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

Chiral phase transition of quark matter in the background of parallel electric and magnetic fields 1 corresp first-author marco.ruggieri@ucas.ac.cn marco.ruggieri@ucas.ac.cn marco.ruggieri@ucas.ac.cn 1 2 corresp gxpeng@ucas.ac.cn gxpeng@ucas.ac.cn gxpeng@ucas.ac.cn College of Physics, University of Chinese Academy of Sciences, Yuquanlu 19A, Beijing 100049, China College of Physics, University of Chinese Academy of Sciences, Yuquanlu 19A, Beijing 100049, China Theoretical Physics Center for Science Facilities, Institute of High Energy Physics, Beijing 100049, China Theoretical Physics Center for Science Facilities, Institute of High Energy Physics, Beijing 100049, China 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, n 5 , produced by axial anomaly on the critical temperature. The equilibration of n 5 allows for the introduction of the chiral chemical potential, μ 5 , 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 n 5 at least if the average μ 5 , at equilibrium is not too large. Chiral chemical potential Nambu-Jona-Lasinio model Chiral phase transition in parallel electric and magnetic fields 1 Introduction 1 Systems with a finite chiral density, n 5 n R - n L , have attracted some interest recently. A medium with chirality imbalance can be obtained dynamically because of quantum anomaly [ 1 1 , 2 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 3 , 4 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 n 5 0 and B 0 might lead to the Chiral Magnetic Effect (CME) [ 5 5 , 6 6 ]. Beside CME other interesting effects related to anomaly and chiral imbalance can be found in [ 7 7 - 21 21 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, μ 5 [ 22 22 - 43 43 22-43 ]. The chiral chemical potential describes a system in which chiral density is in thermody namical equilibrium; naming τ the time scale in which n 5 equilibrates, one might assume that μ 5 describes a system in thermodynamical equilibrium with a fixed value of n 5 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 44 - 54 54 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 36 , 37 37 ]. In our context, n 5 is produced thanks to the quantum anomaly by E B 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, μ 5 , conjugated to the value of n 5 at equilibrium. In our study, we compute the value of μ 5 self-consistently by coupling the gap equation to the number equation. As a consequence, μ 5 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 T c [ 22 22 , 23 23 , 28 28 - 33 33 28-33 ]. Therefore, it is interesting to compute the response of T c to the simultaneous presence of μ 5 and fields to check if the inverse catalysis scenario obtained at μ 5 = 0 still persists at μ 5 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 1 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 55 - 58 58 55-58 ] with a local interaction kernel. The set up of the gap equation has been presented in great detail in [ 54 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 <math display="block" id="M1"><mrow><mi>L</mi><mo>=</mo><mover accent="true"><mi>ψ</mi><mo>¯</mo></mover><mrow><mo>(</mo><mrow><mi>i</mi><mi>D</mi><mtext>​</mtext><mtext>​</mtext><mtext>​</mtext><mtext>​</mtext><mo>/</mo><mo>−</mo><msub><mi>m</mi><mn>0</mn></msub></mrow><mo>)</mo></mrow><mi>ψ</mi><mo>+</mo><mi>G</mi><mrow><mo>[</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mover accent="true"><mi>ψ</mi><mo>¯</mo></mover><mi>ψ</mi><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow><mo stretchy="false">(</mo><mover accent="true"><mi>ψ</mi><mo>¯</mo></mover><mi>i</mi><msub><mi>γ</mi><mn>5</mn></msub><mi>τ</mi><mi>ψ</mi><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow><mo>]</mo></mrow><mn>,</mn></mrow></math> with ψ being a quark field with Dirac, color, and flavor indices, m 0 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 <math id="M2"><mrow><mi>D</mi><mtext>​</mtext><mtext>​</mtext><mtext>​</mtext><mtext>​</mtext><mo>/</mo><mo>=</mo><mo stretchy="false">(</mo><msub><mo>∂</mo><mi>μ</mi></msub><mo>−</mo><mi>i</mi><msub><mi>A</mi><mi>μ</mi></msub><mover accent="true"><mi>q</mi><mo>^</mo></mover><mo stretchy="false">)</mo><msub><mi>γ</mi><mi>μ</mi></msub></mrow></math> , where γ μ denotes the set of Euclidean Dirac matrices and <math id="M3"><mover accent="true"><mi>q</mi><mo>^</mo></mover></math> is the quark electric charge matrix in flavor space. In this work, we use the gauge A μ = ( iEz ,0,- Bx ,0). Introducing the auxiliary field <math id="M4"><mrow><mi>σ</mi><mo>=</mo><mo>−</mo><mn>2</mn><mi>G</mi></mrow></math> , and, within the mean field approximation, the thermodynamic potential can be written as <math display="block" id="M5"><mrow><mi>Ω</mi><mo>=</mo><mfrac><mrow><msup><mrow><mo stretchy="false">(</mo><msub><mi>M</mi><mi>q</mi></msub><mo>−</mo><msub><mi>m</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow><mrow><mn>4</mn><mi>G</mi></mrow></mfrac><mo>−</mo><mfrac><mn>1</mn><mrow><mi>β</mi><mi>V</mi></mrow></mfrac><mtext>Tr</mtext><mi>log</mi><mi>β</mi><mo stretchy="false">(</mo><mi>i</mi><mi>D</mi><mtext>​</mtext><mtext>​</mtext><mtext>​</mtext><mtext>​</mtext><mo>/</mo><mo>−</mo><msub><mi>M</mi><mi>q</mi></msub><mo stretchy="false">)</mo><mn>,</mn></mrow></math> where the constituent quark mass is <math id="M6"><mrow><msub><mi>M</mi><mi>q</mi></msub><mo>=</mo><msub><mi>m</mi><mn>0</mn></msub><mo>−</mo><mn>2</mn><mi>G</mi><mo>〈</mo><mo>〉</mo></mrow></math> , β =1/ T , and βV corresponds to the Euclidean quantization volume. The constituent quark mass differs from m 0 because of spontaneous chiral symmetry breaking, the latter being related to a nonvanishing chiral condensate, <math id="M7"><mrow><mo>〈</mo><mover accent="true"><mi>ψ</mi><mo>¯</mo></mover><mi>ψ</mi><mo>〉</mo><mo>≠</mo><mn>0</mn></mrow></math> . 