Quark Wigner distribution of the pion meson in light-cone quark model

We study the quark Wigner distributions of the pion meson which are the phase-space distributions containing the most general one-parton information. Using the pion wave functions deduced from a light-cone quark model, we calculate the Wigner distributions of the unpolarized, longitudinally polarized and transversely polarized valence quarks inside the pion meson via the Fock-state overlap representation, respectively. We present the numerical results of the transverse Wigner distributions in which the longitudinal momentum fraction is integrated out. The mixed Wigner distributions as functions of $b_y$ and $k_x$ are also presented. As an intermediate step, we provide the analytic results for the generalized transverse-momentum dependent distributions of the pion meson in the same model.

Figure 1

The transverse Wigner distributions $\rho ({\mathit{b}}_{\perp },{\mathit{k}}_{\perp })$ of the pion meson. The upper, central and lower panels show the Wigner distributions of the unpolarized quark, the longitudinal polarized quark and the transversely polarized quark inside the pion meson, respectively. The left panels plot the distribution in the impact-parameter space with fixed transverse momentum ${\mathit{k}}_{\perp }=k_{\perp }{\widehat{\mathit{e}}}_y$ and $k_{\perp }=0.3\mathrm{GeV}$, while the right panels plot the distribution in the transverse-momentum space with fixed impact parameter ${\mathit{b}}_{\perp }=b_{\perp }{\widehat{\mathit{e}}}_y$ and $b_{\perp }=0.3\mathrm{fm}$.

Figure 2

Quadrupole distortions as functions of $b_{\perp }$ (left panel) and $k_{\perp }$ (right panel), respectively.

Figure 3

The mixed transverse Wigner distributions $\tilde{\rho }(b_y,k_x)$ for the pion meson. The left, central and right panels show the mixed Wigner distributions of the unpolarized quark, the longitudinal polarized quark and the transversely polarized quark inside the pion meson, respectively.