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

The vacuum effective potential and phase diagram for the three (2+1) flavor quark-meson model have been computed and compared in an extended mean-field approximation (e-MFA) where the model parameters are fixed by using different renormalization prescriptions after including quark one-loop vacuum fluctuations. When the vacuum one-loop divergence is regularized in the minimal subtraction scheme and the curvature masses of the scalar and pseudo-scalar mesons are used for fixing the parameters, the setting of the quark-meson model with the vacuum term (QMVT) turns out to be inconsistent as one notes that the curvature masses are defined by the evaluation of self-energy at zero momentum. This work constitutes the first application of the consistent on-shell parameter fixing scheme to the three flavor quark-meson (QM) model. In this setting of the renormalized quark-meson (RQM) model, the physical (pole) masses of the $\pi, K, \eta$ and $\eta^{\prime}$ pseudo-scalar mesons and the scalar $\sigma$ meson,the pion decay constant and kaon decay constant are put into the relation of the running mass parameter and couplings by using the on-shell and the minimal subtraction renormalization schemes. The nonstrange direction normalized vacuum effective potential plots for both the RQM model and QMVT model, are exactly identical for the $m_\sigma=$ 658.8 MeV while the nonstrange direction order parameter temperature variations and phase diagrams for both the models RQM and PQMVT are identical when the $m_\sigma$ value is smaller by 10 MeV i.e. $m_\sigma=$ 648 MeV. This happens because the normalized vacuum effective potential variation in the nonstrange direction is somewhat influenced by its variation in the strange direction.