$π$- and $K$-Mesons Properties for Large $N_f$

TL;DR

This work investigates how increasing the number of light-quark flavors $N_f$ affects dynamical chiral symmetry breaking and confinement in QCD-like theories by studying pion and kaon properties. The authors employ a flavor-dependent, symmetry-preserving vector-vector contact interaction (FCI) within the Schwinger-Dyson equation (SDE) and homogeneous Bethe-Salpeter equation (BSE) framework under rainbow-ladder truncation. They show that as $N_f$ increases toward a critical value $N_f^c \approx 8$, dynamical quark masses, in-meson condensates, and decay constants decrease and the confinement scale diverges, signaling chiral restoration and deconfinement; the Goldstone pion becomes a resonant state and bound-state masses $m_{\pi}$ and $m_{K}$ dissociate at $N_f^d \approx 8.2$. For $N_f=2$, their results align with experimental and other SDE–BSE studies, and GMOR relations remain satisfied across the flavor range, providing insights into light-hadron structure in the conformal window.

Abstract

The restoration of dynamical chiral symmetry for a higher number of light-quark flavors $N_f$ implies suppression of the dynamically generated quark mass. The study of various larger values of $N_f$ may have a greater impact on the internal structure of light hadrons. In this work, we study the properties of the $π$- (pion) and $K$-meson (kaon), such as the mass, condensate, and leptonic decay constant, for various $N_f$. We use the symmetry-preserving vector-vector flavor-dependent contact interaction model of quark. The dynamical quark masses are calculated using the Schwinger-Dyson equation (SDE). The masses of pion ($m_π$) and kaon ($m_{K}$) for different values of $N_f$ and are determined using the homogeneous Bethe-Salpeter equation. For fixed $N_c=3$ and $N_f$ is increased, the dynamically generated quark mass $M_{u/d}$ ( mass of up and down quarks), strange quark mass ($M_s$), meson in-condensate $κ^{1/3}_{(π,K)}$, and decay constant $f_{(π, K)}$. monotonically decrease as a function of $N_f$, except for the pion and kaon mass $m_{(π, K)}$, which increase above a critical value of $N_f$ around $8$. This is the region where chiral symmetry is restored and the pion and kaon behave as free particles, similar to thier behavior in the the presence of a heat bath. The results obtained for fixed $N_f=2$ and $N_c=3$ are fairly in decent agreement with experimentally calculated statistics and previous model calculations based on the Schwinger-Dyson equation (SDE) and Bethe-Salpeter equation (BSE).

Figures (7)

• Figure 1: In the chiral limit, behavior of the dressed quark mass $M_0$, the masses of the Goldstone boson $m^{0}_{GB}$ (massless pions) , the leptonic decay constant $f_{0}$, the quark-antiquark condensate $-<\bar{q}q>^{1/3}$, and the confinement scale $\tilde{\tau}$ for various number of flavors $N_f$. All parameters, except for the confinement scale and the massless bosons, vanish when $N_f$ reaches or exceeds $N^{c}_{f}\approx8$, indicating the restoration of dynamical chiral symmetry. The confinement scale $\tilde{\tau}$ diverges, signaling a transition to deconfinement. In the chiral limit, $N_f$-dependence of dressed quark mass $M_0$, Goldstone boson (massless pion) masses $m^{0}_{GB}$, leptonic decay constant $f_{0}$, the quark-antiquark condensate ($-<\bar{q}q>^{1/3}$) and the confinement scale $\tilde{\tau}$ small figure inside. All the parameters except the confinement scale and massless boson, vanishes at and above $N^{c}_{f}\approx8$, where the dynamical chiral symmetry is restored. The confinement scale $\tilde{\tau}^{0}$ diverges, signaling deconfinement.
• Figure 2: The behavior of the $N_f$-dependent invariant BSAs $E^{0}$ and $F^{0}$ in the chiral limit is examined. While $F^{0}$ approaches zero at the chiral symmetry restoration point $N^{c}_{f}$, $E^{0}$ retains non-zero values in this region, normalized around $N_f = 12.5$ as a function of $N_f$.
• Figure 3: Behavior of the dressed quark mass $M_{u/d}$ (with $m_{u=d}=0.007$) GeV, and dressed strange quark mass $M_s$ (with bare mass $m_s=0.17$ GeV), and the average dressed mass $(M_{u/d}+M_{s})/2$, for various number $N_f$. All the parameters monotonically decrease as we increase $N_f$.
• Figure 4: The behavior of the leptonic pion decay constant $f_{\pi}$, the kaon decay constant $f_K$, the in-pion condensate $\kappa^{1/3}_{\pi}$ and the in-kaon condensate $\kappa^{1/3}_{K}$ as a function $N_f$ is analyzed with bare quark masses. All the parameters monotonically decrease as $N_f$ increases, indicting a partial restoration of dynamical chiral symmetry above $N^{c}_{f}\approx 8.2$.
• Figure 5: Behavior of the invariant BSAs for pion $E_{\pi}$, $F_{\pi}$ and kaon $E_{K}$, $F_{K}$ , with bare quark masses as a function of $N_f$.