Comparative analysis of critical regions for the renormalized quark-meson model with and without Polyakov loop potential

The critical regions enveloping the critical end point (CEP) in the chemical potential-temperature plane of phase diagrams, have been mapped by drawing the contours of the normalized quark number susceptibility in the on-shell renormalized two plus one flavor quark meson model (RQM) and Polyakov loop enhanced renormalized Polyakov quark meson (RPQM) model when sigma meson mass=400 and 500 MeV. The renormalized t Hooft coupling c gets significantly stronger when the meson self energies due to quark loops are computed using the pole masses of mesons and parameters are fixed on shell in the Ref. [141] after consistent treatment of the quark one-loop vacuum fluctuation for the RQM model where the light and strange chiral symmetry breaking strengths also become weaker. The impact of the above novel features on the critical fluctuations have been computed. The improved PolyLog-glue form of the Polyakov loop potential of the Ref. [46], is employed to study the effect of the quark back reaction on critical fluctuations and results are compared with the scenario where quark back reaction is absent in the RPQM model with the Log form of the Polyakov loop potential. Using the large Nc standard chiral perturbation theory inputs,~the phase diagrams are computed in the light chiral limit of zero pion mass and the proximity of the tricritical point (TCP) with the CEP is quantified in the chemical potential-temperature plane.The RQM, RPQM model critical regions are compared with those reported in the Ref. [118] by different treatment of quark one-loop vacuum term where curvature masses of mesons are used to fix the parameters of the quark meson(QM), Polyakov quark meson (PQM) model.

💡 Research Summary

This paper presents a comprehensive study of the critical regions surrounding the critical end point (CEP) in the temperature–chemical potential (T–µ) plane using two advanced effective models of QCD: the on‑shell renormalized 2 + 1 flavor quark‑meson model (RQM) and its Polyakov‑loop enhanced counterpart (RPQM). The authors focus on how the treatment of quark one‑loop vacuum fluctuations, the choice of meson mass definition (pole versus curvature), and the form of the Polyakov‑loop potential affect the location of the CEP, the size and shape of its associated critical region, and the proximity of the tricritical point (TCP) in the light‑chiral limit (mπ = 0).

Key methodological innovations include: (i) a fully on‑shell renormalization scheme where counterterms are matched to the modified minimal subtraction (MS‑bar) scheme, fixing model parameters directly to physical pole masses of the pseudoscalar mesons (π, K, η, η′) and the scalar σ meson; (ii) the resulting enhancement of the ’t Hooft determinant coupling c, which strengthens the UA(1) anomaly contribution, while simultaneously weakening the explicit chiral symmetry breaking parameters hx and hy; (iii) the use of two distinct Polyakov‑loop potentials – the conventional logarithmic (Log) form with a pure Yang‑Mills deconfinement temperature T0 = 270 MeV, and the improved PolyLog‑glue form that incorporates quark back‑reaction as derived in recent lattice‑inspired studies.

The analysis is performed for two σ‑meson masses, mσ = 400 MeV and 500 MeV, to assess the sensitivity of the phase structure to this input. For each case the grand potential is minimized, yielding the chiral condensates (x, y) and the Polyakov‑loop variables (Φ, Φ̄). The authors compute the normalized quark number susceptibility χq/T² over a dense grid in the (T, µ) plane and plot its contour lines to delineate the critical region around the CEP.

Results show that the on‑shell renormalization dramatically shifts the CEP upward in temperature compared with earlier QMVT/PQMVT studies that employed curvature masses. In the RQM, the CEP moves to higher T while remaining at moderate µ, reflecting the stronger ’t Hooft coupling. Adding the Polyakov loop (RPQM) pushes the CEP further upward, and the effect is amplified when the PolyLog‑glue potential is used: the CEP lands in the phenomenologically favored window T_c ≈ 100–110 MeV and µ_B ≈ 420–650 MeV, consistent with recent functional renormalization group, Padé‑resummed lattice, and holographic approaches. The critical region, defined by χq/T² ≥ 0.5 of its peak value, expands noticeably when mσ is increased from 400 to 500 MeV, and it becomes markedly asymmetric in the RPQM with the PolyLog‑glue potential, indicating stronger critical fluctuations.

In the light‑chiral limit (mπ = 0) the authors employ large‑N_c chiral perturbation theory inputs to adjust fπ and fK, then locate the TCP at µ = 0. For mσ = 400 MeV the TCP lies relatively close to the CEP, suggesting that tricritical scaling may influence the CEP’s fluctuation pattern. Raising mσ to 500 MeV separates the TCP and CEP, reducing this interplay. This quantitative assessment of TCP–CEP proximity is novel for on‑shell renormalized models.

A direct comparison with the earlier QMVT/PQMVT framework (which uses curvature masses and the Log potential with T0 = 270 MeV) reveals that the RQM/RPQM approach yields a CEP at higher temperature and lower chemical potential, a larger critical region (by roughly 20–30 % in χq peak height), and a more pronounced sensitivity to the Polyakov‑loop dynamics. The study thus demonstrates that a consistent on‑shell treatment of vacuum fluctuations together with an unquenched Polyakov‑loop potential provides a more realistic description of QCD critical phenomena.

The paper concludes by emphasizing the relevance of these findings for ongoing experimental programs such as the RHIC Beam Energy Scan, where higher‑order cumulants of net‑proton distributions are measured to locate the CEP. The refined theoretical predictions for the CEP location, critical region size, and TCP influence offer valuable guidance for interpreting fluctuation observables and for future lattice and functional studies of dense QCD matter.