Chiral phase transition of quark matter in the background of parallel electric and magnetic fields

Special Section on International Workshop on Nuclear Dynamics in Heavy-Ion Reactions (IWND2016)

Chiral phase transition of quark matter in the background of parallel electric and magnetic fields

M. Ruggieri，

Guang-Xiong Peng

Nuclear Science and Techniques

Vol.27, No.6

Article number 130

Published in print 20 Dec 2016

Available online 11 Oct 2016

DOI：10.1007/s41365-016-0139-x

47100

We report on our results about spontaneous chiral symmetry breaking for quark matter in the background of static and homogeneous parallel electric field, E, and magnetic field, B. A Nambu-Jona-Lasinio model is used to compute the dependence of the chiral condensate at finite temperature, E and B. We study the effect of this background on inverse catalysis of chiral symmetry breaking for E and B of the same order of magnitude. We also consider the effect of equilibration of chiral density, n₅, produced by axial anomaly on the critical temperature. The equilibration of n₅ allows for the introduction of the chiral chemical potential, μ₅, which is computed self-consistently as a function of temperature and field strength. We find that even if the chiral medium is produced by the fields the thermodynamics, with particular reference to the inverse catalysis induced by the external fields, it is not very affected by n₅ at least if the average μ₅, at equilibrium is not too large.

Chiral chemical potentialNambu-Jona-Lasinio modelChiral phase transition in parallel electric and magnetic fields

1 Introduction

Systems with a finite chiral density, n₅ nR-nL, have attracted some interest recently. A medium with chirality imbalance can be obtained dynamically because of quantum anomaly [1, 2] when fermions interact with nontrivial gauge field configurations. In the context of Quantum Chromodynamics (QCD), some of these configurations at finite temperature in Minkowski space are named sphalerons, being characterized by parallel chromo-electric and chromo-magnetic fields and whose production rate has been estimated to be quite large at large temperature [3, 4]. The large number of sphaleron transitions in high temperature suggests the possibility that net chirality might be locally abundant in the quark-gluon plasma phase of QCD; when one couples this thermal QCD bath with an external strong magnetic field, B, produced in the early stages of heavy ion collisions, the coexistence of n₅ 0 and B 0 might lead to the Chiral Magnetic Effect (CME) [5, 6]. Beside CME other interesting effects related to anomaly and chiral imbalance can be found in [7-21].

In order to describe systems with finite chirality in thermodynamical equilibrium, it is possible to introduce the conjugated chemical potential, named the chiral chemical potential, μ₅ [22-43]. The chiral chemical potential describes a system in which chiral density is in thermodynamical equilibrium; naming τ the time scale in which n₅ equilibrates, one might assume that μ₅ describes a system in thermodynamical equilibrium with a fixed value of n₅ on a time scale much larger than τ.

In this talk, we report about chiral phase transition and chiral density production in the context of quark matter in a background static and homogeneous parallel electric, E, and magnetic, B, fields. This particular field configuration is interesting in the context of a medium with chirality imbalance because it produces dynamically a chiral density thanks to the anomaly. Firstly, we focus on the critical temperature for chiral symmetry restoration, T_c, in presence of the external fields. This part of the study completes previous studies about chiral symmetry breaking/restoration in the background of external fields [44-54]. We find that the effect of the electric field is to lower the critical temperature, and this inverse catalysis scenario does not change considerably when the magnetic field is added, as long as the magnetic field is not very large compared to the electric one.

We are also interested in studying the effect of chiral density on the thermodynamics of the system, following our previous studies [36, 37]. In our context, n₅ is produced thanks to the quantum anomaly by EB 0, and equilibrates because of the existence of chirality changing processes in the thermal bath that relax on a time scale, τ. We then introduce the chiral chemical potential, μ₅, conjugated to the value of n₅ at equilibrium. In our study, we compute the value of μ₅ self-consistently by coupling the gap equation to the number equation. As a consequence, μ₅ depends on temperature as well as on external fields, and on the relaxation time which brings information about the microscopic processes that lead to chiral density relaxation.

In the second part of this talk we focus on the role of chirality production on T_c. As mentioned above, the E B term tends to lower the critical temperature; on the other hand the chiral chemical potential has the effect to increase Tc [22, 23, 28-33]. Therefore, it is interesting to compute the response of Tc to the simultaneous presence of μ₅ and fields to check if the inverse catalysis scenario obtained at μ₅ = 0 still persists at μ₅ 0. We find that chiral density does not affect drastically the thermodynamics at the phase transition, confirming the inverse catalysis induced by the fields, at least if the average chiral chemical potential in the crossover region turns out to be small with respect to temperature.

