Sensitivity of future lepton colliders and low-energy experiments to charged lepton flavor violation from bileptons

The observation of charged lepton flavor violation is a clear sign of physics beyond the Standard Model (SM). In this work, we investigate the sensitivity of future lepton colliders to charged lepton flavor violation via on-shell production of bileptons, and compare their sensitivity with current constraints and future sensitivities of low-energy experiments. Bileptons couple to two charged leptons with possibly different flavors and are obtained by expanding the general SM gauge invariant Lagrangians with or without lepton number conservation. We find that future lepton colliders will provide complementary sensitivity to the charged-lepton-flavor-violating couplings of bileptons compared with low-energy experiments. The future improvements of muonium-antimuonium conversion, lepton flavor non-universality in leptonic {\tau} decays, electroweak precision observables and the anomalous magnetic moments of charged leptons will also be able to probe similar parameter space.


I. INTRODUCTION
The observation of neutrino oscillations and thus non-zero neutrino masses clearly established the existence of lepton flavor violation in the neutrino sector. We also expect the existence of charged lepton flavor violation (CLFV) which occurs in short-distance processes without neutrinos in the initial or final state. In the Standard Model (SM) with three massive neutrinos, the rates of CLFV processes are suppressed by G 2 F m 4 ν 10 −50 due to the unitarity of the scattering matrix and thus beyond the sensitivity of any current or planned experiments. Hence, the observation of any CLFV process implies the existence of new physics beyond the SM with three massive neutrinos. The CLFV is predicted by many different new physics models (see Refs. [1,2] for recent reviews), including neutrino mass models such as the inverse seesaw model [3] and radiative neutrino mass models [4].
It may also arise in other extensions of the SM such as the multi-Higgs doublet models [5] or the minimal supersymmetric SM via gaugino-slepton loops with off-diagonal terms in the slepton soft mass matrix [2,6].
As CLFV induces rare processes, they are generally searched at low-energy experiments with high intensity. See Refs. [7,8] for a list of constraints on the effective CLFV operators obtained from several low-energy precision measurements. The CLFV processes may also be searched for at high-energy colliders. The Large Electron-Positron Collider (LEP) sets upper limits on the branching ratio of Z boson rare decays [9], i.e. Z → induced by loop diagrams, and still provides the most stringent constraint on the branching ratios of Z → τ e(µ) as 0.98(1.2) × 10 −5 up to now. The ATLAS experiment currently sets the most stringent limit of 7.3 × 10 −7 on BR(Z → eµ) [10] and comparable limit of 5.8(2.4) × 10 −5 on BR(Z → τ e(µ)) [11]. The future Z factories could improve the sensitivity by about four orders of magnitude [12]. The CLFV can also occur in Higgs boson decay through the dimension-6 operator H † HLe R H in SM effective field theory [13]. The Large Hadron Collider (LHC) recently improved its limit on the effective τ couplings to the level of (1 − 2) × 10 −3 [14,15] and the proposed Higgs boson factories are expected to be sensitive to CLFV couplings down to the order of 10 −4 [12].
Besides these rare decays, CLFV can also be probed through scattering processes at colliders. A hadron collider is sensitive to effective operators with two colored particles and two leptons in processes such as qq → [16] and gg → [17,18]. A lepton collider may probe effective operators with four charged leptons via e + e − → . These searches can also be interpreted in terms of simplified models. CLFV processes at the lepton collider can be described by seven bileptons [19], which are scalar or vector bosons coupled to two leptons via a renormalizable coupling. In particular, off-shell bileptons can mediate the processes e + e − → whose potential observation at future lepton colliders has recently been studied by us [20]. See also Ref. [21] and Refs. [22,23] for related studies of electroweak doublet and triplet scalar bileptons, respectively. Another promising probe for CLFV is through the on-shell production of a bilepton X together with two charged leptons with different flavors, i.e. e + e − → X . This production scenario only depends on a single CLFV coupling in each production channel and thus can be directly compared with other constraints. On-shell production has been studied in Ref. [21] for an electroweak doublet scalar and Refs. [22,23] for an electroweak triplet scalar.
The main aim of this work is to explore the sensitivity reach to the CLFV couplings for all seven bileptons through the on-shell production of a bilepton X in association with two charged leptons at proposed future lepton colliders. We compare the sensitivities of future lepton colliders with the existing constraints and future sensitivities of other experiments.
Currently, the most relevant constraints are from the anomalous magnetic moments (AMMs) of electrons and muons, muonium-antimuonium conversion, lepton flavor universality (LFU) in leptonic τ decays, W boson mass measurement, and previous collider searches at the LEP and the LHC experiments. Our analysis here goes beyond the previous work by extending the study to all possible bileptons and including additional constraint from the violation of lepton flavor universality in leptonic τ decays and W boson mass measurement as well as a discussion of neutrino trident productions. We also improve the calculation of muoniumantimuonium conversion for the bileptons.
The paper is outlined as follows. In Sec. II we describe the general SM extensions with CLFV couplings. Then we discuss the relevant existing constraints on the CLFV couplings in Sec. III. In Sec. IV we present the sensitivity of neutrino trident production, future lepton colliders, and a new state-of-the-art muonium-antimuonium conversion experiment to the CLFV couplings of bileptons and compare it with the existing low-energy constraints. Our conclusions are drawn in Sec. V.

