Model independent analysis of New Physics effects on B_{c}\to(D_{s},\,D^{\ast}_{s})\,\mu^{+}\mu^{-} decay observables

# Model independent analysis of New Physics effects on Bc→(Ds,D∗s)μ+μ− decay observables

Rupak Dutta National Institute of Technology Silchar, Silchar 788010, India
###### Abstract

Motivated by the anomalies present in neutral current decays, we study the corresponding decays within the standard model and beyond. We use a model independent effective theory formalism in the presence of vector and axial vector new physics operators and study the implications of the latest global fit to the data on various observables for the decays. We give predictions on several observables such as the differential branching ratio, ratio of branching ratios, forward backward asymmetry, and the longitudinal polarization fraction of the meson within standard model and within various new physics scenarios. These results can be tested at the Large Hadron Collider and, in principle, can provide complimentary information regarding new physics in neutral current decays.

## I Introduction

Although standard model (SM) of particle physics is successful in explaining various experimental observations, it, however, can not accommodate several long standing issues such as dark matter, dark energy, neutrino mass, matter antimatter asymmetry in the universe etc. It is indeed certain that physics beyond the SM exists. There are two ways to determine the nature of new physics (NP). One is direct detection of new particles and their interactions and another is indirect detection through their effects on various low energy processes. In this respect, flavor physics can, in principle, be the ideal platform to look for indirect evidences of NP. In fact, various anomalies with the SM prediction have been reported by dedicated experiments such as BABAR, Belle, and more recently by LHCb. In particular, measurement of various observables in charged current interactions and in neutral current interactions already provided hints of NP. We will focus here on anomalies present in meson decays mediated via neutral current interactions. The most important observables are the lepton flavor universality (LFU) ratios and , various angular observables in decays, and the branching ratio of decays. The experimental results confirming these anomalies are listed below.

A significant deviation from the SM expectation is observed in the LFU ratios and defined as

 R(∗)K=B(B→K(∗)μμ)B(B→K(∗)ee). (1)

The first LHCb measurement of  Aaij:2014ora () in the low bin deviates from the SM prediction  Hiller:2003js (); Bobeth:2007dw (); Bordone:2016gaq () at level. Very recently, the earlier measurement was superseded by LHCb Collaboration and it is reported to be  Aaij:2019wad (). Although it moves closer to the SM value, the deviation with the SM prediction still stands at level. Similarly, the measured value of and in the dilepton invariant mass and  Aaij:2017vbb () deviate from the SM prediction of  Capdevila:2016ivx (); Serra:2016ivr () at approximately and , respectively. Very recently, Belle collaboration has reported the values of in multiple bin but with a much larger uncertainties Abdesselam:2019wac (). The other notable deviation is the deviation observed in the angular observable in decays DescotesGenon:2012zf (); Descotes-Genon:2013vna (). LHCb Aaij:2013qta (); Aaij:2015oid () and ATLAS Aaboud:2018krd () have measured the value of the angular observable in the range and the deviation from the SM prediction is found to be more than  Aebischer:2018iyb (). Belle Abdesselam:2016llu () and CMS cms () have also measured this observable in the bin and , respectively. Although the Belle measured value differs from the SM expectation at level, the measured value by CMS is consistent with the SM expectation at level. Similarly, there is a systematic deficit in the measured value of branching ratio of  Aaij:2013aln (); Aaij:2015esa () decays as compared to the SM prediction Aebischer:2018iyb (); Straub:2015ica (). Currently the deviation with the SM prediction stands at around . If it persists and is confirmed by future experiments, it could unravel new flavor structure beyond the SM physics. Various global fits Capdevila:2017bsm (); Altmannshofer:2017yso (); DAmico:2017mtc (); Hiller:2017bzc (); Geng:2017svp (); Ciuchini:2017mik (); Celis:2017doq (); Alok:2017sui (); Alok:2017jgr (); Alok:2019ufo () to the data have been performed and it was suggested that some of these anomalies can be resolved by modifying the Wilson coefficients (WCs).

If these anomalies are due to NP, this will show up in other transition decays as well. In this paper, we analyze decays mediated via neutral current transitions within the SM and in several NP scenarios. LHCb has already measured the ratio of branching ratio in decays. Detection and measurement of various observables pertaining to meson decaying to other mesons via neutral current interactions will be feasible once more and more data will be accumulated by LHCb. It is worth mentioning that the study of such modes is complimentary to the study of decays and it can, in principle, provide useful information regarding different NP Lorentz structures. Moreover, study of these decay modes both theoretically and experimentally can act as a useful ingredient in maximizing future sensitivity to NP.

