Probing the Doubly Charged Higgs Boson with a Muonium to Antimuonium Conversion Experiment

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 conversion 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. It turns out that MACE can neglect the momentum dependence in the spinors, the mass provide additional information in this hybrid model on the mixing element is

055023-2 PROBING THE DOUBLY CHARGED HIGGS BOSON WITH A … PHYS. REV. D 103, 055023 (2021) Z 3 2 3 10−3 GMM¯ d p ðμÞ α ðeÞ ðμÞ ðeÞ the Wilson coefficient as GMM¯ =GF < × at 90% C.L. m ¯ ¼ 16 × pffiffiffi fðpÞ ½u¯ γ u v¯ γαv : MM 2 ð2πÞ3 L L L L The MACE experiment is expected to improve the sensi- −5 tivity by at least 2 orders with G ¯ =G ≲ Oð10 Þ, though ð4Þ MM F the magnetic field correction factor is unknown yet and depends strongly on the design of detection systems. Here, the integral of fðpÞ in the module jj2 is the spatial wave function at zero distance so that jj2 ¼ 2 jΨð0Þj =ð2memμÞ. The latter spinor product can be sim- III. THE MUONIUM-ANTIMUONIUM CONVERSION IN SEESAW MODELS plified into 2memμ in terms of their normalizations and spin combinations. Therefore, the mass mixing element Now we explore the charged lepton flavor violation in arrives at the type-II seesaw model with the M − M¯ transition. Especially, we would forecast the prospects of the Ψ 0 2 μα 3 GMM¯ j ð Þj GMM¯ ð Þ MACE experiment to detect such a muonium-to-antimuo- m ¯ ¼ 16 × pffiffiffi ¼ 16 × pffiffiffi ; ð5Þ MM 2 2 2 2π nium conversion and constrain the parameter space in this popular neutrino mass model. memμ with the reduced mass μ¼ ≈me. The mass eigenstates The type-II seesaw model, which is an extension of the meþmμ ffiffiffi ¯ p standard model with a weak-scale triplet Higgs boson, is are then the simple combinationsffiffiffi jM1i¼ðjMiþjMiÞ= 2 ¯ p capable of generating small neutrino masses naturally. First and jM2i¼ðjMi − jMiÞ= 2, and the difference in mass of all, we briefly review the model in which the triplet eigenvalues is given by Δm ≡ m1 − m2 ¼ 2m ¯ .In MM Higgs boson possesses a weak scale mass, concentrating on experiments, muonium oscillation time is much longer how small neutrino masses are produced [25,26]. The than their decay time. So one can only hope to observe the triplet is arranged into an SU 2 scalar multiplet denoted mixing phenomenon by measuring the probability that a ð Þ − by Δ with a hypercharge Y 1, state that starts as a muonium (μþe ) decays as an ¼ pffiffiffi antimuonium, where final states include a high energy ξþ= 2 ξþþ electron and a low energy positron. Therefore, the state that Δ ¼ pffiffiffi : ð9Þ ξ0 −ξþ 2 starts as jMi at t ¼ 0 [denoted it by jMðtÞi] evolves as = follows: The standard model gauge symmetry allows the Yukawa ν T −im1t −im2t interaction between the lepton doublet l ¼ð L;eLÞ and M t M1 M1 M e M2 M2 M e ; 6 j ð Þi¼j ih j i þj ih j i ð Þ the triplet Δ, and will decay as a M¯ with a total probability, 1 L ¼ − ðy Þ l¯cεΔl þ H:c: Z Yuk 2 N ij i j ∞ Δ τ 2 ¯ dt −t=τ ¯ 2 ð m Þ 1 1 P M → M e M M t c 0 c c ð Þ¼ τ jh j ð Þij ¼ 2 1 Δ τ 2 ¼ − ðy Þ ν¯ P ν ξ − pffiffiffi ðν¯ P e þ e¯ P ν Þξþ 0 ð þð m Þ Þ 2 N ij i L j 2 i L j i L j 1 ≈ Δ τ 2 ð m Þ ; ð7Þ − ¯c ξþþ 2 ei PLej þ H:c:; ð10Þ τ where is the muon lifetime. Equation (7) does not take where ðy Þ are Yukawa coupling constants and the Latin into account the effect of static electromagnetic fields in N ij indices i, j represent generations and ε ≡ iσ2. materials, which break the degeneracy mMM ¼ mM¯ M¯ and further suppresses the probability [22–24], an important It is easy to get ðyNÞij ¼ðyNÞji and find that two terms of ξþ are actually equal. The final Yukawa terms for effect in some experiments. In any case, we see from Eq. (7) Δ that the conversion probability is in general very small and arrive at 2 proportional to jG ¯ j , 1 MM L ffiffiffi ξþνc T ξ− ¯ νc Yuk ¼ p ½ LðUν yNÞeL þ eLðyNUνÞ L 2 6 6 2 2 64G α m τ G ¯ 2 PðM → M¯ Þ¼ F e MM 1 π2 G þ ½ξþþec ðy Þe þ ξ−−e¯ ðy Þec F 2 L N L L N L 2 −5 GMM¯ 1 ¼ 2.64 × 10 : ð8Þ 0 c 0 c − ½ξ ν ðy ÞνL þ ξ νLðyNÞν GF 2 L N L 1 − νc ν νc T ν The above formula is the transition probability without the 2 ½ LðmνÞ L þ Lðv3UνyNUν Þ L: ð11Þ magnetic field effect. In the magnetic field, the probability will be reduced by another factor of 0.35 based on Table II In addition to nonzero neutrino masses, the theory predicts of Ref. [20]. It is easy to transform the PSI limit to that on the existence of three neutral, one charged, and one doubly

