Combined analysis of Bc → D(∗)s µ+µ− and Bc → D

(∗)s νν decays within Z ′ and leptoquark

new physics models

Manas K. Mohapatra1,∗ N Rajeev2,† and Rupak Dutta2‡

1Department of Physics, Indian Institute of Technology Hyderabad, Kandi - 502285, India2National Institute of Technology Silchar, Silchar 788010, India

We investigate the exclusive rare semileptonic decays Bc → D(∗)s (``, νν) induced by neutral cur-

rent transition b → s(``, νν) in the presence of non-universal Z′, scalar and vector leptoquark newphysics models. We constrain the new physics parameter space by using the latest experimentalmeasurements of RK(∗) , P ′5, B(Bs → φµ+µ−) and B(Bs → µ+µ−). Throughout the analysis, wechoose to work with the particular new physics scenario Cµµ9 (NP ) = −Cµµ10 (NP ) where both Z′,S3

1/3 and U3−2/3 leptoquarks satisfy the condition. Using these new coupling parameters we scruti-

nize the several physical observables such as differential branching fraction, the forward backwardasymmetry, the lepton polarization asymmetry, the angular observable P ′5 and the lepton flavor uni-versal sensitive observables including the ratio of branching ratio R

D(∗)s

and the few Q parameters

in the Bc → D(∗)s µ+µ− and Bc → D

(∗)s νν decay processes.

I. INTRODUCTION

The hint of new physics (NP) in the form of new interactions which demands an extension of the standard model(SM) of particle physics are witnessed not only in the flavor changing neutral current decays of rare beauty particlesof the form b → s`+`− and b → sνν but also in the flavor changing charged current decays which undergo b → c`νquark level transitions. The rare weak decays of several composite beauty mesons such as Bd, Bs, and Bc whichare forbidden at the tree level in SM appear to follow loop or box level diagrams. Theoretically, the radiative andsemileptonic decays of B → K(∗) and Bs → φ processes have received great attention and are studied extensively bothwithin the SM and beyond. The sensitivity of new physics possibilities in these decays require very good knowledge ofthe hadronic form factors more specifically for B → V transitions. It requires informations from both the light conesum rule (LCSR) and lattice QCD (LQCD) methods to compute the form factors respectively at low and high q2

regions which eventually confine the whole kinematic region [1]. Currently, we do have the very precise calculationsof the form factors that have very accurate SM predictions of the differential branching fractions and various angularobservables in b → s`+`− decays. Similarly, the family of neutral decays proceeding via b → sνν transitions equallyprovide the interesting opportunity in probing new physics signatures to that of b → s`+`− transitions. Howeverso far no experiments have directly addressed any anomalies except the upper bounds of the branching fractions ofB → K(∗)νν decay processes. In principle, under the SU(2)L flavor symmetry both the charged leptons and neutralleptons are treated equally and hence one can extract a close relation between both b → s`+`− and b → sνν decaysin beyond the SM scenarios. In addition, the decays with νν final state are well motivated for several interestingfeatures since these decays are considered to be theoretically cleaner as they do not suffer from hadronic uncertaintiesbeyond the form factors such as the non-factorizable corrections and photonic penguin contributions.

Experimentally, several measurements in b→ s`+`− transitions such as RK∗ = B(B → K∗ µ+µ−)/B(B → K∗e+e−)from LHCb [2, 3] and Belle [4] at q2 ∈ [0.045, 1.1] and q2 ∈ [1.1, 6.0] show 2.1 − 2.4σ deviation from the SMexpectations [5, 6]. Similarly, the angular observable P ′5 in B → K∗µ+µ− from in the bins q2 ∈ [4.0, 6.0], [4.3, 6.0]and [4.0, 8.0] from ATLAS [7], LHCb [8, 9], CMS [10], Belle [11] respectively deviate at 3.3σ, 1σ and 2.1σ from the SMexpectations [12–14]. The recent updates in the measurements of RK = B(B → Kµ+µ−)/B(B → Ke+e−) [15, 16]in q2 ∈ [1.0, 6.0] and the branching fraction of B(Bs → φµ+µ−) [17–19] in q2 ∈ [1.1, 6.0] region from LHCb stillindicate 3.1σ in RK [5, 6] and 3.6σ in B(Bs → φµ+µ−) [1, 20] from the SM expectations. Similarly, the measurementspertaining to b→ sνν transitions, the upper bound measured by the Belle collaboration in the branching fraction ofB → K(∗)νν decays are B(B → Kνν) < 1.6 × 10−5 and B(B → K∗νν) < 2.7 × 10−5 [21] respectively. There also

∗Electronic address: manasmohapatra12@gmail.com†Electronic address: rajeev rs@phy.nits.ac.in‡Electronic address: rupak@phy.nits.ac.in

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exist the BaBar measurement on B(B → K∗νν) < 4×10−5 [22]. Very recently, the Belle II updated the upper boundof B(B → Kνν) < 4.1× 10−5 in 2021 [23].

There exist several other decay channels similar to the B → K(∗) and Bs → φ undergoing b → s`+`− quark leveltransitions [23–37]. If any new physics present in B → K(∗) and Bs → φ decays can in principle be reflected in several

other decays as well. In that sense we choose to study the explicit rare decays Bc → D(∗)s µ+µ− and Bc → D

(∗)s νν

which undergo similar b → s neutral transition. The particular decay modes Bc → D(∗)s µ+µ− have been studied

previously in SM using various form factors which include relativistic quark model (RQM) [38], the light front andconstituent quark model [39, 40], the three point QCD sum rules approach [41] and the covariant quark model [42]. Inaddition, as far as beyond SM analysis are concerned these decays have also been analysed within model independentand dependent new physics as well. The model independent study within the effective field theory approach was donein Ref. [43] under various 1D and 2D NP scenarios. In Ref. [44], the decay mode was studied within non-universal Z ′

model with the NP contribution coming from only the right handed currents. Similarly, the contribution of Z ′ wasanalysed by considering the UTfit inputs of left-handed coupling of Z ′ boson and the different values of new weak

phase angle in the Ref. [45]. Similarly, the SM results pertaining to the Bc → D(∗)s νν decays have been addressed in

Refs. [38, 40, 46].Even from the experimental point of view after the discovery of Bc meson at CDF via Bc → J/Ψ`ν [47], the study of

Bc decays were found to be very interesting. Unlike the weak decays of other B mesons, the Bc mesons are interestingas it is composed of both heavy b and c quarks that allows broader kinematic range which eventually involve largenumber of decays. In addition, the upcoming LHC run can produce around 108 − 1010 Bc mesons which offer very

rich laboratory for the associated Bc weak decays. Hence, exploring the new physics in Bc → D(∗)s (µ+µ−, νν) decays

are well motivated both theoretically as well as experimentally.

