Searching for Lepton Flavour (Universality) Violation and Collider Signals from a Singly-Charged Scalar Singlet

A singly charged $SU(2)_L$ singlet scalar ($\phi^\pm$) can have only flavour off-diagonal couplings to neutrinos and charged leptons, therefore necessarily violating lepton flavour (universality). In fact, it gives a (necessarily constructive) tree-level effects in $\ell\to\ell^\prime\nu\nu$ processes, while contributing to charged lepton flavour violating only at the loop-level. Therefore, it can provide a common explanation of lepton flavour universality violation in $\tau\to\mu\nu\nu/\tau(\mu)\to e\nu\nu$ and the Cabibbo Angle Anomaly. For the best fit region we predict ${\rm Br }[\tau\to e\gamma]$ to be of the order of a few times $10^{-11}$ while ${ \rm Br}[\tau\to e\mu\mu]$ can of the order of $10^{-9}$ for order one couplings. Furthermore, we derive a coupling-independent lower limit on the scalar mass of $\approx 200\,$GeV, by recasting LHC slepton searches. In the scenario preferred by low energy precision data, the lower limit is even strengthened to $\approx300\,$GeV. Finally, we point out that our scenario can be tested by re-interpreting DM mono-photon searches at future $e^+e^-$ colliders.

Interestingly, not only the CAA can be explained by a constructive NP contribution to the SM µ → eν µ νe amplitude, but also the analogous tau decays τ → µν τ νµ prefer a constructive NP effect at the 2 σ level [45].Such an effect can be most naturally generated at tree-level, as loop effects are strongly constrained by LEP and LHC data.Furthermore, as data require NP to interfere constructively with the SM, there are only four possible NP candidates 1 : vector-like leptons [39], a left-handed vector SU (2) L triplet [46], a left-handed Z with flavour violating couplings [47] and a singly charged SU (2) L singlet scalar.Interestingly, the last option even gives a necessarily constructive effect and due to hermiticity of the Lagrangian automatically violates lepton flavour (universality).Furthermore, as a singly charged scalar cannot couple to quarks and only generates charged lepton flavour violation at the loop level, it is weakly constrained experimentally by other processes and can therefore potentially explain the CAA and the hints for LFU violation in τ decays.This letter is thus dedicated to the study of the phenomenology of the singly charged SU (2) L singlet scalar in the light of the hints for LFU violation.
Singly charged scalars have been proposed within the Babu-Zee model [48,49] and studied in Refs.[50][51][52][53][54][55][56][57][58][59] as part of a larger NP spectrum, mostly with the aim of generating neutrino masses at loop-level.Here, we focus on the SM supplemented only by the singly charged scalar (which constitutes a UV complete model) and perform a comprehensive analysis of flavour and collider constraints in the context of the existing hints for LFU violation.

II. MODEL AND OBSERVABLES
As motivated in the introduction, we supplement the SM by a SU (2) L × SU (3) C singlet φ + with hypercharge +1.Interestingly, this allows only for Yukawa-type interaction with leptons but not with quarks.Here L is the left-handed SU (2) L lepton doublet, c stands for charge conjugation, a and b are SU (2) L indices, i and j are flavour indices and ε ab is the two-dimensional anti-symmetric tensor.Note that without loss of generality, ξ ij can be chosen to be anti-symmetric in flavour space, ξ ji = −ξ ij , such that ξ ii = 0 and our free parameters are ξ 12 , ξ 13 and ξ 23 .
In addition, there can be a coupling to the SM Higgs doublet λH † Hφ + φ − which contributes to the mass m φ but otherwise only has a significant impact on h → γγ.

A.
→ νν The SM decay of a charged lepton into a lighter one and a pair of neutrinos is modified at tree-level in our model.Applying Fierz identities (see e.g.Ref. [60]) one can remove the charge conjugation and transform the amplitude to the V −A structure of the corresponding SM amplitude.Taking only into account interfering effects with the SM we have This has to be compared to [45] A(τ → µν ν) A(µ → eν ν) EXP = 1.0029 (14), with the correlations also given in Ref. [45].Furthermore, the effect in A(µ → e ν µ ν e ) leads to a modification of the Fermi constant which enters not only the electroweak (EW) precision observables but also the determination of V ud from beta decays.Super-allowed beta decays provide the most precise determination of This value of V β us can now be compared to V us from kaon [64] and tau decays [45] 2 Alternative determinations can be found in Refs.[43,61].In addition, there is the possibility of "new nuclear corrections" (NNCs) [62,63].However, as this issue is debated, we will not consider them here for the sake of argument (i.e.pointing out the potential NP implications).
which are significantly lower.This is what constitutes the CAA.The tension can be alleviated by the NP effect given by where V L us(ud) is the value appearing in the CKM matrix.As G F enters also the calculation of the EW gauge boson masses and Z pole observables, a global fit is necessary.Adding the determinations of the CKM elements to the these standard EW observables (see e.g.Ref. [65] for details on our input and implementation) calculated by HEPfit [66] we find δ(µ → eνν) = 0.00065 (15) . B.