055023-3 HAN, HUANG, TANG, and ZHANG PHYS. REV. D 103, 055023 (2021) charged physical Higgs scalar particles. Their masses and GMM¯ μ 5 5 H ¼ pffiffiffi ½μγ¯ ð1 − γ Þe½μγ¯ μð1 − γ Þe; with couplings with leptons and quarks depend crucially on the eff 2 mechanism used to break the global U(1) symmetry ðyNÞeeðyNÞμμ associated with the conservation of lepton number L. In G ¯ ¼ pffiffiffi : ð15Þ MM 16 2 2 one realization is that the lepton number L is conserved by mþþ the Higgs potential VðΦ; ΔÞ, while the U(1) global sym- The integrated probability that the muonium Mðμþe−) metry associated with the conservation of L is broken − 0 decay as μ rather than μþ is spontaneously as ξ develops a nonzero vacuum expectation value (VEV). At this time, a linear combination of the 3π2α3 2 6 2 P → ¯ 643 me GMM¯ imaginary parts of the neutral components of the Higgs ðM MÞ¼ 2 : ð16Þ doublet field Φ will act as a physical massless neutral scalar GFmμ mμ GF particle called a Majoron, which is almost ruled out in experiments. So we do not take this case into account. According to the latest experimental upper bound from PSI, P → ¯ ≤ 2 0 105 2 Interestingly, the other possibility is to assign two units of L we can obtain ðM MÞ . × GMM¯ at a 90% con- to the Higgs triplet Δ (LΔ ¼ −2) so that the Yukawa fidence level. coupling terms (l¯cεΔl) would conserve the lepton number. The form of the U(1) symmetry breaking leading to IV. CONSTRAINTS ON SEESAW MODELS ξ0 ≠ 0 h i and L nonconservation is determined by the The seesaw models considered in the present paper are Φ Δ assumed properties of the Higgs potential Vð ; Þ of strongly constrained by the lepton flavor violation (LFV) the theory. The details of Higgs potential and the gauge processes involving the charged lepton sector, such as sector can be found in the Appendix. l → l∓ll l → lγ a b c d and a b . In both type-II seesaw and In addition to the triplet Higgs boson, we also consider hybrid models, the leading-order contribution to la → the possibility that the neutrino masses get contributions ∓ l ll is given by the tree-level diagram induced by the from the other particles. In particular, we introduce a right- b c d doubly charged scalar ξþþ. With the Yukawa couplings handed neutrino ν with additional couplings, R defined in Eq. (10), the corresponding branching ratios are given as follows [27]: L ⊃− ¯ Φν νc ν yνiLi R þ MR R R: ð12Þ † 2 ∓ 1 jðy Þ ðy Þ j B l → l ll N ab N cd ; ð a b c d Þ¼8 1 δ 2 4 ð17Þ Then part of the neutrino mass matrix could be generated ð þ cdÞ GFmþþ from the type-I seesaw mechanism. We take this model as δ the hybrid seesaw model. where the Kronecker delta cd accounts for identical 0 While ξ develops a vacuum expectation value v3, the leptons in the final states. In particular, the most precise total Majorana neutrino masses are channel in this class is provided by the decay μþ → eþe−eþ with the branching ratio given by [28–36] 2 v † 2 m y v3 yν yν : 13 jðy Þμ ðy Þ j ð NÞij ¼ð NÞij þ i j 2 ð Þ B μþ → þ − þ N e N ee MR ð e e e Þ¼ 16 2 4 ; ð18Þ GFmþþ The diagonalized neutrino masses is obtained by unitary which can be compared with the current best upper bound transformations, as follows [37]:

