$\pi$- and $K$-Mesons Properties for Large $N_f$

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 $\pi$- (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 ($m_{\pi}$) 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 $\kappa^{1/3}_{(\pi,K)}$, and decay constant $f_{(\pi, K)}$. monotonically decrease as a function of $N_f$, except for the pion and kaon mass $m_{(\pi, K)}$, 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).