# Lepton-Number Violating Decays of Heavy Mesons

###### Abstract

The experimental observation of lepton-number violating processes would unambiguously indicate the Majorana nature of neutrinos. Various = 2 processes for pseudoscalar meson decays to another pseudoscalar meson and two charged leptons , () have been studied extensively. Extending the existing literature on the studies of these kinds of processes, we consider the rare decays of heavy mesons to a vector meson or a pseudoscalar meson. These processes have not been searched for experimentally, while they may have sizable decay rates. We calculate their branching fractions and propose to search for these decay modes in the current and forthcoming experiments, in particular at the LHCb.

###### pacs:

13.25.Ft, 13.25.Hw, 14.40.Lb, 14.40.Nd, 14.60.St^{†}

^{†}preprint: APS/123-QED

## I Introduction

The neutrino oscillation experiments have proved that neutrinos are massive Kam (); SNO (); NEMO (); Barger (). However, the nature of neutrino masses is still one of the main puzzles in contemporary particle physics, i.e., are neutrinos Dirac or Majorana particles? As we all known that the Majorana mass term violates lepton number by two units ( = 2). Thus, the unambiguous answer to the question above is the experimental observation of a lepton-number violating (LV) process.

Various = 2 processes have been studied in the literature Ali (); Tao (); Atre (); Claudio (); Rodejohann (). Among them Atre et al. Tao () have studied 36 LV processes from , and decays, generically written by:

(1) |

where and denote charged pseudoscalar mesons and leptons, respectively.

Most of these processes have been searched for and the non-observation in the current experiments set the bounds on branching fractions. In turn, they led to some stringent constrains on the mixing parameters between Majorana neutrino and charged lepton directly. However, there are still more LV heavy meson decay modes that have not been studied experimentally, that may have sizable branching fractions in theory. In particular, heavy mesons (with flavors) are easier to identify and the LHCb experiments will provide us with a large data sample. So as an extension of current existing calculations, we explore some new decay modes in this paper. We mainly consider the rare decays of heavy mesons and to vector meson final states. Since the LV heavy meson decay modes under our consideration have no experimental results, we cannot extract the mixing parameters through these decay channels like Atre et al. did.

However, those processes considered by Ali Ali () and Atre Tao () are clearly correlated with the decay modes under our consideration, with the same mixing parameters specified by the charged lepton flavors. We thus adopt the numerical values of mixing parameters extracted from Ref. Tao (), and the decay widths and branching fractions of heavy mesons for our processes can be predicted correspondingly. We choose the strongest constrains on mixing parameters from Ref. Tao () as input in our study, in order to be conservative.

we mainly consider the heavy pseudoscalar meson and to vector meson final states. From theoretical point of view, the decays of vector mesons may have different and uncorrelated rates from that of the pseudoscalars if there is other type of new physics, like a heavy particle exchange of either a pseudoscalar/scalar or a vector boson. Thus, it is well motivated to carry out the complementary searches for all of the availabel final states.

The paper is organized as follows. In Sec. II we outline the useful formulas to set the general notation. We list the constraints on mixing parameters and give the Monte Carlo sampling of the branching fractions as a function of the heavy neutrino mass in Sec. III, and draw our conclusion in Sec. IV.

## Ii The General Formalism For Lepton-Number Violating Decay

The simplest renormalizable extension of the standard model (SM) to generate neutrino Majorana masses is to introduce right-handed SM singlet neutrinos . Therefore, the complete neutrino mass sector is composed of both Dirac masses that produced via the Yukawa couplings to the Higgs doublet in the SM, and possible heavy Majorana mass term .

In terms of the mass eigenstates, the gauge interaction lagrangian of the charged currents now has the following form:

(2) |

where , and are the mass eigenstates, is the mixing matrix between the light flavor and light neutrinos, and is the mixing matrix between the light flavor and heavy neutrinos.

The basic process with shown in Fig. 1 with exchange of two virtual SM bosons can be generically expressed by:

(3) |

where . The process can occur only if neutrinos are Majorana particles. Unfortunately, the transition rate of this process encounters a severe suppression either due to the small neutrino mass like , or due to the small mixing . However, when the heavy neutrino mass is kinematically accessible, the process may undergo a resonant production of the heavy neutrino, thus substantially enhancing the transition rate. The resonant contributions of heavy Majorana neutrinos to processes involving two charged leptons and another pseudoscalar meson have been considered in the Ref. Tao (). In this paper, we extend the study of the heavy Majorana neutrinos to processes involving two charged leptons and a vector meson final state.

