Medium effects on the electromagnetic form factors of the $ρ$ meson

TL;DR

The paper develops a covariant NJL model with Schwinger proper-time regularization to study how a nuclear medium alters the ρ^+ meson's internal structure and electromagnetic form factors. By solving the Bethe-Salpeter equation and computing in-medium quark propagators, it predicts that the ρ mass and the light-quark mass decrease with density, while the charge radius and quadrupole moment increase and the magnetic moment decreases. The work provides density-dependent predictions for $G_C^*(Q^2)$, $G_M^*(Q^2)$, and $G_Q^*(Q^2)$, along with static quantities like $\mu_ρ^*$ and $\mathcal{Q}_ρ^*$, offering a consistent in-medium picture that can be tested by future experiments and lattice QCD. These results contribute to understanding EMC-like effects in hadrons and guide explorations of spin-1 mesons in dense environments.

Abstract

The dynamics of partons inside the light $ρ$ 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 $ρ^+$ 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 $ρ^+$ meson electromagnetic form factors. To this end, we analyze the spacelike electromagnetic form factors of the $ρ^+$ meson in free space and in a nuclear medium. We find that the charge radius and quadrupole moment of the $ρ^+$ meson increase with increasing nuclear matter density, while the magnetic moment decreases, in agreement with the existing previous theoretical predictions. The enhancement of the $ρ^+$ meson charge radius at normal density relative to that in free space is about 11\% (0.08 fm), while the reduction of $ρ^+$ meson magnetic moment is about 8\% (0.20 $μ_N$). Our predictions for the charge radius, magnetic moment, and quadrupole moment of the $ρ^+$ meson in both free space and nuclear medium, remain challenging to be verified experimentally.

Figures (7)

• Figure 1: Diagrammatic representations of the electromagnetic interaction with the $\rho$ meson. In each panel, the BSE vertices are represented by the filled pair circles (purple), while the quark-photon vertex is represented by a filled oval (red), where we can assume that in the left figure, the photon hits the quark and, in the right figure, the photon hits the antiquark.
• Figure 2: Results for (a) $\rho$ meson effective mass, (b) dynamical quark effective mass, (c) $\rho$-quark coupling constant, and (d) nucleon effective mass as a function of $\rho_B/\rho_0$, where $\rho_B$ and $\rho_0\,\, (=0.16\,\, \text{fm}^{-3})$ are respectively the baryon and saturation densities. Note that the in-medium $\rho$ meson and dynamical effective quark masses represented by the filled asterisk (red) data are taken from Ref. deMelo:2018hfw. Note that, as mentioned in the text, the results of Ref. deMelo:2018hfw used $\rho_0 = 0.15$ fm$^{-3}$ and thus, for an exact comparison with the present results ($\rho_0 = 0.16$ fm$^{-3}$), we adjust the horizontal values of the filled asterisk (red) data multiplying by a factor of (0.15/0.16) = 0.9375. Additionally, the "hybrid" model of the $\rho$-meson in Ref. deMelo:2018hfw can form the bound state $\rho$ meson up to around $\rho_B/\rho_0 \simeq 0.9 \times 0.9375 = 0.84375$ in the present $\rho_B/\rho_0$ units. Thus, for larger nuclear matter densities, the in-medium $\rho$ meson properties could not be calculated within the "hybrid model" in Ref. deMelo:2018hfw.
• Figure 3: Results for the BSE dressed quark form factors for different nuclear matter densities, (a) BSE dressed form factor of the down quark, $F_{\mathrm{1D}}^*(Q^2)$, and (b) BSE dressed form factor of the up quark, $F_{\mathrm{1U}}^* (Q^2)$ for different densities as a function of $Q^2$.
• Figure 4: Results of the vector body form factors, (a) $f_{1}^{* V} (Q^2)$, (b) $f_{2}^{* V} (Q^2)$, and (c) $f_{3}^{* V} (Q^2)$ for different nuclear matter densities as a function of $Q^2$.
• Figure 5: Results of the $\rho^+$ meson EMFFs for different densities (a) $F_{1}^{* \rho} (Q^2)$, (b) $F_2^{* \rho} (Q^2)$, and (c) $F_{3}^{* \rho} (Q^2)$ for different nuclear matter densities as a function of $Q^2$.