Combined analysis of Bc→Ds(*)μ+μ− and Bc→Ds(*)νν¯ decays within Z′ and leptoquark new physics models

Combined analysis of B

_c

→ D

^ðÞs

μ

⁺

μ

⁻

and B

_c

→ D

^ðÞs

ν¯ ν decays within Z

⁰

and leptoquark new physics models

Manas K. Mohapatra ,^1,* N. Rajeev,^2,† and Rupak Dutta^2,‡

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

2National Institute of Technology Silchar, Silchar 788010, India

(Received 29 September 2021; accepted 20 May 2022; published 16 June 2022)

We investigate the exclusive rare semileptonic decaysB_c→D^ðÞs ðll;ν¯νÞ induced by neutral current transition b→sðll;ν¯νÞ in the presence of nonuniversal Z⁰, scalar and vector leptoquark new physics models. We constrain the new physics parameter space by using the latest experimental measurements of R_KðÞ, P⁰₅,BðB_s→ϕμ^þμ⁻Þ and BðB_s→μ^þμ⁻Þ. Throughout the analysis, we choose to work with the particular new physics scenarioC^μμ₉ ðNPÞ ¼−C^μμ₁₀ðNPÞwhere bothZ⁰,S³₁₌₃, andU³₋₂₌₃leptoquarks satisfy the condition. Using these new coupling parameters, we scrutinize the several physical observables such as differential branching fraction, the forward backward asymmetry, the lepton polarization asymmetry, the angular observable P⁰₅, and the lepton flavor universal sensitive observables, including the ratio of branching ratioR_DðÞ

s and the fewQparameters in theB_c→D^ðÞs μ^þμ⁻andB_c→D^ðÞs ν¯νdecay processes.

DOI:10.1103/PhysRevD.105.115022

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 particles of the form b→sl^þl⁻ but also in the flavor changing charged current decays proceeding viab→clν quark level transitions. The rare weak decays of several composite beauty mesons such asB_d,B_s, andB_c, which are forbidden at the tree level in SM, appear to follow loop or box level diagrams. Theoretically, the radiative and semi- leptonic decays of B→K^ðÞ and B_s→ϕ processes have received greater attention and are studied extensively both within the SM and beyond. The sensitivity of new physics possibilities in these decays requires very good knowledge of the hadronic form factors, more specifically forB→V transitions. It requires information from both the light cone sum rule (LCSR) and lattice QCD (LQCD) methods to compute the form factors, respectively, at low and highq² regions, which eventually confine the whole kinematic region [1]. Currently, we do have the very precise

calculations of the form factors that have very accurate SM predictions of the differential branching fractions and various angular observables in b→sl^þl⁻ decays.

Similarly, the family of neutral decays proceeding via b→sνν¯ transitions equally provide the interesting oppor- tunity in probing new physics signatures to that of b→sl^þl⁻ transitions. However, so far no experiments have directly addressed any anomalies except the upper bounds of the branching fractions of B→K^ðÞν¯ν decay processes. In principle, under theSUð2ÞLgauge symmetry, both the charged leptons and neutral leptons are treated equally, and hence, one can extract a close relation between bothb→sl^þl⁻ and b→sνν¯ decays in beyond the SM scenarios. In addition, the decays withν¯νfinal state are well motivated for several interesting features since these decays are considered to be theoretically cleaner as they do not suffer from hadronic uncertainties beyond the form fac- tors such as the nonfactorizable corrections and photonic penguin contributions.

Experimentally, several measurements in b→sl^þl⁻ transitions, such asR_K¼BðB→Kμ^þμ⁻Þ=BðB→Ke^þe⁻Þ from LHCb[2,3]and Belle[4]atq²∈½0.045;1.1andq²∈

½1.1;6.0show2.1–2.4σdeviation from the SM expectations [5,6]. Similarly, the angular observableP⁰₅inB→Kμ^þμ⁻ from in the binsq²∈½4.0;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 SM expectations [12–14]. The recent updates in the measurements ofR_K¼ BðB→Kμ^þμ⁻Þ=BðB→Ke^þe⁻Þ[15,16]inq²∈½1.0;6.0 and the branching fraction ofBðBs→ϕμ^þμ⁻Þ [17–19]in q²∈½1.1;6.0region from LHCb still indicate 3.1σ inR_K

