Interplay between the chiral and deconfinement transitions from a Curci-Ferrari-based Polyakov loop potential

TL;DR

The paper presents a PNJL-like framework that couples a two-flavor NJL model to a center-symmetric Curci-Ferrari glue potential, yielding a minimal, one-parameter description of the glue sector. By extremizing the combined potential, it maps the interplay between chiral restoration and deconfinement across the $T$-$\mu$ plane, locating a CEP and detailing the role of the net quark number response as a diagnostic beyond Polyakov loops. The zero-temperature analysis highlights the Silver-Blaze property for the constituent mass up to $|\mu|<M_0$ and reveals how Polyakov loops acquire or lose sensitivity beyond $|\mu|=M_0$, depending on the glue model. Thermodynamic observables show qualitative agreement with lattice data and illuminate the impact of the CF gluon mass on approaching the Stefan–Boltzmann limit, with the net quark number observables offering insight into phase structure and CEP connections. Overall, the work demonstrates that a minimally parameterized glue sector, combined with chiral dynamics, provides a coherent and predictive picture of QCD thermodynamics at finite temperature and density, while suggesting concrete avenues for refinement and extension.

Abstract

We couple the two-flavor Nambu--Jona-Lasinio model to a gluonic background corresponding to the gauge-field expectation value in the center-symmetric Landau gauge. Low-energy features in this gauge are captured by a center-symmetric extension of the Curci-Ferrari model and provide a good grasp on key aspects of the confinement/deconfinement transition. Within this framework, we can investigate the interplay between the chiral and deconfinement transitions. Compared to other approaches based on multi-parameter Ansätze of the Polyakov loop potential fixed from comparison to finite-temperature lattice data, the modeling of the glue sector in the present set-up depends on only one phenomenological parameter that can be fixed by comparison to lattice data in the vacuum. We detail the structure of the phase diagram, with special emphasis on the finite density axis, and compute thermodynamical observables relevant for applications. We also highlight the properties of the recently introduced net quark number response of the medium as a sensible probe of the phases of QCD, in particular as a tool to disambiguate the nature of certain regions of the phase diagram where the use of the Polyakov loops could lead to misinterpretations. Finally, we critically assess the sensitivity of our results to the various parameters, both in the glue sector and in the chiral sector.

Figures (17)

• Figure 1: Order parameters as functions of the temperature $T$ at various chemical potentials $\mu$. Here $T_0\simeq186\,$MeV stands for $(T_{\rm pc}^\sigma+T_{\rm pc}^\ell)/2$ at ${\mu=0}$, see the main text for the definitions of $T_{\rm pc}^\sigma$ and $T_{\rm pc}^\ell$, and $\mu_{\rm CEP}\simeq324\,$MeV.
• Figure 2: Comparison of the temperature dependence of the renormalized chiral condensate $\langle\bar{\psi}\psi\rangle_R$ (dark) and LO and NLO Polyakov loop $\ell_R$ (light) found in this work (lines, ${N_f=2}$) to continuum extrapolated lattice data (bands), and data from a single lattice size with temporal extent $N_t=8$ (triangles), both from Ref. Borsanyi:2010ordPar (${N_f=2+1}$). The temperature has been rescaled by the pseudocritical temperature for each observable, see the discussion in the main text. The thin lines around the lattice data bands (horizontal error) show the 2% systematic temperature uncertainty due to the scale setting.
• Figure 3: Temperature dependence of the order parameters at the chemical potential ${\mu=M_0\simeq336\,}$MeV corresponding to the vacuum constituent quark mass, which crosses through the spinodal region. Thin lines represent metastable and unstable extrema. The filled points mark the first-order transition, and the open points mark additional inflection points, see the text and App. \ref{['app:inflections']} for more details.
• Figure 4: Phase diagram of the csCF-PNJL model in the $(\mu,T)$-plane for the parameters given in Sec. \ref{['sec: framework']}. The dashed lines mark the crossovers of the various order parameters $\sigma$, $\ell$ and $\bar{\ell}$ (lighter, dotted), the black dot labels the critical endpoint, and the solid lines mark the spinodal region and first-order transition, equal for the three order parameters, see App. \ref{['app:CEP']}.
• Figure 5: The zero-temperature potential $V(M;\mu)-V(M_0;0)$ normalized by $M_0^4$ for increasing values of $|\mu|$. The plain curves of decreasing darkness correspond to the potential at the special values of ${|\mu|\in\{|M_+|,|M_-|,M_0,\mu_{s1},\mu_c,\mu_{s2},\Lambda\}}$, see the main text for the definitions, and the dotted, light gray line between $(-M_+,M_+)$ shows the potential at ${\mu=0}$ in the range where it deviates (slightly) from the one at $M_+$.