# Majorana neutrinos with effective interactions in decays.

###### Abstract

We investigate the possible contribution of Majorana neutrinos to meson decays in an effective interaction formalism, in the mass range GeV GeV. We study the decay of the meson via at LHCb, which is a signal for leptonic number violation and the presence of Majorana neutrinos, and put bounds on different new physics contributions, characterized by their Dirac-Lorentz structure. We also study the bounds imposed by the radiative decay () results from Belle. The obtained bounds are more restrictive than previous values found for dimension 6 four-fermion contact vectorial and scalar Majorana neutrino interactions in the context of the Left-Right symmetric model for higher Majorana masses at the LHC, showing that the direct calculation of the effective interactions contribution to different processes can help to put more stringent bounds to different UV-complete models parameterized by an effective Lagrangian.

## I Introduction.

The search for particles beyond the standard model (SM) content has been extensive in the past few years, among them sterile Majorana neutrinos , which appear as a natural consequence in several SM extensions. The discovery of neutrino oscillations suggests that the standard neutrinos are massive particles. One of the possible ways to generate their mass is the seesaw mechanism Minkowski:1977sc (); Mohapatra:1979ia (); Yanagida:1980xy (); GellMann:1980vs (); Schechter:1980gr (), which introduces at least one right handed singlet and produces Majorana neutrinos. In this way one obtains masses for the standard neutrinos of order compatible with current oscillation data, assuming sufficiently heavy Majorana masses ( GeV) and convenient Yukawa couplings of order . On the other hand, for smaller Yukawa couplings of order , sterile neutrinos with masses around GeV could exist. However, in the simplest Type-I seesaw scenarios, a major drawback is that the left-right mixing parameters need to be negligibly small in order to account for light masses Cai:2017mow (); Atre:2009rg (). The mixings weight the coupling of the heavy with the SM particles, in particular with charged leptons through the interaction

(1) |

so this leads to the decoupling of the Majorana neutrinos. However, the observation of any lepton number violating (LNV) process would point to the Majorana nature of the exchanged fermion. Recent approaches consider a toy-like model in which the SM is extended by incorporating a massive Majorana sterile fermion, assumed to have non-negligible mixings with the active states, without making any hypothesis on the neutrino mass generation mechanism Abada:2017jjx (); Pascoli:2018heg (). Such a minimal SM extension leads to contributions to LNV observables which are already close, or even in conflict, with current data from meson and tau decays, for Majorana masses below GeV (see Abada:2017jjx (); Abada:2018nio () and the references therein). So, also from the experimental point of view, the simple SM extensions which attribute LNV only to the mixing between heavy Majorana states and the active neutrinos are facing increasingly stringent constraints.

As suggested in delAguila:2008ir (), the detection of Majorana neutrinos () would be a signal of physics beyond the minimal seesaw mechanism, and its interactions could be better described in a model independent approach based on an effective theory. One can think of an alternative treatment and consider the Majorana neutrino interactions as originating in new physics described by an unknown underlying renormalizable theory valid at a higher energy (UV) scale and parametrized at low energies by a model independent effective Lagrangian. In this approach, we consider that the sterile interacts with the SM particles by higher dimension effective operators, taking these interactions to be dominant in comparison with the mixing with light neutrinos through the Yukawa couplings, which we neglect Peressutti:2011kx (); Peressutti:2014lka (); Duarte:2014zea (); Duarte:2015iba (); Duarte:2016miz (); Duarte:2016caz (); Duarte:2018xst (); Duarte:2018kiv (). We depart from the usual viewpoint in which the mixing with the standard neutrinos is assumed to govern the production and decay mechanisms. Here, for simplicity, we consider a scenario with only one Majorana neutrino and negligible mixing with the .

The different operators in the effective Lagrangian, with distinct Dirac-Lorentz structure, parameterize a wide variety of UV-complete new physics models, like extended scalar and gauge sectors as the Left-Right symmetric model, vector and scalar leptoquarks, etc. Thus, discerning between the possible contributions given by them to specific processes gives us a hint on what kind of new physics at a higher energy regime could be responsible for the observed interactions.

