Lepton Flavor Violation Induced by a Neutral Scalar at Future Lepton Colliders

Many new physics scenarios beyond the Standard Model often necessitate the existence of a (light) neutral scalar $H$, which might couple to the charged leptons in a flavor violating way, while evading all existing constraints. We show that such scalars could be effectively produced at future lepton colliders, either on-shell or off-shell depending on their mass, and induce lepton flavor violating (LFV) signals, i.e. $e^+ e^- \to \ell_\alpha^\pm \ell_\beta^\mp (+H)$ with $\alpha\neq \beta$. We find that a large parameter space of the scalar mass and the LFV couplings can be probed, well beyond the current low-energy constraints in the lepton sector. In particular, a scalar-loop induced explanation of the longstanding muon $g-2$ anomaly can be directly tested in the on-shell mode.

Introduction.-The observation of neutrino oscillations [1] suggests that the lepton family numbers are violated. It also calls for an extension of the Standard Model (SM) to include neutrino mass terms, which necessarily induce charged lepton flavor violation (cLFV). In the minimal extension of the SM with Dirac neutrinos, cLFV rates are highly suppressed due to small neutrino masses (compared to the electroweak scale). This makes the experimental searches for cLFV all the more interesting, because any observable effect must come from physics beyond the minimally extended SM related to the origin of neutrino mass.
There are various theoretical models of new physics which lead to cLFV effects at an observable level [2,3]. They generally involve extending the Higgs sector, which allows flavor-violating Yukawa couplings of new neutral scalars beyond the SM. In particular, if any of the new neutral scalars (call it H) is (almost) hadrophobic, it could remain sufficiently light and contribute sizably to cLFV, while easily evading the direct searches at hadron colliders, as well as the low-energy quark flavor constraints, such as the rare flavor-changing decays and oscillations of K and B mesons. Some well-motivated examples include supersymmetric models with leptonic Rparity violation [4], left-right symmetric models [5], mirror models [6], and two-Higgs doublet models [7], where the cLFV coupling might arise at tree or loop level [8].
In this letter, we show that such scenarios of neutral scalar-induced cLFV can be effectively probed in a model-independent way at future lepton colliders, such as the Circular Electron-Positron Collider (CEPC) [15], International Linear Collider (ILC) [16], Future Circular Collider (FCC-ee) [17] and Compact Linear Collider (CLIC) [18]. Compared to the hadron colliders, the lepton colliders are generally very "clean" and the SM processes therein are well understood, which render them primary facilities to search for new physics via the cLFV signals e + e − → ± α ∓ β + X (with α, β = e, µ, τ and α = β). Previous studies of LFV at lepton colliders have either been performed in the framework of effec-tive four-fermion couplings [19,20] or in the context of flavor-violating SM Higgs decays [21] and tau decays [22] or with doubly-charged scalars [23]. Here we include both on and off-shell production of the new neutral scalar H (including resonance) at lepton colliders, which enables us to derive the LFV sensitivity as a function of the mass m H for a direct comparison with the current bounds from low-energy experiments. Moreover, for m H small compared to the center-of-mass energy, the effective theory approximation does not work.
Without loss of generality, we can write the effective Yukawa couplings of H to the charged leptons as Here for simplicity we have assumed the couplings are all real and chirality-independent and thus symmetric. The scalar H may or may not be responsible for symmetry breaking and/or mass generation of other particles in realistic models, where it could be part of a singlet, doublet or triplet scalar field. We assume that it is CP even and its mixing with and/or coupling to the SM Higgs is small. If the scalar is CP-odd, the limits and prospects derived in this Letter would not change significantly. Even though there are all varieties of stringent low-energy cLFV constraints, such as α → β γ, , only a few of them are directly relevant to the LFV prospects discussed below. With an ab −1 level of integrated luminosity, a large parameter space of m H and h αβ could be probed, well beyond the current cLFV constraints and complementary to the projected low-energy constraints from future experiments at the intensity frontier [24]. In addition, the Lagrangian (1) also gives rise to a one-loop contribution to the lepton anomalous magnetic moment. In particular, the longstanding muon g − 2 discrepancy [1] could also be tested directly at lepton colliders.