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 M q for simplicity, keeping in mind that whenever we discuss about the chiral phase transition in terms of M q the decrease of the latter is related to the decreasing of magnitude of the chiral condensate. The main technical task is to compute M q at finite temperature and in presence of the external fields: this is achieved by requiring the physical value of M q minimizes the thermodynamic potential, and this in turn implies that M q satisfies the gap equation, ∂Ω/∂ M q = 0, namely <math display="block" id="M8"><mrow><mfrac><mrow><msub><mi>M</mi><mi>q</mi></msub><mo>−</mo><msub><mi>m</mi><mn>0</mn></msub></mrow><mrow><mn>2</mn><mi>G</mi></mrow></mfrac><mo>−</mo><mfrac><mn>1</mn><mrow><mi>β</mi><mi>V</mi></mrow></mfrac><mtext>Tr</mtext><mi>S</mi><mo stretchy="false">(</mo><mi>x</mi><mn>,</mn><msup><mi>x</mi><mo>'</mo></msup><mo stretchy="false">)</mo><mo>=</mo><mn>0,</mn></mrow></math> 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 54 ], namely <math display="block" id="M9"><mtable columnalign="left"><mtr><mtd><mfrac><mrow><msub><mi>M</mi><mi>q</mi></msub><mo>−</mo><msub><mi>m</mi><mn>0</mn></msub></mrow><mrow><mn>2</mn><mi>G</mi></mrow></mfrac><mo>=</mo><msub><mi>M</mi><mi>q</mi></msub><mfrac><mrow><msub><mi>N</mi><mi>c</mi></msub></mrow><mrow><mn>4</mn><msup><mi>π</mi><mn>2</mn></msup></mrow></mfrac><mstyle displaystyle="true"><munder><mo>∑</mo><mi>f</mi></munder></mstyle><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mn>0</mn><mi>∞</mi></msubsup></mrow></mstyle><mfrac><mrow><mi>d</mi><mi>s</mi></mrow><mrow><msup><mi>s</mi><mn>2</mn></msup></mrow></mfrac><msup><mi>e</mi><mrow><mo>−</mo><msubsup><mi>M</mi><mi>q</mi><mn>2</mn></msubsup><mi>s</mi></mrow></msup><mi>F</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mo>+</mo><msub><mi>M</mi><mi>q</mi></msub><mfrac><mrow><msub><mi>N</mi><mi>c</mi></msub><msub><mi>N</mi><mi>f</mi></msub></mrow><mrow><mn>4</mn><msup><mi>π</mi><mn>2</mn></msup></mrow></mfrac><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mrow><mn>1</mn><mo>/</mo><msup><mi>Λ</mi><mn>2</mn></msup></mrow><mi>∞</mi></msubsup></mrow></mstyle><mfrac><mrow><mi>d</mi><mi>s</mi></mrow><mrow><msup><mi>s</mi><mn>2</mn></msup></mrow></mfrac><msup><mi>e</mi><mrow><mo>−</mo><msubsup><mi>M</mi><mi>q</mi><mn>2</mn></msubsup><mi>s</mi></mrow></msup><mn>,</mn></mtd></mtr></mtable></math> where we have defined <math display="block" id="M10"><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>θ</mi><mn>3</mn></msub><mrow><mo>(</mo><mrow><mfrac><mi>π</mi><mn>2</mn></mfrac><mn>,</mn><msup><mi>e</mi><mrow><mo>−</mo><mo>|</mo><mi>A</mi><mo>|</mo></mrow></msup></mrow><mo>)</mo></mrow><mfrac><mrow><msub><mi>q</mi><mi>f</mi></msub><mi>e</mi><mi>B</mi><mi>s</mi></mrow><mrow><mi>tanh</mi><mo stretchy="false">(</mo><msub><mi>q</mi><mi>f</mi></msub><mi>e</mi><mi>B</mi><mi>s</mi><mo stretchy="false">)</mo></mrow></mfrac><mfrac><mrow><msub><mi>q</mi><mi>f</mi></msub><mi>e</mi><mi>E</mi><mi>s</mi></mrow><mrow><mi>tan</mi><mo stretchy="false">(</mo><msub><mi>q</mi><mi>f</mi></msub><mi>e</mi><mi>E</mi><mi>s</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>−</mo><mn>1</mn></mrow></math> with θ 3 ( x , z ) being the third elliptic theta function, and <math display="block" id="M11"><mrow><mi>A</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><msub><mi>q</mi><mi>f</mi></msub><mi>e</mi><mi>E</mi></mrow><mrow><mn>4</mn><msup><mi>T</mi><mn>2</mn></msup><mi>tan</mi><mo stretchy="false">(</mo><msub><mi>q</mi><mi>f</mi></msub><mi>e</mi><mi>E</mi><mi>s</mi><mo stretchy="false">)</mo></mrow></mfrac><mn>.