of heavy ion collisions [436, 623, 624].

Recently, a configuration in which static and non-uniform parallel electric and magnetic fields was considered [625]. Such configuration is applied to a 3D Dirac material and SUSY-QM is used to solve the eigenvalue problem, finding bound states for hyperbolic and trigonometric profiles. It is worth mentioning that the zero-energy bound state has an associated probability current due to the chiral symmetry conservation, which is perpendicular to the fields but in the same plane, thus defining a chiral planar Hall effect (PHE). This phenomenon has been also addressed in Weyl semimetals with perturbations, such as strains, in other words, the Weyl semimetals are in the presence of electromagnetic pseudofields, and an in-plane transverse voltage appears in the material [626]. Furthermore, in the particular case of bilayer graphene, a non-linear PHE was studied, as a consequence of an orbital effect from the in-plane magnetic field on the electrons without spin-orbit coupling [627].

## 22.3. Green Functions

In many physical situations, finding the wave functions for electronic states it is equally useful as to know the corresponding propagator. However, because the asymptotic in- and out-states in the background of a classical electromagnetic field do not correspond to plane waves, representing the two-point function is cumbersome, making it nearly impossible to express the propagator in a closed form except in a few specific cases, such as for uniform electric or magnetic fields (either parallel or perpendicular) and plane wave electromagnetic fields [615]. To address this, alternative representations have been developed. Among these, the Schwinger method [615] and the spectral representation, the Ritus method [628, 629, 630] which was pedagogically discussed in [631], provide ways to express the propagator in a closed form.
We focus our attention to the expansion of the propagator in the basis of Ritus functions which correspond to the eigenfunctions of the operator $(\gamma \cdot \Pi)^2$ where $\Pi_\mu = p_\mu + eA_\mu$ is the canonical momentum operator that includes the effect of the external magnetic field through minimal coupling (with $A_\mu$ denoting the corresponding vector potential and $e$ is the elementary charge) and $\gamma^\mu$ denote the Dirac matrices. Considering a general situation in which a static magnetic field of general spatial profile points perpendicularly to the plane of motion of Dirac fermions, working in a Landau-like gauge, we introduce an electromagnetic potential $A^\mu = (0, 0, W_0(x), 0)$, where $W_0(x)$ is a scalar function such that $W'_0(x) = \partial_x W_0(x)$ defines the profile of the field. In these circumstances, the fermion propagator cannot be diagonalized on a plane wave basis because, as mentioned before, the asymptotic states of these fermions in a background field do not correspond to the kinematical momentum eigenfunctions. Motivated by this observation, we notice that the Green function for Dirac particles, $G(z, z')$, satisfies

$$((\gamma \cdot \Pi) - m)G(z, z') = \delta^{(3)}(z - z'), \qquad (22.22)$$

with $z^\mu = (t, x, y, z)$, $\gamma^\mu$ denoting the Dirac matrices and $\Pi_\mu$ is the canonical momentum. We omit the Lorentz index in the vectors to keep a shorthand notation when necessary. Since $G(z, z')$ commutes with $(\gamma \cdot \Pi)^2$, we expand the propagator on the basis of the eigenfunctions of the later, namely,

$$(\gamma \cdot \Pi)^2 \mathbb{E}_p(z) = p^2 \mathbb{E}_p(z), \qquad (22.23)$$

where the eigenvalue $p^2$ can be any real number corresponding, as we shortly will see, to the magnitude squared of the vector $p^\mu$ (or simply $p$ to avoid cumbersome notation) that labels the functions $\mathbb{E}_p(z)$. We refer to the functions $\mathbb{E}_p(z)$ as the Ritus eigenfunctions [628, 629, 630]. It can be directly verified that these functions $\mathbb{E}_p(z)$ fulfill the closure and completeness relations

$$\int d^4 z \bar{\mathbb{E}}_{p'}(z) \mathbb{E}_p(z) = \mathbb{I} \hat{\delta}(p - p'), \qquad (22.24a)$$

$$\int d^4 p \mathbb{E}_p(z') \bar{\mathbb{E}}_p(z) = \mathbb{I} \hat{\delta}(z - z'), \qquad (22.24b)$$

with $\bar{\mathbb{E}}_p(z) = \gamma^0 \mathbb{E}_p^*(z) \gamma^0$ and $\mathbb{I}$ is the unit matrix. The hat over the $\delta$ function refers to the situation when quantum numbers induced by the external fields should be considered discrete labels, whereas those that remain unafected are continuous variables.

In order to construct the Ritus eigenfunctions, we notice that the operator

$$(\gamma \cdot \Pi)^2 = \gamma^\mu \gamma^\nu \Pi_\mu \Pi_\nu = \Pi^2 + \frac{e}{2} \sigma^{\mu\nu} F_{\mu\nu}, \qquad (22.25)$$

where $F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$ is the electromagnetic field strength tensor and $\sigma^{\mu\nu} = i[\gamma^\mu, \gamma^\nu]/2$. For a static magnetic field pointing perpendicularly to the plane, the only non-vanishing components of these tensors are

$$F_{12} = -F_{21} = W'_0(x), \quad \sigma^{12} = \Sigma_3. \qquad (22.26)$$

Then, the eigenvalue Eq. (22.23) becomes

$$(\Pi^2 + e\Sigma_3 W'_0(x))\mathbb{E}_p(z) = p^2\mathbb{E}_p(z), \qquad (22.27)$$

from where we observe that the Ritus eigenfunctions are actually block matrices, depending upon the dimensionality of the representation of the Dirac matrices $\gamma^\mu$. Notice that the Ritus functions are labeled by a subscript $p$, which is the
shorthand notation of the vector $p^\mu = (p_0, p_2, p_3, k)$ is a vector that contains the eigenvalues of the operators $i\partial_t, -i\partial_y, \partial_z$ and $\mathcal{H}_\sigma$, respectively, and whose norm corresponds to the eigenvalue in Eq. (22.23). That is, the components of the vector $p^\mu$ are the numbers such that

$$i\partial_t \mathbb{E}_p(z) = p_0 \mathbb{E}_p(z), \quad i\partial_y \mathbb{E}_p(z) = -p_2 \mathbb{E}_p(z), \quad i\partial_z \mathbb{E}_p(z) = p_3 \mathbb{E}_p(z), \quad \mathcal{H}_\sigma \mathbb{E}_p(z) = k \mathbb{E}_p(z), \qquad (22.28)$$

with $\mathcal{H}_\sigma = -(\gamma \cdot \Pi)^2 + \Pi_0^2 + \Pi_3^2$. These eigenvalues allow us to write the scalar functions as

$$E_{p,\sigma}(x) = e^{-i(p_0 t - p_2 y - p_3 z)} F_{k,p_2,\sigma}(x), \qquad (22.29)$$

where $\sigma = \pm 1$ are the eigenvalues of $\sigma_3$ and the functions $F_{k,p_2,\sigma}(x)$ satisfy

$$[-\partial_x^2 + (p_2 + eW_0(x))^2 - e\sigma W'_0(x)] F_{k,p_2,\sigma} = (k + p_3^2) F_{k,p_2,\sigma}, \qquad (22.30)$$

which corresponds to a Pauli equation for a particle with mass $m = 1/2$ and gyromagnetic factor $g = 2$. This equation possesses a supersymmetric structure as we will briefly discuss below. Thus, $F_{k,p_2,\sigma}(x)$ are the solutions of the equations in (22.30) associated to each of the supersymmetric-partner potentials

$$V_0^\sigma(x) = (p_2 + eW_0(x))^2 - e\sigma W'_0(x). \qquad (22.31)$$

From now on, we fix the value $\sigma = 1$. Then, we have the required ingredients to construct the Ritus eigenfunctions from a first-order supersymmetric formalism.

Therefore, we can use these functions to diagonalize the fermion propagator $S(z, z')$ in momentum space in the same way plane waves are used to define the Fourier transform,

$$S(z, z') = \int d^4 p \, d^4 p' \mathbb{E}_p(z) \, S_F(p, p') \bar{\mathbb{E}}_{p'}(z'). \qquad (22.32)$$

Inserting this Green's functions in Eq. (22.22), using the property [628]

$$(\gamma \cdot \Pi) \mathbb{E}_p(z) = \mathbb{E}_p(z) (\gamma \cdot \bar{p}), \qquad (22.33)$$

where $\bar{p}$ is the shorthand notation to define the three-momentum vector $\bar{p}^\mu = (p_0, 0, \sqrt{k}, p_3)$ that satisfies $\bar{p}^2 = p^2 = p_0^2 + p_3^2 - k$ [631] and the properties (22.24a), the propagator in momentum space takes the form

$$S_F(p) = \frac{1}{\gamma \cdot \bar{p} - m}, \qquad (22.34)$$

similar to the free-particle propagator, but the momentum $\bar{p}$, which carries the quantum numbers induced on the dynamics of Dirac fermions by in the presence of the external field. In the configuration space, we write the propagator as

$$\begin{aligned} S(z, z') &= \int d^4 p \, d^4 p' \mathbb{E}_p(z) \left[ \frac{1}{\gamma \cdot \bar{p} - m} \right] \bar{\mathbb{E}}_{p'}(z') \\ &= \int d^4 p \, d^4 p' \mathbb{E}_p(z) \left[ \frac{\gamma \cdot \bar{p} + m}{p^2 - m^2} \right] \bar{\mathbb{E}}_{p'}(z'). \end{aligned} \qquad (22.35)$$

From this expression we can find the value of the vacuum current density

$$j^\mu = \mathrm{Tr}[\gamma^\mu S(z, z)]. \qquad (22.36)$$

The propagator has been widely explored for constant magnetic fields in the context of the phenomenon of magnetic catalysis (see, for example, [632, 633, 634, 635, 636, 637]). Extensions to exponentially decaying magnetic fields were
considered in [638] exploring the local electric current correlation function. In low dimensions, this propagator has been discussed for uniform and exponentially decaying fields [631, 639]. It has also been discussed for second-order SUSY-generated magnetic fields generated from these profiles as seed fields [640]. Furthermore, the same Ritus basis has been used to derive the Foldy-Wouthuysen transformation in external magnetic fields, rendering it to a free from with the quantum numbers induced by the field [641].

## 23. Quark condensate nonequilibrium dynamics under a strong magnetic field

In this section, we address the non-equilibrium dynamics of the quark condensate under a strong magnetic field. Time evolution is a common feature of strongly interacting matter in settings such as the early universe, collapsing stars, and heavy-ion collisions. A complete understanding of the condensate dynamics in these settings requires solving a complex nonequilibrium quantum many-body problem in QCD. Although solving such a problem is beyond our present capabilities, we can nevertheless gain insight into it by focusing on some general aspects that transcend features peculiar to a specific setting. Within such a perspective, we employ a Langevin field equation to describe the dynamics [642]. Langevin field equations are widely used in field theory treatments of dynamical phase transitions [643, 644]; they are either motivated phenomenologically or derived from a microscopic model for the system of interest—for an extensive list of references on these approaches, see Ref. [642]. In this communication, we discuss a recent derivation [642] of a Langevin field equation from the linear sigma model [645] with quarks (LSMq) coupled to an external magnetic field. We also present new results [646], not contemplated in the original publication [642], regarding the long-time evolution of condensate and its thermalization.

The Langevin equation we discuss here was derived in Ref. [642] using the Schwinger-Keldysh closed-time path (CTP) effective action [647, 648] and the Feynman-Vernon influence functional (IF) [649] formalisms of nonequilibrium quantum field theory [650, 651]. More specifically, Ref. [642] built on the semiclassical approach developed in Ref. [652] to include magnetic-field effects. The derived Langevin equation contains magnetic field-dependent damping and noise kernels that reflect the condensate's interactions with a locally equilibrated magnetized quark medium. These kernels determine time scales associated with the chiral crossover dynamics in the presence of a magnetic field.

The explicit numerical results presented here relate to a quench scenario of dynamical phase transitions, using parameters relevant to heavy-ion collisions. In this scenario, the thermodynamic state of the system changes much more rapidly than any variation in an external parameter or field. A classic example is a temperature quench in a spin system, where a sudden drop in temperature drives the system irreversibly from a spin-disordered phase into a spin-ordered phase. Although our results focus on parameters pertinent to heavy-ion collisions, the derived Langevin equation is also applicable to other contexts, such as the early universe and collapsing stars.

### 23.1. Langevin equation from the LSMq

We outline the derivation in Ref. [642] of the Langevin field equation for the condensate. Due to the lack of space, we cannot go into much detail, but the reader is directed to that reference for an in-depth discussion. To make the discussion self-contained and set the notation, we write the LSMq Lagrangian density used to derive the Langevin equation describing
the quark condensate dynamics:

$$ \mathcal{L} = \bar{q}[i\not\partial - g(\sigma + i\gamma_5 \boldsymbol{\tau} \cdot \boldsymbol{\pi})]q + \frac{1}{2}[\partial_{\mu}\sigma\partial^{\mu}\sigma + \partial_{\mu}\boldsymbol{\pi}\cdot\partial^{\mu}\boldsymbol{\pi}] - \frac{\lambda}{4}(\sigma^2 + \boldsymbol{\pi}^2 - v^2)^2 - h_q\sigma - U_0, \quad (23.1) $$

where $q = (u, d)^T$ is a fermion isodoublet representing the light u and d quark Dirac fields, and $\boldsymbol{\pi}$ and $\sigma$ are respectively a pseudoscalar-isotriplet and scalar-isoscalar fields that Yukawa-couple to the quarks with strength $g$. Finally, $U_0$ is an arbitrary constant that sets to zero of the classical ground-state energy. We use the metric signature $g^{\mu\nu} = (1, -1, -1, -1)$ and the Bjorken-Drell [653] conventions for the Dirac $\gamma^\mu$ matrices, for which $\{\gamma^\mu, \gamma^\nu\} = 2g^{\mu\nu}$. One can fit the parameters of the model to chiral physics observables—a fit at the classical level, for example, sets the parameters as: $h_q = f_\pi m_\pi^2$, $v^2 = f_\pi^2 - m_\pi^2/\lambda$, $m_\sigma^2 = 2\lambda f_\pi^2 + m_\pi^2$, and $m_q = g\langle\sigma\rangle$. Here $f_\pi$ and $m_\pi$ are the pion weak decay constant and mass, $m_\sigma$ the $\sigma$-meson mass, and $m_q$ the constituent-quark mass. The external magnetic field is introduced by the minimal substitution, in that $\partial_\mu \to D_\mu = \partial_\mu + iqA_\mu$, where $A_\mu$ the electromagnetic vector field and $q$ is the quark charge. In this study pions are neglected; therefore, the $\sigma \leftrightarrow \bar{q}q$ processes are the only source of dissipation, as we discuss at the end of this section. The parameters of the model were set so that in the mean-field approximation and $B=0$ it leads to a crossover transition; the values are [652] $g=3.3$ and $\lambda=20$. These lead to $m_\sigma = 604$ MeV and $m_q = 290$ MeV. The crossover temperature is $T_{pc} \simeq 150$ MeV, close to the lattice QCD value [654], $T_{pc} = 160$ MeV. Although the LSMq in the mean-field approximation does not reproduce lattice QCD results close to the chiral transition when $B \neq 0$, it nevertheless provides a great deal of insight in many aspects of magnetic field effects in hadron physics. In addition, going beyond the mean-field approximation by e.g. including it can be improved to self-coupling and boson-fermion coupling.

