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

Abstract

$q\bar{q}$ scattering phase shift in the ${\pi }^0$ channel ${\Phi }_{{\pi }^0}({\omega }^2,{\mathbf{k}}_{\perp }^2,k_3^2)$ and ${\pi }^0$ meson spectral function ${\rho }_{{\pi }^0}({\omega }^2,{\mathbf{k}}_{\perp }^2,k_3^2)$ under external magnetic field $eB$ and finite meson momentum ${\mathbf{k}}_{\perp }^2,k_3^2$ 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 $k_3^2$ 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.

Comments: 15 pages, 5 figures