The Feynman diagram for the LV decay of heavy meson into two charged leptons and another meson :

(4) |

is shown in Fig. 2. According to narrow-width approximation, the tree level decay amplitude when is a pseudoscalar meson is given as Tao ():

(5) |

when is a vector meson, the tree level decay amplitude can be written as Tao ():

(6) |

where is Fermi constant; is the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements; is the decay constant for meson ; are the momenta of mesons and leptons , respectively. Here we consider the case when only one heavy Majorana neutrino is kinematically accessible and denote it by , with the corresponding mass and mixing with charged lepton flavors . is the total decay width of the heavy Majorana neutrino, summing over all accessible final states. Then the partial decay width and the normalized branching fraction for the LV process Eq. (4) can be calculated by the decay amplitude. Following the approach of Ref. Tao (), we will take the mixing parameter and the mass as phenomenological parameters.

## Iii Monte Carlo Sampling For Lepton-Number Violating Decays

Decay mode | Mixing element | Range of (MeV) | Range of |
---|---|---|---|

776 - 1869 | 0.002 - 0.998 | ||

892 - 1869 | 0.002 - 0.998 | ||

776 - 1968 | 0.023 - 0.992 | ||

892 - 1968 | 0.022 - 0.992 | ||

1870 - 5278 | 0.236 - 0.992 | ||

1969 - 5278 | 0.231 - 0.992 | ||

2011 - 5278 | 0.231 - 0.992 | ||

2113 - 5278 | 0.232 - 0.992 | ||

882 - 1863 | 0.002 - 0.993 | ||

1975 - 5173 | 0.28 - 0.971 | ||

2074 - 5173 | 0.28 - 0.971 | ||

2116 - 5173 | 0.28 - 0.971 | ||

2218 - 5173 | 0.327 - 0.971 | ||

776 - 1869 | 0.02 - 0.497 | ||

892 - 1869 | 0.023 - 0.497 | ||

776 - 1868 | 0.02 - 0.497 | ||

892 - 1968 | 0.023 - 0.497 | ||

1870 - 5278 | 0.242 - 0.49 | ||

1969 - 5278 | 0.236 - 0.49 | ||

2011 - 5278 | 0.236 - 0.49 | ||

2113 - 5278 | 0.236 - 0.49 |

The key step to calculate decay widths and branching fractions of the LV heavy meson decays is to determine the limits on the mixing parameters and neutrino mass in Eq. (5) and Eq. (6). Generally speaking, one can determine limits on the mixing parameters from the LV heavy meson decay modes which have the current experimental limits on branching fractions and determine the mass of neutrino by kinematics. However, as mentioned in the introduction, since the LV decay modes which we studied with vector meson and several pseudoscalar meson final states are missing in directly experimental searches, so we cannot yet get information of mixing parameters from those decays. We thus propose the direct searches for those modes in the existing and forth coming experiments such as in CLEO, -Factories, and the LHCb. On the other hand, there are direct experimental results on the processes that may share common mixing parameters with those under our consideration. These decay modes given by Ref. PDG04 () and Ref. CLEO1 () have been summarized and translated into the direct bounds in Ref. Tao (). We thus adopt them for consistency and carry out our analyses. We first obtain the consistent limits on mixing parameters from some decay modes which have the current experimental bounds on the branching fractions for the heavy mesons with Ref. Tao (). We then translated the limits on mixing parameters to the relevant decays modes of the heavy mesons , and in the Table I. Depending on the flavors of the final state leptons, the mixing parameters probed are , and , corresponding to the decay modes:

(7) |

respectively. We list the most stringent limits on , and for the 21 new decay modes in Table I. The ranges of heavy neutrino mass in the Table I are determined by the kinematics accessible.

