Disentangling new physics contributions in lepton flavour violating tau decays

The possibility to discriminate between different operators contributing to lepton flavour violating tau decays is discussed within an effective field theory framework. Correlations among decay rates in different channels as well as differential distributions in many-body decays are considered. Recent developments in the determination of the hadronic form factors for $\tau \rightarrow \ell \pi \pi$ ($\ell = e, \mu$) decays are incorporated in the analysis. The above issues are exemplified by considering a Higgs-like boson with lepton flavour violating couplings. Implications of the search for lepton flavour violating Higgs decays performed recently by the CMS collaboration are also discussed.


Introduction
The observation of charged lepton flavour violating (LFV) transitions would be a clear indication of physics beyond the Standard Model. We will be concerned here with LFV τ decays. The Belle and BaBar collaborations have stopped collecting data, putting bounds on the branching ratio (BR) of these transitions at the level of 10 −7 -10 −8 . In the future, Belle-II is expected to bring the search for LFV τ decays to a new level of sensitivity. The LHCb collaboration could also play an important role for some of these processes, like τ → 3µ.
Nothing guarantees that LFV τ decay rates are within the reach of current and/or future experimental facilities. It is however well known that many new physics models predict large rates for charged lepton flavour violating transitions, which could be observed experimentally [1]. In case these transitions are observed in the future it will be crucial to address the following question: 1

Speaker
• How can we discriminate different kinds of new physics in LFV τ decays? This is discussed in Secs. 2 and 3 by considering correlations among different LFV τ decay rates and differential distributions in three-body decays. The approach taken here is to consider an effective Lagrangian describing τ-µ LFV transitions. A series of benchmark scenarios in which only one type of effective operator is relevant will then be defined to frame the discussion. We do not consider τ-e transitions but the arguments presented here also hold in general for these processes. As a specific new physics scenario giving rise to different operators at the low energy scale, we consider the possibility that the recently discovered Higgs boson has sizable LFV couplings. We show that the pion invariant mass spectrum in τ → µπ + π − decays provides a useful handle to unravel the underlying dynamics in this case. A proper determination of the hadronic form factors near the resonance region, as implemented here, turns out to be crucial for such purpose [2,3]. With this scenario in mind, we also explore the complementarity between the energy and intensity frontiers in probing for LFV phenomena by addressing: • What are the implications of the recent search for LFV Higgs decays performed by the CMS collaboration?
This is discussed in Sec. 4. We conclude in Sec. 5.
The results presented in this talk have been obtained in Refs. [3,4].

Effective Lagrangian
We consider the following effective Lagrangian describing τ-µ transitions where the effective dipole terms are contained in and the four-fermion operators are included in The final piece of the effective Lagrangian contains gluonic effective operators Here P L,R = (1 ∓ γ 5 )/2 are the usual chiral projectors, G a µν is the gluon field strength tensor and G a µν its dual. Similarly F ρν denotes the electromagnetic field strength tensor. The Fermi coupling constant is denoted by G F , β L /(4α s ) = −9α s /(8π) and Λ represents the high energy scale of the LFV dynamics.

Discriminating effective operators in LFV τ decays
Different new physics scenarios are expected to generate distinctive patterns for the low-energy Wilson coefficients of the effective Lagrangian describing τ − µ LFV transitions in Eq. (1). Explicit examples of how these effective operators arise after integrating out heavy degrees of freedom have been provided, for example, within the framework of Supersymmetric models [2,[11][12][13], Leptoquark models [10,14] and leftright symmetric models [15].
To explore the discriminating power to different kinds of operators in LFV τ decays, we consider a series of benchmark models where only one type of operator is present or relevant. For simplicity, we also restrict the analysis to the case in which the outgoing muon has a definite chirality. The benchmark models analyzed here are: • Gluonic model (Parity-even): In this model only the Parity-even gluonic operator is assumed to be relevant • Z-penguin model: Dominance of an effective Zpenguin LFV vertex is assumed. The relative size of the Vector couplings is given in this case by with where sin 2 θ W 0.223 is the weak mixing angle.
• Scalar model: The four-fermion scalar operator dominates with a Yukawa-like flavour structure • Dipole model: The dipole operator is assumed to dominate. We set in this scenario Having defined the different benchmark models, we can now proceed to discuss how correlations between LFV τ decay rates and differential distributions in threebody decays provide valuable information about the underlying LFV dynamics. Observables involving polarized τ decays [16] or searches for µN → τX conversion with high-intensity muon beams [17,18] also constitute a complementary handle to unravel the origin of LFV, though they will not be discussed here.

