Upper limits on branching ratios of the lepton-flavor-violating decays $\tau \to \ell \gamma\gamma$ and $\tau \to \ell X$

From analysis of data produced by the BABAR experiment, the first upper bounds (90% C.L.) were obtained on the branching ratios $Br(\tau \to e \gamma\gamma)<2.5 \times 10^{-4}$ and $Br(\tau \to \mu \gamma\gamma)<5.8 \times 10^{-4}$. In addition, improved upper bounds (95% C.L.) were found on branching ratios $Br(\tau \to e X)<1.4 \times 10^{-3}$ and $Br(\tau \to \mu X)<2.0 \times 10^{-3}$, where $X$ is an undetected weakly interacting boson with mass $m_X<1.6$ GeV/$c^2$.


I. INTRODUCTION
The violation of lepton family number has been firmly established by the observation of neutrino oscillations, which also implies charged lepton family (flavor) number violation (CLFV).Although no CLFV has been observed yet, it is of fundamental interest, and searches for CLFV processes continue to be pursued.In the Standard Model (SM) extended to include massive neutrinos (generically denoted the νSM), the branching ratios for CLFV decays such as µ → eγ, µ → eγγ, µ → eeē, τ → ℓγ, and τ → ℓℓ ′ l′ , where ℓ = e, µ and ℓ ′ = e, µ, are many orders of magnitude below the level where they could be observed in existing or planned experiments.This means that searches for these decays and similar CLFV processes are of great interest as probes of physics beyond the νSM (BSM).
Current upper bounds 1 on some CFLV τ decay modes are listed in Table I.In this paper, we used existing data to set the first upper limits on the branching ratios of the CLFV decays τ → eγγ and τ → µγγ.We also examined the branching ratios for the decays τ → ℓX.

II. THEORETICAL BACKGROUND A. CLFV in the νSM
To accommodate the observed neutrino oscillations and associated violation of lepton family number in the neutrino sector, the (renormalizable) SM Lagrangian can be modified by adding a number n s of electroweak-singlet neutrino fields ν i,R , i = 1, ..., n s , conventionally written as right-handed chiral fermions.With these, Yukawa terms are formed with the left-handed lepton doublets which, via the vacuum expectation values of the Higgs field, yield Dirac-type mass terms for neutrinos.The electroweak-singlet neutrinos also generically lead to Majorana mass terms of the form i,j . The diagonalization of this combination of Dirac and Majorana mass terms yields the neutrino mass eigenstates.The resultant unitary transformation relating the lefthanded chiral components of the mass eigenstates of the neutrinos, ν i,L , to the weak eigenstates, ν a,L , is given by (2.1) The property that U (ν) is different from the identity gives rise to neutrino oscillations and the associated violation of lepton family number in the neutrino sector.(There is, in general, also violation of total lepton number in the νSM, due to the presence of Majorana mass terms.) The diagonalization of the charged lepton mass matrix involves another unitary matrix U (ℓ) , and the product of (the adjoint of) U (ℓ) and U (ν) determines the form of the weak charged current: where U is the lepton mixing matrix, As an example of CLFV in the νSM, the branching ratio for ℓ a → ℓ b + γ is [15,16] where α em = e 2 /(4π) is the fine structure constant, and a is the family or generation index, with ℓ 1 ≡ e, ℓ 2 ≡ µ, and ℓ 3 ≡ τ .Using current data on neutrino masses and lepton mixing, the resultant SM predictions for the branching ratios for the decays µ → eγ, τ → eγ, and τ → µγ have values < ∼ 10 −53 , far below a level that could be observed in any existing or planned experiment.In passing, we recall the current upper limits on CLFV muon decays, Br(µ → eγ) < 4.2 × 10 −13 from the MEG experiment at PSI [17], Br(µ → eγγ) < 0.72 × 10 −10 from the Crystal Box experiment at LAMPF [18], and Br(µ → eeē) < 1.0 × 10 −12 from the SINDRUM experiment at SIN/PSI [19].Since the decay ℓ a → ℓ b γγ involves emission of a second photon, as compared with ℓ a → ℓ b γ, it follows that for the νSM, up to logarithmic terms, Although CLFV processes are predicted to be unobservably small in the νSM, there are many models of physics beyond the νSM that generically predict CLFV at observable rates.While