</mn></mrow></math> 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/Λ 2 ; 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 57 ], namely Λ = 1086 MeV and G =3.78/Λ 2 . The presence of the 1/tan( q f eEs ) 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 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 59 , 60 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 1 3.1 Zero temperature 2 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 Fig. 1 , we plot M q 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 M q 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 E B 0 as we specify later), which manifests in a non-monotonic behaviour of M q 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 44 - 48 48 44-48 ]. The behaviour of M q for small values of the fields can be understood quantitatively at T =0 and m 0 =0: in fact, in this case, we can find an analytical solution for the gap equation by writing M q = M 0 + δ m , where M 0 corresponds to the solution of the gap equation for in the zero field case. We find <math display="block" id="M13"><mrow><mi>δ</mi><mi>m</mi><mo>=</mo><mfrac><mn>1</mn><mrow><mn>2</mn><msub><mi>N</mi><mi>f</mi></msub><mo>|</mo><msub><mi>E</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mo>−</mo><msubsup><mi>M</mi><mn>0</mn><mn>2</mn></msubsup><mo>/</mo><msup><mi>Λ</mi><mn>2</mn></msup><mo stretchy="false">)</mo><mo>|</mo></mrow></mfrac><mo stretchy="false">(</mo><msub><mi>ϒ</mi><mn>1</mn></msub><mo>+</mo><msub><mi>ϒ</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mn>,</mn></mrow></math> where <math display="block" id="M14"><mrow><msub><mi>ϒ</mi><mn>1</mn></msub><mo>=</mo><mfrac><mrow><msubsup><mi>q</mi><mi>u</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>q</mi><mi>d</mi><mn>2</mn></msubsup></mrow><mrow><mn>3</mn><msubsup><mi>M</mi><mn>0</mn><mn>3</mn></msubsup></mrow></mfrac><msub><mi>I</mi><mn>1</mn></msub><mn>,</mn></mrow></math> <math display="block" id="M15"><mrow><msub><mi>ϒ</mi><mn>2</mn></msub><mo>=</mo><mo>−</mo><mfrac><mrow><msubsup><mi>q</mi><mi>u</mi><mn>4</mn></msubsup><mo>+</mo><msubsup><mi>q</mi><mi>d</mi><mn>4</mn></msubsup></mrow><mrow><mn>45</mn><msubsup><mi>M</mi><mn>0</mn><mn>7</mn></msubsup></mrow></mfrac><mo stretchy="false">(</mo><msubsup><mi>I</mi><mn>1</mn><mn>2</mn></msubsup><mo>+</mo><mn>7</mn><msubsup><mi>I</mi><mn>2</mn><mn>2</mn></msubsup><mo stretchy="false">)</mo><mn>,</mn></mrow></math> with <math id="M16"><mrow><msub><mi>I</mi><mn>1</mn></msub><mo>≡</mo><msup><mrow><mo stretchy="false">(</mo><mi>e</mi><mi>B</mi><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>−</mo><msup><mrow><mo stretchy="false">(</mo><mi>e</mi><mi>E</mi><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow></math> , <math id="M17"><mrow><msub><mi>I</mi><mn>2</mn></msub><mo>≡</mo><mo stretchy="false">(</mo><mi>e</mi><mi>E</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>e</mi><mi>B</mi><mo stretchy="false">)</mo></mrow></math> ; moreover, <math id="M18"><mrow><msub><mi>E</mi><mi>i</mi></msub></mrow></math> denotes the exponential integral function, <math id="M19"><mrow><msub><mi>E</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mrow><mo>−</mo><mi>x</mi></mrow><mi>∞</mi></msubsup></mrow></mstyle><mi>d</mi><mi>s</mi><msup><mi>e</mi><mrow><mo>−</mo><mi>s</mi></mrow></msup><mo>/</mo><mi>s</mi></mrow></math> . From Eq. (7) we notice that for B =0, <math id="M20"><mrow><mi>δ</mi><mi>m</mi><mo>∝</mo><mo>−</mo><msup><mi>E</mi><mn>2</mn></msup><mo>/</mo><msubsup><mi>M</mi><mn>0</mn><mn>3</mn></msubsup></mrow></math> at the lowest order; the curvature of <math id="M21"><mrow><mi>δ</mi><mi>m</mi></mrow></math> versus eE does not change as long as eE > eB . For E = B one has to take into account the contribution O ( E 2 B 2 ) which still shows <math id="M22"><mrow><mi>δ</mi><mi>m</mi><mo>∝</mo><mo>−</mo><msup><mi>E</mi><mn>2</mn></msup><msup><mi>B</mi><mn>2</mn></msup><mo>/</mo><msubsup><mi>M</mi><mn>0</mn><mn>7</mn></msubsup></mrow></math> leading to a decreasing M q . Finally, for eB > eE the catalysis sets in, at least for small values of the fields, eventually leading to <math id="M23"><mrow><mi>δ</mi><mi>m</mi><mo>∝</mo><msup><mi>B</mi><mn>2</mn></msup><mo>/</mo><msubsup><mi>M</mi><mn>0</mn><mn>3</mn></msubsup></mrow></math> for E =0. 3.2 Finite temperature 2 The situation depicted in the previous subsection is qualitatively unchanged at finite temperature. In Fig. 2 Fig. 2 , we plot M q 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 <math id="M24"><mrow><mi>e</mi><mi>E</mi><mo>=</mo><msubsup><mi>m</mi><mi>π</mi><mn>2</mn></msubsup></mrow></math> , orange dotted line to <math id="M25"><mrow><mi>e</mi><mi>E</mi><mo>=</mo><mn>8</mn><msubsup><mi>m</mi><mi>π</mi><mn>2</mn></msubsup></mrow></math> , and green dashed line to <math id="M26"><mrow><mi>e</mi><mi>E</mi><mo>=</mo><mn>15</mn><msubsup><mi>m</mi><mi>π</mi><mn>2</mn></msubsup></mrow></math> . 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 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. 3.3 Phase diagram 2 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 m 0 ≠0, there is no a chiral phase transition, but a smooth crossover, see data in Fig. 2 Fig. 2 , therefore we have to chose a criterion to define a pseudo-critical temperature, T c : in our work we identify T c with the temperature at which the maximum of | dM q /dT | occurs (in the case m 0 =0 T c is uniquely defined by the temperature at which the condensate is zero). In Fig. 3 Fig. 3 we plot T c versus eE (measured in units of <math id="M30"><mrow><msubsup><mi>m</mi><mi>π</mi><mn>2</mn></msubsup></mrow></math> ) for several values of the external magnetic field: black squares correspond to B =0, red diamonds to <math id="M31"><mrow><mi>e</mi><mi>B</mi><mo>=</mo><mn>5</mn><msubsup><mi>m</mi><mi>π</mi><mn>2</mn></msubsup></mrow></math> and green triangles to <math id="M32"><mrow><mi>e</mi><mi>B</mi><mo>=</mo><mn>10</mn><msubsup><mi>m</mi><mi>π</mi><mn>2</mn></msubsup></mrow></math> . 