Correlations in the decay rates
Assuming that only one type of effective operator dominates, correlations among different LFV τ decay modes arise. In case only the effective dipole operator is relevant, what would correspond to our Dipole model, LFV τ decay rates will be fixed relative to the radiative decay mode τ → µγ. Fig. 1 shows upper bounds on the BR for different LFV τ decay modes extracted from the non-observation of these transitions, in each of the benchmark models introduced previously. Also shown in Fig. 1 are the current experimental upper bounds in each decay channel considered (blue triangles) as well as an estimate of the limits that could be obtained in future factories. We assume an order of magnitude improvement of the sensitivity at Belle-II (black diamonds). The bound expected from a Super Tau-Charm Factory on BR(τ → µγ) is taken from Ref. [19] (purple square). In the Gluonic model the most sensitive LFV decay mode is τ → µπ + π − , the BR is given in this case by In the Z-penguin model, the most restrictive measurement is that of τ → µρ as can be seen from Fig. 1. The BR for τ → µπ + π − is given by For the Scalar model we obtain The strongest constraint is coming in this case from the present limits on τ → µπ + π − . Finally, for the Dipole model the most sensitive channel is naturally τ → µγ for which: The message of this subsection is clear. If LFV τ decays are observed in the future, a combination of measured LFV τ decay rates together with upper bounds on other non-observed LFV τ decay channels will provide the main tool to discriminate different types of new physics. For a more complete discussion see Ref. [4] and references therein.

Differential distributions in three-body decays
Assuming LFV τ decays are observed, we would like to gain as much information as possible about the underlying new physics. Besides the information provided by a comparison of different LFV τ decay channels, discussed in the previous subsection, a natural step forward would be to exploit differential distributions in LFV three-body decays. Of course, we assume here that such transitions would be observed.
The discrimination of different effective operators in three-body leptonic decays like τ → 3µ has been discussed in detail in Refs. [4,5]. In this case a Dalitz plot analysis could be used to determine the dominant operators. The main obstacle for such analysis would be the low number of collected events, triggering the interest in observables involving polarized τ decays which might be more useful having a small sample of events [16].
Semileptonic decays τ → π + π − also contain information about the underlying new physics in the pion invariant mass spectrum. Counting the number of events as a function of the di-pion invariant mass can be used to disentangle different effective operators. The di-pion invariant mass s = (p π + + p π − ) 2 is kinematically limited in these decays to This raises an important issue. The pion invariant mass can reach values √ s ∼ 1 GeV which are well above the regime of validity of Chiral Perturbation Theory (ChPT) as a low energy effective theory of QCD. In other words, a determination of the hadronic form factors entering in τ → ππ decays based on ChPT alone is not reliable in all the accessible kinematical range. Claims that large deviations from the ChPT predictions are not to be expected in the chiral limit [10], miss the point that even in this limit, ChPT is inadequate to describe the hadronic dynamics for large invariant masses of the pion pair. A proper estimation of the decay rate and the pion invariant mass spectrum can be obtained by a combination of ChPT and dispersive techniques as done in Refs. [2][3][4].

Figs. 2, 3 and 4 show the pion invariant mass spec-
trum in τ → µπ + π − decays for the Gluonic, Scalar and Dipole models respectively. The pink (short-dashed) and gray (long-dashed) bands in these figures denote the experimental cuts on the pion invariant mass used to search for τ → µρ and τ → µ f 0 respectively [20,21]. For the Gluonic and Scalar models we simply show the differential BR as a function of √ s, a peak around the f 0 hadronic resonance is clearly observed in both cases. In Fig. 4 we have plotted for convenience the ratio Note that all the dependence on C D /Λ 2 cancels in dR π + π − . In the Dipole model, the pion invariant mass spectrum is determined by the pion vector form factor and peaks around the ρ mass (m ρ ∼ 770 MeV).

LFV Higgs couplings and semileptonic τ decays
We consider in this section the possibility that the recently discovered Higgs boson with mass around 125 GeV has sizable flavour violating coupling to leptons [22][23][24][25][26][27][28][29][30][31][32][33][34], Such couplings could arise from an extended Higgs sector or from effective operators of dimension six encoding details of the high energy dynamics, see Ref. [3] and references therein. Effective dipole operators appear at low energy via Higgs mediated loop diagrams as that shown in Fig. 5. Scalar and gluonic operators also appear due to the diagrams shown in Fig. 6.  To describe Higgs mediated τ → ππ decays it is crucial to take into account: • The Higgs coupling with strange quarks and the effective coupling to gluons induced by heavy quarks.
• A proper determination of the hadronic form factors in all the kinematical range, specially in the resonance region.
All these points have been considered for the first time in Ref. [3]. Effective gluonic interactions induced by heavy quarks were not being included in previous analyses of Higgs mediated τ → ππ decays. These effects are known to play an important role in the context of µ-e conversion in Nuclei [35,36] and very light Higgs decays [37]. Similarly, effective gluonic interactions play a crucial role for τ → ππ decays [3]. Fig. 7 illustrates the role of the pion invariant mass spectrum in unraveling the origin of LFV in τ decays. In this figure we show the differential decay width for τ → µπ + π − as a function of √ s. The Higgs mediated contribution is shown in blue (dashed) while the photon mediated one is shown in orange (continuous-thick). The total differential rate is shown in red (continuousthin). We have fixed in this case the Higgs mass to 125 GeV, the LFV couplings to |Y h µτ | 2 + |Y h τµ | 2 = 1 and the Higgs diagonal couplings have been taken to be SM-like. The photon mediated contribution includes Figure 7: Pion invariant mass spectrum in τ → µπ + π − decays mediated by a Higgs boson with LFV couplings. The Higgs mass is taken to be M h = 125 GeV, the LFV Higgs couplings are fixed to |Y h µτ | 2 + |Y h τµ | 2 = 1 and its diagonal couplings are assumed to be SM-like. Figure taken from Ref. [3].

