Form factors and differential distributions in rare radiative leptonic Bdecays
Abstract
We study rare radiative leptonic decays and within relativistic quark model. In addition to previous analysis we give the estimations of the branching ratios for four values of the minimal photon energy, which correspond to photon selection criteria of the Belle and LHCb detectors. We find out that the branching ratios only slightly change. The highest values corresponding to are and . We present the distribution of the forwardbackward asymmetry.
Form factors and differential distributions in rare radiative leptonic Bdecays
Anastasiia Kozachuk^{a}^{a}aemail: anastasiia.kozachuk@cern.ch \@andlastauthor Nikolai Nikitin
1D. V. Skobeltsyn Institute of Nuclear Physics, M. V. Lomonosov
Moscow State University, 119991, Moscow, Russia
\@textsuperscript2M. V. Lomonosov Moscow State University, Faculty of Physics, 119991 Moscow, Russia
\@textsuperscript3A. I. Alikhanov Institute for Theoretical and Experimental Physics, 117218 Moscow, Russia
\@textsuperscript4National Research Nuclear University MEPhI, 115409 Moscow, Russia

Abstract.
1 Introduction
At the quark level rare radiative leptonic decays correspond to transitions, which proceed through flavorchanging neutral currents (FCNCs) forbidden at tree level in the Standard Model (SM). These currents are described by penguin and box diagrams containing loops and thus lead to small values of the branching ratios. The typical order is – (see e.g. [1]). Possible contributions of new particles to the loops make these decays sensitive to New Physics. Therefore there have been a lot of studies in this sector of flavor physics and as a result at the moment we have several deviations from the SM of the order of (see discussion in [2, 3, 4]). We list some of them here:

same was observed for , in the range the discrepancy for the branching ratio is more than [14].
Indeed, more tensions come from angular analysis of performed by LHCb [15] and Belle [16], and from measurements of the ratios [17, 18, 19].
Finally, the value of the branching ratio of is 25% lower than the SM predicts, but only at . For the branching ratios of and decays the following relation takes place
(1) 
were the squared ratio of masses means that the radiative decay does not have the chirality constraint, comes from the photon emission and in the denominator is the difference between three and twoparticle phase space. For muons one can easily get the estimate , which means that the branching ratios are approximately equal . In fact, is a little bit larger due to additional dynamical effects, such as resonant contributions.
The paper is organized as follows: In Section 2 we discuss the contributions to the decay amplitude . In Section 3 we calculate the transition formfactors via dispersion approach based on constituent quark picture. In Section 4 we give numerical predictions for the branching ratios, differential distributions of the decay rates and forwardbackward asymmetry.
2 The effective Hamiltonian and the amplitude
The effective Hamiltonian describing the () weak transition has the form ([21, 22])
(2) 
where is the Fermi constant, are the scaledependent set of Wilson coefficients, and are the basis operators. For decays the scale parameter is approximately equal to GeV. The amplitudes of the basis operators between the initial and the final states may be parameterized in terms of Lorentzinvariant form factors. The calculation of these form factors is the main challenge of this work because they contain nonperturbative QCD effects.
2.1 Direct emission of the real photon from Bmeson valence quarks
The most important part which contains nonperturbative QCD contributions corresponds to the cases, when the real photon is directly emitted from the valence or quarks, and the pair is coupled to the penguin. The effective Hamiltonian in this case takes the form^{1}^{1}1Our notations and conventions are: , , , , .
(3) 
and the corresponding diagrams are shown in Fig. 1. The coefficient includes longdistance effects related to resonances in the channel [26, 27, 28]. The transition form factors of the basis operators in (2.1) are defined according to [25]
(4)  
The penguin form factors are defined as functions of two variables: is the momentum of the photon emitted from the penguin, and is the momentum of the photon emitted from the valence quark of the meson. We calculate the form factors in the framework of the dispersion approach based on constituent quark picture, the details are presented in Section 3.
2.2 Direct emission of the virtual photon from Bmeson valence quarks
Another process contributing to the amplitude is that with the real photon emitted from the penguin, whereas one of the valence quarks directly emits the virtual photon which then goes into the final pair.
This process is described by the diagrams of Fig. 2. The corresponding amplitude has the same structure as the part of the amplitude in 2.1 with replaced by . The form factors at the necessary timelike momentum transfers are not known. The difficulty is connected with appearance of neutral light vector meson resonances, and for decays and for decays, in the physical decay region. We calculate the form factors for with the use of gaugeinvariant version [29, 30] of the vector meson dominance [31, 32, 33]
(5) 
where and are the mass and the width of the vector meson resonance, are the transition form factors, defined according to the relations
and calculated in [34, 35] via relativistic quark model. The leptonic decay constant of a vector meson is given by
(6) 
2.3 Bremsstrahlung
Fig. 3 represents diagrams describing bremsstrahlung. The corresponding contribution to the amplitude reads
(7) 
.
2.4 Weak annihilation contribution
3 Transition form factors
We calculate the transition form factors in the framework of relativistic quark model, which is a dispersion approach based on constituent quark picture [34, 35]. All hadron observables are given by dispersion representations in terms of hadronic relativistic wave functions and spectral densities of corresponding feynman diagrams with constituent quarks in the loops. For the wave functions we use a Gaussian parametrization . The simplest relation can be obtained for a pseudoscalar or vector meson decay constant
(9) 
were is the meson relativistic wave function and is the spectral density. The latter is obtained as a direct result of feynman rules for the corresponding feynman diagram.
The example of the diagram for a Bmeson decay constant is given in Fig. 5. In this work we consider mesontophoton transitions; the corresponding form factors may be obtained in the form of the spectral representation
(10) 
were and are momenta of the emitted photons. The form factors were calculated in [37]. We now perform the calculation of the form factors . Each of these form factors contains two contributions corresponding to the cases when the photon is emitted from or quarks of the Bmeson:
(11) 
The spectral representations of the form factors in (3) have the form
(12) 
where is the mass of the quark, which emits the photon, is the mass of the spectator, and
(13) 
The model contains only a few parameters such as the constituent quark masses and the parameter of the wave function . These parameters were fixed in [37] using relations (9) for meson decay constants so that our results reproduce the predictions from QCD sum rules and lattice QCD.
4 Numerical results
4.1 Branching ratios
For numerical estimates we use the following values of Wilson coefficients at GeV:
, , , [22]. The evolution including ccresonances is taken from [26, 27, 28].
We obtained several distributions of the differential branching ratios, they are shown in Fig. 6 and 7.
80 MeV  100 MeV  500 MeV  1000 MeV  