− −12 T Bðμþ → eþe eþÞ ≤ 1.0 × 10 : ð19Þ νL → UννL;Uν mNUν ¼ diagfm1;m2;m3g ≡ mν; ≥ 0 mi : ð14Þ For the type-II seesaw model, the LFV decay process μ → eγ is generated at a one-loop level with the help of the Without a loss of generality, we can work in a basis doubly and singly charged scalars. As a result, the partial – where charged lepton masses are diagonalized. Then Uν is width for this process is given by [27,29,31 36] the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) mixing † 2 2 α jðy y Þ μj 1 8 matrix in charge-current interactions. B μ → γ ≃ N N e ð e Þ 768π 2 2 þ 2 ; ð20Þ Given the Yukawa interaction in Eq. (10), it is easy to GF mþ mþþ draw the leading-order contribution to the muonium- antimuonium conversion. As described in [9], if the usual where α refers to the electromagnetic fine structure con- Yukawa coupling of the lepton is assumed to be diagonal, stant, and in our derivation, we have ignored the depend- we can obtain ence on the internal lepton masses since they are assumed

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2 l l0 1 l ≠ l0 to be much smaller than the mþ and mþþ. On the other where k ¼ for ¼ and k ¼ for P . Assuming ξ B ξ → hand, for the hybrid model, the heavy singlet right-handed , almost all decay to leptons, i.e., l¼e;μ;τ ð 2 neutrino would mix with the SUð ÞL active neutrinos and ll0Þ ≃ 1, we can get the branch ratios, give rise to an additional one-loop contribution to μ → eγ – – 2 [34 36,38 48]. Even though this new LFV mode would be kðy Þ 0 B ξ → ll0 P N ll enhanced by the breaking of the Gloashow-Iliopoulos- ð Þ¼ 2 : ð23Þ l μ τkðy Þ 0 Maina mechanism in the SM [38–48], it can be shown [34– ¼e; ; N ll 36,47,48] that the corresponding amplitude is subdominant compared with those mediated by the charged scalars V. NUMERICAL RESULTS ξþ; ξþþ and thus, can be safely ignored. Therefore, the dominant contribution to μ → eγ remains to be the same as In this section, we present our numerical results by that of the type-II seesaw in Eq. (20). Nowadays, the most sampling in the model parameter space. For simplicity, we stringent constraint on μ → eγ is given by the MEG assume all the model parameters are real. In our scanning, Collaboration with the upper bound as follows [49]: we selected models that predict the neutrino mass squared differences and mixing angles to be within the experimen- Bðμ → eγÞ < 4.2 × 10−13: ð21Þ tally allowed ranges for both normal ordering (NO) and inverted ordering (IO) at the 3σ confidence level obtained Later, we will use it to constrain the parameter spaces of from the existing neutrino oscillation data as shown Table I. both models in our numerical scanning. Although the recently released data from T2K and NOνA δ The seesaw models considered in the present paper also Collaborations prefer a nonzero Dirac CP phase CP, this δ suffer constraints from the μ → e conversion in nuclei, result is still not conclusive. Thus, we still allow CP to be which can arise from both short-range and long-range varied within ½0; 2π. We also take into account the contributions. However, explicit calculations [32,48] have cosmological limit on the sum of the neutrino masses so shown that the current μ → e conversion sensitivity cannot that the lightest neutrino mass is sampled uniformly within place any useful constraints on the parameter spaces, given the range of [0,0.05 eV]. The value of the triplet Higgs already severe limits given by μ → eγ and μþ → eþe−eþ. VEV is adopted to be from 10−2 eV to 1 GeV, while the Furthermore, the lepton number conservation in both mass doubly charged Higgs ξ is allowed to be within models is broken by two units, and the obtained active [100 GeV, 10 TeV]. Furthermore, we require that the neutrino masses are Majorana in nature. Thus, these models selected models should satisfy the stringent bounds for can be constrained by the lepton-number violating proc- cLFV processes, such as μ → eγ and μþ → e−eþeþ.For esses like neutrinoless double beta (0νββ) decays in nuclei, the remaining model parameters, we compare their pre- which so far have not yet been found. In the type-II seesaw dicted muonium-to-antimuonium conversion probability model, there are two contributions to this process: one is the PðM ↔ M¯ Þ with the existing PSI bound and the expected ordinary long-range