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

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.

💡 Research Summary

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This paper investigates how to improve chiral‑limit studies of the 2 + 1‑flavor quark‑meson (QM) model by incorporating inputs from large‑(N_c) standard (U(3)) chiral perturbation theory (ChPT). The authors use the scaling relations for the pion decay constant (f_\pi), the kaon decay constant (f_K) and the squared η‑meson mass (M_\eta^2) as functions of the pseudoscalar masses ((m_\pi,m_K)). Two distinct sets of inputs are considered: (i) the large‑(N_c) standard (U(3)) ChPT results, which define the “RQM‑S” (standard) framework, and (ii) the infrared‑regularized (U(3)) ChPT results, defining the “RQM‑I” (infrared) framework.

The underlying effective theory is the on‑shell renormalized quark‑meson (RQM) model. Its Lagrangian contains a Yukawa coupling between quarks and the nine scalar/pseudoscalar mesons assembled in a complex (3\times3) matrix (M). Spontaneous chiral symmetry breaking (SχSB) is driven by non‑zero vacuum expectation values of the scalar fields, while explicit breaking is introduced through external fields (h_x) and (h_y). The model also includes the ’t Hooft determinant term with coupling (c), which encodes the (U_A(1)) anomaly.

A key technical step is the on‑shell renormalization: the pole masses of the pseudoscalar mesons ((\pi, K, \eta, \eta’)) and the scalar (\sigma) are used to fix the running couplings and the mass parameter. This procedure automatically strengthens the ’t Hooft coupling and weakens the explicit breaking parameters, leading to a more realistic description of the vacuum.

Parameter fixing proceeds from the physical point ((m_\pi^{\rm phys},m_K^{\rm phys})=(138,496)) MeV. By feeding the ChPT scaling relations into the RQM model, the authors continuously vary ((m_\pi,m_K)) towards the chiral limit ((0,0)) while keeping the on‑shell conditions satisfied. They repeat this for several values of the bare (\sigma) mass, (m_\sigma=400,500,600,750,800) MeV, thereby generating a family of models that can be compared across a wide range of scalar masses.

The main observable is the Columbia plot, i.e. the phase‑diagram in the ((m_\pi,m_K)) plane (or equivalently in the quark‑mass plane ((m_{ud},m_s)) and the temperature–kaon‑mass plane ((\mu,m_K))). The plot distinguishes regions of first‑order, second‑order and crossover chiral transitions, and the line separating first‑order from second‑order is the tricritical line. Its intersection with the (\mu=0) axis defines the tricritical point (TCP).

Results for the RQM‑S (standard) framework

• For (m_\sigma=400,500,600) MeV the tricritical line in the ((\mu,m_K)) plane quickly becomes flat after a modest increase in (\mu). This saturation is physically expected because the 2 + 1‑flavor tricritical line should connect smoothly to the two‑flavor tricritical point at larger chemical potential.
• The TCP moves towards lower kaon mass as (m_\sigma) grows: the ratio (m_{TCP}^K/m_K^{\rm phys}) drops from ≈ 0.44 at (m_\sigma=400) MeV to ≈ 0.30 at (m_\sigma=800) MeV.
• The critical pion mass (m_c^\pi) (the maximal pion mass for which the transition remains first order) also decreases, from ≈ 134 MeV at (m_\sigma=400) MeV to ≈ 82 MeV at (m_\sigma=800) MeV.
• When (m_\sigma) is pushed to 750–800 MeV the first‑order region shrinks dramatically, essentially disappearing, while the tricritical line remains a tiny, well‑behaved segment.

Results for the RQM‑I (infrared) framework

• The tricritical line does not saturate. For (m_\sigma=500) MeV it rises steeply, and for (m_\sigma=600) MeV it diverges dramatically once (m_K) exceeds ≈ 500 MeV.
• Consequently the TCP moves to much smaller kaon masses (ratios down to ≈ 0.11 for (m_\sigma=800) MeV) and the model predicts an unphysical, divergent first‑order region.
• This pathological behaviour persists for all examined (\sigma) masses, indicating that the infrared‑regularized ChPT inputs do not provide a stable mapping onto the RQM parameter space.

Comparison with other approaches
The RQM‑S Columbia plots are in good quantitative agreement with functional‑renormalization‑group (FRG) results obtained in the local‑potential approximation (LPA) for a QM model with (m_\sigma\approx 530) MeV. Both approaches show a modest first‑order region that shrinks as the scalar mass increases. In contrast, the RQM‑I results differ sharply, displaying a runaway tricritical line not seen in FRG or in extended‑mean‑field (e‑MF) studies.

Physical interpretation
The success of the large‑(N_c) standard (U(3)) ChPT inputs can be traced to two factors: (1) the scaling relations respect the underlying large‑(N_c) counting, ensuring that the anomaly term and decay constants evolve consistently with the quark masses; (2) the on‑shell renormalization guarantees that the meson pole masses used to fix the model are the same quantities that appear in the ChPT formulas, eliminating the mismatch that plagued earlier “fixed‑UV” schemes.

Conclusions
The paper demonstrates that employing large‑(N_c) standard (U(3)) ChPT scaling relations to fix the parameters of an on‑shell renormalized 2 + 1‑flavor quark‑meson model yields a robust framework (RQM‑S) for exploring the chiral limit. The resulting Columbia plots exhibit physically sensible tricritical lines, realistic shifts of the TCP, and a systematic reduction of the first‑order region with increasing (\sigma) mass. By contrast, the infrared‑regularized ChPT inputs lead to divergent behaviour (RQM‑I) and are therefore unsuitable for precise chiral‑limit investigations. The study thus provides a clear prescription for future effective‑theory analyses of QCD phase structure near the chiral limit.