A The neutral collective modes

# Chiral phase transition and Schwinger mechanism in a pure electric field

## Abstract

We systematically study the chiral symmetry breaking and restoration in the presence of a pure electric field in the Nambu–Jona-Lasinio (NJL) model at finite temperature and baryon chemical potential. In addition, we also study the effect of the chiral phase transition on the charged pair production due to the Schwinger mechanism. For these purposes, a general formalism for parallel electric and magnetic fields is developed at finite temperature and chemical potential for the first time. In the pure electric field limit , we compute the order parameter, the transverse-to-longitudinal ratio of the Goldstone mode velocities, and the Schwinger pair production rate as functions of the electric field. The inverse catalysis effect of the electric field to chiral symmetry breaking is recovered. And the Goldstone mode is find to disperse anisotropically such that the transverse velocity is always smaller than the longitudinal one, especially at nonzero temperature and baryon chemical potential. As expected, the quark-pair production rate is greatly enhanced by the chiral symmetry restoration.

###### pacs:
11.30.Qc, 05.30.Fk, 11.30.Hv, 12.20.Ds

## I Introduction

Chiral symmetry and its spontaneous breaking is of fundamental importance for the quantum chromodynamics (QCD) as it explains the dynamical origin of the masses of hadrons. About two decades ago, it was revealed that the presence of a magnetic field would enhance the chiral condensate at zero temperature and zero quark chemical potentials (1); (2); (3); (4); (5) — a phenomenon later known as the magnetic catalysis of chiral symmetry breaking (CSB) (6); (7); (8), see recent review (9). Quite recently, the lattice QCD simulations showed that in the temperature region near the critical temperature of the chiral phase transition, the effect of the magnetic field on the CSB is very different from that at zero temperature: the presence of the magnetic field tends to restore rather than break the chiral symmetry (10); (11); (12). This inverse magnetic catalysis of CSB near seems very surprising and attracts a lot of theoretical interests, but it is still not fully understood, see e.g. Refs. (13); (14); (15); (16); (17); (18); (19); (20); (21); (22). At zero temperature but finite quark chemical potential, analogous inverse magnetic catalysis was also found (23).

Where can strong magnetic fields be generated? In nature, the neutron stars especially the magnetars may have surface magnetic fields of the order Gauss (24); (25). In experiments, recently, it was revealed that very strong magnetic fields can be generated in high-energy peripheral heavy-ion collisions (HICs) (26); (27); (28); (29): the numerical studies showed that the magnetic field in RHIC Au + Au collisions at GeV can reach while in LHC Pb + Pb collisions at TeV can reach where is the pion mass. These strong magnetic fields in HICs may drive the charge separation with respect to the reaction plane and the splitting of the elliptic flows of the charged pions through the underlying chiral magnetic and separation effects (30); (31); (32); (33); (34), see Refs. (35); (36); (37); (38) for review.

In the HICs, the electric fileds can also be generated owing to the event-by-event fluctuations (28); (29); (39); (40) or in asymmetric collisions like Cu + Au collision (41); (42); (43), and the strength of the electric fields can be roughly of the same order as the magnetic fields. These strong electric fields can lead to anomalous transport phenomena in HICs as well, that is, chiral electric separation effect (44); (45); (46); (47) and other novel observations like the charge dependence of the directed flow in Cu + Au collisions (41); (43).

The strong electric fields in HICs naturally inspire us to consider the effect of the electric field on the chiral phase transition. In this paper, we will systematically study the effect of a pure electric field on the chiral symmetry breaking and restoration in the framework of the Nambu–Jona-Lasinio (NJL) model. In fact, the effect of electric field had been previously studied at zero temperature many years ago and it was discovered that the electric field always tends to restore the chiral symmetry (1). The underlying mechanism is simple: the electric field always tends to break the quark-antiquark pair constituting the chiral condensate which triggers the chiral symmetry breaking. Another work that concerned about the effect of the second Lorentz invariant also found that the presence of the electric field in would suppress the chiral condensate at zero temperature (48). More recently, a detailed study of the stability of a chiral symmetry breaking system in electric field was performed in the framework of the chiral perturbation theory (49). As had been illuminated in Schwinger’s seminal work, the electric field would induce pair production and the production rate is closely related to the relative magnitude of field strength and the charged particle mass (50). The Schwinger mechanism has been explored in different physical contexts, see e.g., Refs. (51); (52); (53); (54); (55), but as far as we know, none of the previous works has combined the chiral phase transition with the pair production mechanism in a single quark model. In our opinion, the presence of the electric field will on one hand modify the QCD vacuum (i.e. suppress the chiral condensate) and on the other hand create quark-antiquark pairs on top of the modified vacuum. Thus, it will be interesting to study how this twofold effect of the electric field works in detail. In addition, when the temperature and the quark chemical potential are finite, richer phenomena are expected to emerge and these, to our best knowledge, have not been addressed so far.