In this context, we study the implication of the latest b → s`+`− data on the Bc → D(∗)s µ+µ− and Bc → D

(∗)s νν

decay processes under the model dependent analysis. We choose in particular, the specific models such as Z ′ and thevarious scalar and vector leptoquarks (LQs) which satisfy CNP9 = −CNP10 new physics scenario. Among various LQs,we opt for the specific LQs in such way that they should have combined NP effects in the form of CNP9 = −CNP10

both in b→ s`+`− and b→ sνν decays. Hence, the main aim of this work is to extract the common new physics thatappear simultaneously in b→ s`+`− and b→ sνν decays.

The layout of the present paper is as follows. In Section II, we add a theoretical framework that includes a briefdiscussion of effective Hamiltonian for b → s`+`− and b → sνν parton level transition. In addition to this, we also

present the differential decay distributions and other q2 dependent observables of Bc → D(∗)s µ+µ− and Bc → D

(∗)s νν

processes. In the context of new physics, we deal with the contributions arising due to the exchange of LQ and Z ′

particles in Section III. In Section IV, we report and discuss our numerical analysis in the SM and in the presenceNP contributions. Finally we end with our conclusion in Section V.

II. THEORETICAL FRAMEWORK

A. Effective Hamiltonian

The effective Hamiltonian responsible for b→ s`` parton level transition in the presence of NP vector operator canbe represented as [48]

Heff = −αGF√2π

VtbV∗ts

[2Ceff

7

q2[sσµνqν(msPL +mbPR)b] (¯γµ`) + Ceff

9 (sγµPLb)(¯γµ`)

+ C10(sγµPLb)(¯γµγ5`) + CNP9 (s γµ PL b )(¯γµ ` ) + CNP10 s γµ PL b ¯γµ γ5 `], (1)

where GF is the Fermi coupling constant, α is the fine structure constant, Vij is the CKM matrix element, and

Ceff7 , Ceff

9 and C10 are the relevant Wilson coefficients (WC) evaluated at µ = mpoleb scale [48]. The Wilson coefficients

CNP9 and CNP10 are the effective coupling constants associated with the corresponding NP operators. The effectiveHamiltonian describing b→ sνν decay processes is given by [49]

Hννeff =GFα

2√

2πVtbV

∗ts

X(xt)

s2W

(sγµPLb)(νγµ(1− γ5)ν) (2)

where X(xt) is the Inami - Lim function which is given in Ref. [50]. In principle, several Lorentz structures in theform of chiral operators can be possible in the NP scenario such as vector, axial vector, scalar, pseudoscalar, and

3

tensor. However, among all the operators scalar, pseudoscalar and tensor are severely constrained by Bs → µµ andb → sγ measurements [51]. Therefore we consider the vector and axial vector contributions only. In our analysis,among possible NP operators we consider only the left chiral ONP9 and ONP10 contributions and the associated Wilsoncoefficients are assumed to be real. The effective Wilson coefficients Ceff

7 and Ceff9 are defined as [48]

Ceff7 = C7 −C5

3− C6

Ceff9 = C9(µ) + h(mc, s)C0 −1

2h(1, s)(4C3 + 4C4 + 3C5 + C6)

− 1

2h(0, s)(C3 + 3C4) +

2

9(3C3 + C4 + 3C5 + C6) , (3)

where s = q2/m2b , mc = mc/mb and C0 = 3C1 + C2 + 3C3 + C4 + 3C5 + C6.

h(z, s) = −8

9lnmb

µ− 8

9ln z +

8

27+

4

9x− 2

9(2 + x)|1− x|1/2

{ln|√

1−x+1√1−x−1

| − iπ , for x ≡ 4z2

s < 1

2 arctan 1√x−1

, for x ≡ 4z2

s > 1(4)

and

h(0, s) = −8

9lnmb

µ− 4

9ln s +

8

27+

4

9iπ. (5)

Here we have included the short distance perturbative contributions to Ceff9 . As we are interested in the q2 bin room

[0.1 - 0.98] and [1.1 - 6] GeV2, the short distance contributions embedding with cc resonance contribution comingfrom J/ψ, ψ(2S)... states have been excluded in our analysis. In fact, we do not include any non local effects in ouranalysis which are important below the charmonium contributions that have been studied in detail in Refs. [52–55]. Inprinciple, these hadronic non local effects are generally neglected in the study of the lepton flavor universality violation(LFUV) in various b → s`` decays. Additionally, the factorizable effects arising due to the spectator scattering maynot affect severely to the LFU ratios and hence these effects are neglected in our present analysis [56, 57].

B. Differential decay distribution and q2 observables in Bc → D(∗)s µ+µ−

In analogy with B → K`+`− decay mode, it is useful to note that the rare semileptonic Bc → Ds`+`− process is

also mediated through b→ s`+`− transition in the parton level. In the standard model, we present the formula of q2

dependent differential branching ratio which is given as follows [58]:

dBR

dq2=τBc~

(2a` +2

3c`), (6)

where the parameters a` and c` are given by

a` =G2Fα

2EW |VtbV ∗ts|2

29π5m3Bc

β`√λ[q2|FP |2 +

λ

4(|FA|2 + |FV |2) + 4m2

`m2Bc |FA|

2

+ 2m`(m2Bc −m

2Ds + q2)Re(FPF

∗A)], (7)

c` = −G2Fα

2EW |VtbV ∗ts|2

29π5m3Bc

β`√λλβ2

`

4(|FA|2 + |FV |2). (8)

Here the kinematical factor λ and the mass correction factor β` given in the above equations are given by

λ = q4 +m4Bc +m4

Ds − 2(m2Bcm

2Ds +m2

Bcq2 +m2

Dsq2),

β` =√

1− 4m2`/q

2. (9)

However, the explicit expressions of the form factors such as FP , FV and FA are given as follows:

FP = −m`C10

[f+ −

M2B −M2

K

q2(f0 − f+)

], (10)

FV = Ceff9 f+ +

2mb

MB +MKCeff

7 fT , (11)

FA = C10f+. (12)

4

The transition amplitude for Bc → D∗s`+`− decay channel can be obtained from the effective Hamiltonian given in

the Eq. (1). The q2 dependent differential branching ratio for Bc → D∗s`+`− process is given as [33]

dBR/dq2 =dΓ/dq2

ΓTotal=τBc~

1

4

[3Ic1 + 6Is1 − Ic2 − 2Is2

], (13)

where the q2 dependent angular coefficients are given in Appendix B. In addition to this, we also define otherprominent observables such as the forward-backward asymmetry (AFB), the longitudinal polarization fraction (FL)and the angular observable 〈P ′5〉 which are given by

FL(q2) =3Ic1 − Ic2

3Ic1 + 6Is1 − Ic2 − 2Is2, AFB(q2) =

3I63Ic1 + 6Is1 − Ic2 − 2Is2

, 〈P ′5〉 =

∫bin

dq2I5

2√−∫bin

dq2Ic2∫bin

dq2Is2

. (14)