Observable effects of the existence of sterile Majorana neutrinos such as lepton number violation have been sought thoroughly in hadron colliders like the LHC, and colliders, low energy high precision experiments as neutrinoless double beta decay searches () among others (for comprehensive reviews see Cai:2017mow (); Deppisch:2015qwa () and references therein). In particular, heavy flavor meson decays could be the place where for the first time the Majorana neutrino effects were observed or, in the absence of a discovery, this fact can be used to set limits for its coupling to SM particles. -mediated lepton number violation in rare meson decays has been studied, for example, in Abada:2017jjx (); Asaka:2016rwd (); Cvetic:2016fbv (); Cvetic:2015naa (); Wang:2014lda (); Cvetic:2010rw (); Zhang:2010um (); Helo:2010cw (); Atre:2009rg (); Ali:2001gsa (), and the references therein. Concerning the resonant production of Majorana neutrinos in semileptonic pseudoscalar meson three-body decays, the recently measured branching ratio for intermediate neutrinos with lifetimes shorter than ps at the LHCb experiment Aaij:2014aba () gives the currently more stringent bounds on the mixing parameter in the case of the minimal SM extension by one Majorana neutrino (e.g. Abada:2017jjx (); Yuan:2017uyq ()) for Majorana masses in the range GeV.

In this paper we aim to exploit the recent -decay data to constrain the possible values of the couplings that weight the contribution of different effective operators to the Majorana-mediated same sign dilepton -decay and the radiative leptonic muon-mode . The LHCb collaboration has presented model independent upper limits on the branching ratio of the first process Aaij:2014aba (); Ossowska:2018ybk (), and the Belle collaboration has set new limits on the integrated differential width of the decay Gelb:2018end (). The obtained bounds (for GeV) are more restrictive than previous values obtained for dimension 6 four-fermion contact vectorial and scalar Majorana neutrino interactions in the context of the Left-Right symmetric model for higher Majorana masses Ruiz:2017nip (), and constrain the perspectives of discovery of Majorana neutrinos with effective interactions with GeV-scale masses by direct production in colliders and meson decays Duarte:2016caz (); Yue:2017mmi (); Yue:2018hci ().

The paper is organized as follows. In Sect. I.1 we introduce the effective Lagrangian formalism. In Sect. II we calculate the branching ratio and the decay in this formalism. In Sect. III we present our results for the obtained bounds, and in Sect.IV we make our final comments.

### i.1 Majorana neutrino effective interactions.

An appropriate way to include the Majorana neutrino into the theory is to extend the SM Lagrangian. In this work we consider an effective Lagrangian in which we include only one relatively light right handed Majorana neutrino as an observable degree of freedom. The new physics effects are parameterized by a set of effective operators constructed with the SM and the Majorana neutrino fields and satisfying the gauge symmetry Wudka:1999ax ().

The effect of these operators is suppressed by inverse powers of the new physics scale . The total Lagrangian is organized as follows:

(2) |

where is the mass dimension of the operator .

Note that we do not include the Type-I seesaw Lagrangian terms giving the Majorana and Yukawa terms for the sterile neutrinos. The dominating effects come from the lower dimension operators that can be generated at tree level in the unknown underlying renormalizable theory.

The dimension 5 operators were studied in detail in Aparici:2009fh (). These include the Weinberg operator Weinberg:1979sa () which contributes to the light neutrino masses, which gives Majorana masses and couplings of the heavy neutrinos to the Higgs (its LHC phenomenology has been studied in Caputo:2017pit (); Graesser:2007yj ()), and the operator inducing magnetic moments for the heavy neutrinos, which is identically zero if we include just one sterile neutrino in the theory.

In the following, as the dimension 5 operators do not contribute to the studied processes -discarding the heavy-light neutrino mixings- we will only consider the contributions of the dimension 6 operators, following the treatment presented in delAguila:2008ir (). We start with a rather general effective Lagrangian density for the interaction of right-handed Majorana neutrinos including dimension 6 operators.