On-shell LFV.-If kinematically allowed, the neutral scalar H can be directly produced at lepton colliders, in association with a pair of flavor-changing leptons through the couplings in Eq. (1) Muonium-antimuonium oscillation: The Feynman diagrams for the H induced muonium oscillation are presented in Fig. S2. The muonium-anti-muonium oscillation probability [S3] with the H-induced mass splitting with ↵ EM the fine-structure constant, and µ = m e m µ /(m e + m µ ) the e↵ective mass. The di-muonium, i.e. the bound state Dm ⌘ (µ + µ ) (not yet found experimentally), could also be a↵ected by the LFV couplings in Eq. (1), e.g. a t-channel H could contribute to the decay Dm ! e + e . Even if Dm is found in the low-energy experiments and the BR into e + e is consistent with the SM prediction, the theoretical uncertainties would imply a bound |h eµ | 2 /m 2 H . GeV 2 [S4-S6], which is much weaker than the muonium oscillation constraint shown in Fig. 2. ⌧ decay: In the limit of m H m ⌧ m e , the partial decay width [S7] (⌧ ! e + e e ) ' with the symmetry factor = 2 accounting for the identical particles in the final state. The decay width (⌧ ! µ e + e ) and (⌧ ! µ + e e ) are quite similar, with the couplings in Eq. (S9) replaced by |h † ee h µ⌧ | and |h † eµ h e⌧ |, and the symmetry factors of 1 and 2 respectively.
The presence of couplings h ee and h e⌧ could induce the e↵ective Wilson operators with the e↵ective coe cients, in the limit of m H m ⌧ m e [S8], As expected, in the large m H limit, the e↵ective coecients are suppressed by the H mass squared. Then the partial decay width The calculation of (⌧ ! µ ) in presence of the couplings h eµ and h e⌧ is quite similar: in the limit of m H m ⌧ m e, µ one has only to replace the couplings in Eq. (S11) by h † eµ h e⌧ . All the current BRs of relevant LFV two-body and three-body ⌧ decays and the corresponding constraints on the |h † h| couplings are collected in Table I.
Lepton Magnetic Dipole Moments: The Feynman diagram for the magnetic dipole moment of electron is shown in Fig. S3, in presence of the coupling h eµ . The one-loop contribution is given by [S10] It is trivial to get similar diagrams with the coupling h e⌧ and also those for the muon g 2, by changing accordingly the flavors of the fermion lines and the couplings for the vertices. If the incoming electron is replaced by β), as shown in Fig. 1 (top panel). e − µ coupling: Here for simplicity we assume the other two h eτ and h µτ are vanishing. It should be emphasized that the amplitudes in Fig. 1 depend only on the LFV couplings h αβ (here αβ = eµ), and thus could be easily made to satisfy the rare lepton decay constraints, such as µ → eee and µ → eγ, which depend on the product |h † ee h eµ |. Similarly, with vanishing or suppressed couplings to the quark sector, the µ − e conversion limits are irrelevant. Finally, for real Yukawa couplings, we do not either have any limits from electric dipole moment. Thus we are left only with the following constraints (summarized in Table I): (i) Muonium-antimuonium oscillation: This could occur in both s and t-channels [8]. The oscillation probability P ∝ |h eµ | 4 /m 4 H . The MACS experiment [25] could then exclude a large parameter space, as shown in the left panel of Fig. 3.
(ii) (g − 2) e : The anomalous magnetic moment of electron a e receives a contribution from the H − µ loop [8]. As a result of the precise measurement of a e [26], the con-TABLE I. Current experimental constraints on the LFV couplings. The ∆ae, µ constraints have an additional logarithmic dependence on the scalar mass [8].
straint on h eµ is comparable to that from muonium oscillation, as shown in the left panel of Fig. 3. To explain the longstanding theoretical and experimental discrepancy of the muon g − 2, i.e. ∆a µ = (2.87 ± 0.80) × 10 −9 [1], the LFV coupling h eµ is required to be larger, as shown in the left panel of Fig. 3 (by the brown line and the green, yellow bands corresponding respectively to the central value and the 1σ, 2σ ranges), which is already excluded by the (g − 2) e data. See however the µτ sector in the right panel of Fig. 3 for an explanation of (g − 2) µ in this setup.
(iii) e + e − → µ + µ − : A t-channel H could mediate the scattering e + e − → µ + µ − , which interferes with the SM diagrams in the s-channel [27]. Both the total cross section and differential distributions would be modified by the presence of H, depending on its mass and the coupling h eµ . In the heavy H limit, the LEP data exclude an effective cutoff scale Λ m H /h eµ [28]. When H is lighter than the center-of-mass energy √ s, the limits on Λ do not apply, and we consider the H propagator: For simplicity we take an average over the scattering angle cos θ 1/2 to interpret the LEP constraints. Then in the limit of m H √ s, the propagator is dominated by the q 2 term, and the ee → µµ limit in Fig. 3 approaches a constant, as expected.
