Model independent analysis of new physics effectson Bc → ðDs;D�

s Þμ + μ− decay observables

Rupak Dutta*

National Institute of Technology Silchar, Silchar 788010, India

(Received 14 June 2019; published 21 October 2019)

Motivated by the anomalies present in b → slþl− neutral current decays, we study the correspondingBc → ðDs;D�

sÞμþμ− decays within the standard model and beyond. We use a model independent effectivetheory formalism in the presence of vector and axial vector new physics operators and study theimplications of the latest global fit to the b → slþl− data on various observables for the Bc →ðDs;D�

sÞμþμ− decays. We give predictions on several observables such as the differential branchingratio, ratio of branching ratios, forward backward asymmetry, and the longitudinal polarization fraction ofthe D�

s meson within standard model and within several new physics scenarios. These results can be testedat the Large Hadron Collider and, in principle, can provide complementary information regarding newphysics in b → slþl− neutral current decays.

DOI: 10.1103/PhysRevD.100.075025

I. INTRODUCTION

Although standard model (SM) of particle physics issuccessful in explaining various experimental observations,it, however, cannot accommodate several longstandingissues such as dark matter, dark energy, neutrino mass,matter antimatter asymmetry in the universe etc. It indi-rectly confirms the existence of a more global theorybeyond the SM. There are two ways to determine thenature of new physics (NP). One is direct detection of newparticles and their interactions and another is indirectdetection through their effects on various low energyprocesses. 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 havebeen reported by dedicated experiments such as BABAR,Belle, and more recently by LHCb. In particular, meas-urement of various observables in b → cτν charged currentinteractions and in b → slþl− neutral current interactionsalready provided hints of NP. We will focus here onanomalies present in B meson decays mediated via b →slþl− neutral current interactions. The most importantobservables are the lepton flavor universality (LFU) ratiosRK and RK� , various angular observables in B → K�μþμ−

decays, and the branching ratio of Bs → ϕμþμ− decays.

The experimental results confirming these anomalies arelisted below.A significant deviation from the SM expectation is

observed in the LFU ratios RK and RK� defined as

Rð�ÞK ¼ BðB → Kð�ÞμμÞ

BðB → Kð�ÞeeÞ : ð1Þ

The first LHCb measurement of RK ¼ 0.745þ0.090−0.074 � 0.036

[1] in the low q2 bin 1 < q2 < 6 GeV2 deviates fromthe SM prediction RK ≈ 1 [2–4] at 2.6σ level. Veryrecently, the earlier measurement was superseded byLHCb Collaboration and it is reported to be RK ¼0.846þ0.060þ0.016

−0.054−0.014 [5]. Although it moves closer to the SMvalue, the deviation with the SM prediction still standsat 2.5σ level. Similarly, the measured value of RK� ¼0.66þ0.11

−0.07 � 0.03 and 0.69þ0.11−0.07 � 0.05 in the dilepton invari-

ant mass q2 ¼ ½0.045; 1.1� GeV2 and ½1.1; 6.0� GeV2 [6]deviate from the SM prediction of RK� ≈ 1 [7,8] at approx-imately 2.1σ and 2.4σ, respectively. Very recently, Bellecollaboration has reported the values of RK� in multiple q2

bin but with a much larger uncertainties [9]. The othernotable deviation is the deviation observed in the angularobservable P0

5 in B → K�μþμ− decays [10,11]. LHCb[12,13] and ATLAS [14] have measured the value of theangular observableP0

5 in the q2 range 4.0 < q2 < 6.0 GeV2

and the deviation from the SMprediction is found to bemorethan 3σ [15]. Belle [16] and CMS [17] have also measuredthis observable in the q2 bin 4.3 < q2 < 8.68 GeV2 and4.3 < q2 < 6.0 GeV2, respectively. Although the Bellemeasured value differs from the SM expectation at 2.6σ

*rupak@phy.nits.ac.in

Published by the American Physical Society under the terms ofthe Creative Commons Attribution 4.0 International license.Further distribution of this work must maintain attribution tothe author(s) and the published article’s title, journal citation,and DOI. Funded by SCOAP3.

PHYSICAL REVIEW D 100, 075025 (2019)

2470-0010=2019=100(7)=075025(13) 075025-1 Published by the American Physical Society

level, the measured value by CMS is consistent with the SMexpectation at 1σ level. Similarly, there is a systematic deficitin the measured value of branching ratio of Bs → ϕμþμ−[18,19] decays as compared to the SM prediction [15,20].Currently the deviation with the SM prediction stands ataround 3.7σ. If it persists and is confirmed by futureexperiments, it could unravel new flavor structure beyondthe SMphysics.Various global fits [21–31] to theb → slþl−data have been performed and it was suggested that some ofthese anomalies can be resolved by modifying the Wilsoncoefficients (WCs).It should be noted that, if these anomalies are due to NP,

this will show up in other similar decay modes as well.There are various similar decay channels with same under-lying quark level transition and the LFU violation can beexplored in these decay modes as well. In this context,B → K�

2μþμ− and B → K1μ

þμ− decay modes have beenanalyzed very recently in Ref. [32] and Ref. [33], respec-tively. In this paper, we analyze Bc → ðDs;D�

sÞμþμ−decays mediated via b → slþl− neutral current transitionswithin the SM and in several NP scenarios. LHCb hasalready measured the ratio of branching ratio RJ=Ψ in Bc →J=Ψlν decays. Detection and measurement of variousobservables pertaining to Bc meson decaying to othermesons via b → slþl− neutral current interactions willbe feasible once more and more data will be accumulatedby LHCb. It is worth mentioning that the study of suchmodes is complimentary to the study of B → ðK;K�Þμþμ−decays and it can, in principle, provide useful informationregarding different NP Lorentz structures. Moreover, studyof these decay modes both theoretically and experimentallycan act as a useful ingredient in maximizing futuresensitivity to NP.Within the SM, Bc → ðDs;D�

