  Observing the mass shifts of mesons immersed in nuclear matter is
interesting, as the changes are expected to shed light on the effects of chiral
symmetry breaking on the origin of hadron masses. At the same time, it is
important to understand the momentum dependence of the masses for spin-1
mesons, as the changes manifest differently across the two polarization modes.
Here, the mass shifts of $K_{1}^{\pm}$ mesons with finite three-momentum in
nuclear medium are studied in the QCD sum rule approach. We find that the mass
of $K_{1}^{+}$($K_{1}^{-}$) meson is increased(decreased) by the non-trivial
momentum effect in both the transverse and longitudinal modes. Specifically,
compared to its rest mass in the nuclear medium, in the transverse mode, the
mass of $K_{1}^{+}(K_{1}^{-})$ is observed to shift by +2(-55) MeV, while in
the longitudinal mode, the mass shift is +13(-11) MeV, all at a momentum of 0.5
GeV. Exploring the medium modifications of $K_{1}$ meson through kaon beams at
J-PARC will provide insights on the partial restoration of chiral symmetry in
nuclear matter.
