Sensitivity of future lepton colliders to the search for charged lepton flavor violation

The observation of lepton flavor violation indicates new physics beyond the Standard Model. Lepton colliders are ideal facilities to probe charged lepton flavor violation (CLFV) signals induced by new physics at high energy. In this work we perform a comprehensive study of the sensitivity of future lepton colliders to charged lepton flavor violation. We consider the most general renormalizable Lagrangian coupling two leptons to new bosonic particles, involving both $\Delta L=0$ and $\Delta L=2$ interactions. The CLFV processes are introduced by the exchange of off-shell new particles at tree-level. We find that CEPC, ILC, FCC-ee, and CLIC each provide a complementary probe of CLFV couplings to low-energy precision experiments for $\tau$ lepton(s) in final states, while low-energy precision experiments are more sensitive in the absence of $\tau$ leptons.


I. INTRODUCTION
The observation of lepton flavor violation (LFV) clearly indicates new physics beyond the Standard Model (SM). In the SM, lepton flavor numbers are conserved because neutrinos are massless. The observation of neutrino oscillations and thus neutrino masses, however, confirms the existence of LFV in the neutrino sector. Thus LFV naturally occurs among charged leptons due to SU(2) L symmetry, that is the charged lepton flavor violation (CLFV) [1,2].
It is defined as processes conserving total lepton number L ≡ L e + L µ + L τ (and baryon number B) but violating the global symmetry group without involving neutrinos [3]. Massive neutrinos themselves can lead to CLFV processes from right-handed Dirac neutrinos in the SM or from Majorana neutrinos induced by the socalled Weinberg operator. However, the induced CLFV processes are currently unobservable because they are suppressed by the unitarity of the leptonic mixing matrix and the smallness of neutrino masses. Many other neutrino mass models, such as the inverse seesaw model (see e.g. [4]) and radiative neutrino mass models [5], predict sizable CLFV processes. CLFV can also arise in many other extensions of the SM such as multi-Higgs doublet models or the minimal supersymmetric standard model via gauginos-slepton loops with off-diagonal terms in the slepton soft mass matrix [6].
The search for CLFV and the corresponding new physics can be traced back to the 1940s and 50s when the first bounds were obtained for µ → eγ [7] and µ-e conversion in nuclei [8]. CLFV has yet to be observed experimentally. Thus upper limits have been derived on the scale of CLFV of the order of 100 (10) TeV for muon (tau) decays [9]. There are also several scenarios of CLFV search at colliders. The ATLAS and CMS experiments searched for CLFV from Higgs decays, i.e. H → τ [10,11], and derived upper bounds on the off-diagonal Yukawa couplings of the order of 10 −3 . The ATLAS experiment currently provides the most stringent limit for Z → eµ [12] with BR < 7.5 × 10 −7 , while experiments at the Large Electron Positron (LEP) collider set the most stringent limits for Z → eτ (Z → µτ ) with BR < 9.8 × 10 −6 [13] (BR < 1.2 × 10 −5 [14]). Lepton colliders with their clean environment and well-understood backgrounds can outperform hadron colliders with less integrated luminosity and thus provide an ideal facility to probe rare CLFV events.
With their large foreseen c.m. energies and luminosities, the search for CLFV at lepton colliders should yield complementary results to those of searches for rare lepton decays. In this work, we illustrate the projected sensitivity of future lepton colliders to the CLFV signal arisen from new physics beyond the SM. Note that we will not include the results of ILC with 2 ab −1 luminosity at 250 GeV, as the c.m. energy of proposed ILC is very close to that of CEPC and FCC-ee and its integrated luminosity is much smaller as a linear collider.
The above ILC case is stated as ILC500 in the following context and figures. We will also consider the option of colliding two electron beams for each of the four proposed lepton colliders, assuming the same center of mass energies but a reduced integrated luminosity of 500 fb −1 .
Previous searches for CLFV at lepton colliders mainly focused on the rare decays of a tau lepton [19], Z boson [20,21] and Higgs boson [22][23][24]. We instead study the scattering CLFV processes mediated by the new particles beyond the SM at lepton colliders. The advantage of this scenario of off-shell channels is that one can directly compare the projected limit of CLFV couplings with current low-energy precision constraints. To apply the results to a broader extent, in the aspect of new physics contribution, we consider the most generic interactions that are allowed by Lorentz invariance and gauge invariance.
The paper is outlined as follows. In Sec. II, we describe the general SM extensions with CLFV couplings. Then, we discuss the low-energy precision constraints on the CLFV couplings in Sec. III. We present the sensitivity of future lepton colliders to the CLFV couplings in Sec. IV, where we also show the comparison to the low-energy constraints. Our conclusions are drawn in Sec. V. Some technical details for low-energy precision constraints are collected in the Appendix.