sÞμþμ− decays have beenstudied previously using the relativistic constituent quarkmodel [34], light-front quark model [35,36], QCD sumrules [37,38], and relativistic quark model [39]. In thispaper, we use the relativistic quark model of Ref. [39] andsupplement the previous analysis by analyzing the effect ofseveral NP on these decay modes in a model independentway. We use an effective theory formalism in the presenceof new vector and axial vector couplings that couples onlyto the muon sector. We give prediction of several observ-ables such as the ratio of branching ratios, lepton sideforward backward asymmetry, and the longitudinal polari-zation fraction of the D�

s meson within the SM and withinvarious NP scenarios.Our paper is organized as follows. In Sec. II, we start

with the effective weak Hamiltonian for b → slþl− decaysin the presence of new vector and axial vector operators.We also discuss the hadronic matrix elements of Bc → Dsand Bc → D�

s and their parametrization in terms of variousmeson to meson transition form factors. In Sec. III, wewrite down the helicity amplitudes for the Bc → Dsμ

þμ−and Bc → D�

sμþμ− decay modes and construct several

observables. In Sec. IV, we give predictions of all theobservables in the SM and in several NP cases obtainedfrom the global fit. We conclude with a brief summary ofour results in Sec. V.

II. FORMALISM

The most general effective weak Hamiltonian in thepresence of new vector and axial vector operators for thejΔBj ¼ jΔSj ¼ 1 transition can be written as

Heff ¼−4GFffiffiffi

2p VtbV�

tsαe4π

�Ceff9 s̄γμPLbl̄γμlþC10s̄γμPLbl̄γμγ5l

−2mb

q2Ceff7 s̄iqνσμνPRbl̄γμlþCNP

9 s̄γμPLbl̄γμl

þCNP10 s̄γ

μPLbl̄γμγ5lþC09s̄γ

μPRbl̄γμl

þC010s̄γ

μPRbl̄γμγ5l

�; ð2Þ

where GF is the Fermi coupling constant, αe is theelectromagnetic coupling constant, Vtb and Vts are therelevant Cabibbo Kobayashi Maskawa (CKM) matrixelements, and PR;L ¼ ð1� γ5Þ=2 are the chiral projectors.All the WCs are evaluated at a renormalization scale ofμ ¼ mpole

b ¼ 4.8 GeV. The b quark mass associated withCeff7 is considered to be running mass in the MS scheme.

In principle, there can be several NP Lorentz structuressuch as vector, axial vector, scalar, pseudoscalar, andtensor. The scalar, pseudoscalar and the tensor NP oper-ators are severely constrained by Bs → μμ and b → sγmeasurements [40–42]. Hence, we consider NP in the formof vector and axial vector operators only. Again, we do notconsider NP in the dipole operator as these are wellconstrained by radiative decays. The non factorizablecorrections coming from electromagnetic corrections tothe matrix elements of purely hadronic operators in theweak effective Hamiltonian are ignored in our analysis.These corrections, however, are expected to be significantat low q2 [43,44]. All the NP WCs CNP

9 , CNP10 , C

09, and C0

10

are assumed to be real for our analysis. In the SM,CNP9 ¼ CNP

10 ¼ C09 ¼ C0

10 ¼ 0. The effective WCs Ceff7

and Ceff9 are defined as

Ceff7 ¼ C7 −

1

3C5 − C6;

Ceff9 ¼ C9 þ yðq2Þ þ yBWðq2Þ; ð3Þ

where the contributions coming from the one loopmatrix elements of the four quark operators are containedin [45]

RUPAK DUTTA PHYS. REV. D 100, 075025 (2019)

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yðq2Þ ¼ h

�mc

mb;q2

m2b

�ð3C1 þ C2 þ 3C3 þ C4 þ 3C5 þ C6Þ −

1

2h

�1;

q2

m2b

�ð4C3 þ 4C4 þ 3C5 þ C6Þ

−1

2h

�0;

q2

m2b

�ðC3 þ 3C4Þ þ

2

9ð3C3 þ C4 þ 3C5 þ C6Þ: ð4Þ

Here

hðz; sÞ ¼ − lnmb

μ−8

9ln zþ 8

27þ 4

9x −

2

9ð2þ xÞj1 − xj1=2

8>>><>>>:

ln

����ffiffiffiffiffiffiffiffiffiffiffi1 − x

p þ 1ffiffiffiffiffiffiffiffiffiffiffi1 − x

p− 1

���� − iπ for x ¼ 4z2s < 1

2 arctan 1ffiffiffiffiffiffix−1

p for x ¼ 4z2s > 1

hð0; sÞ ¼ 8

27− ln

mb

μ−4

9ln sþ 4

9iπ: ð5Þ

The phenomenological parameter yBWðq2Þ involves thelong distance effects coming from the cc̄ resonancecontributions coming from J=Ψ, Ψ0 etc. In particular, theseresonances provide large peaked contributions in the q2

bins that are close to these charmonium resonance masses.The corresponding q2 bins are not considered in ouranalysis. The values of masses of charm and bottom quarkin these expressions are defined in pole mass scheme. TheWCs that contains the short distance contribution can be

calculated perturbatively, whereas, for the calculation of thelong distance contributions contained in the matrix ele-ments of local operators between initial and final hadronstates, it requires a nonperturbative approach. The hadronicmatrix elements can be expressed in terms of various mesonto meson transition form factors.The hadronic matrix elements for the Bc → Dsμ

