Lepton flavor violation by two units

Charged lepton flavor violation arises in the Standard Model Effective Field Theory at mass dimension six. The operators that induce neutrinoless muon and tauon decays are among the best constrained and are sensitive to new-physics scales up to $10^7\,$GeV. An entirely different class of lepton-flavor-violating operators violates lepton flavors by two units rather than one and does not lead to such clean signatures. Even the well-known case of muonium--anti-muonium conversion that falls into this category is only sensitive to two out of the three $\Delta L_\mu = - \Delta L_e = 2$ dimension-six operators. We derive constraints on many of these operators from lepton flavor universality and show how to make further progress with future searches at Belle II and future experiments such as $Z$ factories or muon colliders.


I. INTRODUCTION
The Standard Model (SM) has accidental symmetries that lead to the conservation of electron, muon, and tauon number [1,2].Neutrino oscillations are proof that these symmetries are ultimately violated in nature, but might be unobservably suppressed by the tiny m 2 ν in the chargedlepton sector in the worst-case scenario [3].Luckily, many SM extensions violate these flavor symmetries and can lead to testable flavor-violating processes, unsuppressed by the neutrino mass [1,2].
To stay model agnostic, we can resort to the Standard Model Effective Field Theory (SMEFT, see Ref. [4] for a recent review), which extends the SM by higher-dimensional effective operators suppressed by powers of some high scale Λ [5,6].Integrating out heavy new particles produces exactly these kind of operators.Ordering the higherdimensional operators by their mass dimension (i.e.powers of 1/Λ), the leading one is Weinberg's LLHH/Λ that generates Majorana neutrino masses [5], solving one of the SM's biggest problems.At 1/Λ 2 , or mass dimension d = 6, there are thousands of operators [7,8], most of which just lead to small corrections to processes that are already allowed in the SM.But some of them violate the SM's accidental symmetries and thus lead to completely different processes that in principle have zero background.
Alas, there are 21 d = 6 operators that violate lepton flavor by two units rather than one, and thus might not give rise to neutrinoless decays.For example, operators of the form µµēē violate ∆L µ = −∆L e = 2 and do not lead to on-shell muon decays.In this particular example, muonium-anti-muonium conversion provides a good alternative signature for two out of three µµēē operators.However, once we have a look at the tauon sector, e.g.τ τ ēē, even the leptonium option is removed and the operators are seemingly unconstrained despite violating lepton flavor and being of low mass dimension.
Here, we will investigate such ∆L α = 2 operators and identify possible ways to constrain them.Weinberg's d = 5 operator already contains ∆L α = 2 pieces, but since they are suppressed by neutrino masses these operators are rendered unobservable, with the possible exception of neutrinoless double beta decay [9].At d = 6, all ∆L α = 2 operators are part of the four-lepton operators [8] L ⊃ a,b,c,d=e,µ,τ where L a is the left-handed lepton doublet of flavor a, ℓ a the right-handed charged lepton of flavor a, and the y = O(Λ −2 ) are the Wilson coefficients of mass dimension −2.UV-complete realizations of these operators involve neutral or doubly-charged bosons and can for example be found in Refs.[10][11][12][13].
We stress that the ∆L α = 2 operators under investigation here are fundamentally distinct from the more familiar ∆L α = 1 operators; since they carry different lepton numbers it is easily possible to impose a symmetry in the charged-lepton sector that would forbid ∆L α = 1 but allow for ∆L α = 2. Examples are U (1) flavor symmetries or their Z N subgroups [3], and lepton flavor triality Z 3 [14][15][16], which arises in many neutrino-mass models based on discrete symmetries such as A 4 [17][18][19] and only allows for operators of the form τ μee, τ ēµµ, and τ τ eµ.Renormalization-group running can turn ∆L α = 1 operators into ∆L α = 2, but not vice versa, rendering dedicated searches for ∆L α = 2 absolutely necessary to cover these blind spots.( All three operators will contribute to muoniumantimuonium conversion [12,[24][25][26][27]. Using the experimental setup of the PSI experiment that provides the strongest limit to date [20], the conversion probability P takes the approximate form [ We can put stronger limits, shown in Tab.I, by setting one of the linear combinations to zero, i.e. forbidding interference between the contributions.Ref. [24] has shown that muonium conversion is actually also affected by y LL µeµe − y RR µeµe through the muonium widths difference, but since these effects are further suppressed by the Fermi constant G F , the resulting limits are only of order (1.2 GeV) −2 , probing scales far below the electroweak scale and thus not particularly relevant for the SMEFT.Future experiments such as the Muonium-to-Antimuonium Conversion Experiment (MACE) at the China Spallation Neutron Source (CSNS) [28] and a new setup at the Japan Proton Accelerator Research Complex (J-PARC) [29] are expected to improve the old PSI bounds by orders of magnitude.
Note that the three operators in Eq. ( 2) carry the same quantum numbers and thus mix via loops or renormalization group equations [30].In the SMEFT energy region above the electroweak scale, this requires insertions of the lepton Yukawa couplings y ℓ ≡ m ℓ /174 GeV, see Fig. 1.For example, the operator with coefficient y LL µeµe − y RR µeµe then generates a y LR µeµe operator of magnitude The tiny prefactor y e y µ /16π 2 ≃ 10 −11 renders these effects small and gives irrelevant limits on y LL µeµe − y RR µeµe .Better limits on the linear combination y LL µeµe − y RR µeµe can be obtained by noticing that the underlying operator contains both neutrinos and charged leptons and thus leads to the muon decay µ − → e − νµ ν e .Since we are looking at ∆L µ = −∆L e = 2 processes, this decay does not interfere with the ∆L µ = ∆L e = 0 SM decay µ − → e − ν µ νe .We find that our operators from Eq. ( 2) generate exactly the same electron energy spectrum as the SM decay, so the Michel spectrum remains unperturbed; only the overall muon lifetime or decay rate is affected: even including radiative QED corrections.Since |y LR µeµe | is already constrained to be tiny from the muonium limits, we can safely neglect it here.To obtain limits on the Wilson coefficients, we employ lepton-flavor-universality tests [31],1 i.e. we calculate Γ µ /Γ τ →eνν , using the definitions and experimental values from Ref. [21], which gives Using instead the ratio Γ µ /Γ SM (τ → µνν) puts a better limit, because the two-decades old measured τ → µνν rate shows a 2σ increase with respect to the SM.This is likely a statistical fluctuation or systematic effect since a recent preliminary measurement of Γ(τ → µνν)/Γ(τ → eνν) at Belle II