To confirm the existence of the lepton universality violation, one can construct additional observables associatedwith the two different families of lepton pair which are quite sensitive to shed light into the windows of NP. Theexplicit expressions are given as below:

〈QFL〉 = 〈FLµ〉 − 〈FLe〉, 〈QAFB 〉 = 〈AFBµ〉 − 〈AFBe〉, 〈Q′5〉 = 〈P ′µ5 〉 − 〈P ′e5 〉. (15)

Also we define the ratio of the branching ratios of µ to e transition in Bc → D(∗)s `+`− decay modes as follows:

RD

(∗)s

(q2) =BR

(Bc → D

(∗)s µ+µ−

)BR

(Bc → D

(∗)s e+ e−

) . (16)

C. Differential decay distribution in Bc → D(∗)s νν

The explicit study of Bc → D(∗)s νν processes involved with b→ sνν transitions are also quite important to search

for NP beyond the SM as they are associated to b → s`+`− parton level by SU(2)L symmetry group. From the

effective Hamiltonian given in Eq. 2, the differential decay rate for Bc → D(∗)s νν decay channel is given by [35]

dBR(Bc → Dsνν)SM

dq2= τBc3|N |2

X2t

s4w

ρDs(q2), (17)

dBR(Bc → D∗sνν)SM

dq2= τBc3|N |2

X2t

s4w

[ρA1(q2) + ρA12(q2) + ρV (q2)

], (18)

where the factor 3 comes from the sum over neutrino flavors, and

N = VtbV∗ts

GFα

16π2

√mBc

3π(19)

is the normalization factor. The relevant rescaled form factors ρi given in the above equations are given below.

ρDs(q2) =

λ3/2Ds

(q2)

m4Bc

[fDs+ (q2)

]2, ρV (q2) =

2q2λ3/2Ds∗

(q2)

(mBc +mDs∗)2m4

Bc

[V (q2)

]2,

ρA1(q2) =

2q2λ1/2Ds∗

(q2)(mB +mDs∗)2

m4Bc

[A1(q2)

]2, ρA12

(q2) =64m2

Ds∗λ

1/2Ds∗

(q2)

m2Bc

[A12(q2)

]2. (20)

where

λ(a, b, c) = a2 + b2 + c2 − 2(ab+ bc+ ac), λDs(∗)(q2) ≡ λ(m2

Bc ,m2Ds(∗)

, q2) . (21)

5

NP model Cµµ9 (NP) Cµµ10 (NP) Cµµν (NP)

S31/3 Ry′µb`q (y′µs`q )∗ −Ry′µb`q (y′µs`q )∗ 1

2Ry′µb`q (y′µs`q )∗

U3−2/3 −Rg′µb`q (g′µs`q )∗ Rg′µb`q (g′µs`q )∗ −2Rg′µb`q (g′µs`q )∗

Z′ −MgbsL gµµL MgbsL g

µµL −MgbsL g

ννL

TABLE I: Contributions of the LQs - S31/3, U

3−2/3, and Z ′ to the Wilson coefficients. The normalization

R(M) ≡ π/(√

2αGFVtbV∗ts(MLQ(MZ′))

2 and MLQ = MZ′ = 1 TeV.

III. NEW PHYSICS ANALYSIS IN THE SCENARIO Cµµ9 (NP ) = −Cµµ10 (NP )

Assuming the NP exist only in the context of µ mode in b → s`+`− transition, it will contribute to more numberof Lorentz structures. The new physics scenarios for the parton level b → sµ+µ− transition that account for NPcontributions are given as [28, 59]

(I) : Cµµ9 (NP ) < 0,

(II) : Cµµ9 (NP ) = −Cµµ10 (NP ) < 0,

(III) : Cµµ9 (NP ) = −C ′µµ9 (NP ) < 0,

(IV) : Cµµ9 (NP ) = −Cµµ10 (NP ) = C ′µµ9 (NP ) = C ′µµ10 (NP ) < 0, (22)

where the unprimed couplings differ from the primed Wilson coefficients by their corresponding chiral operator asdiscussed in the previous section. Keeping in mind, as from the Ref. [60], only three out of ten leptoquarks such asS3, U1 and U3 can explain the b→ sµ+µ− data as they have good fits under certain scenario. Among S3, U3 and U1

leptoquarks, the U1 leptoquark has no contribution to the couplings corresponding to the NP operator responsiblefor b → sνν processes whereas other two LQs are differentiated with a definite contributions to it. Hence the effectfrom U1 LQ is not taken into account in the present analysis. It is important to say that the remaining S3 and U3

LQs do not satisfy the scenarios I and III. On the other hand, in Ref. [61] it is also reported that Z ′ can contributeto both scenario I and II whereas a vast majority of this model use the scenario II. Hence the feasible environmentto study both LQs and Z ′ simultaneously will be the scenario II : Cµµ9 (NP) = −Cµµ10 (NP). Many works have beenstudied in these scenarios in LQs [62–70] and in the presence of Z ′ [62, 71–76]. Therefore, the purpose of this workis to concentrate on the scenario II : Cµµ9 (NP ) = −Cµµ10 (NP ) [61].

A. Leptoquark contribution

There are 10 different leptoquark multiplets under the SM gauge group SU(3)C ×SU(2)L×U(1)Y in the presenceof dimension ≤ 4 operators [77] in which five multiplets include scalar (spin 0) and other halves are vectorial (spin 1)in nature under the Lorentz transformation. Among all, both the scalar triplet S3 (Y= 1/3) and vector isotriplet U3

(Y= -2/3) can explain the b → sµ+µ− and b → sνν processes simultaneously. The relevant Lagrangian is given asfollows [60]

LS = y′`q¯cLiτ2−→τ qLS3

1/3 + h.c,

LV = g′`q¯Lγµ−→τ qLU3

−2/3 + h.c, (23)

where the fermion currents in the above Lagrangian include the SU(2)L quark and lepton doublets “qL” and “`L”respectively, and τ represent the Pauli matrices. Most importantly the parameters y′`q and g′`q are the quark - lepton

couplings associated with the corresponding leptoquarks. In particular for this analysis y′µb(s)`q is the coupling of the

leptoquark S31/3 to the left-handed µ or νµand a left-handed fermion field b (s). Similarly g′

µb(s)`q is the coupling

correspond to the leptoquark U3−2/3. On the other hand, for the parton level b→ sνν transitions both S3

1/3 and U3−2/3

LQs contribute differently as reported in Table I. Hence the Wilson coefficient CL associated with b → sνν can be

obtained by replacing CL → CL +CLQL . In our paper, we consider the couplings y′µb`q (y′µs`q )∗ and g′µb`q (g′µs`q )∗ as real for

the S31/3 and U3

−2/3 LQs respectively with the assumption of same mass for both the leptoquarks.

