Polarization-dependent mass modifications of $\phi$ meson with finite momentum in nuclear matter

We investigate the in-medium properties of the $\phi$ meson with finite momentum, going beyond the commonly studied case at rest. In a nuclear medium, Lorentz invariance is broken, leading to distinct longitudinal and transverse polarization modes that evolve differently with density and momentum. Within an effective Lagrangian approach, we calculate the polarization-dependent mass shifts and width modifications of the $\phi$ meson arising from kaon loops and mean-field interactions. The divergent loop integrals are regulated using two different schemes: a covariant form factor and dimensional regularization. Our results show that the mass shift of the transverse polarization is independent of the $\phi$-meson momentum, whereas that of the longitudinal polarization decreases quadratically with momentum. This difference originates from the coupling of the longitudinal mode to the vector mean field and derivative-type interactions in the self-energy. These effects have direct implications for experimental observables, especially for upcoming measurements at J-PARC, and provide a new prediction for experiments studying hadron dynamics in dense matter.