The paper is organized as follows. In Sec.II, we establish a general formalism for systems with parallel electric and magnetic fields at finite temperature and baryon chemical potential. Due to the non-renormalizability of NJL model, a proper regularization scheme is introduced to both the gap equation and the expansion coefficients in Sec.II.3. In Sec.III, we present our numerical calculations for the cases with vanishing temperature and finite temperature, respectively. Finally, a summary is given in Sec.IV.

## Ii Formalism

### ii.1 NJL model in the case B∥E

In order to study the chiral symmetry breaking and restoration of quark matter, we adopt the NJL model which has the same approximate chiral symmetry as QCD. We will consider a background with constant parallel electric and magnetic fields and with a baryon chemical potential . It is more convenient to express the Lagrangian density in Euclidean space,

 L=¯ψ(i⧸D−m0−iμγ4)ψ+G[(¯ψψ)2+(¯ψiγ5τψ)2], (1)

where is the two-flavor quark field, is the current quark mass, is the coupling constant with dimension GeV and are pauli matrices in flavor space. Here,  () is the covariant derivative with the electric charge matrix in flavor space and the vector potential in Euclidean space chosen as which stands for electric and magnetic fields both along -direction without loss of generality. In order to study the ground state of the system, we introduce four auxiliary fields and , and the Lagrangian density becomes

 L = ¯ψ[i⧸D−m0−σ−iγ5(τ3π0+τ±π±)−iμγ4]ψ (2) −σ2+π20+π∓π±4G,

where the physical iso-vector fields are related to the auxiliary fields as , and are the raising and lowering operators in flavor space, respectively.

The presence of the electromagnetic field breaks the isospin symmetry explicitly, that is, . The order parameters for the spontaneous chiral symmetry breaking and -isospin symmetry breaking can be chosen respectively the expectation values of the physical fields and . We choose for simplicity 1 and because we will consider vanishing isospin chemical potential. Then, by taking the relationship between the chiral condensate and the dynamical mass into account and integrating out the quark degrees of freedom, the partition function can be expressed in a bosonic degree of freedom only,

 Z = ∫[D^σ][D^π0][D^π±]exp{−∫dx[(m−m0)2+^σ2+^π20+^π2±4G] (3) +Trln[i⧸D−m−^σ−iγ5(τ3^π0+τ±^π±)−iμγ4]},

where the fields with hat denote the bosonic fluctuations and the trace is taken over the quark spin, flavor, color, and the space-time coordinate spaces. In mean field approximation, the thermodynamic potential can be directly obtained as

 Ω(m)=(m−m0)24G−1βVTrln[i⧸D−m−iμγ4], (4)

where is the inverse temperature and is the volume of the system. Then, the gap equation can be formally derived by the extremal condition ,

 m−m02G−1βVTrS(x,x′)=0, (5)

where is the fermion propagator which is consistent with that defined in Schwinger’s work (50).

The bosonic fluctuations also contribute to the thermodynamic potential and their propagators satisfy

 iD−1M(x,x′) = δ(x−x′)2G+ΠM(x,x′) (6) = eiqM∫xx′Adx2Gδ(x−x′)+TrS(x,x′)ΓMS(x′,x)Γ∗M,

where are polarization functions and the interaction vertices are given by

 ΓM=⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩1,M=^σiγ5τ+,M=^π+iγ5τ−,M=^π−iγ5τ3,M=^π0;Γ∗M=⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩1,M=^σiγ5τ−,M=^π+iγ5τ+,M=^π−iγ5τ3,M=^π0. (7)

It is easy to verify that there is no mixing between different collective modes because of the absence of pion condensate. For neutral modes, that is, and , the total Wilson lines vanish; thus, the propagators only depend on the relative variables and can be transformed to energy-momentum space. Then, the dispersions of the collective modes can be obtained from the poles of the propagators,

 i~D−1M(q)=12G+~ΠM(q)=12G+Tr~S(p+q)Γi~S(p)Γ∗i=0, (8)

where is the effective fermion propagator in energy-momentum space which will be given later with Schwinger’s approach and the trace is now taken over the quark spins, flavors, colors, and energy-momenta. It is easy to verify that in the chiral limit , is the Goldstone mode for chiral symmetry breaking even with the presence of the electromagnetic field. However, the charged modes are no longer the Goldstone modes and we will not focus on them in the present study.