6

B. Non-universal Z′ contribution

The extension of SM by an extra minimal U(1)′ gauge symmetry produces a neutral gauge boson the so calledZ ′ boson. It is the most obvious candidate which represent to b → sµ+µ− in the NP scenario. However the mainattraction of this model includes the flavor changing neutral current (FCNC) transition in the presence of new non-universal gauge boson Z ′ [78–80] which can contribute at tree level. After integrating out the heavy Z ′ the effectiveLagrangian for 4 fermion operator is given as

FIG. 1: The tree level contribution of the LQs and the Z ′ for Bc → D(∗)s (µ+µ−, νν)

LeffZ′ = − 1

2M2Z′J ′µJ

µ′, (24)

where the new current is given as

Jµ′ = −gµµLLLγµPLL+ gµµL(R)µγ

µPL(R)µ+ gijL ψiγµPLψj + h.c. (25)

where i and j are family index, PL(R) is the projection operator of left (right) chiral fermions and gijL denote the leftchiral coupling of Z ′ gauge boson. Now relevant interaction Lagrangian is given as

LeffZ′ = − gbsLm2Z′

(sγµb)(µγµ(gµµL PL + gµµR PR)µ). (26)

Now, the modified Wilson coefficients in the presence of Z ′ model can be written as [25]

Cµµ9 (NP ) = −[

π√2GFαVtbV ∗ts

]gbsL (gµµL )

m2Z′

,

Cµµ10 (NP ) =

[π√

2GFαVtbV ∗ts

]gbsL (gµµL )

m2Z′

, (27)

where gbsL is the coupling when b quark couple to s quark and gµµL is the µ+ − µ− coupling in the presence of newboson Z ′ and we have assumed gµµR = 0. Similarly for b → sνν transition, the NP contribution arising due to Z ′

is CNPL = Cµµ9 (NP) = −Cµµ10 (NP)[81]. From the neutrino trident production, gµµL = 0.5 has been considered in ourpaper [25, 60, 61]. The new parameter gbsL is taken to be real in our analysis.

IV. NUMERICAL ANALYSIS AND DISCUSSIONS

A. Input parameters

In this subsection we report all the necessary input parameters used for our computational analysis. We reportin Table II the masses of mesons, quarks, Fermi coupling constant in the unit of GeV and lifetime of Bc meson inthe unit of second. We employ the Wilson coefficients at the renormalization scale µ = 4.8 GeV as reported in theTable III. We have adopted the relativistic quark model based on quasipotential approach for the form factors ofBc → Ds and Bc → D∗s transitions. The form of the form factors are given as follows

F (q2) =

F (0)(1− q2

M2

)(1− σ1

q2

M2B∗s

+ σ2q4

M4B∗s

) , for F = {f+, fT , V, A0, T1}

F (0)(1− σ1

q2

M2B∗s

+ σ2q4

M4B∗s

) , for F = {f0, A1, A2, T2, T3} (28)

7

Here M = MBs for A0(q2) whereas M = MB∗sis considered for all other form factors. We use MB∗s

= 5.4254 GeV fromthe Ref. [82]. All the related form factor input parameters are reported in the Table IV. We employ 10% uncertaintyin the form factor at zero recoil momentum i.e F (0).

Parameter Value Parameter Value Parameter Value Parameter Value Parameter Value

mBc 6.2751 mB+ 5.27963 mB0 5.27963 mBs 5.3668 mDs 1.96834mD∗s 2.112 mK+ 0.493677 mK0 0.892 mφ 1.01946 fBs 0.225

mMSb 4.2 mMS

c 1.28 mpoleb 4.8 τBc 0.507× 10−12 τB+ 1.638× 10−12

τB0 1.520× 10−12 τBs 1.515× 10−12 GF 1.1663787× 10−5 α 1/133.28 |VtbV ∗ts| 0.04088(55)

TABLE II: Input parameters for numerical results [82]

C1 C2 C3 C4 C5 C6 Ceff7 Ceff9 C10

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

TABLE III: Wilson coefficients evaluated at scale of µ = 4.8 GeV [83]

f+ f0 fT V A0 A1 A2 T1 T2 T3

F (0) 0.129 0.129 0.098 0.182 0.070 0.089 0.110 0.085 0.085 0.051

σ1 2.096 2.331 1.412 2.133 1.561 2.479 2.833 1.540 2.577 2.783

σ2 1.147 1.666 0.048 1.183 0.192 1.686 2.167 0.248 1.859 2.170

TABLE IV: The form factors of Bc → Ds and Bc → D∗s transition at q2 = 0 and the corresponding fittedparameters σ1 and σ2[38]

B. Fit Results

To obtain the NP parameter space in the presence of Z ′ and LQs we perform a naive χ2 analysis with the availableb → s`` experimental data. In the fit we consider specifically the LHCb measurements of five different observablessuch as RK , RK∗ , P

′5, BR(Bs → φµµ) and BR(Bs → µ+µ−). Our fit include the latest measurements of RK ,

BR(Bs → φµµ) and BR(Bs → µ+µ−) as reported from LHCb in 2021. For our theoretical computation of theunderlying observables we refer to the lattice QCD form factors [58] for RK and the form factors obtained from thecombined analysis of LCSR+LQCD for B → K∗ and Bs → φ decay processes [1]. We define the χ2 as

χ2(CNPi ) =

∑i

(Othi (CNP

9,10)−Oexpi

)2

(∆Oexpi )2 + (∆Osm

i )2, (29)

where Othi represent the theoretical expressions including the NP contributions and Oexp

i are the experimental centralvalues. The denominator includes 1σ uncertainties associated with the theoretical and experimental results. Fromour analysis we obtained the NP coupling strengths associated with Z ′, S3

1/3 and U3−2/3 LQs respectively as follows:

Z ′ : gµµbs = 1.74× 10−3,

S31/3 : y′µb`q (y′µs`q )∗ = −8.70× 10−4,

U3−2/3 : g′µb`q (g′µs`q )∗ = 8.70× 10−4. (30)

8

C. Interpretation of Bc → D(∗)s (µ+µ−, νν) decays in standard model and beyond

1. Bc → D(∗)s µ+µ− decays

We perform NP studies of Bc → D(∗)s µ+µ− decays in the presence of Z ′, S3

1/3 and U3−2/3 LQs which satisfy

CNP9 = −CNP10 NP scenario. Although the NP coupling strengths associated with the Z ′, S31/3 and U3

−2/3 LQs are

different from each other, the contribution from the CNP9 = −CNP10 new Wilson coefficients in b→ s`+`− decays are

same. Hence we expect similar NP signature from Z ′, S31/3 and U3

−2/3 LQs in the underlying Bc → D(∗)s µ+µ− decays.