none of these models has been confirmed by experiment, they remain of interest since they address incomplete aspects of the SM.One such aspect concerns the Higgs mass.There is a fine-tuning problem associated with this quantity since one-loop corrections to the Higgs mass squared are quadratically sensitive to the highest mass scale in an ultraviolet completion of the SM, such as a grand unified theory.Two early ideas for BSM physics that addressed this problem were supersymmetry (SUSY) and dynamical electroweak symmetry breaking (EWSB), and both of these generically predicted CLFV (as well as a number of flavor-changing neutral-current processes) at observable levels.For example, supersymmetric extensions of the SM predicted the decay µ → eγ to occur at observable levels [20][21][22][23], and this is also true of τ → ℓγ.Early SUSY models with light neutralinos χ allowed the decay µ → e χ χ, which would be distinct from SM µ decay [24].Substantial contributions to µ → eγγ and τ → ℓγγ would also be expected in such SUSY theories.Although searches for supersymmetric particles at the Fermilab Tevatron and at the CERN Large Hadron Collider (LHC) have yielded null results so far, there still remains the possibility of supersymmetry characterized by a SUSY-breaking scale that is larger than the electroweak scale.
Dynamical EWSB models also predict CLFV processes at possibly observable levels [25][26][27].A relevant property of reasonably ultraviolet-complete dynamical EWSB models is the generic presence of sequential stages of breaking of an asymptotically free chiral gauge symmetry in the ultraviolet.The feature that the third generation is associated with the lowest of these scales could give rise to enhanced CLFV procees involving the τ lepton [28].Modern versions of dynamical EWSB models typically involve quasi-conformal behavior, which can result naturally from an approximate infrared fixed point of the renormalization group equations describing the strongly coupled vectorial gauge interaction [29][30][31].In general, in these models, the observed Higgs is a composite state.These dynamical EWSB models are tightly constrained by precision electroweak data, the observed agreement of the Higgs boson with SM predictions, and, more generally, the non-observation of any BSM Higgs properties at the LHC.
A large variety of other BSM theories predict CLFV effects at potentially observable levels.These could have the potential to alter the νSM relation (2.5).For example, in theories with doubly charged leptons, the ratio of branching ratios Br(τ → ℓγγ)/Br(τ → ℓγ) can be substantially enhanced relative to the O(α em ) relation in Eq. (2.5), just as was true of the ratios Br(µ → eeē)/Br(µ → eγ) and Br(µ → eγγ)/Br(µ → eγ) [32]; some recent studies of theories with doubly charged leptons [33] provide experimental constraints.These theories could also lead to an enhancement of τ → ℓπ 0 , which, via the π 0 → γγ decay, could contribute to a ℓγγ final state and hence to an overall τ → ℓγγ decay.
Of particular interest for τ → ℓX decays are models with a light pseudo-Nambu-Goldstone boson (NGB) or a massless NGB that can couple to fermions in a flavorviolating manner [34,35].These arise in models that hypothesize a "horizontal" symmetry mixing SM fermions transforming in the same manner under the SM gauge group, G SM = SU(3) c ⊗ SU(2) L ⊗ U(1) Y , namely the sets (e, µ, τ ) L , (e, µ, τ ) R , and so forth for the neutrinos and quarks.With the hypothesized generational (i.e., family) symmetry taken to be global, a consequence would be that the spontaneous breaking of the flavor symmetry would lead to massless, spinless NGB(s) (often called familons).In the presence of some explicit breaking of the generational symmetry, the spontaneous breaking yields light NGB(s), with mass(es) determined by the relative sizes of explicit and spontaneous symmetry breaking.These NGBs are often called "axion-like particles" (ALPS).CLFV effects may also be associated with spontaneous breaking of total lepton number and resultant majorons [36][37][38][39].Some more recent studies and reviews include [40][41][42][43][44][45][46][47][48][49].