II. GENERAL LAGRANGIAN FOR CHARGED LEPTON FLAVOR VIOLATION
In this work we consider all possible 1 scalar and vector bileptons with possible CLFV couplings [19]. They are obtained by expanding the most general SM gauge invariant Lagrangian in terms of explicit leptonic fields. The bileptons fall in two categories depending whether they carry lepton number L or not. The most general SM invariant Lagrangian of ∆L = 0 bileptons has four terms where to the presence of CLFV processes. Below we focus on the off-diagonal elements of the couplings which induce CLFV on-shell production of a bilepton X with two different flavor charged leptons, although we present the general results for all possible bilepton interactions.
Models with new massive vector bosons generally require the introduction of a new Higgs boson with the exception of an Abelian vectorial symmetry where the mass of the gauge boson can be generated via the Stückelberg mechanism [32]. This may lead to new contributions mediated by the components of the new Higgs boson. However, the processes which we are considering do not suffer from any theoretical problems like the violation of perturbative unitarity. Thus, to remain as model-independent as possible, we will restrict ourselves to the Lagrangians in Eqs. (1) and (2) for the rest of the paper.

III. CONSTRAINTS
In this section we summarize relevant constraints on the CLFV couplings from anomalous magnetic moments of leptons, muonium-antimuonium conversion, lepton flavor universality in leptonic τ decays, W boson mass measurement, and the existing collider searches. Note that, although we give the analytical results for general coupling matrices, in this and the following sections we assume all the CLFV couplings are real and symmetric in our numerical analysis.

A. Anomalous magnetic moments
The muon magnetic dipole moment has ∼ 3.7σ discrepancy between the SM prediction [33,34] and experimental measurements [9,35] For the electron g−2, Refs. [36,37] recently presented a precise measurement with a 2.4σ Apparently, the muon (electron) AMM requires a positive (negative) new physics contribution to explain the discrepancy between the theoretical SM prediction and the experimental to leading order in the charged lepton mass. For the scalars ∆ 1,3 and H 2 , the new AMMs are given by to leading order in the charged lepton masses. One can see that, apart from H 2 and H 3 , each new contribution to the anomalous magnetic moment has a definite sign. For H 2 in the limit of degenerate scalar masses m h 2 = m a 2 , the anomalous magnetic moment is not enhanced proportional to the mass of the τ lepton in the loop and the contribution obtains a definite negative sign Similarly, if all scalars apart from the CP-even neutral scalar h 2 are decoupled, the contri- anomalies. In this work we do not attempt to explain the deviations from the SM but rather derive a constraint on the LFV couplings described in Eqs. (1) and (2). In order to derive a constraint, we demand that the new physics contribution deviates from the experimental observation by at most 3σ for the electron and 4σ for the muon in order to account for the discrepancies in both measurements. The constraints from the AMMs are summarized in Table I.