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 . 3.4 Chiral density effects on the critical temperature 2 The electric-magnetic background considered in our study is unstable because of the Schwinger pair production [ 60 60 , 59 59 ]. Once we define the chiral density, n 5 = n R - n L where n L/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 n 5 [ 61 61 ] <math display="block" id="M34"><mrow><mfrac><mrow><mi>d</mi><msub><mi>n</mi><mn>5</mn></msub></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><msubsup><mi>q</mi><mi>f</mi><mn>2</mn></msubsup><mo stretchy="false">(</mo><mi>e</mi><mi>E</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>e</mi><mi>B</mi><mo stretchy="false">)</mo></mrow><mrow><mn>2</mn><msup><mi>π</mi><mn>2</mn></msup></mrow></mfrac><msup><mi>e</mi><mrow><mo>−</mo><mfrac><mrow><mi>π</mi><msup><mi>M</mi><mn>2</mn></msup></mrow><mrow><mo>|</mo><msub><mi>q</mi><mi>f</mi></msub><mi>e</mi><mi>E</mi><mo>|</mo></mrow></mfrac></mrow></msup><mn>.</mn></mrow></math> If evolution of n 5 was given only by the above equation, then treating the fields as external, quantum anomaly would lead to an ethernal production of n 5 , 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 n 5 . In order to take into account of these processes we add a relaxation term on the right hand side of the above equation, <math display="block" id="M35"><mrow><mfrac><mrow><mi>d</mi><msub><mi>n</mi><mn>5</mn></msub></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><msubsup><mi>q</mi><mi>f</mi><mn>2</mn></msubsup><mo stretchy="false">(</mo><mi>e</mi><mi>E</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>e</mi><mi>B</mi><mo stretchy="false">)</mo></mrow><mrow><mn>2</mn><msup><mi>π</mi><mn>2</mn></msup></mrow></mfrac><msup><mi>e</mi><mrow><mo>−</mo><mfrac><mrow><mi>π</mi><msup><mi>M</mi><mn>2</mn></msup></mrow><mrow><mo>|</mo><msub><mi>q</mi><mi>f</mi></msub><mi>e</mi><mi>E</mi><mo>|</mo></mrow></mfrac></mrow></msup><mo>−</mo><mfrac><mrow><msub><mi>n</mi><mn>5</mn></msub></mrow><mi>τ</mi></mfrac><mn>.</mn></mrow></math> For t > > τ the solution of Eq. (11) relaxes to the equilibrium value <math display="block" id="M36"><mrow><msubsup><mi>n</mi><mn>5</mn><mrow><mtext>eq</mtext></mrow></msubsup><mo>=</mo><mfrac><mrow><msubsup><mi>q</mi><mi>f</mi><mn>2</mn></msubsup><mo stretchy="false">(</mo><mi>e</mi><mi>E</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>e</mi><mi>B</mi><mo stretchy="false">)</mo></mrow><mrow><mn>2</mn><msup><mi>π</mi><mn>2</mn></msup></mrow></mfrac><msup><mi>e</mi><mrow><mo>−</mo><mfrac><mrow><mi>π</mi><msup><mi>M</mi><mn>2</mn></msup></mrow><mrow><mo>|</mo><msub><mi>q</mi><mi>f</mi></msub><mi>e</mi><mi>E</mi><mo>|</mo></mrow></mfrac></mrow></msup><mi>τ</mi><mo>;</mo></mrow></math> 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 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 n 5 . Equation (12) shows that on a time scale larger than the relaxation time an equilibrium value of n 5 , that we name <math id="M37"><mrow><msubsup><mi>n</mi><mn>5</mn><mrow><mtext>eq</mtext></mrow></msubsup></mrow></math> , is produced. The existence of <math id="M38"><mrow><msubsup><mi>n</mi><mn>5</mn><mrow><mtext>eq</mtext></mrow></msubsup></mrow></math> means it is possible to introduce a chemical potential for the chiral charge, the chiral chemical potential μ 5 , conjugated to <math id="M39"><mrow><msubsup><mi>n</mi><mn>5</mn><mrow><mtext>eq</mtext></mrow></msubsup></mrow></math> . A self-consistent computation of μ 5 , given the value of <math id="M40"><mrow><msubsup><mi>n</mi><mn>5</mn><mrow><mtext>eq</mtext></mrow></msubsup></mrow></math> in Eq. (12), requires a canonical ensemble calculation in which the gap equation for M q is solved self-consistently with the number equation, namely <math display="block" id="M41"><mrow><msubsup><mi>n</mi><mn>5</mn><mrow><mtext>eq</mtext></mrow></msubsup><mo>=</mo><mo>−</mo><mfrac><mrow><mo>∂</mo><mi>Ω</mi></mrow><mrow><mo>∂</mo><msub><mi>μ</mi><mn>5</mn></msub></mrow></mfrac><mn>,</mn></mrow></math> with μ 5 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 μ 5 as well as small fields. See Ref. [ 36 36 ] for more details. Taking into account of the equilibrium value of the chiral density is important because it is known that μ 5 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 μ 5 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 μ 5 on T c . In the upper panel of Fig. 4 Fig. 4 we plot M q versus temperature for the case <math id="M42"><mrow><mi>e</mi><mi>E</mi><mo>=</mo><mi>e</mi><mi>B</mi><mo>=</mo><mn>8</mn><msubsup><mi>m</mi><mi>π</mi><mn>2</mn></msubsup></mrow></math> and n 5 =0 (green dots), and we compare it to the case in which n 5 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 Fig. 4 we plot the equilibrium value of n 5 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 n 5 depends on the quark flavor because electric charge is different, see Eq. (12). Finally, in the lower panel of Fig. 4 Fig. 4 we show the value of μ 5 corresponding to n 5 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 M q is pushed towards larger values in comparison with the case μ 5 =0. Compare dots and the dashed line in the upper panel of Fig. 4 Fig. 4 . When we compute the derivative of M q with respect to temperature, we also find that T c is slightly increased by μ 5 . However, the net increase of T c , due to μ 5 , is very small in comparison with the lowering of T c induced by the fields. As a consequence, the equilibrated chiral density does not affect drastically the thermodynamics and the phase diagram. 