Models of the muonium to antimuonium transition

Muonium is a bound state composed of an antimuon and an electron, and it constitutes a hydrogenlike atom. Because of the absence of the hadronic matter in the bound state, the muonium is a useful probe to explore new physics being free from the hadronic uncertainties. The process of the muonium-to-antimuonium transition is considered to be effective to identify fundamental interactions which relate to the lepton flavor and lepton number violation. New experiments are being planned at J-PARC in Japan and CSNS in China, and it is expected to attract more attention in the near future. In this paper, we will study what kind of model can be verified in the next generation of the muonium-to-antimuonium transition search experiments while escaping the limitations from other experiments. Though the transition probability is strongly suppressed by the lepton flavor conservation in the standard model, it can be much larger by the exchanges of neutral and doubly charged bosons, and by box loop diagrams in new physics beyond the standard model. We study the neutrino models with heavy Majorana neutrinos at TeV scale, a type-II seesaw model, left-right models, and models for radiative neutrino masses such as the Zee-Babu model in particular, in addition to other possible models to induce the sizable transition probability, which can be tested in the forthcoming experiments.

Figure 1

The box loop diagrams to generate the transition operator $Q_1$. There are two types of contribution: the momentum parts ($\cancel{p}$) of the numerators of neutrino propagators are picked (left), and the mass parts of the propagators are picked (right). The lepton flavors are changed at the vertices in the left diagram, while the lepton numbers are violated in the neutrino masses in the right diagram.

Figure 2

The upper bound of $|G_1|/G_F$ as a function of the heavy neutrino mass. For the neutrino mass to be less than $\sim 30\mathrm{TeV}$, the $\mathrm{Mu}\text{−}\mathrm{to}\text{−}\overline{\mathrm{Mu}}$ transition is bounded by the $\mu \to e\gamma$ constraint.

Figure 3

The tree-level exchange of a doubly charged scalar boson ${\mathrm{\Delta }}^{++}$ to induce the $\mathrm{Mu}\text{−}\mathrm{to}\text{−}\overline{\mathrm{Mu}}$ transition.

Figure 4

Contour plots of upper bound of ${\log }_{10}(|G_1|/G_F)$ from LFV decays as a function of $\delta$ and ${\alpha }_2$ (in radian) for normal mass ordering (left) and inverted mass ordering (right). In the dark blue region, there is no solution to make ${\kappa }_{e\mu }^L\to 0$.

Figure 5

The plot of $|G_1|/G_F$ allowed by LFV constraints versus the total neutrino mass for normal mass ordering (left) and inverted mass ordering (right). The vertical red line shows $\sum m_{\nu }=0.12\mathrm{eV}$.

Figure 6

The upper bound of $|G_2|/G_F$ as a function of $M_{W_R}$ and the heavy neutrino mass $M_N$.

Figure 7

The diagram to induce the neutrino masses in the Zee-Babu model. The symbol $⟨H⟩$ stands for the vev of the SM Higgs boson.

Figure 8

The contour plots of $|G_2|/G_F$ (left), $\mathrm{Br}(\tau \to 3\mu )$ (right) as functions of the Dirac phase $\delta$ and the Majorana phase ${\alpha }_2$. The choices of parameters in the Zee-Babu model are given in the text.

Figure 9

A diagram to induce the neutrino mass in the cocktail model. The charged scalar in the inert doublet $\eta$ and the $SU(2{)}_L$ singlet scalar $S^{+}$ are mixed to be $H_{1,2}^{+}$. By splitting the masses of the real and imaginary parts ($H^0$, $A^0$) of the neutral scalar in $\eta$, neutrino masses are generated. The symbol $⟨H⟩$ stands for the vev of the SM Higgs boson.

Figure 10

The relation between $\delta$ (radian) and ${\theta }_{23}$ for various ${\theta }_{12}$ in the case of $(M_{\nu }{)}_{ee,e\mu }\to 0$. We use central values for the 1–3 neutrino mixing and mass squared differences: ${\theta }_{13}=8.57°$, $\mathrm{\Delta }{m}_{\mathrm{sol}}^2=7.42\times {10}^{-5}{\mathrm{eV}}^2$, and $\mathrm{\Delta }{m}_{\mathrm{atm}}^2=2.52\times {10}^{-3}{\mathrm{eV}}^2$ [63].

Figure 11

The diagram to generate the neutrino mass radiatively in the model without the Dirac neutrino masses. The symbol $⟨H⟩$ stands for the vev of the SM Higgs boson.

Figure 12

The box loop diagrams to generate the $\mathrm{Mu}\text{−}\mathrm{to}\text{−}\overline{\mathrm{Mu}}$ transition via the charged Higgs bosons.

Figure 13

A diagram to induce the neutrino mass in the KNT model. The symbol $⟨H⟩$ stands for the vev of the SM Higgs boson.

Figure 14

A diagram to induce the neutrino mass in the AKS model. The symbol $⟨H⟩$ stands for the vev of the SM Higgs boson.

Figure 15

The tree-level exchange of a neutral scalar boson ($H^0$, here) to induce the $\mathrm{Mu}\text{−}\mathrm{to}\text{−}\overline{\mathrm{Mu}}$ transition.

Figure 16

The tree-level exchange of the doubly charged gauge boson to induce the $\mathrm{Mu}\text{−}\mathrm{to}\text{−}\overline{\mathrm{Mu}}$ transition.

Figure 17

The tree-level exchange of a neutral gauge boson to induce the $\mathrm{Mu}\text{−}\mathrm{to}\text{−}\overline{\mathrm{Mu}}$ transition.

Figure 18

The contribution to the electron $g-2$ ($\mathrm{\Delta }{a}_e$) is shown as a function of the $U(1)$ charge $a$ for the right-handed charged leptons when the $\mathrm{Mu}\text{−}\mathrm{to}\text{−}\overline{\mathrm{Mu}}$ transition is assumed to be just the same as the experimental upper bound.