Inverse magnetic catalysis in the linear sigma model: a beyond mean field approach

TL;DR

This work investigates chiral symmetry restoration in a linear sigma model with quarks under strong magnetic fields by going beyond mean-field through ring-diagram resummation and self-consistent boson masses. In a two-step approach, tree-level masses first yield magnetic catalysis (MC) and a critical endpoint (CEP), while a self-consistent curvature-based mass determination in the lowest Landau level produces inverse magnetic catalysis (IMC) and shifts the CEP, aligning qualitatively with lattice QCD results. The method removes the need for externally prescribed magnetic couplings, capturing collective effects via dynamical masses and ring resummation. The findings demonstrate IMC and a CEP in the T-|eB| phase diagram, highlighting the importance of self-consistent mass generation for accurate thermomagnetic predictions in QCD-inspired effective theories.

Abstract

We explore the restoration of chiral symmetry in the linear sigma model coupled to quarks under the influence of strong magnetic fields and finite temperature, incorporating screening effects through ring diagrams. While previous studies using tree-level thermal masses lead to magnetic catalysis across all temperature ranges, in tension with lattice QCD results, we go beyond this limitation by computing the bosonic masses self-consistently within the lowest Landau level (LLL) approximation. The self-consistent approach modifies the effective potential and allows us to accurately track the thermal evolution of the order parameter. Our results reveal the emergence of a critical end point (CEP) in the $T-|eB|$ phase diagram and, notably, exhibit inverse magnetic catalysis (IMC) behavior: the (pseudo)critical temperature decreases with increasing magnetic field strength. This is in contrast to the magnetic catalysis behavior found when non-self-consistent masses are used. To the best of our knowledge, this is the first time that self-consistent boson masses have been implemented in this context, offering a new framework for exploring the QCD phase diagram using effective models.

Figures (8)

• Figure 1: One-loop self-energy diagrams for the bosonic degrees of freedom. Dashed lines represent the sigma meson; single and continuous lines correspond to neutral pions; double and continuous lines denote charged pions; and continuous lines with arrows represent quark fields. The coefficients multiplying each diagram are the corresponding combinatorial factors.
• Figure 2: Purely thermal part of the bosonic one-loop contribution to the effective potential as a function of the vacuum expectation value $v$. The dashed blue line corresponds to the numerical solution, while the solid red line represents the high-temperature approximation. Panel $(a)$ shows the result for a low temperature, $T=0.1$ GeV; panel $(b)$ illustrates the validity region for an intermediate temperature, $T=0.6$ GeV; and panel $(c)$ displays the result at high temperature, $T=1.1$ GeV. The parameters used are $\lambda=13.32$, $g=2.58$ and $a=0.309$ GeV, obtained from the relations in Eq. (\ref{['fixedparameters']}).
• Figure 3: Purely thermomagnetic part of the bosonic one-loop contribution to the effective potential as a function of the vacuum expectation value $v$. The dashed blue line corresponds to the numerical solution, while the solid red line represents the high-temperature approximation. Panel $(a)$ shows the result for a low temperature, $T=0.1$ GeV; panel $(b)$ illustrates the validity region for an intermediate temperature, $T=0.6$ GeV; and panel $(c)$ displays the result at high temperature, $T=1.1$ GeV. The parameters used are $\lambda=13.32$, $g=2.58$ and $a=0.309$ GeV, obtained from the relations in Eq. (\ref{['fixedparameters']}).
• Figure 4: Purely thermomagnetic part of the fermionic one-loop contribution to the effective potential as a function of the vacuum expectation value $v$. The dashed blue line corresponds to the numerical solution, while the solid red line represents the high-temperature approximation. Panel $(a)$ shows the result for a low temperature, $T=0.1$ GeV; panel $(b)$ illustrates the validity region for an intermediate temperature, $T=0.6$ GeV; and panel $(c)$ displays the result at high temperature, $T=1.1$ GeV. The parameters used are $\lambda=13.32$, $g=2.58$ and $a=0.309$ GeV, obtained from the relations in Eq. (\ref{['fixedparameters']}).
• Figure 5: Minimum of the effective potential Eq. (\ref{['fulleffectivepotentialN']}) as a function of the temperature, for five different values of the magnetic field $|eB|= 1- 9 \ \text{GeV}^2$. The parameters used are $\lambda=13.32$, $g=2.58$ and $a=0.309$ GeV, obtained from the relations in Eq. (\ref{['fixedparameters']}).