The first operators subset includes those with scalar and vector bosons (SVB),

(3) |

and a second subset includes the baryon-number conserving four-fermion contact terms (4-f):

(4) |

where , , and , denote, for the family labeled , the right handed singlet and the left-handed doublets, respectively. The field is the scalar doublet. Also are the Dirac matrices, and is the antisymmetric symbol.

One can also consider operators generated at one-loop (1-loop) order in the underlying full theory, whose coefficients are naturally suppressed by a factor delAguila:2008ir (); Arzt:1994gp ():

(5) |

Here and represent the and field strengths respectively, and is the Dirac tensor.

The effective operators above can be classified by their Dirac-Lorentz structure into scalar, vectorial and tensorial.

In this paper we will consider the decays in Sect. II.2 and in Sect. II.3, mediated by an on-shell Majorana neutrino . We can thus take into account the following effective Lagrangian terms involved in the and processes (from eqs. (3) and (I.1)):

(6) |

The couplings are associated to specific operators:

(7) |

For the decay, the considered Lagrangian terms come from one-loop level generated tensorial operators:

(8) |

where is the 4-momentum of the outgoing photon, and are the sine and cosine of the weak mixing angle, and a sum over the neutrino family index is understood. The couplings and correspond respectively to the operators in (5).

After spontaneous electroweak symmetry breaking, taking the scalar doublet as , with being the Higgs field and its vacuum expectation value, we can write the Lagrangian (6) terms involved in our calculation (and its charge conjugate), as

(9) | |||||

Here the quark fields are flavor eigenstates with family . In order to find the contribution of the effective Lagrangian to the and decays, we must write it in terms of the massive quark fields. Thus, we consider that the contribution of the dimension 6 effective operators to the Yukawa Lagrangian are suppressed by the new physics scale with a factor , and neglect them, so that the matrices that diagonalize the quark mass matrices are the same as in the pure SM.

Writing with a prime symbol the flavor fields, we take the matrices and to diagonalize the SM quark mass matrix in the Yukawa Lagrangian. Thus the left- and right- handed quark flavor fields (subscript ) are written in terms of the massive fields (subscript ) as:

(10) |

With this notation, the SM mixing matrix corresponds to the term appearing in the charged SM current .

## Ii mediated decays.

We first consider the lepton number violating decay shown in Fig. 1. This LNV process is strictly forbidden in the SM and when mediated by a Majorana neutrino it allows to probe masses up to GeV. Also, the radiative muon-mode shown in Fig. 4 is well suited to probe this mass range, as the channel dominates the total decay with for Majorana masses up to GeV Duarte:2015iba ().

We calculate the decay of the meson for the two studied processes in two stages. Firstly we obtain the decay of to a muon and a Majorana neutrino . Secondly, we calculate the decay of to another muon and a pion and the radiative decay .

The decay width of the meson is obtained in both cases in the following way

(11) |

(12) |

and

(13) |

where is the total decay width for the Majorana neutrino. This is equivalent to calculating the whole decay process assuming an on-shell intermediate Majorana neutrino. For the decay width we include all the kinematically allowed channels for a Majorana neutrino of mass in the range GeV which are depicted in Fig.2. In our calculation we keep all the final-state masses.

The details of the calculation of the total decay width are described in Duarte:2015iba (); Duarte:2016miz (). In Figs.(a)a and (b)b we present the results for the total width for the different sets of effective couplings, as will be described in Sect. III.1.

### ii.1 The decay.

We start with the calculation of the decay width in (11). The Lagrangian terms contributing to the decay can be explicitly displayed in terms of the massive quark fields as:

(14) | |||||

where we have written the muon and massive up and b quark fields. From this Lagrangian we find the amplitude

(15) | |||||

The first term in the amplitude corresponds to the -mediated diagram which includes a SM vertex, giving the contribution. In the last term, we need to rearrange the field operators in order to put together the quark fields in a sandwich and the lepton fields in another. So we make a Fierz transformation taking into account a minus sign from the permutation of fermions, and then we replace it by

