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

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 $T_{\mathrm{l}\mathrm{a}\mathrm{b}}=300$ 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 ${}^3{\mathrm{G}}_3$.