Study of the effects of external imaginary electric field and chiral chemical potential on quark matter

The behavior of quark matter with both external electric field and chiral chemical potential is theoretically and experimentally interesting to consider. In this paper, the case of simultaneous presence of imaginary electric field and chiral chemical potential is investigated using the lattice QCD approach with $N_f=1+1$ dynamical staggered fermions. We find that overall both the imaginary electric field and the chiral chemical potential can exacerbate chiral symmetry breaking, which is consistent with theoretical predictions. However we also find a non-monotonic behavior of chiral condensation at specific electric field strengths and chiral chemical potentials. In addition to this, we find that the behavior of Polyakov loop in the complex plane is not significantly affected by chiral chemical potential in the region of the parameters consider in this paper.


Introduction
In the quark-gluon-plasma phase chirality imbalance of QCD is expected [1][2][3], so it is of interest to study how chirality imbalance modifies the structure of QCD.A convenient way to treat quark matter with net chirality is to introduce an chiral (axial) chemical potential, µ 5 , conjugated to chiral density, the difference between the number of right-and lefthanded fermions, n 5 = n R − n L .The chiral chemical potential can be introduced to study the topological changing transitions [4][5][6][7][8], where the effect of topological charges can be mimicked by an effective θ angle, and µ 5 can be interpreted as the time derivative of the θ angle as µ 5 = θ ′ (t)/(2N f ), where t is the time coordinate, and N f is the number of fla-vors [9,10].One of its most important applications is the chiral magnetic effect (CME) [9,11].
Various effective models such as the Nambu-Jona-Lasinio (NJL) model [12], the Polyakov loop extended NJL model [13,14], the linear sigma model coupled to the Polyakov loop [10,15] and chiral perturbation theory [16] are used to investigate effects of chiral chemical potential, including the effects of chiral chemical potential on chiral magnetic effect, chiral phase transition, critical end point in quark matter, etc.The chirality imbalanced hot and dense strongly interacting matter is also studied by means of the Dyson-Schwinger equations [17].Catalysis effect of dynamical chiral symmetry breaking by chiral chemical potential is observed and chiral charge density generally increases with temperature, quark number chemical potential and chiral chemical potential.
The effect of an external electric field on the QCD vacuum structure is also a very interesting subject because of academic and realistic reasons.In the relativistic heavy-ion collisions, the electric fields can be generated owing to the event-by-event fluctuations or in asymmetric collisions like Cu + Au collision, and the strength of the electric fields can be roughly of the same order as the magnetic fields [18][19][20][21][22][23].It has been shown that, the electric field restores the chiral symmetry [24][25][26][27][28][29][30][31], and n 5 can be affected by the external electric field and magnetic field.The impact of electric and magnetic fields on chirality has also been studied using lattice approach [32].
In lattice QCD approach, the chiral chemical potential was introduced to study the CME [33], and has been studied in various previous works [34][35][36][37][38][39].It is confirmed that n 5 is proportional to the magnetic field and to the chiral chemical potential.In the case that only µ 5 is presented, both string tension and chiral susceptibility grow with the chiral chemical potential.In this paper, we consider the case where both the electric and chiral chemical potential are presented.
The remainder of this paper is organized as follows, in Sec. 2 the action on the lattice is briefly introduced, the numerical results are shown in Sec. 3, the Sec. 4 is a summary.