The calculation of the leptonic matrix elements is straightforward,

(16) |

In order to calculate the hadronic matrix elements, we have to rely on the symmetries Campbell:2008um (); Shanker:1982nd (). The matrix element is a Lorentz 4-vector because the meson is a pseudoscalar and is a pseudo 4-vector. The meson state is described solely by its four momentum , since it has zero spin. Therefore, is the only 4-moment on which the matrix element depends and it must be proportional to . Thus, we can write

(17) |

On the other hand, for the same reason, the matrix elements of the 4-vector, the tensor and pseudo-tensor are zero

(18) |

In the case of the matrix element of the scalar or pseudo-scalar interactions, we have to use the Dirac equations of motion, and we obtain the relations for the current matrix elements

(19) | |||||

(20) |

Putting it all together and integrating over the 2-body phase space, we obtain

(21) |

with in (15) giving

(22) | |||||

where

(23) |

The effective couplings in , as the subscript indicates, correspond to vectorial and scalar interactions.

The result is

(24) | |||||

where .

### ii.2 The decay.

Let us now calculate the decay . According to the Lagrangian (6), the amplitude for this process can be written as

(25) | |||||

The last term in (25) also needs to be modified by means of a Fierz transformation. After some algebra, it is written as

In order to calculate the various factors in (25), we make use of the definition for the pion form factor

(26) |

and from this equation we obtain the following expressions

(27) |

The contribution of the pseudo-scalar quark current to the matrix element of the ordinary pion decay (27) is enhanced in comparison with the standard chirality suppressed contribution and it is expected to be severely constrained by the experimental data. We finally have for the squared amplitude

(28) | |||||

where ^{1}^{1}1Again, the effective couplings in correspond to vectorial and scalar interactions.

(29) |

from which we obtain the decay width for

(30) | |||||

where

### ii.3 The decay.

The SM radiative leptonic decays have been extensively studied in the literature Beneke:2011nf (); Wang:2016qii (); DescotesGenon:2002mw (); Korchemsky:1999qb (); Wang:2018wfj (); Beneke:2018wjp (), as they are a means of probing the strong and weak SM interactions in a heavy meson system. The measurement of pure leptonic decays is very difficult due to helicity suppression and the fact of having only one detected final state particle. On the other hand the radiative modes, with an extra real final photon, can be even larger than the pure leptonic modes as they escape helicity suppression and are also easier to reconstruct.

The Belle collaboration has recently released an analysis of the full Belle experiment dataset Gelb:2018end () using new theoretical inputs Beneke:2018wjp () for the QCD calculations and new algorithms prepared for the Belle II experiment. They obtain the experimental bound for the integrated partial branching ratio of the muon-mode radiative decay.

We consider the SM and the effective contribution coming from the followed by reaction, and use the Belle bound to set limits on the one-loop generated effective couplings involved in this last decay mode, as will be discussed in Sect. III.1.

The SM differential decay width in the meson rest frame can be parameterized as Wang:2018wfj ()

(32) |

with form factors depending on the final photon energy . Here and are the fine structure and Fermi couplings. In order to perform the energy integration, we estimate the values for the form factors taking the central values presented in figure (8) of reference Wang:2018wfj () ^{2}^{2}2These values are also consistent with the central values given in figures (7) and (8) of reference Beneke:2018wjp (), for the inverse moment of the leading twist light cone distribution amplitude value given by Belle Gelb:2018end ()..

We call to the integrated partial branching ratio in the energy range ,

(33) |

Here for kinematic reasons and the minimal photon energy infrared cutoff is such that the theoretical QCD treatment remains valid. As we will use the latest Belle results for the experimental limit , we take GeV, as in ref. Gelb:2018end ().
The value we obtain for our estimation of the partial branching ratio in the SM is , which is of the order of the values recently considered in ref. Zuo:2018sji ().

Now we calculate the contribution of the Majorana-mediated decay in the effective Lagrangian formalism we want to probe. As we discussed in Sect. II we consider the process with an intermediate on-shell Majorana neutrino in the Narrow Width Approximation. This process is shown in Fig. 4. Under these conditions the phase space needs to be organized in order to apply the approximation

(34) | |||||