To be specific, we consider two benchmark configurations for future lepton colliders: i.e. the CEPC [15] and ILC [16], with the center-of-mass energies √ s, integrated luminosities, and the nominal cuts on the leptons (implemented by using CalcHEP [29]) summarized in Table II. The total cross sections σ(ee → α β (+H)) in the light H limit (and m H = 100 GeV for the on-shell production) are also presented in the table, with a conservative efficiency of 60% for the τ lepton [16]. The systematic uncertainties such as initial state radiation, beamstrahlung, and the electron and muon efficiencies lead only up to a few percent correction to the total cross sections [15,16].
After being produced, H could decay back into the charged lepton pairs or other SM particles. Reconstructing the H peak from the decay products could improve further the significance of the LFV signals, which are however rather model-dependent. To work in a modelindependent way, we consider three benchmark values, where 1%, 10% or 100% of the decay products of H are visible and can be reconstructed. The corresponding LFV prospects are shown in the left panel of Fig. 3, where we have assumed a minimum of 10 signal events at both CEPC and ILC. It is clear from Fig. 3 that with a BR of 10%, a large region of m H and |h eµ | can be probed in future lepton colliders, which extends the limits well beyond what is currently available. e − τ coupling: Turning now to the coupling h eτ , the most stringent limit comes from the electron g − 2, which is similar to the case of h eµ except for the enhancement by the τ mass [cf. Eq. (S13)], as shown by the pink region in the middle panel of Fig. 3. The LEP e + e − → τ + τ − limit is slightly stronger than the muon case [28], as shown by the shaded purple region in Fig. 3. The reconstruction of τ lepton is more challenging than µ, and thus the prospects of h eτ are somewhat weaker than h eµ , but there is still ample parameter space to probe at both CEPC and ILC, as long as the effective BR is 10%. µ − τ coupling: Turning now to the coupling h µτ , there are currently no experimental limits, except for the muon g − 2 discrepancy. This could be explained in presence of H when it couples to muon and tau, as shown by the brown line and the green and yellow bands in the right panel of Fig. 3, while the shaded region is excluded by the current muon g − 2 data at the 5σ level. As µτ can only be produced in e + e − collider in the s-channel in Fig. 1, the production cross section is smaller than those of eµ and eτ . From Eq. (S13) (with the couplings and lepton masses changed accordingly), the (g−2) µ anomaly can be directly tested at CEPC up to a scalar mass of 100 GeV, as shown in Fig. 3, as long as there is a sizable BR of H into visible states. With a larger luminosity being planned [17], FCC-ee could do even better.
Off-shell (& resonant) LFV.-The LFV signals could also be produced from an off-shell H, i.e. e + e − → ± α ∓ β ,as shown in Fig. 1 (bottom panel). This could occur in both the s and t channels; in the s-channel H is on-shell if the colliding energy √ s m H (resonance). Different from the on-shell case, the off-shell production amplitudes have a quadratic dependence on the Yukawa couplings (some of them might be flavor conserving), and thus largely complementary to the on-shell LFV searches.
nel. In the τ lepton sector, the LFV decay constraints are comparatively much weaker. In the parameter space of interest m H m τ , the limits on |h † h|/m 2 H are almost constants, as in effective field theories with superheavy mediators. These constraints are all presented in Fig. 4, with the shaded regions excluded. The analytic formulae and calculation details are given in [8]. As for the on-shell case above, the couplings h eβ (β = e, µ, τ ) are constrained respectively by the LEP e + e − → + − data. Thus we can set upper limits on the couplings |h † ee h eτ | and |h † eµ h eτ |, as shown in Table I and the left and right panels of Fig. 4, which get weaker for lighter H, as in the on-shell case.
Given h ee and h eτ , the electron g −2 receives both contributions from the H loops with an e/τ in the intermediate state, and for a fixed value of |h † ee h eτ |, the weakest (g −2) e constraint occurs when h ee m e ∼ h eτ m τ , with the two loops contributing almost equally. Similarly, one can obtain the (g − 2) e limit on |h † eµ h eτ |, which induces the µ/τ -mediated diagrams. Both the constraints are presented in the left and right panels of Fig. 4. Note that the muon g −2 can not be used to set unambiguous limits on the combinations |h † ee h µτ | and |h † eµ h eτ |, although the couplings h µτ and h eµ could contribute to (g − 2) µ by themselves.