The model
In this paper, we consider the case of an external uniform electric field at z direction and with the chiral chemical potential.The electric gauge field can be written as A EM µ = (−E z z, 0, 0, 0) in the axial gauge, the superscript 'EM' is added to distinguish with the QCD gauge field.The Lagrangian with one massless fermion is where the last term corresponds to the chiral chemical potential.The fermion action is, where γ j=1,2,3 = iγ E j , and γ E 4 = γ 0 .The chiral chemical potential tends to break the chiral symmetry.Meanwhile, the electric field tends to break chiral condensation whether the chiral imbalance is taken into account or not.Apart from that, chirality imbalance can be affected when homogeneous parallel electric field and magnetic field are presented [31].Therefore, the case of quark matter with chiral chemical potential and external electric field is interesting.
It is known that, using lattice approach, there is notorious 'sign problem' in the case of external real electric field, except for the case that u and d quarks have opposite charges [40].To void the 'sign problem', analytical continuation is often used to study the case of electric field, which is found to be reliable [41][42][43][44][45][46][47][48][49].
Except for the boundaries, in axial gauge the presence of the external electric field can be viewed as a stacking of volumes with different imaginary chemical potentials µ = Q q eE z z extending the z-axis.The case of a homogeneous imaginary chemical potential and the corresponding Roberge-Weiss (R-W) transition [50] has been studied [51][52][53][54][55][56].Similar R-W transition is also found in the case of electric field [49] (similar behavior is also found in the case of magnetic field [57]).In this paper, the behavior of Polyakov loop at different z coordinate in the complex plane is also studied.
On the lattice, the discretized action with staggered fermions, and without the electric field and the chiral chemical poten-tial can be written as [58][59][60], where a is the lattice spacing, β = 2N c /g 2 YM with g YM the coupling strength of the gauge fields to the quarks, m is the fermion mass, We will introduce the electric field and the chiral chemical potential in the following subsections.

Electric field
To avoid the 'sign problem', in this paper a Wick rotation is performed to the electric field, in other words, a substitution is applied.The result of the substitution corresponds to an imaginary electric field, which is also known as an 'Euclidean' or 'classical' electric field.
The electric field is introduced as a U(1) phase in this paper, i.e., a U(1) phase corresponds to the external electric field is added to the gauge links connecting bilinear terms of χ, the fermion action without chiral chemical potential is then, where To ensure gauge invariance of the external electric field and the translational invariance at the same time, a twisted boundary condition is applied [43,[61][62][63], in this paper we use ( where L µ is the extent at direction µ. In this paper, the origin of the axis is set to be the middle of the spatial volume and at n τ = 1.Therefore, V µ (n) = 1 except for The strength of electric field is quantized and is decided by L z × L τ .In this paper, the extents of the lattice is Besides, it is also required that exp(−iQ q a 2 eE z ) to be an approximation of 1 − iQ q a 2 eE z , as a consequence, for a 2 eE z ∼ O(1) or larger, the result suffer from strong discretization errors.To keep our result qualitatively reliable, we use where k is an integer satisfying 0 ≤ k ≤ 8.

Chiral chemical potential
By using the definition of the staggered quarks, Ψ αa (h) = 65], and χ = a 2 ψ/ √ 2a, it can be shown that [66] (2a where ⃗ s is an offset vector pointing to diagonals of elementary cubes in the x − y − z space.The sum over h in the left hand side of Eq. ( 7) is a sum over even sites (2 4 hypercubes).
The U,V (n, n+⃗ s) are Wilson lines of SU(3) and U(1) gauge fields connecting n and n +⃗ s.For a three dimensional cube, there are six shortest paths connecting n and n +⃗ s, similar as in Refs.[35,36], we use the average of the Wilson lines along the six paths.In this paper, the µ 5 term is introduced in a linear form in the action, instead of exponential factors on the links.In this case, there will be µ 5 -dependent additive divergences in the observables.The case of exponential form is treated using Taylor-expansion and is discussed in the study of the anomalous transport phenomena [67].
The fermion action with both the electric field and chiral chemical potential considered is then S q = S EM q + S µ 5 q .

Numerical results
In this paper, we simulate on a lattice with N x × N y × N z × N t = 8 × 8 × 24 × 6, and with N f = 1 + 1 staggered fermions with am = 0.1.The dynamic fermions with same masses are charged as Q u = 2/3 and Q d = −1/3.For one flavour, the 'forth root trick' is applied to remove the taste degeneracies, and the rational hybrid Monte Carlo [68] is used.For each β , at first, 200 + 3000 × 9 trajectories are simulated.