Within the SM, decays have been studied previously using the relativistic constituent quark model Faessler:2002ut (), light-front quark model Geng:2001vy (); Choi:2010ha (), QCD sum rules Azizi:2008vy (); Azizi:2008vv (), and relativistic quark model Ebert:2010dv (). In this paper, we use the relativistic quark model of Ref. Ebert:2010dv () and supplement the previous analysis by analyzing the effect of various NP on these decay modes in a model independent way. We use an effective theory formalism in the presence of new vector and axial vector couplings that couples only to the muon sector. We give prediction of various observables such as the ratio of branching ratios, lepton side forward backward asymmetry, and the longitudinal polarization fraction of the meson within the SM and within various NP scenarios.

Our paper is organized as follows. In section II, we start with the effective weak Hamiltonian for decays in the presence of new vector and axial vector operators. We also discuss the hadronic matrix elements of and and their parametrization in terms of various meson to meson transition form factors. In section III, we write down the helicity amplitudes for the and decay modes and construct several observables. In section. IV, we give predictions of all the observables in the SM and in several NP cases obtained from the global fit. We conclude with a brief summary of our results in section. V.

## Ii Formalism

The most general effective weak Hamiltonian in the presence of new vector and axial vector operators for the transition can be written as

 Heff = −4GF√2VtbV∗tsαe4π[Ceff9¯sγμPLb¯lγμl+C10¯sγμPLb¯lγμγ5l−2mbq2Ceff7¯siqνσμνPRb¯lγμl (2) +CNP9¯sγμPLb¯lγμl+CNP10¯sγμPLb¯lγμγ5l+C′9¯sγμPRb¯lγμl+C′10¯sγμPRb¯lγμγ5l],

where is the Fermi coupling constant, is the electromagnetic coupling constant, and are the relevant Cabibbo Kobayashi Maskawa (CKM) matrix elements, and are the chiral projectors. All the WCs are evaluated at a renormalization scale of . The quark mass associated with is considered to be running mass in the scheme. In principle, there can be several NP Lorentz structures such as vector, axial vector, scalar, pseudoscalar, and tensor. The scalar, pseudoscalar and the tensor NP operators are severely constrained by and measurements Alok:2010zd (); Alok:2011gv (); Bardhan:2017xcc (). Hence, we consider NP in the form of vector and axial vector operators only. Again, we do not consider NP in the dipole operator as these are well constrained by radiative decays. The non factorizable corrections coming from electromagnetic corrections to the matrix elements of purely hadronic operators in the weak effective Hamiltonian are ignored in our analysis. These corrections, however, are expected to be significant at low  Beneke:2001at (); Beneke:2004dp (). All the NP WCs , , , and are assumed to be real for our analysis. In the SM, . The effective WCs and are defined as

 Ceff7=C7−13C5−C6, Ceff9=C9+y(q2)+yBW(q2), (3)

where the contributions coming from the one loop matrix elements of the four quark operators are contained in Buras:1994dj ()

 y(q2) = h(mcmb,q2m2b)(3C1+C2+3C3+C4+3C5+C6)−12h(1,q2m2b)(4C3+4C4+3C5+C6) (4) −12h(0,q2m2b)(C3+3C4)+29(3C3+C4+3C5+C6).

Here

 h(z,s)=−lnmbμ−89lnz+827+49x−29(2+x)|1−x|1/2⎧⎪⎨⎪⎩ln∣∣√1−x+1√1−x−1∣∣−iπforx=4z2s<12arctan1√x−1forx=4z2s>1 h(0,s)=827−lnmbμ−49lns+49iπ. (5)

The phenomenological parameter involves the long distance effects coming from the resonance contributions coming from , etc. In particular, these resonances provide large peaked contributions in the bins that are close to these charmonium resonance masses. The corresponding bins are not considered in our analysis. The values of masses of charm and bottom quark in these expressions are defined in pole mass scheme. The WCs that contains the short distance contribution can be calculated perturbatively, whereas, for the calculation of the long distance contributions contained in the matrix elements of local operators between initial and final hadron states, it requires non perturbative approach. The hadronic matrix elements can be expressed in terms of various meson to meson transition form factors.