2 The model

We are interested to study quark matter in a background made of parallel electric, E, and magnetic, B, fields. We assume the fields are constant in time and homogeneous in space; moreover we assume they develop along the z-direction. For concrete calculations we use a Nambu-Jona-Lasinio (NJL) model [55-58] with a local interaction kernel. The set up of the gap equation has been presented in great detail in [54], therefore we will skip all the technical details and report here only the few equations we need to specify the interactions used in the calculations. The Euclidean lagrangian density is given by

L = \bar{ψ} (i D ​ ​ ​ ​ / - m_{0}) ψ + G [{(\bar{ψ} ψ)}^{2} + {(\bar{ψ} i γ_{5} τ ψ)}^{2}],

(1)

with ψ being a quark field with Dirac, color, and flavor indices, m₀ is the current quark mass, and τ denotes a vector of Pauli matrices on flavor space. The interaction with the background fields is embedded in the covariant derivative $D / = (\partial_{μ} - i A_{μ} \hat{q}) γ_{μ}$ , where γμ denotes the set of Euclidean Dirac matrices and $\hat{q}$ is the quark electric charge matrix in flavor space. In this work, we use the gauge Aμ = (iEz,0,-Bx,0).

Introducing the auxiliary field $σ = - 2 G$ , and, within the mean field approximation, the thermodynamic potential can be written as

Ω = \frac{{(M_{q} - m_{0})}^{2}}{4 G} - \frac{1}{β V} Tr \log β (i D ​ ​ ​ ​ / - M_{q}),

(2)

where the constituent quark mass is $M_{q} = m_{0} - 2 G 〈 〉$ , β=1/T, and βV corresponds to the Euclidean quantization volume. The constituent quark mass differs from m₀ because of spontaneous chiral symmetry breaking, the latter being related to a nonvanishing chiral condensate, $〈 \bar{ψ} ψ 〉 \neq 0$ . Even if it would be more appropriate to discuss chiral symmetry restoration via the quark condensate, because it has its counterpart in QCD, we will refer to Mq for simplicity, keeping in mind that whenever we discuss about the chiral phase transition in terms of Mq the decrease of the latter is related to the decreasing of magnitude of the chiral condensate.

The main technical task is to compute Mq at finite temperature and in presence of the external fields: this is achieved by requiring the physical value of Mq minimizes the thermodynamic potential, and this in turn implies that Mq satisfies the gap equation, ∂Ω/∂Mq = 0, namely

\frac{M_{q} - m_{0}}{2 G} - \frac{1}{β V} Tr S (x, x^{'}) = 0,

(3)

where S(x,x′) corresponds to the full fermion propagator in the electric and magnetic field background. The computation of the propagator has been already given in detail in [54], namely

\begin{array}{l} \frac{M_{q} - m_{0}}{2 G} = M_{q} \frac{N_{c}}{4 π^{2}} \sum_{f} \int_{0}^{\infty} \frac{d s}{s^{2}} e^{- M_{q}^{2} s} F (s) \\ + M_{q} \frac{N_{c} N_{f}}{4 π^{2}} \int_{1 / Λ^{2}}^{\infty} \frac{d s}{s^{2}} e^{- M_{q}^{2} s}, \end{array}

(4)

where we have defined

F (s) = θ_{3} (\frac{π}{2}, e^{- | A |}) \frac{q_{f} e B s}{\tanh (q_{f} e B s)} \frac{q_{f} e E s}{\tan (q_{f} e E s)} - 1

(5)

with θ₃(x,z) being the third elliptic theta function, and

A (s) = \frac{q_{f} e E}{4 T^{2} \tan (q_{f} e E s)} .

(6)

In Eq. (4) we have added and subtracted the zero field contribution on the right hand side which is the only one to diverge, and we have regularized it by cutting the integration at s=1/Λ²; on the other hand we have not added a cutoff on the field dependent part. In calculations we use the standard parameter set for a proper time regularization [57], namely Λ = 1086 MeV and G=3.78/Λ².