Models featuring extra Z ′ vector bosons with flavornon-diagonal couplings can yield CLFV effects at observable levels (e.g., [50,51]).These could contribute to CLFV decays such as τ → ℓγγ and τ → ℓγ.CLFV processes have also been studied in models with extra (spatial) dimensions and fermion fields having localized wave functions in these extra dimensions [52,53].An appeal of these models is that they can produce a strong hierarchy in SM fermion masses via moderate separation of fermion wave function centers in the extra dimensions [54,55].Connections between reported anomalies in B meson decays, e − µ universality violation, and models with CFLV have been discussed in a number of studies and are reviewed, e.g., in [56].Although dark matter is, in principle, independent of CFLV, there may be connections between these in certain models [57].

III. τ → ℓγγ
In the following we discuss the angular distribution expected for the decay products τ → ℓγγ.Then, we use existing data on searches for τ → eγ and τ → µγ from the BABAR experiment at SLAC [58] to derive the first upper limits on the branching ratios for these decays, τ → eγγ and τ → µγγ.(See also the recent result from Belle [59], which improves slightly on the upper limit on B(τ → µγ) in [58].)In abstract notation, these decays are of the form ℓ a → ℓ b γγ with the generational indices a = 3 and b = 1, 2, respectively.
For our analysis, we do not assume a particular BSM theory, but instead use an effective field theory method.In general, for a decay of the form ℓ a → ℓ b γγ, the operators that contribute to leading order to the effective Lagrangian L ef f,ℓaℓ b γγ are lepton bilinears contracted with F αβ F αβ or F αβ F αβ (with coefficients given in Eq. (3.2) below), where F αβ is the electromagnetic field strength tensor and F αβ = (1/2)ǫ αβλρ F λρ is its dual.At more suppressed levels, there are additional operators involving derivatives.In d spacetime dimensions, the mass dimension of an operator O comprised of a lepton bilinear, a product of F F or F F , and where cO is dimensionless and Λ O denotes a scale of BSM physics responsible for the appearance of the operator O.
Since in both of the decays τ → ℓγγ with ℓ = e or ℓ = µ, m ℓ ≪ m τ , the only mass that enters into the phase space kinematics of the decay is m τ .Because the derivatives yield factors of momenta in the amplitude, and the sizes of these momenta are set (in the τ rest frame) by m τ , it follows that the contribution of an operator with n ∂ derivatives is suppressed by the factor (m τ /Λ O ) n ∂ .The agreement of the νSM with current data implies that the scales Λ O are much larger than m W,Z , and hence (m τ /Λ O ) n ∂ ≪ 1.Therefore, operators with derivatives are expected to make a negligible contribution to the amplitude for τ → ℓγγ.The effective Lagrangian for τ → ℓγγ can then be written, retaining non-negligible terms, as where the subscript LR on c τ ℓγγ,LR;F F refers to the chirality structure in the associated lepton bilinear, [ lL τ R ], and similarly for the other coefficients.Without loss of generality, we will introduce a single effective mass scale Λ τ ℓγγ to characterize the CFLV physics responsible for the decay τ → ℓγγ; any differences in the actual mass scales characterizing different operators O are absorbed into the values of the dimensionless coefficients cO .Then Eq. (3.1) reads for each of the coefficients c τ ℓγγ,LR;F F , c τ ℓγγ,RL;F F , c τ ℓγγ,LR;F F , and c τ ℓγγ,RL;F F in L ef f,τ ℓγγ .Let us denote the four-momenta of the τ , the final-state charged lepton ℓ, and the two photons as p τ , p ℓ , k 1 , and k 2 , respectively, and Lorentz-scalar products of two fourvectors as p τ • p ℓ , etc. Let us further denote the matrix element for this decay as M τ →ℓγγ .As usual, the amplitude is Bose-symmetrized with respect to the interchange of the identical bosons (photons) in the final state.With the above input L ef f,τ ℓγγ , the square of the amplitude has a kinematic factor (p , and the differential decay rate is where E ℓ , E γ1 , and E γ2 are the energies of the daughter lepton ℓ and the two photons, respectively, and θ γγ is the angle between the 3-momenta of the photons (i.e., cos We note that a two-photon final state also arise as a radiative correction to the decay τ → ℓγ, via emission of the second photon from the initial τ or from the final-state ℓ, where ℓ = e or µ.An event of this type would have an angular distribution different from that of an event in which the two photons originated directly as a consequence of the BSM physics, and the associated L ef f,τ ℓγγ in Eq. (3.2).Events in which a second photon is emitted as a radiative correction to a τ → ℓγ decay were considered by the BABAR experiment [58], were modelled by the event simulation programs used in that experiment, and were taken into account in their upper limits on Br(τ → ℓγ).
