  The constructions of the new high-intensity muon beamlines are progressing in
facilities around the world, and new physics searches related to the muons are
expected. The facilities can observe the transverse positron polarization of
the polarized $\mu^+$ decay to test the standard model. The transition of
muonium into antimuonium (Mu-to-$\bar{\text{Mu}}$ transition), which is one of
the interesting possibilities in the models beyond the standard model, can be
also tested. The near-future observation of the transition gives us a great
impact since it indicates that there is an approximate discrete symmetry in the
lepton sector. If the Mu-to-$\bar{\text{Mu}}$ transition operator is generated,
a new muon decay operator can exist and it may interfere with the standard
model muon decay operator to induce the corrections to the transverse positron
polarization in the $\mu^+$ decay. In this paper, we examine the possibility
that the Mu-to-$\bar{\text{Mu}}$ transition and the correction to the
transverse positron polarization are related, and we show that those two are
related in the model of a neutral flavor gauge boson. We also investigate the
models to generate the Mu-to-$\bar{\text{Mu}}$ transition, such as an inert
$SU(2)_L$ doublet, a $SU(2)_L$ triplet for the type-II seesaw model, a dilepton
gauge boson, and a left-right model. The non-zero value of the transverse
polarization for one of the two directions, $P_{\rm T_2}$, violates the
time-reversal invariance, and the experimental constraint of the electron
electric dipole moment can provide a severe constraint on $P_{\rm T_2}$
depending on the model.
