Neutrinoless double beta decay and the muonium-to-antimuonium transition in models with a doubly charged scalar

The lepton number and flavor violations are important possible ingredients of the lepton physics. The neutrinoless double beta decay and the transition of the muonium into antimuonuim are related to those violations. The former can give us an essential part of fundamental physics, and there are plenty of experimental attempts to observe the process. The latter has also been one of the attractive phenomena, and the experimental bound will be updated in planned experiments at new high-intensity muon beamlines. In models with a doubly charged scalar, not only can those two processes be induced, but also the active neutrino masses can be induced radiatively. The flavor violating decays of the charged leptons constrain the flavor parameters of the models. We study how the muonium-to-antimuonium transition rate can be as large as the current experimental bound, and we insist that the updated bound of the transition rate will be useful to distinguish the models to generate the neutrinoless double beta decay via the doubly charged scalar. We also study a possible extension of the model to the left-right model which can induce the neutrinoless double beta decay.

Figure 1

Cocktail diagram to induce the $0\nu 2\beta$ decay. The $\times$ mark stands for the insertion of the electroweak vev of the Higgs boson. The $h^{-}$ and $k^{--}$ fields are $SU(2{)}_L$ singlet scalars, and ${\eta }^{-}$, ${\eta }^0$ and $A^0$ are contained in the $SU(2{)}_L$ doublet scalar which does not acquire a vev.

Figure 2

Diagram to induce the neutrino mass in the model with $Z_2$-charge A (Liu-Gu model).

Figure 3

Diagram to induce the neutrino mass in the model with $Z_2$-charge B (Zee-Babu model).

Figure 4

Plot of the $0\nu 2\beta$ half-lives of Xe for various scalar masses. 1. $m_{h_1^{+}}=300\mathrm{GeV}$, $m_H=450\mathrm{GeV}$, $m_A=415\mathrm{GeV}$, 2. $m_{h_1^{+}}=250\mathrm{GeV}$, $m_H=450\mathrm{GeV}$, $m_A=415\mathrm{GeV}$, 3. $m_{h_1^{+}}=250\mathrm{GeV}$, $m_H=400\mathrm{GeV}$, $m_A=360\mathrm{GeV}$, 4. $m_{h_1^{+}}=250\mathrm{GeV}$, $m_H=300\mathrm{GeV}$, $m_A=240\mathrm{GeV}$. Horizontal axis is for the mass of $h_2^{+}$. The horizontal dotted line gives the current bound of the half-life by Kamland-Zen.

Figure 5

$|G_2|/G_F$ as a function of $R$. The solid line shows the bound in the case where the Zee-Babu contribution is zero (the $\ell \ell h$ coupling is zero). For $R<O(1)$, the LFV decay constrains $|G_2|/G_F$. Even though $g_{\mu \mu }$ becomes larger for smaller $R$, $|G_2|/G_F$ cannot become larger. The circles “o” are plotted if the corresponding parameters to realize the neutrino mass matrix can satisfy the LFV constraints by switching on the $\ell \ell h$ coupling. The asterisks “$*$” are plotted if the doubly charged scalar $k^{++}$ needs to be heavier than 1 TeV to satisfy the LFV data. The horizontal dotted line shows the current experimental bound from the Mu-to-$\overline{\mathrm{Mu}}$ transition. See the text for more detail on the setup.

Figure 6

Box loop diagram to induce the $0\nu 2\beta$ decay. The $\times$ mark stands for the insertion of the electroweak vev, and the $\otimes$ mark stands for the mass insertion of $N$.

Figure 7

Ratio of the amplitudes of $A_{\text{cocktail}}/g_{ee}$ and $A_{\mathrm{box}}/{\kappa }^2$ for various scalar masses. 1. $m_{h_1^{+}}=300\mathrm{GeV}$, $m_H=450\mathrm{GeV}$, $m_A=415\mathrm{GeV}$, 2. $m_{h_1^{+}}=250\mathrm{GeV}$, $m_H=450\mathrm{GeV}$, $m_A=415\mathrm{GeV}$, 3. $m_{h_1^{+}}=250\mathrm{GeV}$, $m_H=400\mathrm{GeV}$, $m_A=360\mathrm{GeV}$, 4. $m_{h_1^{+}}=250\mathrm{GeV}$, $m_H=300\mathrm{GeV}$, $m_A=240\mathrm{GeV}$. We choose $m_k^2/m_{hhk}=1\mathrm{TeV}$ and $m_{h_2^{+}}=1\mathrm{TeV}$. The horizontal axis is the mass of $N$.

Figure 8

Diagram to induce the $0\nu 2\beta$ decay by the $W_R$ currents with the exchange of the right-handed neutrino $N$. The $\otimes$ mark stands for the Majorana mass insertion.

Figure 9

Diagram to induce the $0\nu 2\beta$ decay by the gauge interaction of the $SU(2{)}_R$ triplet $\mathrm{\Delta }$.

Figure 10

Plots of $(16{\pi }^2{)}^2{L}_{\mathrm{LG}}$ (left) and $(16{\pi }^2{)}^2{L}_{\mathrm{ZB}}$ (right). We choose $m_k=1\mathrm{TeV}$. For $L_{\mathrm{ZB}}$, we choose $m_{h_1^{+}}=300\mathrm{GeV}$.

Figure 11

Two-loop diagram which can generate the $\mu \to e\gamma$ decay in the model with the $h^{-}–{\eta }^{-}$ mixing in the $Z_2$-charge B. The photon line can be attached to the lines of the charged particles.