On electric fields in hot QCD: infrared regularization dependence

TL;DR

This paper resolves a fundamental tension in the electric-field response of hot QCD/QED plasmas by dissecting how infrared regularization and the order of limiting procedures affect the electric susceptibility. Using an exact finite-volume, finite-temperature fermion propagator in a background electric field and contrasting with Weldon’s gradient-based perturbative approach, the authors show that the discrepancy between $\\xi_S$ and $\\xi_W$ arises from non-commuting IR and spatial-averaging limits and from the choice of thermodynamic ensemble (canonical vs grand canonical). They map out the limit orderings that yield each susceptibility and demonstrate where each description applies, including a finite-volume analysis that isolates microscopic medium response from macroscopic charge rearrangements. As a concrete low-temperature application, they compute the electric and magnetic susceptibilities in a hadron resonance gas and find good agreement with lattice QCD, strengthening the physical interpretation of these susceptibilities as observable, ensemble-dependent quantities. The work has broad implications for modeling electromagnetic responses in heavy-ion collisions and intense-field laser experiments, and it provides a concrete framework for consistent IR regularization in finite-temperature gauge theories.

Abstract

We study the impact of background electric fields on a hot plasma of charged particles -- a setting relevant for the early stages of heavy-ion collisions as well as laser pulse experiments. Historically, the electric susceptibility -- encoding the behavior of the hot medium for weak fields -- has been defined within two different formalisms, leading to two distinct results at nonzero temperature. With the help of an exact fermion propagator in a homogeneous electric background field at nonzero temperature and finite volume on the one hand, and an improved perturbative result on the other, we identify the origin of this disagreement. The equilibrium conditions for the system are discussed and the role of the thermodynamic ensemble used to describe the system is highlighted. Finally, we construct the electric susceptibility in a simplified hadron resonance gas model, relevant for the strongly interacting medium in the low-temperature regime.

Figures (4)

• Figure 1: The negative of the electric susceptibility $\xi$ and the magnetic susceptibility $\chi$ of a simple hadron resonance gas model, consisting only of the pions compared to continuum extrapolated lattice results Endrodi:2023wwf.
• Figure 2: The IR regulated electric susceptibility in the canonical ensemble -- obtained via the difference of Eqs. (\ref{['eq:2legsuscont']}) and (\ref{['eq:0legsuscont']}), along with the normalization of Eq. (\ref{['eq:normalization']}) -- for nonzero momentum $k$ averaged over a finite volume $L_3$ for $m/T=1$. The point in the bottom left corresponds to the limit of a homogeneous electric field in the infinite volume. The result for the susceptibility depends on the angle used to approach this limiting point.
• Figure 3: A visualized charge distribution $\langle\bar{\psi}(x)\gamma^0\psi(x)\rangle$ on the left and a visualized charge current $\langle\bar{\psi}(x)\gamma^3\psi(x)\rangle$ on the right as a function of the local chemical potentials $\bar{\mu}$ and $\bar{\mu}_3$ for even $N_e$.
• Figure 4: The contour integration required for the analytic continuation is shown in five steps. The red crosses symbolize the poles of the function, for which the contour integration is performed.