PHYSICAL REVIEW D 103, 055023 (2021) Probing the doubly charged Higgs boson with a muonium to antimuonium conversion experiment † Chengcheng Han,1 Da Huang ,2,3,4,* Jian Tang ,1, and Yu Zhang 5,6 1School of Physics, Sun Yat-Sen University, Guangzhou 510275, China 2National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012, China 3School of Fundamental Physics and Mathematical Sciences, Hangzhou Institute for Advanced Study, University of Chinese Academy of Sciences, Hangzhou 310024, China 4International Center for Theoretical Physics Asia-Pacific, Beijing/Hangzhou 10010, China 5Institutes of Physical Science and Information Technology, Anhui University, Hefei 230601, China 6School of Physics and Materials Science, Anhui University, Hefei 230601, China (Received 8 February 2021; accepted 10 March 2021; published 30 March 2021) The spontaneous muonium-to-antimuonium conversion is one of the interesting charged lepton flavor violation processes. The Muonium-to-Antimuonium Conversion Experiment (MACE) is the next generation experiment to probe such a phenomenon. In models with a triplet Higgs boson to generate neutrino masses, such as the type-II seesaw and its variant, this process can be induced by the doubly charged Higgs boson contained in it. In this article, we study the prospect of MACE to probe these models via the muonium-to-antimuonium transitions. After considering the limits from μþ → eþγ and μþ → eþe−eþ, we find that MACE could probe a parameter space for the doubly charged Higgs boson, which is beyond the reach of LHC and other flavor experiments. DOI: 10.1103/PhysRevD.103.055023 I. INTRODUCTION and for μþ → eþγ by MEG-II [2]. Furthermore, the coherent muon-to-electron conversions (μ−N → e−N) will The observation of neutrino oscillations has indicated be searched for by COMET [3] at J-PARC and Mu2e [4] at that neutrinos have very tiny but nonzero masses, which is FNAL, both of which are still under construction. one of the direct evidences towards the new physics (NP) Another equally important but less studied cLFV chan- beyond the Standard Model (SM). Traditionally, the nel is the muonium-to-antimuonium conversion, which was explanation of the measured neutrino masses and mixings first brought forward by Pontecorvo [5] more than half a involves the seesaw mechanism, where the smallness of century ago. Muonium M is a hydrogenlike bound state neutrino masses is allowed with the existence of heavy − (e μþ) formed by an electron and an antimuon, while an eigenstates. One of the most striking predictions of the antimuonium M¯ is the corresponding antiparticle bounded seesaw models is the existence of charged lepton flavor − by a positron and a muon (eþμ ). Experimentally, one can violation (cLFV) processes. Indeed, the neutrino oscilla- produce muonium atoms by colliding slow muons into a tions can be viewed as a manifestation of the neutral lepton target material and then trying to see the spontaneous flavor violation (LFV). Therefore, it is natural to think conversion from muoniums into antimuoniums. In the SM, about the corresponding LFV in the charged lepton such a conversion is forbidden by the lepton flavor channels. Currently, there are many ongoing and upcoming symmetry. Hence, the detection of this process can be experiments searching for the cLFV processes all over the viewed as a probe to the physics beyond the SM. In light of world. The accelerator muon beam experiments at PSI are its potential significance, there have been many theoretical μþ → þ þ − searching for e e e by the Mu3e Collaboration [1] studies on the muonium-to-antimuonium transition in some well-motivated models, such as the type-I [6–8] and type-II seesaw [9] models, the Z8 model with more than four *Corresponding author. [email protected] lepton generations [10], the minimal 331 model [11], the † Corresponding author. R-parity violated SUSY model [12], and the left-right [email protected] symmetric model [13], as well as the low-energy effective operator framework [14]. The complementary study Published by the American Physical Society under the terms of between low-energy and high-energy phenomenonology the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to in LFVs was well explored in terms of a doubly charged the author(s) and the published article’s title, journal citation, scalar in Refs. [15,16], while more attention was paid to and DOI. Funded by SCOAP3. the type-II seesaw scalar triplet model at a high-energy 2470-0010=2021=103(5)=055023(11) 055023-1 Published by the American Physical Society HAN, HUANG, TANG, and ZHANG PHYS. REV. D 103, 055023 (2021) proton-proton collider [17,18]. More recently, the M − M¯ parameter regions, which cannot be reached by other conversion was explored theoretically in a model- experiments. independent way in Ref. [19]. This article is organized as follows: in Sec. II, we briefly At