Process
( < 0.26 Table 1: Current experimental limits on different LFV τ decays and bounds extracted on possible LFV Higgs couplings. The Higgs mass is fixed at 125 GeV and the Higgs diagonal couplings are taken to be SM-like [3].
two-loop diagrams of the Barr-Zee type calculated in Ref. [41] and recently discussed in Ref. [30]. If these decays are observed, the pion invariant mass spectrum would allow to disentangle the scalar and photon mediated contributions to this process. Current bounds on LFV Higgs couplings from τ → µ decays are summarized in Table 1. Certainly, the strongest bound is coming from the radiative mode τ → µγ. Note however that the decay rate for this process receives very large contributions from two-loop diagrams and is therefore very sensitive to details of the UV completion of the theory. Additional particles with LFV couplings can enter at the same level than the Higgs boson causing interfering contributions. The same happens for τ → 3µ for which the same kind of two-loop diagrams are present, by attaching the photon to a µ + µ − pair. In this sense, we consider the bound coming from τ → µπ + π − as a complementary handle that should not to be neglected in phenomenological discussions. The fact that the extracted limit on the LFV couplings is the same from the µπ + π − and µρ modes is a mere coincidence.
After the Higgs discovery, phenomenological studies analyzed the possibility to observe LFV Higgs decays at the LHC [30,31,42]. Recently, the CMS collaboration has performed a search for LFV Higgs decays h → τµ [43,44], setting the following upper bound at 95% CL., It is important to stress a couple of points when interpreting this bound as limits on the LFV Higgs couplings: • All the diagonal Higgs couplings are taken to be SM-like, meaning that g hVV = (g hVV ) SM and g hf f = (g hf f ) SM . Here VV = (W + W − , ZZ) and f stands for any of the SM fermions. The only exotic Higgs couplings are those given in Eq. (16).
• The decay h → τµ is assumed to be the only relevant non-SM decay channel of the Higgs boson. That is, the total width is When working with a specific new physics model none of these assumptions will remain valid in general. In this sense, the bound on |Y h µτ | 2 + |Y h τµ | 2 presented previously should be interpreted with care. To have an idea of the complementarity between the search for LFV Higgs decays and the search for LFV τ decays, we translate the bounds on BR(h → τµ) into limits over the low energy transitions: Where again, we have kept the diagonal Higgs couplings to their SM-value. Two-loop contributions of the Barr-Zee type have been taken into account. See discussions in Ref. [3] for more details on the evaluation of the decay rate for these processes. Fig. 8 shows a comparison between the limits on LFV Higgs couplings extracted from h → τµ with those from τ → µπ + π − . Predictions for BR(τ → µπ + π − ) are shown with the same conventions than in Fig. 7: Higgs mediated (blue), photon mediated (orange) and total rate (red). The main message is that Higgs mediated LFV τ → µ transitions are strongly constrained by the CMS limit on BR(h → τµ) [43,44]. The observation of LFV Higgs decays at the LHC remains an interesting possibility allowed by low energy constraints. If nonzero rates for these decay modes are measured at some point, the search for CP violating effects would also reveal features of the underlying LFV dynamics [45].

Conclusions
In case LFV τ decays are observed in the future, correlations between the decay rates will provide the main handle for the determination of the underlying dynamics. Additionally, differential distributions in three-body decays like τ → 3µ or τ → µππ provide a complementary handle to discriminate different kinds of new physics. Finally, with the discovery of a Higgs boson around 125 GeV, a new window opens in the search for LFV phenomena through Higgs decays. Low energy constraints on LFV Higgs couplings still allow for sizable effects to be observed at the LHC or at a future Higgs factory.
Commission [Grants FPA2011-23778 and CSD2007-00042 (Consolider Project CPAN)]. The work of V.C. and E.P. is supported by the DOE Office of Science, Nuclear Physics program.