4 Conclusion 1 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, T c , is lowered by the simultaneous presence of E B . We have then considered the effect of equilibration of chiral density, n 5 , produced dynamically by axial anomaly on the critical temperature. The equilibration of n 5 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 n 5 . We have focused here on a constant value of τ =1 fm/c in line with the results of [ 37 37 ]. Because this dynamical system reaches a thermodynamical equilibrium state for t ≫ τ , with a specified value of <math id="M44"><mrow><msub><mi>n</mi><mn>5</mn></msub><mo>=</mo><msubsup><mi>n</mi><mn>5</mn><mrow><mtext>eq</mtext></mrow></msubsup></mrow></math> depending on the actual values of the field and of the temperature, it is possible to introduce the chiral chemical potential, μ 5 , conjugated to <math id="M45"><mrow><msubsup><mi>n</mi><mn>5</mn><mrow><mtext>eq</mtext></mrow></msubsup></mrow></math> at equilibrium. The value of μ 5 has been computed by coupling the gap equation to the number equation, solving both within a small μ 5 approximation. The chiral chemical potential acts as a catalyzer of chiral symmetry breaking, therefore, in principle the background of n 5 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 μ 5 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 T c , due to μ 5 , is much smaller than the lowering induced by the fields. This conclusion might be no longer valid in the case of large μ 5 , 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 μ 5 approximation, that will be the subject of our near future work. References S. L. Adler , Axial vector vertex in spinor electrodynamics , Phys. Rev . 177 , 2426 ( 1969 ). doi: 10.1103/PhysRev.177.2426 http://doi.org/10.1103/PhysRev.177.2426 Axial vector vertex in spinor electrodynamics J. S. Bell and R. Jackiw , A PCAC puzzle: π0 → γγ in the sigma model , Nuovo Cim. A 60 , 47 ( 1969 ). doi: 10.1007/BF02823296 http://doi.org/10.1007/BF02823296 A PCAC puzzle: π0 → γγ in the sigma model G. D. Moore , Do we understand the sphaleron rate?, hep-ph/0009161 . Do we understand the sphaleron rate?, hep-ph/0009161 G. D. Moore and M. Tassler , The Sphaleron Rate in SU(N) Gauge Theory , JHEP 1102 , 105 ( 2011 ). doi: 10.1007/JHEP02(2011)105 http://doi.org/10.1007/JHEP02(2011)105 The Sphaleron Rate in SU(N) Gauge Theory D. E. Kharzeev , L. D. McLerran and H. J. Warringa , The Effects of topological charge change in heavy ion collisions: ’Event by event P and CP violation’ , Nucl. Phys. A 803 , 227 ( 2008 ). doi: 10.1016/j.nuclphysa.2008.02.298 http://doi.org/10.1016/j.nuclphysa.2008.02.298 The Effects of topological charge change in heavy ion collisions: ’Event by event P and CP violation’ K. Fukushima , D. E. Kharzeev and H. J. Warringa , The Chiral Magnetic Effect , Phys. Rev. D 78 , 074033 ( 2008 ). doi: 10.1103/PhysRevD.78.074033 http://doi.org/10.1103/PhysRevD.78.074033 The Chiral Magnetic Effect D. T. Son and P. Surowka , Hydrodynamics with Triangle Anomalies , Phys. Rev. Lett . 103 , 191601 ( 2009 ). doi: 10.1103/PhysRevLett.103.191601 http://doi.org/10.1103/PhysRevLett.103.191601 Hydrodynamics with Triangle Anomalies N. Banerjee , J. Bhattacharya , S. Bhattacharyya , S. Dutta , R. Loganayagam and P. Surowka , Hydrodynamics from charged black branes , JHEP 1101 , 094 ( 2011 ). doi: 10.1007/JHEP01(2011)094 http://doi.org/10.1007/JHEP01(2011)094 Hydrodynamics from charged black branes K. Landsteiner , E. Megias and F. Pena-Benitez , Gravitational Anomaly and Transport , Phys. Rev. Lett . 107 , 021601 ( 2011 ). doi: 10.1103/PhysRevLett.107.021601 http://doi.org/10.1103/PhysRevLett.107.021601 Gravitational Anomaly and Transport D. T. Son and A. R. Zhitnitsky , Quantum anomalies in dense matter , Phys. Rev. D 70 , 074018 ( 2004 ). doi: 10.1103/PhysRevD.70.074018 http://doi.org/10.1103/PhysRevD.70.074018 Quantum anomalies in dense matter M. A. Metlitski and A. R. Zhitnitsky , Anomalous axion interactions and topological currents in dense matter , Phys. Rev. D 72 , 045011 ( 2005 ). doi: 10.1103/PhysRevD.72.045011 http://doi.org/10.1103/PhysRevD.72.045011 Anomalous axion interactions and topological currents in dense matter D. E. Kharzeev and H. U. Yee , Chiral Magnetic Wave , Phys. Rev. D 83 , 085007 ( 2011 ). doi: 10.1103/PhysRevD.83.085007 http://doi.org/10.1103/PhysRevD.83.085007 Chiral Magnetic Wave M. N. Chernodub , Chiral