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The constructions of the new high-intensity muon beamlines are progressing infacilities around the world, and new physics searches related to the muons areexpected. The facilities can observe the transverse positron polarization ofthe polarized $\mu^+$ decay to test the standard model. The transition ofmuonium into antimuonium (Mu-to-$\bar{\text{Mu}}$ transition), which is one ofthe interesting possibilities in the models beyond the standard model, can bealso tested. The near-future observation of the transition gives us a greatimpact since it indicates that there is an approximate discrete symmetry in thelepton 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 standardmodel muon decay operator to induce the corrections to the transverse positronpolarization in the $\mu^+$ decay. In this paper, we examine the possibilitythat the Mu-to-$\bar{\text{Mu}}$ transition and the correction to thetransverse positron polarization are related, and we show that those two arerelated in the model of a neutral flavor gauge boson. We also investigate themodels 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 dileptongauge boson, and a left-right model. The non-zero value of the transversepolarization for one of the two directions, $P_{\rm T_2}$, violates thetime-reversal invariance, and the experimental constraint of the electronelectric dipole moment can provide a severe constraint on $P_{\rm T_2}$depending on the model.

Transverse positron polarization in the polarized $\mu^+$ decay related with the muonium-to-antimuonium transition