The starting point for the derivation of the Langevin field equation for $\sigma$ is the semi-classical effective action:

$$ \Gamma[\sigma, S] = \Gamma_{\text{cl}}[\sigma] + i \text{Tr} \ln S - i \text{Tr} (i \not{D} - m_0) S + \Gamma_2[\sigma, S], \quad (23.2) $$

where $\Gamma_2[\sigma, S]$ contains the 2PI diagrams, for which it was taken the fermion tadpole:

$$ \Gamma_2[\sigma, S] = g \int_C d^4 x \text{tr} [S^{++}(x, x)\sigma^{+}(x) + S^{--}(x, x)\sigma^{-}(x)], \quad (23.3) $$

where $\sigma^\pm$ mean fields and the quark propagators $S_+, S_-$, and $S_\pm$ are defined on the Schwinger-Keldysh contour $C$ shown in Fig. 23.1, and tr means trace over color, flavor, and Dirac indices. The two fields $\sigma^\pm(x)$ are not independent;

<figure><img src="image_79.png" /><figcaption>Figure 23.1: Schwinger-Keldish contour.</figcaption></figure>

they couple through the CTP boundary condition $\sigma^+(T, \boldsymbol{x}) = \sigma^-(T, \boldsymbol{x})$ where $T \to +\infty$ (borders of the CTP contour) so that there is a single mean field $\sigma(x)$ given by the equation of motion motion [650]:

$$ \left. \frac{\delta \Gamma[\sigma, S]}{\delta \sigma^{+}(x)} \right|_{\sigma^{+} = \sigma^{-} = \sigma} = \left. \frac{\delta \Gamma[\sigma, S]}{\delta \sigma^{-}(x)} \right|_{\sigma^{+} = \sigma^{-} = \sigma} = 0. \quad (23.4) $$

Next, quarks are integrated out by using their equation of motion obtained by varying the action w.r.t. to the components of $S$:

$$ \frac{\delta \Gamma[\sigma, S]}{\delta S^{ab}(x, y)} = 0, \quad (23.5) $$
which leads to

$$ (i\not D - g\sigma_0(x)) S^{ab}(x, y) - \int_C d^4 z \frac{\delta\Gamma_2[\sigma, S]}{\delta S^{ac}(x, z)} S^{cb}(z, y) = i\delta^{ab}\delta^{(4)}(x-y). \quad (23.6) $$

The above equations of motion are very difficult to solve, even numerically. A tractable approximate scheme to solve those equations is to “expand around the local equilibrium”, in a sense similar to the hydrodynamic description of many-particle systems. That is, the full $\sigma$ and $S$ are written as a sum of terms with the leading contribution being the corresponding equilibrium quantities $\sigma_0$ and $S_{\text{thm}}$ at fixed values of temperature $T$ and magnetic field. Specifically:

$$ \sigma^a(x) = \sigma_0^a(x) + \delta\sigma^a(x), \quad (23.7) $$

$$ S^{ab}(x, y) = S_{\text{thm}}^{ab}(x, y) + \delta S^{ab}(x, y) + \delta^2 S^{ab}(x, y) + \dots, \quad (23.8) $$

in which the leading terms are solutions of

$$ \frac{\delta\Gamma_{\text{cl}}}{\delta\sigma_0^a(x)} = -g\text{Tr}S^{aa}(x, x), \quad (23.9) $$

$$ [i\not D - m_0 - g\sigma_0(x)] S_{\text{thm}}^{ab}(x, y) = -i\delta^{ab}\delta^{(4)}(x-y), \quad (23.10) $$

and the terms $\delta S^{ab}(x, y)$ and $\delta^2 S^{ab}(x, y)$ are constructed iteratively [652, 642]. Equation (23.10) was solved in Ref. [642] for a constant magnetic field and using the lowest Landau-level (LLL) approximation, which is valid only for very strong magnetic fields. The equation is easily solved in momentum space. Taking the magnetic field pointing in the $z$-direction, one can define transverse and longitudinal momentum components $\mathbf{p}_{\perp}^2 = p_x^2 + p_y^2, p_{\parallel}^2 = p_0^2 - p_z^2$, in terms of which the individual $S_{\text{thm}}^{ab}$ components of the thermomagnetic quark propagator are given by [642]:

$$ S_{\text{thm}}^{++}(p) = e^{-\mathbf{p}_{\perp}^2/q_f B} A(p) \left[ \frac{i}{p_{\parallel}^2 - m_q^2 + i\epsilon} - 2\pi n_F(p_0) \delta(p_{\parallel}^2 - m_q^2) \right], \quad (23.11) $$

$$ S_{\text{thm}}^{+-}(p) = e^{-\mathbf{p}_{\perp}^2/q_f B} A(p) 2\pi\delta(p_{\parallel}^2 - m_q^2) [\theta(-p_0) - n_F(p_0)], \quad (23.12) $$

$$ S_{\text{thm}}^{-+}(p) = e^{-\mathbf{p}_{\perp}^2/q_f B} A(p) 2\pi\delta(p_{\parallel}^2 - m_q^2) [\theta(p_0) - n_F(p_0)], \quad (23.13) $$

$$ S_{\text{thm}}^{--}(p) = e^{-\mathbf{p}_{\perp}^2/q_f B} A(p) \left[ \frac{-i}{p_{\parallel}^2 - m_q^2 - i\epsilon} - 2\pi n_F(p_0) \delta(p_{\parallel}^2 - m_q^2) \right], \quad (23.14) $$

where $A(p) = (p_{\parallel} + m_q) [1 + i\gamma^1\gamma^2\text{sign}(qB)]$ with $p_{\parallel} = \gamma^0 p_0 - \gamma^3 p_z$ and $m_q = g\sigma_0, q_u = 2e/3, q_d = -e/3$, where $e = 1/\sqrt{137}$, and $n_F(p_0)$ is the Fermi-Dirac distribution:

$$ n_F(p_0) = \frac{1}{e^{|p_0|/T + 1}}. \quad (23.15) $$

The asymmetry between the dependencies $\mathbf{p}_{\perp}$ and $p_z$ in these propagator components imposes an asymmetry in the Langevin field equation, as we shall discuss in the following.

The traditional method of varying the action $\Gamma[\sigma, S]$ w.r.t. $\sigma^{\pm}$ to obtain an e.o.m. is not well defined because the action is complex. The Feynman-Vernon trick is used to identify the imaginary part with a noise source coupling linearly to the field such that a real action is obtained and the variation w.r.t. $\sigma$ leads to a Langevin equation for $\sigma$. The resulting Langevin equation features dissipation and noise kernels with memory. The presence of memory in those kernels obstructs an analytical treatment, and further simplification is necessary to deal with such effects analytically. A common way to proceed is to use the so-called linear harmonic approximation, in that the memory dynamics is
effectively taken into account by soft-mode (long wavelengths) harmonic oscillations around a mean field—for details, see Refs. [652, 655, 656, 642]. This leads to the following Langevin equation in momentum-space [642]:

$$ \frac{\partial^2 \sigma(t, \boldsymbol{p})}{\partial t^2} + \boldsymbol{p}^2 \sigma(t, \boldsymbol{p}) + \eta_{\sigma}(\boldsymbol{p}_{\perp}) \frac{\partial \sigma(t, \boldsymbol{p})}{\partial t} + F_{\sigma}(t, \boldsymbol{p}) = v_{\sigma}(t, \boldsymbol{p}), \qquad (23.16) $$

where $\eta_{\sigma}(\boldsymbol{p}_{\perp})$ is the transverse-momentum-dependent dissipation coefficient

$$ \eta_{\sigma}(\boldsymbol{p}_{\perp}) = g^2 \frac{N_c}{4\pi} \left[1 - 2n_F(E_{\sigma}(\boldsymbol{p}_{\perp})/2T)\right] \frac{1}{E_{\sigma}^2(\boldsymbol{p}_{\perp})} \sqrt{E_{\sigma}^2(\boldsymbol{p}_{\perp}) - 4m_q^2} \sum_{f=u,d} |q_f B| e^{-\boldsymbol{p}_{\perp}^2/2|q_f B|}, \qquad (23.17) $$

with $E_{\sigma}(\boldsymbol{p}_{\perp}) = (\boldsymbol{p}_{\perp}^2 + m_{\sigma}^2)^{1/2}$. The noise field $v_{\sigma}(t, \boldsymbol{p})$ has zero mean, $\langle v_{\sigma}(t, \boldsymbol{p}) \rangle_v = 0$, and its correlation is also controlled by $\eta_{\sigma}(t, \boldsymbol{p}_{\perp})$, namely:

$$ \langle v_{\sigma}(t, \boldsymbol{p}) v_{\sigma}(t, \boldsymbol{p}) \rangle_v = (2\pi)^3 \delta(\boldsymbol{p} + \boldsymbol{p}') \delta(t - t') N_{\sigma}(\boldsymbol{p}_{\perp}), \qquad (23.18) $$

where

$$ N_{\sigma}(\boldsymbol{p}_{\perp}) = \eta(\boldsymbol{p}_{\perp}) E_{\sigma}(\boldsymbol{p}_{\perp}) \coth(E_{\sigma}(\boldsymbol{p}_{\perp})/2T), \qquad (23.19) $$

a feature reflecting the fluctuation-dissipation theorem. Finally, $F_{\sigma}(t, \boldsymbol{p})$ is given by

$$ F_{\sigma}(t, \boldsymbol{p}) = \int d^3 x e^{-i\boldsymbol{p} \cdot \boldsymbol{x}} \left[ \lambda \sigma(t, \boldsymbol{x}) (\sigma^2(t, \boldsymbol{x}) - v^2) + g \rho_s(\sigma_0) + h_q \right], \qquad (23.20) $$

with the scalar density $\rho_s(\sigma_0)$ given by the sum $\rho_s(\sigma_0) = \rho_s^{BT}(\sigma_0) + \rho_s^B(\sigma_0)$, where $\rho_s^{BT}(\sigma_0)$ depends on B and T and $\rho_s^B(\sigma_0)$ depends only on B:

$$ \rho_s^{BT}(\sigma_0) = -\frac{N_c}{\pi^2} m_q (|q_u B| + |q_d B|) \int_0^{\infty} dp_z \frac{n_F(E_q(p_z))}{E_q(p_z)}, \qquad (23.21) $$

$$ \rho_s^B(\sigma_0) = -\frac{N_c}{2\pi^2} m_q \sum_{f=u,d} |q_f B| \left[ \ln \Gamma(x_f) - \frac{1}{2} \ln 2\pi + x_f - \frac{1}{2}(2x_f - 1) \ln x_f \right], \qquad (23.22) $$

where $N_c = 3$ is the number of colors, $n_F(E_q)$ is the Fermi-Dirac distribution given in Eq. (23.15) with $E_q = \sqrt{p_z^2 + m_q^2}$, $x_f = m_q^2/(2|q_f B|)$, and $\Gamma(x)$ the Euler gamma function. Equation (23.16) displays the $(\boldsymbol{p}_{\perp}, p_z)$ asymmetry imposed by the magnetic field, as mentioned above. More specifically, the asymmetry comes through the $\boldsymbol{p}_{\perp}$ dependence of the dissipation coefficient—Ref. [652] deals with zero magnetic fields and obtains a momentum-independent $\eta_{\sigma}$, namely:

$$ \eta_{\sigma}(eB = 0) = g^2 \frac{2N_c}{4\pi} \left[1 - 2n_F(m_{\sigma}/2T)\right] \frac{1}{m_{\sigma}^2} (m_{\sigma}^2 - 4m_q^2)^{3/2}. \qquad (23.23) $$

Equation (23.16) describes the nonequilibrium dynamics of the condensate, the field $\sigma$. Compared to the traditional mean field equation of motion for $\sigma$, Eq. (23.16) is modified by two important physical effects: dissipation and fluctuation, the first is described by the first-order time derivative and the second by the noise field $v$. Both effects are driven by the dissipation coefficient $\eta_{\sigma}$, which is sourced by the $\sigma \leftrightarrow \bar{q}q$ processes; from Eq. (23.17), one sees that if $E_{\sigma} < 2m_q$, $\eta_{\sigma}$ vanishes. The magnetic field affects both the sigma mass and the quark mass, and the temperature dependence of $\eta_{\sigma}$ might be very different from that of $B = 0$, as we discuss shortly. If pions were included, they would also contribute to dissipation through the $\sigma \leftrightarrow \pi\pi$ processes.

## 23.2. Explicit results

Before presenting explicit numerical solutions of the Langevin equation (23.16) and analyzing the effect of the magnetic field, it is instructive to recall well-known qualitative facts about the solutions at short and long times. We consider
a high-to-low temperature quench, such that at $t = 0$ the temperature $T \gg T_{pc}$ suddenly drops to a value $T < T_{pc}$. Right after the temperature quench, the field $\sigma$ is still small and one can neglect the term $\sigma^3$ in $F_\sigma$ and analytically solve (23.16) for $\sigma(t, \mathbf{p})$ of the field, since the equation becomes linear in $\sigma(t, \mathbf{p})$. The solution for the field average $\langle \sigma'(t, \mathbf{p}) \rangle_v \equiv \bar{\sigma}(t, \mathbf{p})$ is of the form:

$$ \bar{\sigma}(t, \mathbf{p})|_{t \approx 0} = A_{-}(\mathbf{p}) e^{\lambda_{+}(\mathbf{p}) t/2} + A_{-}(\mathbf{p}) e^{\lambda_{-}(\mathbf{p}) t/2}, \quad \lambda_{\pm}(\mathbf{p}) = -\eta_{\sigma} \pm \sqrt{\eta_{\sigma}^2 + 4(\lambda v^2 - \mathbf{p}^2)}, \quad (23.24) $$

where $A_\pm(\mathbf{p})$ are determined by initial conditions and are irrelevant for the following discussion. This result shows the well-known short-time “explosion” of an order parameter after a temperature quench [657, 658]. The explosion, i.e., the short-time exponential growth, is driven by long-wavelength fluctuations, i.e., fluctuation modes with $\mathbf{p}^2 < \lambda v^2$, for which $\omega_+ > 0$, with the zero mode $\mathbf{p} = 0$ giving the most important contribution. However, short-wavelength fluctuations, for which $\mathbf{p}^2 > \lambda v^2$, decay exponentially with time (for $\mathbf{p}^2 > \eta_\sigma^2/4 + \lambda v^2$ they are damped oscillations). The time scales involved in this short-time behavior depend on the values of $\eta_\sigma(\mathbf{p}_\perp)$ and, of course, on $\lambda v^2$, which is related to the $\sigma$ mass, as we discuss shortly. As time increases, the linear equation cannot be used and one needs to solve the full, nonlinear equation. The initial explosion of the explosion is stopped by the term $\sigma^3$ in $F_\sigma$. For long times, the field eventually thermalizes, that is, it reaches an equilibrium value. Denoting $\sigma'(t, \mathbf{p})$ the fluctuations around the equilibrium value $\sigma_{eq}$, the Langevin equation for $\sigma'(t, \mathbf{x})$ can, again, be linearized, and the resulting equation for $\sigma'(t, \mathbf{x})$ is in the form of a stochastically driven damped harmonic oscillator. The average of $\sigma'(t, \mathbf{x})$ is given by the homogeneous solution of this equation, namely:

$$ \bar{\sigma}'(t, \mathbf{p})|_{t \to \infty} = e^{-\eta_{\sigma} t/2} \left[ A_{-}(\mathbf{p}) e^{i\omega(\mathbf{p})t} + A_{-}(\mathbf{p}) e^{-i\omega(\mathbf{p})t} \right], \quad \omega(\mathbf{p}) = \sqrt{\mathbf{p}^2 + \lambda(3\sigma_{eq}^2 - v^2) - \eta_{\sigma}^2/4}. \quad (23.25) $$

For the parameter values used in this work, the frequencies $\omega(\mathbf{p})$ are real for any value of $\mathbf{p}^2$, a characteristic that implies that the long-time behavior of the field is in the form of damped oscillations, as should be, since the system has to reach equilibrium for a nonzero value of $\eta_\sigma$.