Heat Wave and mixing of Magnetic, Vortical and Heat waves in chiral media , JHEP 1601 , 100 ( 2016 ). doi: 10.1007/JHEP01(2016)100 http://doi.org/10.1007/JHEP01(2016)100 Chiral Heat Wave and mixing of Magnetic, Vortical and Heat waves in chiral media M. N. Chernodub and M. Zubkov , Intrinsic chiral magnetic effect in Dirac semimetals due to dislocations , arXiv:1508.03114 [cond-mat.mes-hall] . Intrinsic chiral magnetic effect in Dirac semimetals due to dislocations M. N. Chernodub , A. Cortijo , A. G. Grushin , K. Landsteiner and M. A. H. Vozmediano , Condensed matter realization of the axial magnetic effect , Phys. Rev. B 89 , no. 8 , 081407 ( 2014 ). doi: 10.1103/PhysRevB.89.081407 http://doi.org/10.1103/PhysRevB.89.081407 Condensed matter realization of the axial magnetic effect V. Braguta , M. N. Chernodub , K. Landsteiner , M. I. Polikarpov and M. V. Ulybyshev , Numerical evidence of the axial magnetic effect , Phys. Rev. D 88 , 071501 ( 2013 ). doi: 10.1103/PhysRevD.88.071501 http://doi.org/10.1103/PhysRevD.88.071501 Numerical evidence of the axial magnetic effect A. V. Sadofyev and M. V. Isachenkov , The Chiral magnetic effect in hydrodynamical approach , Phys. Lett. B 697 , 404 ( 2011 ). doi: 10.1016/j.physletb.2011.02.041 http://doi.org/10.1016/j.physletb.2011.02.041 The Chiral magnetic effect in hydrodynamical approach A. V. Sadofyev , V. I. Shevchenko and V. I. Zakharov , Notes on chiral hydrodynamics within effective theory approach , Phys. Rev. D 83 , 105025 ( 2011 ). doi: 10.1103/PhysRevD.83.105025 http://doi.org/10.1103/PhysRevD.83.105025 Notes on chiral hydrodynamics within effective theory approach Z. V. Khaidukov , V. P. Kirilin , A. V. Sadofyev and V. I. Zakharov , On Magnetostatics of Chiral Media , arXiv:1307.0138 [hep-th] . On Magnetostatics of Chiral Media V. P. Kirilin , A. V. Sadofyev and V. I. Zakharov , Anomaly and long-range forces , doi: 10.1142/9789814616850_0014 http://doi.org/10.1142/9789814616850_0014 arXiv:1312.0895 [hep-th] . Anomaly and long-range forces A. Avdoshkin , V. P. Kirilin , A. V. Sadofyev and V. I. Zakharov , On consistency of hydrodynamic approximation for chiral media , Phys. Lett. B 755 , 1 ( 2016 ). doi: 10.1016/j.physletb.2016.01.048 http://doi.org/10.1016/j.physletb.2016.01.048 On consistency of hydrodynamic approximation for chiral media M. Ruggieri and G. X. Peng , Critical Temperature of Chiral Symmetry Restoration for Quark Matter with a Chiral Chemical Potential , arXiv:1602.03651 [hep-ph] . Critical Temperature of Chiral Symmetry Restoration for Quark Matter with a Chiral Chemical Potential M. Ruggieri and G. X. Peng , Chiral Symmetry Restoration with a Chiral Chemical Potential: the Role of Momentum Dependent Quark Self-energy , arXiv:1602.05250 [hep-ph] . Chiral Symmetry Restoration with a Chiral Chemical Potential: the Role of Momentum Dependent Quark Self-energy R. Gatto and M. Ruggieri , Hot Quark Matter with a chiral Chemical Potential , Phys. Rev. D 85 , 054013 ( 2012 ). doi: 10.1103/PhysRevD.85.054013 http://doi.org/10.1103/PhysRevD.85.054013 Hot Quark Matter with a chiral Chemical Potential K. Fukushima , M. Ruggieri and R. Gatto , Chiral magnetic effect in the PNJL model , Phys. Rev. D 81 , 114031 ( 2010 ). doi: 10.1103/PhysRevD.81.114031 http://doi.org/10.1103/PhysRevD.81.114031 Chiral magnetic effect in the PNJL model M. N. Chernodub and A. S. Nedelin , Phase diagram of chirally imbalanced QCD matter , Phys. Rev. D 83 , 105008 ( 2011 ). doi: 10.1103/PhysRevD.83.105008 http://doi.org/10.1103/PhysRevD.83.105008 Phase diagram of chirally imbalanced QCD matter M. Ruggieri , The Critical End Point of Quantum Chromodynamics Detected by Chirally Imbalanced Quark Matter , Phys. Rev. D 84 , 014011 ( 2011 ). doi: 10.1103/PhysRevD.84.014011 http://doi.org/10.1103/PhysRevD.84.014011 The Critical End Point of Quantum Chromodynamics Detected by Chirally Imbalanced Quark Matter L. Yu , H. Liu and M. Huang , The effect of the chiral chemical potential on the chiral phase transition in the NJL model with different regularization schemes , arXiv:1511.03073 [hep-ph] . The effect of the chiral chemical potential on the chiral phase transition in the NJL model with different regularization schemes L. Yu , J. Van Doorsselaere and M. Huang , Inverse Magnetic Catalysis in the three-flavor NJL model with axial-vector interaction , Phys. Rev. D 91 , no. 7 , 074011 ( 2015 ). doi: 10.1103/PhysRevD.91.074011 http://doi.org/10.1103/PhysRevD.91.074011 Inverse Magnetic Catalysis in the three-flavor NJL model with axial-vector interaction M. Frasca , Nonlocal Nambu-Jona-Lasinio model and chiral chemical potential , arXiv:1602.04654 [hep-ph] . Nonlocal Nambu-Jona-Lasinio model and chiral chemical potential V. V. Braguta , E.-M. Ilgenfritz , A. Y. Kotov , B. Petersson and S. A. Skinderev , Study of QCD Phase Diagram with Non-Zero Chiral Chemical Potential , arXiv:1512.05873 [hep-lat] . Study of QCD Phase Diagram with Non-Zero Chiral Chemical Potential V. V. Braguta , V. A. Goy , E.