The BABAR experiment searches for τ → ℓγ decays [58] were performed at the SLAC PEP-II e + e − storage rings, primarily using center-of-mass (c.m.) energy √ s ≃ 10.6 GeV at the Υ(4S) resonance.The BABAR detector is described in Ref. [60].Charged particles were reconstructed as tracks with a silicon vertex tracker and a drift chamber inside a 1.5 T solenoidal magnet.A CsI(Tl) electromagnetic calorimeter identified electrons and photons, and a ring imaging Cherenkov detector identified charged pions and kaons.The flux return of the solenoid was instrumented with resistive plate chambers, and limited streamer tubes were used to identify muons.
Events ascribed to the reaction e + e − → τ + τ − were selected, and events of the form τ ± → ℓ ± γ, were identi-fied by a ℓ, γ pair with an invariant mass and total energy in the c.m. frame close to m τ = 1.777GeV/c 2 and √ s/2, respectively.Another τ ± decay in the opposite detector hemisphere was used as a tag.Important backgrounds arose from the reaction e + e − → τ + τ − γ yielding a hard photon when one τ underwent a SM decay to an ℓ and a neutrino anti-neutrino pair.Other backgrounds for the τ → ℓγ search arose from the reaction e + e − → ℓ + ℓ − γ and from hadronic τ decays with particle mis-identification.
The signal-side hemisphere was required to contain one photon with c.m. energy > 1 GeV, with no other photon with energy > 100 MeV in the laboratory frame.The signal had to contain one track identified as an electron or muon within the calorimeter acceptance with c.m. momentum less than 0.77 √ s/2.Muons were also required to have momentum greater than 0.7 GeV/c in the laboratory frame.In addition, the cosine of the opening angle between the signal track and signal photon was required to be less than 0.786 characterizing the back-to-back distribution of τ → ℓγ events in the τ rest frame.Neural net cuts were also applied to the BABAR data.
Signal decays were identified by two kinematic variables: the energy difference ∆E = E c.m. ℓγ − √ s/2, where ℓγ is the c.m. energy of the ℓγ pair, and the beam energy constrained τ mass (mEC), obtained from a kinematic fit after requiring the c.m. τ energy to be √ s/2; the origin of the γ candidate was assigned to the point of closest approach of the signal lepton track to the e + e − collision axis [58].
Limits on the decays τ → eγγ and τ → µγγ were obtained using the results of the BABAR experiment searching for τ → eγ and τ → µγ decays.Using Eq. (3.4), we simulated 1×10 7 events for each τ → ℓγγ process applying momentum and energy resolutions (smearing) for the charged track and photons as reported by Ref. [60] and applying the cuts indicated above (except for the neural net cuts) to select events.Then, without the resolution effects applied, we constructed the mEC and ∆E variables for the τ → ℓγγ events which passed the cuts and were within the BABAR detector acceptance.The mEC and ∆E variables were then smeared according to their reported resolutions [58].