II. GENERAL LAGRANGIAN FOR CHARGED LEPTON FLAVOR VIOLATION
In this section we construct the most general Lagrangian which couples two charged leptons to either a scalar or a vector boson. We consider both lepton number conserving bilinears¯ Γ (∆L = 0) and lepton number violating bilinears¯ c Γ (∆L = 2), where Γ denotes different γ-matrix structures. The most general Lagrangians for ∆L = 0 and ∆L = 2 bilinears are given by The subscript of the new bosonic fields, i.e. 1, 2 or 3, indicates the SU(2) L representation, singlet, doublet or triplet, respectively. The terms in the Lagrangians are obtained by expanding the most general SM gauge invariant Lagrangian in terms of explicit leptonic fields [25].
In the ∆L = 0 Lagrangian, H is a complex neutral scalar. We thus have to consider both scalar and pseudoscalar parts of H 0 2 . The couplings y ( ) 1 may arise from new gauge interactions with a LFV Z , and y 2 can naturally appear in two Higgs doublet models. Note that the interaction with an SU(2) L triplet gauge boson, i.e.Lγ µ P L LH 3µ , gives the same charged lepton interaction as H 0 1µ and thus we do not list it separately.
The ∆L = 2 coupling λ 1 may originate from the SU(2) L singlet field in the Zee-Babu model which only couples to right-handed charged leptons [26,27] and λ 3 may come from the SU(2) L triplet field in the Type II Seesaw model which only interacts with left-handed charged leptons [28][29][30][31][32]. The coupling λ 2 can arise after the breaking of a unified gauge model where the lepton doublet L and charge-conjugate of the charged lepton singlet c reside in the same multiplet. One example is an SU(3) c ×SU(3) L ×U(1) Y model [33].
Two examples of ultraviolet(UV)-complete models are a LFV two Higgs doublet model whose leptonic Yukawa term is described by and the Type II Seesaw model with the charge conjugation matrix C = iγ 2 γ 0 . Besides the charged lepton terms in Eqs. (2) and (3), as shown below, there are additional contributions to low-energy precision constraints in the above UV-complete models, which we provide in the Appendix. In the rest of this paper, however, we will restrict ourselves to the Lagrangians in Eqs. (2) and (3).

III. CONSTRAINTS
We discuss three different classes of constraints on the CLFV parameters in the models introduced in the previous section. In the analysis we restrict ourselves to one new boson at a time. The first subsection discusses low-energy precision constraints from tree-level rare decays, radiative decays, the anomalous electromagnetic moments of leptons, and muonium antimuonium conversion. The second one presents existing constraints from the DELPHI experiment at the Large Electron Positron (LEP) collider and searches at the Large Hadron Collider (LHC), while the third subsection briefly describes any additional constraints which arise from embedding the particles in complete SM multiplets.