B. Muonium-antimuonium conversion
Muonium is the bound state of µ + and e − and antimuonium is that of µ − and e + . If there is a mixing of muonium (M = (µ + e − )) and antimuonium (M = (µ − e + )), the lepton flavor conservation of electron and muon must be violated and thus it is a sensitive probe for CLFV.
The probability of muonium-antimuonium conversion has been firstly calculated in Refs. [39,40]. Following the discussions in Refs. [39][40][41][42], we use the density matrix formalism to calculate the probability of muonium to antimuonium conversion. In contrast to previous calculations [42], we include off-diagonal elements in the Hamiltonian H MM which mediates muonium-antimuonium conversion, and expand to the first order in the interaction Hamiltonian H MM . This is generally a good approximation for B 0.1 µT assuming at most weak-scale interaction strength for H MM .

Muonium is described by the Hamiltonian
where H 0 denotes the non-relativistic Hamiltonian for a hydrogen-like system, i.e. a bound state of two particles via a Coulomb interaction. The hyperfine splitting of the 1s state is [43,44], where S e,µ are the spins of the electron and muon, respectively. Finally, H Z = −( µ e + µ µ ) · B describes the Zeeman effect with external magnetic field B. The magnetic moments for electron and muon are defined as µ e = −g e µ B S e and µ µ = g µ µ B S µ m e /m µ with two g-factors g e,µ 2 (1 + α/2π) and the Bohr magneton µ B = e/2m e .
In the uncoupled basis |↑↑ , |↑↓ , |↓↑ , |↓↓ , the Hamiltonian can be expressed as in terms of the ground state energy E 0 = −α 2 m red /2, the fine structure constant α, the reduced mass of the two-body system m red = m e m µ /(m e + m µ ) m e and two functions which parameterize the Zeeman effect. The energy eigenstates with their eigenenergies are thus given by and the mixing is described by For a vanishing magnetic field s = c = 1/ √ 2 and thus λ The interaction Hamiltonian H MM inducing the muonium-antimuonium conversion may have different forms. We are particularly interested in the following vector and scalar inter-actions with equal and opposite chirality leptons where the matrix representation on the right-hand side is in the basis λ The corresponding interaction Hamiltonians in the |λ i basis are Diagrams contributing to the muonium-antimuonium conversion by the Lagrangians in Eqs. (1) and (2). with Note the non-trivial dependence on the magnetic field in case of the electroweak doublet scalar H 2 . In the limit of degenerate scalar masses m h 2 = m a 2 , the effective Lagrangian and the interaction Hamiltonian induced by H 2 simplify to which are consistent with our results in Ref. [20].
The probability to observe a µ + decay instead of a µ − decay starting from an unpolarized muonium is is the density matrix of the initial state muonium and PM = i λ is the projection operator onto the final state antimuonium. For the interaction Hamiltonians of interest, there is only mixing between the second and third state as seen from Eqs. (17) and (18), and thus the probability can be explicitly written as and in particular for vanishing magnetic field, B = 0, we find We compared our result with the analytic expression in Ref. [42] and the numerical values in Table II of Ref. [45] and found good agreement numerically, although the contributions from the off-diagonal entries in the interaction Hamiltonians H MM were not included in Ref. [42]. These additional contributions vanish given no external magnetic field B and are generally subdominant at finite external magnetic field. They are suppressed by the factor of γ 2 /(γ 2 + b 2 (1 + X 2 )) compared with the dominant contribution, because the weak decay rate γ is much smaller than the hyperfine splitting and the Zeeman effect, i.e. γ b, bX, bY .
Typically there are magnetic fields in the experimental setup. They suppress the conversion probability, because the degeneracy of the energy levels in M andM is lifted. For the Hamiltonians with same chirality vector currents H LL(RR) and opposite chirality vector currents H LR , the suppression factors of the probability at a finite magnetic field B are For the Lagrangians described in Eqs. (1) and (2), we obtain their probabilities as follows where A and C are defined in Eq. (27). Note the non-trivial dependence of P (H 2 ) on the magnetic field. For real symmetric Yukawa couplings, the probability for H 2 can be written as which simplifies to The search for muonium-antimuonium conversion at the Paul Scherrer Institut (PSI) placed a constraint on the probability to observe the decay of the muon in antimuonium decay instead of the decay of the antimuon in muonium with a magnetic field of B = 0.1 T, that is P (B = 0.1 T) ≤ 8.3 × 10 −11 [45]. This bound can be used to obtain the constraints on the CLFV couplings of the bileptons which we summarize in Table II.

