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

Abstract

$q\bar{q}$ scattering phase shift in the $π^0$ channel $Φ_{π^0}(ω^2,\mathbf{k}_\perp^2,k^2_3)$ and $π^0$ meson spectral function $ρ_{π^0}(ω^2,\mathbf{k}_\perp^2,k^2_3)$ 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 $π^0$ channel $Φ_{π^0}$ is closely related to $π^0$ spectral function $ρ_{π^0}$. We consider three situations, chiral broken phase ($T=μ=0$), chiral restoration phase ($T>T_{pc},\ μ=0$) and chiral restoration phase ($T=0,\ μ>μ_{pc}$). For $T=μ=0$ and $T>T_{pc},\ μ=0$ cases, $π^0$ meson spectral function $ρ_{π^0}$ shows a delta peak, several Breit-Wigner peaks and several non-Breit-Wigner peaks. The delta peak indicates the bound state of $π^0$ meson, and the Breit-Wigner peak means the resonant state of $π^0$ meson. For $T=0,\ μ>μ_{pc}$ 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 $π^0$ channel $Φ_{π^0}$ shows a jump from $0$ to $π$ when $π^0$ meson is in bound state. When $π^0$ meson is in resonant state, $Φ_{π^0}$ has the value $π/2$ and changes continuously. In large $ω$ region, at the starting and end points of wide peaks of spectral function, $Φ_{π^0}$ jumps abruptly (from $π$ 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 $ρ_{π^0}$ and scattering phase shift $Φ_{π^0}$, which demonstrates the anisotropy in the system induced by external magnetic field.