þμ−decays can be parametrized in terms of three invariant formfactors. Those are

hDsjs̄γμbjBci ¼ fþðq2Þ�pμBc

þ pμDs

−M2

Bc−M2

Ds

q2qμ�þ f0ðq2Þ

M2Bc

−M2Ds

q2qμ;

hDsjs̄σμνqνbjBci ¼ifTðq2Þ

MBcþMDs

½q2ðpμBc

þ pμDsÞ − ðM2

Bc−M2

DsÞqμ�: ð6Þ

Similarly, for the Bc → D�sμ

þμ− decays, the hadronic matrix elements can be parametrized in terms of seven invariant formfactors, i.e.,

hD�s js̄γμbjBci ¼

2iVðq2ÞMBc

þMD�s

ϵμνρσϵ�νpBcρpD�

s σ;

hD�s js̄γμγ5bjBci ¼ 2MD�

sA0ðq2Þ

ϵ� · qq2

qμ þ ðMBcþMD�

sÞA1ðq2Þ

�ϵ�μ −

ϵ� · qq2

qμ�

− A2ðq2Þϵ� · q

MBcþMD�

s

�pμBc

þ pμD�

s−M2

Bc−M2

D�s

q2qμ�;

hD�s js̄iσμνqνbjBci ¼ 2T1ðq2Þϵμνρσϵ�νpBcρ

pD�s σ;

hD�s js̄iσμνγ5qνbjBci ¼ T2ðq2Þ½ðM2

Bc−M2

D�sÞϵ�μ − ðϵ� · qÞðpμ

Bcþ pμ

D�s�

þ T3ðq2Þðϵ� · qÞ�qμ −

q2

M2Bc

−M2D�

s

ðpμBc

þ pμD�

sÞ�; ð7Þ

where qμ ¼ ðpB − pDs;D�sÞμ is the four momentum transfer and ϵμ is polarization vector of the D�

s meson. For the Bc → Ds

and Bc → D�s transition form factors we follow the relativistic quark model adopted in Ref. [39]. It was mentioned in

Ref. [39] that in the limit of infinitely heavy quark mass and large energy of the final meson, the form factor results obtainedin this approach are consistent with all the model independent symmetry relations [46,47]. We refer to Ref. [39] for all theomitted details.

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III. HELICITY AMPLITUDES AND DECAYOBSERVABLES

For the helicity amplitudes, we pattern our analysis afterthat of Ref. [39] and, indeed, adopt a common notation. We

use the helicity techniques of Refs. [48,49] and write thehadronic helicity amplitudes for Bc → Dslþl− decays inthe presence of vector and axial vector NP operators asfollows:

HðiÞ� ¼ 0;

Hð1Þ0 ¼

ffiffiffiffiffiffiλ

q2

s �ðCeff

9 þ CNP9 þ C0

9Þfþðq2Þ þ Ceff7

2mb

MBcþMDs

fTðq2Þ�

Hð2Þ0 ¼

ffiffiffiffiffiffiλ

q2

sðC10 þ CNP

10 þ C010Þfþðq2Þ;

Hð1Þt ¼ M2

Bc−M2

Ds

q2ðCeff

9 þ CNP9 þ C0

9Þf0ðq2Þ;

Hð2Þt ¼ M2

Bc−M2

Ds

q2ðC10 þ CNP

10 þ C010Þf0ðq2Þ ð8Þ

Similarly, for Bc → D�slþl− decays, the hadronic helicity amplitudes are

Hð1Þ� ¼ −ðM2

Bc−M2

D�sÞ�ðCeff

9 þ CNP9 − C0

9ÞA1ðq2Þ

MBc−MD�

s

þ 2mb

q2Ceff7 T2ðq2Þ

�

�ffiffiffiλ

p �ðCeff

9 þ CNP9 þ C0

9ÞVðq2Þ

MBcþMD�

s

þ 2mb

q2Ceff7 T1ðq2Þ

�;

Hð2Þ� ¼ ðC10 þ CNP

10 − C010Þ½−ðMBc

þMD�sÞA1ðq2Þ� � ðC10 þ CNP

10 þ C010Þ

ffiffiffiλ

p

MBcþMD�

s

Vðq2Þ;

Hð1Þ0 ¼ −

1

2MD�s

ffiffiffiffiffiq2

p �ðCeff

9 þ CNP9 − C0

9Þ�ðM2

Bc−M2

D�s− q2ÞðMBc

þMD�sÞA1ðq2Þ −

λ

MBcþMD�

s

A2ðq2Þ�

þ 2mbCeff7

�ðM2

Bcþ 3M2

D�s− q2ÞT2ðq2Þ −

λ

M2Bc

−M2D�

s

T3ðq2Þ�

Hð2Þ0 ¼ −

1

2MD�s

ffiffiffiffiffiq2

p ðC10 þ CNP10 − C0

10Þ�ðM2

Bc−M2

D�s− q2ÞðMBc

þMD�sÞA1ðq2Þ −

λ

MBcþMD�

s

A2ðq2Þ�

Hð1Þt ¼ −

ffiffiffiffiffiλ

q2

sðCeff

9 þ CNP9 − C0

9ÞA0ðq2Þ;

Hð2Þt ¼ −

ffiffiffiffiffiλ

q2

sðC10 þ CNP

10 − C010ÞA0ðq2Þ; ð9Þ

where

λ ¼ M4Bc

þM4Ds;D�

sþ q4 − 2ðM2

BcM2

Ds;D�sþM2

Ds;D�sq2 þM2

Bcq2Þ: ð10Þ

Using the helicity amplitudes, the three body Bc → Dslþl− and Bc → D�slþl− differential decay rate can be written as [39]

dΓdq2

¼ G2F

ð2πÞ3�αejVtbV�

tsj2π

�2 λ1=2q2

48M3Bc

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 −

4m2l

q2

s �Hð1ÞH†ð1Þ

�1þ 4m2

l

q2

�þHð2ÞH†ð2Þ

�1 −

4m2l

q2

�þ 2m2

l

q23Hð2Þ

t H†ð2Þt

�; ð11Þ

where ml denotes the mass of lepton and

HðiÞH†ðiÞ ¼ HðiÞþ H†ðiÞ

þ þHðiÞ− H†ðiÞ

− þHðiÞ0 H†ðiÞ

0 : ð12Þ

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We define the differential ratio of branching ratio asfollows:

RDs;D�sðq2Þ ¼ dΓ=dq2ðBc → ðDs;D�

sÞμþμ−ÞdΓ=dq2ðBc → ðDs;D�

sÞeþe−Þ: ð13Þ

We also construct observables such as the forward back-ward asymmetry of the lepton pair AFB and the longitudinalpolarization fraction of the D�

s meson FL as a function ofdilepton invariant mass q2. The forward backward asym-metry AFBðq2Þ is given by [39]