We study various observables such as the differential branching ratio, the forward backward asymmetry, the leptonpolarization fraction, the LFU sensitive observables including the ratio of branching ratio R

D(∗)s

and the difference

of the observables associated with Q parameters such as QFL , QAFB and Q′5 in the presence of SM as well as newphysics. In Table V we report the central values and the corresponding standard deviation for all the observables inboth SM and Z ′/LQ new physics. Similarly in Fig. 2 and Fig. 3 we display the corresponding q2 distribution plotsas well as q2 integrated bin wise plots for Bc → Dsµ

+µ− and Bc → D∗sµ+µ− processes respectively. For the binned

plots we choose different bin sizes which are compatible with the LHCb experiments starting from [0.1, 0.98], [1.1,2.5], [2.5, 4.0], [4.0, 6.0] and also [1.1, 6.0]. The SM central curve is obtained by considering only the central valuesof all the input parameters and the corresponding 1σ error band is obtained by varying the form factors and the

CKM matrix element within 1σ. In SM we obtain the branching fraction to be O(10−7) for Bc → D(∗)s µ+µ− decay

channels. The detailed observations of our study are as follows:

• The q2 dependency of the differential branching fraction for Bc → D(∗)s µ+µ− decays are shown in the top - left

panel of Fig. 2 and Fig. 3 respectively. We notice that the differential branching ratio is reduced in the presenceof Z ′/LQ new physics and lies away from the SM 1σ uncertainty band for Bc → Ds µ

+µ− decay. Althoughthe Z ′/LQ new physics contribution in Bc → D∗sµ

+µ− decay deviate from SM central curve but it can notbe distinguished beyond the SM uncertainty. However, slight more deviation can be found for q2 > 4GeV2.Similarly, in the top - right panel of Fig. 2 and bottom middle panel Fig. 3 we display the corresponding binnedplots respectively for both the decay modes. We observe that for Bc → Ds µ

+µ− decay in the bins [0.045,0.98] and [1.1, 2.5] the new physics contribution stand at almost 1σ away from the SM while in other binsthe deviation are not that significant and show < 1σ from the SM contribution. For the explicit decay chanelBc → D∗sµ

+µ− however, the NP central values differ from the SM but no such significant observations can bemade.

• The ratio of branching ratio RD

(∗)s

(q2) is constant over the range q2 ∈ [0.1, 6.0] and is approximately equal to

∼ 1. The uncertainty associated with this observable is almost zero. The NP contribution from Z ′/LQ is easilydistinguishable from the SM at more than 5σ except for the range q2 ∈ [0.1, 0.98] for the case of RD∗s (q2). This

is true in the case of each q2 bin as shown in Fig. 2 and Fig. 3 respectively for both the decays.

• The q2 distribution of the forward backward asymmetry AFB(q2) have a zero crossing at ∼ 2.2GeV2 in SMwhich is different from the Z ′/LQ new physics contribution crossing nearly at ∼ 2.5GeV2 as shown in Fig. 3.Similarly, in the binned plots we observe that except for the bin [0.1,0.98], the AFB values are shifted to highervalues as compared to the SM estimations due to the Z ′/LQ new physics contribution. However, in fact in allq2 bins the Z ′/LQ new physics spans less than 1σ deviation from the SM.

The Z ′/LQ new physics contribution in the longitudinal polarization fraction FL(q2) has shifted from the SMfor q2 < 2GeV2 while in the rest of q2 region the NP contributions coincides with the SM contribution. Noimportant observations can be drawn from FL(q2).

For the angular observable P ′5(q2), in the region q2 ∈ [1.1, 2.5] the Z ′/LQ new physics contribution can beclearly distinguished from the SM however it lies within the SM error band. We do observe the zero crossing forP ′5(q2). In SM we get the zero crossing at ∼ 1.2GeV2 which is different from the Z ′/LQ new physics contributionobserved at ∼ 1.4GeV2.

• The observables 〈QFL〉, 〈QAFB 〉 and 〈Q′5〉 are purely sensitive to test the lepton flavor universality violation.The NP contribution in the Q observables can be clearly visualized. This is because of the reason that all Q’s arezeros in SM and hence any non-zero contribution due the NP obviously justifies the beyond SM effects. Fromthe Fig. 3, for 〈QFL〉 we see that the uncertainties associated with the Z ′/LQ new physics contribution in thelower q2 bins such as [0.1, 0.98] and [1.1, 2.5] are huge and hence the deviation reduces nearly to 2σ whereas,for q2 > 2.5GeV2 the Z ′/LQ new physics contributions are clearly distinguishable at more than 5σ. In the case

9

of 〈QAFB 〉, the first three bins [0.1, 0.98], [1.1,2.5] and [2.5, 4.0] however show upto 3σ deviation from the SM,the last bin [4.0, 6.0] is quite interesting with > 3σ deviation. Similarly, for 〈Q′5〉 except for the bin [4.0, 6.0]rest of the bins are significantly distinguishable at more than 5σ from the SM predictions.

Observable [0.10, 0.98] [1.1, 2.5] [2.5, 4.0] [4.0, 6.0] [1.1, 6.0]

Bc → Ds µ+ µ−

BR× 10−7SM 0.024± 0.004 0.045± 0.008 0.057± 0.012 0.096± 0.020 0.200± 0.037

LQ/Z′ 0.018± 0.003 0.035± 0.006 0.044± 0.009 0.075± 0.014 0.154± 0.032

〈RµeDs〉SM 0.992± 0.024 1.000± 0.007 1.000± 0.004 1.000± 0.002 1.000± 0.002

LQ/Z′ 0.771± 0.018 0.777± 0.005 0.778± 0.003 0.780± 0.003 0.777± 0.003

Bc → D∗s µ+ µ−

BR× 10−7SM 0.018± 0.003 0.017± 0.007 0.029± 0.009 0.070± 0.024 0.116± 0.031

LQ/Z′ 0.017± 0.002 0.013± 0.005 0.022± 0.008 0.053± 0.013 0.088± 0.028

〈FL〉SM 0.332± 0.122 0.707± 0.111 0.586± 0.129 0.454± 0.101 0.525± 0.093

LQ/Z′ 0.270± 0.099 0.682± 0.113 0.593± 0.095 0.461± 0.082 0.528± 0.101

〈AFB〉SM 0.163± 0.026 0.077± 0.060 −0.193± 0.054 −0.361± 0.064 −0.254± 0.051

LQ/Z′ 0.159± 0.017 0.137± 0.085 −0.151± 0.037 −0.341± 0.049 −0.220± 0.045

〈P ′5〉SM 0.528± 0.082 −0.477± 0.124 −0.869± 0.100 −0.936± 0.085 −0.842± 0.094

LQ/Z′ 0.573± 0.085 −0.337± 0.134 −0.825± 0.084 −0.924± 0.087 −0.803± 0.070

〈RµeD∗s 〉SM 0.979± 0.011 0.988± 0.005 0.990± 0.001 0.993± 0.000 0.992± 0.001

LQ/Z′ 0.924± 0.042 0.783± 0.020 0.752± 0.004 0.753± 0.004 0.757± 0.003

〈QFL〉 LQ/Z′ −0.057± 0.027 −0.018± 0.009 0.010± 0.001 0.008± 0.001 0.006± 0.001

〈QAFB 〉 LQ/Z′ −0.023± 0.012 0.058± 0.015 0.043± 0.017 0.020± 0.007 0.0.034± 0.010

〈Q′5〉 LQ/Z′ 0.073± 0.009 0.132± 0.016 0.038± 0.015 0.008± 0.006 0.032± 0.009

TABLE V: The SM central value and the corresponding 1σ standard deviation of various physical observables in

