# Effective-field theory analysis of the decays

###### Abstract

The rare decays, which are suppressed by -parity in the Standard Model (SM), can be sensitive to the effects of new interactions. We study the sensitivity of different observables of these decays in the framework of an effective field theory that includes the most general interactions between SM fields up to dimension six, assuming massless neutrinos. Owing to the strong suppression of the SM isospin breaking amplitudes, we find that the different observables would allow to set constraints on scalar interactions that are stronger than those coming from other low-energy observables.

###### pacs:

13.15.+g ,12.15.-y, 14.60.Lm## I Introduction

Rare processes are suppressed decay modes of particles originated by approximate symmetries of the SM. They provide an ideal place to look for new physics because their suppressed amplitudes can be of similar size as the (virtual) effects due to new particles and interactions. It turns out that having a good control of SM uncertainties is crucial to disentangle the effects of such New Physics contributions in precision measurements at flavor factories.

In this paper we study the rare decays, which will be forbidden if parity Lee:1956sw () were an exact symmetry of the SM (, with
the charge conjugation operation and the components of the isospin rotation operators). This process was suggested long ago Leroy:1977pq () as a clean test of Second Class Currents
(SCC) following a classification proposed by Weinberg Weinberg:1958ut () for strangeness-conserving interactions. According to this classification, SCC must have quantum numbers as opposite to (first class) currents in the SM which have .
Since isospin is only a partial symmetry of strong interactions, parity gets broken by the quark mass and electric
charge differences and decays can occur, although at a suppressed rate. This suppression makes interesting these decays to study the effects of
genuine SCC, (i. e. not induced by isospin breaking effects), such as the ones induced by the exchange of charged Higgs Branco:2011iw (); Jung:2010ik () or leptoquark bosons Becirevic:2016yqi () ^{1}^{1}1Genuine SCC can also be searched for in nuclear decays, although having a good control of
isospin breaking effects, which is a challenge in these processes Severijns:2006dr () (see Triambak:2017jpw () for a recent analysis).. We study these processes in the framework of an
effective Lagrangian where the effects of New Physics are encoded in the most general Lagrangian involving dimension-six operators with left-handed neutrino fields.

Our study focuses on different partial and total integrated observables on decays, as they can exhibit different sensitivities to the various effective
couplings. Previous studies (including specific beyond the SM approaches) have focused mainly in the estimates of the branching fractions in the ()
range for the () decay channels BRs () , as well as on the invariant mass distribution Vienna (); Orsay (); Escribano:2016ntp (). An important source of uncertainty in most of these
estimates arises from the predictions used for the scalar form factor contribution. Of course, a good knowledge of the scalar form factor is necessary in order to assess the possible contributions
of beyond SM effects. Once the decays have been observed at future superflavor factories, we expect that detailed studies
of the different observables will be very useful to disentangle the New Physics effects from the SM isospin-violating contributions ^{2}^{2}2Dedicated studies of backgrounds specific for these
SCC decays have been carried out recently in Refs. Guevara:2016trs (); Hernandez-Tome:2017pdc ()..

The current experimental limits for the SCC tau branching ratios of are: Br , CL (BaBar delAmoSanchez:2010pc ()), , CL (Belle Hayasaka:2009zz ()) and , CL (CLEO Bartelt:1996iv ()) collaborations, respectively. Those upper limits lie very close to the SM estimates based on isospin breaking BRs (); Vienna (); Orsay (); Escribano:2016ntp (). The corresponding BaBar limit for the decays is , CL Aubert:2008nj (), while Belle obtained , CL Hayasaka:2009zz () (CLEO set the earlier upper bound , CL Bergfeld:1997zt ()). Future experiments at the intensity frontier like Belle II Abe:2010gxa (), which will accumulate tau lepton pairs in the full dataset, are expected to provide the first measurements of the SCC decays B2TIPReport ().

This paper is organized as follows: in section II we set our conventions for the effective field theory analysis of the decays, to be used in the remainder of the article. In section III, we discuss the different effective weak currents contributing to the considered decays and define their corresponding hadronic form factors. The tensor form factor within low-energy QCD is computed in section IV. In section V we discuss the different observables that can help elucidating non-SM contributions to the decays and in section VI we state our conclusions.