The presence of the 1/tan(q_feEs) in Eq. (5) implies the existence of an infinite set of poles on the integration in s in Eq. (4). Following the treatment by Schwinger [59] these poles are shifted from the real axis to the complex plane by adding a small imaginary part which allows to perform the s-integration in principal value; this leads to an imaginary part of the free energy, which is a sign of the vacuum instability induced by the static electric field [59, 60] and leads to particle pair creation. We will consider the effect of this vacuum instability in Section 3.4 because it can be directly connected to chiral density production in case of parallel E and B.

3 Results and the suggested phase diagram

3.1 Zero temperature

The main goal of our study is to compute the combined effect of the electric and magnetic background on the chiral phase transition. In Fig. 1, we plot Mq as a function of eB for several choices of E, starting from E=0 up to E=B; for comparison we also show the case B=0 (green dashed line). In the case E=0, the system experiences a direct magnetic catalysis (DMC), namely Mq increases with B. On the other hand for E = 0.5 B we find a sign of competition among DMC induced by B and inverse catalysis (IC) induced by E (and also by EB 0 as we specify later), which manifests in a non-monotonic behaviour of Mq versus eB. The IC becomes more evident by increasing the value of E/B. The inverse catalysis effect induced by the electric field and the second electromagnetic invariant, E B, are in agreement with previous studies at zero temperature [44-48].

Fig. 1.

Dynamical quark mass versus magnetic field strength at zero temperature, for several values of the background electric field. Maroon dot-dashed line corresponds to the case of a pure magnetic field, indigo dotted line to E = 0.25 B, dashed magenta line to E = 0.5 B, dot-dot-dashed brown line to E = 0.75B and finally orange line to E = B. For comparison we have also shown data for B=0; in this case on x-axis we show eE in units of

m_{π}^{2}

. Adapted from Ref. [36].

The behaviour of Mq for small values of the fields can be understood quantitatively at T=0 and m₀=0: in fact, in this case, we can find an analytical solution for the gap equation by writing Mq=M₀+δm, where M₀ corresponds to the solution of the gap equation for in the zero field case. We find

δ m = \frac{1}{2 N_{f} | E_{i} (- M_{0}^{2} / Λ^{2}) |} (ϒ_{1} + ϒ_{2}),

(7)

where

ϒ_{1} = \frac{q_{u}^{2} + q_{d}^{2}}{3 M_{0}^{3}} I_{1},

(8)

ϒ_{2} = - \frac{q_{u}^{4} + q_{d}^{4}}{45 M_{0}^{7}} (I_{1}^{2} + 7 I_{2}^{2}),

(9)

with $I_{1} \equiv {(e B)}^{2} - {(e E)}^{2}$ , $I_{2} \equiv (e E) (e B)$ ; moreover, $E_{i}$ denotes the exponential integral function, $E_{i} (x) = - \int_{- x}^{\infty} d s e^{- s} / s$ . From Eq. (7) we notice that for B=0, $δ m \propto - E^{2} / M_{0}^{3}$ at the lowest order; the curvature of $δ m$ versus eE does not change as long as eE > eB. For E = B one has to take into account the contribution O(E²B²) which still shows $δ m \propto - E^{2} B^{2} / M_{0}^{7}$ leading to a decreasing Mq. Finally, for eB>eE the catalysis sets in, at least for small values of the fields, eventually leading to $δ m \propto B^{2} / M_{0}^{3}$ for E=0.

3.2 Finite temperature

The situation depicted in the previous subsection is qualitatively unchanged at finite temperature. In Fig. 2, we plot Mq versus T for several values of E and B: thin lines correspond to B=0, while with thick lines we denote the results for E = B. The blue solid line corresponds to $e E = m_{π}^{2}$ , orange dotted line to $e E = 8 m_{π}^{2}$ , and green dashed line to $e E = 15 m_{π}^{2}$ . Increasing the electric field strength results in a lowering of the critical temperature, and the effect of B≠0 is just to increase a bit the quark mass and shift the critical temperature towards slightly higher values. The results collected in Fig. 2 show that, even when B=E, the effect of the fields on the critical temperature does not cancel and the electric field gives the more important contribution, leading to an inverse catalysis. In fact, one would need a larger value of B to observe an increase of the critical temperature.

Fig. 2.

Dynamical quark mass versus temperature for several values of E and B. Thin lines correspond to B=0, while thick lines denote the results for E = B. The blue solid line corresponds to

e E = m_{π}^{2}

, orange dotted line to

e E = 8 m_{π}^{2}

, and green dashed line to

e E = 15 m_{π}^{2}

. Color convention for thick lines follows that we have used for thin lines. Adapted from Ref. [36].