SM and in the presence of Z ′/LQs for Bc → D(∗)s µ+µ− decays

2. Bc → D(∗)s νν decays

We know that the neutral semileptonic decays with the neutrinos in the final states are interesting due to the reducedhadronic uncertainties beyond the form factors. In fact, the SU(2)L flavor symmetry which treats the charged leptons(µ+µ−) and neutral leptons (νν) to be analogous that invites one to examine b → s νν decays in the presence ofvarious beyond the SM scenarios with the implications of available b → s`+`− experimental data. Since we areinterested to find out the combined new physics solution which appear both in b → s (`+`−, νν) decays, we study

Bc → D(∗)s νν decays in SM and also in the presence of Z ′, S3

1/3 and U3−2/3 LQs which satisfy CNP9 = −CNP10 NP

scenario in b → s`+`− decays. This particular CNP9 = −CNP10 NP scenario under similar Z ′/LQ models have been

discussed for the Bc → D(∗)s µ+µ− decays in the previous section. The new physics contribution to the left handed

WC CνL associated with operator Oν in b → s νν decays are related to the corresponding semileptonic WCs such asCNP9 and CNP10 . This contribution is different for Z ′, S3

1/3 and U3−2/3 LQs as mentioned in Table I. Since we look

for the new physics effects associated with left handed neutrinos, the longitudinal polarization fraction appearing inBc → D∗sνν decays have no effects beyond the SM. Hence we give predictions for the differential branching fraction in

Bc → D(∗)s νν decays both in SM and in the presence of several NP models. We obtain the branching fraction for the

underlying decays in SM of the O(10−6). In Table VI we report the corresponding branching ratios integrated overdifferent q2 bins. Similarly, in Fig. 4 we display the q2 dependency of the differential branching ratio in SM, Z ′, S3

1/3

and U3−2/3 LQs. The SM central values are obtained by considering the central values of each input parameters and

10

SM

SM (1σ)

LQ

0 1 2 3 4 5 60.00

0.02

0.04

0.06

0.08

q2[GeV2]dB

R/d

q2 ×10

7

SM

LQ

0 1 2 3 4 5 60.00

0.02

0.04

0.06

0.08

0.10

0.12

q2[GeV2]

BR×

10-

7

SM

LQ

0 1 2 3 4 5 60.0

0.2

0.4

0.6

0.8

1.0

1.2

q2[GeV2]

RD

s

FIG. 2: The q2 dependency and the bin wise distribution of the branching ratio and the ratio of branching ratio inBc → Ds µ

+µ− decays in SM and in the presence of Z ′/LQs.

the corresponding 1σ uncertainty is obtained by varying the form factors and CKM matrix element within 1σ. Forthe different NP models which are constrained by the latest b→ s`+`− data, we modify the SM WC CνL in b → s ννdecays accordingly as reported in Table V. The detailed observations of our study are as follows:

• The q2 dependency of Bc → D(∗)s νν decays for the whole kinematic range are displayed in the left panel of

Fig. 4. We observe from the plots that the differential branching ratios are enhanced for all the NP contributionsand very interestingly the U3

−2/3 LQ show significant deviation from the SM curve and lie away from the SM

error band. This is because of the reason that the CνL(NP )− the left handed new physics WC in b → s ννdecays for U3

−2/3 LQ is rescaled to two times the CNP9 = −CNP10 contribution i.e., CνL(NP ) = 2CNP9 = −2CNP10 .

Similarly, for the S31/3 LQ the CνL(NP ) = (1/2)CNP9 = −(1/2)CNP10 . On the other hand, for the Z ′ contribution

it is simply CνL(NP ) = CNP9 = −CNP10 without any enhancement or reduction. To this end we see that the S31/3

LQ lie close to the SM whereas Z ′ contribution lie at the 1σ boundary of the SM error band. Similarly, on theright panel of Fig. 4 we display the bin wise distribution of the branching ratios with upto q2 = 6GeV2 in thesimilar bin sizes as reported earlier. In Table VI we also have additional bin predictions which are not shown infigure. In all the bins we do expect > 1σ deviation for U3

−2/3 LQ and < 1σ in rest of the two models.

• The interesting fact about the new physics contribution in the longitudinal polarization fraction of Bc → D∗s ννdecay is that it is sensitive only to the right handed currents. Since in our analysis the new physics arising fromZ ′, S3

1/3 and U3−2/3 LQs include only the left handed contributions, the polarization fraction will not exhibit

any additional new physics effects. This is because of the reason that when we define FL as

FL = FL(SM)|CL|2 + |CR|2 − 2CLCR|CL|2 + |CR|2 − κCLCR

(31)

where, the CL(R) are the Wilson coefficients associated with left (right) handed operators in b → s νν decays.In the above equation for CR = 0 we obtain FL(NP ) = FL(SM) and hence any new contribution in CL will becancelled. Therefore in this section we report only the SM predictions for the FL. The q2 dependency of thelongitudinal polarization fraction FL(q2) for Bc → D∗s νν decay in the whole kinematic range is displayed in thethird row of Fig. 4. In the figure we have shown only the central curve and the corresponding 1σ error band forSM. Similarly, in the Table VII we report the SM mean and the corresponding standard deviation in variousbins including from zero to maximum q2 for which we obtain FL = 0.301± 0.040.