## Ii Effective theory analysis of

The effective Lagrangian with invariant dimension six operators at the weak scale contributing to low-energy charged current processes^{3}^{3}3The most general effective
Lagrangian including SM fields was derived in Refs. Buchmuller:1985jz (); Grzadkowski:2010es (). can be written as Bhattacharya:2011qm (); Cirigliano:2009wk ()

(1) |

with the dimensionless new physics couplings, which are for an scale .

The low-scale effective Lagrangian for semi-leptonic () strangeness and lepton-flavor conserving transitions ^{4}^{4}4Strangeness-changing processes are
discussed in an EFT framework in Refs. Chang:2014iba (); Gonzalez-Alonso:2016etj (); Gonzalez-Alonso:2016sip (). involving only left-handed neutrino fields is given by
(subscripts refer to left-handed (right-handed) chiral projections)

(2) | |||||

where
stands for the tree-level definition of the Fermi constant, , and gives the SM Lagrangian.
In the Lagrangian above, as usual, Higgs, , and boson degrees of freedom have been integrated out, as well as , and quarks. Since
we will be considering only CP-even observables, the effective couplings , , and characterizing New Physics^{5}^{5}5These couplings, as functions of the couplings of the SM electroweak
gauge invariant weak-scale operators, can be found in appendix A of Ref. Bhattacharya:2011qm (). can be taken real.

In terms of equivalent effective couplings^{6}^{6}6The physical amplitudes are renormalization scale and scheme independent. However, the individual
effective couplings and hadronic matrix elements do depend on the scale. As it is conventionally done, we choose GeV in the scheme. ( and ) we have the following form of the
semileptonic effective Lagrangian^{7}^{7}7The factor 2 in the tensor contribution originates from the identity .
(particularized for ):

(3) | |||||

where for . This factorized form is useful as long as conveniently normalized rates allow to cancel the overall factor . Keeping terms linear in the small effective couplings, the ’s reduce to the expression in Ref. Bhattacharya:2011qm ().

## Iii Semileptonic decay amplitude

Let us consider the semileptonic decays. Owing to the parity of pseudoscalar mesons, only the vector, scalar and tensor currents give a non-zero
contribution to the decay amplitude, which reads ^{8}^{8}8The short-distance electroweak radiative corrections encoded in Erler:2002mv () do not affect the scalar and tensor contributions. However, the error made by
taking as an overall factor in eq. (4) is negligible.:

(4) | |||||

where we have defined the following leptonic currents

(5) | |||||

In eq. (4) we have defined the following vector, scalar and tensor hadronic matrix elements

(6) | |||||

(7) | |||||

(8) |

where we have defined , , and , ; the constants , denote Clebsch-Gordan flavor coefficients. In the case ( remains to be ). For simplicity we have not written the labels in the form factors, which are different for specific hadronic channels.

The divergence of the vector current relates the and form factors via

(9) |

Since ChPT ()

(10) |

where and MeV Aoki:2016frl (), it is seen –by using MeV– that . Thus, basically inherits the strong isospin suppression of .

Observe that the scalar contribution in eq. (7) can be ‘absorbed’ into the vector current amplitude by using the Dirac equation and eq. (9). This can be achieved by replacing

(11) |

in the second term of eq. (6). We will see in the next section that the remaining contribution to eq. (4), given by the tensor current (), is also suppressed in low-energy QCD.

## Iv Hadronization of the tensor current

The hadronization of the tensor current, eq. (8), is one of the most difficult inputs to be reliably estimated. In the tau lepton decays under consideration, the momentum transfer ranges within , which is the kinematic region populated by light resonances. Here we will neglect the -dependence, namely , and we will estimate its value using Chiral Perturbation Theory Weinberg:1978kz (); Gasser:1983yg (); Gasser:1984gg (); Colangelo:1999kr (). We do not consider tensor current contributions at the next-to-leading chiral order in order to keep predictability.

