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

Abstract

We investigate the in-medium properties of the $φ$ 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 $φ$ 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 $φ$-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.

Figures (8)

• Figure 1: Self-energy of the $\phi$ meson through: (top) a $K\bar{K}$ loop term and (bottom) a contact four-point coupling term.
• Figure 2: In-medium mass and energy of a kaon at rest in nuclear matter calculated using the QMC model Tsushima:1997df. The kaon mass shown with green dashed line is reduced by the scalar mean field $V_\sigma$, while the kaon--antikaon energy splitting arises from the vector mean field. The kaon energy exhibits a slight repulsion, consistent with experimental observations.
• Figure 3: In-medium mass and width of the $\phi$ meson at rest as functions of nuclear density, computed using two different regularization schemes. Both approaches exhibit qualitatively similar density dependencies.
• Figure 4: In-medium mass of the longitudinal-mode $\phi$ meson as a function of nuclear density at momenta $p = 0, 1, 2,$ and $3~\mathrm{GeV}$, computed using two different regularization scheme. The transverse-mode mass remains identical to that of the $\phi$ meson at rest.
• Figure 5: Momentum dependence $p$ of the in-medium $\phi$ meson mass ratio $R \equiv \Delta m_\phi^*(p)/m_\phi^*(0)$ (in percentage), where $\Delta m_\phi^*(p) = m_\phi^*(p) - m_\phi^*(0)$, at normal nuclear density for the longitudinal and transverse modes.