Wilson coefficient
Upper limit Process Violated quantum numbers  (combined with older measurements) is perfectly compatible with the SM [22]: The difference in the limits from τ → eνν and τ → µνν will therefore probably decrease with dedicated measurements of the tauon branching ratios at Belle II.In the meantime, we construct one-sided 95% C.L. confidence intervals from the above to find a limit on |2y LL µeµe | of 1.8/TeV 2 , which can also be written as As y LL µeµe + y LL µeµe is already constrained to be ∼ 20 times smaller than the first term we obtain an upper bound on the Wilson-coefficient linear combination that is unconstrained by the muonium of |y LL µeµe − y RR µeµe | < 1.8/TeV 2 (Tab.I).This is the strongest limit on the remaining Wilson coefficients, corresponding to new-physics scales of 0.74 TeV, above the electroweak scale and thus perfectly applicable to our SMEFT ansatz.Since the uncertainties on the flavoruniversality ratios are dominated by the tau lifetime and branching ratios, these are the quantities that need to be measured more precisely in order to improve the bound on |y LL µeµe − y RR µeµe |; Belle II will likely achieve this in the near future [22].
Although not currently relevant, let us mention some other experiments and signatures that could play a future role in constraining the y µeµe Wilson coefficients.μμee operators involving neutrinos can induce mixed-flavor neutrino-trident effects, e.g.ν µ X → ν e µ + e − X or ν e X → ν µ e + µ − X, which could be probed in future neutrino detectors such as DUNE [33][34][35][36].Due to the non-interference with SM amplitudes the effects are expected to be small, roughly which is at most 10 −2 given the above-derived constraints.
Because of this, tridents will be at most useful at constraining the weakest linear combination, |y LL µeµe − y RR µeµe |.For a pure ν µ beam, the µ + e − appearance would be an unambiguous sign of lepton flavor violation and thus a background-free signature, but realistic neutrino beams will have admixtures of ν e and νµ that induce indistinguishable SM processes such as ν e X → ν µ µ + e − X, rendering trident searches for ∆L µ = 2 difficult.Still, tridents might eventually become a probe competitive with lepton flavor universality violation.
The μμee operators could also be probed at future lepton colliders via the background-free e − e − → µ − µ − , µ + e − → µ − e + , or µ + µ + → e + e + .A setup for the latter two initial states was recently proposed as µTRISTAN [37], a high-energy lepton collider using the ultra-cold antimuon technology developed at J-PARC [38] that could run in the µ + e − mode with √ s = 346 GeV, and later in the µ + µ + mode [39] with √ s = 2 TeV or even higher.Judging by the analyses of similar four-lepton operators in Refs.[40,41], we can expect µTRISTAN to probe all |y µeµe | down to (O(10) TeV) −2 , superseding all current limits.The high centre-of-mass energy might even allow for a direct production of the mediators underlying our d = 6 operators, see Ref. [42][43][44][45] for such studies.While we have focused on µTRISTAN here, other collider designs could also provide good reach for μμee operators as long as they collide µ ± µ ± , e ± e ± , or µ ± e ∓ .µ − -e − scattering, e.g. at MuonE [46], has much weaker sensitivity to our ∆L µ = −∆L e = 2 opera- These operators do not immediately lead to neutrinoless tau decays, nor do we have any tauonium-antitauonium conversion experiments at our disposal.For y LL and y LR , we can once again calculate the decay rates and electron spectra of τ − → e − ντ ν e and compare them to experimental data.Just like for the muon decay, these operators generate the same electron spectrum as SM tauon decays, so the partial width is simply rescaled compared to the SM: Again, we use lepton-flavor-universality tests to obtain limits.We calculate Γ τ →eνν /Γ µ→eνν and compare to the experimental values [21]: A stronger limit can be achieved using preliminary Belle II data [22] (combined with older measurements) for Γ τ →eνν /Γ τ →µνν : Using one-sided confidence intervals, we obtain a 95% C.L. limit on |2y LL τ eτ e | and |y LR τ eτ e | of 2.2/TeV2 (Tab.I).These are viable SMEFT limits that can be improved with future Belle II data.
y RR τ eτ e remains unconstrained here since it does not involve any neutrinos.Closing SM loops to generate neutrinos -equivalent to mixing the three operators, see Eq. ( 4 and Fig. 1 -requires chirality flips and is thus heavily suppressed.To avoid such suppressions we can consider Z → τ τ ēē decays, see Fig. 2. Neglecting lepton masses, the branching ratio for this process is where M Z (Γ Z ) is the Z mass (width), and s W (c W ) the sine (cosine) of the weak mixing angle.Currently, we do not have any experimental constraints on this decay channel; demanding the branching ratio to be less than one gives the weak bound |y RR τ eτ e | < 15.5/GeV 2 .The total Z width agrees very well with the SM prediction [47], which can be translated into a 2σ upper bound of 2×10 −3 on any non-SM Z branching ratio. 2 This improves the bound to |y RR τ eτ e | < 0.7/GeV 2 (Tab.I).A dedicated LHC search for this decay could realistically reach branching ratios of order 10 −5 , corresponding to a limit |y RR τ eτ e | < 4.89×10 4 /TeV 2 .Z decays could conceivably be measured more extensively in the future at a so-called Z factory [48], producing trillions of Z bosons.Assuming an optimistic reach of 10 −12 for the above branching ratios would probe |y RR τ eτ e | < 15.5/TeV 2 , just barely above the electroweak scale.This is likely our best shot at providing EFT limits on y RR τ eτ e .Better limits on explicit UV completions of this operator are of course possible and probe complementary parameter space, see Ref. [11,16,49].Notice that four-lepton Z decays are also useful for many other SMEFT coefficients [50].
Yet another probe of the RR coupling can be found in tauon decays involving an off-shell tauon in the final state, see Fig. 3.These are additionally suppressed by G F and phase space, but can be competitive due to the large number of collected tauon decays compared to Z.For massless final states we find where ℓ is an electron or muon; τ − → e − e − π + ντ has a similar rate.Probing scales above the electroweak scale is clearly out of the question even at Belle II, but at least we can confirm the current Z-decay limit from above.For this we notice that CLEO has long ago observed the SM decay τ − → e − e − e + ν τ νe [51] with branching ratio (2.8 ± 1.5) × 10 −5 [51], compatible with the SM prediction 4.2 × 10 −5 [51][52][53].This yields a 95% C.L. limit of 4.4/GeV 2 on |y RR τ eτ e |, slightly worse than the Z-decay limit but with entirely different assumptions.Belle II can significantly improve this limit.