### ii.2 Explicit expressions with Schwinger’s approach

In order to obtain an explicit formulism for further calculations, the most important mission is to find the explicit forms of quark propagators. In a constant electromagnetic field, the quark propagators can be given explicitly by using the Schwinger’s approach (50) as

where the baryon chemical potential is introduced by considering the effective vector potential , , with the tensor matrices, and and are the two Lorentz invariants for electromagnetic field. It is important to point out that the chemical potential is a prior constrained by certain periodicity condition at finite temperature and thus cannot be canceled out beforehand by the gauge transformation. The integration involved in the Wilson line is chosen along a straight line here and we will neglect it in the following discussion as it will not affect either the gap equations or the dispersions of neutral collective modes. Then, we can define the effective propagators of quarks which only depends on the relative displacement :

 ~Sf(x−x′) = −i(4π)2∫∞0dss2e(x4−x′4)μ−(qfs)2I2Imcosh(iqfs(I1+2iI2)1/2)[−12γ(qfFcoth(qfFs)+qfF)(x−x′)+m] (10) ×exp{−im2s+i4(x−x′)qfFcoth(qfFs)(x−x′)+i2qfσFs}.

As can be seen, the effective propagators are also gauge invariant. According to the finite temperature quantum theory, the effective propagators should be (anti-)periodic in imaginary-time. This will be automatically satisfied if we take the following Fourier transformation:

 ~Sf(~p) = ∫d4(x−x′)~Sf(x−x′)e−ip(x−x′) (11) = −i∫∞0dss2det[iqfFcoth(qfFs)]−1/2−(qfs)2I2Imcosh(iqfs(I1+2iI2)1/2)[−γ(1+qfFqfFcoth(qfFs))~p+m] ×exp{−im2s−i~p1qfFcoth(qfFs)~p+i2qfσFs},

where with the fermionic Matsubara frequency . The reason why the integration over can be completed is that the effective integral variables are now which vary from to near the most significant point . Usually, the matrix can not be guaranteed to be positive definite under the transformation , thus this expression is only formal. On the other hand, because of the introduction of the proper time , the collective modes can only be adequately evaluated in energy-momentum space when . Therefore, the integrations in energy-momentum space should also be done formally to arrive at correct results. To make the procedure mathematically operable, we define

 ∫∞−∞e−ax2dx = |a|1/2a1/2∫∞−∞e−|a|x2dx, T∞∑n=−∞e−aω2n = |a|1/2a1/2T∞∑n=−∞e−|a|ω2n, (12)

for any real parameter . In this way, we are able to recover the correct gap equations at , which can also be evaluated directly in coordinate space and thus free from the non-positive-definite problem. Thus the integration in Eq. (11) should be understood in the same sense as in Eq. (II.2).

We now evaluate the effective propagators of quarks, then the gap equations and the inverse propagators of neutral collective modes explicitly in the presence of parallel electric and magnetic fields. In Euclidean space, the electromagnetic field strength tensor can be easily evaluated to have the following anti-diagonal form:

 F=⎛⎜ ⎜ ⎜⎝000−iE00−B00B00iE000⎞⎟ ⎟ ⎟⎠. (13)

It is easy to check that is diagonal, thus the variable transformation matrix can be evaluated as

 qfFcoth(qfFs) = 1s[1+∞∑n=122nB2n(qfFs)2n(2n)!]=⎛⎜ ⎜ ⎜ ⎜⎝qfEcoth(qfEs)0000qfBcot(qfBs)0000qfBcot(qfBs)0000qfEcoth(qfEs)⎞⎟ ⎟ ⎟ ⎟⎠, (14)

where the Taylor expansion form of the matrix has been used and is the Bernoulli number. The diagonal form of this matrix means no mixing between different space-time components and would make the following reductions much easier. The exponential term can also be evaluated explicitly by utilizing the general property  (50):

 exp(i2qfσFs) = ∑t=±[cosh(iqfs√I1+2itI2)1−tγ52+12σFsinh(iqfs√I1+2itI2)√I1+2itI21−tγ52)] (15) = cos(qfBs)cosh(qfEs)+isin(qfBs)sinh(qfEs)γ5+sin(qfBs)cosh(qfEs)γ1γ2+icos(qfBs)sinh(qfEs)γ4γ3.

