On-shell versus curvature mass parameter fixing schemes in the quark-meson model and its phase diagrams

We compute and compare the effective potential and phase structure for the quark-meson model in an extended mean-field approximation when vacuum one-loop quark fluctuations are included and the model parameters are fixed using different renormalization prescriptions. When the quark one-loop vacuum divergence is regularized under the minimal subtraction scheme, the fixing of the model parameters using the curvature masses of the scalar and pseudoscalar mesons has been termed as the quark-meson model with the vacuum term (QMVT). However, this prescription becomes inconsistent when we notice that the curvature mass is akin to defining the meson mass by the self-energy evaluation at vanishing momentum. In this work, we apply the recently reported exact prescription of the on-shell parameter fixing to that version of quark-meson model where the two quark flavors are coupled to the eight mesons of the $S{U}_L(2)\times S{U}_R(2)$ linear sigma model with isosinglet $\sigma$ ($\eta$), isotriplet $\overrightarrow{a_0}$ ($\overrightarrow{\pi }$) scalar (pseudoscalar) mesons. The model then becomes the renormalized quark-meson (RQM) model where physical (pole) masses of mesons and pion 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 vacuum effective potential plots, the phase diagrams and the order parameter temperature variations for both the RQM model and the QMVT model are exactly identical for the $m_{\sigma }=616\mathrm{MeV}$. The vacuum effective potential, when the $m_{\sigma }<616\mathrm{MeV}$, is deepest for the QMVT model. An interesting trend reversal is observed for the $m_{\sigma }>616\mathrm{MeV}$ when the effective potential of the RQM model becomes deepest. We find similar $m_{\sigma }$ dependent differences in the nature of the RQM and QMVT model phase diagrams and the order parameter temperature variations. Furthermore, $S{U}_A(2)$ chiral and $U_A(1)$ axial symmetry breaking/restoration and their interplay can also be investigated in this framework.

Figure 1

(a) One loop self-energy diagrams for sigma particle. (b) One loop self-energy diagrams for the $a_0$.

Figure 2

(a) One-loop self-energy diagrams for the eta. (b) One-loop self-energy diagrams for the pion.

Figure 3

Counterterm for the two-point functions of the scalar $\sigma$ and ${\mathbf{a}}_{\mathbf{0}}$ meson.

Figure 4

Counterterm for the two-point functions of the pseudoscalar $\eta$ and $\pi$ meson.

Figure 5

One-point diagram for the sigma particle and its counterterm.

Figure 6

Blue dotted line, solid red line, and green dashed line, respectively, depict the QM, RQM, and QMVT model result. (a) Effective potential for $m_{\sigma }=500\mathrm{MeV}$. (b) Effective potential for $m_{\sigma }=616\mathrm{MeV}$. (c) Effective potential for $m_{\sigma }=700\mathrm{MeV}$.

Figure 7

Blue dotted line, solid red line and green dashed line present the respective temperature variation in the QM, RQM, and QMVT model. (a) Chiral order parameter $m_{\sigma }=500\mathrm{MeV}$. (b) Chiral order parameter $m_{\sigma }=616\mathrm{MeV}$. (c) Chiral order parameter $m_{\sigma }=700\mathrm{MeV}$.

Figure 8

Blue dotted line, solid red line, and green dashed line present the respective temperature variation in the QM, RQM, and QMVT model. (a) Phase diagram for the $m_{\sigma }=500\mathrm{MeV}$. (b) Phase diagram for the $m_{\sigma }=616\mathrm{MeV}$. (c) Phase diagram for the $m_{\sigma }=700\mathrm{MeV}$.

Figure 9

Phase diagram for the sigma masses of 400, 500, 600, 616, and 700 MeV in the RQM model.