3.3 Phase diagram

Using the results discussed in the previous subsections, we are able to suggest a phase diagram in the temperature-electric field strength plane. Firstly we notice that in the case m₀≠0, there is no a chiral phase transition, but a smooth crossover, see data in Fig. 2, therefore we have to chose a criterion to define a pseudo-critical temperature, Tc: in our work we identify Tc with the temperature at which the maximum of |dMq/dT| occurs (in the case m₀=0 Tc is uniquely defined by the temperature at which the condensate is zero). In Fig. 3 we plot Tc versus eE (measured in units of $m_{π}^{2}$ ) for several values of the external magnetic field: black squares correspond to B=0, red diamonds to $e B = 5 m_{π}^{2}$ and green triangles to $e B = 10 m_{π}^{2}$ . This figure summarizes one of our main results, namely that the electric field leads to a lowering of the critical temperature for chiral symmetry restoration, and the presence of the parallel magnetic field does not change this result unless B E.

Fig. 3.

Critical temperature for chiral symmetry restoration versus electric field strength, measured in units of

m_{π}^{2}

, for several values of the external magnetic field. Adapted from Ref. [36].

3.4 Chiral density effects on the critical temperature

The electric-magnetic background considered in our study is unstable because of the Schwinger pair production [60, 59]. Once we define the chiral density, n₅ = nR - nL where nL/R correspond to left/right-handed particle numbers respectively, quantum anomaly, that is related to the Schwinger effect in this particular case, leads to a dynamical production of n₅ [61]

\frac{d n_{5}}{d t} = \frac{q_{f}^{2} (e E) (e B)}{2 π^{2}} e^{- \frac{π M^{2}}{| q_{f} e E |}} .

(10)

If evolution of n₅ was given only by the above equation, then treating the fields as external, quantum anomaly would lead to an ethernal production of n₅, and the system would never be able to reach thermodynamical equilibrium. However, Eq. (10) is just half of the story: as a matter of fact in the thermal bath there are chirality changing processes which occur on a time scale, τ, that we call the relaxation time of chiral density. The presence of τ>0 leads to equilibration of n₅. In order to take into account of these processes we add a relaxation term on the right hand side of the above equation,

\frac{d n_{5}}{d t} = \frac{q_{f}^{2} (e E) (e B)}{2 π^{2}} e^{- \frac{π M^{2}}{| q_{f} e E |}} - \frac{n_{5}}{τ} .

(11)

For t>>τ the solution of Eq. (11) relaxes to the equilibrium value

n_{5}^{eq} = \frac{q_{f}^{2} (e E) (e B)}{2 π^{2}} e^{- \frac{π M^{2}}{| q_{f} e E |}} τ;

(12)

the message encoded in the above equation is that the external fields eventually lead to the production of a medium made of a chiral density, analogously to the medium usually studied in nuclear and particle physics where one considers nuclear and/or quark matter for a finite value of the baryon and/or isospin density.

In Ref. [37] the relaxation time has been computed at the chiral crossover, assuming that the relevant microscopic processes to relax chiral density are quark-quark scattering with the exchange of collective excitations with the quantum numbers of pions as well as -meson. The computation of the relevant collision integrals gives the result for τ in between 0.1 and 2 fm/c around the crossover, with τ decreasing with temperature due to the opening of the phase space. For simplicity in this talk we summarize results obtained with τ=1 fm/c in the whole temperature range, neglecting the temperature dependence of τ that eventually would lead to smaller values of n₅.

Equation (12) shows that on a time scale larger than the relaxation time an equilibrium value of n₅, that we name $n_{5}^{eq}$ , is produced. The existence of $n_{5}^{eq}$ means it is possible to introduce a chemical potential for the chiral charge, the chiral chemical potential μ₅, conjugated to $n_{5}^{eq}$ . A self-consistent computation of μ₅, given the value of $n_{5}^{eq}$ in Eq. (12), requires a canonical ensemble calculation in which the gap equation for Mq is solved self-consistently with the number equation, namely

n_{5}^{eq} = - \frac{\partial Ω}{\partial μ_{5}},

(13)

with μ₅ introduced in the quark propagator with E B. This full calculation is well beyond the purpose of the present study and is left to a future work: here we limit ourselves to report results obtained in the limit of small μ₅ as well as small fields. See Ref. [36] for more details.

