  The dispersion relation of the $\phi$ meson in nuclear matter is studied in a
QCD sum rule approach. In a dense medium, longitudinal and transverse modes of
vector particles can have independently modified dispersion relations due to
broken Lorentz invariance. Employing the full set of independent operators and
corresponding Wilson coefficients up to operator dimension 6, the $\phi$ meson
QCD sum rules are analyzed with changing densities and momenta. The non-trivial
momentum dependence of the $\phi$ meson mass is found to have opposite signs
for the longitudinal and transverse modes. Specifically, the mass is reduced by
5 MeV for the longitudinal mode, while its increase amounts to 7 Mev for the
transverse mode, both at a momentum scale of 1 GeV. In an experiment which does
not distinguish between longitudinal and transverse polarizations, this could
in principle be seen as two separated peaks at large momenta. Taking however
broadening effects into account, the momentum dependence will most likely be
seen as a small but positive effective mass shift and an increased effective
width for non-zero momenta.