AFBðq2Þ ¼3

4

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 −

4m2l

q2

s 8<: ReðHð1Þ

þ H†ð2Þþ Þ − ReðHð1Þ

− H†ð2Þ− Þ

Hð1ÞH†ð1Þ ð1þ 4m2l

q2 Þ þHð2ÞH†ð2Þ ð1 − 4m2l

q2 Þ þ2m2

lq2 3Hð2Þ

t H†ð2Þt

9=;: ð14Þ

Similarly, the longitudinal polarization fraction of the D�s

meson can be written as [39]

FLðq2Þ

¼Hð1Þ

0 H†ð1Þ0 ð1þ 4m2

lq2 ÞþHð2Þ

0 H†ð2Þ0 ð1− 4m2

lq2 Þþ

2m2l

q2 3Hð2Þt H†ð2Þ

t

Hð1ÞH†ð1Þ ð1þ 4m2l

q2 ÞþHð2ÞH†ð2Þ ð1− 4m2l

q2 Þþ2m2

lq2 3H

ð2Þt H†ð2Þ

t

ð15Þ

It should be noted that the forward backward asymmetryobservable AFBðq2Þ for the Bc → Dsμ

þμ− decay mode iszero in the SM as the helicity amplitudes Hi

� ¼ 0. It isworth mentioning that it can have a nonzero value only if itreceives contribution from scalar, pseudoscalar, or tensorNP operators. Since we consider NP in vector and axialvector operators only, we do not discuss AFBðq2Þ for theBc → Dsμ

þμ− decay mode in Sec. IV.

IV. RESULTS AND DISCUSSION

A. Inputs

For definiteness, we first report all the inputs that areused for the computation of all the decay observables. Weemploy a renormalization scale of μ ¼ 4.8 GeV throughoutour analysis. For the meson masses, we use MBc

¼6.2751 GeV, MDs

¼1.968GeV, and MD�s¼ 2.1122 GeV,

as given in Ref. [50]. For the lepton masses, we use me ¼0.5109989461 × 10−3 GeV and mμ ¼ 0.1056583715 GeV

from Ref. [50]. Similarly, the mean life time of Bc mesonand the Fermi coupling constant are taken to be τBc

¼0.507 × 10−12 s and GF ¼ 1.1663787 × 10−5 GeV−2, asreported in Ref. [50]. For the quark masses, we usembðMSÞ ¼ 4.2 GeV, mcðMSÞ ¼ 1.28 GeV, and mpole

b ¼4.8 GeV [51]. For the electromagnetic coupling constant,we use α−1e ¼ 133.28. We use jVtbVtsj ¼ 0.0401� 0.0010as given in Ref. [52]. The WCs in our numerical estimates,taken from Refs. [53], are reported in Table. I. A relativisticquark model based on quasipotential approach was adoptedin Ref. [39] to determine various Bc → Ds and Bc → D�

s

transition form factors. Various form factors at q2 ¼ 0 andthe fitted parameters σ1 and σ2, taken from Ref. [39], arereported in Table II. It was shown in Ref. [39] that the q2

dependence of the form factors can be well parametrizedand reproduced in the form:

Fðq2Þ ¼ Fð0Þð1 − q2

M2Þ1 − σ1

q2

M2B�sþ σ2

q4

M4B�s

� ð16Þ

for Fðq2Þ ¼ fþðq2Þ; fTðq2Þ; Vðq2Þ; A0ðq2Þ; T1ðq2Þ.Whereas, for Fðq2Þ¼ f0ðq2Þ; A1ðq2Þ; A2ðq2Þ;T2ðq2Þ;T3ðq2Þ, it can be well approximated by

Fðq2Þ ¼ Fð0Þ1 − σ1

q2

M2B�sþ σ2

q4

M4B�s

� ; ð17Þ

TABLE I. Wilson coefficients evaluated at renormalization scale of μ ¼ 4.8 GeV from Ref. [53].

C1 C2 C3 C4 C5 C6 Ceff7

C9 C10

−0.248 1.107 0.011 −0.026 0.007 −0.031 −0.313 4.344 −4.669

TABLE II. Bc → Ds and Bc → D�s form factors at q2 ¼ 0 and the fitted parameters σ1 and σ2 from Ref. [39].

fþ f0 fT V A0 A1 A2 T1 T2 T3

F0 0.129 0.129 0.098 0.182 0.070 0.089 0.110 0.085 0.085 0.051σ1 2.096 2.331 1.412 2.133 1.561 2.479 2.833 1.540 2.577 2.783σ2 1.147 1.666 0.048 1.183 0.192 1.686 2.167 0.248 1.859 2.170

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where M ¼ MBsfor A0ðq2Þ and M ¼ MB�

sfor all other

form factors. We use MBs¼ 5.36689 GeV and MB�

s¼

5.4154 GeV from Ref. [50]. The form factors describe thehadronization of quarks and gluons: these involve QCD inthe nonperturbative regime and are a significant source oftheoretical uncertainties. To gauge the effect of the formfactor uncertainties on various observables, we have used�5% uncertainty in Fð0Þ, σ1 and σ2.