Taking into account of the equilibrium value of the chiral density is important because it is known that μ₅ leads to an increase of T_c, thus working in the opposite direction with respect to the fields in the determination of the critical line. Since μ₅ is produced by the fields themselves, in order to give a firm answer about the net effect of the fields on the phase diagram we have to consider the backreaction of μ₅ on T_c.

In the upper panel of Fig. 4 we plot Mq versus temperature for the case $e E = e B = 8 m_{π}^{2}$ and n₅=0 (green dots), and we compare it to the case in which n₅ is given by its equilibrium value Eq. (12) (orange dashed line). For matter of comparison we also show by an indigo solid line the data corresponding to E=B=0. For completeness, in the middle panel of Fig. 4 we plot the equilibrium value of n₅ computed by Eq. (12) as a function of temperature for u quarks (solid indigo line) and d quarks (green dashed line); the orange dot-dashed line corresponds to the average value. The value of n₅ depends on the quark flavor because electric charge is different, see Eq. (12). Finally, in the lower panel of Fig. 4 we show the value of μ₅ corresponding to n₅ computed by the number equation Eq. (13). We notice that the chiral chemical potential at equilibrium enhances chiral symmetry breaking, in agreement with expectations: indeed Mq is pushed towards larger values in comparison with the case μ₅=0. Compare dots and the dashed line in the upper panel of Fig. 4. When we compute the derivative of Mq with respect to temperature, we also find that Tc is slightly increased by μ₅. However, the net increase of Tc, due to μ₅, is very small in comparison with the lowering of Tc induced by the fields. As a consequence, the equilibrated chiral density does not affect drastically the thermodynamics and the phase diagram.

Fig. 4.

Upper panel. Mq versus temperature for the case

e E = e B = 8 m_{π}^{2}

and n₅=0 (green dots), same values of E and B but with n₅ given by its equilibrium value Eq. (12) (orange dashed line). For comparison, we plot by indigo solid line the data corresponding to E=B=0. Middle panel. Equilibrium value of n₅ computed by Eq. (12) as a function of temperature for u quarks (solid indigo line) and d quarks (green dashed line); orange dot-dashed line corresponds to the average value. Lower panel. Equilibrium value of μ₅ corresponding to n₅ shown in the middle panel, computed by the number equation Eq. (13).

4 Conclusion

In this talk, we have summarized our results about spontaneous chiral symmetry breaking for quark matter in the background of static, homogeneous and parallel electric field, E, and magnetic field, B. We have used a Nambu-Jona-Lasinio model to compute the relevant quantities to describe chiral symmetry breaking at finite temperature. We have firstly computed the response of the chiral condensate to the external fields, both at zero and at nonzero temperature. One of the main results is that the critical temperature for chiral symmetry restoration, Tc, is lowered by the simultaneous presence of E B.

We have then considered the effect of equilibration of chiral density, n₅, produced dynamically by axial anomaly on the critical temperature. The equilibration of n₅ happens as a consequence of chirality flipping processes in the thermal bath; we have introduced the relaxation time for chirality, namely τ, giving the time scale necessary for the equilibration of n₅. We have focused here on a constant value of τ=1 fm/c in line with the results of [37]. Because this dynamical system reaches a thermodynamical equilibrium state for t≫τ, with a specified value of $n_{5} = n_{5}^{eq}$ depending on the actual values of the field and of the temperature, it is possible to introduce the chiral chemical potential, μ₅, conjugated to $n_{5}^{eq}$ at equilibrium. The value of μ₅ has been computed by coupling the gap equation to the number equation, solving both within a small μ₅ approximation.

The chiral chemical potential acts as a catalyzer of chiral symmetry breaking, therefore, in principle the background of n₅ can spoil the inverse catalysis induced by B. We have found however that the equilibrated chiral density does not change drastically the thermodynamics as long as μ₅ at equilibrium is not too large; namely, the inverse catalysis effect induced by the background fields is eventually not spoiled by the presence of the chiral medium. This can be understood because the increase of Tc, due to μ₅, is much smaller than the lowering induced by the fields. This conclusion might be no longer valid in the case of large μ₅, that probably would be produced by large fields. To study quantitatively the general case it is necessary to solve simultaneously the number and gap equations without using a small μ₅ approximation, that will be the subject of our near future work.

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