√
2τ ind [60], where σ jk is statistical error calculated using 'jackknife' method, and 2τ ind is the separation of molecular dynamics time units (TU) such that the two configurations can be regarded as independent.τ ind is calculated by using 'autocorrelation' with S = 1.5 [69] on the bare chiral condensation of u quark.A brief introduction of the calculation of τ int is established in Appendix A.
The breaking/restoration of chiral symmetry phase transition is also studied on a 12 3 × 6 lattice [49].The pseudo critical temperature is T = 229 MeV which corresponds to β = 5.34.Therefore, in this paper, the case of β = 5.3 corresponds to chiral symmetry breaking phase, and β = 5.4 corresponds to chiral symmetry restoring phase, and both cases are close to the pseudo critical temperature.
In Ref. [49], a transition related to the R-W phase is observed.There is a phases transition at high temperature and with large electric field, where the winding number of the function of Polyakov loop along z coordinate at the complex plane changes from none-zero to zero, and this is accompanied by the phenomena that the chiral condensation start to oscillate along the z-axis, and the local charge density becomes non-zero.However, this transition does not occur at β = 5.3 (T = 202.5 MeV), nor does it occur at β = 5.4 (T = 270.5 MeV) and a 2 eE z ≤ π/3.Therefore, for the parameters used in this paper, this transition is irrelevant.

Chiral condensation
The chiral condensation is related to the chiral symmetry.It has been shown in Ref. [49] that, the imaginary electric field will increase the chiral condensation.Meanwhile, the chiral chemical potential will also increase the chiral condensation [35,36,38].The behavior of the chiral condensation with both the electric field and chiral chemical potential is of interest.Besides, how the chiral charge density n5 = n 5 /V is affected by the presences of electric field and chiral chemical potential is also investigated, where n5 = ⟨ Ψ γ 5 γ 4 Ψ ⟩/V with V denotes the volume.
The measured quantities correspond to ⟨ ΨΨ ⟩ and n5 are where four in the denominator is the number of taste In Ref. [49], it has been found that at high temperatures the chiral condensation has an oscillation over the z direc-tion.Meanwhile, at high temperatures, there is also imaginary charge condensation which depends on z coordinate induced by imaginary electric field [47][48][49].However, the temperature and strength of the electric field used in this paper is not high enough to see the oscillation and charge condensation, therefore we do not consider the z dependence of the condensations, and the charge condensation is not considered.The chiral condensation as a function of a 2 eE z and µ 5 for different quarks at different temperatures are shown in Fig. 2. It can be shown that, the imaginary electric field breaks the chiral symmetry.The behavior indicates that, after Wick rotation back to the case of real electric field, at a small a 2 eE z , the real electric field will restore the chiral symmetry.Apart from that, it is also shown that the chiral chemical potential breaks the chiral symmetry.Both the results consistent with the theoretical predictions.The results for real electric field will be presented in next subsection.However, in the case of the β = 5.3, T = 202.5 MeV, a nonmonotonic behavior is found at small chiral chemical potential aµ 5 = 0.06 and electric field strength a 2 eE z = π/24.Apart from that, it can be found that, the effect of µ 5 is much smaller at a higher temperature.
It is interesting to see the change of chiral condensation with different influences.Denoting c q (E z , µ 5 ) as the chiral condensation at a certain strengths of electric field and chiral chemical potential, then we use, where the superscript 'E' and 'CCP' stand for the changes caused by electric field and chiral chemical potential, respectively.∆ c CCP q are orders of magnitudes smaller than ∆ c E q , indicating that the effect of chiral chemical potential is much smaller.The effect of chiral chemical potential is almost the same for u and d quarks, however, with large electric field there is a small difference between u and d quarks.Note with large electric field c u > c d , so the above feature indicates that the more severely the chiral symmetry is broken, the greater the effect of chiral chemical potential.It can be also observed that the dependence of the effect of chiral chemical potential on the electric field is not monotonic.
In both Figs. 2 and 4, it can be found that, at high temperatures the case of a 2 eE z = 0 is different from the case of a 2 eE z ̸ = 0, i.e., the change of chiral condensation implies a discontinuity at non-zero imaginary electric field, no matter whether the chiral chemical potential presents or not.These discontinuities are discussed at length in Ref. [73].Only 100 configurations are discarded seems too little.However, on the one hand we incrementally increase the electric field strength/chiral chemical potential, i.e., when using new electric field strengths and chiral chemical potential, we change from the previous value and start the simulation from the configuration at the previous value, as shown in Fig. 1.On the other hand, the 8 × 8 × 24 × 6 lattice is a relatively small lattice with a relatively large fluctuation, which also makes it easier to reach equilibrium.Taking the case of β = 5.3 and µ 5 = 0 with growing E z , and the case of β = 5.3 and E z = 0 with growing µ 5 as examples, the history of chiral condensation with TU are shown in Fig. 6.In addition, the results for τ int in Tables 1 and 2.