A comment is in order with respect to neglecting resonance contributions in the hadronization of the tensor current, as it couples to the
resonances, being the its lightest representative. In principle, one should expect a contribution from these resonances to the considered decays,
providing an energy-dependence to and increasing its effect in the observables that we study. The will contribute very little to the decay
mode, owing to kinematical constraints, and the contributions of and will be damped by phase space and their wide widths. Thus, it is quite justified to
assume . Our previous reasoning does not apply to the vector resonance
contribution to , however. It is predicted by large- arguments that vector resonances couple to the tensor current with
a strength only a factor smaller than to the vector current Cata:2008zc () (which is also supported by lattice evaluations
Becirevic:2003pn (); Braun:2003jg (); Donnellan:2007xr ()). Consequently, the contribution to should not be negligible
(the vector current contribution of the state to the branching ratio is , according to
Ref. Escribano:2016ntp ()). As a result, our limits on the allowed values of obtained from the decay mode, which are
presented in the next section, could be made stronger including this missing contribution. However, as we will see, the main point of this article is that
decays are competitive setting limits on non-standard scalar interactions in charged current decays, while they are
not in tensor interactions ^{9}^{9}9As we discuss at the end of section VI, our upper limit on is ,
while the level is reached in radiative pion decays. Our educated guess for the contribution through the tensor current to the
decays (based on its contribution through the vector current) is that with a good understanding of the former we could probably
reach , but not the level..
This main conclusion is not affected by our assumption .
Therefore our analyses (right panel in figures 5 and 6) involving the tensor source with a constant form factor should be simply viewed as a benchmark to compare with those with the scalar source, and not as a full fledged and theoretically sound computation.

According to Ref. Cata:2007ns (), there are only four operators at the leading chiral order, , that include the tensor current. Only the operator with coefficient contributes to the decays we
are considering ^{10}^{10}10We note that although flavor symmetry was considered in Ref. Cata:2007ns (), extending it
to U(3) (for a consistent treatment of the meson) does not bring any extra operator at this order, as this extension entails the appearance
of a factor, which adds to the chiral counting, belonging thus to the next-to-leading order Lagrangian that we do not
consider. Also, odd-intrinsic parity sector operators including the tensor source first appear at Cata:2007ns ().:

(12) |

where and stands for a trace in flavor space. The chiral tensors entering eq. (12) are , including the left- and right-handed sources and , the (chiral) tensor sources, and its adjoint; and , including the field-strength tensors for and .

The non-linear representation of the pseudoGoldstone bosons is given by , where

with and the light and strange quark components of the mesons, respectively ( is the pseudoGoldstone having the flavor quantum numbers of the Gell-Mann matrix, which coincides with the neglecting isospin breaking). These constants describing the mixing are given by ChPTLargeN ()

(13) |

and the corresponding values of the pairs of decay constants and mixing angles are 2anglemixing ()

(14) |

with MeV being the pion decay constant.

We recall Cata:2007ns () that the tensor source () is related to its chiral projections ( and ) by means of

(15) |

with as the tensor current.

Taking the functional derivative of eq. (12) with respect to , putting all other external sources to zero, expanding and taking the suitable matrix element, it can be shown that in the limit of isospin symmetry

(16) |

Once isospin symmetry breaking is taken into account, the leading contributions to the tensor hadronic matrix elements are given by:

(17) | |||||

(18) |

For the numerical values of the isospin breaking mixing parameters we will take the determinations and Escribano:2016ntp (). To our knowledge, there is no phenomenological or theoretical information on . However, appearing in the Lagrangian eq. (12) was predicted –using QCD short-distance constraints– in Ref. Mateu:2007tr () to be

(19) |

where we took from Aoki:2016frl (). This yields , which is consistent with the chiral counting proposed in
Ref. Cata:2007ns (). As a conservative estimate ^{11}^{11}11We note that the operators with coefficients and in eq.
(12) share the same chiral counting order Cata:2007ns ()., we will assume in our analysis.
This, in turn, results in and (we note that, according to our definition in eq. (8), includes the factor . If, instead, the tilded
form factors of Ref. Escribano:2016ntp () are used, then GeV).
Our uncertainty in the sign of translates in the corresponding lack of knowledge for the interference between tensor and scalar or vector contributions.
We finally note that the overall suppression given by the factors in eq. (18), together with our
estimate of , make decays not competitive with the radiative pion decay in setting bounds on non-standard
tensor interactions.