When performing the calculations, the input parameters for the CKM matrix elements and the decay constants of pseudoscalar and vector mesons are chosen as follows PDG04 (); PDG08 (); CLEO2 (); MILC (); Ebert ():

(8) |

We note that there may be some errors in determining the decay constants Error (), but they would not result in any qualitative difference for our predictions for the SM-forbbiten modes.

With these parameters and the limits on mixing parameters and corresponding mass ranges in the Table I, the decay widths and branching fractions of the heavy mesons , and are calculated correspondingly. We perform a Monte Carlo sampling of the branching fractions and the mass of the heavy neutrino, i.e., we plot the excluded region of the branching fractions as a function of , as shown in Fig. 3 Fig. 5 for the modes in Eq. (7). The regions inside the curve are excluded by the direct experimental searches for the various LV decay modes of heavy mesons as obtained in Ref. Tao (). The theoretical allowed regions are below the curve, i.e., the regions below the curve are the currently allowed branching fractions for those LV heavy meson decay modes in the Table I.

From the figures, one can see that, if the heavy neutrino mass is located in the range from GeV to GeV, even with the most stringent constraints on mixing parameters, our theoretical predictions of upper bound of the branching fractions can be large, for example, , and , since about CLEO () events and CLEO3 () events have been collected by CLEO collaboration, these mentioned decay modes can be analyzed in the current experiment, which will provide us a strong information of GeV to GeV neutrino mass. But if the heavy neutrino mass is heavier, around GeV to GeV, we cannot use the or decay modes to detect the LV processes because of the kinetic limited, and the decay modes are favored, whereas our predicted branching fractions of the LV decay channels are lower than , which cannot be detected by current -factories, but in the forthcoming LHC experiments, around meson events are expected Chen (), all the LV decays modes we studied can be effectively searched for by LHCb.

As a final remark, we would like to reiterate the advantage of our treatment in searching for the resonant production and decay of a Majorana neutrino in vector meson decay. Although kinematically limited, the signal rate for the rare meson decay is substantially enhanced due to the resonant nature. In contrast to the similar decay channels as discussed in Ref. Ali (), where the intermediate Majorana neutrinos are far off mass-shell, the signal would be much weaker. Other contributions such as the box diagrams etc in Ref. Ali () are all of similar nature and much smaller than those considered here.

## Iv Conclusions

We extended the existing literature to consider the rare decays of heavy mesons and to a vector or pseudoscalar meson final state. Since there have not been any direct experimental searches on these LV heavy meson decay modes, we calculated their decay branching fractions and proposed to search for them in the existing and forth coming experiments.

We first re-evaluated the limits on the mixing parameters , and from some decay modes which have experimental limits on the branching fractions for the heavy mesons, and obtained full agreement with those in Ref. Tao (). We then translated the limits on mixing parameters and corresponding mass ranges to the relevant decays modes of and of our current interests as summarized in Table I. Finally, we calculated the decay widths and branching fractions for various LV decay modes by the limits on the mixing parameters and the heavy neutrino mass. We sampled the constraints on branching fractions as a function of the heavy neutrino mass as shown in the figures.

Although the prevailing theoretical prejudice prefers Majorana neutrinos, the unambiguous signature to prove the Majorana nature of neutrinos is the experimental detection of a LV process. A detection in one of the LV heavy meson decay modes studied in our analysis would imply LV and hence the existence of a Majorana neutrino. At present, about CLEO () events and CLEO3 () events have been collected by CLEO collaboration. So these decay modes , and et al. which we studied in the Table I might show up in the current experiments if the mass of the heavy neutrino is in the range . But those decay modes for the range of the heavy neutrino is cannot be detected presently due to small branching fraction. Fortunately, in the forthcoming LHC experiments, around meson events are expected Chen (), which will provide us with chances of discovering all the LV decays modes we studied. Therefore, we may have the opportunity to discover the LV process of heavy mesons , and via the distinctive channels of like-sign dilepton production with no missing energy. Hadron colliders may serve as the discovery machine for the mysterious Majorana neutrinos.

ACKNOWLEDGMENTS

We would like to thank Tao Han for his suggestions to carry out this research, for providing the FORTRAN codes Hanlib for the calculations, and careful reading of the manuscript. This work was supported in part by the National Natural Science Foundation of China (NSFC) under Grant No. 10875032 and in part by SRF for ROCS, SEM.

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