Analytical continuation of chiral condensation
Since the imaginary electric field is used, to compare with the real physics, one needs to Wick rotate the electric back to a real electric field.The result of analytical continuation  depends the ansatz to used.Following the study of the imaginary chemical potential, we use the following three different ansatzes and take their differences as systematic errors [74].We assume that, at a given µ 5 , the chiral condensation is a function of E z , where α, β and ρ are parameters to be fitted.Due to the non-monotonic feature of the chiral condensation, the chiral condensations at a 2 eE z = π/24 and π/12 are ignored.The fit of the results are shown in the Appendix B. The non-monotonic of the chiral condensation appears as an oscillation damped by a large imaginary electric field.After the electric field is Wick rotated back to the real electric field, the chiral condensation are shown in Fig. 7.For larger real a 2 eE z , the systematic error is large, therefore only chiral condensations for a 2 eE z < 0.5 are shown.The results indicate that the real electric field will restore the chiral symmetry which is consistent with the theoretical predictions.Meanwhile, a large chiral chemical potential breaks the chiral symmetry which is also consistent with the theoretical predictions.However, for a small chiral chemical potential, the non-monotonic feature still exists.

Charge density
Another quantity of interest is the charge density n5 .n5 as a function of a 2 eE z and aµ 5 for different quarks at different temperatures are shown in Fig. 5.The results indicates that n5,u > 0, n 5,d < 0 and n5,u + n5,d ≈ 0. It can be seen that n5 is sensitive to µ 5 at low temperatures.This phenomenon is compatible with the previous one, since the more severely the chiral symmetry is broken, the greater the effect of the chiral chemical potential.However, we also note that, the n5 is only sensitive to µ 5 with weak electric fields.Another interesting phenomenon is that, at a 2 eE z = π/24 and β = 5.3, n5 is not a monotonic function of µ 5 .This phenomenon matches the behavior of chiral condensation.

Polyakov loop
It has been found in Ref. [49] that, there is R-W transition when imaginary electric field is applied, and the Polyakov loop oscillates along the z-axis.In this paper, we measure the ⟨P(z)⟩, where P(z) is defined as, where the product over n τ is ordered in the τ direction.
It was pointed out that, the Polyakov loop as a function of z can be fitted using ⟨P(z)⟩ = A p +∑ q=u,d C q exp(L τ Q q iazeE z ), where A p is a complex number, C q are real numbers depending on temperature.In this paper, we find that the ansatz can be further simplified and one can remove another degree of freedom to use where A p is a complex number, B p is a real number, both depending on temperature and the chiral chemical potential.The oscillation of ⟨P⟩ for the case with smaller electric field is more significant, therefore we focus on the cases that a 2 eE z = π/24 and a 2 eE z = π/12.For different chiral chemical potential, the results at β of the complex plane are none-zero.Meanwhile, the dependency of A p and B p on chiral chemical potential is negligible.
The effect of the chiral chemical potential on the Polyakov loop is also studied in Refs.[36,37].In this paper, the case of β = 5.3 corresponds to T = 202.5 and µ 5 ≤ 583.2 MeV, the case of β = 5.4 corresponds to T = 270.5 and µ 5 ≤ 779.0 MeV, respectively.For the parameters in this range, the effect of the chiral chemical potential on the Polyakov loop is small, however, it can be expected that the effect of the chiral chemical potential is much more significant for µ 5 ∼ 1 GeV [36,37].