4.6  4.6  4.6  4.6  
1.5  1.5  1.5  1.4  
18.5  18.5  18.4  18.3  
11.9  11.8  11.6  11.5 
In Table 1 we present the results for different values of the minimal photon energy . Numerical values of are given in the meson rest frame and are different in others because energy is not Lorentzinvariant. The values in the range correspond to different photon selection criteria at the Belle II detector [40], while the interval is relevant for those at the LHCb detector [38, 39]. We only take typical values as in LHCb reference frame Bmesons have a sufficiently wide energy distribution. However, from Table 1 it is clear that the branching ratios only slightly depend on the particular choice of , and the related error is smaller then that of the model. We also use the value to compare the results with those of the previous work [41].
4.2 Forwardbackward asymmetry
We obtained distribution of the forwardbackward asymmetry, defined by the relation
(14) 
where , is the angle between and . The distribution is presented in Fig. 8. The measurement of the forwardbackward asymmetry seems to be a hard if not impossible task, because the final state does not carry any information about the flavor of the decaying Bmeson. In addition, the signs of the asymmetries corresponding to and mesons are opposite. In the absence of flavor cut the total asymmetry equals zero aside from CPviolating effects. It appears that such selection is impossible at LHCb, but at Belle II one can use the fact that neutral Bmesons are produced in an entangled state. Thus, if one of the Bmesons decays to a state with a certain flavor, the other Bmeson which decays to has the opposite flavor. Now, if the interval between the decays is less than half of the of oscillation period, one can claim that the flavor of the second Bmeson is known with sufficient probability. For each selection procedure one can also account for the oscillations contribution and therefore improve the prediction accuracy. A method for such calculations was developed in [44].
The decay rates and the forwardbackward asymmetry were previously calculated in several works [23, 24, 25, 41, 43, 44]. In [23] and [24] not all the contributions were taken into account, and in [23, 24, 25, 41] the transition form factors were estimated from symmetry considerations coming from LEET. We made direct calculation of the form factors in the framework of the relativistic quark model. Our results agree nicely with [24]. The results [23, 43] are based on not fully consistent models for the form factors and therefore do not seem to us convincing.
5 Conclusions
We obtained predictions for the differential distributions and the branching ratios for the and decays taking into account the following contributions to the amplitude of the process: photon emission from the and the valence quarks of the meson, weak annihilation, and bremsstrahlung. The corresponding form factors were calculated in the framework of the relativistic quark model. We represented the numerical estimates for several values of the minimal photon energy corresponding to photon selection criteria of the Belle and LHCb detectors and found out that the branching ratios almost do not change with the photon energy cut.
6 Acknowledgements
The work was supported by grant 161210280 of the Russian Science Foundation. The authors are thankful to Dmitri Melikhov for useful discussions.
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