Improved results of chiral limit study with the large $N_c$ standard U(3) ChPT inputs in the on-shell renormalized quark-meson model

TL;DR

The authors develop an on-shell renormalized 2+1 flavor quark-meson model (RQM) that incorporates ChPT-based inputs to fix parameters away from the physical point and study the chiral limit along a β-path. By comparing two ChPT input schemes—large-Nc standard U(3) ChPT (S) and infrared-regularized U(3) ChPT (I)—they compute Columbia plots for varying σ-meson masses m_σ and show that the S scheme yields a more reliable, saturating tricritical structure in the μ–m_K and m_π–m_K planes, maintaining spontaneous chiral symmetry breaking across m_σ up to 800 MeV. The study maps meson observables to quark masses (m_ud, m_s) and finds that the S inputs produce smaller, more physically consistent first-order regions and critical masses (e.g., m_π^c≈134 MeV at μ=0 for several m_σ), while the I inputs can lead to divergent tricritical lines and weaker chiral transitions. Overall, the large-Nc ChPT-based prescriptions provide a clearer, more predictive route for chiral-limit analyses in the RQM framework, with implications for interpreting the phase structure of QCD and guiding comparisons with lattice results.

Abstract

When the $f_π, f_{K} \text{ and } M_η^2$ given by the $ (m_π,m_{K}) $ dependent scaling relations of the large $N_{c}$ standard U(3) Chiral perturbation theory (ChPT) and the infrared regularized U(3) ChPT,~are used in the on-shell renormalized 2+1 flavor quark-meson (RQM) model to find its parameters in the path to the chiral limit $(m_π,m_{K}) \rightarrow 0$ away from the physical point $ (m_π^{\text{\tiny{Phys}}},m_{K}^{\text{\tiny{Phys}}})=(138,496) $ MeV,~one gets the respective framework of RQM-S model and RQM-I model.~Computing and comprehensively comparing the RQM-S and I model Columbia plots for the $m_σ=400,500\text{ and }600$ MeV,~it has been shown that the use of large $N_c$ standard U(3) ChPT inputs give the better and improved framework for the Chiral limit studies as the RQM-S model tricritical lines show the expected saturation pattern after becoming flat in the vertical meson (quark) mass $μ-m_{K}$ ($μ-m_{s}$) plane for all the cases of $m_σ$ whereas the divergent RQM-I model tricritical line for the $m_σ=500$ MeV becomes strongly divergent when the $m_σ=600$ MeV.

Figures (11)