channel by exchanging the light active MACE sensitivity. In the following, we shall give the final neutrinos ν, while the other is the short-range mode scanning results for the type-II and hybrid seesaw models, mediated by the doubly charged scalar ξ. However, respectively. these channels are severely suppressed either by the tiny light neutrino masses or by the doubly charged scalar mass A. Type II seesaw scale [35,50] and thus, can be neglected. The inclusion of In this case, we assume all of the neutrino mass comes one heavy singlet right-handed neutrino in the hybrid from the triplet Higgs boson, while the contribution from model does not alleviate the problem since the short-range the right-handed neutrino can be negligible. In Fig. 1,we mode induced by it suffers from a significant suppression arising from the extremely small heavy-light neutrino TABLE I. The allowed ranges of the neutrino mass square mixing. In summary, both models are insufficient to differences and mixing angles at 3 σ confidence level for normal generate observable 0νββ signals and thus, cannot be ordering and inverted ordering used in our numerical analysis 2 2 constrained by the limits from the existing experiments, [54]. Note that Δm3l ≡ Δm31 > 0 for normal ordering and 2 2 like KamLAND-Zen [51] and GERDA [52]. Δm3l ≡ Δm32 < 0 for inverted ordering. The search for doubly charged Higgs boson has been presented at the LHC with the ATLAS detector [53],in Normal ordering Inverted ordering ξ → 2 which the analysis focused on the decays e e , sin θ12 0.269–0.343 0.269–0.343 2 ξ → e μ , and ξ → μ μ . The partial decay width of sin θ23 0.407–0.618 0.411–0.621 2 ξ to leptons is given by sin θ13 0.02034–0.02430 0.02053–0.02436 Δ 2 m21 6.82–8.04 6.82–8.04 2 10−5 eV2 ðy Þ 0 Δ 2 0 N ll m3l þ2.431– þ 2.598 −2.583– − 2.412 Γðξ → l l Þ¼k m ; ð22Þ −3 2 16π þþ 10 eV

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FIG. 1. Upper: Scatter plots of the doubly charged Higgs mass mþþ and the conversion probability P without magnetic field correction in the type-II seesaw with v3 ¼ 0.2 eV (blue points) and v3 ¼ 1 eV (red points) for normal ordering (left) and inverted ordering (right). 800 Lower: Scatter plots of the VEV v3 and the conversion probability P in the type-II seesaw with mþþ ¼ GeV (blue points) and 3 mþþ ¼ TeV (red points) for normal ordering (left) and inverted ordering (right). The PSI bound is transformed into the one at zero magnetic field at 90% C.L. All the plotted points are allowed by the flavor and collider constraints. show the scan result by requiring the neutrino mass conversion probability less than 10−15, which is beyond the satisfying the observation. Since there is an ambiguity in reach of the future MACE experiment. the neutrino mass ordering, we split our analysis in these two cases. The left panels show the preferred parameter space for B. Hybrid seesaw neutrino mass normal ordering and the right panels that of inverted ordering. For the top two plots, we fixed the vacuum As shown in previous model, the strongest limit arises value of the triplet Higgs boson at 1 eV (red) and 0.2 eV from the cLFV processes, such as μþ → eþγ and μþ → − þ þ (blue). We find for v3 ¼ 1 eV, the doubly charged Higgs e e e . The reason is that from the PMNS matrix boson should be heavier than around 1.5 TeV to evade we can see the neutrino mixing is relatively large, and large the limit from flavor constraints, particularly from off diagonal terms in the neutrino mass matrix are needed. þ − þ þ μ →e e e . In the case of v3 ¼ 0.2 eV, the limit on Since the neutrino mass matrix mainly originates from the the doubly charged Higgs boson is stronger since a larger triplet Higgs boson, it requires a large flavor changing Yukawa coupling is needed to generate the neutrino mass for coupling between the triplet Higgs boson and the leptons, a smaller vacuum value of Δ. We note that the preferred inducing a large flavor changing effect. In the hybrid seesaw parameter space behaves as a band due to the uncertainty model, the flavor constrains could be weaker due to the of the neutrino mass parameter. Finally, we find all the contribution of off diagonal terms from a right-handed survived parameter space predicts a muonium-antimuonium neutrino. Specifically, here we consider that all off diagonal