Then the explicit forms of the effective propagators of quarks are given by

 ~Sf(~p) = i∫∞0dsexp{−im2s−itanh(qfEs)qfE(~p24+p23)−itan(qfBs)qfB(p21+p22)}[−γ~p+m+itanh(qfEs)(γ4p3−γ3~p4) (16) Missing or unrecognized delimiter for \Big

We find that the electric and magnetic fields couple with the coordinate indices and , separately. Furthermore, the effective propagators are actually invariant under the following combined transformations:

 E↔iB, γ4(γ3)↔γ1(γ2), ~p4(p3)↔p1(p2), (17)

which shows the duality between the electric and magnetic fields.

Armed with the effective propagators of quarks, it is straightforward but a little tedious to deduce the inverse propagators of the neutral collective modes. We put the details to Appendix.A and give directly the final results here,

 i~D−1M(q) = 12G−4Nc∑f=u,d∑~p∫∞0ds∫∞0ds′exp{−im2s−ifs[(~p4+q4)2+(p3+q3)2]qfE−igs[(p1+q1)2+(p2+q2)2]qfB (18) Missing or unrecognized delimiter for \Big ×(1−f2s)(1−f2s′)(1−gsgs′)−[(p2+q2)p2+(p1+q1)p1](1+g2s)(1+g2s′)(1+fsfs′)},

where for simplicity, , , and for mode. The expansion coefficients of around small momenta at zero energy are:

 ξ⊥ = −Nc4π2∑f=u,d∫∞0ds∫∞0ds′e−im2(s+s′)qfBi(gs+gs′)qfEi(fs+fs′)ϑ3⎛⎝π2+iμ2T,e−|iqfE4(fs+fs′)T2|⎞⎠{igsgs′qfB(gs+gs′) (19) ×[m2fs′(fs+fs′)(1−gsgs′)−iqfB(gs′−gsgs(gs+gs′)−gs′(1−gsgs′))(1−f2s′)(1+g2s)(1−gsgs′)] +2gsgs′(gs+gs′)2(1+g2s)(1+g2s′)(1+fsfs′)}, ξ3 = −Nc4π2∑f=u,d∫∞0ds∫∞0ds′e−im2(s+s′)qfBi(gs+gs′)qfEi(fs+fs′)ϑ3⎛⎝π2+iμ2T,e−|iqfE4(fs+fs′)T2|⎞⎠{ifsfs′qfE(fs+fs′)[m2fs′(fs+fs′) (20) Missing or unrecognized delimiter for \Big +fs′fs+fs′(1−f2s)(1−f2s′)(1−gsgs′)},

where and correspond to the transverse and longitudinal motions relative to the direction of electric field respectively and is the third Jacobi theta function obtained by working out the summation over the Matsubara frequency. It is worth mentioning that the expansion coefficient around small can only be correctly obtained when we sum over firstly in technique at finite temperature. Thus, the expansion coefficient around small can not be effectively evaluated by taking Taylor expansion beforehand as for the momenta. In the chiral limit, and are proportional to sound velocities along the transverse and longitudinal directions. Thus, their relative ratio will reflect whether any direction is more favored to the other in the electromagnetic field and deserves to study.

The dynamical mass involved in the formulas should be determined by the gap equation which can be obtained by following Eq. (5), that is,

 m−m02G = 4imNc∑f=u,d∑~p∫∞0dsexp{−im2s−itanh(qfEs)qfE(~p24+p23)−itan(qfBs)qfB(p21+p22)} (21) = −imNc4π2∑f=u,d∫∞0dse−im2sϑ3(π2+iμ2T,e−|iqfE4tanh(qfEs)T2|)qfBtan(qfBs)qfEtanh(qfEs),

where the property of gamma matrices: has been used. This result is consistent with that given in Ref. (6); (7); (8); (1); (48) in the pure magnetic field case and the pure electric field case respectively at zero temperature limit. It should be pointed out that there is no Lorentz invariance in the system now because the finite temperature breaks it; but gauge invariance is always guaranteed. In the physical vacuum, that is, , the gap equation is reduced to

 m−m02mG=−iNcI24π2∑f=u,dq2f∫∞0dse−im2sRecosh[iqfs√I1−2iI2]Imcosh[iqfs√I1−2iI2], (22)

which is shown to be explicitly gauge and Lorentz invariant as expected.