Next, we present numerical results of the full Langevin equation and make contact with the short- and long-time behaviors just discussed. We first discuss the impact of a strong magnetic field on the dissipation coefficient $\eta_\sigma$, which is a key parameter in the time evolution of $\sigma$, as seen above. We limit our discussion to the contribution of the zero mode $\mathbf{p}_\perp = 0$ to $\eta_\sigma$, since it is the most important mode in the dynamics of $\sigma$, as seen above. Therefore, from Eq. (23.17), one sees that the value of $\eta_\sigma$ is nonzero only when $m_\sigma > 2m_q$. For $eB = 0$, the study in [652] has shown that $\eta_\sigma$ vanishes below the pseudocritical temperature $T_{pc} = 150$ MeV. For $eB \neq 0$, the situation changes substantially [642]: $m_\sigma$, $m_q$ and the transition temperature increase (magnetic catalysis) [659], $m_\sigma$ and can be greater than $2m_q$ and so $\eta_\sigma$ can be nonzero even below the new critical temperature. In Table 23.1 we show selected results for $\eta_\sigma$ from [642] for two values of temperature and $eB$ —we recall that we are restricted to use relatively high values of $eB$ due to the use of the LLL approximation. We have chosen the temperature values $T = 50$ MeV and $T = 100$ MeV to emphasize a quench scenario for the simulation results we present shortly. The results shown in the table reveal that, for a given temperature, $\eta_\sigma$ increases with $eB$. This result is driven partially by the linear dependence on $eB$ shown in Eq. (23.17), which is a direct consequence of the dimensional reduction brought about by the magnetic field: the dimensional reduction replaces the $(m_\sigma^2 - 4m_q^2)^{3/2}$ in the zero field expression for $\eta_\sigma$ in Eq. (23.23) by $(m_\sigma^2 - 4m_q^2)^{1/2} eB$. On the other hand, for a given value of $eB$, $\eta_\sigma$ decreases with $T$, because both $m_\sigma$ and $m_q$ decrease with $T$.

We numerically solved the Langevin equation in coordinate space. We solved the equation in the $(x, y)$ plane, perpendicular to the magnetic field; in doing so, we took the field independent of the $z$ coordinate. We used a leapfrog
<table><caption>**Table 23.1:** Values of the zero-mode dissipation coefficient, $\eta_{\sigma}(\mathbf{p}_{\perp} = 0)$.</caption><thead><tr><th>$eB/m_{\pi}^2$</th><th>$T$ [MeV]</th><th>$m_{\sigma}$ [MeV]</th><th>$m_q$ [MeV]</th><th>$\eta_{\sigma}$ [fm<sup>-1</sup>]</th></tr></thead><tbody><tr><td rowspan="2">0</td><td>50</td><td>601</td><td>307</td><td>0</td></tr><tr><td>100</td><td>448</td><td>271</td><td>0</td></tr><tr><td rowspan="2">10</td><td>50</td><td>726</td><td>342</td><td>1.12</td></tr><tr><td>100</td><td>690</td><td>339</td><td>0.62</td></tr><tr><td rowspan="2">15</td><td>50</td><td>810</td><td>370</td><td>1.89</td></tr><tr><td>100</td><td>775</td><td>367</td><td>1.49</td></tr></tbody></table>

algorithm to evolve the field in discrete time and a finite differences to discretize the spatial derivatives [646, 660, 661]. Figure 23.2 displays the results for the spatial average of $\sigma$, that is, the average of $\sigma$ over the lattice, with the lattice extent being 10 fm. The results displayed are for two temperature quenches, $T_{qch} = 50$ MeV and $T_{qch} = 100$ MeV, and two values of the magnetic field, $eB = 50$ MeV and $eB = 100$ MeV, the same as those in Table 23.1.

<figure><img src="image_80.png" /><figcaption>**Figure 23.2:** Time dependence of average of the quark condensate corresponding to the temperature and $eB$ values in Table 23.1.</figcaption></figure>

Both the short-time fast growth and the long-time damped oscillations discussed above are clearly visible in both panels of the figure. Short-time growth occurs within a time interval of about of 1 fm/c after the quench in all cases. The growth is controlled by $\lambda_{+}(0) = \sqrt{4\lambda\nu^2 + \eta_{\sigma}^2 - \eta_{\sigma} > 0}$, which grows with $T$ because $\eta_{\sigma}$ decreases with $T$ for the parameter values used in this work. Thermalization of the field occurs within a time interval of 10 fm/c for both temperature quenches and $eB = 15 m_{\pi}^2$, whereas for $eB = 10 m_{\pi}^2$, only for $T_{qch} = 50$ MeV it thermalized within that time interval. We recall from Eq. (23.25) that the thermalization of the field is determined mainly by the value of $\eta_{\sigma}$ which, for $eB = 10 m_{\pi}^2$ for $T_{ech} = 50$ MeV is the smallest among all the values considered in Table 23.1.

## 23.3. Conclusions and perspectives

We addressed the non-equilibrium dynamics of the quark condensate under a strong magnetic field within the framework developed in Ref. [642]. That framework bases the dynamics on a mean-field Langevin equation derived from the linear sigma model using the Schwinger-Keldysh closed-time path effective action and the Feynman-Vernon influence functional formalisms of non-equilibrium quantum field theory. We summarized the main steps involved in the derivation of the
Langevin field equation and presented new results regarding the long-time evolution of the condensate and discussed its thermalization, results that were not contemplated in the original publication [642]. Although the results presented have omitted important physical effects, they served the purpose of getting insight and developing an intuition on what to expect in a more complete calculation. Among other effects, a more complete calculation must include those due to pions and take into account the time dependence of the magnetic field. In addition, to address phenomenological consequences of the time evolution of the quark condensate in a heavy-ion setting, one needs to address the expansion and magnetohydrodynamics of the medium. We close by remarking that the framework developed in Ref. [642] can be adapted to study magnetic field effects on the QCD phase transition in the early universe and in the interior of magnetized compact stars (magnetars).
## Dense Magnetized Matter and Phase Diagram

### 24. Introduction

Quantum chromodynamics (QCD) encompasses multiple phases characterized by different degrees of freedom [662]. Quarks and gluons, though fundamental to the theory, are not directly observable experimentally; instead, we observe their bound states, namely mesons and baryons. It is well known that phenomena at low energies are primarily governed by the dynamics of the lightest mesons. Considerable effort has been devoted to exploring QCD under extreme conditions such as very high temperatures and densities [663]. A significant challenge lies in understanding the physics governing quark-gluon plasma (QGP), a novel state of matter observed experimentally as a thermally equilibrated, deconfined phase of nuclear matter. Numerous experiments involving heavy-ion collisions (HIC) are actively investigating this unique phase of QCD matter to provide insights into constructing an accurate QCD phase diagram.

From a theoretical perspective, substantial efforts have been directed towards elucidating the QCD phase diagram. However, our understanding remains limited primarily due to the necessity of performing QCD calculations in the non-perturbative regime, which is currently impractical. Challenges in the ab initio lattice QCD approach, particularly in regions of moderately high densities due to the sign problem, further complicate this endeavor [664, 665]. Consequently, much of our current understanding of the QCD phase diagram derives from effective models, which offer predictions for phenomena inaccessible through lattice techniques.

Recent years have seen a notable focus on the generation of strong magnetic fields during noncentral heavy-ion collisions, reaching strengths up to $10^{20}$ G. These fields, emerging immediately after the collision, can significantly impact the phases of QCD, often exceeding or matching the $\Lambda_{QCD}$ scale. The study of strong magnetic fields in systems described by QCD has seen significant growth in various fields, including magnetars, non-central heavy-ion collisions, neutron star mergers, and early universe physics. Recent reviews provide comprehensive insights into the phase structure and transitions of QCD matter under strong magnetic fields [193, 194, 30, 666, 339]. The number of works investigating quark/nuclear matter under intense magnetic fields has expanded greatly. Studies using lattice QCD simulations have explored the effects of strong magnetic fields on hadron spectra and scenarios of chiral symmetry restoration, yielding nontrivial results such as inverse magnetic catalysis [14] at high temperatures and unexpected behaviors of neutral mesons [121].

In this Part, we delineate recent theoretical advancements that enhance the continuous exploration of the magnetic field effects on the QCD phase diagram and dense magnetized matter. In section 25 a new mechanism for accelerating proto-neutron stars involving the chiral separation effect is proposed, which induces an axial vector current in dense environments. This study focuses on neutrino scattering interactions with the background axial vector current of electrons. It has become clear that the presence of anisotropy, whether in the magnetic field or density within momentum space, is essential for generating a nonzero recoil effect. This phenomenon, termed "chiral anisotropy conversion", requires conditions such as a strong magnetic field and moderate anisotropy. Under these conditions, the results suggest that chiral anisotropy conversion can lead to velocities of the order of typical pulsar kicks, comparable to velocities observed in nature.

In Section 26, we examine the impact of incorporating ring diagrams into the four-quark interaction on the quark
gap equation within a chiral model. This in-medium coupling is shown to decrease the chiral transition temperature in certain chiral models and has the potential to induce inverse magnetic catalysis at finite temperatures. Furthermore, the influence of confining forces on this phenomenon will also be analyzed.

In Sec. 27 we apply finite energy sum rules (FESR) in dense nuclear matter and in the presence of a constant and uniform external magnetic field to obtain information on different hadronic and QCD parameters. Nucleon coupling currents and hadronic continuum thresholds can be obtained from the nucleon-nucleon correlator. The hadronic threshold signals confinement or deconfinement, so it is the main parameter provided in this formalism. The objective of this work is the study of the nuclear axial coupling constant, which can also be obtained through the correlation of nucleons and axial-vector current. This scenario of dense, magnetized matter emulates the interior of magnetars, so it may give an idea of what to expect there. The direct signal is the neutrino emission through the Urca process, which is a function of nuclear axial coupling. As a result, it is confirmed that dense matter tends to deconfine, while the magnetic field promotes confinement. It is also confirmed that the axial coupling at nuclear saturation density is ~ 1. Apparently, in this order of density values, the magnetic field has no important relevance, but it does for the most dilute or extremely dense nuclear matter.

In Sec. 28 we investigate the momentum diffusion coefficients of heavy quarks in a hot, magnetized medium, examining a wide spectrum of external magnetic field strengths. Our methodology involves systematically integrating the magnetic field's effects by using effective gluon and quark propagators designed for such a medium. To achieve gauge independence and derive analytical form factors applicable to all Landau levels, we utilize the hard thermal loop approach to the effective gluon propagator. By applying this resummed gluon propagator along with the generalized Schwinger quark propagator, we are able to analytically determine both longitudinal and transverse momentum diffusion coefficients for charm and bottom quarks, going beyond the static approximation.

In Sec. 29 the exploration of the magnetized QCD phase diagram is of considerable interest due to its potential to replicate fundamental properties of matter found in magnetars, the primordial universe, and peripheral ultrarelativistic heavy-ion collisions. Effective theories and models are commonly employed in the non-perturbative QCD regime because of their simplicity, which facilitates the investigation of conditions involving high temperatures, densities, and magnetic fields. Additionally, novel environments and effects could provide connections between effective models and lattice QCD. In this context, the quark anomalous magnetic moment (AMM) presents an opportunity to probe new phenomena within the magnetized QCD phase diagram. Current literature, using the Nambu–Jona-Lasinio model, predicts notable effects such as potential first-order phase transitions induced by strong magnetic fields, inverse magnetic catalysis at both zero and finite temperatures, and various non-physical oscillations in quark condensates. These phenomena are often associated with regularization procedures that intertwine magnetic field and vacuum contributions within the thermodynamic potential. This study will elucidate why these effects result from regularization issues and how the vacuum magnetic regularization (VMR) scheme can effectively address these challenges.
# 25. Axial-vector current and implications to the phase diagram and the pulsar kick

## 25.1. Induced vector and axial-vector currents

It is a widely accepted idea that the magnetic field coupled with the vector (axial vector) field induces the axial vector (vector) current, which is concisely summarized as $j_A^\mu \propto \epsilon^{\mu\nu\rho\sigma}(\partial_\nu\phi_V)F_{\rho\sigma}$ and $j_V^\mu \propto \epsilon^{\mu\nu\rho\sigma}(\partial_\nu\phi_A)F_{\rho\sigma}$. The best known examples include the chiral magnetic effect (CME) and the chiral separation effect (CSE) [667, 668], that is,

$$ \dot{j}_V = \frac{\mu_5 e \mathbf{B}}{2\pi^2}, \qquad \dot{j}_A = \frac{\mu e \mathbf{B}}{2\pi^2} \qquad (25.1) $$

for single Dirac fermion with the electric charge $e$. Here, $\mu$ is the chemical potential coupled to the fermion number (if the fermion represents a single-flavor quark, $\mu$ is the quark chemical potential), while $\mu_5$ is the chiral chemical potential. We note that a finite mass should modify $j_A$, while the CME current $j_V$ is robust with respect to infrared scales such as the mass and the temperature.

Usually, these vector expectation values are considered as realization of topological transport. Of course, this interpretation is completely legitimate. Nevertheless, it is strange that almost nobody had taken account of $j_V$ and $j_A$ for the phase diagram research of magnetized quark matter. Actually, it is a common approach to postulate an effective interaction of four fermions, that is,

$$ \mathcal{L}_{\text{int}} = -\sum_{i} G_i \bar{\psi} \Gamma_i \psi \bar{\psi} \Gamma_i \psi. \qquad (25.2) $$

The interaction bases should contain the scalar, the pseudo-scalar, the vector, the axial-vector, and the spin (anti-symmetric tensor) channels. Therefore, the mean-field interaction from $\Gamma = \gamma^\mu$ (and $\gamma^\mu\gamma^5$) has a contribution proportional to the squared quantity of $j_V$ (and $j_A$, respectively). Let us specifically consider the CME current only. Then, the mean-field interaction reads [669]:

$$ \mathcal{L}_{\text{intMF}} = -G_V \dot{j}_V \cdot \dot{j}_V + 2G_V \dot{j}_V \bar{\psi} \gamma \psi. \qquad (25.3) $$

The first term is the mean-field energy and the second term is regarded as the gauge coupling. That is, in view of the Dirac kinetic term, $\bar{\psi}i\gamma_\mu\partial^\mu\psi$, the above second term can be incorporated in the replacement of $\partial^\mu \rightarrow D^\mu = \partial^\mu - iA^\mu$ with $A = -2G_Vj_V$.

Now, it is straightforward to write down the grand potential. For the concrete setup, let us take the magnetic field direction along the $z$ axis. The grand potential should be

$$ \Omega_{j_V}/V = G_V(j_V^z)^2 + \Omega_A/V = G_V(j_V^z)^2 - \frac{|eB|}{2\pi} \sum_{s,n} \alpha_{s,n} \int \frac{dp^z}{2\pi} \omega_s(p), \qquad (25.4) $$

where the dispersion relation is [329]

$$ \omega_s^2 = m^2 + [|\mathbf{p}| + \operatorname{sgn}(p^z)s\mu_5]^2, \qquad |\mathbf{p}|^2 = (p^z + A^z)^2 + 2|eB|n \qquad (25.5) $$

with the Landau level $n$. The spin degeneracy factor is $\alpha_{s,n} = 1$ for $n \neq 0$ and $\alpha_{s,0} = \delta_{s,+}$ for $s = \pm$. In the ordinary case of equilibrated quark matter, the chiral chemical potential $\mu_5$ should be vanishing, and the phase diagram is not modified unless $\mu_5 \neq 0$ is introduced by hand.