IV. τ τ µµ
In this section we consider the d = 6 operators with which have a similar phenomenology as the τ τ ee operators from the previous section.y LL τ µτ µ and y LR τ µτ µ give rise to τ − → µ − ν µ ντ , once again just rescaling the SM decay rate.We compute Γ τ →µνν /Γ µ→eνν and compare it to the experimental results, obtaining A better limit can be achieved using preliminary Belle II data (combined with older measurements) [22] for Γ τ →µνν /Γ τ →eνν : Using one-sided confidence intervals, one can get 95% C.L. limits on |2y LL τ µτ µ | and |y LR τ µτ µ | of 2.5/TeV 2 (Tab.I).Just like in the previous section, the last Wilson coefficient y RR τ µτ µ can be constrained using Z → τ τ μμ decays (Tab.I).Since we are considering tauons, muons, and electrons to be massless we obtain the same branching ratio as in Eq. ( 16) and the same current and future limits.The τ τ µµ operators could also be probed at µTRISTAN via µ + µ + → τ + τ + , with sensitivity to all |y τ µτ µ | down to (O(10) TeV) −2 .
Using one-sided confidence intervals, one can get 95% C.L. limits of 7.9/TeV  22) to Eq. ( 23): By comparing to the experimental data from Eq. ( 9) we get a limit on the following linear combination: This allows us to improve the limit on |y LR eτ µτ |.Combining equations (24) and (27) we have: Again, utilizing one-sided confidence intervals we recover a slightly improved limit on |y LR eτ µτ | of 3.3/TeV 2 (Tab.I).Three of the four Wilson coefficients are thus constrained from universality ratios.Similarly to sections III and IV one can obtain limits on |y RR eτ µτ | from Z decays.The prefactor of the branching ratio is a factor of 2 smaller here since two final state particles are not the same anymore: This change makes the limit on |y RR eτ µτ | a factor of √ 2 weaker compared to |y RR τ eτ e | and |y RR τ µτ µ | (Tab.I).Five-body tauon decays τ − → e − µ − ℓ + ν ℓ ντ also have a factor-2 smaller branching ratio than Eq. ( 17): CLEO provides a 90% C.L. limit on the SM-allowed decay BR(τ − → µ − e − e + ν τ νµ ) < 3.2 × 10 all of which give rise to the clean lepton-flavor-violating decay τ + → e + e + µ − with rate assuming vanishing electron and muon mass and hence no interference terms.Dedicated searches for this decay mode at Belle [23] yield the strong limits |y LL µeτ e |, . . ., |y RR τ eµe | < 0.0096/TeV 2 (Tab.I).Belle II is expected to reach BR(τ − → µ + e − e − ) < 2.3 × 10 −10 with 50 ab −1 [54,55], which can probe the Wilson coefficients down to (29.0 TeV) −2 .

