Functional renormalization group study of rho condensate at a finite isospin chemical potential in the quark meson model

TL;DR

This paper addresses rho-vector-meson condensation under finite isospin chemical potential within a two-flavor quark-meson model. It employs the non-perturbative functional renormalization group to include fluctuations beyond mean field, revealing that the onset of rho condensation occurs near $μ_I \,\approx\, m_\pi$ due to fluctuations and can dominate the order parameters at large $μ$ and strong vector coupling. The chiral phase boundary shifts to lower temperatures and chemical potentials with increasing $μ_I$, and the vector coupling $g_\rho/m_\rho$ modulates the boundary and the magnitude of the rho condensate. Overall, the work highlights the significant role of isovector interactions and fluctuations in shaping the QCD-like phase diagram and has implications for dense hadronic matter in heavy-ion and astrophysical contexts.

Abstract

We investigate the effect of an isospin chemical potential ($μ_{I}$) within the quark-meson model, which approximates quantum chromodynamics (QCD) by modeling low energy phenomena such as chiral symmetry breaking and phase structure under varying conditions of temperature and chemical potential. Using the functional renormalization group (FRG) flow equations, we calculate the phase diagram in the chiral limit within the two-flavor quark-meson model in a finite $μ_{I}$ with $ρ$ vector meson interactions. Fluctuation effects significantly decrease the critical chemical potential from the mean-field (MF) value $μ_{I, MF} > m_ρ$ to lower value, at which point the $ρ$ vector meson condensates alongside the chiral condensate once the isospin chemical potential exceeds the critical value $μ_{I}^{\text{crit}}$. This $ρ$ condensation is investigated numerically for different meson coupling strengths. The $ρ$ meson dominated region is delineated from other phases by a second-order phase transition at lower $μ_{I}$ and a first-order transition at slightly higher $μ_{I}$.

Figures (9)

• Figure 1: The FRG ${T-\mu}$ chiral phase diagram with different vector couplings. The solid lines show the first-order phase transition, and the dashed lines show the second-order phase transition. The stars show the TCPs. The parameters are set as: $f_{\pi}$ = 93 MeV, $g_{s}$ = 3.2, $\lambda$ = 8, the ultraviolet cutoff $\Lambda_{FRG}$ =500 MeV, ${\mu_I}$ = 200 MeV.
• Figure 2: The FRG ${T-\mu}$ chiral phase diagram with different $\mu_{I}$. The solid lines represent the first-order phase transition, and the dashed lines represent the second-order phase transition. The stars show the TCPs. The parameters are set as: $f_{\pi}$ = 93 MeV, $g_{s}$ = 3.2, $\lambda$ = 8, the ultraviolet cutoff $\Lambda_{FRG}$ =500 MeV, the coupling constant ${g_{\rho}}$$m^{-1}_{\rho} = 0.006 \ \mathrm{MeV}^{-1}$.
• Figure 3: Chiral condensates as a function of quark chemical potential under different isospin chemical potentials, calculated at $T=10$ MeV, ${g_{\rho}}/m_{\rho} = 0.006~\mathrm{MeV}^{-1}$. The different colored lines correspond to different isospin chemical potentials.
• Figure 4: The chiral and $\rho$ condensates as a function of isospin chemical potential, shown for (a) $\frac{{g_{\rho}}}{m_{\rho}} = 0.002$ MeV$^{-1}$, (b) $\frac{{g_{\rho}}}{m_{\rho}} = 0.006$ MeV$^{-1}$, (c) $\frac{{g_{\rho}}}{m_{\rho}} = 0.008$ MeV$^{-1}$ and (d) $\frac{{g_{\rho}}}{m_{\rho}} = 0.010$ MeV$^{-1}$. The chemical potential is $\mu = 100$ MeV, temperature is $T = 10$ MeV and the ultraviolet cutoff $\Lambda_{FRG}$ =500 MeV
• Figure 5: The chiral and $\rho$ condensates $\rho$ as a function of isospin chemical potential, shown for (a) $\frac{{g_{\rho}}}{m_{\rho}} = 0.002$ MeV$^{-1}$, (b) $\frac{{g_{\rho}}}{m_{\rho}} = 0.006$ MeV$^{-1}$, (c) $\frac{{g_{\rho}}}{m_{\rho}} = 0.008$ MeV$^{-1}$ and (d) $\frac{{g_{\rho}}}{m_{\rho}} = 0.010$ MeV$^{-1}$. The chemical potential is $\mu = 200$ MeV, temperature is $T = 10$ MeV and the ultraviolet cutoff $\Lambda_{FRG}$ =500 MeV