C. Lepton flavor universality
The interactions of leptons with neutrinos lead to new contributions to effective operators with two leptons and two neutrinos. In the absence of light right-handed neutrinos, there are only two types of effective operators The second term in the expression for g LL,SM originates from W boson exchange, while the other ones are due to Z boson exchange. The new physics contributions to the two different sets of Wilson coefficients are given by and thus lepton flavor universality in lepton decays provides an interesting probe to the CLFV interactions for the bileptons. The relevant decay width for 1 → 2 ν 1ν2 is [46][47][48][49][50]   where m 1,2 is the mass of the charged lepton 1,2 and α(µ) is the running fine structure constant at scale µ with the result at one-loop order as The most sensitive probes of LFU are the ratios Thus, at the 2σ level, the relevant constraints are −0.0015 < Re(g τ µµτ LL,N P − g τ eeτ LL,N P ) < 0.0049 , −0.0017 < Re(g τ eeτ LL,N P − g µeeµ LL,N P ) < 0.0039 .
Assuming the dominance of a single operator at a time, these bounds can be translated to the constraints on the CLFV couplings as shown in Table IV. Only the couplings of H 0 1 and the charged components of H 3 and ∆ 3 can be constrained as they contribute to the g LL coefficients.

D. W boson mass
A modification of the Fermi constant G F extracted from muon decay leads to a change in the theoretical prediction of the W boson mass, which has been measured very precisely to be m W,exp = (80.379 ± 0.012) GeV [9]. The current SM prediction for the W boson mass is m W,SM = (80.363 ± 0.020) GeV [9]. Adding the errors in quadrature, we obtain i.e. the experimental measurement of the W boson mass is consistent with the SM prediction at 1σ. We follow Ref. [51] and obtain the correction to the SM prediction of the W boson mass in the (G F , m Z , α) scheme where we expressed the Weinberg angle in terms of the input parameters. For H 0 1 and ∆ 3 the correction is always negative and moves the theory prediction further away from the experimental measurement, while the H 3 contribution may have either sign. Using the input values in Table III, we obtain  Table V, assuming there is only one bilepton at a time.

E. Existing collider constraints
The DELPHI collaboration interpreted their searches for e + e − → + − in terms of 4lepton operators [52] which are defined by the effective Lagrangian where Λ denotes the scale of the effective operator, g is the coupling and η ij parameterizes which operators are considered at a given time and the relative sign of the operators in order to distinguish constructive (destructive) interference with the SM contribution. Conservative limits on the new physics scalars are obtained by setting the coupling to g 2 = 4π and are summarized in the Table 30 of Ref. [52].   As we have demonstrated in Ref. [20], the analysis of contact interactions in Ref. [52] does not directly apply to ∆ ++ 2µ , because the induced effective interactions do not fall into any of the types of effective interactions considered in Ref. [52]. Similarly for H 2 , the analysis only applies in the limit of degenerate neutral (pseudo)scalar masses (m h 2 = m a 2 ) and in the absence of one of the diagonal entries y ee, There are searches for doubly-charged scalars produced via electroweak pair production in both ATLAS and CMS experiments. The most stringent limits for decays to e ± e ± , µ ± µ ± , e ± µ ± pairs are set by the ATLAS experiment [55]. It excludes masses m ∆ ++