B. SM prediction of Bc → ðDs;D�s Þμ+ μ− decay

observables

Now let us proceed to discuss our results in the SM. InTable III, we report our q2 bin averaged values of variousobservables for the Bc → Dsμ

þμ− and Bc → D�sμ

þμ−decays. We restrict our analysis to low dilepton invariantmass region and consider seven q2 bins ranging fromð0.045–6.0Þ GeV2. The central values are obtained using thecentral values of all the input parameters. For the uncertain-ties, we have performed a naive χ2 analysis defined as

χ2 ¼Xi

ðOi −O0i Þ2

Δ2i

; ð18Þ

whereOi ¼ ðjVtbV�tsj; Fð0Þ; σ1; σ2Þ. HereO0

i represents thecentral values of all the parameters and Δi represents 1σuncertainty associated with each parameter. To find out theuncertainties in each observable, We impose χ2 ≤ 2.156 forthe Bc → Dsμ

þμ− decays and χ2 ≤ 8.643 for the Bc →D�

sμþμ− decays. In the SM, we find the branching ratios of

Bc → ðDs;D�sÞμþμ− decays to be of Oð10−8Þ which might

bewithin the experimental sensitivity ofLHCbbecause of thelarge number of Bc mesons that are being produced atthe LHC. We also obtain the LFU ratios to be RDs;D�

s≈ 1

in the SM. It is observed that in the q2 bin ranging fromð0.045–2Þ GeV2, the hAD�

sFBi observable assumes negative

values,whereas, forq2 > 2 GeV2, it assumes positivevalues.It should be noted that the uncertainty associated with theLFU ratiosRDs

andRD�sare quite negligible in comparison to

the uncertainties present in the branching ratio, the forwardbackward asymmetry hAD�

sFBi and the longitudinal polarization

fraction of theD�s meson hFD�

sL i.Measurements of these ratios

in future will be crucial in determining various NP Lorentzstructures.We have shown in Fig. 1 the q2 dependence of differ-

ential branching ratios, forward backward asymmetry, andlongitudinal polarization fraction ofD�

s meson in the low q2

region 0.045 ≤ q2 ≤ 6 GeV2. The line corresponds to thecentral values of all the input parameters, whereas, the bandcorresponds to the uncertainties associated with the CKMmatrix element and the form factor inputs. In the SM, wefind the zero crossing in AFBðq2Þ of Bc → D�

sμþμ− decays

at q2 ¼ 2.2� 0.2 GeV2. Our results are quite similar to thevalues reported in Ref. [39]. Slight deviations may occurdue to different choices of input parameters.

C. New physics analysis

Our main objective is to determine the effect of NP onBc → ðDs;D�

sÞμþμ− decay observables in a model inde-pendent way. To this end, we use an effective theoryformalism in the presence of new vector (V) and axialvector (A) couplings in our analysis. Although there canbe other NP Lorentz structures such as scalar (S), pseu-doscalar (P), and tensor (T), they are severely constrainedby Bs → μμ and b → sγ data. Hence we omit any dis-cussion regarding these NP operators. Global fits of NP tothe b → slþl− data have been carried out by several groups[21–31]. In Ref. [30], the authors perform a global fit ofCNP9 , CNP

10 , C09, and C0

10 by using the constraints comingnot only from RK , RK� , P0

5, and BðBs → ϕμþμ−Þ but alsofrom BðBs → μþμ−Þ, differential branching ratios ofB0;þ → K0;þ�

μþμ−, B0;þ→K0;þμþμ−, and B → Xsμþμ−,

angular observables in B0 → K0�μþμ− and Bs → ϕμþμ−decays. Two different scenarios were considered inRef. [30]. In 1D scenario, the best solutions to theseanomalies were obtained for CNP

9 , CNP10 , C

NP9 ¼ −CNP

10 andCNP9 ¼ −C0

9. Similarly, for 2D scenario, where NP con-tributes to two WCs, the best solutions were obtained forðCNP

9 ; CNP10 Þ, ðCNP

9 ; C09Þ, and ðCNP

9 ; C010Þ. There are other

possibilities with different WCs exist that give rise tosimilar fits. We, however, consider only seven of them: fourfrom 1D scenario and three from 2D scenario. The best fit

TABLE III. q2 bin (in GeV2) averaged values of various observables of Bc → Dsμþμ− and Bc → D�

sμþμ− decays in the SM. The

uncertainties in each observable corresponds to the uncertainties associated with the meson to meson transition form factors and theCKM matrix elements.

Observable=q2 bin [0.045–1.0] [1.0–2.0] [2.0–3.0] [3.0–4.0] [4.0–5.0] [5.0–6.0] [1.0–6.0]

107 × BðBc → DsμμÞ 0.025� 0.001 0.030� 0.002 0.034� 0.002 0.038� 0.002 0.043� 0.003 0.049� 0.004 0.194� 0.013RDs

1.006� 0.008 1.007� 0.002 1.005� 0.001 1.004� 0.001 1.003� 0.001 1.003� 0.001 1.004� 0.001107 × BðBc → D�

sμμÞ 0.024� 0.001 0.011� 0.001 0.014� 0.002 0.020� 0.002 0.028� 0.003 0.039� 0.004 0.113� 0.012

hAD�s

FBi −0.064� 0.002 −0.076� 0.012 0.033� 0.006 0.110� 0.009 0.160� 0.011 0.194� 0.013 0.123� 0.009

hFD�s

L i 0.266� 0.033 0.711� 0.034 0.662� 0.032 0.566� 0.035 0.488� 0.036 0.430� 0.037 0.526� 0.035

RD�s

0.999� 0.005 0.993� 0.002 0.992� 0.001 0.993� 0.001 0.994� 0.001 0.995� 0.001 0.994� 0.001

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values and the corresponding Δχ2 values of all these NPWCs for 1D and 2D scenarios, taken from Ref. [30], arereported in Table IV. It should be noted that NP contribu-tions to ðCNP

9 ; C09Þ and ðCNP

9 ; C010Þ are the most favored ones

from the 2D scenario and NP in CNP9 ¼ −C0

9 is the mostfavored one from 1D scenario.In Appendix, we report q2 bin averaged values of various

observables such as the branching ratio, ratio of branchingratio, forward backward asymmetry, and longitudinalpolarization fraction of the D�

s meson for the Bc →Dsμ

þμ− and Bc → D�sμ

þμ− decays in the presence ofthese NP WCs. In each q2 bin, the branching ratio of Bc →Dsμ

þμ− is smaller in each NP scenarios than in the SMexcept for CNP

9 ¼ −C09. It remains SM like for CNP

9 ¼ −C09.