A. Low-energy precision constraints
The new bosons contribute to low-energy precision observables at either tree-level or loop-level. In particular, there are tree-level contributions to trilepton decays and muonium- The trilepton decay constraints are obtained by a straightforward tree-level calculation and are given as a ratio of the three-body decay branching ratio from new physics contribution and the dominant leptonic SM branching ratio. We neglect loop-level contributions to trilepton decays, because they are generally suppressed. Box diagrams become relevant and similar in size to the tree-level contribution, if one of the couplings is close to its perturbative process experimental constraint future sensitivity lepton flavor symmetry µ + → e + γ BR ≤ 4.3 × 10 −13 [34] 6 × 10 −14 [35] |∆(L e − L µ )| = 2  limit ∼ 4π. Although this leads to an O(1) correction to the branching ratio, the tree-level calculation still provides a good estimate and is sufficiently precise for the following discussion. In some cases additional trilepton decay modes are induced via box diagrams. For example, with non-zero couplings y eµ 1 y eτ 1 , the τ decay to 3µ is induced in addition to the treelevel decay τ − → µ − e + e − . Given the comparable experimental sensitivity of leptonic LFV τ decays, the loop-induced decays do not provide stronger constraints. For the radiative LFV decays and the anomalous magnetic moments we use the general formulas provided by Lavoura in Ref. [43]. The experimental measurement of the muon anomalous magnetic moment differs from the SM prediction by more than 3σ [44] and the electron magnetic moment shows deviations at 2.5σ [45]. Hence we demand the muon anomalous magnetic moment to be only within 4σ of the experimental measurement, while we require the anomalous magnetic moment of the electron to be within 3σ of the experimental measurement.
The muonium-antimuonium conversion probability in the absence of magnetic fields is given by  [40,49] LFV Z-boson decays do not lead to competitive constraints: They are several orders of magnitude weaker, because they are induced at loop-level and the current experimental limits are of order 10 −5 for Z → τ . Although the sensitivity will be improved by 3-6 orders of magnitude at the CEPC [50], the future sensitivity of Belle II to trilepton τ decays will be more sensitive for our parameters of interest.
From the experimental constraints in Table I Tables II and III, respectively. Future sensitivities by Belle II [36], the MEG upgrade [51], and Mu3E [38] are indicated inside square brackets. The detailed formulas for low-energy precision constraints are collected in the Appendix. Note that, for the numerical results in this and the next sections, we assume all the CLFV couplings to be real and symmetric. For H 0 2 we additionally assume equal masses for the scalar h 2 and the pseudoscalar a 2 ,

B. Current collider constraints
There are already constraints from experiments at the LEP and the LHC colliders. In Ref. [52] the DELPHI collaboration interpreted their searches for e + e − → + − in terms of leptonic dimension 6 operators 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. The analysis sets the coupling to g 2 = 4π to obtain conservative limits on the new physics scale, which are summarized in Tab. 30 of Ref. [52]. We list the translated limits for masses well above the center-of-mass energy of LEP, √ s ∼ 130 − 207 GeV, in Tab. IV. The analysis of contact interactions in Ref. [52] does not directly apply to ∆ ++ 2µ , because the induced effective interactions do not fall into the class of effective interactions considered in Ref. [52]. Note that these limits are only valid when the new particle mass is much greater than √ s. To make it valid for any masses, we should replace the mass in Tab. IV by (s cos θ/2 + m 2 ) −1/2 after averaging over the scattering angle cos θ 1/2. Nevertheless, the LEP limits are less restrictive than the low-energy precision constraints. The ATLAS and CMS experiments searched for electroweak pair production of doublycharged scalars with subsequent decay to e ± e ± , µ ± µ ± , e ± µ ± pairs. Currently the most stringent limits by the ATLAS [53] are m ∆ ++

C. Other constraints
Although we focus on the models defined in Eqs. (2) and (3)  and ∆ ++ 2µ are accompanied by a singly-charged scalar which are searched for at colliders. In particular the singly-charged scalar of a second electroweak doublet has been searched for at the LHC under the assumption that it couples to quarks and thus these searches do not apply in this case.
The additional particles in the multiplets also contribute to the radiative LFV decay and may interfere constructively like for ∆ 3 or destructively like for H 2 . As the LFV tree-level trilepton decays are more stringent and do not receive any additional contributions, we will only show the limits from LFV tree-level trilepton decays in the following.

