$\rho$ meson generalized parton distributions in the Nambu–Jona-Lasinio model

In this paper, both the unpolarized and the polarized $\rho$ meson generalized parton distributions are investigated in the framework of the Nambu–Jona-Lasinio model using the proper time regularization scheme. The symmetry properties of $\rho$ meson generalized parton distributions are checked. The three independent distribution functions in deep inelastic scattering, $F_1(x)$, $b_1(x)$, and $g_1(x)$, and the Sachs-like charge, magnetic, and quadruple form factors $G_C(t)$, $G_M(t)$, and $G_Q(t)$, which are the first Mellin moments of unpolarized generalized parton distributions, are obtained. In addition, the $u$ quark axial vector form factors ${\tilde{F}}_1^u(t)$ and ${\tilde{F}}_2^u(t)$ related to the axial currents from the polarized generalized parton distributions are studied. The impact parameter dependent parton distribution functions, which are the two-dimensional Fourier transform of generalized parton distributions are studied, too. The obtained $\rho$ meson generalized parton distributions satisfy the required properties well.

Figure 1

Feynman diagrams representing the ${\rho }^{+}$ meson GPDs.

Figure 2

The unpolarized $u$ quark GPDs of $\rho$ meson. Left: $H_1^u(x,0,t)$; right: $H_1^u(x,0.5,t)$.

Figure 3

The unpolarized $u$ quark GPDs of $\rho$ meson. Left: $H_2^u(x,0,t)$; right: $H_2^u(x,0.5,t)$.

Figure 4

The unpolarized $u$ quark GPDs of a $\rho$ meson. Left: $H_3^u(x,0,t)$; right: $H_3^u(x,0.5,t)$.

Figure 5

The polarized $u$ quark GPDs of a $\rho$ meson. Left: ${\tilde{H}}_1^u(x,0,t)$; right: ${\tilde{H}}_1^u(x,0.5,t)$.

Figure 6

The polarized $u$ quark GPDs of a $\rho$ meson. Left: ${\tilde{H}}_2^u(x,0,t)$; right: ${\tilde{H}}_2^u(x,0.5,t)$.

Figure 7

$H_1^u(x,\xi ,t)$ at $t=0{\mathrm{GeV}}^2$ (left panel) and $t=-5{\mathrm{GeV}}^2$ (right panel). $\xi =0$ (dotted purple), 0.2 (dot-dashed black), 0.4 (dashed red), 0.6 (solid green), 0.8 (thick-dotted blue).

Figure 8

$H_2^u(x,\xi ,t)$ at $t=0{\mathrm{GeV}}^2$ (left panel) and $t=-5{\mathrm{GeV}}^2$ (right panel). $\xi =0$ (dotted purple), 0.2 (dot-dashed black), 0.4 (dashed red), 0.6 (solid green), 0.8 (thick-dotted blue).

Figure 9

$H_3^u(x,\xi ,t)$ at $t=0{\mathrm{GeV}}^2$ (left panel) and $t=-5{\mathrm{GeV}}^2$ (right panel). $\xi =0$ (dotted purple), 0.2 (dot-dashed black), 0.4 (dashed red), 0.6 (solid green), 0.8 (thick-dotted blue).

Figure 10

${\tilde{H}}_1^u(x,\xi ,t)$ at $t=0{\mathrm{GeV}}^2$ (left panel) and $t=-5{\mathrm{GeV}}^2$ (right panel). $\xi =0$ (dotted purple), 0.2 (dot-dashed black), 0.4 (dashed red), 0.6 (solid green), 0.8 (thick-dotted blue).

Figure 11

${\tilde{H}}_2^u(x,\xi ,t)$ at $t=0{\mathrm{GeV}}^2$ (left panel) and $t=-5{\mathrm{GeV}}^2$ (right panel). $\xi =0$ (dotted purple), 0.2 (dot-dashed black), 0.4 (dashed red), 0.6 (solid green), 0.8 (thick-dotted blue).

Figure 12

The $u$ quark DIS distribution function $F_1^u(x)$.

Figure 13

The $u$ quark DIS distribution function $b_1^u(x)$.

Figure 14

The $u$ quark DIS distribution function $g_1^u(x)$.

Figure 15

The dressed Sachs-like charge (dashed red), magnetic (dotted green), and quadruple (dot-dashed purple) form factors.

Figure 16

The $u$ quark axial form factors, ${\tilde{F}}_1^u(t)$ (solid red), $-{\tilde{F}}_2^u(t)$ (dotted green), of the $\rho$ meson.

Figure 17

The impact parameter dependent PDF $q_C(x,{\mathit{b}}_{\perp })$ with different $|{\mathit{b}}_{\perp }|$; 0.15 fm (dotted purple), 0.25 fm (dot-dashed black), 0.35 fm (dashed red), 0.45 fm (solid green).

Figure 18

The impact parameter dependent PDF $q_M(x,{\mathit{b}}_{\perp })$ with different $|{\mathit{b}}_{\perp }|$; 0.15 fm (dotted purple), 0.25 fm (dot-dashed black), 0.35 fm (dashed red), 0.45 fm (solid green).

Figure 19

The impact parameter dependent PDF $q_Q(x,{\mathit{b}}_{\perp })$ with different $|{\mathit{b}}_{\perp }|$; 0.15 fm (dotted purple), 0.25 fm (dot-dashed black), 0.35 fm (dashed red), 0.45 fm (solid green).