Columbia plot based on symmetry-improved CJT formalism in linear sigma model

TL;DR

This work analyzes the chiral phase structure of QCD in a three-flavor linear sigma model using symmetry-improved CJT (SI-CJT) to address NG theorem violations in standard CJT. The SI-CJT framework enforces 1PI Ward-Takahashi identities, preserving low-energy theorems and threshold behavior while solving self-consistent gap equations for meson propagators. The study finds a genuine first-order region and a tricritical point on the $m_s$ axis with $m_s^{tri}/m_s^{phys} = 0.175$, and determines $m_pi^c = 52.4$ MeV and $T_c \approx 51.7$ MeV in the symmetric three-flavor limit, while eliminating spurious first-order regions that appear in conventional CJT. Overall, the SI-CJT approach yields a more robust qualitative picture of the Columbia plot and a pathway to reliable exploration of QCD-like critical phenomena beyond mean-field approximations.

Abstract

We study the Columbia plot for the chiral phase transition in the framework of a three-flavor linear sigma model based on the Cornwall-Jackiw-Tomboulis (CJT) formalism. The conventional CJT approach with the Hartree truncation suffers from artificial chiral breaking, leading to the violation of the Nambu-Goldstone theorem and the (anomalous) chiral Ward-Takahashi identities. We apply the symmetry-improved CJT formalism to resolve this issue. We observe a first-order phase transition and a tricritical point in the light-quark mass regime, which is fairly insensitive to the size of the sigma meson, in contrast to the conventional CJT approach. The tricritical point, found on the $m_s$ axis, is at $m_s^{\rm tri}/m_s^{\rm phys.} = 0.175$ with $m_s^{\rm phys.}$ being the physical strange quark mass in real-life QCD. The critical pion mass in the three-flavor symmetric limit, on the second-order boundary, is measured at $m_π\sim 52.4$ MeV, with the critical temperature $T_c \sim 51.7$ MeV.

Figures (8)

• Figure 1: Feynman diagrams in the Hartree approximation contributing to $V_2$ in Eq. \ref{['eq:V2UnderHartreeMainText']}. The solid lines represent the scalar meson propagators $S_{ab}$, and the doubled solid lines the pseudoscalar meson propagators $P_{ab}$. The black blob stands for the bare vertices, which are denoted in the text as $\mathcal{F}^{abcd}$ and $\mathcal{H}^{abcd}$, respectively.
• Figure 2: A schematic sketch of the SICJT formalism. The figure is drawn in the $(\phi,\Delta)$-plane where $\phi$ is the sourced field VEV and $\Delta$ denotes the sourced propagator, describing the variables of the effective actions. The black blobs (from right to left) represent the solution obtained from the full-quantum action $\Gamma$, the truncated 1PI effective action $\Gamma_{\rm tr}^{\rm 1PI}$, and the truncated 2PI effective action $\Gamma_{\rm tr}^{\rm 2PI}$. The black-dashed and -dotted curve respectively denote the constraint trajectories from the 1PI or the 2PI formalism, and the blue-solid line denotes the gap equation of the propagators $\Delta$ with arbitrary $\phi$. The red square mark points to the solution realized by the SICJT formalism.
• Figure 3: Plots of $T$-dependences on $\bar{\Phi}_{1,3}$ (left panel) and masses of scalar (middle) and pseudoscalar mesons (right), at the physical point, in the cases of the conventional CJT (upper panels) and SICJT (lower panels) formalisms applied to the present LSM with the parameter set I in Sec. \ref{['Para-I']}.
• Figure 4: The same as Fig. \ref{['fig:PhysicalPointCJTvsSICJT']}, for the chiral limit case, with the plots of the scalar and pseudoscalar meson masses combined into a single one, which has been summarized in the right panel. The red-dashed line in the bottom-right panel denotes the degenerate masses of all mesons in the chirally symmetric phase below the critical temperature.
• Figure 5: The conventional CJT result projected onto the Columbia plots with the parameter sets I in Sec. \ref{['Para-I']} (top-left panels) and II in Sec. \ref{['para-II']} (top-right panel), in comparison with the SICJT results (bottom-left and -right panels) on the same parameter setup. The orange-solid curves denote the second-order phase boundaries, the blue-shaded areas stand for the first-order domains, and blank areas represent the crossover regions. The three-flavor symmetric limit (the case with $m_l=m_s$) has been drawn by the dotted curves, and the red-diamond marker corresponds to the physical point where $(m_c, m_s) = (m_l^{\rm phys.},m_s^{\rm phys.})$. The $m_l$ and $m_s$ axes have been normalized by $m_l^{\rm phys.}$.