Still, it is an interesting question how $G_V$ may have impact to $j_V$. The mean-field, $j_V$, should be determined from the self-consistency condition, i.e.,

$$ \frac{\partial(\Omega_{j_V}/V)}{\partial j_V} = 0. \qquad (25.6) $$
We shall treat the induced gauge field as a perturbation, and then the perturbative expansion gives the following expression:

$$j_V^z \approx \left. \frac{\partial(\Omega_A/V)}{\partial A^z} \right|_{A^z=0} + \left. \frac{\partial^2(\Omega_A/V)}{\partial A^{z2}} \right|_{A^z=0} A^z = \frac{|eB|}{2\pi^2} \operatorname{sgn}(eB)\mu_5 - 2G_V\chi_V^{(0)}j_V^z. \quad (25.7)$$

Here, the correction from the vector interaction is proportional to the susceptibility defined by

$$\chi_V^{(0)} = \left. \frac{\partial^2(\Omega_A/V)}{\partial A^{z2}} \right|_{A^z=0} = \frac{|eB|}{2\pi^2} \left( 1 + \frac{2\Lambda^2}{3|eB|} \right). \quad (25.8)$$

It should be noted that the first term appears from the Lowest Landau Level contribution (that is the derivative of the CME current) and the second term from the higher Landau levels has scheme dependence with the ultraviolet cutoff $\Lambda$. In the above expression, $\Lambda$ is the four-vector cutoff, that is, $\Lambda^2 = 2|eB|n + \Lambda_n^2$ with the longitudinal momentum cutoff $\Lambda_n$. The numerical value of the coefficient in front of $\Lambda^2$ may differ for other regularization such as the proper-time regularization. In fact, $\chi_V^{(0)}$ is nothing but the mass of the gauge field $A^z$, and it must be zero if $|eB| \to 0$. Therefore, the term $\propto \Lambda^2$ should be an artifact from the explicit breaking of gauge symmetry due to naive momentum cutoff. This implies that the properly renormalized value of $\chi_V$ should be $\chi_V = |eB|/2\pi^2$. Then, finally, we can conclude that the self-consistent CME current should take the following form:

$$\dot{j}_V = \frac{1}{1 + 2G_V \chi_V} \frac{\mu_5 e \mathbf{B}}{2\pi^2}. \quad (25.9)$$

This very simple exercise tells us a stimulating lesson. If we impose a condition that the CME formula should hold regardless of the interaction, one possible interpretation of the above result is that the input variables receive in-medium modification. Because $|eB|$ is renormalization invariant, it is unlikely that $\mathbf{B}$ is screened. Thus, we see a renormalization of the chiral chemical potential as

$$\mu_5 \rightarrow \frac{\mu_5}{1 + 2G_V \chi_V} = \frac{\mu_5}{1 + G_V |e\mathbf{B}| / \pi^2}. \quad (25.10)$$

We should be aware that $\mu_5$ can be medium modified just like $\mu$ shifted by the vector interaction. It is noteworthy that the effective $\mu_5$ is significantly suppressed if $G_V$ is positive and large. This is qualitatively understandable. Suppose that the magnetic field is sufficiently strong to justify the dimensional reduction along the longitudinal dynamics. Then, in such a (1+1)-dimensional system, the spatial component of the vector current is identified as the temporal component of the axial-vector current. A finite $\mu_5$ leads to a nonzero chirality $n_5$, so that the axial-vector interaction gives rise to renormalization of $\mu_5$.

A more interesting question, which is potentially relevant to the phase diagram research, is the effect of the axial-vector current from the CSE at finite density and magnetic field. In the same way as the previous discussions, we can consider the mean-field interaction as

$$\mathcal{L}_{\text{intML}} = -G_A \dot{j}_A \cdot j_A + 2G_A \dot{j}_A \bar{\psi} \gamma \gamma_5 \psi. \quad (25.11)$$

This extra term causes the axial gauge field, $-A_5 \cdot \bar{\psi} \gamma \gamma_5 \psi$, with $A_5 = -2G_A \dot{j}_A$. It is a nontrivial observation that this induced axial gauge field results in a finite density as

$$\dot{j}_{\text{ind}}^{0} = -\frac{1}{2\pi^2} A_5 \cdot e\mathbf{B}. \quad (25.12)$$

Then, a natural extension we may want to ask is whether an axial-vector current is further induced or not. The short answer is yes, but the final expression depends on the cutoff and renormalization scheme again. We can parametrize it with an unknown coefficient $\alpha$. Then, the same argument as before leads to

$$\dot{j}_{\text{A}}^{z} \approx \frac{|eB|}{2\pi^2} \mu - 2G_{\text{A}} \chi_{\text{A}}^{(0)} \dot{j}_{\text{A}}^{z} \quad (25.13)$$
with the susceptibility given by

$$ \chi_A^{(0)} = \left. \frac{\partial^2(\Omega_{A_5}/V)}{\partial A_5^2} \right|_{A_5^2=0}, \quad \chi_A = \alpha \frac{|eB|}{2\pi^2}. \qquad (25.14) $$

Here again, the susceptibility has the divergence from the Landau sum, that is, the naive expression involves $\alpha = 1 + 2 \sum_{n>0}$ with $n$ the nonzero Landau level, which could be somehow renormalized. In such a way, after all, the in-medium CSE current is modified into the following screened form:

$$ \dot{j}_A = \frac{1}{1 + 2G_A \chi_A} \frac{\mu e B}{2\pi^2}. \qquad (25.15) $$

We have seen that these induced currents may be affected by the interaction effects in extreme environments. So far, there are not many theoretical works to study the in-medium modification effects, but definitely, the question deserves further investigations in the future. Now, does it make sense to think of an opposite possibility that the medium properties are influenced by these currents?

## 25.2. Inhomogeneous condensation and the axial-vector interaction

Let us deepen our thinking about the physical meaning of the axial-vector interaction. The low-energy effective theory in general would accommodate the axial-vector interaction like

$$ \mathcal{L}_{\text{int}} = -G_A \bar{\psi} \gamma_\mu \gamma_5 \psi \bar{\psi} \gamma^\mu \gamma_5 \psi. \qquad (25.16) $$

Using the Dirac representation, it is easy to take the nonrelativistic limit; $\psi = (\varphi, \boldsymbol{\sigma} \cdot \mathbf{p}/(E+m)\varphi)^\mathrm{T}$ in which the lower component is negligible in the nonrelativistic treatment. In this approximation, we immediately see, $\bar{\psi}\gamma\gamma_5\psi \simeq \varphi^*\boldsymbol{\sigma}\varphi$. Therefore, the above interaction, $\mathcal{L}_{\text{int}} \sim G_A\varphi^*\boldsymbol{\sigma}\varphi \cdot \varphi^*\boldsymbol{\sigma}\varphi$, represents the spin-spin interaction in the scattering of heavy fermions.

Such interaction vertices are also known in nuclear physics since long time ago. In the conventional notation, the short-range Landau-Migdal interaction is parametrized as

$$ f_{NN} + g_{NN} \boldsymbol{\sigma}_1 \cdot \boldsymbol{\sigma}_2 + f'_{NN} \boldsymbol{\tau}_1 \cdot \boldsymbol{\tau}_2 + \left( \frac{f_{\pi}^2}{m_{\pi}^2} \right) g'_{NN} (\boldsymbol{\sigma}_1 \cdot \boldsymbol{\sigma}_2)(\boldsymbol{\tau}_1 \cdot \boldsymbol{\tau}_2), \qquad (25.17) $$

where $\boldsymbol{\sigma}_i$ and $\boldsymbol{\tau}_i$ are the spin and the isospin operators, respectively. For the p-wave pion condensation, the most important term is the last one with $g'_{NN}$ together with the counterparts with $\Delta$, i.e., $g'_{N\Delta}$ and $g'_{\Delta\Delta}$. Although $\Delta$ is a spin-isospin 3/2 baryon, the generalization is straightforward with the replacement of $\boldsymbol{\sigma}_i$ and $\boldsymbol{\tau}_i$ by the transition spin-isospin operators. The task for the investigation of the pion condensation is the computation of the pion dispersion relation $D_{\pi}^{-1}(\omega, \mathbf{k})$ with increasing baryon density. The propagator inverse involves the pion self-energies (which are dominated by the p-wave $\pi N$ interaction) as well as the diagrams with the Landau-Migdal interaction. Then, one may find $D_{\pi}^{-1}(\omega = 0, k = k_c) = 0$ at a threshold density $\rho = \rho_c$ depending on the values of $g'_{NN}$, $g'_{N\Delta}$, and $g'_{\Delta\Delta}$. Experimentally, $g'_{NN}$ and $g'_{N\Delta}$ are constrained, while $g'_{\Delta\Delta}$ is undetermined; see Ref. [670] for a review. Many nuclear theorists are not in favor of the pion condensation based on the analysis with the universality ansatz, $g'_{NN} \sim g'_{N\Delta} \sim g'_{\Delta\Delta} = 0.6-0.8$, but the case is not closed yet. These calculations and interpretations are usually presented in the nonrelativistic language, but the full relativistic translation can be easily done with the covariant form of the axial-vector current interaction. It should be worth approaching the possibility of the pion condensation from the higher density where the relativistic formulation is more appropriate.

Surprisingly, what is discussed in the context of quark matter has remarkable similarity. In a sense, if quark-hadron phases are continuously connected, quark matter may well have a dual of the pion condensed phase. Let us simplify the
story by assuming an effective (1+1)-dimensional system. The dimensional reduction could occur in the strong magnetic field limit or even near the Fermi surface at very high density. Then, in the chiral limit, the chemical potential can be absorbed in the fermion basis rotation as

$$ \bar{\psi}[(\partial_4 - \mu)\gamma^4 + \partial_z\gamma^z]\psi = \bar{\psi}'(\partial_4\gamma^4 + \partial_z\gamma^z)\psi', \qquad (25.18) $$

where

$$ \psi = e^{\mu\gamma^z\gamma^4}z\psi', \qquad \bar{\psi} = \bar{\psi}'e^{\mu\gamma^z\gamma^4}z. \qquad (25.19) $$

At first glance, the chemical potential is just eliminated from the theory, and physical observables seem to be independent of $\mu$. If this is true, the derivative of the grand potential with respect to $\mu$ is naturally zero, meaning that the density is zero even though the chemical potential was originally finite. Is it really possible to delete all the fermions by the basis rotation only? This plausible conclusion is drawn from careless momentum shift implied by the basis rotation. We may indeed split the fermion into two components according to the eigenvalues of $\gamma^z\gamma^4$. Because of the anti-commutative nature, $\gamma^z\gamma^4$ is an anti-Hermitian matrix, and its eigenvalues are $\pm i$. Thus, we can introduce two eigenstates; $\gamma^z\gamma^4\psi_+ = +i\psi_+$ and $\gamma^z\gamma^4\psi_- = -i\psi_-$. The basis rotation takes a simple form of $e^{\pm i\mu z}$, which results in the momentum shift by $\pm\mu$.

We can express the grand potential in the form of the phase space integration of the zero-point oscillation energy. In the chiral limit, the energy dispersion relation is simply $\varepsilon(p) = |p|$. If the original theory has a mass term, $\bar{\psi}\psi$, it is not invariant under the basis rotation. Nevertheless, we may still consider a situation that some interaction yields a finite mass in terms of $\psi'$. If this is the case, the dispersion relation is $\varepsilon(p) = \sqrt{p^2 + m^2}$. Then, $\psi_\pm$ bases are mixed and the grand potential is written as

$$ \Omega(\mu)/V = - \int_{-\Lambda+\mu}^{\Lambda-\mu} \frac{dp}{2\pi} \frac{\varepsilon(p)}{2} - \int_{-\Lambda-\mu}^{\Lambda+\mu} \frac{dp}{2\pi} \frac{\varepsilon(p)}{2}, \qquad (25.20) $$

Interestingly, for $\Lambda \gg \mu$, we can extract the finite-density correction as

$$ \Omega(\mu)/V = \Omega(\mu=0)/V - \frac{\mu^2}{2\pi}, \qquad (25.21) $$

from which we derive the density as $n_{\text{rotated}} = -\partial(\Omega/V)/\partial\mu = \mu/\pi$. We note that the density is mass independent even though the energy dispersion relation explicitly contains a finite mass. This mass independence is quite important. Let us presume a system of (1+1)-dimensional fermion without any basis rotation at all. Then, with a dynamically generated mass, $M$, the density should be $n_{\text{non-rotated}} = p_F/\pi\theta(\mu - M)$ with $p_F = \sqrt{\mu^2 - M^2}$. It is obvious that $n_{\text{rotated}} > n_{\text{non-rotated}}$ for $M \neq 0$, and so the free energy with the basis rotation should be smaller than the non-rotated case. This energy consideration explains why the basis rotation to eliminate the $\mu$ term from the Lagrangian is more favored. After the change of the basis, a condensate $\langle\bar{\psi}'\psi'\rangle$ is formed, which generates a dynamical mass but does not suppress the density due to the anomalous nature as we have elaborated above. Suppose that we have a homogeneous condensate, $\Sigma = \langle\bar{\psi}'\psi'\rangle \neq 0$, this state has a helical condensate once expressed in terms of the original basis. That is, it is easy to see [671]:

$$ \langle\bar{\psi}\psi\rangle = \Sigma \cos(2\mu z), \qquad \langle\bar{\psi}\gamma^z\gamma^4\psi\rangle = -\Sigma \sin(2\mu z), \qquad (25.22) $$

which is conventionally called the chiral spirals. In this simple (1+1)-dimensional system, $\gamma^z\gamma^4$ could be identified as a counterpart of $\gamma_5$, and then, the above expresses an alternative helical condensate with $\sigma$ and $\eta$ mesons. Interestingly enough, such a structure is analogous to what is called the Alternating-Layer-Spin (ALS) structure of neutral pion condensate [672], which is characterized by $\pi^0 \sim \cos(k_c z)$.
<figure><img src="image_81.png" /><figcaption>**Figure 25.1:** Phase boundaries calculated in the holographic QCD model (Sakai-Sugimoto model) with the dimensionless temperature $T$ and $\mu$. The solid curve represents the onset of the spatially modulated phase at $B = 0$, which is pushed up toward higher density with increasing $B$. Figure is adapted from Ref. [674].</figcaption></figure>

This type of helical condensate should naturally be affected by the Landau-Migdal-type interaction, specifically, $g$ for this example. Of course, not in the (1+1)-dimensional case but in the (3+1)-dimensional case, one can generalize the helical condensate with $\sigma$ and $\pi^0$ with the isospin matrix inserted in the condensate, which is called the dual chiral density wave state [673]. For such a system, a counterpart of $g'$ should be taken into account and the axial-vector iso-vector interaction like

$$ \mathcal{L}_{\text{int}} = -G'_{A} \bar{\psi} \gamma_{\mu} \gamma_{5} \boldsymbol{\tau} \psi \cdot \bar{\psi} \gamma^{\mu} \gamma_{5} \boldsymbol{\tau} \psi. \qquad (25.23) $$

would be necessary. In the neutron star environment, isospin symmetry is broken and it is expected that the iso-nonsinglet axial-vector should arise. For example, the component coupled to $\tau_3$ is nothing but the difference between the axial-vector current of $u$ quarks and that of $d$ quarks. It is a quite nontrivial question whether this axial-vector iso-vector interaction in neutron star matter (nearly pure neutron matter) would favor or disfavor the helical condensate structures in quark matter similar to the ALS in nuclear matter.

A systematic investigation is desirable in the future, but for the moment, we introduce a suggestive result from the holographic QCD model as shown in Fig. 25.1; see Ref. [674] for details. Here, the solid curve represents the onset of the spatially modulated phase at $B = 0$. In the region below this solid curve and above the dotted curve with the label, “Homogeneous Chiral Phase Boundary”, only the inhomogeneous condensate is formed. According to the previous arguments, stronger $B$ would render the system closer to the (1+1)-dimensional situation, and the chiral spirals should be preferred. However, surprisingly, the onset curve is pushed up toward higher density with increasing $B$ as seen in Fig. 25.1. One feasible account for this observation is that the background field of the axial-vector current enhances the effect of the axial-vector interactions that destroy the inhomogeneous condensates. Although the holographic QCD model is a powerful approach including nonperturbative interactions, it is difficult to diagnose the microscopic mechanism behind the resulting answer. To clarify the role played by the axial-vector interaction with strong magnetic field at finite density, model studies would provide us with useful insights.
<figure><img src="image_82.png" /><figcaption>**Figure 25.2:** Neutrino scattering processes in the presence of the background field of the axial-vector current. The dotted region schematically represents the neutrino sphere. Figure is adapted from Ref. [676].</figcaption></figure>

## 25.3. Axial-vector background scattering for the pulsar kick

Finally, let us take one more example to demonstrate the implication of the mean-field background of the axial-vector current with magnetic field at finite density. The proto-neutron star has strong B and the neutrino density is so high that the mean-free-path of the neutrinos can be less than 1m. In cores of the proto-neutron star, the baryon density is high, and the neutrinos cannot escape outside, which defines the neutrino sphere from where the neutrinos can be emitted. Inside of the neutrino sphere, scatterings makes the neutrinos thermally equilibrated with the electrons.

