A Closer Look at and
The measurement of (), the ratio of the branching fraction of to that of , shows deviation from its Standard Model (SM) prediction. The combined deviation is at the level of according to the Heavy Flavour Averaging Group (HFAG). We perform an effective field theory analysis (at the dimension 6 level) of these potential New Physics (NP) signals assuming gauge invariance. We first show that, in general, and are theoretically independent observables and hence, their theoretical predictions are not correlated. We identify the operators that can explain the experimental measurements of and individually and also together. Motivated by the recent measurement of the polarisation in decay, by the Belle collaboration, we study the impact of a more precise measurement of (and a measurement of ) on the various possible NP explanations. Furthermore, we show that the measurement of in bins of , the square of the invariant mass of the lepton neutrino system, along with the information on polarisation, can completely distinguish the various operator structures.
A Closer Look at and
Debjyoti Bardhan††thanks: Speaker. ††thanks: The speaker wishes to thank the co-authors on the paper for helping with the talk. Further, thanks is due to Prof. Amol Dighe and Prof. Gautam Bhattacharyya for valuable suggestions before the talk.
Tata Institute of Fundamental Research
The quantity is defined as the following ratio between two branching ratios:
where . This quantity, being a ratio, is a ‘clean’ observable devoid of the systematic uncertainties that plague individual measurements of branching ratios. Experimental measurements of these two quantities - for and for  - don’t match with the theoretical predictions from the Standard Model (SM) - for and for . This corresponds to deviations and significance for and respectively, while the discrepancy for the two taken together is quite large 111It is worth noting that even though the quoted results suggest a large deviation, a recent measurement of by the Belle collaboration  is consistent with the SM value, although the measurement is quite imprecise.. This might well be a signal for new physics and we perform a model-independent analysis of the process using six-dimensional operators; in this analysis, we assume that any NP only affects the third leptonic generation.
Besides and , we also consider the binned value of and , the polarisation of the final state lepton, and and the forward-backward asymmetry in the two processes, and 222In principle, various differential distributions are also sensitive to the different NP Lorentz structures, see e.g., .. While a recent measurement of has been reported by Belle for the first time (although with large errors) , none of the other quantities have been experimentally measured as yet. The definitions of the observables is given below:
The branching ratio can be written as
where and is the angle between the lepton and -meson in the lepton-neutrino centre-of-mass frame.
The decay amplitude for the process can be factorised into two parts - the hadronic part an the leptonic part. The hadronic part of the decay amplitude cannot be calculated exactly and is parameterised using form factors. These form factors are calculated in some theoretical and numerical framework and, in this work, we choose to simply borrow those results.
2 Lagrangian and Operator Basis
The effective six-dimensional Lagrangian for we use for the analysis is given by:
and the set of Wilson Coeffients (WCs) corresponding to these operators are defined at the renormalization scale .
In the SM, we would have . We wish to go beyond the SM, but we shall respect the full gauge invariance of the SM and consequently only consider the operators listed on the left in 2. Further, since it is difficult to build a microscopic model with tensor interaction, we neglect its contribution in this note.(For the study of tensor operators, refer to the Appendix of ).
3 Form Factors
3.1 For decay
The non-zero hadronic matrix elements for transition (ignoring the tensor) are parameterized by
Calculations for the form factors and are known in a lattice framework . The axial vector and the pseudoscalar matrix elements are zero from symmetry considerations and thus only the WCs and contribute to this decay.
3.2 For decay
The non-zero hadronic matrix elements for transition are parametrised by
While no lattice calculations exist for the form factors in this case, they have been calculated in a Heavy Quark Effective Theory (HQET) framework  and we borrow those results. In this case, symmetry dictates that the scalar current is zero and thus there is no contribution to the decay width from .
3.3 Independence of and
We see that while and contribute to the decay process, , and contribute to the other one. Thus, given the independence of the WCs, the two processes are independent of each other since they depend of different sets of WCs. In other words, and are theoretically independent measurements and allow for separate explanations.
4 Explaining Alone
The quantities , and (in 1) can be calculated for a particular helicity of the final state lepton using a helicity amplitude approach. The complete expressions are given in  and it is not repeated here. Since only and are relevant, we can plot as a variation of the two WCs and note the range of values for which it satisfies the experimental bounds.
This is done in Fig. 1, where the red (brown) band corresponds to the () value on the experimental measurement.
5 Explaining Alone
We can carry out a similar treatment for the case of decay. In this case, three WCs - , and - contribute. The plots of as a function of the different WCs are given in Fig. 4 As before, the () bands are indicated by the red (brown) bands.
The prediction for the binned is given in Fig. 5. In this case, we do have a measurement of , but it is quite imprecise. In the left plot of Fig. 6, the size of the errors indicated for the Belle measurement is a projection with 20 data, which is expected to be collected by the year 2021; the central value indicated is the current central value. As a matter of completion, we also plot the prediction for on the right of Fig. 6, although no measurement of this quantity exists as yet.
We can combine the predictions for the binned restricted to the highest bin, and to construct three planes. When plotted in these three planes, the regions of the allowed values of the WCs all separate out nicely as shown in Fig. 7. A future measurement of any two of these three observables would help in restricting us to a particular region, thus limiting the scope of any NP model.
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