IV. SENSITIVITY OF FUTURE LEPTON COLLIDERS TO THE CLFV
The CLFV processes can happen through the scattering of either opposite-sign [56][57][58] or same-sign [59,60]  We apply basic cuts p T > 10 GeV and |η| < 2.5 on the leptons in final state and assume 10 discovered signal events. The total width of the mediating particle is set to be 10 GeV [56].
If we take tau efficiency of 60% [61], the sensitivity to the combination of Yukawa couplings in the scattering amplitude will become weaker by 77% and 60% for final states with one and two tau leptons, respectively. Note that we do not intend to discriminate the chiral CLFV channels for probing coupling y ij y i j or λ ij λ i j via ∆L = 0 or ∆L = 2 interaction at e + e − collider, while e ↔ µ for µ + µ − collider. In the left panels of Fig. 4, we show the lepton collider sensitivity to the detection of |y ee 1 y eµ(eτ ) 1 | (top left) through e + e − → e ± µ ∓ (e ± τ ∓ ), and |y ee 1 y µτ 1 | (center left) and |y eµ 1 y eτ 1 | (bottom left) through e + e − → µ ± τ ∓ , as a function of m H 1 for the H 0 1 interactions. The reachable limits for the couplings in H 0 2 interaction are given in the right panels of Fig. 4, assuming m h 2 = m a 2 . As stated before, the e and µ labels of the couplings in these and following figures should be swapped to obtain the couplings which can be probed at a muon collider. In summary, the |y ee 1(2) y eµ 1(2) | and |λ ee 1(2) λ eµ 1(2) | couplings are highly constrained by the treelevel decay µ → 3e and hence the accessible regions for lepton colliders have already been excluded. In contrast, CLFV couplings involving τ flavor are less constrained and a larger parameter space can be probed by future lepton colliders. Comparing different machines, | through e + e − → e ± µ ∓ (e ± τ ∓ ), and |λ eµ 1 λ eτ 1 | through e + e − → µ ± τ ∓ , as a function of m ∆ 1 for ∆ ++ 1 interaction. Right: Sensitivity to |λ ee 2 λ eµ(eτ ) 2 | through e + e − → e ± µ ∓ (e ± τ ∓ ), and |λ eµ 2 λ eτ 2 | through e + e − → µ ± τ ∓ , as a function of m ∆ 2 for ∆ ++ 2 interaction.
the FCC-ee with the highest integrated luminosity in proposal can explore smaller couplings in the low mass region. In the high mass region, however, the ILC500 and CLIC with larger c.m. energies are able to reach a broader parameter space.

B. Same-sign lepton collision
We show the CLFV channels and the explored couplings in either ∆L = 0 or ∆L = 2 Lagrangian at a same-sign lepton collider in Table VI. When same-sign leptons collide, the CLFV process led by ∆L = 0 and ∆L = 2 interactions only emerges in t and s channel, respectively. This type of collider can also probe identical lepton final states, such as µ − µ − , τ − τ − , which are forbidden in opposite-sign lepton collisions. As a result, single couplings such as y eµ 1(2) or y eτ 1(2) can be measured through a t channel process for ∆L = 0 interactions.
As the luminosity for an e − e − collider will be much smaller than that of e + e − machines, we assume an integrated luminosity of 500 fb −1 below for the same-sign lepton colliders. The left panel of Fig. 6 displays the sensitivity to |y ee 1 y eµ(eτ ) 1 | through e − e − → e − µ − (e − τ − ), and |y eµ 1 y eτ 1 | through e − e − → µ − τ − , as a function of m H 1 for H 0 1 interaction. The sensitivity to |y eµ 1 y eµ 1 |(|y eτ 1 y eτ 1 |) through the process with identical final states, i.e. e − e − → µ − µ − (τ − τ − ), is weakened by a factor of two. The right panel is for H 0 2 couplings. Smaller couplings of vector H 0 1 can be reached in t channel, compared with scalar H 0 2 . We display the explored couplings for ∆ ++ 1 and ∆ ++ 2 through the same scattering as above, but in s channel, in Fig. 7.
The sensitivity results for processes with identical final states are weaker by a factor of two.

