# as a source of background on the Lepton Flavour Violating decay

###### Abstract

We calculate the decay of in the framework of Resonance Chiral Theory (). By demanding the high energy constraints from QCD on the related form factors, we could predict the various physical observables of without any free parameter. Our results show that for a realistic cut on the photon energy (around MeV) this mode gives a branching ratio of roughly that should have already been detected at the heavy-flavour factories. Another interesting subject we have studied based on our calculation of the decay is the experimental background estimation of the lepton flavour violation process . We point out that although the description of radiation that PHOTOS provides -which has been used by BaBar and Belle collaborations to estimate this source of background- is in excellent agreement with the theoretical expectations in the low energy region in decay, this is not the case in the high energy region, precisely where it is easier that this decay mimics the process .

## 1 Hadronic decays of the lepton

The decays of the lepton into hadrons are a very interesting subject in Particle Physics that has been a very active area of research during the
last 25 years [1]. The is privileged because its mass allows it to decay hadronically while being a lepton, which means a clean place to scrutinize
hadronization effects at low-energies, where QCD becomes nonperturbative. One can exploit this advantage to a great extent through its inclusive decays [2].
However we will focus in what follows in one of its exclusive decays.

The decay amplitude for the one meson radiative decays of the includes an internal bremsstrahlung (IB) component, that is given by QED, and thus can
be calculated unambiguously to any desired order in perturbation theory. In addition, one has the structure dependent (SD) part, dominated by the effects
of the strong interaction and the interference (INT) between both. Lorentz symmetry determines that there are two independent structures, the so-called vector
(V) and axial-vector (A) form factors that encode our lack of knowledge of the precise mechanism responsible for hadronization.

We consider the process ^{1}^{1}1The decay , which is
related to it via chiral symmetry, is discussed in Ref. [3], in which this proceeding is based.. The kinematics of this decay is equivalent
to that of the radiative pion decay [4]. We will use . In complete analogy to that case [5], the
matrix element for the decay of can be written as the sum of four contributions:

(1) |

with ^{2}^{2}2Definitions and conventions differ with respect to Ref. [6] (), in which we have detected [3] several typoes.

where is the electric charge of the positron and is the polarization vector of the photon. and are the so called form factors. Finally and are lepton currents defined by

The notation introduced for the amplitudes describes the four kinds of contributions: is the bremsstrahlung off the tau, (Figure 1(a)); is the sum of the bremsstrahlung off the (Figure 1(b)), and the seagull diagram (Figure 1(c)); is the vector contribution (Figure 1(d)) and the axial-vector contribution (Figure 1(e)). Our ignorance of the exact mechanism of hadronization is parametrized in terms of the two form factors and (t). In fact, these form factors are the same functions of the momentum transfer as those in the radiative pion decay, the only difference being that now varies from up to rather than just up to .

The dimensionless variables

(2) |

allow to measure the photon and pion energies in units of in the tau rest frame. Their kinematical boundaries are given by

(3) |

where

(4) |

so that the photon spectrum will extend up to . In the decay the photon energy would be fixed to so one could expect some contamination of the decay to , with the misidentified as a . The invariant mass of the - system will vary in the range when the other independent kinematical variable is taken as , with .

## 2 Theoretical setting

Tau lepton decays probe QCD in its non-perturbative regime where standard expansions in powers of the coupling constant are no longer applicable. However, the chiral symmetry of the massless theory allows to build an effective field theory dual
to QCD in the light-meson sector, Chiral Perturbation Theory () [7]. This will not be enough, since can reach , which is far beyond the region of applicability
of . Still, the low-energy region of semileptonic tau decays is well described by the results [8]. Then,
one should envisage a way of enlarging the range of applicability of while respecting its low-energy behaviour. The limit of a large number of
colors () in QCD [9] is a useful tool in the development of Resonance Chiral Theory () [10], a Lagrangian formulation including the resonances as
dynamical degrees of freedom that preserves assymptotic QCD properties [11], [12]. Remarkably, was capable of predicting
the low-energy constants (LECs) of in terms of resonance masses and couplings. The phenomenological application of the theory (in the limit)
to study two-[13] and three-meson decays of the [14] has been successful and several Green-functions [15], [16] and associated form factors
have been studied within it over the years.

The leading action of the Lagrangian includes the in the even-intrinsic parity sector and the leading
(given by the Wess-Zumino-Witten term [17]) in the odd-intrinsic parity sector. Higher-order pieces in the chiral expansion are assumed to be generated
by the integration of the resonances (this was checked to occur at for the couplings of [10]).

Next, one adds all pieces including resonances (R) and chiral tensors with low enough chiral order () to not violate high-energy conditions or to force fine-tuned cancellations
among the relevant couplings to fulfil the short distance constraints. In the odd-intrinsic-parity sector, that contributes to the vector form factor, this amounts to include all terms of
and . For the even-intrinsic-parity operators that are contributing to the axial-vector form factors, these are the terms of . Since previous
analysis showed the relevance of the with negative intrinsic parity we will consider them here, as well. All mentioned pieces of the Lagrangian
can be found in Ref. [3], as well as the expressions for the form factors in decays. In Figs. 2
and 3 the Feynman diagrams contributing to these form factors are displayed. The thick dots represent strong vertices.