present, the best upper constraint on the muonium-to- describe quantum mechanics of the muonium-to-antimuo- antimuonium conversion has been given by the PSI experi- nium transition based on the effective field theory and then ment in 1999, which presented the upper limit on the estimate the MACE sensitivity to this process. In Sec. III, conversion probability to be PðM ↔ M¯ Þ ≲ 8.3 × 10−11 at we give a brief introduction to the type-II and hybrid the 90% confidence level [20]. During the past two seesaw models. Then we detail the leading-order calcu- decades, there has not been any experimental improvement lation of the M − M¯ conversion probability in these two in this important cLFV channel. However, this situation is models. Section IV is devoted to the discussion of existing expected to soon change in the near future due to the advent constraints on the type-II and hybrid seesaw models, of the Muonium-to-Antimuonium Conversion Experiment including those from the measured neutrino masses and (MACE) in China [21]. MACE is the next-generation mixings, other cLFV processes, and collider searches on experiment specialized to measure this exceptional cLFV the doubly charged scalar. We present our numerical results process, which will be operated at the upgraded 500 kW in Sec. V. Finally, we summarize and conclude in Sec. VI. pulsed proton accelerator running at the China Spallation Neutron Source (CSNS). By taking advantage of the high- II. MACE PROSPECTS ON THE MUONIUM- 8 quality intense slow muon sources with more than 10 μþ ANTIMUONIUM TRANSITIONS produced per second and the beam spread smaller than 5% at CSNS, together with the high-efficiency muonium Muonium jMi is a nonrelativistic Coulombic bound state μþ − ¯ formation targets, the high-precision magnetic spectrom- of and e , and antimuonium jMi is a similar bound state − þ ¯ eter, and the optimized detector setup by locating the of μ and e . The nontrivial mixing between jMi and jMi detector at the front end of a future muon collider, implies the nonvanishing lepton flavor violation (LFV) − þ þ − MACE is expected to enhance the sensitivity to the amplitude for e μ → e μ . M − M¯ conversion probability by more than 2 orders of Currently, the most precise measurement of the conver- sion probability of the M − M¯ conversion was given by the magnitude from the existing PSI experiment. Therefore, we ¯ expect that MACE can play an important role in probing PSI Collaboration two decades ago, with PðM ↔ MÞ < −11 and constraining the parameter space of NP models to 8.3 × 10 at 95% C.L. The MACE Collaboration at CSNS generate the neutrino masses. attempts to improve the sensitivity of this conversion to the ¯ −14 In light of promising experimental developments pro- level of PðM ↔ MÞ ∼ Oð10 Þ. ¯ jected by MACE, it is timely to investigate the muonium- Microscopically, the transition between M and M can be to-antimuonium conversion in more detail, paying attention described by the following local effective Hamiltonion to its interplay with other flavor and collider searches in density [7]: probing the model parameter space of interest. In the present paper, we shall explore this aspect in the type-II H GMffiffiffiM¯ μ¯ γα 1 −γ5 μ¯ γ 1 − γ5 eff ¼ p ½ ðxÞ ð ÞeðxÞ½ ðxÞ αð ÞeðxÞ; ð1Þ seesaw model and its variant as a case study. The traditional 2 type-II seesaw model generates the small active neutrino 2 where G ¯ is the corresponding Wilson coefficient. Due to masses by introducing a SUð ÞL triplet scalar, in which the MM ¯ doubly charged scalar component can induce the muonium- the LFV transition, jMi and jMi are not mass eigenstates. to-antimuonium conversion at tree level. Note that, due to In this basis, the mass matrix has nondiagonal components, the model structure in the type-II seesaw, the predicted Z − ¯ 3 H ⃗ ¯ M M conversion probability is intimately correlated to mMM¯ ¼hMj d x effðxÞjMi; ð2Þ the neutrino oscillation data and other cLFV processes, such as μþ → e−eþeþ and μþ → eþγ. As a result, it will be which can be transformed into the form of its constituents seen below that MACE is not sensitive to the relevant with a momentum distribution f p , parameter space surviving after imposing the existing cFLV ð Þ constraints. In other words, if MACE can observe the Z 3 d p † † positive signal on the muonium-to-antimuonium transition, M 0 f p a p; s bμ −p; s 0 ; j ð Þi ¼ 2π 3 ð Þ eð Þ ð Þj i ð3Þ it means that the simplest type-II seesaw model is ruled out, ð Þ and one is required to consider some extensions to it. In our † − discussion, we shall provide such an example by incorpo- where aeðp; sÞ creates a fermion e with energy þEp and † rating in the type-II seesaw model only a single heavy momentum p⃗and bμð−p; sÞ creates an antiparticle μþ with right-handed neutrino, which will be called the hybrid the opposite momentum and the same energy þEp.Ifwe seesaw model in the following.