Since the electrons are charged particles, in response to the external magnetic field, they immediately develop the axial-vector current following the CSE formula. There is no direct coupling between the magnetic field and the neutrino (except for the magnetic moment), but in the hydrodynamic regime, the axial-vector background may contain the neutrino component as well as the electrons. If this is true, the CSE formula gives an estimate for the neutrino transport in the proto-neutron star along the magnetic direction. Because the neutrino emission has a preferred direction reflecting the presence of the background current, the recoil accelerates the proto-neutron star. In this way, the pulsar-kick mechanism can have a novel origin from the CSE current [423, 675].

This scenario has a hypothesis of the hydrodynamic regime, however. Even though the neutrinos are trapped in the neutrino sphere due to scatterings, it is not so conceivable that scatterings bend the neutrino motion coherently along the magnetic direction. It is thus a challenging question what microscopic processes can realize the macroscopic transport of the neutrinos along the axial-vector current of the electrons. This can be studied in a simplified treatment of the physical system in which the axial-vector current of the electrons is given as a background field [676]. The weak interaction describes the interaction between the electrons and the neutrinos, and we introduce a mean-field approximation into those electron-neutrino vertices. More specifically, the mean-field effective interaction is found as

$$ \mathcal{L}_{\text{eff}} = \frac{3G_F}{2\sqrt{2}} j_{A,e}^{\mu} \bar{\nu}_e \gamma_{\mu} (1 - \gamma_5) \nu_e. \qquad (25.24) $$

Using these vertices, we can consider the scattering processes such as $\nu_e + j_{A,e} \rightarrow \nu_e$ and $\bar{\nu}_e + j_{A,e} \rightarrow \bar{\nu}_e$. Now, the background field has a preferred direction proportional to B, and it is a natural anticipation that the scatterings parallel to the current and anti-parallel to the current may exhibit different amplitudes. In other words, the weak interaction explicitly breaks the parity symmetry. Because the magnetic field is externally applied, the neutrino emission and absorption must have anisotropy.

Concrete calculations conclude that there is no difference between the upward scattering and downward scattering as schematically sketched in Fig. 25.2. This is understandable from the weak interaction. For the anisotropic emission of the
neutrino, the spin flip of the nucleus is necessary [677]. However, the background field of the axial-vector current cannot achieve the spin flipping and thus there is no anisotropic scattering. Now, we learn a lesson; the scattering between the electron axial-vector current and the neutrino is not sufficient to justify the hydrodynamic regime with the neutrino axial-vector current. One should then deal with the scattering processes of the electron and the neutrino under the magnetic field. Such a setup itself constitutes an alternative scenario for the pulsar-kick mechanism.

Although no kick is derived from the scattering between the electron axial-vector current and the neutrino, the presence of the current at least tends to focus the scattering directions and some beaming effects are expected. The problem is that the neutrinos are emitted equally in both directions parallel and anti-parallel to the magnetic field. Since the total energy of emitted neutrinos is huge, only a small imbalance would suffice to accelerate the pulsar. If we see the supernovae simulations, the distributions of the magnetic field and the electron density profile, which both contribute to the axial-vector current, are not isotropic at all. Therefore, it should be an acceptable assumption that the background field of the axial-vector current has spatial anisotropy. Then, we can make a statement that the scattering on top of the spatially distorted axial-vector background is an efficient process to convert the spatial anisotropy to the acceleration (i.e., the chiral anisotropy conversion) [676].

It should really be a challenging attempt to model the proto-neutron star evolution including the full dynamics of the neutrino and the background field effect of the electron axial-vector current. It is always a subtle question how much effects can survive outside of the neutrino sphere where the density is small. So far, the implication of the CSE current has been discussed based on the order of magnitude estimates. To go beyond the order of magnitude estimates, the numerical simulations should be refined in the future studies.
# 26. Driving chiral phase transition with ring diagram

## 26.1. Introduction

Understanding the properties of QCD matter under extreme conditions, such as those created in ultra-relativistic heavy-ion collisions or in the cores of neutron stars, demands a reliable description of chiral symmetry restoration in a medium of partially deconfined quarks and gluons. Effective models are a robust exploratory tool for studying such dynamical systems. One advantage of these models is their ability to include or suppress specific interactions or diagrams, allowing for isolated examination of their effects. Additionally, they provide insights into the values of phenomenological parameters and their connections to the properties of underlying constituents. This approach is valuable for interpreting results from first-principle methods such as lattice QCD (LQCD).

Magnetic field provides an additional handle to probe the QCD phase diagram. While most effective chiral models can capture the magnetic catalysis in vacuum, i.e. chiral condensate increases in magnitude at $T = 0$ on increasing $B$, they predict the opposite trend from LQCD calculations [13, 16] on the magnetic field dependence of the chiral transition temperature [533, 678].

There is yet another problem: The standard NJL model lacks a description of confinement. A common remedy is to couple quarks to the Polyakov loop, resulting in the Polyakov-NJL (PNJL) model [679, 680]. However, this class of models tends to overestimate the pseudo-critical temperature, predicting $T_{ch} \approx 220$ MeV instead of the LQCD result of $T_{ch} \approx 156.5$ MeV. Currently there is no satisfactory solution to this problem, and one has to resort to an ad hoc rescaling of the deconfinement transition temperature $T_d$ in the gluon potential from its physical value of $T_d \approx 270$ MeV to about $T_d \approx 200$ MeV [681, 682, 683]. Such an adjustment is far from ideal and indicates missing interactions in the original model.

In Ref. [684], we proposed a natural resolution to the aforementioned problems by including the polarization (ring) diagram. This approach drives the screening of the four-quark interaction, which lowers the $T_{ch}$ at $B = 0$ and leads to inverse magnetic catalysis at $B \neq 0$. A key merit of this method is that it introduces no new parameters and leaves vacuum properties unaffected, as the ring diagram vanishes in vacuum. In field-theoretical terms, this represents an improved truncation scheme beyond the standard mean-field approximation, specifically beyond the self-consistent Hartree-Fock method.

## 26.2. Chiral Quark Model

We review an effective chiral quark model motivated by the Coulomb Gauge QCD [685, 686, 684]. The Lagrangian density reads:

$$ \mathcal{L}(x) = \bar{\psi}(x) (i\not\partial_x - m) \psi(x) - \frac{1}{2} \int d^4 y \rho^a(x) V^{ab}(x, y) \rho^b(y) \qquad (26.1) $$

where $m$ is the current quark mass, $\rho^a(x) = \bar{\psi}(x)\gamma^0 T^a \psi(x)$ is the color quark current, and $T^a$ is a generator of the $SU(N_c)$ symmetry group, with $a = 1, 2, \dots, N_c^2 - 1$. For the class of models where the interaction potential $V$ is instantaneous and color-diagonal, i.e.,

$$ V^{ab}(x, y) \rightarrow \delta^{ab} \times \delta(x^0 - y^0) V(x - y), \qquad (26.2) $$
the gap equation for the dynamical quarks has been derived [686]. It can be summarized as follows:

$$S^{-1}(p) = \not p - m - \Sigma(p) \qquad (26.3)$$

where

$$\Sigma(p) = C_F \int \frac{d^4 q}{(2\pi)^4} V(\boldsymbol{p}-\boldsymbol{q}) i \gamma^0 S(q) \gamma^0. \qquad (26.4)$$

The constant $C_F = \frac{N_c^2-1}{2N_c}$ is introduced via the quadratic Casimir operator. A particularly transparent case is a contact interaction

$$V(\boldsymbol{p}-\boldsymbol{q}) \rightarrow V_0. \qquad (26.5)$$

The solution to the gap equation (26.3) becomes

$$\begin{aligned} M &= m + \text{Tr}(\Sigma)/\text{Tr}(I) \\ &= m + C_F V_0 \int \frac{d^3 q}{(2\pi)^3} \frac{M}{2E} \\ & \qquad \times (1 - 2N_{\text{th}}(E)), \end{aligned} \qquad (26.6)$$

where $E = \sqrt{q^2 + M^2}$ and $N_{\text{th}}(E) = (e^{\beta E} + 1)^{-1}$ is the Fermi-Dirac distribution. Equation (26.6) is of the same form as the familiar result for quark mass ($N_f$ flavors) in the model of Nambu-Jona and Lasinio (NJL) [687].

The in-medium dressing of the interaction potential $V_0$ by the polarization tensor $\Pi_{00}$ is implemented via [686]

$$\tilde{V}_0^{-1} = V_0^{-1} - \frac{1}{2} N_f \Pi_{00} \qquad (26.7)$$

where the polarization is given by

$$\Pi_{00}(p^0, \boldsymbol{p}) = \frac{1}{\beta} \int \mathrm{Tr} (\gamma^0 S(q) \gamma^0 S(q+p)). \qquad (26.8)$$

Here $\int \mathrm{Tr}$ denotes a Matsubara sum over the fermionic frequencies ($\omega_n = (2n + 1)\pi/\beta$), and an integral over the momenta $d^3q$. Equation (26.7) describes the screening of the gluon propagator by the Debye mass. The factor of $1/2$ in Eq. (26.7) comes from the color structure, i.e. $\text{Tr} T^a T^b = \frac{1}{2} \delta^{ab}$, and is essential to reproduce the known result of the perturbative Debye mass for QCD, rather than for QED.

For simplicity we only consider the screening by the static limit of $\Pi_{00}$. The vacuum part vanishes exactly. The matter part is related to the Debye (electric) mass via

$$\begin{aligned} \hat{m}_{el}^2 &= -\frac{1}{2} N_f \Pi_{00}(p^0 = 0, \boldsymbol{p} \rightarrow \boldsymbol{0}) \\ &= \frac{1}{2} N_f \times \int \frac{d^3 q}{(2\pi)^3} 4\beta N_{\text{th}}(1 - N_{\text{th}}). \end{aligned} \qquad (26.9)$$

In Fig. 26.1 we demonstrate a numerical calculation of the Debye mass Eq. (26.9) at fixed value of $M = 0.18$ GeV (a typical value near the transition), and $N_f = 2$ flavors. The results at finite $B$ are also shown.

The following limits help interpret the results:

1. At $B = 0$ and $M \to 0$ (or large $T$):

$$\hat{m}_{el}^2 \approx \frac{1}{2} N_f \times \frac{T^2}{3}; \qquad (26.10)$$
<figure><img src="image_83.png" /><figcaption>Figure 26.1: The electric mass (Eq. (26.9)) for the 2-flavor case at fixed quark mass $M = 0.18$ GeV, as a function of temperature.</figcaption></figure>

2. At $B \neq 0$, considering the contribution from the lowest Landau level (LLL):

$$ \hat{m}_{el}^2 \approx \frac{1}{2} \frac{|e_f|B}{4\pi} \int \frac{dq_z}{2\pi} \frac{4\beta e^\beta \sqrt{q_z^2+M^2}}{(e^\beta \sqrt{q_z^2+M^2}+1)^2}, \qquad (26.11) $$

which for massless quarks reduces to:

$$ \hat{m}_{el}^2 \rightarrow \frac{1}{2} \frac{|e_f|B}{2\pi^2}. \qquad (26.12) $$

Observe how the exact numerical result follows the LLL at low temperature and eventually reaches the limit in Eq. (26.10) at high temperature. Note that the limit in Eq. (26.12) is not realized.

Following the dressing of the four-quark coupling in Eq. (26.7), it is intuitive to see the effect of the ring diagram: the coupling is substantially weakened at high temperatures, allowing a transition at lower $T_{ch}$. Additionally, at a fixed temperature, the coupling is weakened by the magnetic field, driving inverse magnetic catalysis. The actual prediction by the model involves a complex interplay between chiral symmetry and confinement. A typical result is shown in Fig. 26.2. See Ref. [684] for details $^{13}$.

## 26.3. Towards Dense Matter

To advance our understanding of chiral quarks at finite baryon density, it is essential to incorporate additional physical effects absent in the current model. Chief among these is restoring the momentum dependence of various quark dressing

<footnote>$^{13}$For completeness we give the model setup here: we use $\tilde{V}_0(T; M, \ell) \approx \tilde{V}_0(T; \langle M \rangle, \ell)$, and fix $\langle M \rangle = 0.136$ GeV. The set of model parameters is given by: $\Lambda = 1.076$ GeV, $G_{NJL} \Lambda^2 = 4.232$, $R_{4D}(q) = e^{-q^8/\Lambda^8}$, and a current quark mass $m = 5$ MeV. In the vacuum, this gives: $f_\pi = 92.9$ MeV, $m_\pi = 137.8$ MeV, and $\langle \bar{\psi}\psi \rangle = -(250 \text{ MeV})^3$ (per flavor).</footnote>
<figure><img src="image_84.png" /><figcaption>**Figure 26.2:** The chiral condensate (with current quark mass contribution subtracted), normalized to the vacuum value, versus the temperature, at zero (left) and finite (right) magnetic field. Dashed lines represent results obtained from a PNJL model without screening effect. The model with dressed coupling is capable of producing the inverse magnetic catalysis at finite temperature.</figcaption></figure>

functions, such as the constituent mass function and wavefunction renormalization, through improved modeling of the confining potential [686]. This would enhance our understanding of deconfinement beyond the statistical confinement picture employed [688, 678]. Additionally, a lattice computation of the gluon Debye mass at low temperature with a finite magnetic field could validate the relevance of ring diagram screening to inverse magnetic catalysis.
<figure><img src="image_85.png" /><figcaption>Figure 27.1: Pac-man contour used in FESR (left). Proton, neutron and axial-vector currents in the hadronic sector (center) and in the QCD sector (right)</figcaption></figure>

# 27. Nucleon axial coupling constant under magnetars conditions

## 27.1. Introduction

The study of dense nuclear matter and high-energy processes focuses the attention of researchers mainly on the study of quantum chromodynamics (QCD) phase transition in relativistic heavy ion collisions and the interior of neutron stars. One of the particular problems for studying QCD under extreme baryon density is that lattice QCD simulations are not applicable under this regime, due to the well-known sign problem [664]. This scenario provides the need to deal with effective models to describe the properties in the QCD medium, in particular at high baryon chemical potential, or baryon density. The particular case of compact stars completely escapes the possibilities of lattice simulation, so the use of effective hadronic or QCD models with non-perturbative effects becomes the only way at the moment. The possibility of extracting information from the hadronic sectors as well as from the quark degrees of freedom makes the QCD sum rules a versatile tool, since it can connect any hadronic model at intermediate energies to the general QCD description with non-perturbative effects parameterized by the operator product expansion (OPE).

Here we show the implementation of the finite energy sum rules (FESR) at high baryon density under the presence of an external magnetic field, considering nucleon correlators. The calculation of the axial coupling constant will be presented in detail.

There are different formalisms of addition rules, all of them with their own advantages and disadvantages. Here FESR will be presented as a practical tool for the determination of various in-medium hadronic and QCD parameters based on quark-hadron duality. Among the different types of sum rules in the literature, FESR establishes a clear criterion for the separation between the hadronic and QCD sectors. Considering a form factor $\Pi(s)$ generated by current correlation, with $s = q^2$, the FESR are obtained by integrating the form factor along the well-known contour *pac-man*, as shown in Fig. 27.1 (left). The hadronic resonances lie on the positive real axis, while the QCD sector lies on the circle. The radius of this circle, $s_0$, is the hadronic continuum threshold, denoted as the energy where the resonances begin to overlap. At finite temperature, $s_0$ acts as an order parameter for the deconfinement phase transition.