The dominant SM backgrounds are from the process e + e − → W + W − → e − τ +ν e ν τ which is expected to be small, if we require the two charged leptons to be back-toback and their reconstructed energy E √ s/2 [19]. The angular distributions of charged leptons can also be used to suppressed the SM W W backgrounds [32]. Assuming 10 signal events as above, the coupling |h † ee h eτ | could be probed up to 6.5 × 10 −5 (6.0 × 10 −4 ) at CEPC (ILC) in the light H limit, as shown in Fig. 4. At the resonance m H √ s, the production cross section can be greatly enhanced by m 2 H /Γ 2 H . To be specific, we have set the width Γ H = 10 (30) GeV at √ s = 240 GeV (1 TeV), where the prospects could be strengthened by roughly one order of magnitude (the dips in Fig. 4). For m H > √ s, the production rate diminishes rapidly as H becomes heavier. An off-shell H could however be probed up to a few-TeV range, as shown in Fig. 4, and ILC is expected to be more promising than CEPC in this mass range, as a result of the higher √ s.
The process e + e − → µ ± τ ∓ could proceed via both the s and t channels, which depend on different couplings, namely |h † ee h µτ | and |h † eµ h eτ |, and are constrained respectively by the rare decays τ − → µ − e + e − and τ − → µ + e − e − . Analogous to the eτ case above, a broad range of m H and |h † ee h µτ | could be probed in the s channel, in particular in vicinity of the resonance, as shown by the middle panel of Fig. 4. In the t channel, the cross sections are comparatively smaller, and the detectable regions are much narrower, as shown by the right panel of Fig. 4.
The future reaches of the LFV couplings in both the on-shell and off-shell production modes are collected in Table III. It is clear that orders of magnitude of the couplings can be probed at future lepton colliders, i.e. from ∼ 10 −4 up to O(0.1) for a scalar mass range of ∼ GeV to 200 GeV at CEPC (900 at ILC) in the on-shell channel, and couplings from ∼ 10 −4 up to O(1) for a mass range from ∼ 100 GeV to few TeV in the off-shell mode.
Conclusion.-We have shown that a hadrophobic neutral scalar H, which is well-motivated in a large class of new physics scenarios, can be probed in an e + e − collider via its LFV couplings to the charged lepton sec- tor. We present a model-independent analysis of how far the LFV coupling strengths and the scalar mass can be probed beyond the existing limits from the low-energy sector. In particular, we find that the full mass and coupling range of the scalar, that can explain the muon g−2 anomaly, can be tested in the future lepton colliders. This is largely complementary to the searches of LFV in the low-energy experiments and hadron colliders.
Acknowledgements.-The work of R.N.M. was supported by the US National Science Foundation under Grant No. PHY1620074. Y.Z. is grateful to the Center for High Energy Physics, Peking University for the hospitality, the local support, and the active discussions during the visit.

Supplemental Material
Example model frameworks RPV SUSY: A natural hadrophobic neutral BSM scalar appears in the minimal supersymmetric standard model with leptonic R-parity violation (RPV) in the form of the bosonic partners of the SM neutrinos, i.e. the sneutrinosν. The relevant RPV term is with α, β and γ flavor indices, L and E c respectively the superfields with respect to the SM left-handed lepton doublets and the right-handed lepton singlets, and λ the RPV coupling. The phenomenological consequences of Eq. (S1) has been studied extensively, see e.g. Refs. [4]. Given Eq. (S1), we can write explicitly the couplings in terms of the four-component Dirac spinors: The first term in Eq. (S2) implies that the sneutrino couples to the charged leptons at the tree level in a LFV way if β = γ. Here either the CP-even or odd component ofν could be identified as the hadrophobic H in Eq. (1) with LFV couplings. There exist various constraints on the λ-couplings [4], but it is still possible to have some of the elements at the O(0.01 − 0.1) level, which could give rise to observable cLFV.