Figure 1 shows a plot of mEC vs. ∆E for simulated τ → eγγ events after the cuts and resolution smearing.Compared with τ → eγ in Ref. [58], the plot of mEC vs. ∆E for τ → eγγ is widely distributed due to the requirement for the second gamma to have E < 100 MeV if in the signal side hemisphere or to be outside the detector acceptance.The red ellipse in Figure 1 represents the signal region for τ →eγ used by the BABAR analysis including the observed shift in position due to radiative effects [58].This elliptical region contains the simulated τ → eγγ events which would have been classified as consistent with the τ → eγ signal representing an efficiency of ǫ eγγ = 1.2 × 10 −4 compared to ǫ eγ = 0.50 for our simulation efficiency for τ → eγ.The estimated uncertainty in the ratio ǫ eγ /ǫ eγγ (used below) is approximately 10%.

IV. τ → ℓX
In this section we obtain new constraints on the decays τ → ℓX where X is a weakly interacting neutral boson that escapes without being detected.The latter condition is satisfied if the lifetime τ X is sufficiently long or if X decays invisibly.Theoretical motivations for searching for such emission were discussed in Section II.
The signature for the decay τ → ℓX is a monochromatic peak in the energy of the daughter lepton ℓ in the τ rest frame at the value where m X is the mass of the X particle.This type of search involves an analysis of the energy or momentum spectrum of the daughter lepton in the τ decay.A different approach to setting an upper limit on Br(τ → eX) and Br(τ → µX) is based on the fact that if such events occurred and were included together with events from the corresponding SM leptonic decays of the τ , they would alter the observed rates of the respective decays.Measurements of the individual branching ratios for τ → ν τ eν e and τ → ν τ µν µ have been carried out, with the results [14] Br(τ → ν τ eν e ) = 0.1782 ± 0.0004 (4.2) and 3) The measured branching ratios (4.2) and (4.3) and the τ lifetime τ τ = (2.903± 0.005) × 10 −13 s [14] can be used to obtain the decay rates to compare with SM calculations.Using the formulation in [73], the calculated values for the branching ratios (denoted by superscript (c)) are Br (c) (τ → ν τ eν e ) = 0.17781 ± 0.00031 and Br (c) (τ → ν τ µν µ ) = 0.17293 ± 0.00030.Then, the ratios of experimental to calculated decay rates are [74,75] S τ →e = Γ τ →e /Γ   These limits are plotted in Fig. 2 along with the previous results from the ARGUS experiment [13].Using the measured τ → ν τ ℓν ℓ branching ratios in Eqs.(4.2) and (4.3), we found Br(τ → eX) < 1.4 × 10 −3 and Br(τ → µX) < 2.0 × 10 −3 .Our new upper bounds (4.8) and (4.9) yield improved lower bounds on the weighted decay constants F τ ℓ , ℓ = e, µ, appearing in the effective Lagrangian for τ → ℓX.For example, in the notation of Table 1 of Ref. [49], at an illustrative mass m X = 0.6 GeV, our bounds increase the lower limit on F τ e from 4.3 × 10 6 GeV to ∼ 7 × 10 6 GeV and increase the lower limit on F τ µ from 3.3 × 10 6 GeV to ∼ 6 × 10 6 GeV.The limits found for τ → ℓX decays also apply to three-body decays of the form τ → ℓXX, for which no previous bounds have been reported.

V. CONCLUSIONS
Using an analysis of data from searches for τ → eγ and τ → µγ performed by the BABAR experiment, we have obtained the first upper limits on the branching ratios Br(τ → eγγ) and Br(τ → µγγ).We have also presented improved upper limits Br(τ → ℓX) where ℓ denotes e or µ and X is a weakly interacting boson with mass m X < 1.6 GeV/c 2 that escapes detection.We expect that these decay modes can be searched for with considerably higher sensitivity at Belle II [77].

Figure 1 .
Figure 1.mEC vs. ∆E for simulated τ → eγγ events.The red ellipse indicates the signal region where events would have passed cuts for τ → eγ [58].