11

q2 bin SM LQ - S3 LQ - U3 Z′

BR(Bc → Dsνν)× 10−6

[0.1− 0.98] 0.014± 0.002 0.015± 0.003 0.019± 0.003 0.016± 0.003

[1.1− 2.5] 0.026± 0.005 0.028± 0.005 0.035± 0.007 0.030± 0.006

[2.5− 4.0] 0.032± 0.006 0.035± 0.007 0.045± 0.009 0.038± 0.008

[4.0− 6.0] 0.058± 0.010 0.058± 0.011 0.074± 0.014 0.063± 0.012

[6.0− 8.0] 0.068± 0.013 0.074± 0.015 0.094± 0.019 0.081± 0.016

[11− 12.5] 0.089± 0.018 0.097± 0.020 0.122± 0.024 0.105± 0.017

[15− q2max] 0.159± 0.027 0.173± 0.057 0.218± 0.030 0.188± 0.037

[1.1− 6.0] 0.113± 0.022 0.122± 0.024 0.154± 0.032 0.133± 0.026

[0− q2max] 0.761± 0.179 0.827± 0.144 1.042± 0.254 0.896± 0.174

BR(Bc → D∗s ν ν)× 10−6

[0.1− 0.98] 0.004± 0.002 0.005± 0.002 0.006± 0.003 0.005± 0.002

[1.1− 2.5] 0.013± 0.004 0.014± 0.004 0.017± 0.005 0.015± 0.004

[2.5− 4.0] 0.022± 0.005 0.024± 0.008 0.031± 0.006 0.026± 0.006

[4.0− 6.0] 0.051± 0.015 0.055± 0.011 0.070± 0.011 0.060± 0.010

[6.0− 8.0] 0.0087± 0.020 0.094± 0.021 0.119± 0.018 0.102± 0.022

[11− 12.5] 0.201± 0.045 0.218± 0.032 0.275± 0.034 0.236± 0.040

[15− q2max] 0.481± 0.085 0.523± 0.092 0.659± 0.089 0.566± 0.100

[1.1− 6.0] 0.087± 0.022 0.094± 0.025 0.119± 0.021 0.102± 0.020

[0− q2max] 1.602± 0.313 1.741± 0.304 2.193± 0.285 1.886± 0.321

TABLE VI: The branching ratios of Bc → D(∗)s νν decays in different q2 bins in SM and in the presence of Z ′, S3

1/3

and U3−2/3 LQs.

q2 bin SM

FL(Bc → D∗sνν)

[0.1− 0.98] 0.825± 0.074

[1.1− 2.5] 0.603± 0.097

[2.5− 4.0] 0.475± 0.105

[4.0− 6.0] 0.389± 0.078

[6.0− 8.0] 0.333± 0.064

[11− 12.5] 0.277± 0.040

[15− q2max] 0.301± 0.011

[1.1− 6.0] 0.444± 0.101

[0− q2max] 0.301± 0.040

TABLE VII: The lepton polarization fraction of Bc → D∗sνν decays in different q2 bins in SM.

12

SM

SM (1σ)

LQ

0 1 2 3 4 5 60.00

0.02

0.04

0.06

0.08

0.10

q2[GeV2]

dBR/d

q2 ×10

7

SM

SM (1σ)

LQ

1 2 3 4 5 60.0

0.2

0.4

0.6

0.8

1.0

1.2

q2[GeV2]

RD

s*(q

2)

SM

SM (1σ)

LQ

0 1 2 3 4 5 6-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

q2[GeV2]

AFB

(q2)

SM

SM (1σ)

LQ

0 1 2 3 4 5 60.0

0.2

0.4

0.6

0.8

q2[GeV2]

FL(q

2)

SM

SM (1σ)

LQ

0 1 2 3 4 5 6-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

q2[GeV2]

P5'(q

2)

SM

LQ

0 1 2 3 4 5 60.00

0.02

0.04

0.06

0.08

0.10

q2[GeV2]

BR×

10-

7

SM

LQ

1 2 3 4 5 60.0

0.2

0.4

0.6

0.8

1.0

1.2

q2[GeV2]

RD

s

SM

LQ

0 1 2 3 4 5 6-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

q2[GeV2]

AFB

SM

LQ

0 1 2 3 4 5 60.0

0.2

0.4

0.6

0.8

q2[GeV2]

FL

SM

LQ

0 1 2 3 4 5 6-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

q2[GeV2]

P5'

SM

LQ

0 1 2 3 4 5 6-0.10

-0.08

-0.06

-0.04

-0.02

0.00

0.02

q2[GeV2]

QF

L

SM

LQ

0 1 2 3 4 5 6

-0.04

-0.02

0.00

0.02

0.04

0.06

0.08

0.10

q2[GeV2]

QA

FB

SM

LQ

0 1 2 3 4 5 6-0.05

0.00

0.05

0.10

0.15

0.20

q2[GeV2]

Q5'

FIG. 3: The q2 dependency and the bin wise distribution of various observables such as the differential branchingfraction, the ratio of branching ratio R

D(∗)s

, the forward backward asymmetry, the lepton polarization asymmetries,

the angular observable P ′5 and the Q parameters for Bc → D∗s µ+µ− decays in SM and in the presence of Z ′/LQs.

V. CONCLUSION

With the experimental data associated with b → s`+`− neutral current transition reported in the semileptonicB → (K,K∗)µ+µ− and Bs → φµ+µ− and also purely leptonic Bs → µ+µ− decay processes, we scrutinize the

Bc → D(∗)s µ+µ− and Bc → D

(∗)s νν decays in the SM followed by the effects in the presence of leptoquark and

Z ′ new physics models. Throughout the analysis, we have concentrated on the particular new physics scenarioCµµ9 (NP ) = −Cµµ10 (NP ) where both the leptoquark and Z ′ models satisfy the following condition. We obtain the Z ′

13

SM

SM (1σ)

LQU3

LQS3

Z'

0 5 10 150.00

0.02

0.04

0.06

0.08

0.10

q2[GeV2]

dBR/d

q2 ×10

6

SM

LQU3

LQS3

Z'

0 1 2 3 4 5 60.00

0.02

0.04

0.06

0.08

0.10

q2[GeV2]

BR×

10-

6

SM

SM (1σ)

LQU3

LQS3

Z'

0 5 10 150.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

q2[GeV2]

dBR/d

q2 ×10

6

SM

Z'

LQS3

LQU3

0 1 2 3 4 5 60.00

0.02

0.04

0.06

0.08

q2[GeV2]

BR×

10-

6SM

SM (1σ)

0 5 10 150.0

0.2

0.4

0.6

0.8

1.0

q2[GeV2]

FL(q

2)

FIG. 4: The q2 dependency and bin wise distribution of the branching ratios of Bc → Dsνν (top) and Bc → D∗sνν(bottom) decays in the whole kinematic range in SM and in the presence of Z ′, S3

1/3 and U3−2/3 LQs.

and LQ coupling strengths by fitting the five LHCb experimental data associated with b → s`+`− decays includingRK(∗) , P ′5, B(Bs → φµ+µ−) and B(Bs → µ+µ−). Notably we include the latest updates of RK , B(Bs → φµ+µ−) andB(Bs → µ+µ−) in our fit analysis.