V. CONCLUSION
We perform a comprehensive study of the sensitivity to both opposite-and same-sign lepton colliders to charged lepton flavor violating interactions. We consider the most general  CLIC with larger c.m. energies are able to explore broader parameter space in the high mass region.

ACKNOWLEDGMENTS
MS thanks Julian Berengut for discussions. Feynman diagrams were created with Tik Z-Feynhand [65]. This work has been supported in part by the Australian Research Council.  [44]. In order to derive a constraint we add experimental and theoretical errors in quadrature and 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 take into account the current discrepancies between experiment and SM theory prediction.

( ) 1
The branching ratio of the trilepton decay − 0 → − 1 + 2 − 3 in the limit of vanishing final state masses is given by in Feynman gauge. For radiative LFV decays we find There is a new (t-channel) contribution to e + e − → + − scattering cross section from H under the assumption that either y ee 1 or y 1 vanish.

H 2
The branching ratio of the trilepton decay − 0 → − 1 + 2 − 3 mediated by neutral scalar exchange in the limit of vanishing final state masses is given by For radiative LFV (taking only the neutral scalar into account) we find for the branching with the fine structure constant α em and in the limit of a vanishing final state mass and to leading order in the fermion mass in the loop. The anomalous magnetic moment of a lepton receives an additional contribution Thus, for real symmetric Yukawa couplings and m h 2 = m a 2 , the contribution is always negative. The relevant interaction Lagrangian for muonium-antimuonium conversion is [(y µe 2 − y eµ * 2 )μe + (y µe 2 + y eµ * 2 )μγ 5 e] 2 (A11) which reduces in the limit m h 2 = m a 2 for real and symmetric couplings to There is a new (t-channel) contribution to e + e − → + − scattering cross section from H ( ) 1µ . In the limit of real symmetric Yukawa couplings and m h 2 = m a 2 we obtain under the assumption that either y ee 2 or y 2 vanish. If the neutral complex scalar originates from a two Higgs doublet model, there is a second contribution to radiative LFV decays from the charged scalar and thus the branching ratio is given by Eq. (A7) by using the following replacements and the anomalous magnetic moment receives an additional contribution (A15) The mass of the charged scalar is denoted m h + . Note that there is a partial cancellation between the contribution of the charged scalar and the neutral scalars. Thus, depending on the masses, the contribution to the anomalous magnetic moment may be positive or negative.

∆ ++ 1
The branching ratio of the trilepton decay − 0 → + 1 − 2 − 3 in the limit of vanishing final state masses is given by .

(A16)
For radiative LFV decays decays we find and the anomalous magnetic moment is given by The relevant interaction Lagrangian for muonium-antimuonium conversion is There is a new (t-channel) contribution to e + e − → + − scattering cross section from ∆ ++ 1 which is described by the effective operator L ef f = 2|λ e 1 | 2 (1 + 3δ e )m 2 ∆ 1ē γ µ P R e¯ γ µ P R .

(A21)
For radiative LFV decays we find in the limit of vanishing final state masses and fermion masses in the loop. Similarly there is a contribution to the anomalous magnetic moment of the lepton The relevant interaction Lagrangian for muonium-antimuonium conversion is There is a new (t-channel) contribution to e + e − → + − scattering cross section from ∆ ++ 3 which is described by the effective operator L ef f = 2|λ e 3 | 2 (1 + δ e )m 2 ∆ 3ē γ µ P L e¯ γ µ P L . (A25) In the Type II Seesaw model there are additional contributions to the radiative decays and the anomalous magnetic moments due to the additional charged scalars. In the limit of vanishing final state masses and fermion masses in the loop we find for radiative LFV decays and the anomalous magnetic moment is changed by where m ∆ + 3 (m ∆ 3 ) denotes the singly (doubly) charged scalar mass of the electroweak triplet scalar.

∆ ++ 2
The branching ratio of the trilepton decay − 0 → + 1 − 2 − 3 in the limit of vanishing final state masses is given by .