From Cauchy's theorem, quark-hadron duality relates both sectors, i.e.

$$ \int_0^{s_0} \frac{ds}{\pi} s^N \mathrm{Im}\Pi^{\mathrm{had}}(s) = - \oint_{s_0} \frac{ds}{2\pi i} s^N \Pi^{\mathrm{QCD}}(s), \qquad (27.1) $$

where the factor $s^N$ is an analytic kernel. To relate these two sectors, non-perturbative effects are included, parameterized in terms of the operator product expansion (OPE) leading to $\Pi(s) = C_0(s) + \sum_{n>0} C_n(s) \langle O_n \rangle / s^n$, where the first term
corresponds to perturbative QCD, and the second term is the sum of operators corresponding to combinations of quark and gluon background fields.

## 27.2. In-medium nucleon-nucleon correlator

In order to describe nucleon currents we use Ioffe's interpolating function [689] for protons

$$ \eta_p(x) = \varepsilon^{abc}[(u^a)^T(x) C \gamma_\mu u^b(x)]\gamma^\mu \gamma_5 d^c(x), \qquad (27.2) $$

where C is the charge conjugation matrix given by $C = i\gamma_0\gamma_2$. The neutron current is the same but exchanging the quark flavors, i.e. $u \leftrightarrow d$. In the hadronic sector $\eta(x)$ is defined by their matrix elements as $\langle 0|\eta_N(x)|N(p, s)\rangle = \lambda_N u_N(p, s)e^{ip \cdot x}$, being $\lambda_N$ the current-nucleon coupling, a phenomenological parameter, and with $u_N$ the nucleon spinor. The interpolating function can then be expressed in terms of the asymptotic nucleon field as $\eta_N(x) = \lambda_N \Psi_N(x)$ which turns to be convenient if we want to calculate in-medium correlators. The two-point function is $\Pi(q) = i \int d^4x e^{iqx} \langle 0|T \eta(x) \bar{\eta}(0)|0\rangle$.

The procedure now consists of treating the interpolating function in terms of nucleon fields and calculating the time-ordered correlators by considering fermion propagators in a magnetic background. Also the same procedure will be used in the QCD sector with quarks. Here we will expand the propagator in power series of the magnetic field $B$, getting a general expression for the Green function $G(x, x') = e^{\Phi(x, x')}S(x - x')$, where $S$ is the local part of the propagator and $\Phi$ is the Schwinger phase.

The information of the presence of a uniform and constant magnetic field is provided by the propagators. In the QCD sector, the correlators can be expanded in powers of $B$, which after contour integration becomes a series in powers of $(eB/s_0)^n$. Considering that $s_0 > 1$ GeV$^2$, the magnetic field can reach easily physical values for systems like relativistic heavy-ion collisions or magnetars.

The general expression for the local part of the fermion propagator in momentum space is [84].

$$ S(p) = \frac{i(\not p + m)}{p^2 - m^2 + i\epsilon} - (qB)\frac{i\sigma_{12}(\not p_\parallel + m)}{(p^2 - m^2 + i\epsilon)^2} + 2i(qB)^2\frac{(\not p_\parallel + m)[p_\perp^2 - \not p_\perp(\not p_\parallel - m)]}{(p^2 - m^2 + i\epsilon)^4} - (\kappa B)\frac{i(\not p + m)\sigma_{12}(\not p + m)}{(p^2 - m^2 + i\epsilon)^2} + \dots, \qquad (27.3) $$

where $\kappa$ is the anomalous magnetic moment of the fermion. In the hadronic sector, it is customary to introduce a perpendicular fermion velocity denoting the symmetry breaking by the background magnetic field, so in this case $p = (p^0, v_\perp p_\perp, p^3)$. This velocity is related with the perpendicular screening mass as $m_\perp = m/v_\perp$.

In vacuum, the nucleon correlator can be decomposed as $\Pi(p) = \not p \Pi_1(p^2) + \Pi_2(p^2)$. However, the magnetic field will introduce new structures which has to be combined with the electromagnetic tensor $F^{\mu\nu}$. The most general decomposition of a correlator is given in germs of the Clifford algebra structures, scalar, pseudoscalar, vector, axial-vector, and tensor

$$ \Pi = \Pi_S + i\gamma_5\Pi_P + \gamma_\mu\Pi_V^\mu + \gamma_\mu\gamma_5\Pi_A^\mu + \sigma_{\mu\nu}\Pi_T^{\mu\nu}. \qquad (27.4) $$

$\Pi_P = 0$ unless topological anomalies be present. So, the most general combinations of the vector, tensor and axial-vector components must involve $p_\mu$, $g_{\mu\nu}$, $\epsilon_{\mu\nu\alpha\beta}$ and $F_{\mu\nu}$, considering that the correlator must be parity even:

$$ \Pi_V^\mu = p_\parallel^\mu \Pi_V^\parallel + p_\perp^\mu \Pi_V^\perp + \tilde{p}_\perp^\mu \tilde{\Pi}_V^\perp, \quad \Pi_A^\mu = \tilde{p}_\parallel^\mu \Pi_A, \quad \Pi_T^{\mu\nu} = \epsilon_\perp^{\mu\nu} \Pi_T^\perp + (p_\parallel^\mu p_\perp^\nu - p_\parallel^\nu p_\perp^\mu) \Pi_T^{\parallel\perp} + (p_\parallel^\mu \tilde{p}_\perp^\nu - p_\parallel^\nu \tilde{p}_\perp^\mu) \tilde{\Pi}_T^{\parallel\perp}, \quad (27.5) $$

where $\tilde{p}_\perp^\mu \equiv \epsilon_\perp^{\mu\alpha} p_\alpha$, $\tilde{p}_\parallel^\mu \equiv \epsilon_\parallel^{\mu\alpha} p_\alpha$, with the anti-symmetric tensors defined as $\epsilon_\perp^{\mu\nu} \equiv \epsilon^{0\mu\nu 3}$ and $\epsilon_\parallel^{\mu\nu} = \epsilon^{\mu 12\nu}$.
One may notice that we now have eight independent structures instead of two. This abundance of information is compensated by the appearance of new condensates, where for example the gluon condensate will now be divided into chromoelectric and chromomagnetic condensate, along or perpendicular to the external magnetic field.

Each independent form factor can now be integrated along the FESR contour in the frame. $p_{\perp} = 0$ with $p_{\perp}^2 = s$. FESR has the advantage of skipping the form factor calculation, because the contour integration can be done before the integration of the internal loop momentum. This simplifies the calculations considerably [690, 86].

In [86] the structures $\Pi_S$, $\Pi_V^\parallel$, $\Pi_V^\perp$ and $\Pi_T^\perp$ were used to calculate the proton and neutron current couplings$\lambda_p$, $\lambda_n$, the hadronic thresholds $s_p$, $s_n$, the mass parameters $m_n$, $m_p$, the perpendicular velocities $v_\perp^p$, $v_\perp^n$ and the spin polarization of the quark condensates $\langle \bar{u}\sigma_{12}u \rangle$ and $\langle \bar{d}\sigma_{12}d \rangle$.

QCD sum rules for baryons in dense baryonic medium has been strongly studied since the first set of papers [691, 692, 693, 694, 695]. The dense medium will break Lorentz symmetry and hence temporal and spatial coordinates will split. This is often parameterized by the introduction of the fourthvector $u = (1, 0, 0, 0)$ denotes as the fourth-velocity in the thermal bath rest frame.

The structure of correlators now changes and new condensates appear. All correlator structures can be separated into even and odd contributions in terms of $p_0$ as $\Pi(p_0, \mathbf{p}^2) = \Pi^e(p_0^2, \mathbf{p}^2) + p_0\Pi^o(p_0^2, \mathbf{p}^2)$. In this case, the contour integration will be done in the frame $\mathbf{p} = 0$ with $s = p_0^2$. In the case of the effective nucleon propagator, assuming a generalized momentum-expanded self-energy, the breaking of the Lorentz symmetry will lead to some correction of the baryon chemical potential, as well as to the appearance of a velocity in the third component of the momentum. This can be summarized in Eq (27.3) by setting $p = (p^0 + \Delta\mu, v_\perp \mathbf{p}_\perp, v_\parallel p^3)$.

## 27.3. Axial-vector coupling

To obtain the nucleon axial coupling we need the following current correlator

$$ \Pi_{\mu}(x, y, z) = -\langle 0 | \mathcal{T} \eta_{p}(x) A_{\mu}(y) \bar{\eta}_{n}(z) | 0 \rangle, \qquad (27.6) $$

where in the QCD sector $A_\mu(y) = \bar{d}(y)\gamma_\mu\gamma_5 u(y)$ and in the hadronic sector it can be expressed in terms of the elements of the axial current matrix involving nucleons

$$ \langle p', s' | A_{\mu}(y) | p, s \rangle = \bar{u}_{p}^{s'}(p') T_{\mu}(q) u_{n}^{s}(p) e^{iq \cdot y}, \qquad (27.7) $$

with $q = p' - p$ and with $T$ is the most general form is

$$ T_{\mu}(q) = G_{A}(t)\gamma_{\mu}\gamma_{5} + G_{P}(t)\gamma_{5}\frac{q_{\mu}}{2m_{N}} + G_{T}(t)\sigma_{\mu\nu}\gamma_{5}\frac{q_{\nu}}{2m_{N}}, \qquad (27.8) $$

with $t = q^2$ and $m_N$ the vacuum nucleon mass. The nucleon Axial coupling constant is defined as $g_A \equiv G_A(0)$. We can write the axial current related with nucleons in terms of the nucleon fields

$$ A_{\mu}(y) = \int d^4 v \bar{\psi}_p(v) \tilde{T}_{\mu}(v - y) \psi_n(v), \qquad (27.9) $$

where $\tilde{T}_\mu(x)$ is the inverse Fourier transformation of $T_\mu(q)$ in configuration space. So now, we can explicitly deal with magnetic field dependent propagators. The hadronic correlator is expressed diagrammatically in Fig. 27.1 (center). The Fourier transformation of the correlator is

$$ \Pi_{\mu}(p, p') = \int d^4 y d^4 z e^{-i(q \cdot y + p \cdot z)} \Pi_{\mu}(0, y, z). \qquad (27.10) $$
which in the hadronic sector is, in vacuum case

$$ \Pi_{\mu}^{\text{had}}(p, p') = \lambda_{n} \lambda_{p} \frac{(\not p + m_{n}) T_{\mu}(q) (\not p' + m_{p})}{(p^{2} - m_{n}^{2})(p'^{2} - m_{p}^{2})}. \qquad (27.11) $$

Now we want to isolate $G_A(t)$ from the other form factors. Tracing the correlator multiplied by a gamma matrix will produce the desired selection in the following way:

$$ \mathrm{tr} [\Pi_{\mu}(p, p') \gamma_{\nu}] = -4i\epsilon_{\mu\nu\alpha\beta}p^{\alpha}p'^{\beta}\Pi(s, s', t), \qquad (27.12) $$

with $s = p^2$, $s' = p'^2$. $\Pi$ in the hadronic sector is

$$ \Pi^{\text{had}}(s, s', t) = \lambda_n \lambda_p \frac{G_A(t) + G_T(t)(m_n - m_p)/m_N}{(s - m_n^2)(s' - m_p^2)}. \qquad (27.13) $$

If we neglect the term $(m_n - m_p)/m_N$, this procedure isolates $G_A$.

The current correlator in the QCD sector is described diagrammatically in Fig 27.1 (right). The same procedure applied to the QCD sector at $t=0$ produces at leading order

$$ \Pi^{\text{QCD}}(s, s', 0) = \frac{s^2 \ln(-s/\mu^2) - s'^2 \ln(-s'/\mu^2)}{(2\pi)^4 (s' - s)} + \text{regular terms}, \qquad (27.14) $$

where $\mu$ is the $\overline{\text{MS}}$ scale. Regular terms means terms without discontinuities on the real axes, nor singularities which would vanish when integrating in the FESR contour. Now we have two variables so a double FESR [696, 87, 697] must be used for each current channel, in this case the momentum associated to nucleon currents.

$$ \int_0^{s_p} \frac{ds'}{\pi} \mathrm{Im}_{s'} \int_0^{s_n} \frac{ds}{\pi} \mathrm{Im}_s \Pi^{\text{had}}(s, s', t) = \oint_{s_p} \frac{ds'}{2\pi i} \oint_{s_n} \frac{ds}{2\pi i} \Pi^{\text{QCD}}(s, s', t), \qquad (27.15) $$

where $\mathrm{Im}_s f(s) \equiv \lim_{\epsilon \to 0} \mathrm{Im} f(s+i\epsilon)$. Applying the double FESR to the proton-axial-neutron currents correlator projected to $G_A$ in the frame $t=0$ we get

$$ g_A \lambda_n \lambda_p = \frac{1}{48\pi^4} [s_n^3 \theta(s_p - s_n) + s_p^3 \theta(s_n - s_p)], \qquad (27.16) $$

where $s_p$ and $s_n$ are the proton and neutron current hadronic thresholds. The values of the current-couplings and the hadronic thresholds are obtained through the nucleon-nucleon correlator FESR [86]. In vacuum, both nucleon thresholds are equal so $g_A = s_0^3/48\pi^4\lambda_N^2$. In [87], this value was calculated in vacuum by fitting the experimental value of $g_A$ and then searching for the allowed values of the quark condensate and the gluon condensate. When medium effects are taken into account, it is expected that the axial-vector contribution to the form factor splits into different components, according to the symmetry breaking induced by external magnetic field and thermal or dense bath

$$ G_A \gamma_{\mu} \rightarrow G_A^{(0)} \gamma_0 + G_A^{(\perp)} \gamma_{\mu}^{\perp} + G_A^{(3)} \gamma_3 + \tilde{G}_A F_{\mu\nu} \gamma^{\nu}. \qquad (27.17) $$

The case in the presence of magnetic field only was obtained in [87], while the effects of both, baryon density and magnetic field was obtained in [697]. There was considered only the leading contribution in Eq. (27.16), medium effects will be present only through the nucleon-current couplings and nucleon hadronic thresholds.

## 27.4. Results

For the combined magnetic field and baryon density problem, we select only six FESR from the neutron-neutron and proton-proton correlators, in order to obtain the current-couplings $\lambda_n$, $\lambda_p$, the hadronic thresholds $s_n$, $s_p$, and the
<figure><img src="image_86.png" /><figcaption>Figure 27.2: In-medium nucleon axial coupling constant as a function of baryon density for different values of the magnetic field (left), and as a function of the magnetic field for different values of the baryon density (right).</figcaption></figure>

condensates $\langle \bar{u}\sigma_{12}u \rangle$ and $\langle \bar{d}\sigma_{12}d \rangle$. The inputs considered for the in-medium condensates depend mainly on the baryon density, except for the quark condensate (see details in [697]). The first thing to be noticed is that, for all the range of values considered both for baryon density and magnetic field, $s_p > s_n$. This means that the axial-vector coupling constant at finite baryon density and magnetic field is then

$$g_A = \frac{1}{48\pi^4} \frac{s_n^3}{\lambda_n \lambda_p}. \qquad (27.18)$$

The medium evolution of the nucleon axial-vector coupling constant can be seen in Fig. 27.2 as function of the baryon density and as function of the magnetic field strength. The axial-vector coupling constant decreases both with the baryon density and with the magnetic field. It is worth to notice that at $B = 0$ and at the nuclear saturation density $\rho_B = \rho_0 = 0.16 \text{ fm}^{-3}$ we get $g_A^* \equiv g_A(\rho_0) \approx 0.92$, in agreement with other approaches which get the result $g_A^* \lesssim 1$ [698, 699, 700, 701, 702, 703, 704, 705, 706, 707].