The hadronic matrix elements for the decays can be parametrized in terms of three invariant form factors. Those are

 = f+(q2)[pμBc+pμDs−M2Bc−M2Dsq2qμ]+f0(q2)M2Bc−M2Dsq2qμ, = ifT(q2)MBc+MDs[q2(pμBc+pμDs−(M2Bc−M2Ds)qμ]. (6)

Similarly, for the decays, the hadronic matrix elements can be parametrized in terms of seven invariant form factors, i.e,

 = 2iV(q2)MBc+MD∗sϵμνρσϵ∗νpBcρpD∗sσ, = 2MD∗sA0(q2)ϵ∗⋅qq2qμ+(MBc+MD∗s)A1(q2)(ϵ∗μ−ϵ∗⋅qq2qμ) −A2(q2)ϵ∗⋅qMBc+MD∗s[pμBc+pμD∗s−M2Bc−M2D∗sq2qμ], = 2T1(q2)ϵμνρσϵ∗νpBcρpD∗sσ, = T2(q2)[(M2Bc−M2D∗s)ϵ∗μ−(ϵ∗⋅q)(pμBc+pμD∗s)] (7) +T3(q2)(ϵ∗⋅q)[qμ−q2M2Bc−M2D∗s(pμBc+pμD∗s)],

where is the four momentum transfer and is polarization vector of the meson. For the and transition form factors we follow the relativistic quark model adopted in Ref. Ebert:2010dv (). It was mentioned in Ref. Ebert:2010dv () that in the limit of infinitely heavy quark mass and large energy of the final meson, the form factor results obtained in this approach are consistent with all the model independent symmetry relations Charles:1998dr (); Ebert:2001pc (). We refer to Ref. Ebert:2010dv () for all the omitted details.

## Iii Helicity amplitudes and decay observables

For the helicity amplitudes, we pattern our analysis after that of Ref. Ebert:2010dv () and, indeed, adopt a common notation. We use the helicity techniques of Refs. Korner:1989qb (); Kadeer:2005aq () and write the hadronic helicity amplitudes for decays in the presence of vector and axial vector NP operators as follows:

 H(i)±=0, H(1)0=√λq2[(Ceff9+CNP9+C′9)f+(q2)+Ceff72mbMBc+MDsfT(q2)] H(2)0=√λq2(C10+CNP10+C′10)f+(q2), H(1)t=M2Bc−M2Dsq2(Ceff9+CNP9+C′9)f0(q2), H(2)t=M2Bc−M2Dsq2(C10+CNP10+C′10)f0(q2) (8)

Similarly, for decays, the hadronic helicity amplitudes are

 H(1)± = −(M2Bc−M2D∗s)[(Ceff9+CNP9−C′9)A1(q2)MBc−MD∗s+2mbq2Ceff7T2(q2)] ±√λ[(Ceff9+CNP9+C′9)V(q2)MBc+MD∗s+2mbq2Ceff7T2(q2)], H(2)± = (C10+CNP10−C′10)[−(MBc+MD∗s)A1(q2)]±(C10+CNP10+C′10)λMBc+MD∗sV(q2), H(1)0 = −12MD∗s√q2{(Ceff9+CNP9−C′9)[(M2Bc−M2D∗s−q2)(MBc+MD∗s)A1(q2)−λMBc+MD∗sA2(q2)] +2mbCeff7[(M2Bc+3M2D∗s−q2)T2(q2)−λM2Bc−M2D∗sT3(q2)]} H(2)0 = −12MD∗s√q2(C10+CNP10−C′10)[(M2Bc−M2D∗s−q2)(MBc+MD∗s)A1(q2)−λMBc+MD∗sA2(q2)] H(1)t = −√λq2(Ceff9+CNP9−C′9)A0(q2), H(2)t = −√λq2(C10+CNP10−C′10)A0(q2), (9)

where

 λ = M4Bc+M4Ds,D∗s+q4−2(M2BcM2Ds,D∗s+M2Ds,D∗sq2+M2Bcq2) (10)

Using the helicity amplitudes, the three body and differential decay rate can be written as Ebert:2010dv ()

 dΓdq2 = G2F(2π)3(αe|VtbV∗ts|2π)2λ1/2q248M3Bc√1−4m2lq2[H(1)H†(1)(1+4m2lq2)+H(2)H†(2)(1−4m2lq2) (11) +2m2lq23H(2)tH†(2)t],

where denotes the mass of lepton and

 H(i)H†(i) = H(i)+H†(i)++H(i)−H†(i)−+H(i)0H†(i)0. (12)