→ γ
The singly charged scalar generates → γ (see Fig. 1).Using the results of Ref. [67] we obtain with Γ µ being the total width of the muon and In what follows we will neglect the mass of the electron and thus c eµ L .Similarly, the expressions for τ → µ(e)γ can be obtained by a straightforward exchange of indices.The current experimental limits at 90% C.L. are [68][69][70]: Note that in principle also contributions to anomalous magnetic moments of charged leptons are generated.However, since the effect in our model is not chirally enhanced, the effect is numerically small and can be safely neglected.Interestingly, note that the φ ± interactions do not generate electric dipole moments (EDMs) (disregarding very small quark and neutrino effects already present in the SM) and therefore automatically agree with the latest very stringent bound on the electron EDM from measurements of Rb atoms [71].

C. → ( ) ( )
The singly charged scalar contributes to three-body decays to charged leptons at the loop level.Here the dominant contribution for sizable couplings ξ is the box diagram shown in Fig. 1.For concreteness, we give the results for τ → 3e and τ → µee, while the other decays can be obtained by an appropriate exchange of the flavour indices: where Γ τ is the total decay width of the tau.Here we did not include the small on-and off-shell photon contributions (they are given in appendix A, together with our results for µ → e conversion) and we did not give the branching ratios for the decays involving more than one flavour change (such as τ → eµe) which must be tiny in our model due to the measured smallness of µ → eγ.The corresponding experimental bounds (95%

D. LHC searches
The singly charged SU (2) L singlet scalar has the same quantum numbers as the right-handed slepton in supersymmetry.Therefore, bounds from direct searches for smuons and selectrons can be recast to set bounds on our model [76][77][78].The dominant contribution is given by Drell-Yan pair production of φ ± , represented by the Feynman diagram in Fig. 2. We assume that interference with the SM background (mostly W + W − production in this case) can be neglected in the limit of a large enough m φ and a narrow φ ± width.
For the reinterpretation of bounds, we consider the most recent ATLAS analysis [79] with 139/fb of data, searching for final states with an oppositely charged lepton pair (e + e − or µ + µ − ) and missing transverse energy.The search targets sleptons decaying into leptons and neutralinos, which corresponds to our setup in the case of a vanishing neutralino mass.The ATLAS bounds on the right-handed slepton mass in this limit is ≈ 425 GeV for both the e + e − and µ + µ − channels and for a 100% branching ratio of the slepton into the given channel.To reinterpret this result, we have simulated the pair production cross-section at leading order with MG5 aMC [80] and rescaled it with a constant K-factor, obtained by matching our values with the production cross section to the one given by ATLAS (for a right-handed slepton mass of 500 GeV).A conservative error of 10% has been added on the cross-section to account for the differences in the simulation procedures.Fig. 3 shows the bounds in the m φ -Br(φ ± → e ± (µ ± )ν) plane extracted from the analysis of the e + e − and µ + µ − channels of ATLAS.The red (green) hatched region is excluded by the e + e − (µ + µ − ) channel.The colored bands indicate the change in the limit obtained by linearly varying the efficiency calculated on the value of the ATLAS bound by ±40% between 200 GeV and 425 GeV for m φ .The solid line corresponds to the estimated limit without taking into account the additional uncertainties discussed above.As due to the anti-symmetry of the couplings the sum of the branching ratio to muons and electrons can never be smaller than 1/2, we can set a coupling independent limit ≈ 200 GeV on m φ .

E. Mono Photon Searches
We can also set a lower limit of |ξ 2  12,13 |/m 2 φ from dark matter (DM) searches at LEP with mono-photon signatures.Using the DELPHI analysis of Refs.[81,82], Ref. [83] was able to exploit the kinematic distributions to obtain a bound of ≈ 480 GeV for zero DM mass on the DM mediator mass for unit coupling strength and vectorial interactions (in the effective theory).Taking into account that we have neutrinos and therefore interference with the SM, this translates into a bound of ≈ 1 TeV.
Assuming that m φ is sufficiently above the LEP production threshold, as suggested by LHC searches discussed above, we can recast these results.Taking into account that we have a left-handed vector current, we find (|ξ 2  12,13 |)/m 2 φ 1/(175 GeV) 2 .This bound would be strengthened if ξ 12 and ξ 13 are simultaneously non zero, but further weakens as m φ approaches the LEP beam energy.Therefore, it is not yet competitive with flavour bounds but could be significantly improved at future e + e − colliders.

