$q\bar{q}$ scattering phase shift in the $\pi^0$ channel and ${\pi}^0$ meson spectral function under external magnetic field and finite meson momentum

$q\bar{q}$ scattering phase shift in the $\pi^0$ channel $\Phi_{\pi^0}(\omega^2,\mathbf{k}_\perp^2,k^2_3)$ and ${\pi}^0$ meson spectral function $\rho_{\pi^0}(\omega^2,\mathbf{k}_\perp^2,k^2_3)$ under external magnetic field $eB$ and finite meson momentum $\mathbf{k}_\perp^2,k^2_3$ are studied in the framework of a two-flavor Nambu-Jona-Lasinio (NJL) model. The $q\bar{q}$ scattering phase shift in the $\pi^0$ channel $\Phi_{\pi^0}$ is closely related to $\pi^0$ spectral function $\rho_{\pi^0}$. We consider three situations, chiral broken phase ($T=\mu=0$), chiral restoration phase ($T>T_{pc},\ \mu=0$) and chiral restoration phase ($T=0,\ \mu>\mu_{pc}$). For $T=\mu=0$ and $T>T_{pc},\ \mu=0$ cases, ${\pi}^0$ meson spectral function $\rho_{\pi^0}$ shows a delta peak, several Breit-Wigner peaks and several non-Breit-Wigner peaks. The delta peak indicates the bound state of $\pi^0$ meson, and the Breit-Wigner peak means the resonant state of $\pi^0$ meson. For $T=0,\ \mu>\mu_{pc}$ case, Pauli blocking effect plays a role, which changes the inner structure of these Breit-Wigner peaks and non-Breit-Wigner peaks. Such multiple peak structure is caused by the external magnetic field. The $q\bar{q}$ scattering phase shift in the $\pi^0$ channel $\Phi_{\pi^0}$ shows a jump from $0$ to $\pi$ when $\pi^0$ meson is in bound state. When $\pi^0$ meson is in resonant state, $\Phi_{\pi^0}$ has the value $\pi/2$ and changes continuously. In large $\omega$ region, at the starting and end points of wide peaks of spectral function, $\Phi_{\pi^0}$ jumps abruptly (from $\pi$ to finite value or from finite value to $0$), and such jumps are caused by the external magnetic field. Finite momentum $\mathbf{k}_\perp^2$ or $k^2_3$ modifies the spectral function $\rho_{\pi^0}$ and scattering phase shift $\Phi_{\pi^0}$, which demonstrates the anisotropy in the system induced by external magnetic field.