-M. Ilgenfritz , A. Y. Kotov , A. V. Molochkov , M. Muller-Preussker and B. Petersson , Two-Color QCD with Non-zero Chiral Chemical Potential , JHEP 1506 , 094 ( 2015 ). doi: 10.1007/JHEP06(2015)094 http://doi.org/10.1007/JHEP06(2015)094 Two-Color QCD with Non-zero Chiral Chemical Potential V. V. Braguta and A. Y. Kotov , Catalysis of Dynamical Chiral Symmetry Breaking by Chiral Chemical Potential , arXiv:1601.04957 [hep-th] . Catalysis of Dynamical Chiral Symmetry Breaking by Chiral Chemical Potential M. Hanada and N. Yamamoto , Universality of phase diagrams in QCD and QCD-like theories , PoS LATTICE 2011 , 221 ( 2011 ). Universality of phase diagrams in QCD and QCD-like theories S. S. Xu , Z. F. Cui , B. Wang , Y. M. Shi , Y. C. Yang and H. S. Zong , Chiral phase transition with a chiral chemical potential in the framework of Dyson-Schwinger equations , Phys. Rev. D 91 , no. 5 , 056003 ( 2015 ). doi: 10.1103/PhysRevD.91.056003 http://doi.org/10.1103/PhysRevD.91.056003 Chiral phase transition with a chiral chemical potential in the framework of Dyson-Schwinger equations M. Ruggieri and G. X. Peng , Quark matter in a parallel electric and magnetic field background: Chiral phase transition and equilibration of chiral density , Phys. Rev. D 93 , no. 9 , 094021 ( 2016 ). doi: 10.1103/PhysRevD.93.094021 http://doi.org/10.1103/PhysRevD.93.094021 Quark matter in a parallel electric and magnetic field background: Chiral phase transition and equilibration of chiral density M. Ruggieri , G. X. Peng and M. Chernodub , Chiral Relaxation Time at the Chiral Crossover of Quantum Chromodynamics , arXiv:1606.03287 [hep-ph] . Chiral Relaxation Time at the Chiral Crossover of Quantum Chromodynamics R. L. S. Farias , D. C. Duarte , G. Krein and R. O. Ramos , Quark matter with a chiral imbalance in the Nambu-Jona-Lasinio model , arXiv:1604.04518 [hep-ph] . Quark matter with a chiral imbalance in the Nambu-Jona-Lasinio model Z. F. Cui , I. C. Cloet , Y. Lu , C. D. Roberts , S. M. Schmidt , S. S. Xu and H. S. Zong , Critical endpoint in the presence of a chiral chemical potential , arXiv:1604.08454 [nucl-th] . Critical endpoint in the presence of a chiral chemical potential A. A. Andrianov , D. Espriu and X. Planells , An effective QCD Lagrangian in the presence of an axial chemical potential , Eur. Phys. J. C 73 , no. 1 , 2294 ( 2013 ). doi: 10.1140/epjc/s10052-013-2294-0 http://doi.org/10.1140/epjc/s10052-013-2294-0 An effective QCD Lagrangian in the presence of an axial chemical potential S. S. Afonin , A. A. Andrianov and D. Espriu , The masses of vector mesons in holographic QCD at finite chiral chemical potential , Phys. Lett. B 745 , 52 ( 2015 ). doi: 10.1016/j.physletb.2015.04.027 http://doi.org/10.1016/j.physletb.2015.04.027 The masses of vector mesons in holographic QCD at finite chiral chemical potential D. Ebert , T. G. Khunjua , K. G. Klimenko and V. C. Zhukovsky , Competition and duality correspondence between chiral and superconducting channels in (2+1)-dimensional four-fermion models with fermion number and chiral chemical potentials , Phys. Rev. D 93 , no. 10 , 105022 ( 2016 ). doi: 10.1103/PhysRevD.93.105022 http://doi.org/10.1103/PhysRevD.93.105022 Competition and duality correspondence between chiral and superconducting channels in (2+1)-dimensional four-fermion models with fermion number and chiral chemical potentials B. Wang , Y. L. Wang , Z. F. Cui and H. S. Zong , Effect of the chiral chemical potential on the position of the critical endpoint , Phys. Rev. D 91 , no. 3 , 034017 ( 2015 ). doi: 10.1103/PhysRevD.91.034017 http://doi.org/10.1103/PhysRevD.91.034017 Effect of the chiral chemical potential on the position of the critical endpoint A. Y. Babansky , E. V. Gorbar and G. V. Shchepanyuk , Chiral symmetry breaking in the Nambu-Jona-Lasinio model in external constant electromagnetic field , Phys. Lett. B 419 , 272 ( 1998 ). doi: 10.1016/S0370-2693(97)01445-7 http://doi.org/10.1016/S0370-2693(97)01445-7 Chiral symmetry breaking in the Nambu-Jona-Lasinio model in external constant electromagnetic field S. P. Klevansky and R. H. Lemmer , Chiral symmetry restoration in the Nambu-Jona-Lasinio model with a constant electromagnetic field , Phys. Rev. D 39 , 3478 ( 1989 ). doi: 10.1103/PhysRevD.39.3478 http://doi.org/10.1103/PhysRevD.39.3478 Chiral symmetry restoration in the Nambu-Jona-Lasinio model with a constant electromagnetic field H. Suganuma and T. Tatsumi , On the Behavior of Symmetry and Phase Transitions in a Strong Electromagnetic Field , Annals Phys . 208 , 470 ( 1991 ). doi: 10.1016/0003-4916(91)90304-Q http://doi.org/10.1016/0003-4916(91)90304-Q On the Behavior of Symmetry and Phase Transitions in a Strong Electromagnetic Field K. G. Klimenko , Three-dimensional Gross-Neveu model at nonzero temperature and in an external magnetic field, Z . Phys. C 54 , 323 ( 1992 ). doi: 10.1007/BF01566663 http://doi.org/10.1007/BF01566663 Three-dimensional Gross-Neveu model at nonzero temperature and in an external magnetic field, Z K. G. Klimenko , Three-dimensional Gross-Neveu model at nonzero temperature and in an external magnetic field, Theor . Math. Phys . 