# A complete determination of New Physics parameters in leptonic decays of

###### Abstract

Recently LHCb and CMS have measured the branching ratio of the rare decay of which is at par with the predicted standard model (SM) value. This result does not predict the absence of new physics (NP), rather it confines the NP parameter-space in a more stringent fashion. In this paper we have used the general loop level Hamiltonian to constrain the parameter space for the NP couplings for from the experimental branching ratio. Using the available parameter space for these couplings, we have explored the angular analysis of the cascade decay of . This analysis shows presence of NP can be identified by the angular analysis and also shows how to isolate the NP contribution from the SM. We believe that in future with the reduced error in the branching ratio measurement and better cone reconstruction technique we will be able to probe NP signal using this angular analysis.

PACS numbers.: 13.20.He, 13.35.Dx, 12.60.-i.

## I I. Introduction

The rare decays of neutral -mesons into 2-body leptonic modes, for can happen in Standard Model(SM) only through flavour changing neutral current(FCNC) processes. They cannot occur through tree level processes and can be mediated by only electro-weak box and Z penguin diagrams and are extremely rare due to helicity suppression by factor where and are the masses of the lepton and respectively. These decays are further suppressed by due to an internal quark annihilation within the meson, where is the decay constant of the meson. In SM, the main uncertainty comes due to the partial knowledge of the decay constant and CKM matrix elements involved in these branching ratios(). Though recently using different lattice QCD methods, error has been reduced in the estimation of these decay constants decay-const (). In SM, sm-br (),

(1) | |||||

Due to smallness of the , verifying it is beyond the scope of present experimental limit. The is the largest of the helicity suppressed purely leptonic decays, since the amplitude is proportional to the mass of the lepton . However, the leptons are difficult to detect making the detection of this mode uniquely challenging. The complications for this mode arise not only from the combinatorial background but also from the fact that at least the two from the lepton decays are undetected and hence the two leptons cannot be fully reconstructed. At present we do not have exact experimental numbers for most of these s, but recently LHCb collaboration exp-b-mumu () has given experimental data on

(2) |

and CMS has given exp-b-mumu-cms ()

(3) |

In this paper we mainly concentrate on channel, except . We consider that NP contribution in can occur only through loop level process like its SM counterpart. With this assumption, the effective Hamiltonian for the process can be written in terms of three distinct Dirac bilinear contributions as (almost same as Hamiltonian ())

(4) |

and the matrix element for the process can be written as matelement ():

(5) |

where,

(6) | |||||

is the axial vector - axial vector type interaction term. It is present in the SM and in general, this type of coupling may be present in NP too. The factor before comes due to helicity suppression. We divide into two terms and . is solely the SM part and is coming due to the NP contribution and we call it as . and are the pseudo scalar - pseudo scalar and pseudo scalar - scalar type interaction terms respectively. can have a possible SM contribution from a neutral Goldstone boson penguin diagram gbpenguin () and can have a possible SM contribution from SM Higgs penguin diagram higgspenguin (). But these contributions from SM higgs and neutral Goldstone bosons to the amplitude are further suppressed by compared to the dominant contribution. That is why the SM contributions in and can be ignored and it can be considered that they are coming from purely NP contribution. For our discussion we distinguish these two terms as and respectively. In this Hamiltonian as both NP and SM are appearing at loop level, we have separated out and CKM elements from the couplings unlike Hamiltonian ().

In SM, coupling is identical for the process for and , whereas in existing NP models like MSSM, 2HDM, top-color assisted LHT models, it has been shown that and get suppression factor mlsup () due to the involvement of various NP particles in box, penguin and fermion self-energy diagrams. But no such effect of suppression factor has been predicted for coupling in existing literature. In our paper, we consider and are same, whereas and have a suppression factor for and in . With this assumption, it can be seen from Eq. (6) that both and will differ for different due to the involvement of different lepton masses.

LHCb and CMS data of the is in the same ballpark of the SM, but it is not ruling out the possibility of the presence of new Physics coming from any one of the three different type of NP interaction terms. We have to wait till the experimental error reduces to have any conclusive remark on the presence of new physics in this leptonic decay mode. There is a possibility that all of , and terms are present but they are cancelling each other in such a fashion that at the end of the day the SM term gives the main contribution in the branching ratio. Another possibility is that any one of the NP and SM terms are mutually cancelling each other and contribution from rest of the two NP terms satisfies the experimental data. Our main intention is to construct such observables which can separate out the effect of the presence of the NP contribution from its SM counterpart or at least identify the prominent presence of NP.

Considering the general Hamiltonian, it has been shown that the term is nothing but the Willson coefficient wc (). We are not taking the value of exactly same as as there is some uncertainty. We constrain the parameter space for from the SM estimated value of and then using this constraint we figure out the allowed parameter space for the modulus of , and from the experimental data of . From these constraints we can figure out the sizes for the modulus of and , whereas we cannot obtain any bound on the phase factor of these couplings. We choose that the phases can vary from for the couplings involved in as in general these couplings can be complex. Using these constraints we figure out the sizes of and . With these allowed sets of and we present an angular analysis of the cascade decay of in a model independent fashion to check whether we can separate out (which is purely NP contribution) from or we can establish a measurable difference between SM and NP effect due to itself. Isolation of will help us further to constrain the scalar sector. Here we would like to mention that it has been shown in literature that NP contribution in can be much larger, 3% - 10% Hamiltonian (),damol () which will open up many more ways to find NP signature.