A similar conclusion can be made for the ratio of branchingratio RDs

as well. For the Bc → D�sμ

þμ− decay, the binaveraged branching ratio and the ratio of branching ratioRD�

sin each q2 bin for each NP scenarios are smaller than

the corresponding SM value. However, the AFB and FLvalues can be either smaller or larger in NP cases than theSM central value. It should be mentioned that the deviationof RDs

, RD�s, and AFB from the SM prediction can be quite

large in some q2 bins.In Fig. 2, we show various q2 dependent observables for

the Bc → ðDs;D�sÞμþμ− decays in the presence of various

NP WCs in 1D scenario. Our observations are as follows:

(i) The differential branching ratio for the Bc →Dsμ

þμ− decays is reduced at all q2 for CNP9 , CNP

10 ,and CNP

9 ¼ −CNP10 , whereas, it remains SM like for

CNP9 ¼ −C0

9. This could very well be understood

from Eq. (8) that Hð1Þ0 and Hð1Þ

t helicity amplitudesfor the Bc → Dsμ

þμ− decay mode depend on thecombination CNP

9 þ C09. Hence the NP contribution

cancels. It should be noted that, except forCNP9 ¼ −C0

9, all the other NP scenarios are distin-guishable from the SM prediction at slightly morethan 1σ significance. Moreover, the deviation from

TABLE IV. Best fit and the corresponding Δχ2 values ofdifferent new vector and axial vector Wilson coefficients in1D and 2D scenarios taken from Ref. [30].

Wilson coefficients Best fit values Δχ2

CNP9

−1.07 37.6

CNP10

þ0.78 27.0

CNP9 ¼ −CNP

10−0.52 36.3

CNP9 ¼ −C0

9−1.11 40.5

ðCNP9 ; CNP

10 Þ ð−0.94;þ0.23Þ 41.8

ðCNP9 ; C0

9Þ ð−1.27;þ0.68Þ 49.4

ðCNP9 ; C0

10Þ ð−1.36;−0.46Þ 52.8

FIG. 1. Differential branching ratio dB=dq2, forward backward asymmetry of lepton pair AFBðq2Þ and longitudinal polarizationfraction of D�

s meson FLðq2Þ for the Bc → Dsμþμ− and Bc → D�

sμþμ− decays in the SM. The band corresponds to the uncertainties in

the transition form factors and the CKM matrix elements as discussed in the text.

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the SM prediction is more pronounced in case ofCNP9 ¼ −CNP

10 ¼ −0.52 NP scenario.(ii) The differential branching ratio for the Bc →

D�sμ

þμ− decays is reduced at all q2 for CNP9 , CNP

10 ,CNP9 ¼ −CNP

10 , and CNP9 ¼ −C0

9. The deviation withthe SM prediction increases as q2 increases for eachNPWCs. It should be mentioned that the differentialbranching ratio obtained in each of these NPscenarios, however, lies within the SM error band.

(iii) For all the NP couplings, the zero crossing in theforward backward asymmetry observable AD�

sFB is

shifted to the higher values of q2 than in the SM.There is, however, one exception. For CNP

10 ¼ 0.78,the zero crossing coincides with the SM predictionalthough the shape of AD�

sFB may slightly vary.

Maximum deviation from the SM prediction isobserved for CNP

9 and CNP9 ¼ −C0

9. We observethe zero crossing of AD�

sFB at q2 ≈ 2.8 GeV2 in case

of CNP9 and CNP

9 ¼ −C09 NP scenarios which are

clearly distinguishable from the SM prediction q2 ≈2.2� 0.2 GeV2 at 3σ significance. Similarly, forCNP9 ¼ −CNP

10 scenario, the zero crossing appears atq2 ≈ 2.4 GeV2 and it is distinguishable from the SMcase at 1σ significance.

(iv) The peak of the longitudinal polarization fraction ofD�

s meson may shift toward a higher values of q2

than in the SM. Although the longitudinal polari-zation fraction FL is reduced at all q2 for CNP

9 ,CNP9 ¼ −CNP

10 , and CNP9 ¼ −C0

9, it may increase withCNP10 for q2 > 1.2 GeV2.

There are other combinations of VA couplings exist in the1D scenario as reported in Ref. [30]. We, however, do notconsider those cases because of their small Δχ2 values.We now consider several NPWCs from the 2D scenarios

having high Δχ2 values from the global fit [30]. The best fitvalues, taken from Ref. [30], are reported in Table IV. Weshow in Fig. 3 various observables such as differentialbranching ratio dB=dq2, forward backward asymmetry oflepton pair AFBðq2Þ, and longitudinal polarization fractionof D�

s meson FLðq2Þ as a function of dilepton invariantmass q2 for the Bc → Dsμ

þμ− and Bc → D�sμ

þμ− decaysin the presence of such NP. Our main observations are asfollows:

(i) The differential branching ratio for theBc → Dsμþμ−

decay is reduced at all q2 for each NP couplings. Thedeviation from the SMprediction is more pronouncedin case of ðCNP

9 ;CNP10 Þ¼ð−0.94;þ0.23Þ. It should be

noted that only ðCNP9 ; CNP

10 Þ NP scenario can be

FIG. 2. Differential branching ratio dB=dq2, forward backward asymmetry of lepton pair AFBðq2Þ and longitudinal polarizationfraction of D�

s meson FLðq2Þ for the Bc → Dsμþμ− and Bc → D�

sμþμ− decays in the SM (red) and for the best fit values of new VA

couplings in 1D scenario. Green, blue, purple, and yellow lines correspond to the best fit values of CNP9 ¼ −1.07, CNP

10 ¼ 0.78,CNP9 ¼ −CNP

10 ¼ −0.52, and CNP9 ¼ −C0

9 ¼ −1.11, respectively. The SM error band is shown with grey.

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distinguished from the SM case at around 2σsignificance.

(ii) Similar to Bc → Dsμþμ−, the differential branching

ratio for theBc → D�sμ

þμ− decay is also reduced at allq2. The deviation with the SM prediction, however,increases with increase in q2. It reaches maximum atq2 ¼ 6 GeV2. The deviation observed in each sce-nario, however, lies within the SM error band.