Left-right symmetric model: The minimal left-right symmetric model provides another natural framework to accommodate a hadrophobic neutral scalar in the form of the neutral component of the SU (2) R -triplet scalar field ∆ R . The Yukawa Lagrangian is given by where Q L, R and ψ L, R are the left-and right-handed quark and lepton doublets respectively (with the heavy right-handed neutrino being the neutral component of ψ R ), Φ is the bidoublet scalar field,Φ = σ 2 Φ * σ 2 with σ 2 the second Pauli matrix, and h ,h , f are independent Yukawa coupling matrices in the flavor space. After symmetry breaking, the CP-even neutral scalar sector of the model consists of three physical scalar fields, namely, h (identified as the SM Higgs boson) and H 1 coming from the bidoublet scalar Φ and H 3 coming from the neutral component of the triplet scalar ∆ R [5]. From Eq. (S3) it is clear that at tree-level, ∆ R (and hence, H 3 ) does not couple to the SM quarks, and hence, is naturally hadrophobic. So H 3 can be identified as the hadrophobic scalar H in Eq. (1). The LFV couplings of H 3 are induced from its mixing with the CP-even neutral components h and H 1 of the bidoublet Φ through the quartic terms α 1 Tr(Φ † Φ)Tr(∆ R ∆ † R ) and α 2 Tr(Φ † Φ)Tr(∆ R ∆ † R ) + H.c.. The couplings of H 1 to the SM charged leptons can be written as m D,αβ H 1¯ α β /v EW with m D the 3 × 3 Dirac mass matrix for the type-I seesaw mechanism and v EW the electroweak VEV. With the scalar mixing with H 1 , the couplings of H to the charged leptons are also proportional to the matrix m D [5]. Thus, for large off-diagonal entries of m D , this will give rise to observable cLFV effects. Such large off-diagonal elements can in principle be motivated from discrete flavor symmetries like A 4 [27] or Z 4 [28], which also explain why some of the other couplings can be zero (in the exact symmetry limit) or very small (generated by perturbations).
Mirror model: Another class of seesaw models for the tiny neutrino masses are the mirror models, where mirror leptons Ψ R are introduced, which are coupled to the SM leptons via a singlet scalar φ: Then the singlet scalar φ could couple to the SM charged leptons either at tree level through the mixing of the SM leptons with the mirror leptons, or at 1-loop level through the Yukawa interaction above and the trilinear scalar coupling, which arises naturally from the quartic terms in the scalar potential. With the flavor structure in the y matrix, the effective coupling of φ to the charged leptons could be flavor-violating [6], thus providing another example of LFV hadrophobic scalar H. Two-Higgs doublet model: There is a class of Z 2 symmetric two-Higgs doublet models (2HDM) called leptonspecific (or type-X) 2HDM, where one of the scalar doublets only couples to leptons: In this case, both CP-even and odd neutral components of Φ 1 are naturally hadrophobic and either of them (or a linear combination) can be identified as H in Eq. (1). The LFV couplings of H can be naturally induced by breaking the lepton-specific structure (e.g. with a soft mass term m 2 12 ), with the additional Yukawa couplings [7] ∆L Y = ξ u αβQαLΦ2 u βR + ξ d αβQαL Φ 2 d βR + ξ αβψαL Φ 1 βR + H.c. . C. Experimental constraints on the LFV couplings and muon g 2 anomaly Muonium-antimuonium oscillation: The Feynman diagrams for the H induced muonium oscillation are presented in Fig. S1. The muonium-anti-muonium oscillation probability [S3] with the H-induced mass splitting with ↵ EM the fine-structure constant, and µ = m e m µ /(m e + m µ ) the e↵ective mass. The di-muonium, i.e. the bound state Dm ⌘ (µ + µ ) (not yet found experimentally), could also be a↵ected by the LFV couplings in Eq. (1), e.g. a t-channel H could contribute to the decay Dm ! e + e . Even if Dm is found in the low-energy experiments and the BR into e + e is consistent with the SM prediction, the theoretical uncertainties would imply a bound |h eµ | 2 /m 2 H . GeV 2 [S4-S6], which is much weaker than the muonium oscillation constraint shown in Fig. 2.
⌧ decay: In the limit of m H m ⌧ m e , the partial decay width [S7] (⌧ ! e + e e ) ' 1 |h † ee h e⌧ | 2 m 5 with the symmetry factor = 2 accounting for the identical particles in the final state. The decay width (⌧ ! µ e + e ) and (⌧ ! µ + e e ) are quite similar, with the couplings in Eq. (S9) replaced by |h † ee h µ⌧ | and |h † eµ h e⌧ |, and the symmetry factors of 1 and 2 respectively.
The presence of couplings h ee and h e⌧ could induce the e↵ective Wilson operators O ⌧ e , (L,R) = e 8⇡ 2 m ⌧ (ē µ⌫ P L,R ⌧ ) F µ⌫ (S10) with the e↵ective coe cients, in the limit of m H m ⌧