We provide the resonances with an adequate energy-dependent width obtained consistently within [18] and consider only one resonance multiplet
per set of quantum numbers (the contribution of spin-zero resonances is suppressed by conservation laws and the fact that there is only one meson in the final
state). Since this process has not been measured yet, one should try the simplest possible description (in order to be predictive) that can eventually be
completed if the data require so.

Next we require a Brodsky-Lepage [11] behaviour to the vector form factor and demand that the axial-vector form factor satisfies a once
subtracted dispersion relation. This results in constraints among the Lagrangian couplings. Noteworthily, the relations found in the
decays are all consistent with those obtained in the decays into two and three mesons. There is only one relation that differs with respect to the study of
the Vector-Vector-Pseudoscalar Green’s function [15] for a coupling whose impact is very mild and the discrepancy is less than .

## 3 Phenomenology in decays

First we have assessed the importance of the model independent contributions, meaning the described by QED and the Wess-Zumino-Witten contribution to the vector
form factor, that is determined from QCD. We have thus switched off the remaining contributions to the vector form factor and the whole axial-vector form factor
in this first step. We find that for a cut on the photon energy of MeV, it amounts to of the non-radiative decay, namely a branching fraction of
. Next, we include also the model dependent contributions, obtained as described in Sect. 2.

In Fig. 4 we show the differential decay width of the process including all contributions as a function of , i.e. the photon energy in the tau rest frame and in Fig. 5 we display the contributions, that are enhanced near the endpoint region, where the process
can contaminate the decay .

## 4 Comparison with PHOTOS

In the previous section, we have shown that for a realistic cut on the photon energy (around MeV) this mode gives a branching ratio of roughly
that should have already been detected at the heavy-flavour factories. Notwithstanding, this decay mode has not been measured yet. This does not
mean that the detection of soft photons at B-factories is not as good as estimated, but that the splitting of the radiative and non-radiative pion decay of
the tau was not considered a priority.

According to the last report by the Belle [19] Collaboration ^{3}^{3}3Although the BaBar report [20] is not that detailed, we assume similar
figures to hold in their case., the main source of background in searches is the process , where the
pion is miss-identified as a muon. This decay represents up to of all background and the pion fake probability to be detected as a muon is .
However, the most important contribution to this background would come from radiation off the pair and not off the or . The background
related to the decay is estimated using PHOTOS [21] that only incorporates the model independent contributions from
QED to , i.e. the IB parts in Eq.(1). For low photon energies, this decay is dominated by the IB parts and is detected
as , since the photon cannot be resolved. However, we have seen in Fig. 4, that for large photon energies the dominant
contribution is given by the terms. Indeed, it is precisely near the endpoint of the photon decay spectrum where it is easier that the miss-ID happens,
since in the two body-decay , , and the correction is , while in
the photon energy spectrum extends essentially to the same value and the correction here is given by , see Eq. (4).

We compare our prediction for the photon spectrum in with what is obtained with PHOTOS [21] and MC-TESTER [22]
runned with fixed first order only (similar results are obtained with exponentiation on), as it is displayed in Fig. 6.

In order to assess the impact of the effects in that background estimation, one needs to compare PHOTOS and our prediction for the case where the
and energies reconstruct the mass up to 9 MeV of difference. We do this in Fig. 7, where this missing mass is distributed evenly
between the and the and only the relevant region near the endpoint is displayed in order to better appreciate the differences. The PHOTOS simulation
was obtained with 100Mevents generated for where corresponds to the radiative decay. We see that in the last six
bins there is only one event. We can estimate -rather conservatively- the difference between our prediction and PHOTOS by taking the integral over these last six
bins, where we will estimate the PHOTOS contribution by our prediction including only . This gives a ratio of , as the underestimation of background
due to the effects we have studied.

Nevertheless, we stress that the total decay width is mildly affected by the parts so that the global description provided by PHOTOS is good. For example, for the total decay width one has

Our estimations (factor of 5 more background due to the considered decay that obtained with PHOTOS) would imply that this source of background could affect
the present upper bounds in Refs. [19], [20]: at , making them even stronger. The new version [23] of
hadronic currents in TAUOLA [24] and the inclusion on PHOTOS of our current in the line of Ref. [25] will be the ideal tools to
estimate reliably this background.

## 5 Conclusions and Outlook

We have analyzed the decay . It turns out essential to measure the photon energy spectrum near the endpoint in order to assess the real background affecting the lepton flavour violating decay , and the upper bound for its branching ratio. Moreover, it would be an interesting tool for lepton universality tests through the ratio /. To our view this decay channel should no longer be regarded as attached to the measurement for energetic enough photons which would allow experimental resolution. Therefore, it would become a golden mode to be searched for by the (super)B and tau-charm factories in the near future.

