Magnetic susceptibility of hot hadronic medium and quark degrees of freedom near the QCD cross-over point

TL;DR

This paper addresses the mismatch between lattice QCD results for the magnetic susceptibility $\\chi_B$ near the QCD crossover and Hadron Resonance Gas (HRG) predictions. It shows that HRG fails to capture the observed near-$T_c$ paramagnetism unless a light paramagnetic sector is present, and it then develops a non-interacting quark–meson model with temperature-dependent quark masses fitted to lattice susceptibilities $\\chi_{BB}$ and $\\chi_{BS}$, including vacuum contributions and quark anomalous magnetic moments. The results demonstrate that the vacuum quark contribution, the thermal quark sector, and the meson sector together can reproduce the lattice behavior of $\\chi_B(T)-\\chi_B(0)$ around $T_c$, while hadronic polarizabilities are negligible and pion–vector-meson loops provide only modest corrections. The findings underscore the necessity of light, paramagnetic degrees of freedom and vacuum effects below $T_c$ to correctly describe the magnetic response of hot QCD matter, with implications for modeling the hadronic phase in strong magnetic fields.

Abstract

We argue that the lattice QCD results for the temperature-dependent magnetic susceptibility of the medium below the cross-over temperature are difficult to reconcile with the available hadronic models. In particular, in the widely used Hadron Resonance Gas model, reproducing well numerous other features of the medium, one observes a substantially too strong diamagnetism compared to the lattice simulations. One thus needs a significant source of paramagnetism below the QCD cross-over temperature, possible to achieve with sufficiently light fermions. We thus consider a quark-meson model, where the temperature-dependent quark masses are fixed from the baryon-baryon and baryon-strangeness susceptibility data from the lattice at zero magnetic field. We show that in such a framework one can describe the magnetic susceptibility, with the vacuum contributions duly incorporated. In the hadronic picture, we also evaluate the contribution of the pion--vector-meson loops to the magnetic susceptibility evaluated via the photon polarization, showing it is small and paramagnetic.

Figures (12)

• Figure 1: Susceptibilities $\chi_{\cal{BB}}$ and $\chi_{\cal{BS}}$ as functions of $T$, compared to the lattice data Bollweg:2021vqf. The red solid (dashed) line represents the HRG result below (above) $T_c$.
• Figure 2: Magnetic susceptibility $\chi_B$ as a function of temperature. The red solid (dashed) line represents the HRG result below (above) $T_c$. The orange and green bands represent the lattice data obtained using two recent methods: the photon polarization method (pp) Bali:2020bcn and the full field method (ff) Brandt:2024blb, respectively.
• Figure 3: Magnetic susceptibility as a function of temperature, calculated in the HRG model including (red) and excluding (blue) the anomalous magnetic moments of hadrons. The lines above $T_c$ are dashed. The orange and green bands show the lattice data as in Fig. \ref{['fig:chiB']}.
• Figure 4: Magnetic susceptibility of several hadron species as functions $T$. The numbers in parenthesis next to labels denote factors by which a given result is multiplied. The hadron labels include the antiparticles and the spin-isospin degeneracy.
• Figure 5: Subtracted magnetization, $\mathcal{M}(B,T)-\mathcal{M}(B,0)$, at $B=0.2~{\rm GeV}^2$, plotted as a function of $T$. The lattice result, obtained from the data from Bali:2014kia, is represented with the orange band. The blue and red curves represent, respectively, the HRG results without and with the anomalous magnetic moments of hadrons.