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FIG. 2. Upper: Scatter plots of the doubly charged Higgs mass mþþ and the conversion probability P without magnetic field correction in the hybrid seesaw model with v3 ¼ 0.2 eV (blue points) and v3 ¼ 1 eV (red points) for normal ordering (left) and inverted ordering 800 (right). Lower: Scatter plots of the VEV v3 and the conversion probability P in the hybrid seesaw model with mþþ ¼ GeV (blue 3 points) and mþþ ¼ TeV (red points) for normal ordering (left) and inverted ordering (right). The PSI bound is transformed into the one at zero magnetic field at 90% C.L. All the plotted points are allowed by the flavor and collider constraints. terms of the neutrino mass matrix originate from the right- VEV v3 region. Due to the relax of cLFV constraints in this handed neutrino; thus, all of Yukawa couplings yν are fixed. hybrid model, the lower bounds on the doubly charged Hence, all the flavor constraints could be removed1. Higgs mass for both NO and IO cases around 600 GeV are The final scanning results are shown in Fig. 2. As before, given by the collider searches for ξ. As a result, it is seen the upper two scatter plots show the muonium-to-anti- that MACE can increase the detection region considerably. ¯ muonium conversion probability PðM ↔ MÞ as a function Concretely, for v3 ¼ 0.2 eV, MACE can detect a doubly of the doubly charged Higgs mass mþþ for both NO (left charged Higgs boson as heavy as 3 TeV. On the other hand, panel) and IO (right panel), while the lower two ones for a fixed m , MACE has an ability to measure the ¯ þþ correspond to parameter spaces in PðM ↔ MÞ vs v3 plane. parameter region with v3 < 1 eV for both orderings, which Compared with the previous type-II seesaw case, it is clear is almost 1 order more sensitive than the PSI detector. The that the allowed parameter spaces for the hybrid seesaw study in this subsection demonstrates the advantage and model are enlarged greatly, extending aggressively into the the necessity of MACE in the probing and constraining the low doubly charged Higgs mass region and the low triplet triplet scalar in the seesaw models.

VI. SUMMARY AND CONCLUSION 1The contribution of the flavor changing observable from the right-handed neutrino can be safely removed assuming the right- In the present paper, we have discussed prospects handed neutrino is heavy enough. of the proposed MACE experiment to search for the

