On-shell versus curvature mass parameter fixing schemes in the three flavor quark-meson model with vacuum fluctuations

The tree-level parameters of the 2 + 1 flavor quark-meson model become inconsistent when the quark one-loop vacuum fluctuation is included in its effective potential. Relating the counterterms in the (MS) over bar scheme to those in the on-shell scheme, the relations between the renormalized parameters of both the schemes have been determined, and the consistent effective potential for the renormalized quark-meson (RQM) model has been calculated using the modified minimal subtraction method where the relations between the physical quantities (i.e. pole masses of the pseudoscalar pi, K, eta, and eta' and the scalar sigma mesons, the pion and kaon decay constants f(pi) and f(K)) and the running couplings have been used as input. The effective potentials, order parameters, and phase diagrams have been computed for the RQM model and compared respectively with the corresponding calculations in the quark-meson model without the vacuum term and the quark-meson model with the vacuum term where the curvature meson masses have been used for fixing the model parameters. The f(pi), f(K), the Yukawa coupling g, and the minimum of the effective potential do not change but the explicit symmetry breaking strength h(x) gets a small reduction equal to the two flavor RQM model correction while the strength h(y) becomes weaker by a relatively large amount in the 2 + 1 flavor RQM model because the pion curvature mass m(pi,c) = 135.95 MeV is 2.05 MeV smaller than its pole mass m(pi) = 138.0 MeV and the kaon curvature mass m(K,c) = 467.99 MeV is 28.01 MeV smaller than its pole mass m(K) = 496.0 MeV after the renormalization. Therefore the 2 + 1 flavor RQM model phase diagrams and the critical end point location differ noticeably from the existing two flavor RQM model results [S. K. Rai and V. K. Tiwari, Phys. Rev. D 105, 094010 (2022)]. The enhanced coefficient of the 't Hooft determinant term in the RQM model modifies the T(mu) dependence of the U-A(1) axial symmetry restoration pattern.