In Section II we constrain the parameter space for (SM), (), () and (), using the experimental branching ratio for . For this purpose we quote the relevant formulae and present the necessary numerical input. Section III.A deals with the cascade decays of . The choice of reference frame for this cascade decay is discussed in detail. The angular analysis for the cascade decay is discussed in Section III.B. Various observables which help to establish the fact that the presence of NP makes a measurable difference from SM are discussed in this section too. We conclude and summarize in Section IV. In Appendix, we present the matrix element square for the cascade decay of .

## Ii II. Constraints from

In this section first we figure out the constraints on from the theoretical branching ratio of , considering all the NP contribution as zero. Now in SM,

(7) |

Where is the lifetime of . Following the general Hamiltonian from Eq. (I) and considering that all the couplings are present, the branching ratio can be written as:

(8) |

where,

Observables | Value | Reference |
---|---|---|

MeV | pdg () | |

MeV | pdg () | |

MeV | pdg () | |

MeV | pdg () | |

MeV | pdg () | |

MeV | fbs () | |

pdg () | ||

ckmfitter () | ||

ckmfitter () | ||

s | pdg () | |

Case | Allowed situation | Range of | |||

(GeV) | when | (GeV) | including | ||

(Max.) | (Max) | ||||

I | SM | 0 | |||

II | SM+ | 0 | |||

III | SM+ | 0 | |||

IV | SM+ | ||||

V | SM++ | 0 | |||

VI | SM++ | ||||

VII | SM++ | ||||

VIII | SM+++ |

The modulus of the couplings , , and has the following relations with the couplings:

(9) |

and

(10) |

where , are the phases of respectively. We choose the phase factor of as zero without any loss of generality as only the relative phase between two couplings is important. Rest of the couplings have the following relations for these decay channels:

(11) | |||||

Here is same for both and and is also same for both and . The magnitude of these couplings have been expressed in some particular NP models mlsup (). In our paper first we obtain bounds on the modulus of the couplings involved in using the experimental branching ratio data from Eq. (2). Table. (1) contains the data used for this calculation. Experimental error has been included in each numerical input used in the branching ratio. With these bounds we restrict the modulus of the couplings involved in decay using Eq. (II). We vary from as there is no way we can constrain these phases.

As stated earlier that the experimental branching ratio may not be fully satisfied by SM only, there is some scope of NP. We explore all the possible scenarios. At this point we would like to mention that in our following discussion whenever we talk about these couplings, we actually talk about their modulus.

First in Case-I, we constrain the parameter space of the from the SM branching ratio of sm-br () considering all NP terms as zero. From this constrain we can estimate the value of which satisfies the range from 19 to 21.5 for this case. Due to the absence of NP, the coefficient is zero here.

Next in Case-II, we include only from all possible NP contributions with our SM contribution and compare the estimated branching ratio with the experimental branching ratio of exp-b-mumu (). It gives bound on which has been used to figure out the allowed range of . Here ranges from . Like that in Case-III, we choose only coupling with SM. In this case ranges from . These large ranges for in these two situations are due to the constructive and destructive interference between the SM and NP couplings. and are individually playing a key role to increase the range of from the sole presence of SM. In both of these situations, remains zero as we can have nonzero only if is present. If we add only with SM which is Case-IV, then the value of remains exactly same as Case-I. This is quite natural as alone cannot affect the value of . This situation is significantly different from Case-II and Case-III as in this case we have nonzero and it ranges between .

After this we consider situations where any two NP couplings are present with SM. In Case-V, we consider the presence of . Here ranges from and remains 0. In this case a very unnatural fine tuning happens between SM, and . It gives unbounded parameter space for and . It happens due to the presence of three interference terms between SM-, SM- and -. In this situation, we take a logistic approach, where we neglect the values of the couplings for which fine tuning among SM, and is less then 20%. This amount of fine tuning is sufficient to show the difference between the presence of NP from SM. This choice of fine tuning is a very common practice in existing literaturefinetune (). In this way we avoid a very unnatural fine tuning between all the couplings.

Similarly we figure out the ranges for and for the cases of simultaneous presence of and which are mentioned in Case-VI and Case-VII respectively. All these values are listed in Table. (2). In Case-VIII we consider that all the NP couplings are present with SM. In that case ranges from and ranges from . In Case-V to Case-VIII, we can notice that when both and are simultaneously present with SM, we get maximum range for almost. This is expected as for these two cases two phases and are simultaneously playing important role in the interference between various couplings. If only or is present then mostly remains between which is quite large compared to the case when only SM is present i.e. Case-I. The role of the phase factors in the interference between different couplings is quite clear from all these situations.