## References

- [1] A. Pich, “QCD Tests From Tau Decay Data,” M. Davier, A. Hocker and Z. Zhang, Rev. Mod. Phys. 78, 1043 (2006). J. Portolés, Nucl. Phys. Proc. Suppl. 169 (2007) 3. A. Pich, Nucl. Phys. Proc. Suppl. 181-182 (2008) 300. S. Actis et al., Eur. Phys. J. C 66, 585 (2010).
- [2] A. Pich, these proceedings; M. Jamin, these proceedings.
- [3] Z. H. Guo and P. Roig, arXiv:1009.2542 [hep-ph].
- [4] S. G. Brown and S. A. Bludman, Phys. Rev. 136 (1964) B1160.
- [5] P. de Baenst and J. Pestieau, Nuovo Cimento 53A, 407 (1968).
- [6] R. Decker and M. Finkemeier, Phys. Rev. D 48 (1993) 4203 [Addendum-ibid. D 50 (1994) 7079].
- [7] S. Weinberg, Physica A 96 (1979) 327. J. Gasser and H. Leutwyler, Annals Phys. 158 (1984) 142; Nucl. Phys. B 250 (1985) 465.
- [8] G. Colangelo, M. Finkemeier and R. Urech, Phys. Rev. D 54 (1996) 4403.
- [9] G. ’t Hooft, Nucl. Phys. B 72 (1974) 461; 75 (1974) 461. E. Witten, Nucl. Phys. B 160 (1979) 57.
- [10] G. Ecker, J. Gasser, A. Pich and E. de Rafael, Nucl. Phys. B 321 (1989) 311. G. Ecker, J. Gasser, H. Leutwyler, A. Pich and E. de Rafael, Phys. Lett. B 223 (1989) 425.
- [11] S. J. Brodsky and G. R. Farrar, Phys. Rev. Lett. 31 (1973) 1153. G. P. Lepage and S. J. Brodsky, Phys. Rev. D 22 (1980) 2157.
- [12] E. G. Floratos, S. Narison and E. de Rafael, Nucl. Phys. B 155 (1979) 115.
- [13] F. Guerrero and A. Pich, Phys. Lett. B 412 (1997) 382. M. Jamin, A. Pich and J. Portolés, Phys. Lett. B 640 (2006) 176; Phys. Lett. B 664 (2008) 78. Z. H. Guo, Phys. Rev. D 78 (2008) 033004.
- [14] D. G. Dumm, A. Pich and J. Portolés, Phys. Rev. D 69 (2004) 073002. D. G. Dumm, P. Roig, A. Pich and J. Portolés, Phys. Lett. B 685 (2010) 158. D. G. Dumm, P. Roig, A. Pich and J. Portolés, Phys. Rev. D 81 (2010) 034031. D. G. Dumm, P. Roig and A. Pich, to appear.
- [15] P. D. Ruiz-Femenía, A. Pich and J. Portolés, JHEP 0307 (2003) 003.
- [16] M. Knecht and A. Nyffeler, Eur. Phys. J. C 21 (2001) 659. G. Amorós, S. Noguera and J. Portolés, Eur. Phys. J. C 27 (2003) 243. V. Cirigliano, G. Ecker, M. Eidemüller, A. Pich and J. Portolés, Phys. Lett. B 596 (2004) 96. V. Cirigliano, G. Ecker, M. Eidemüller, R. Kaiser, A. Pich and J. Portolés, JHEP 0504 (2005) 006; Nucl. Phys. B 753 (2006) 139.
- [17] J. Wess and B. Zumino, Phys. Lett. B 37 (1971) 95. E. Witten, Nucl. Phys. B 223 (1983) 422.
- [18] D. G. Dumm, A. Pich and J. Portolés, Phys. Rev. D 62 (2000) 054014.
- [19] K. Hayaska et al., Phys. Lett. B 666 (2008) 16; K. Inami, these Proceeedings.
- [20] B. Aubert et al., Phys. Rev. Lett. 104 (2010) 021802; A. Cervelli, these Proceeedings.
- [21] P. Golonka and Z. Was, Eur. Phys. J. C 45 (2006) 97; Eur. Phys. J. C 50 (2007) 53; G. Nanava and Z. Was, Eur. Phys. J. C 51 (2007) 569.
- [22] P. Golonka, T. Pierzchala and Z. Was, Comput. Phys. Commun. 157 (2004) 39; N. Davidson, P. Golonka, T. Przedzinski and Z. Was, arXiv:0812.3215 [hep-ph].
- [23] P. Roig, O. Shekhovtsova and Z. Was. Work in progress.
- [24] S. Jadach, Z. Was, R. Decker and J. H. Kühn, Comput. Phys. Commun. 76 (1993) 361. Z. Was, N. Davidson, G. Nanava, T. Przedzinski, E. Richter-Was, P. Roig, O. Shekhovtsova, Q. Xu, these Proceedings.
- [25] G. Nanava, Q. Xu and Z. Was, arXiv:0906.4052 [hep-ph].