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

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 $μ^+$ 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 $μ^+$ 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.