VIII. CONCLUSIONS
Tests of the SM's predicted lepton flavor conservation are among the best probes of new physics, notably in neutrinoless decays of muons and tauons.Not all lepton flavor violation comes with such clean signatures though: effective operators with ∆L α = 2 are much harder to probe, even though they already arise at mass dimension d = 6 in the SMEFT and, of course, violate lepton flavor.8 of these 21 operators give rise to the clean neutrinoless decays τ − → e − e − µ + , µ − µ − e + , while two more induce muoniumantimuonium conversion.The remaining 11 operators are rarely discussed, presumably because they are harder to detect.We have shown that 8 of them can be tested through lepton-flavor-universality violations, i.e. by comparing leptonic decay rates involving neutrinos, see Tab.I.This leaves only 3 operators that are currently unconstrained, or at least with such weak constraints that the use of effective field theory is questionable; these require future colliders for unambiguous tests, either in the form of a Z factory or like-sign electron or muon colliders.Most of the other operators will be tested more thoroughly at Belle II via searches for τ − → e − e − µ + , µ − µ − e + as well as improved measurements of τ → eνν and τ → µνν that feed into lepton-flavor-universality tests.Upcoming muonium experiments such as MACE take care of two μμee operators.The present study serves as a reminder that lepton flavor violation could still be hidden at low scales in comparably murky observables.

FIG. 1 :
FIG. 1: Example Feynman diagram that shows the conversion of an lµγα ℓe lµγαℓe operator (filled circle) to a Lµγ α Le lµγαℓe operator at loop level using the SM Higgs interactions.

TABLE I :
Current limits on the magnitudes of the 21 ∆Lα = 2 dimension-six Wilson coefficients as well as the corresponding processes and the violated quantum numbers.Details are given in the text.