Susceptibilities and Taylor coefficients of magnetic QCD from perturbation theory

TL;DR

The paper addresses the QCD equation of state in strong magnetic fields by computing the Taylor coefficients $c_2(T,B)$ and $c_4(T,B)$ of the pressure expansion in $\hat{\mu}_B=\mu_B/T$ at two-loop order within perturbative QCD, restricted to the lowest-Landau level regime $m_s \ll T \ll \sqrt{eB}$. It incorporates running couplings and masses with $\bar{\Lambda}=\sqrt{(2\pi T)^2+eB}$ and derives the total two-loop pressure from quark and gluon contributions $P_{free}^G$, $P_{2}^G$, $P_{free}^{LLL}$, and $P_{exch}^{LLL}$. The results include excess pressure $\Delta P$, baryon density $n_B$, and baryon-number susceptibility $\chi_B$ as functions of $\hat{\mu}_B$, as well as susceptibilities in the $(\mu_B,\mu_Q,\mu_S)$ basis, with direct comparison to lattice QCD showing compatibility at high $T$ and large $eB$. The study finds that perturbative results exhibit narrow scale bands and that $c_4$ is strongly suppressed at large magnetic fields, providing a perturbative baseline for magnetic QCD thermodynamics and guiding future lattice and model-based explorations.

Abstract

We compute the coefficients $c_2(T,B)$ and $c_4(T,B)$ of the Taylor expansion for the pressure in powers of $μ_B/T$ in the presence of a large magnetic field within perturbative QCD at finite temperature and baryon density up to two-loops for $N_f=3$ flavors with physical quark masses. We also present results for the excess of pressure, baryon density and baryon number susceptibility as functions of $μ_B$, as well as susceptibilities as functions of the temperature in the $\{ μ_B,μ_Q,μ_S \}$ basis. Our results can be directly compared to recent lattice QCD data. Even though current lattice results do not overlap with its region of validity, perturbative results seem to be compatible with those obtained on the lattice for large temperatures.

Figures (7)

• Figure 1: Coefficient $c_2$ as a function of the temperature for $eB=1.2$ GeV$^2$. Lattice data (lower, red band) from Ref. Astrakhantsev:2024mat and bands on the right from pQCD up to one and two loops. Bands in the perturbative part represent the renormalization-scale dependence in the standard range between half the central scale $\bar{\Lambda}$ and twice its value.
• Figure 2: Coefficient $c_4$ as a function of the temperature for $eB=1.2$ GeV$^2$. Lattice data (left band) from Ref. Astrakhantsev:2024mat and bands on the right from pQCD up to one and two loops. Bands in the perturbative part represent the renormalization-scale dependence in the standard range between half the central scale $\bar{\Lambda}$ and twice its value.
• Figure 3: Coefficient $c_2$ as a function of the magnetic field for $T=500$ MeV. Bands from pQCD up to one and two loops, which represent the renormalization-scale dependence in the standard range between half the central scale $\bar{\Lambda}$ and twice its value.
• Figure 4: The excess of pressure, $\Delta P$, as function of $\hat{\mu}_B$ for two values of temperature, $T= 152, 160$ MeV. Here, $eB=1.2$$\rm{GeV}^2$. Lattice data from Ref. Astrakhantsev:2024mat. Bands in the perturbative results represent the renormalization-scale dependence in the standard range between half the central scale $\bar{\Lambda}$ and twice its value.
• Figure 5: Baryon density as a function of $\hat{\mu}_B$ for two values of temperature, $T= 152, 160$ MeV. Here, $eB=1.2$$\rm{GeV}^2$. Lattice data from Ref. Astrakhantsev:2024mat. Bands in the perturbative results represent the renormalization-scale dependence in the standard range between half the central scale $\bar{\Lambda}$ and twice its value.