$\rho$ meson form factors and parton distribution functions in impact parameter space

In this paper, we investigate the form factors and impact parameter space parton distribution functions of the $\rho$ meson derived from the generalized parton distributions within the framework of the Nambu--Jona-Lasinio model,employing a proper time regularization scheme. We compare the charge $G_C$, magnetic $G_M$, and quadrupole $G_Q$ form factors with lattice data. The dressed form factors, $G_C^D$ and $G_M^D$, exhibit good agreement with lattice results; however, $G_Q^D$ is found to be harder than what is observed in lattice calculations. The Rosenbluth cross section for elastic electron scattering on a spin-one particle can be expressed through the structure functions $A(Q^2)$ and $B(Q^2)$. Additionally, the tensor polarization $T_{20}(Q^2,\theta)$ can also be formulated in terms of these form factors. We analyze the structure functions $A(Q^2)$, $B(Q^2)$ and tensor polarization function $T_{20}(Q^2,\theta)$; our findings quantitatively align with predicted values across various limits. In impact parameter space, we examine parton distribution functions along with their dependence on longitudinal momentum fraction $x$ and impact parameter $\bm{b}_{\perp}$. The width distributions in impact parameter space reveal that the range of the charge distribution $q_C(x,\bm{b}_{\perp}^2)$ is the most extensive. In contrast, the transverse magnetic radius falls within a moderate range, while the quadrupole distribution $q_Q(x,\bm{b}_{\perp}^2)$ demonstrates the narrowest extent.