As can be seen in Fig. 27.2, for $\rho_B \sim \rho_0$ there is no significant variation with the magnetic field, except for lower values of the baryonic density. For $\rho \gtrsim 1.25\rho_0$, $g_A$ appears to increase with the magnetic field and there is a flip in the behavior of the curves. This behavior can be enhanced for higher values of baryon density and magnetic field, for example, near the core of a magnetar. At present, these are the only results in the literature considering these two effects. We hope that similar studies performed with other models and techniques can reproduce these results.
## 28. Dynamics of Heavy quarks in a magnetised medium

### 28.1. Introduction

Heavy quarks have been extensively studied within the heavy-ion community as a prominent signature for characterising the properties of hot and dense quark matter. Considering the non-centrality of the HIC experiments and the subsequent generation of strong magnetic fields, studies related to heavy quarks have recently been extended to magnetised medium. Unlike most of the previous heavy quark (HQ) dynamical studies, in the present study we go beyond those limiting scenarios to tackle the most general case of arbitrary valued external magnetic fields.

The HQ momentum diffusion coefficients are crucial theoretical quantities needed to describe the evolution of heavy quarks and significantly impact the phenomenology of heavy quarks, influencing theoretical predictions for relevant experimental observables [708]. We use the widely adopted Langevin equations which assumes that heavy quarks receive random "kicks" from the thermal partons in the surrounding medium. In absence of the magnetic field and within the nonrelativistic static limit of HQ (i.e. $p \approx 0$, $M \gg T$, $p$ and $M$ being the HQ momentum and mass respectively) there is no anisotropy imposed on the system resulting in a single diffusion coefficient $\kappa$. Several studies have evaluated this $\kappa$ using various techniques [709, 710, 711]. Moving beyond the static limit involves attributing a finite velocity, where $\gamma v \lesssim 1$ (i.e. $p = \gamma Mv \lesssim M$) to the heavy quark (HQ). This introduces an anisotropy in the system due to the HQ's motion in a preferred direction. Consequently, the momentum diffusion coefficient $\kappa$ splits into longitudinal and transverse components, denoted as $\kappa_L$ and, $\kappa_T$, which also has been extensively studied [712, 713, 714, 715, 716].

The presence of an external magnetic field introduces additional anisotropy into the system, raising intriguing questions about how to incorporate the scale $eB$. Most current studies have adopted the Lowest-Landau-Level (LLL) approximation [717, 718, 719], or the weak magnetic field approximation [720]. The validity of these approximations depends on the strength and time-dependence of the magnetic field generated in non-central heavy-ion collisions (HICs). However, extending calculations to arbitrary external magnetic fields eliminates these constraints, freeing us from limitations imposed by the scale of $eB$.

To compute HQ momentum diffusion coefficients, which crucially depend on the HQ scattering rate, in the presence of arbitrary magnetic fields, it is necessary to generalise the effective gluon propagator for a hot magnetised medium beyond the LLL approximation. This involves using hard thermal loop (HTL) approximations to calculate the form factors of the effective generalised gluon propagator across all Landau levels. Recent approaches have also incorporated the magnetic field effect differently in calculations of HQ potential and HQ energy loss, where the medium's influence is assumed to be encapsulated solely through the medium-dependent Debye mass [721, 722, 723]. Within the static limit, comparing results from these methods provides insights into the limitations of using the simplified approximation of a medium-modified Debye mass.

### 28.2. Formalism

We examine a dynamic heavy quark (HQ) system (denoted as $(P \equiv (M, p) \equiv (E, v)$)) interacting with an anisotropic magnetic field $B = B\hat{z}$, deriving the complete expressions for longitudinal and transverse momentum diffusion coefficients for charm and bottom quarks. Despite the heavy quark mass $M \gg \sqrt{eB}$, $T$ being the dominant scale, we do not impose additional scale hierarchies relative to $eB$ and $T$, unlike the LLL or weak field approximations. Following Refs. [717, 719], we utilize the HTL approximation with the condition $\alpha_s eB \ll T^2$ (where $\alpha_s$ denotes the QCD running coupling), allowing
us to neglect soft self-energy corrections of quarks and gluons during scattering rate calculations.

When the HQ moves in the presence of an external magnetic field, focusing on two distinct scenarios is beneficial: $v \parallel B$ and $v \perp B$. In the former case $v \parallel B$, two distinct diffusion coefficients $\kappa_L$ and $\kappa_T$ arise, which are linked to the HQ scattering rate as

$$ \kappa_T(p) = \frac{1}{2} \int d^3 q \frac{d \Gamma(v)}{d^3 q} q_{\perp}^2, \quad \kappa_L(p) = \int d^3 q \frac{d \Gamma(v)}{d^3 q} q_z^2. \qquad (28.1) $$

On the contrary, $v \perp B$ case generates three different diffusion coefficients $\kappa_j$'s ($j = \{x, y, z\} \equiv \{1, 2, 3\}$), i.e.

$$ \kappa_j(p) = \int d^3 q \frac{d \Gamma(v)}{d^3 q} q_j^2. \qquad (28.2) $$

It's clear that when we consider the static limit ($v \to 0$) in a magnetized medium, only one anisotropy remains, dictated by the direction of $B$, which causes the scenario $v \perp B$ to vanish.

From Eqs. 28.1 and 28.2, it's evident that to calculate the momentum diffusion coefficients, we first need to determine the scattering rate or interaction rate of 2 $\leftrightarrow$ 2 scatterings between the light quark/gluon and the heavy quark (i.e. $qH \leftrightarrow qH$ and $gH \leftrightarrow gH$). To evaluate these scattering rates, we employ an effective method initially introduced by Weldon [724]. In this approach, we express the t-channel 2 $\leftrightarrow$ 2 scatterings involving the heavy quark as the cut/imaginary parts of the heavy quark self-energy, which can be formulated as

$$ \Gamma(P) = -\frac{1}{2E} \frac{1}{1 + e^{-E/T}} \mathrm{Tr}[(\not{P} + M) \mathrm{Im} \, \Sigma(p_0 + i\epsilon, p)], \qquad (28.3) $$

where $\Sigma(P)$'s represent the effective HQ self energy with an HTL resummed effective gluon propagator which duly incorporates the soft contributions. The detailed calculation of $\Gamma(P)$ and hence $\kappa_i$'s has been done in Ref. [725].

## 28.3. Results

<figure><img src="image_87.png" /><figcaption>**Figure 28.1:** The variation of the scaled momentum diffusion coefficients with respect to the magnetic field strength $eB$ is illustrated for longitudinal (solid lines) and transverse (dashed lines) diffusion in the static limit, considering both charm (red curves) and bottom (blue curves) quarks. This comparison is presented for two distinct temperatures: $T = 0.4$ GeV (left panel) and $T = 0.6$ GeV (right panel). The masses of charm and bottom quarks, denoted as $M$, are specified in the accompanying text.</figcaption></figure>

We start with our most general static limit results for magnetized medium in Fig. 28.1, by varying the ratio $\kappa_{L/T}^{(s)}/\kappa^{(s)}$ with respect to the external magnetic field. $\kappa^{(s)}$ is the zero magnetic field value of the single isotropic momentum
diffusion coefficient. We opted for two relatively higher temperatures, specifically $T = 0.4$ and $T = 0.6$ GeV, to maintain consistency with the HTL approximation used throughout our calculations. Observations from the plots reveal that at lower values of $eB$, both the longitudinal (solid curves) and transverse (dashed curves) momentum diffusion coefficients exhibit a more pronounced rate of increase compared to higher $eB$ values. This effect is particularly prominent for charm quarks (red curves), leading to a crossover with the bottom quark (blue) curves. Throughout the magnetic field range depicted in Fig. 28.1, the values of $\kappa_L$ consistently surpass those of $\kappa_T$ for both static charm and bottom quarks. Finally, as temperature increases, the overall ratio decreases, as expected due to the competing influences of $eB$ and $T$ scales.

<figure><img src="image_88.png" /><figcaption>**Figure 28.2:** The plots depict the variation of the ratio between the Debye mass approximated results ($\kappa'$) and the exact results ($\kappa$) with respect to the magnetic field strength $eB$. This comparison is shown for both longitudinal (solid lines) and transverse (dashed lines) momentum diffusion coefficients within the static limit, considering charm (red curves) and bottom (blue curves) quarks. The analysis is conducted at two distinct temperatures: $T = 0.4$ GeV (left panel) and $T = 0.6$ GeV (right panel).</figcaption></figure>

In the present study, we also explore an alternative method to incorporate the influence of magnetic fields on heavy quark (HQ) scattering rates and momentum diffusion coefficients. Here, all medium effects are encapsulated through the medium-modified Debye screening mass. In Fig. 28.2, we present comparisons between results obtained from the exact calculation ($\kappa$) and those from the Debye mass approximation ($\kappa'$) which illustrates the variation of the ratio $\kappa'/\kappa$ with respect to the magnetic field strength for two distinct temperatures, $T = 0.4$ GeV and $T = 0.6$ GeV. For both longitudinal and transverse components of charm and bottom quarks, the general trend is similar: the Debye mass approximated results consistently underestimate the exact results for larger values of $eB$ and overestimate them for smaller values of $eB$. These discrepancies are more pronounced for bottom quarks due to their heavier mass ($M_b = 4.18$ GeV) compared to charm quarks ($M_c = 1.27$ GeV).

Additionally, it is noticeable that the ratio of transverse components ($\kappa'_T/\kappa_T$) is consistently larger than that of the longitudinal components ($\kappa'_L/\kappa_L$) across the entire range of $eB$ considered. This observation aligns with the findings in Fig. 28.1. The dominance of $\kappa_L$ over $\kappa_T$ can be attributed to the predominant gluonic contributions in the $t$-channel scatterings studied in this context.

Next we present our estimated results for longitudinal and transverse momentum diffusion coefficients beyond the static limit, while maintaining confinement within the small energy transfer regime and the scale hierarchy $M \ge p \gg T$. We fix the HQ momentum at $p = 1$ GeV and consider temperatures $T = 0.4$ GeV and $T = 0.6$ GeV, consistent with the applied HTL approximation.

Firstly, in the $v \parallel B$ case, we examine the variation of longitudinal and transverse momentum diffusion coefficients
<figure><img src="image_89.png" /><figcaption>**Figure 28.3:** $v \parallel B$ case: The longitudinal (solid curves) and transverse (dashed curves) momentum diffusion coefficients are depicted for charm (red curves) and bottom (blue curves) quarks as functions of the external magnetic field strength. These results are shown for two distinct temperatures: $T = 0.4$ GeV (left panel) and $T = 0.6$ GeV (right panel). The magnetized momentum diffusion coefficients are normalized relative to their values at $eB = 0$.</figcaption></figure>

<figure><img src="image_90.png" /><figcaption>**Figure 28.4:** $v \perp B$ case: The variation of the transverse components $\kappa_1$ (solid curves), $\kappa_2$ (dashed curves), and the longitudinal component $\kappa_3$ (dotted curves) of the momentum diffusion coefficient is illustrated for charm quarks (left panel) and bottom quarks (right panel) as functions of the external magnetic field. These results are presented at a fixed temperature of $T = 0.4$ GeV. The magnetized momentum diffusion coefficients are normalized relative to $T^3$.</figcaption></figure>
with external magnetic field in Fig.28.3 for the aforementioned fixed temperatures. The coefficients are scaled relative to their respective values at eB = 0. Similar to the static limit, we observe that for higher values of eB, the rate of change of $\kappa_{L/T}$ becomes relatively flat, especially for charm quarks (red curves). Interestingly, $\kappa_T$ (dashed curves) dominates over $\kappa_L$ (solid curves) throughout the range of eB considered for charm quarks. Conversely, for bottom quarks (blue curves), $\kappa_L$ is larger than $\kappa_T$ until a higher threshold of eB, beyond which a crossover occurs. These crossovers reflect the interplay among the three competing scales M, T, and eB. Bottom quarks, with their higher mass, require a correspondingly higher eB to exhibit similar behaviors in $\kappa_{L/T}$ as observed for charm quarks. The right panel of Fig. 28.3 demonstrates this trend more clearly, where the absence of an immediate crossover indicates that bottom quarks would require an even higher eB for similar behavior.

Figure 28.4 illustrates the results for the perpendicular case, $v \perp B$. At eB = 0, no such configuration exists, thus we scale the values of $\kappa_j$ with $T^3$ here. Transverse components $\kappa_1$ (solid curves) and $\kappa_2$ (dashed curves) dominate over the longitudinal component $\kappa_3$ (dotted curves) for both charm and bottom quarks. The specific choice of $p_\perp$ ensures $\kappa_1$ is larger than $\kappa_2$. Unlike previous cases, no saturation behavior is observed for any of the momentum diffusion coefficients at higher eB values; instead, their rates of change visibly increase with increasing eB. Additionally, the values of momentum diffusion coefficients for bottom quarks (right panel) are approximately an order of magnitude lower compared to those for charm quarks (left panel).

Combining the results from both cases, we observe that for lower eB values, the effect of the external magnetic field on various momentum diffusion coefficients is more pronounced when the HQ moves parallel to the magnetic field direction with higher momentum. This effect tends to saturate as eB increases. Conversely, when the HQ moves perpendicularly to the external magnetic field, the effect of magnetic field on the momentum diffusion coefficients increases continuously with higher eB values.

## 28.4. Summary

In this section, we have discussed the momentum diffusion coefficients of heavy quarks (HQ) within a magnetized medium. We have explored both general cases: (a) arbitrary values of the external magnetic field and (b) beyond the static limit by considering a finite HQ velocity. By comparing our exact results with an alternative Debye mass approximated procedure within the static HQ limit, we have clearly identified the limitations of the alternate approach. We emphasize the necessity of incorporating the full structure of gluon two-point correlation functions in a hot and magnetized medium. Across both static and non-static scenarios (particularly when the HQ moves parallel to the magnetic field), we observe a consistent pattern in the dependence of momentum diffusion coefficients on eB: rapid increase at lower eB values followed by saturation, especially pronounced for charm quarks. Conversely, when the HQ moves perpendicularly to the magnetic field, we observe increasing changes in momentum diffusion coefficients with higher eB values. Within the static limit, soft gluon scatterings dominate the longitudinal diffusion coefficient over the transverse counterpart, reflecting the dynamics of t-channel processes at leading order of strong coupling. However, beyond the static limit, we observe an opposite trend. The interplay among scales such as M, p, eB, and T is clearly manifested in our findings. This study opens avenues for future research directions, such as investigating the in-medium evolution of HQ and its implications on experimental observables like directed and elliptic flow of open heavy flavor mesons.
# 29. The role of quark anomalous magnetic moment in magnetized quark matter

## 29.1. Introduction

The description of the magnetized QCD (Quantum Chromodynamics) phase diagram has remarkable developments in the past few years, with intensive efforts from both lattice QCD (LQCD) and effective model communities. In LQCD at zero density, it is well known the magnetic catalysis (MC), which is the increasing of the chiral condensate as function of the magnetic field [194]. However, close to the pseudocritical temperature it is observed the inverse magnetic catalysis (IMC) [14], which is the non-monotonic behavior of the quark condensate as a function of the magnetic field. Both effects are very important in order to describe the basic physics of the strongly interacting quark matter, but, unfortunately, just the MC effect is well described by effective models in the mean field approximation [30]. These results have motivated several possibilities in effective approaches, as one can see in the references therein Ref. [193]. Particularly, in the case of the Nambu-Jona-Lasinio model [529, 726] one can explore beyond mean field approximations [727], fittings of the coupling [57, 193, 17, 85, 22] and, also including the anomalous magnetic moment (AMM) of the quarks [728, 729, 730, 76, 78, 731, 115, 732, 733, 734, 735, 736, 737, 200, 738, 739, 116, 740, 741, 742, 743, 744, 745, 746, 747, 748, 749, 750].