Summary
For theoretical and experimental reasons, the case of simultaneous chiral chemical potential and external electric field is a scenario worth investigating.In this paper, this case is studied using lattice QCD approach with N f = 1 + 1 dynamic staggered fermions.The electric field is introduced as an imaginary electric field to avoid the sign problem.
It is confirmed that, generally, the chiral chemical potential and the imaginary electric field break the chiral symmetry, which is consistent with the results obtained by various theoretical studies.However, in the case of the β = 5.3, T = 202.5 MeV, a non-monotonic behavior is observed at small chiral chemical potential aµ 5 = 0.06 and electric field strength a 2 eE z = π/24.It is also found that, the chiral chemical potential can suppress the effect of electric field at lower temperatures, and slightly enhance the effect of electric field at high temperatures.The dependence of n 5 on a 2 eE z and µ 5 is also investigated.
The Polyakov loop is also studied.It is found that, the behavior of Polyakov is barely affected by the chiral chemical potential.This should be due to the range of parameters taken in this paper (i.e.temperature and µ 5 ).Besides, we find a simplification of the ansatz for the Polyakov loop used in Ref. [49], when the electric field strength is small, the ansatz ⟨P(z)⟩ = A p +B p ∑ q=u,d Q q exp(L τ Q q iazeE z ) fits well.

Fig. 2 :
Fig. 2: Chiral condensation as a function of a e E z and µ 5 at different temperatures for different quarks.The first row is for the case of β = 5.3, T = 202.5 MeV, where the left panel shows the u quark and the right panel shows the d quark.The second row is for the case of β = 5.4,T = 270.5 MeV, where the left panel shows the u quark and the right panel shows the d quark.

Fig. 5 :
Fig. 5: n5 as functions of E z and µ 5 at different temperatures and for different quarks.The first row is for the case of β = 5.3, T = 202.5 MeV, where the left panel shows the u quark and the right panel shows the d quark.The second row is for the case of β = 5.4,T = 270.5 MeV, where the left panel shows the u quark and the right panel shows the d quark.

Fig. 6 :
Fig. 6: The history of chiral condensation with growing TU in the case of β = 5.3 and µ 5 = 0 with growing E z (the top panel), and the case of β = 5.3 and E z = 0 with growing µ 5 (the bottom panel).The first 200 configurations of the case β = 5.3 and µ 5 = 0 are discarded and therefore not shown.The configurations correspond to thermalization are marked as yellow.

Fig. 7 :
Fig. 7: Chiral condensations as functions of real electric field strength and µ 5 .The value is taken as medians among different ansatzes, and the error bars represent the maximums and minimums of different ansatzes.The first row is for the case of β = 5.3, T = 202.5 MeV, where the left panel shows the case of c u and the right panel shows the case of c d .The second row is for the case of β = 5.4,T = 270.5 MeV, where the left panel shows the case of c u and the right panel shows the case of c d .

Fig. 8 :
Fig. 8: A p and B p in Eq. (12) as functions of the chiral chemical potential at different temperatures and electric fields.The first row is for the case of β = 5.3, T = 202.5 MeV, where the left panel shows the case of a 2 eE z = π/24 and the right panel shows the case of a 2 eE z = π/12.The second row is for the case of β = 5.4,T = 270.5 MeV, where the left panel shows the case of a 2 eE z = π/24 and the right panel shows the case of a 2 eE z = π/12.
Fig. 9: c u as a function of a 2 eE z fitted with three different ansatzes in Eq. (10) at β = 5.3, T = 202.5 MeV with different chiral chemical potentials.