All these cases and their outcomes are presented in Table. (2), Fig. (1) and Fig. (2). At this point we want to mention that we have verified the allowed parameter space of , , , and using the time integrated SM value of branching ratio for tism () too. We have not found any significant difference in the estimated values of and .

Our main motivation is to isolate the NP from the SM so that it can be probed via experiment or at least establish a process of analysis such that the presence of any kind of NP can be identified prominently. It leads us to the obvious question that for what values of and we should continue our further analysis. Table. (2) gives us the maximum allowed value for is 81.6 and is 22.8. The minimum is 0.2 and 0 respectively as mentioned in Table. (2). In next section we discuss about the way to isolate NP from SM.

## Iii III. The decay

It is now clear that though experimental branching ratio of is within the same regime of SM, the possibility of the presence of NP is not ruled out. However, the main challenge is how to experimentally observe the presence of NP. For this purpose we are exploring a good old technique of angular analysisangular-analysis (). For such angular analysis we will choose not the as further decays of muons is not suitable for doing angular analysis to isolate the NP from SM. The decay of is also governed by the same effective Hamiltonian involved in decay except for the fact that mass is replaced by . Then we consider the further decays of ’s. It makes as an interesting choice to perform angular analysis. In this section our primary motivation is to present a technique to distinguish experimentally the various NP effects from the SM contribution through angular analysis of this cascade decay channel.

We divide this section into two parts. For the cascade decay, we provide the information about different reference frames, momentums of various particles involved, assumptions of ignoring negligible masses, constraints from energy-momentum conservation relations etc. in the first part. Second part involves detail analysis. Where we show that how NP can be isolated from SM via this analysis. However, one very essential and remarkable thing which we establish there is that if any kind of NP is present then that situation is significantly different from the sole presence of SM in our angular analysis, which can be tested experimentally at LHCb, using sophisticated technique.

### iii.1 III. A. Relevant formulae for the cascade decay of

We begin by considering the decay . And then consider, the and to decay further with and . The produced is considered to decay into and , with four-momentum and respectively. We first define the kinematics of the process by expressing the four momentum vectors of all the particles in the decay process. We describe the decay in the rest frame, where, the and decay back to back. The axis is defined to be along the direction of the as shown in Fig. 3.

The momenta and the angles involved in the decay are related in the rest frame as follows:

where we have used to obtain

(12) |

The remaining three energy-momentum conservation relations, , , and , give us the freedom to eliminate in terms of the rest of the momentums. The on-shell conditions for the initial meson, the final state pions and the neutrinos are , and respectively. We also require the intermediate leptons and meson to be on shell, which is imposed by using and respectively. Energy-momentum conservation gives , resulting in a relation for :

(13) |

The and momentum can be easily written in rest frame since and are back to back. Neglecting the masses of pions and neutrinos, their respective four momentums are as follows :

(14) | |||||

We note that and are angles described in the rest frame. Similarly, in rest frame, and are back to back and their respective four momentums are as given in terms of rest frame angles and as follows:

(15) |

With the same logic the and are back to back in the rest frame. Considering along z axis and neglecting masses of pions and neutrinos, their respective four momentums are described in terms of the rest frame angles and as follows:

(16) | |||||

The angles of decay products defined in the rest frame can be expressed in terms of the angles defined in the respective rest frames of (, ), (, ) and (, ). We always have the freedom to choose either or as zero as the relative angle between any two azimuthal angles is the only relevant quantity. Without any loss of generality we choose . The relations between the other two azimuthal angles are simplified to and with this choice. The remaining relations are,

The partial decay width for can be written as

(17) | |||||

At this point we would like to mention that rest frame can be chosen as lab frame and angles in this frame are measurable quantities. Matrix element for this cascade decay is expressed in terms of these lab angles in Table.(III) - (VI), whereas from Eq. (17) it is clear that , , rest frame angles are necessary to numerically estimate the total decay width for this cascade decay, though these angles cannot be measured. For this purpose we have provided the relations between lab and individual rest frame angles in this section. The final matrix element square for the cascade decay of is given in Appendix.

### iii.2 III. B. Analysis

This cascade decay involves total five independent angles, 3 polar which can be measured without any ambiguity and two azimuthal angles which can be measured with two fold ambiguity. With these angle information, we prescribe five observables as . These observables are functions of and , hence, sensitive to various NP. Using the constraints obtained for and from Table. (2), we numerically estimate the sizes of these partial decay widths with the help of Vegasvegas (). We have estimated that at 14TeV, with 50 total integrated luminosity, LHCb can generate events for our allowed range of and . So there will be sufficient number of events available to verify our analysis.

involves both SM and NP terms whereas involves purely NP term. We divide our analysis into two separate parts. In the first part, our main motivation is to prescribe a way to detect the the presence of NP. After that we try to show that if NP is present then “how do we determine it’s Lorentz structure?”. In the second part of our analysis, we explore the region where these partial decay widths are sensitive to for a fixed value of . This sensitivity gives us a hope to isolate a pure NP term from SM experimentally.