Medium effects on the electromagnetic form factors of the $\rho$ meson

The dynamics of partons inside the light $\rho$ meson is found to be essential for its properties and internal structure, both in free space and in the nuclear medium. In this paper, we systematically investigate the in-medium structure changes of $\rho^+$ mesons within the covariant Nambu-Jona-Lasinio (NJL) model, utilizing the Schwinger proper-time regularization scheme. We solve the Bethe-Salpeter equations to guarantee the bound meson-state condition. At the quark level, the nuclear medium effects are also derived within the same NJL model to maintain a consistent approach with the in-medium $\rho^+$ meson electromagnetic form factors. To this end, we analyze the spacelike electromagnetic form factors of the $\rho^+$ meson in free space and in a nuclear medium. We find that the charge radius and quadrupole moment of the $\rho^+$ meson increase with increasing nuclear matter density, while the magnetic moment decreases, in agreement with the existing previous theoretical predictions. The enhancement of the $\rho^+$ meson charge radius at normal density relative to that in free space is about 11\% (0.08 fm), while the reduction of $\rho^+$ meson magnetic moment is about 8\% (0.20 $\mu_N$). Our predictions for the charge radius, magnetic moment, and quadrupole moment of the $\rho^+$ meson in both free space and nuclear medium, remain challenging to be verified experimentally.