# Studying ΔL = 2 Lepton Flavor Violation with Muons

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Muonium: The Simplest Bound State

## 3. Muonium–Antimuonium Oscillations

#### 3.1. Phenomenology of Muonium Oscillations

#### 3.2. The Mass Difference x

**Para-muonium.**The matrix elements of the spin-singlet states can be obtained from Equation (6) using the definitions of Equation (26),

**Ortho-muonium**. Computing the relevant matrix elements for the vector ortho-muonium state, we obtain the matrix elements

#### 3.3. The Lifetime Difference y

**Para-muonium**. The relevant matrix elements of the spin-singlet state can be read off the Equation (28). Recalling the definitions in Equations (24) and (27), we obtain an expression for the lifetime difference ${y}_{P}$ for the para-muonium state,

**Ortho-muonium**. Employing the matrix elements for the spin-triplet state computed in Equation (30), the lifetime difference for the vector muonia is

#### 3.4. Experimental Studies of Muonium Oscillations

#### 3.5. Constraints on Explicit Models of New Physics

**Heavy new physics.**Let us consider a model which contains a doubly-charged Higgs boson [27,28,29]. Such states often appear in the context of left-right models [30,31], where an additional Higgs triplet is introduced to introduce neutrino masses

**Light new physics.**Let us consider a model with light axion-like particles that couple derivatively to the lepton current [16,35]. For the flavor off-diagonal interactions of the ALP a, the Lagrangian would contain a term

- $\Delta {L}_{e}=\Delta {L}_{\mu}=0$. In such models, the interactions do not violate lepton flavor quantum numbers. The muonium oscillations are induced at one-loop order by the mass terms of the fields that transform as singlets under the SM gauge group and have ${L}_{e,\mu}=\pm 2$. Examples of such models include constructions with Majorana neutrinos [13,37].
- $\Delta {L}_{e}=\pm 2,\Delta {L}_{\mu}=0$ and $\Delta {L}_{e}=0,\Delta {L}_{\mu}=\pm 2$. In such models, the lepton flavor quantum numbers are separately broken. The mediators of such interactions, sometimes called dilepton bosons, have electric charge $2e$ and lepton number equal to two. The mediators could be scalar or vector bosons. Examples of such models include the doubly-charged Higgs model considered above [24,25,27,28,29,32,33] and many other constructions [37]. Muonium oscillations can be generated at tree level in such models.
- $\Delta {L}_{e}=-\Delta {L}_{\mu}=\pm 1$. In such models, interaction terms violate both lepton flavor numbers. The mediators of such interactions are electrically neutral and could be both scalar and vector bosons. Examples of such models include models with flavor-violating Higgs boson [38], and many others [37]. Muonium oscillations can also be generated at the tree level in such models. These models can also be probed in muon conversion experiments unless the mediator is introduced such that it only interacts with the leptons.
- $\Delta {L}_{e}=\pm 1,\Delta {L}_{\mu}=0$ and $\Delta {L}_{e}=0,\Delta {L}_{\mu}=\pm 1$. In such models, effective operators mediating muonium oscillations are generated at one-loop order. These models can also be probed in other muon transitions, such as $\mu \to 3e$.

## 4. Muonium Decays

## 5. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Note

1 |

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**Figure 1.**A diagram, whose imaginary part, denoted by the dotted line, represents the lifetime difference y. A white square represents a ${\mathcal{H}}_{\mathrm{eff}}^{\Delta {L}_{\mu}=2}$, while a black dot is the SM contribution of Equation (3).

Operator | Interaction Type | ${\mathit{S}}_{\mathit{B}}\left({\mathit{B}}_{0}\right)$ (from [18]) | Scale $\mathbf{\Lambda}$, TeV |
---|---|---|---|

${Q}_{1}$ | $(V-A)\times (V-A)$ | 0.75 | $5.4$ |

${Q}_{2}$ | $(V+A)\times (V+A)$ | 0.75 | $5.4$ |

${Q}_{3}$ | $(V-A)\times (V+A)$ | 0.95 | $5.4$ |

${Q}_{4}$ | $(S+P)\times (S+P)$ | 0.75 | $2.7$ |

${Q}_{5}$ | $(S-P)\times (S-P)$ | 0.75 | $2.7$ |

${Q}_{6}$ | $(V-A)\times (V-A)$ | 0.75 | $0.58\times {10}^{-3}$ |

${Q}_{7}$ | $(V+A)\times (V-A)$ | 0.95 | $0.38\times {10}^{-3}$ |

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Petrov, A.A.; Conlin, R.; Grant, C.
Studying Δ*L* = 2 Lepton Flavor Violation with Muons. *Universe* **2022**, *8*, 169.
https://doi.org/10.3390/universe8030169

**AMA Style**

Petrov AA, Conlin R, Grant C.
Studying Δ*L* = 2 Lepton Flavor Violation with Muons. *Universe*. 2022; 8(3):169.
https://doi.org/10.3390/universe8030169

**Chicago/Turabian Style**

Petrov, Alexey A., Renae Conlin, and Cody Grant.
2022. "Studying Δ*L* = 2 Lepton Flavor Violation with Muons" *Universe* 8, no. 3: 169.
https://doi.org/10.3390/universe8030169