(iii) The zero crossing in the forward backward asym-metry observable AFBðq2Þ is shifted to higher valuesof q2 than in the SM for each NP WCs. Themaximum deviation from the SM prediction isobserved for ðCNP

9 ; C010Þ ¼ ð−1.36;−0.46Þ which

is shown with a purple line in Fig. 3. The zero ofAFB is distinguishable from the SM case at morethan 4σ significance in case of ðCNP

9 ; C09Þ and

ðCNP9 ; C0

10Þ NP scenarios, whereas, the significanceis around 2.5σ for ðCNP

9 ; CNP10 Þ NP scenario. Hence, a

measurement of the zero of the AFB will be a goodcandidate to probe new physics structures that isresponsible for lepton flavor universality violation inb → slþl− sector.

(iv) The longitudinal polarization fraction of the D�s

meson FLðq2Þ decreases once we include the NP

WCs. It is observed that the peak of the FLðq2Þdistribution reduces and shifted toward slightlyhigher q2 than in the SM. Maximum deviation fromthe SM prediction is observed for ðCNP

9 ; C010Þ ¼

ð−1.36;−0.46Þ which is shown with a purple linein Fig. 3.

V. CONCLUSION

Motivated by the anomalies present inB → ðK;K�Þμþμ−decays, we have analyzed Bc → ðDs;D�

sÞμþμ− decaysmediated via b → slþl− neutral current transitions usingthe Bc → ðDs;D�

sÞ transition form factors obtained in therelativistic quark model. We use a model independenteffective theory formalism and include NP effects comingfrom new vector and axial vector operators only. We discardscalar, pseudoscalar, and tensor operators as they areseverely constrained by Bs → μμ and b → sγ data.Several authors have performed global fits to the b →slþl− data and proposed two types of NP scenarios, namely,1D and 2D scenarios. In the 1D scenario, we chose four NPcases in which the NP contribution is coming from only oneNP WCs at a time. In the 2D scenario, we chose three NPcases in which the NP contribution is coming from two NPWCs at a time.

FIG. 3. Differential branching ratio dB=dq2, forward backward asymmetry of lepton pair AFBðq2Þ and longitudinal polarizationfraction of D�

s meson FLðq2Þ for the Bc → Dsμþμ− and Bc → D�

sμþμ− decays in the SM (red) and for the best fit values of new VA

couplings in 2D scenario. Green, blue and purple lines correspond to the best fit values of ðCNP9 ; CNP

10 Þ ¼ ð−0.94;þ0.23Þ, ðCNP9 ; C0

9Þ ¼ð−1.27;þ0.68Þ and ðCNP

9 ; C010Þ ¼ ð−1.36;−0.46Þ, respectively. The SM error band is shown with grey.

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We give predictions on several observables such asbranching ratio, ratio of branching ratio, forward backwardasymmetry, and the longitudinal polarization fraction of theD�

s meson in the SM and in several NP cases. We observethat for most of the NP cases, the branching ratio for boththe decay modes is reduced at all q2. In most cases, the zeroof AFBðq2Þ parameter is shifted to the higher value of q2

than in the SM. However, with only CNP10 , the zero crossing

is SM like. Similarly, for the longitudinal polarizationfraction of the D�

s meson, the peak of the distribution isreduced at all q2 and it is slightly shifted toward highervalue of q2 than in the SM. It is worth emphasizing that thezero of forward backward asymmetry parameter AFBðq2Þcan be clearly distinguished from the SM case at more than4σ significance for ðCNP

9 ; C09Þ and ðCNP

9 ; C010Þ NP scenarios.

Similarly, it can be distinguished from the SM case at 2.5σand 3σ significance for ðCNP

9 ; CNP10 Þ, CNP

9 and CNP9 ¼ −C0

9

NP scenarios, respectively. These results are quite interest-ing because measurement of the zero of AFB for the Bc →D�

sμþμ− decays in future can put additional constraint on

the NP Lorentz structures that are responsible for leptonflavor universality violation in b → slþl− sector. Similarly,the lepton flavor universal ratios RDs

and RD�sare excep-

tionally clean observables with theoretical errors being atthe level of only 1% making them an ideal candidate toprobe new physics.Although there is a hint of NP in b → slþl− transition

decays, NP is not yet established. Unlike B →ðK;K�Þμþμ− decays which are rigorously studied boththeoretically and experimentally, the Bc → ðDs;D�

sÞμþμ−decays mediated via same underlying quark level transitionreceived significantly less attention. Measurement of vari-ous observables for the Bc → ðDs;D�

sÞμþμ− decays and atthe same time improved estimates of various Bc → Ds andBc → D�

s transition form factors in future will be crucial inidentifying the true nature of NP. Again, to enhance thesignificance of various measurements related to b → slþl−decays and to disentangle genuine NP effects from variousstatistical and systematic uncertainties, more data samplesare needed.

APPENDIX: PREDICTION OF OBSERVABLES IN THE SM AND IN SEVERAL NP CASES

In Tables (V–X), we report, the q2 bin averaged values of all the observables for the Bc → ðDs;D�sÞμþμ− decays in the

SM and in several NP cases.