055023-7 HAN, HUANG, TANG, and ZHANG PHYS. REV. D 103, 055023 (2021) muonium-to-antimuonium conversion process in the type- APPENDIX A: HIGGS POTENTIAL II and hybrid seesaw models, the latter of which is an We consider the general Higgs potential for a doublet extension of the type-II seesaw by including a single heavy 0 Φ ¼ðhþ;h ÞT and a triplet Higgs Δ [55], right-handed neutrino. Note that the leading-order contributions to the M − M¯ conversion probability in both Φ Δ 2 Φ†Φ λ Φ†Φ 2 2 Δ†Δ models are induced at tree level by the doubly charged Vð ; Þ¼m ð Þþ 1ð Þ þ MΔTrð Þ 2 λ Δ†Δ 2 λ ΔΔ Δ†Δ† scalar originated in the SUð ÞL triplet scalar. Thus, MACE þ 2½Trð Þ þ 3Trð ÞTrð Þ; can effectively measure the parameter space related to this † † † † þ λ4ðΦ ΦÞ½TrðΔ ΔÞ þ λ5ðΦ Δ ΔΦÞ triplet scalar. By utilizing the high-quality slow muon 1 sources at CSNS, together with other significant advances λ Φ†ΔΔ†Φ μ ΦTεΔ†Φ þ 6ð Þþ2 ð ÞþH:c: ðA1Þ in the detector technologies, MACE has been shown to be able to improve the sensitivity to the conversion probability Φ Δ by almost 3 orders of magnitude compared with the The breaking can be explicit when V( , ) is supposed to ΦTΔΦ existing PSI experiment accomplished more than 20 years contain the trilinear L-nonconserving term ( ). In this ago. Therefore, a large uncharted model parameter space is case, the masses of the new Higgs particles MΔ in the model can be arbitrary large. Collect the following soft expected to be probed by MACE, which is complementary Δ to direct collider searches, neutrino oscillations, and other lepton number violating trilinear interaction between and Φ cLFV channels such as μþ → eþγ and μþ → e−eþeþ.Asa from the Higgs potential: result, it is shown that MACE is not sensitive to the 1 parameter space of interest surviving from the collider and L − μΦTεΔ†Φ soft ¼ 2 þ H:c:; flavor limits. On the other hand, for the hybrid seesaw 1 pffiffiffi model, MACE can reach unexplored parameter region and ¼ − μ½ðhþÞ2ξ−− − 2hþh0ξ− − ðh0Þ2ξ0† thus provides useful information on the triplet scalar 2 complementary to other cLFV experiments like MEG-II þ H:c:; ðA2Þ and Mu3e. In other words, if no positive signals are observed by MACE in the future, we shall be able to set where μ is an undetermined parameter. The minimum of a much more stringent upper bound on the mass of the Higgs potential is presented in the following: doubly charged scalar around 1.2 TeV for v3 ¼ 1 eV in the case of inverted ordering, which will surpass the corre- 1 2 2 2 2 1 4 4 1 2 2 V Φ; Δ m v M v λ1v λ2v λ4v v sponding LHC limits even by taking into account the minð Þ¼2 1 þ Δ 3 þ 4 1 þ 3 þ 2 1 3 uncertainties involving the neutrino mass ordering or the 1 1 λ 2 2 μ 2 VEV of the triplet scalar. Even for the NO, the prospect of þ 2 6v1v3 þ 2 v1v3: ðA3Þ MACE can reach a doubly charged Higgs boson as heavy as 3 TeV for v3 ¼ 1 eV. In summary, with the advent of After the spontaneous symmetry breaking (SSB), solving ∂ Φ Δ MACE, the muonium-to-antimuonium conversion will Vminð ; Þ the equation ∂ ¼ 0 leads to become one of golden channels to study the cLFV physics, vi which, together with other flavor and collider searches, will 0 v1 0 hh i¼pffiffiffi ; hξ i¼v3; shed light on the mystery of the neutrino masses. 2 2 μ 2 ACKNOWLEDGMENTS 2 − m − v1 v1 ¼ ;v3 ¼ 2 2 2 ; ðA4Þ λ1 2ð2M þ v λ4 þ v λ6Þ This work was supported in part by National Natural 1 1 Science Foundation of China under Grant No. 12075326, and 2 2 where μ < 0, m < 0 and M > 0. When jμj is small Guangdong Basic and Applied Basic Research Foundation Δ enough compared with the weak scale, the smallness of under Grant No. 2019A1515012216, Original innovation neutrino masses is attributed to the tiny μ, which is project of Chinese Academy of Sciences (CAS) under Grant estimated at the eV scale in the case, where y ∼ Oð1Þ No. ZDBS-LY-SLH009 and CAS Center for Excellence N and MΔ ∼ v1. For completeness, the charged Higgs masses in Particle Physics (CCEPP). JT appreciates fruitful are listed below, discussions with the accelerator group of EMuS. C. H. is supported by SYSU startup funding. D. H. is supported by 1 λ λ National Science Foundation of China (NSFC) under Grant 2 λ 5 þ 6 2 mþ ¼ MΔ þ 2 4 þ 2 v1 ðA5Þ No. 12005254. Y.Z. is supported by National Science Foundation of China (NSFC) under Grants No. 11805001 1 and No. 11935001. We finally acknowledge Dr. Sampsa 2 2 m ¼ M þ ðλ4 þ λ5Þv : ðA6Þ Vihonen’s help in English proofreading. þþ Δ 2 1