In the decays involving the charged leptons as a final state, we have performed a detailed study of various observablessuch as the differential branching fraction, the forward-backward asymmetry, the lepton polarization asymmetry, the

angular observable P ′5 and the ratio of branching ratios for Bc → D(∗)s µ+µ− decays in SM and in the presence of

Z ′/LQ new physics. Simultaneously, the similar new physics contributions from Z ′ and LQs have been inspected in the

branching ratios of Bc → D(∗)s νν decay processes. We observe from our analysis that the branching ratio is reduced

due to Z ′/LQ in the decays which include the charged leptons as a final state whereas in the processes involvingneutrinos in the final state the branching ratio is increased for Z ′, S3

1/3 and U3−2/3 LQs. In fact more significant

deviation from the SM is found for U3−2/3 particularly in Bc → D

(∗)s νν decays. Moreover, the zero crossing of the

forward-backward asymmetry in Bc → D∗sµ+µ− process is shifted to higher q2 value in the presence of Z ′/LQ new

physics. Similarly, the LFUV sensitive observables including RD

(∗)s

and the Q parameters have significant deviations

at more than 5σ from the SM in most of the q2 bins. In addition, it is important to note that the NP contributions

from Z ′, S31/3 and U3

−2/3 LQs in Bc → D(∗)s µ+µ− decays are indistinguishable whereas in the Bc → D

(∗)s νν case all

the three new physics contributions are clearly distinguished from one another. Having said that the decay modes

Bc → D(∗)s (µ+µ−, νν) mediated by b → s(`+`−, νν) transition have received very less attention than the current

ongoing study in B(s) → (K,K∗, φ)`+`− processes. Hence, the combined study of particular decays Bc → D(∗)s µ+µ−

and Bc → D(∗)s νν will certainly help us in identifying the possible new physics signatures in both b → s`+`−

and b → sνν decays. Moreover, the improved estimations of the various form factors corresponding to Bc → Ds

and Bc → D∗s transitions will be crucial in near future to understand the nature of NP. In addition to this, more

data samples from the experiments are also required to visualize various observables in Bc → D(∗)s (`+`−, νν) decay

processes and in particular the more experimental studies pertaining to b → sνν decays can assist to identify thevarious new physics Lorentz structures.

14

Acknowledgments

MKM would like to acknowledge DST INSPIRE fellowship programme for financial support. NR would like tothank CSIR for the financial help in this work.

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Appendix A: Form factors for Bc → D(∗)s ``(` = e, µ)

The hadronic matrix elements for the exclusive Bc → Ds transition in terms of form factors is given by [43]

Jµ = < Ds|sγµb|Bc >= f+(q2)[pµBc + pµDs −

M2Bc−M2

Ds

q2qµ]

+ f0(q2)M2Bc−M2

Ds

q2qµ ,

JTµ = < Ds|sσµν qνb|Bc >=i fT (q2)

MBc +MDs

[q2(pµBc + pµDs − (M2

Bc −M2Ds)q

µ], (A1)

where q = pBc − pDs and the form factors given above the expression satisfy the following relations:

f+(0) = f0, f0(q2) = f+(q2) +q2

m2Bc−m2

Ds

f−(q2). (A2)

17

Similarly, for the Bc → D∗s transition, the hadronic matrix elements can be given in terms of the form factors as

< D∗s |sγµb|Bc > =2 i V (q2)

MBc +MD∗s

εµνρσε∗ν pBcρ pD∗sσ ,

< D∗s |sγµγ5 b|Bc > = 2MD∗sA0(q2)

ε∗ · qq2

qµ + (MBc +MD∗s)A1(q2)

(ε∗µ

− ε∗ · qq2

qµ)

−A2(q2)ε∗ · q

MBc +MD∗s

[pµBc + pµD∗s −

M2Bc−M2

D∗s

q2qµ],

< D∗s |s i σµν qνb|Bc > = 2T1(q2) εµνρσε∗ν pBcρ pD∗sσ ,

< D∗s |s i σµν γ5 qνb|Bc > = T2(q2)[(M2

Bc −M2D∗s

)ε∗µ

− (ε∗ · q)(pµBc + pµD∗s )]

+T3(q2) (ε∗ · q)[qµ − q2

M2Bc−M2

D∗s

(pµBc + pµD∗s )], (A3)

where qµ = (pµB − pµDs

) is the four momentum transfer and εµ is polarization vector of the D∗s meson.

Appendix B: Angular coefficients

The q2 dependent angular coefficients required for Bc → D∗s ``(` = µ) processes are given as follows:

Ic1 =

(|AL0|2 + |AR0|2

)+ 8

m2l

q2Re

[AL0A

∗R0

]+ 4

m2l

q2|At|2,

Ic2 = −β2l

(|AL0|2 + |AR0|2

),

Is1 =3

4

[|AL⊥|2 + |AL‖|2 + |AR⊥|2 + |AR‖|2

](1− 4m2

l

3q2

)+

4m2l

q2Re

[AL⊥A

∗R⊥ +AL‖A

∗R‖

],

Is2 =1

4β2l

[|AL⊥|2 + |AL‖|2 + |AR⊥|2 + |AR‖|2

],

I3 =1

2β2l

[|AL⊥|2 − |AL‖|2 + |AR⊥|2 − |AR‖|2

],

I4 =1√2β2l

[Re

(AL0A

∗L‖

)+Re

(AR0A

∗R‖

)],

I5 =√

2βl

[Re

(AL0A

∗L⊥

)−Re

(AR0A

∗R⊥

)],

I6 = 2βl

[Re

(AL‖A

∗L⊥

)−Re

(AR‖A

∗R⊥

)],

I7 =√

2βl

[Im

(AL0A

∗L‖

)− Im

(AR0A

∗R‖

)],

I8 =1√2β2l

[Im

(AL0A

∗L⊥

)+ Im

(AR0A

∗R⊥

)],

I9 = β2l

[Im

(AL‖A

∗L⊥

)+ Im

(AR‖A

∗R⊥

)], (B1)

18

where β` =√

1− 4m2`/q

2. According to Ref. [57], the transversity amplitude in terms of form factors and Wilsoncoefficients are given as

AL0 = N1

2mD∗s

√q2

{(Ceff9 − C10)

[(m2

Bs −m2D∗s− q2)(mBs +mD∗s

)A1 −λ

mBs +mD∗s

A2

]+

2mb Ceff7

[(m2

Bs + 3m2D∗s− q2)T2 −

λ

m2Bs−m2

D∗s

T3

]},

AL⊥ = −N√

2

[(Ceff9 − C10)

√λ

mBs +mD∗s

V +

√λ 2mb C

eff7

q2T1

],

AL‖ = N√

2

[(Ceff9 − C10)(mBs +mD∗s

)A1 +2mb C

eff7 (m2

Bs−m2

D∗s)

q2T2

],

ALt = N(Ceff9 − C10)

√λ√q2A0 , (B2)

where λ = (m4Bc

+m4Ds

+ q4 − 2 (m2Bcm2Ds

+m2Dsq2 + q2m2

Bc) and N , the normalization constant which is defined as

N =

[G2Fα

2em

3 · 210π5m3Bs

|VtbV ∗ts|2q2√λ

(1− 4m2

l

q2

)1/2]1/2

. (B3)

The right chiral component ARi of the transversity amplitudes can be obtained by replacing ALi by ALi|C10→−C10(i =

0, ‖,⊥, t).