TABLE V. q2 bin averaged values of 107 × BðBc → Dsμþμ−Þ in the SM and in several NP cases from 1D and 2D scenarios.

q2 bin (GeV2) SM CNP9 CNP

10 CNP9 ¼ −CNP

10 CNP9 ¼ −C0

9 ðCNP9 ; CNP

10 Þ ðCNP9 ; C0

9Þ ðCNP9 ; C0

10Þ[0.045, 1.0] 0.025 0.020 0.021 0.020 0.025 0.019 0.022 0.022[1.0, 2.0] 0.030 0.024 0.025 0.023 0.030 0.023 0.026 0.026[2.0, 3.0] 0.034 0.027 0.028 0.026 0.034 0.026 0.030 0.029[3.0, 4.0] 0.038 0.030 0.032 0.030 0.038 0.029 0.033 0.033[4.0, 5.0] 0.043 0.034 0.036 0.034 0.043 0.033 0.038 0.037[5.0, 6.0] 0.049 0.039 0.042 0.039 0.049 0.038 0.043 0.042[1.0, 6.0] 0.193 0.154 0.163 0.152 0.193 0.149 0.171 0.166

TABLE VI. q2 bin averaged values of RDsin the SM and in several NP cases from 1D and 2D scenarios.

q2 bin (GeV2) SM CNP9 CNP

10 CNP9 ¼ −CNP

10 CNP9 ¼ −C0

9 ðCNP9 ; CNP

10 Þ ðCNP9 ; C0

9Þ ðCNP9 ; C0

10Þ[0.045, 1.0] 1.006 0.798 0.846 0.788 1.006 0.770 0.884 0.859[1.0, 2.0] 1.007 0.803 0.845 0.790 1.007 0.774 0.888 0.867[2.0, 3.0] 1.005 0.802 0.844 0.788 1.005 0.773 0.886 0.865[3.0, 4.0] 1.004 0.801 0.844 0.788 1.004 0.772 0.885 0.863[4.0, 5.0] 1.003 0.800 0.845 0.789 1.003 0.772 0.884 0.861[5.0, 6.0] 1.003 0.799 0.847 0.790 1.003 0.772 0.884 0.859[1.0, 6.0] 1.004 0.801 0.845 0.789 1.004 0.773 0.885 0.863

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TABLE VII. q2 bin averaged values of 107 × BðBc → D�sμ

þμ−Þ in the SM and in several NP cases from 1D and 2D scenarios.

q2 bin (GeV2) SM CNP9 CNP

10 CNP9 ¼ −CNP

10 CNP9 ¼ −C0

9 ðCNP9 ; CNP

10 Þ ðCNP9 ; C0

9Þ ðCNP9 ; C0

10Þ[0.045, 1.0] 0.024 0.024 0.023 0.023 0.023 0.023 0.023 0.023[1.0, 2.0] 0.011 0.010 0.009 0.009 0.009 0.010 0.010 0.009[2.0, 3.0] 0.014 0.012 0.011 0.011 0.011 0.012 0.011 0.011[3.0, 4.0] 0.020 0.017 0.016 0.015 0.016 0.016 0.016 0.015[4.0, 5.0] 0.028 0.023 0.022 0.022 0.022 0.022 0.022 0.021[5.0, 6.0] 0.039 0.032 0.031 0.030 0.030 0.030 0.030 0.028[1.0, 6.0] 0.113 0.095 0.090 0.087 0.088 0.089 0.088 0.085

TABLE VIII. q2 bin averaged values of RD�sin the SM and in several NP cases from 1D and 2D scenarios.

q2 bin (GeV2) SM CNP9 CNP

10 CNP9 ¼ −CNP

10 CNP9 ¼ −C0

9 ðCNP9 ; CNP

10 Þ ðCNP9 ; C0

9Þ ðCNP9 ; C0

10Þ[0.045, 1.0] 0.999 0.995 0.994 0.957 0.961 0.977 0.975 0.970[1.0, 2.0] 0.993 0.915 0.801 0.815 0.834 0.860 0.855 0.830[2.0, 3.0] 0.992 0.859 0.776 0.770 0.790 0.804 0.797 0.767[3.0, 4.0] 0.993 0.834 0.781 0.760 0.778 0.783 0.774 0.744[4.0, 5.0] 0.994 0.820 0.790 0.760 0.772 0.774 0.763 0.733[5.0, 6.0] 0.995 0.812 0.799 0.763 0.768 0.769 0.756 0.726[1.0, 6.0] 0.994 0.834 0.791 0.768 0.786 0.786 0.776 0.746

TABLE IX. q2 bin averaged values of hAD�s

FBi in the SM and in several NP cases from 1D and 2D scenarios.

q2 bin (GeV2) SM CNP9 CNP

10 CNP9 ¼ −CNP

10 CNP9 ¼ −C0

9 ðCNP9 ; CNP

10 Þ ðCNP9 ; C0

9Þ ðCNP9 ; C0

10Þ[0.045, 1.0] −0.064 −0.069 −0.056 −0.062 −0.072 −0.066 −0.072 −0.072[1.0, 2.0] −0.076 −0.124 −0.079 −0.103 −0.138 −0.121 −0.141 −0.150[2.0, 3.0] 0.033 −0.028 0.035 0.006 −0.033 −0.020 −0.043 −0.052[3.0, 4.0] 0.110 0.055 0.117 0.092 0.055 0.065 0.043 0.037[4.0, 5.0] 0.160 0.112 0.168 0.148 0.116 0.123 0.104 0.099[5.0, 6.0] 0.194 0.153 0.201 0.186 0.159 0.164 0.147 0.144[1.0, 6.0] 0.123 0.072 0.129 0.107 0.074 0.082 0.062 0.056

TABLE X. q2 bin averaged values of hFD�s

L i in the SM and in several NP cases from 1D and 2D scenarios.

q2 bin (GeV2) SM CNP9 CNP

10 CNP9 ¼ −CNP

10 CNP9 ¼ −C0

9 ðCNP9 ; CNP

10 Þ ðCNP9 ; C0

9Þ ðCNP9 ; C0

10Þ[0.045, 1.0] 0.266 0.213 0.232 0.215 0.181 0.208 0.186 0.177[1.0, 2.0] 0.711 0.620 0.725 0.671 0.558 0.630 0.567 0.554[2.0, 3.0] 0.662 0.613 0.697 0.661 0.548 0.627 0.566 0.558[3.0, 4.0] 0.566 0.541 0.594 0.573 0.476 0.552 0.498 0.493[4.0, 5.0] 0.488 0.475 0.508 0.496 0.413 0.482 0.435 0.432[5.0, 6.0] 0.430 0.423 0.444 0.437 0.365 0.428 0.387 0.385[1.0, 6.0] 0.526 0.503 0.546 0.529 0.440 0.511 0.461 0.457

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