III. PHENOMENOLOGY
Let us start our phenomenology by considering the NP effect in τ → µνν and µ → eνν.The currently preferred regions (at the 1 σ level) for δ(τ → µνν) and δ(µ → eνν) is shown in Fig. 4 as the orange and red regions, respectively, while the combined region is shown in green.Note that for any point within the combined region, ξ 13 must be vanishingly small in order not to violate the bounds from µ → eγ or µ → e conversion.Therefore, we can neglect its effect in the following.
This means that in this setup (ξ 13 0) and we have Br(φ + → µ + ν) = 0.5, which leads to a bound of ≈ 300 GeV from the µ + µ − channel.This bound could be further improved at the HL-LHC [84] (around 30% more as shown in Fig. 3, rescaling the ATLAS bound for an integrated luminosity of 3/ab) or at the FCC-hh [85] where, considering the projections for other scenarios in absence of a dedicated analysis, we estimate a potential improvement up to a factor of few [86].Furthermore, we can correlate δ(τ → µνν) and δ(µ → eνν) directly to τ → eγ, as indicated by the magenta lines in Fig. 4, while Br[τ → eγ] ≈ 0. The predicted branching ratio for τ → eγ is of the order of a few times 10 −11 .Furthermore, we can also obtain correlations with τ → 3e and τ → eµµ.Since the branching ratio of the latter is predicted to be larger (for the region preferred by data), we depict it in Fig. 4 as black lines.However, here the correlations is not direct since it depends on m φ and we find Br[τ → eµµ] ≈ 10 −10 m 4 φ /(5 TeV) 4 .Interestingly, this lies within the reach of BELLE II [87] or the FCCee [88].We also depict constant values of |ξ 2  12 |/m 2 φ as dashed blue lines.Even though their values are significantly below the LEP bounds discussed above, future e + e − colliders like ILC [89], CLIC [90], CEPC [91] or FCC-ee [92] could test the predicted mono-photon signature.In particular, the ILC can improve the bound on the Wilson coefficient by a factor 50 [93], CEPC by a factor 40 [94] and even bigger improvements could be expected CLIC, and FCC-ee for which a dedicated study is stongly motivated.

IV. CONCLUSIONS
In this article we studied the phenomenology of the singly charged SU (2) L singlet scalar in the light of the hints for LFU violation encoded in the measured values for τ → µνν/τ (µ) → eνν and V us from super allowed beta decays (using CKM unitarity) compared to V us from Kaon and tau decays.Interestingly, the singly charged scalar has only three free couplings (due to hemiticity of the Lagrangain) and is therefore very predictive: it necessarily violates LF(U) and leads to a positive definite effect in → νν as preferred by data while there is no (pure) NP contribution to the otherwise so stringently constraining electron EDM.
Recasting ATLAS searches for right-handed sleptons we derive a coupling independent limit of m φ ≈ 200 GeV.In the region preferred by LFU violation in tau decays and the CAA, ξ 13 ≈ 0 is required by µ → eγ, leading to an LHC bound of m φ ≈ 300 GeV.Concerning LFV, we predicted Br[τ → eγ] to be of order of a few times 10 −11 and Br[τ → eµµ] ≈ 10 −10 m 2 φ /(5 TeV) 2 .Furthermore, we pointed out that our model can be tested not only by future experiments searching for these LFV decays, but also via direct searches at the HL (HE) LHC and FCC-hh and by mono photon searches at future e + e − colliders.In particular, FCC-hh could improve the bound on m φ and push the predicted value for Br[τ → eµµ] towards the region observable by BELLE II and FCC-ee, providing a prime example of complementarity between low energy precision experiments and direct searches for NP. and for the latter we use [96] Γ capt Au = 8.7 × 10 −18 GeV , Γ capt Al = 4.6 × 10 −19 GeV .(A7) The experimental limit on µ → e conversion is [68] Γ conv Au Γ capt Au < 7.0 × 10 −13 SINDRUM II . (A8) Adding the on-shell (see Eq. 9) and off-shell (see Eq. A2) photon contributions to the τ -decays of Eq.

FIG. 1 .
FIG.1.Feynman diagrams showing the contribution of φ ± to µ → eνµ νe, µ → eγ and τ → µee.The corresponding diagrams for analogous processes with different flavours are not depicted but can be deduced by straightforward substitutions.

FIG. 2 .
FIG. 2. Diagram showing the Drell-Yan pair production of singly charged scalars.Their decay necessarily give rise to a signal with an oppositely charged lepton pair and missing transverse energy.