• Figure 1: The curvature masses of the $\pi \text{ and } \sigma$ mesons in the RQM model when its parameters are fixed using the large $N_c$ standard U(3) ChPT inputs (the infrared regularized U(3) ChPT inputs) are termed as $m_{\pi,c}-\text{S(I)} \text{ and } m_{\sigma,c}-\text{S(I)}$. The temperature variations of the $m_{\pi,c}-\text{S} \text{ and } m_{\sigma,c}-\text{S}$ for both the physical point $\beta=1$ and the $\beta_{\text{\tiny CEP}}$ together with the $m_{\pi,c}-\text{I} \text{ and } m_{\sigma,c}-\text{I}$ temperature variations for the $\beta_{\text{\tiny CEP}}$ at $\mu=0$ when the $m_{\sigma}=500 \text{ and } 600$ MeV, are presented in the left panel (a). The right panel (b) shows the $\mu=0$ temperature variations of the $m_{\pi,c}-\text{S} \text{ and } m_{\sigma,c}-\text{S}$ for the physical point $\beta=1$ and $\beta_{\text{\tiny CEP}}$ when the $m_{\sigma}=750 \text{ and } 800$ MeV. The $m_{\pi,c}-\text{S(I)} \text{ and } m_{\sigma,c}-\text{S(I)}$ temperature variations for the $\beta_{\text{\tiny CEP}}$ when $m_{\sigma}=400$ MeV are also plotted in the right panel where $\mu=0$. The fraction $\beta=\frac{m_{\pi}^*}{m_{\pi}}= \frac{m_{K}^*}{m_{K}}$ .
• Figure 2: The chiral transitions for the $m_{\pi}=0$ and $\mu=0$ are depicted respectively in the $\mu-m_{K}$ and $m_{\pi}-m_{K}$ plane. The results for the RQM-S model when its parameters are fixed using the large $N_c$ standard U(3) ChPT inputs, are presented in the left panel (a) whereas the right panel (b) shows the results for the RQM-I model where its parameters are fixed using the infrared regularized U(3) ChPT inputs. The second order $Z(2)$ critical line in solid black color that separates the crossover from the first order transition region in the Fig.(a) and (b), intersects the $m_{\pi}=m_{K}$ green dash line respectively at the critical pion mass $m_{\pi}^{c}=134.51$ and $m_{\pi}^{c}=134.16$ MeV and it terminates respectively at the terminal pion mass $m_{\pi}^{t}=169.25$ and $m_{\pi}^{t}=168.39$ MeV. Solid blue line of the tricritical points which starts respectively at the blue dot $m_{K}^{\text{\tiny TCP}}=248.8$ and $m_{K}^{\text{\tiny TCP}}=242.6$ MeV in the Fig.(a) and (b), separates the second and first order regions. The $h_{y0}$ is negative in the dashed area below the dash dot blue line for the strange chiral limit $h_{y0}=0$. The black dashed vertical line at the physical point in the Fig(a) and (b), shows the crossover transition which ends at the critical end point in blue triangle and the solid red line shows the first order transition. The scalar $\sigma$ mass $m_{\sigma}=400$ MeV.
• Figure 3: The chiral transitions in the $\mu-m_{K}$ plane for the $m_{\pi}=0$ and the $m_{\pi}-m_{K}$ plane for the $\mu=0$ when the $m_{\sigma}=500$ MeV are presented in the left panel (a) for the RQM-S model and the right panel (b) for the RQM-I model. The points, lines, crossover, first and second order chiral transition regions and features are defined similar to the Fig.(\ref{['fig:mini:fig2']}). The critical quantities ($m^{\text{\tiny TCP}}_{K},\ m_{\pi}^{t}, \ m_{\pi}^{c}$) in the RQM-S and RQM-I model are (244.1, 164.65, 130.71) MeV and (240.1, 164.02, 130.84) MeV in respective order.
• Figure 4: Chiral transitions in the $\mu-m_{K}$ plane at $m_{\pi}=0$ and the $m_{\pi}-m_{K}$ plane at $\mu=0$ for the $m_{\sigma}=600$ MeV, are presented in the left panel (a) for the RQM-S model and the right panel (b) for the RQM-I model. The points, lines, crossover, first and second order chiral transition regions and features are defined similar to the Fig.(\ref{['fig:mini:fig2']}). The critical quantities ($m^{\text{\tiny TCP}}_{K},\ m_{\pi}^{t}, \ m_{\pi}^{c}$) are respectively (218.5, 147.66, 117.26) MeV and (207.05, 146.35, 118.01) MeV in the RQM-S and RQM-I model.
• Figure 5: The left panel (a) depicts the RQM-S model chiral transition in the horizontal plane $m_{\pi}-m_{K}$ at $\mu=0$ and the vertical $\mu-m_{K}$ plane at $m_{\pi}=0$ when the $m_{\sigma}=750$ MeV. The horizontal plane $m_{\pi}-m_{K}$ at $\mu=0$ in the right panel (b) presents the chiral transition in RQM-S model for the case of $m_{\sigma}=800$ MeV. The points, lines, crossover, first and second order chiral transition regions and features are defined similar to the Fig.(\ref{['fig:mini:fig2']}). The tricritical line demarcating the first and second order colored regions, is computed only upto the point $(m_{K},\mu)=(280,386.8)$ MeV in the Fig. (a) because the chemical potential becomes very high for a significantly smaller kaon mass in the $\mu-m_{K}$ plane. The RQM-S model critical quantities ($m^{\text{\tiny TCP}}_{K},\ m_{\pi}^{t}, \ m_{\pi}^{c}$) are respectively (159.85, 110.40, 87.90) MeV and (148.2, 103.19, 82.07) MeV when $m_{\sigma}=750 \text{ and } 800$ MeV.