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

TL;DR

This study investigates how an external imaginary electric field and a chiral chemical potential $\mu_5$ influence quark matter using lattice QCD with $N_f=1+1$ staggered fermions. The electric field is implemented as an imaginary Euclidean field to avoid the sign problem, with $E$ oriented along $z$ and twisted boundary conditions, while $\mu_5$ is introduced linearly in the fermion action; observables include the chiral condensate, the chiral charge density $\bar{n}_5$, and the Polyakov loop in the complex plane. The main findings are that both $E$ and $\mu_5$ generally enhance chiral symmetry breaking, though a non-monotonic region appears at small $a^2 eE_z$ and low temperature, and the Polyakov loop shows limited sensitivity to $\mu_5$ within the studied range. Analytically continuing to real $E$ suggests that larger real electric fields restore chiral symmetry, while substantial $\mu_5$ tends to break it; the Polyakov loop's $z$-dependent oscillations are well described by a simplified fit and remain largely unaffected by $\mu_5$ in the explored parameter space.

Abstract

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.

Figures (20)

• Figure 1: The parameters in the simulation. In the case of $a^2eE_z=a\mu_5$, $3200$ trajectories are simulated and the last $2900$ configurations are measured, for the other cases, $3000$ trajectories and the last $2900$ configurations are measured. At first, $200+3000\times 9$ trajectories are simulated with sequentially growing $a^2eE_z=k\pi/24$ for $k=0,1,2,\ldots, 8$, which corresponds to the horizontal line. Then, starting with the configuration at $a^2eE_z=k\pi/24$, $3000\times 8$ trajectories are simulated with sequentially growing $a\mu_5=0.06n$ for $n=1,2,3,\ldots, 8$, which correspond to the vertical lines.
• Figure 2: Chiral condensation as a function of $a^eE_z$ and $\mu _5$ at different temperatures for different quarks. The first row is for the case of $\beta = 5.3, T=202.5\;{\rm MeV}$, where the left panel shows the $u$ quark and the right panel shows the $d$ quark. The second row is for the case of $\beta = 5.4, T=270.5\;{\rm MeV}$, where the left panel shows the $u$ quark and the right panel shows the $d$ quark.
• Figure 3: $\Delta c_q^E$ as functions of $\mu _5$ for the case of $\beta=5.3, T=202.5\;{\rm MeV}$ (the left panel) and $\beta=5.4, T=270.5\;{\rm MeV}$ (the right panel).
• Figure 4: $\Delta c_q^{\rm CCP}$ as functions of $\mu _5$ for the case of $\beta=5.3, T=202.5\;{\rm MeV}$ (the left panel) and $\beta=5.4, T=270.5\;{\rm MeV}$ (the right panel).
• Figure 5: $\bar{n}_5$ as functions of $E_z$ and $\mu _5$ at different temperatures and for different quarks. The first row is for the case of $\beta = 5.3, T=202.5\;{\rm MeV}$, where the left panel shows the $u$ quark and the right panel shows the $d$ quark. The second row is for the case of $\beta = 5.4, T=270.5\;{\rm MeV}$, where the left panel shows the $u$ quark and the right panel shows the $d$ quark.