We define the differential ratio of branching ratio as follows:

 RDs,D∗s(q2)=dΓ/dq2(Bc→(Ds,D∗s)μ+μ−)dΓ/dq2(Bc→(Ds,D∗s)e+e−). (13)

We also construct observables like the forward backward asymmetry of the lepton pair and the longitudinal polarization fraction of the meson as a function of dilepton invariant mass . The forward backward asymmetry is given by  Ebert:2010dv ()

 AFB(q2) = 34√1−4m2lq2{Re(H(1)+H†(2)+)−Re(H(1)−H†(2)−)H(1)H†(1)(1+4m2lq2)+H(2)H†(2)(1−4m2lq2)+2m2lq23H(2)tH†(2)t}. (14)

Similarly, the longitudinal polarization fraction of the meson can be written as Ebert:2010dv ()

 FL(q2) = (15)

It should be noted that the forward backward asymmetry observable for the decay mode is zero in the SM as the helicity amplitudes . It is worth mentioning that it can have a non zero value only if it receives contribution from scalar, pseudoscalar or tensor NP operators. Since we consider NP in vector and axial vector operators only, we do not discuss for the decay mode in section. IV.

## Iv Results and discussion

### iv.1 Inputs

For definiteness, we first report all the inputs that are used for the computation of all the decay observables. We employ a renormalization scale of throughout our analysis. For the meson masses, we use , , and , as given in Ref. Tanabashi:2018oca (). For the lepton masses, we use and from Ref. Tanabashi:2018oca (). Similarly, the mean life time of meson and the Fermi coupling constant are taken to be and , as reported in Ref. Tanabashi:2018oca (). For the quark masses, we use , , and   Altmannshofer:2008dz (). For the electromagnetic coupling constant, we use . We use as given in Ref. Bona:2006ah (). The WCs in our numerical estimates, taken from Refs. Ali:1999mm (), are reported in Table. 1.

A relativistic quark model based on quasipotential approach was adopted in Ref. Ebert:2010dv () to determine various and transition form factors. Various form factors at and the fitted parameters and , taken from Ref. Ebert:2010dv (), are reported in Table. 2.

It was shown in Ref. Ebert:2010dv () that the dependence of the form factors can be well parametrized and reproduced in the form:

 F(q2) = F(0)(1−q2M2)(1−σ1q2M2B∗s+σ2q4M4B∗s) (16)

for . Whereas, for , it can be well approximated by

 F(q2) = F(0)(1−σ1q2M2B∗s+σ2q4M4B∗s), (17)

where for and for all other form factors. We use and from Ref. Tanabashi:2018oca (). The form factors describe the hadronisation of quarks and gluons: these involve QCD in the non-perturbative regime and are a significant source of theoretical uncertainties. To gauge the effect of the form factor uncertainties on various observables, we have used uncertainty in , and .

### iv.2 SM prediction of Bc→(Ds,D∗s)l¯l decay observables

Now let us proceed to discuss our results in the SM. In Table. 3, we report our bin averaged values of various observables for the and decays. We restrict our analysis to low dilepton invariant mass region and consider seven bins ranging from . The central values are obtained using the central values of all the input parameters. For the uncertainties, we have performed a naive analysis defined as

 χ2=∑i(Oi−O0i)2Δ2i, (18)

where . Here represents the central values of all the parameters and represents uncertainty associated with each parameter. To find out the uncertainties in each observable, We impose for the decays and for the decays.

In the SM, we find the branching ratios of decays to be of which might be within the experimental sensitivity of LHCb because of the large number of meson that is being produced at the LHC. We also obtain the LFU ratios to be in the SM. It is observed that in the bin ranging from , the observable assumes negative values, whereas, for , it assumes positive values. It should be noted that the uncertainty associated with the LFU ratios and are quite negligible in comparison to the uncertainties present in the branching ratio, the forward backward asymmetry and the longitudinal polarization fraction of the meson . Measurements of these ratios in future will be crucial in determining various NP Lorentz structures.

We have shown in Fig. 1 the dependence of differential branching ratios, forward backward asymmetry, and longitudinal polarization fraction of meson in the low region . The line corresponds to the central values of all the input parameters, whereas, the band corresponds to the uncertainties associated with the CKM matrix element and the form factor inputs. In the SM, we find the zero crossing in of decays at . Our results are quite similar to the values reported in Ref. Ebert:2010dv (). Slight deviations may occur due to different choices of input parameters.