The separate quark flavor contributions to the pion and kaon valence quark distribution functions are studied, along with the corresponding electromagnetic form factors in the space-like region. The calculations are made using the solution of the Bethe-Salpeter equation for the model of Nambu and Jona-Lasinio with proper-time regularization. Both the pion and kaon form factors and the valence quark distribution functions reproduce many features of the available empirical data. The larger mass of the strange quark naturally explains the empirical fact that the ratio $u_{K^{+}}\left(x\right)/u_{{\pi }^{+}}\left(x\right)$ drops below unity at large $x$, with a value of approximately $M_u^2/M_s^2$ as $x\to 1$. With regard to the elastic form factors we report a large flavor dependence, with the $u$-quark contribution to the kaon form factor being an order of magnitude smaller than that of the $s$-quark at large $Q^2$, which may be a sensitive measure of confinement effects in QCD. Surprisingly though, the total $K^{+}$ and ${\pi }^{+}$ form factors differ by only 10%. In general we find that flavor breaking effects are typically around 20%.

The charge form factor and weak decay constant of the pion as well as the pion-quark coupling constant in symmetric nuclear matter are explored in the framework of the Nambu–Jona-Lasinio model, where the pion is described as a bound state of a dressed quark-antiquark pair obtained by the Bethe-Salpeter equation. For the in-medium current quark properties, we adopt the quark-meson coupling model, which describes successfully many hadron properties in a nuclear medium. The pion decay constant and the pion-quark coupling constant are found to decrease with increasing density as well as the magnitude of the light quark condensate. But the pion mass is found to be insensitive to density up to 1.25 times the normal nuclear density. The pion charge form factor in the spacelike region is also explored and is found to have a $Q^2$ dependence similar to the form factor in vacuum, showing $1/Q^2$ behavior in the large $Q^2$ region, where $Q^2$ is the negative of the four-momentum transfer squared. The modifications of the charge radius of the charged pion in nuclear matter are then estimated and the root-mean-square radius at the normal nuclear density is predicted to be larger than that in vacuum by about 20%.

Charge symmetry breaking (CSB) effects associated with the $u$ and $d$ quark mass difference are investigated in the quark distribution functions and spacelike electromagnetic form factors of the pion and kaon. We use a confining version of the Nambu–Jona-Lasinio model, where CSB effects at the infrared scale associated with the model are driven by the dressed $u$ and $d$ quark mass ratio, which because of dynamical chiral symmetry breaking is much closer to unity than the associated current quark mass ratio. The pion and kaon are given as bound states of a dressed quark and a dressed antiquark governed by the Bethe-Salpeter equation, and exhibit the properties of Goldstone bosons, with a pion mass difference given by $m_{{\pi }^{+}}^2-m_{{\pi }^0}^2\propto {(m_u-m_d)}^2$ as demanded by dynamical chiral symmetry breaking. We find significant CSB effects for realistic current quark mass ratios ($m_u/m_d\sim 0.5$) in the quark flavor-sector electromagnetic form factors of both the pion and kaon. For example, the difference between the $u$ and $d$ quark contributions to the ${\pi }^{+}$ electromagnetic form factors is about 8% at a momentum transfer of $Q^2\simeq 10{\mathrm{GeV}}^2$, while the analogous effect for the light quark sector form factors in the $K^{+}$ and $K^0$ is about twice as large. For the parton distribution functions we find CSB effects which are considerably smaller than those found in the electromagnetic form factors.

A possibility of explaining the anomalies in the semileptonic $B$-meson decay $B\to K^{*}\mu \overline{\mu }$ has been explored in the framework of the gauged $U(1{)}_{\mu -\tau }$ symmetry. Apart from the muon anomalous magnetic moment and neutrino sector, we formulate the model starting with a valid Lagrangian and consider the constraints from the neutral meson mixings, the bounds on direct detection and the relic density of the bosonic dark matter candidate augmented to collider constraints. We search the parameter space, which accommodate the size of the anomaly of the $B\to K^{*}\mu \overline{\mu }$ decay, to satisfy all experimental constraints. We found the allowed region on the plane of the dark matter and $Z^{\prime }$ masses is rather narrow compared to the previous analysis.

In-medium valence-quark distributions of ${\pi }^{+}$ and $K^{+}$ mesons in symmetric nuclear matter are studied by combining the Nambu–Jona-Lasinio model and the quark-meson coupling model. The in-medium properties of the current quarks, which are used as inputs for studying the in-medium pion and kaon properties in the Nambu–Jona-Lasinio model, are calculated within the quark-meson coupling model. The light-quark condensates, light-quark dynamical masses, pion and kaon decay constants, and pion- and kaon-quark coupling constants are found to decrease as nuclear density increases. The obtained valence quark distributions in vacuum for both the ${\pi }^{+}$ and $K^{+}$ could reasonably describe the available experimental data over a wide range of Bjorken-$x$. The in-medium valence $u$-quark distribution in the ${\pi }^{+}$ at $Q^2=16{\mathrm{GeV}}^2$ is found to be almost unchanged compared to the in-vacuum case. However, the in-medium to in-vacuum ratios of both the valence $u$-quark and valence $s$-quark distributions of the $K^{+}$ meson at $Q^2=16{\mathrm{GeV}}^2$ increase with nuclear matter density but show different $x$-dependence. Namely, the ratio for the valence $u$-quark distribution increases with $x$, while that for the valence $s$ quark decreases with $x$. These features are enhanced at higher density regions.

We investigate the in-medium properties of pseudoscalar and vector mesons with the light-light and heavy-light quarks in a light-front quark model, using the in-medium quark properties computed by the quark-meson coupling model. Both models are constructed on an equal footing with the constituent quark degree of freedom. Here, we particularly focus on the weak decay constants and distribution amplitudes (DAs) of the mesons in symmetric nuclear matter. We find that the weak decay constants decrease as nuclear density increases for $\pi$, $K$, $D$, and $B$ pseudoscalar as well as $\rho$, $K^{*}$, $D^{*}$, and $B^{*}$ vector mesons, where their properties in free space have good agreement with the available experimental and lattice QCD data. A larger reduction is found for the light-light quark pseudoscalar mesons, while a smaller reduction is found for the heavy-light quark vector mesons, in particular, with the bottom quark. We discuss the effect of the vector potential on the weak decay constants, and present our predictions for the in-medium modifications of DAs. Also, a comparison with the free space lattice QCD data is made.