### ii.3 Regularization in the pure electric field case

Although the above formalism is general, we will hereafter focus on the pure electric field case. For the convenience of the numerical calculations, we first take a Wick rotation of the proper time integral and then transform the variable to for all formulas. On the other hand, due to the non-renormalizability of NJL model, a proper regularization scheme should be introduced to take care of the nonphysical divergence. The regularization scheme we will adopt is similar to that developed in Ref. (19); (57) in which the Goldstone theorem is self-consistently guaranteed with a single cutoff. Then the regularized gap equation in the pure electric field limit becomes

 m−m02G = Ncm2π2[Λ√1+Λ2m2−mln(Λm+√1+Λ2m2)]−Ncmπ2∑s=±∫∞0p2dp1E(p)21+e(E(p)+sμ)/T (23) Missing or unrecognized delimiter for \Big

where . With this in hand, the explicit form of the thermodynamic potential can be obtained by integrating the gap equation over , that is,

 Ω(m) = ∫dm{m−m02G−Ncm2π2[Λ√1+Λ2m2−mln(Λm+√1+Λ2m2)]+Ncmπ2∑s=±∫∞0p2dp1E(p)21+e(E(p)+sμ)/T (24) Missing or unrecognized delimiter for \Big = (m−m0)24G−Ncm34π2[Λ(1+2Λ2m2)√1+Λ2m2−mln(Λm+√1+Λ2m2)]+2Ncπ2T∑s=±∫∞0p2dpln(1+e(E(p)+sμ)/T) Missing or unrecognized delimiter for \Big

The last term is still logarithmically divergent for the integral domain around . This is not a real problem because what we really care is the difference for different phases which is of course convergent. It should be noticed that the pure magnetic correspondence of has the same magnetic field dependent part as that in Ref. (58) in the zero temperature and chemical potential limit, which justifies our present regularization scheme in the presence of electromagnetic field.

For pure electric field case, we find that there are infinite poles in the thermodynamic potential and the gap equation due to the presence of the electric-field-related tangent term which are from the contribution of Schwinger pair production. These poles render the thermodynamic potential to be complex with its real part determining the ground state while the imaginary part giving the Schwinger pair production rate. Hereafter we will simply call the real part the thermodynamic potential. We give explicit expression for the Schwinger production rate, that is, the probability per unit time and per unit volume for pair production (50),

 Γ=−2ImΩ(m)=∑f=u,d∞∑n=1Nc(qfE)24πe−nπm2/|qfE|(nπ)2, (25)

which does not depend on the temperature or chemical potential explicitly. However, since the quark mass depends on both and as can be seen in the gap equation, they will affect the pair production rate implicitly through . The numerical simulations will be presented in next section.

As well known, in the chiral symmetry breaking phase, and modes are massive and massless collective excitations, respectively. The mass of mode is very hard to determine with Schwinger’s approach because of the difficulty in summing out the Matsubara frequency analytically as we have mentioned and the awful feature of the function in the exponential. Therefore, we merely care about the expansion coefficients of the inverse propagator of mode around small and their relative ratio with each other. Then, in the case of pure electric field, by taking the limit , the inverse propagator of can be easily derived from Eq. (18) as the following:

 i~D−10(q) = 12G−4Nc∑f=u,d∑~p∫∞0ds∫∞0ds′exp{−m2(s+s′)−s[(p1+q1)2+(p2+q2)2]−s′(p21+p22) (26) −tan(qfEs)[(~p4+q4)2+(p3+q3)2]qfE−tan(qfEs′)(~p24+p23)qfE}{[m2+(p2+q2)p2+(p1+q1)p1] Missing or unrecognized delimiter for \big

First, we regularize the divergence of with the three-momentum cutoff as mentioned in Ref. (19); (57) and then by taking into account the gap equation, the inverse propagator becomes

 i~D−10(q) = −4Nc∑f=u,d∑~p∫∞0ds∫∞0ds′exp{−m2(s+s′)−s[(p1+q1)2+(p2+q2)2]−s′(p21+p22) (27) −tan(qfEs)[(~p4+q4)2+(p3+q3)2]qfE−tan(qfEs′)(~p24+p23)qfE}{[m2+(p2+q2)p2+(p1+q1)p1] Missing or unrecognized delimiter for \big Missing or unrecognized delimiter for \Big

which we have verified to be vanishing at numerically thus the Goldstone theorem is satisfied. In dimensions, this regularized inverse propagator is still divergent at finite . To keep consistent with the Goldstone theorem, we shift the divergence to an electric-field-independent part which is then evaluated with the three-momentum cutoff, that is,

 i~D−10(q) = [ΠE0(q)−Π00(q)]+Π00(q,Λ). (28)

The expansion of around small was evaluated before (59) and has a simple form:

 Π00(q,Λ)=ζ4q24+ζiq2+o(q3), ζ4 = Nc∫Λd3p(2π)31E3(p)−Nc∑