In the recent literature, the quark AMM has been used as a new set of parameters in the NJL model, included in the Lagrangian by the Schwinger *ansatz*. These parameters are obtained in order to fix the magnetic moment of proton and neutron [730]. Then, they can be used to explore several properties of the magnetized QCD phase diagram. Some of the modern predictions are claiming that, at low temperatures (zero temperature, in some cases), it is possible to obtain IMC and first-order phase transitions [730, 76, 115, 731, 78, 737, 740]. However, it is well established that these effects are not predicted by LQCD [14, 203]. Also, all the phase transitions obtained in the region of interest, i.e., $eB < 1.0 \text{ GeV}^2$, relevant to the heavy ion collisions, indicate crossover transitions [751], with the possibility of first-order phase transitions at very strong magnetic fields, i.e., $eB \sim 8 \text{ GeV}^2$ [203]. In this way, recent works [200, 747] have been calling the attention that the use of inappropriate regularization procedures can be the source of nonphysical phenomena. At zero AMM case, the methods called MFIR (magnetic field independent regularization) [131] are claimed to eliminate nonphysical oscillations in quark condensate as a function of the magnetic field and also in several other quantities [198]. The main idea of these methods is to separate the contribution of the magnetic field from the natural divergences of the model, which must be regularized. In Ref. [200], the authors adapt the Heisenberg-Weisskopf effective lagrangian, used in the context of the electrons with anomalous magnetic moment [752], for quarks in the NJL model. Then the vacuum magnetic regularization (VMR) scheme [198], which is a step-forward of the MFIR scheme, is applied. The results indicate very different behavior of the effective quark masses as function of the magnetic field, with no indication of first-order phase transition in the region $eB \le 0.3 \text{ GeV}^2$ and just a very small window of IMC when considers sizable values of the quark AMM parameters.

The solution for the difference between the model results is then detailed explained in Ref. [747], where one observes that the non-MFIR procedures hidden mass-dependent terms in the ultraviolet limit of the model which are inducing first-order phase transitions. These transitions can be obtained by both high enough magnetic fields or quark AMM values in the model, when it is considered as a free control parameter. If one uses the full lagrangian with the inclusion of the Schwinger *ansatz*, it is possible to show that in the limit of $eB/M_0^2 \ll 1$, one can reproduce the Heisenberg-Weisskopf effective lagrangian, changing the electron mass for $M_0$, the effective quark mass in the vacuum. Then, the
mass-dependent terms can be avoided in the UV limit, eliminating the possible first-order phase transitions.

In this section we revisit the main ideas explored in the recent literature, some of the basic results and how and what can still be done. The work has the following structure: In Section 29.2 we present the basic analytical structure of the NJL model in a constant magnetic field with quark AMM. In Section 29.3 we define the two main regularization procedures we want to study. The discussion of the basic results are given in Section 29.4 and the conclusions in Section 29.5.

## 29.2. Nambu–Jona-Lasinio SU(2) with quark AMM in a constant magnetic field

The lagrangian density of the Nambu–Jona-Lasinio (NJL) model with two quark flavors possessing anomalous magnetic moments (AMM) in an external electromagnetic field environment is given by [730]

$$ \mathcal{L} = \bar{\psi} (i \not{D} - \hat{m} + \frac{1}{2} \hat{a} \sigma_{\mu\nu} F^{\mu\nu}) \psi + G [(\bar{\psi} \psi)^2 + (\bar{\psi} i \gamma_5 \tau \psi)^2], \qquad (29.1) $$

where $\psi$ is the quark fermion field $\psi = (\psi_u \ \psi_d)^T$, the bare quark mass matrix is $\hat{m} = \text{diag}(m_u, m_d)$, which is in the isospin limit $m_u = m_d \equiv m$, where $\hat{m} = m \mathbf{1}$. The AMM matrix is $\hat{a} = \text{diag}(a_u, a_d)$. The phenomenological Pauli term $\frac{1}{2}\hat{a}\sigma_{\mu\nu}F^{\mu\nu}$ couples each quark AMM flavor with the electromagnetic tensor. At one-loop approximation level, we have the following definitions

$$ a_f = q_f \alpha_f \mu_B, \quad \alpha_f = \frac{\alpha_e q_f^2}{2\pi}, \quad \alpha_e = \frac{1}{137}. \qquad (29.2) $$

The electromagnetic gauge field is $A^\mu$ and the electromagnetic tensor is $F^{\mu\nu} = \partial^\mu A^\nu - \partial^\nu A^\mu$. Here, we adopt $Q$ as the diagonal quark charge matrix$^{14}$ $Q = \text{diag}(q_u = 2/3, q_d = -1/3)$. The covariant derivative is given by $D^\mu = (\partial^\mu + ieQA^\mu)$ and $\not{D} = \gamma_\mu D^\mu$. The coupling constant is given by $G$.

In a mean field approximation, the lagrangian becomes

$$ \mathcal{L} = \bar{\psi} (i \not{D} - M + \frac{1}{2} \hat{a} \sigma_{\mu\nu} F^{\mu\nu}) \psi - \frac{(M-m)^2}{4G}, \qquad (29.3) $$

where the effective quark mass is defined as

$$ M = m - 2G \langle \bar{\psi} \psi \rangle, \qquad (29.4) $$

where $\langle \bar{\psi} \psi \rangle$ is the chiral condensate. Working with constant AMM values, we can define a Bohr magneton as $\mu_B = e/2M$. As we are interested in constant values of quark AMM, in our case we have adopted $\mu_B = e/2M_0$, where $M_0$ is the vacuum effective quark mass. Choosing the Landau gauge, $A_\mu = \delta_{\mu 2} x_1 B$, in order to satisfy $\nabla \times A = B \hat{e}_3$ and $\nabla \cdot A = 0$, we can work with a constant magnetic field in the z-direction.

### 29.2.1. Thermodynamic potential with quark AMM

The general structure of the thermodynamic potential at zero temperature with the quark AMM in a constant magnetic field environment is given by [200, 747, 730]

$$ \Omega = \frac{(M - m)^2}{4G} + \Omega^{\text{mag}}(M, B), \qquad (29.5) $$

<footnote>$^{14}$We are using the Gaussian natural units, i.e., $1 \text{ GeV}^2 = 1.44 \times 10^{19} \text{ G}$ and $e = 1/\sqrt{137}$.</footnote>
where $\Omega^{\text{mag}}$ is written as the magnetic field term, given by

$$ \Omega^{\text{mag}}(M, B) = -N_c \sum_{f=u,d} |B_f| \sum_{n=0}^{\infty} \sum_{s=\pm 1} \int_{-\infty}^{\infty} \frac{dp_3}{4\pi^2} E_{n,s}^f, \qquad (29.6) $$

in which $N_c = 3$ is the number of colors; $B_f \equiv q_f eB$, $s = \pm 1$ is the spin index, $n$ are the Landau levels and the quark energy dispersion relation is defined as

$$ E_{n,s}^f = \sqrt{p_3^2 + (M_{n,s}^f - s a_f B)^2} \qquad (29.7) $$

where we define $M_{n,s}^f = \sqrt{|B_f|(2n + 1 - sfs) + M^2}$ and $s_f = \text{sign}(q_f)$ as the charge sign function.

Since Eq. (29.6) is divergent, one need to adopt a regularization procedure.

## 29.3. Regularization

There are several choices used in the literature, which can be divided in two central regularization procedures, the VMR (vacuum magnetic regularization) [198], which is a MFIR-type (magnetic field independent regularization) [131] technique, and non-MFIR procedures, mostly based in Form-Factor functions [131]. Recent works are using the VMR/MFIR regularization [737, 200, 749, 744, 747] and Form-Factor procedures [730, 115, 116, 740, 745, 736, 739, 742, 746, 738] to deal with the AMM of the quark in the NJL model. Here we will discuss some of the basic properties of the most common regularization methods applied with the inclusion of quark AMM.

### 29.3.1. Form-Factor regularization

The idea of form-factor (FF) regularization resides in use some function to regularize the ultraviolet divergent integrals. We can multiply the vacuum contribution in Eq. (29.6) by a FF the function used in Ref. ([730])

$$ f_{n,s}^f(\Lambda, eB) = \frac{1}{1 + \exp\left(\frac{((p_z^2 + |q_f eB|)(2n+1-ss_f))^{1/2} - \Lambda)}{A}\right)} \qquad (29.8) $$

where $\Lambda$ is the cutoff of the model and $A$ is a new parameter of regularization. We will call this type of FF function as $FF_1$.

The second FF function we will use is the $FF_2$, from Ref. [740, 115]

$$ f_{n,s}^f(\Lambda, eB) = \frac{\Lambda^N}{\Lambda^N + (p_z^2 + |q_f eB|(2n+1-ss_f))^{N/2}} \qquad (29.9) $$

where $N$ is a new regularization parameter. These two regulators should be sufficient to explore the main effects that we are interested. However, there are a variety of different regularization prescriptions that are non-MFIR procedures with different FF functions that we will discuss briefly later.

### 29.3.2. VMR regularization

The magnetic field contribution of the thermodynamic potential Eq. (29.6) at zero temperature in the VMR regularization is written in the Schwinger proper-time method as [747]
$$ \Omega^{\text{mag}}(M, B) = \frac{N_c}{8\pi^2} \sum_{f=u,d} \int_0^\infty \frac{d\tau}{\tau^3} e^{-\tau M^2} F_f(\tau), \qquad (29.10) $$

where the function $F_f(\tau)$ is given by

$$ F_f(\tau) = e^{-\tau(a_f B)^2} \tau |B_f| \left[ s_f \sinh(\tau 2a_f BM) + F_f^{(2)}(\tau) \right], \qquad (29.11) $$

and the function $F^{(2)}(\tau)$ is

$$ F_f^{(2)}(\tau) = \sum_{k=0}^{\infty} \frac{(\tau 2a_f BM)^{2k}}{(2k)!} \sum_{n=0}^{k} \binom{k}{n} (-1)^n \left(\frac{|B_f|}{M^2}\right)^n \frac{d^n}{d(\tau|B_f|)^n} \coth(|B_f|\tau). \qquad (29.12) $$

Now, we have to apply the subtraction of divergences in the thermodynamic potential, Eq. (29.10). The integral diverges when $\tau \to 0$. When considering the Taylor expansion of the function $F_f(\tau)$ around $\tau = 0$ up to the order $O(\tau^2)$, we obtain

$$ F_f^0(\tau) = 1 + (a_f B)^2 \tau + R_f(B_f, M) \tau^2 + O(\tau^3), \quad \tau \ll 1, \qquad (29.13) $$

in the last expression, the coefficient of $\tau^2$ is given by

$$ R_f(B_f, M) = \frac{|B_f|^2}{3} - \frac{(a_f B)^4}{6} + 2(a_f B)^2 M^2 + s_f 2|B_f|(a_f B)M. \qquad (29.14) $$

Then the prescription of VMR can be applied as

$$ \begin{aligned} \Omega^{\text{mag}}(B, M) &= [\Omega^{\text{mag}}(B, M) - \Omega^{\text{VD}}(B, M)] + \Omega^{\text{VD}}(B, M) \\ &\to \Omega_R^{\text{mag}}(B, M) + \Omega^{\text{VM}}(B, M), \end{aligned} \qquad (29.15) $$

where $\Omega_R^{\text{mag}} = \Omega^{\text{mag}}(B, M) - \Omega^{\text{VD}}(B, M)$ is the magnetic part of the regularized thermodynamic potential, $\Omega^{\text{VD}}(B, M)$ is the vacuum-divergent contribution and $\Omega^{\text{VM}}(B, M)$ is the vacuum-magnetic contribution, which must be regularized.

It is possible to show that, in the limit of $eB/M_0^2 \ll 1$, we can expand the $F_f(\tau)$ function in the thermodynamical potential in order to obtain

$$ F_f(\tau) = e^{-\tau(a_f B)^2} \tau |B_f| \left[ \frac{\cosh((\alpha_f + 1)|B_f|\tau)}{\sinh(|B_f|\tau)} \right]. \qquad (29.16) $$

The expansion of $F_f^0(\tau)$ is then, given by

$$ F_f^0(\tau) = 1 + \frac{(B_f \tau)^2}{6} (3c_f^2 - 1) + O(\tau^3). \qquad (29.17) $$

which has no dependence on the effective quark masses. The thermodynamical potential can be written as
$$ \Omega = \frac{N_c}{8\pi^2} \sum_f \int_0^\infty \frac{d\tau}{\tau^3} e^{-\tau} \kappa_{0,f}^2 \left[ \tau |B_f| \frac{\cosh(c_f |B_f| \tau)}{\sinh(|B_f| \tau)} \right], \quad (29.18) $$

in the last expressions we have defined $\kappa_{0,f} = \sqrt{M^2 + (a_f B)^2}$ and $c_f = a_f + 1$. This result is the same obtained in [200] where the one-loop Schwinger-Weisskopf lagrangian in [752] is adapted to the NJL model. The regularization of this expression is given by applying the prescription Eq. (29.15).

## 29.4. Recent Achievements

The vast literature focuses in the case of magnetized Nambu–Jona-Lasinio model in the mean field approximation with constant values of quark AMM, based in the ideas developed in Ref. [730]. Then, the main difference between most works concerns in the regularization procedures. Several different properties of the magnetized QCD phase diagram have been explored, as the thermodynamics [742, 737, 743, 749], transport coefficients [746] meson masses [115, 116, 76, 78, 734] and dilepton production [731].

Recently the study with quark AMM has been extended to non-constant values. In Ref. [740], three cases were considered, in which $\kappa_{u,d}$ = constant, $\kappa_{u,d} \propto \sigma$ and $\kappa_{u,d} \propto \sigma^2$, where $\sigma$ is the chiral condensate, that depends on the magnetic field and temperature in the context of the FF$_2$. The results with $\kappa_{u,d}$ = constant reproduces well the forthcoming discussion.

Besides the FF regularization, there are other regularizations that can be applied in the group of non-MFIR procedures, as is the case of Pauli-Villars (PV) regularization. In Ref. [198], the zero AMM case is explored and, in fact, PV regularization shows very similar result for the average quark condensates as a function of the magnetic field for both MFIR and non-MFIR groups. One of the clear advantages of using PV regularization in this case is to avoiding the nonphysical oscillations previously discussed. The non-zero AMM case with PV regularization in a non-MFIR procedure has been in the context of chiral and deconfinement phase transitions [741, 732]. In the case of the magnetic field dependent three momentum cutoff, some of the results are qualitative the same when compared to the Form-Factor regularization, as the decreasing of the pseudocritical temperature as a function of the magnetic field and oscillations in the effective quark masses [76, 731].

It is possible to study the behavior of the quark AMM as a function of temperatures and magnetic fields in the NJL model as derived in Ref. [736], by evaluating the photon-quark-antiquark vertex function, which is out of scope of the present work. However, the study focuses in the use of non-MFIR procedures, and future developments with MFIR/VMR schemes could be useful in order to provide comparisons between different methods.

### 29.4.1. Form-Factor procedures

We can summarize the basic set of parametrizations used in FF$_1$ and FF$_2$ as

<table><caption>Table 1: In FF$_1$ we follow Ref. ([730]). In FF$_2$ we follow Ref. ([115]).</caption><thead><tr><th>Regularization</th><th>Λ[MeV]</th><th>m<sub>0</sub>[MeV]</th><th>GΛ<sup>2</sup></th><th>Extra Parameters</th></tr></thead><tbody><tr><td>FF<sub>1</sub></td><td>664.30</td><td>5.000</td><td>2.440</td><td>A=0.05Λ</td></tr><tr><td>FF<sub>2</sub></td><td>681.38</td><td>4.552</td><td>1.860</td><td>N=10.00</td></tr></tbody></table>