IV. SENSITIVITY OF FUTURE EXPERIMENTS TO CLFV
A. Sensitivity from neutrino trident production Neutrino trident production, the production of a charged lepton pair from a neutrino scattering off the Coulomb field of a nucleus, provides an interesting signature to search for new physics beyond the SM [57][58][59]. So far, only the muonic trident has been measured with the results of σ exp /σ SM = 1.58 ± 0.64 at CHARM-II [60], σ exp /σ SM = 0.82 ± 0.28 at CCFR [57] and σ exp /σ SM = 0.72 +1.73 −0.72 at NuTeV [61]. While CHARM-II and CCFR achieved an accuracy of the level of 35% [62], their measurements agree with the SM prediction and no signal has been established at NuTeV. This will be improved by a measurement at the near detector of the Deep Underground Neutrino Experiment (DUNE), which can reach an accuracy of 25% [62]. See also Ref. [63] for a related study. The DUNE near detector is expected to measure three neutrino trident channels: ν µ N → ν µ e + e − N , ν µ N → ν µ µ + µ − N and ν µ N → ν e e + µ − N . The third one is not sensitive to new physics in a scheme where the Fermi constant G F is determined by muon decay, as it is directly related to muon decay by crossing symmetry. We calculate the cross sections of the former two channels in presence of the new contributions to the effective operators in Eq. (40), using the code provided by Ref. [62]. Assuming a precision of 25% for the cross section measurements, one can translate the expectations of the Wilson coefficients in Eqs. (42) and (43) into the sensitivities to the CLFV couplings quoted in Table VII. Note that all new physics contributions to the trident process ν µ N → ν µ e + e − N in principle result in two disconnected allowed regions of parameter space if no signal is observed at DUNE.
However, some of them are not accessible by interactions of the bileptons and only one of the two regions is theoretically reasonable. We find the two reasonable regions for H 0 1 and The CLFV channels via ∆L = 0 or ∆L = 2 interaction at an e + e − collider for probing the couplings y ij , λ ij are given in Table VIII. The processes with one electron or position in final states occur through both s and t channels mediated by Z/γ * . The processes without e ± in final states only occur in s channel.
In order to estimate the lepton collider sensitivity to the CLFV couplings, we create UFO model files using FeynRules [68] and interface them with MadGraph5 aMC@NLO [69] to generate signal events. We apply basic cuts p T > 10 GeV and |η| < 2.5 on the leptons in final states and assume 10 discovered signal events. For simplicity, the new bosons are assumed to be reconstructed 100% and their decay branching fraction is taken to be unity. If the branching fraction is 10%, the sensitivity reach to the CLFV couplings will be weakened by a factor 3.2. We assume a tau efficiency of 60% [66] and thus the sensitivity reach is weakened by a factor 1.3 for the channels with one tau lepton in the final state compared with the reach for the eµ channel.
We show the sensitivity to ∆L = 0 and ∆L = 2 couplings in Figs. 3 and 4, respectively.
Note that in this work we do not expect to distinguish the chiral nature of the couplings of the mediating particles. Thus, the following results for vector H 0 1,3 and scalar ∆ ++ couplings. The constraints from the lepton AMMs vary with different mass spectra of the new particles and is relatively weak unless there is only one visible neutral scalar in the H 2 case. Finally, the LEP constraints from e + e − → + − scattering are generally weaker than the constraints from low-energy precision experiments. The neutrino trident cross section measurement at the DUNE near detector is not expected to be able to probe new parameter space.
A future dedicated muonium-antimuonium conversion experiment may be able to improve the sensitivity to the Wilson coefficient of the effective operator by one order of magnitude [70]. This directly translates to an improvement in sensitivity by one order of magnitude compared with the constraints listed in Table II or about a factor of 3 in terms of the CLFV couplings as shown in Fig. 3. Note, although muonium-antimuonium conversion can not probe the CLFV couplings of ∆L = 2 bileptons, it is sensitive to combinations of flavor-diagonal couplings.

V. CONCLUSION
We studied the sensitivity of on-shell production of a bilepton with charged-lepton-flavor- Future experiments will improve the sensitivity to several of these observables. In particular, we expect that a future muonium-antimuonium conversion experiments will lead to a factor of 3 improvement for the y eµ coupling. Furthermore, the measurement of neutrino trident scattering at the DUNE near detector (and other neutrino detectors) will provide independent probes. Despite of the expected success and the increase in sensitivity of low-energy precision experiments, the search for on-shell production of a bilepton at future lepton colliders will provide a complementary probe of CLFV couplings. The FCC-ee with the highest integrated luminosity is the most sensitive machine and CEPC is the second most sensitive one in the low mass region. The CLIC and ILC with larger c.m. energy can probe the high mass region for the new bileptons.
In summary, future lepton colliders provide complementary sensitivity to the CLFV couplings of bileptons compared with low-energy experiments. The future improvements of muonium-antimuonium conversion, lepton flavor universality in leptonic τ decays, the W boson mass determination and the anomalous magnetic moments of charged leptons will probe similar parameter space.