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a a APPENDIX B: GAUGE SECTOR Dμξ ¼ ∂μ − ig1YξBμ − ig2AμT 0 1 μ 3 þþ þ þ Now let us move to gauge couplings DμξD ξ. Since ð∂μ −iBμg1 − iAμg2Þξ − ig2Wμ ξ B C there is a triplet scalar, we have to use a three-dimension B þ − þþ þ 0 C ¼ ð∂μ −iBμg1Þξ − ig2Wμ ξ − ig2Wμ ξ ðB3Þ representation of the SU(2) group, @ A 3 0 − þ ð∂μ −iBμg1 þ iAμg2Þξ − ig2Wμ ξ 0 1 1 0 1 0 pffiffi 0 0 − piffiffi 0 2 2 B C B C 1 1 B pffiffi 0 pffiffi C B piffiffi 0 − piffiffi C T1 ¼ @ 2 2 A;T2 ¼ @ 2 2 A; 0 1ffiffi 0 0 iffiffi 0 1 p p 1 2 2 2 Wμ ¼ pffiffiffi ðAμ ∓ iAμÞ 0 1 2 10 0 B C ≡ θ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffig1 B 00 0C sw sin w ¼ 2 2 T3 ¼ @ A ðB1Þ g1 þ g2 00−1 ≡ θ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffig2 cw cos w ¼ 2 2 0 pffiffiffi 1 g1 þ g2 0 2 0 3 0 θ θ B ffiffiffi C Aμ ¼ Zμ cos w þ Aμ sin w þ 1 2 p T ¼ T þ iT ¼ @ 00 2 A; 0 Bμ ¼ −Zμ sin θ þ Aμ cos θ : ðB4Þ 00 0 w w 0 1 000 ffiffiffi − 1 2 B p C After spontaneous symmetry breaking, Δ develops a T ¼ T − iT ¼ @ 2 00A ðB2Þ pffiffiffi vacuum expectation value. If physical vector bosons are 0 2 0 introduced, we have

0 1 −μ − 0μ μ μ 0μ μ −− iW ξ g2 þ½ig2ðZ c þ A s Þþig1ðA c − Z s Þþ∂ ξ B w w w w C μ ξ ξ μ − þμ −− −μ 0 μ 0μ − D ð þh iÞ ¼ @ iA ξ cwg1 þ ig2ðW ξ þ W ξ Þþð∂ − iZ g1swÞξ A: ðB5Þ þμ − 0μ μ μ 0μ μ 0 iW ξ g2 þ½−ig2ðZ cw þ A swÞþig1ðA cw − Z swÞþ∂ ξ

They will contribute to gauge boson masses,

2 þ −μ 2 2 0 0μ 2 2 2 2 2 jDμhξij ¼ Wμ W g2v3 þ ZμZ ðcwg2 þ 2cwg1swg2 þ g1swÞv3 0 μ 2 2 2 2 2 2 2 2 2 þ ZμA ð−g1g2cw − g1swcw þ g2swcw þ g1g2sw − g1g2cw − g1swcw þ g2swcw þ g1g2swÞv3 μ 2 2 2 2 2 þ AμA ðcwg1 − 2cwg2swg1 þ g2swÞv3 2 2 2 2 þμ − g2v3 0μ 0 ¼ g2v3W Wμ þ 2 Z Zμ: ðB6Þ cos θw

In addition, the traditional SM contains gauge mass terms, As jμj is small enough so that v3 ≪ v1, ρ ≈ 1 meets the requirement from electroweak precision experiments 1 1 g2v2 L 2 2 þ −μ 2 1 0 0μ whereas detectable lepton flavor violating processes are GM ¼ 4 g2v1Wμ W þ 8 2 ZμZ ; ðB7Þ μ 0 cos θw expected. Since the lepton number is restored for ¼ ,it may be natural to have a small μ as a consequence of a tiny which violates the custodial symmetry such as lepton number violation. In fact, there is a tree-level lepton flavor violating process e−μþ → eþμ− mediated by ξþþ 2 1 2 2 2 2 2 c þþ M g2v1 þ g2v3 v due to the interaction term e¯ P e ξ [9] just as the related ρ W 4 1 − 3 : B8 i L j ¼ 2 2θ ¼ 1 2 2 2 2 ¼ 1 2 2 2 ð Þ term of Eq. (10). MZcos w 4 g2v1 þ g2v3 4 v1 þ v3

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