## V Decay observables

Most of the existing studies of decays have focused on the branching ratio BRs () and only a few of them have provided predictions for the spectra in the invariant mass of the hadronic system Vienna (); Orsay (); Escribano:2016ntp (). Once these parity forbidden decays have been discovered at Belle II, the next step will be to characterize their hadronic dynamics and to look for possible effects of genuine SCC (New Physics). This will require the use of more detailed observables like the hadronic spectrum and angular distributions or Dalitz plot analyses. In this section we focus in the decay observables that can be accessible in the presence of New Physics characterized by the effective weak couplings described in Section II.

In the rest frame of the lepton, the differential width for the decay is

(20) |

where is the unpolarized spin-averaged squared matrix element, is the invariant mass of the system (taking values within ) and with kinematic limits given by , and

(21) |

where the Kallen function is defined as .

### v.1 Dalitz plot

The unpolarized spin-averaged squared amplitude in the presence of New Physics interactions is given by

(22) |

where , and originate from the scalar, vector and tensor contributions to the amplitude respectively, and , , are their corresponding interference terms. Their expressions are

(23) | |||||

where we have defined , .

New Physics effects can appear in the distribution of Dalitz plots, with a large enhancement expected towards large values of the hadronic invariant mass (note eq. (11)). The first line of figure 1 shows the square of the matrix element obtained using the SM prediction for form factors Escribano:2016ntp (); it can be appreciated that the dynamics is mainly driven by the scalar resonance with mass GeV (other two most populated spots in the Dalitz plot correspond to effects of the vector form factor, around the peak, in the channel). In the first line of figure 2 we show the squared matrix element for two representative values of the set of parameters that are consistent with current upper limits on the . A comparison of the plots in the first line of figure 1 (left panel) and figures 2 show that the Dalitz plot distribution is sensitive to the effects of tensor interactions but rather insensitive to the scalar interactions. For these, the most probable area around the peak gets thinner, while the one corresponding to the state gets wider, compared to the SM case. In the case of tensor interactions, the effect of the is diluted and the effect is also less marked than in the standard case. Given the fact that the contribution to these processes is much better known than that of the , observing a weak meson effect in the Dalitz plot could be a signature of non-standard interactions, either of scalar or tensor type. Uncertainties on the scalar form factor prevent, at the moment, distinguishing between both new physics types by this Dalitz plot analyses.

In the case of decays the vector form factor contributes negligibly. Then, a comparison of the first rows of figures 1 (right panel) and 3 (where the representative allowed values of differ from those taken for the channel) shows almost no change for scalar new physics. Tensor current contributions would decrease the effect compared to the SM. However, uncertainties on the scalar form factor will prevent drawing any strong conclusion from this feature.

### v.2 Angular distribution

The hadronic mass and angular distributions of decay products are also modified by the effects of New Physics contributions and can offer a different sensitivity to the scalar and tensor interactions. For this purpose it becomes convenient to set in the rest frame of the hadronic system defined by . In this frame, the pion and tau lepton energies are given by and . The angle between the three-momenta of the pion and tau lepton is related to the invariant variable by , where and .

The decay distribution in the variables in the framework of the most general effective interactions is given by

(24) | |||||

When the effective couplings of new interactions are turned off, we recover the usual expressions for this observable in the SM Beldjoudi:1994hi (). It is interesting to observe that no new angular dependencies appear owing to the presence of new interactions, although the coefficients of terms get modified by terms that increase with the hadronic invariant mass . In this respect, it is interesting to point out that the last term of eq. (24), which is linear in cos, would allow to probe the relative phase between the scalar and vector contributions in the absence of new physics. We note that similar modifications to the angular and hadronic-mass distributions are expected for allowed decays, although the effects of scalar and tensor interactions should be very small in those cases.