Peripheral nucleon-nucleon scattering with explicit vector mesons in chiral perturbation theory

Abstract

Background: In chiral perturbation theory (ChPT), heavy degrees of freedom are integrated out, with their contributions absorbed into low-energy constants. Keeping the $\rho$ and $\omega$ explicit within ChPT retains the finite-range momentum dependence associated with these mesons. Purpose: We study how explicit $\rho$ and $\omega$ mesons affect peripheral $NN$ phase shifts. Methods: We incorporate the vector mesons by counting their masses as hard and the momentum they carry as soft. The next-to-next-to-leading-order nuclear force includes, in addition to one- and two-pion exchange, one-vector-meson exchange with dressed propagators and vertex functions. The vector propagators and vertex functions are fixed from the mesonic sector and the nucleon electromagnetic form factors. We calculate neutron--proton scattering with $J=3$--$5$ up to $T_{\rm lab}=300$ MeV and examine the sensitivity to one-pion-exchange iteration and the spectral-function regularization scheme. Results: The vector-meson contribution falls rapidly with $J$. In several $J=3$ waves it moves the phase shifts toward PWA93, while in other waves the change is small or moves the phase shift away. In ${}^{3}\mathrm{G}_{3}$ the change toward PWA93 persists under variations of the spectral cutoff, and its size is nearly independent of the treatment of one-pion exchange. The ${}^{3}\mathrm{D}_{3}$ wave is strongly cutoff dependent, and there the $\omega$ contribution outweighs the opposing $\rho$ contribution. The tensor and spin-spin components of the $\rho$ interaction play the dominant roles in moving $\epsilon_3$ and ${}^{1}\mathrm{F}_{3}$ toward PWA93, respectively. Conclusions: The effect of the explicit vector mesons on the peripheral phase shifts is channel dependent. The $\rho$ meson gives the dominant shifts toward PWA93, particularly in $\epsilon_3$ and ${}^{3}\mathrm{G}_{3}$.