QCD matter at a finite magnetic field and nonzero chemical potential

TL;DR

This study develops a hybrid equation of state by smoothly interpolating a hadron resonance gas at low temperature with an ideal parton gas at high temperature to analyze QCD matter under finite magnetic field $eB$ and nonzero chemical potential $\mu$. The HRG and IPG sectors incorporate Landau quantization for charged states, and a fixed-$T_c$ tanh crossover provides thermodynamic continuity, from which we compute $s/T^3$, $P/T^4$, $\varepsilon/T^4$, $\Delta$, $C_V/T^3$, $c_s^2$, and quadratic fluctuations of conserved charges. The results show that $\mu$ enhances these observables in both phases, while $eB$ suppresses them at low $T$ and enhances them at high $T$, with $c_s^2$ displaying a nontrivial response near $T_c$. Comparisons with lattice QCD indicate good agreement for small $eB$ (0 and 0.04 GeV$^2$) but underestimation at stronger fields ($eB=0.14$ GeV$^2$), highlighting missing physics such as anomalous magnetic moments and QGP interactions in strong fields. Overall, the framework captures key qualitative features of QCD thermodynamics under magnetic fields and density, while pointing to avenues for refinement in the strong-field regime.

Abstract

We construct a hybrid equation of state (EoS) by smoothly interpolating the EoS in the hadron resonance gas at low temperatures to that in the ideal parton gas at high temperatures, and employ it to study the properties of the quantum chromodynamics (QCD) matter at a finite magnetic field and nonzero chemical potential. We find that dimensionless observables such as the entropy density $s/T^3$, the pressure $P/T^4$, the energy density $\varepsilon/T^4$, the trace anomaly $Δ= (\varepsilon - 3P)/T^4$, and the specific heat at constant volume $C_V/T^3$ are sensitive to both finite magnetic field and chemical potential. As the chemical potential increases from zero, these quantities rise in both the hadronic and quark-gluon plasma phases. In contrast, introducing a magnetic field suppresses them at low temperatures but enhances them at high temperatures. Furthermore, nonzero chemical potential and magnetic field introduce nontrivial modifications to the squared speed of sound $c_s^2$. Both effects increase $c_s^2$ close to the critical temperature while reducing it at lower temperatures. When the chemical potential and magnetic field are present simultaneously, their influences superimpose, leading to more intricate changes in the thermodynamic behavior. Finally, we compare our results with the lattice QCD data for the quadratic fluctuations of conserved charges and their correlations. The model successfully reproduces the temperature dependence of these observables at $eB=0$ and 0.04 GeV$^2$. However, at the stronger field strength $eB=0.14$ GeV$^2$, the model underestimates the magnitudes while still capturing the overall temperature trend.

Figures (3)

• Figure 1: Upper panels in (a) ((b), (c), (d), (e), and (f)): $s/T^3$ ($P/T^4$, $\varepsilon/T^4$, $\Delta$, $C_V/T^3$, and $c_s^2$) as a function of $T/T_c$. The dashed red and solid black curves correspond to the cases with $eB =10\ m_{\pi}^2$ and $eB = 0$ GeV$^2$, respectively. Lower panels: the ratio between the thermodynamic quantities at $eB =10\ m_{\pi}^2$ and $eB = 0$ GeV$^2$.
• Figure 2: Upper panels in (a) ((b), (c), (d), (e), and (f)): $s/T^3$ ($P/T^4$, $\varepsilon/T^4$, $\Delta$, $C_V/T^3$, and $c_s^2$) as a function of $T/T_c$. The solid black, solid blue, dashed red, and dotted purple curves correspond to the cases with $eB = 0$ GeV$^2$ and $\mu=0$ GeV, $eB = 0$ GeV$^2$ and $\mu\neq 0$ GeV, $eB = m_{\pi}^2$ and $\mu= 0$ GeV, as well as $eB = m_{\pi}^2$ and $\mu\neq 0$ GeV, respectively. Lower panels: the ratios between the thermodynamic quantities denoted by legend.
• Figure 3: Upper panels in (a) ((b), (c), (d), (e), and (f)): $\hat{\chi}_2^B$ ($\hat{\chi}_2^Q$, $\hat{\chi}_2^S$ , $\hat{\chi}_{11}^{BQ}$, -$\hat{\chi}_{11}^{BS}$, and $\hat{\chi}_{11}^{QS}$) as a function of temperature. The solid black, red, and blue curves correspond to the cases with $eB = 0$, 0.04, and 0.14 GeV$^2$, respectively. The LQCD results at these magnetic fields LQCD_5LQCD_6 are represented by the empty triangles, circles, and squares, respectively. Lower panels: the ratios between the fluctuations or correlations calculated with our model at $eB = 0.14$ and 0 GeV$^2$.