90 , 1 ( 1992 ) Three-dimensional Gross-Neveu model at nonzero temperature and in an external magnetic field, Theor K. G. Klimenko , Three-dimensional Gross-Neveu model at nonzero temperature and in an external magnetic field, Theor [ Teor. Mat. Fiz. 90 , 3 ( 1992 )]. doi: 10.1007/BF01018812 http://doi.org/10.1007/BF01018812 Three-dimensional Gross-Neveu model at nonzero temperature and in an external magnetic field, Theor I. V. Krive and S. A. Naftulin , Dynamical symmetry breaking and phase transitions in a three-dimensional Gross-Neveu model in a strong magnetic field," Phys . Rev. D 46 , 2737 ( 1992 ). doi: 10.1103/PhysRevD.46.2737 http://doi.org/10.1103/PhysRevD.46.2737 Dynamical symmetry breaking and phase transitions in a three-dimensional Gross-Neveu model in a strong magnetic field," Phys V. P. Gusynin , V. A. Miransky and I. A. Shovkovy , Catalysis of dynamical flavor symmetry breaking by a magnetic field in (2+1)-dimensions , Phys. Rev. Lett . 73 , 3499 ( 1994 ), Catalysis of dynamical flavor symmetry breaking by a magnetic field in (2+1)-dimensions V. P. Gusynin , V. A. Miransky and I. A. Shovkovy , Dimensional reduction and dynamical chiral symmetry breaking by a magnetic field in (3+1)-dimensions , Phys. Lett. B 349 , 477 ( 1995 ). doi: 10.1016/0370-2693(95)00232-A http://doi.org/10.1016/0370-2693(95)00232-A Dimensional reduction and dynamical chiral symmetry breaking by a magnetic field in (3+1)-dimensions G. Cao and X. G. Huang , Electromagnetic triangle anomaly and neutral pion condensation in QCD vacuum , Phys. Lett. B 757 , 1 ( 2016 ). doi: 10.1016/j.physletb.2016.03.066 http://doi.org/10.1016/j.physletb.2016.03.066 Electromagnetic triangle anomaly and neutral pion condensation in QCD vacuum M. D’Elia , M. Mariti and F. Negro , Susceptibility of the QCD vacuum to CP-odd electromagnetic background fields , Phys. Rev. Lett . 110 , no. 8 , 082002 ( 2013 ). doi: 10.1103/PhysRevLett.110.082002 http://doi.org/10.1103/PhysRevLett.110.082002 Susceptibility of the QCD vacuum to CP-odd electromagnetic background fields G. Cao and X. G. Huang , Chiral phase transition and Schwinger mechanism in a pure electric field , Phys. Rev. D 93 , no. 1 , 016007 ( 2016 ). doi: 10.1103/PhysRevD.93.016007 http://doi.org/10.1103/PhysRevD.93.016007 Chiral phase transition and Schwinger mechanism in a pure electric field Y. Nambu and G. Jona-Lasinio , Dynamical Model of Elementary Particles Based on an Analogy with Superconductivity. 1 ., Phys. Rev . 122 , 345 ( 1961 ). doi: 10.1103/PhysRev.122.345 http://doi.org/10.1103/PhysRev.122.345 Dynamical Model of Elementary Particles Based on an Analogy with Superconductivity. 1 Y. Nambu and G. Jona-Lasinio , Dynamical Model Of Elementary Particles Based On An Analogy With Superconductivity. Ii , Phys. Rev . 124 , 246 ( 1961 ). doi: 10.1103/PhysRev.124.246 http://doi.org/10.1103/PhysRev.124.246 Dynamical Model Of Elementary Particles Based On An Analogy With Superconductivity. Ii S. P. Klevansky , The Nambu-Jona-Lasinio model of quantum chromodynamics, Rev . Mod. Phys . 64 , 649 ( 1992 ). doi: 10.1103/RevModPhys.64.649 http://doi.org/10.1103/RevModPhys.64.649 The Nambu-Jona-Lasinio model of quantum chromodynamics, Rev T. Hatsuda and T. Kunihiro , QCD phenomenology based on a chiral effective Lagrangian , Phys. Rept . 247 , 221 ( 1994 ). doi: 10.1016/0370-1573(94)90022-1 http://doi.org/10.1016/0370-1573(94)90022-1 QCD phenomenology based on a chiral effective Lagrangian J. S. Schwinger , On gauge invariance and vacuum polarization , Phys. Rev . 82 , 664 ( 1951 ). On gauge invariance and vacuum polarization W. Heisenberg and H. Euler , Consequences of Dirac’s theory of positrons , Z. Phys . 98 , 714 ( 1936 ). Consequences of Dirac’s theory of positrons W. Heisenberg and H. Euler , Consequences of Dirac’s theory of positrons See also arXiv:physics/0605038 for an English translation by W. Korolevski and H. Kleinert . doi: 10.1007/BF01343663 http://doi.org/10.1007/BF01343663 Consequences of Dirac’s theory of positrons H. J. Warringa , Dynamics of the Chiral Magnetic Effect in a weak magnetic field , Phys. Rev. D 86 , 085029 ( 2012 ). doi: 10.1103/PhysRevD.86.085029 http://doi.org/10.1103/PhysRevD.86.085029 Dynamics of the Chiral Magnetic Effect in a weak magnetic field M. Ruggieri would like to acknowledge fruitful discussions with M. Chernodub and M. Frasca. 10.1007/s41365-016-0139-x This work was supported by the CAS President’s International Fellowship Initiative (No. 2015PM008), and the National Natural Science Foundation of China (Nos. 11135011 and 11575190) . 2016-08-15 2016-08-30 2016-09-01 2016-10-11 2016-12-20 2021-04-11 © China Science Publishing & Media Ltd. (Science Press), Shanghai Institute of Applied Physics, the Chinese Academy of Sciences, ChineseNuclear Society and Springer Nature Singapore Pte Ltd. 2016 This is a post-peer-review, pre-copyedit version of an article published in Nuclear Science and Techniques. The final authenticated version is available online at: http://dx.doi.org/10.1007/s41365-016-0139-x http://dx.doi.org/10.1007/s41365-016-0139-x . 2016 Nuclear Science and Techniques 06 27