*manasmohapatra12@gmail.com

†rajeev_rs@phy.nits.ac.in

‡rupak@phy.nits.ac.in

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[5,6] and 3.6σ in BðBs→ϕμ^þμ⁻Þ [1,20] from the SM expectations. Similarly, the measurements pertaining tob→ sν¯ν transitions, the upper bound measured by the Belle Collaboration in the branching fraction of B→K^ðÞν¯ν decays areBðB→KννÞ¯ <1.6×10⁻⁵andBðB→Kν¯νÞ<

2.7×10⁻⁵ [21], respectively. There also exist theBABAR measurement on BðB→KννÞ¯ <4×10⁻⁵ [22]. Very recently, the Belle II updated the upper bound of BðB→ Kν¯νÞ<4.1×10⁻⁵in 2021[23].

There exist several other decay channels similar to the B→K^ðÞ andB_s→ϕundergoingb→sl^þl⁻quark level transitions[23–39]. If any new physics present inB→K^ðÞ andB_s→ϕdecays can in principle be reflected in several other decays as well. In that sense, we choose to study the explicit rare decays B_c→D^ðÞs μ^þμ⁻ and B_c→D^ðÞs ν¯ν, which undergo similar b→s neutral transition. The par- ticular decay modes B_c →D^ðÞs μ^þμ⁻ have been studied previously in SM using various form factors that include relativistic quark model (RQM) [40], the light front and constituent quark model[41,42], the three point QCD sum rules approach[43], the covariant quark model[44], and very recently, using the lattice QCD form factors only for B_c →D_s transitions [45]. In addition, as far as beyond SM analysis are concerned, these decays have also been analyzed within model independent and dependent new physics as well. The model independent study within the effective field theory approach was done in Ref.[46]under various 1D and 2D NP scenarios. In Ref. [47], the decay mode was studied within a nonuniversalZ⁰ model with the NP contribution coming from only the right-handed currents.

Similarly, the contribution ofZ⁰was analyzed by considering the UTfit inputs of left-handed coupling ofZ⁰boson and the different values of new weak phase angle in the Ref.[48].

Similarly, the SM results pertaining to the B_c→D^ðÞs ν¯ν decays have been addressed in Refs.[40,42,49].

Even from the experimental point of view after the discovery ofB_c meson at CDF viaB_c→J=Ψlν[50], the study ofB_cdecays were found to be very interesting. Unlike the weak decays of other B mesons, the B_c mesons are interesting as it is composed of both heavybandcquarks that allows a broader kinematic range, which eventually involve large number of decays. In addition, the upcoming LHC run can produce around 10⁸–10¹⁰ B_c mesons [51–55], which offer very rich laboratory for the associatedB_cweak decays.

At LHC with a luminosity10³⁴ cm⁻²s⁻¹, one could expect around2×10¹⁰B_cevents per year[56,57]. Hence, exploring the new physics inB_c →D^ðÞs μ^þμ⁻ decays are experimen- tally well motivated at the LHCb experiments.

For the experimental scope of theB_c →D^ðÞs νν¯channels concerns, as we know, they are experimentally challenging because of the dineutrinos in the final state, which leave no information in the detector. However, recently, Belle II at SuperKEK uses a novel and independent inclusive tagging

approach and measures the upper limit of the branching fraction ofB→Kνν¯ decays [58]. This novel method has benefited with larger signal efficiency of about 4%, at the cost of higher background level [58]. Since Belle and Belle II mainly work atϒð4SÞ andϒð5SÞresonances, no B_c mesons are produced. Hence, study of B_c →D^ðÞs νν¯ channels would be difficult at the Belle II experiment.

Moreover, as there are more number of B_c mesons produced at LHC, it can, in principle, be feasible to predict an upper limit of the branching fraction of B_c →D^ðÞs νν¯ decays in their future prospects.

In this context, we study the implication of the latest b→sl^þl⁻ data on theB_c→D^ðÞs μ^þμ⁻ andB_c →D^ðÞs νν¯ decay processes under the model dependent analysis. We choose in particular, the specific models such asZ⁰and the various scalar and vector leptoquarks (LQs), which satisfy C^μμ₉ ðNPÞ ¼−C^μμ₁₀ðNPÞ new physics scenario. Among various LQs, we opt for the specific LQs in such a way that they should have combined NP effects in the form of C^μμ₉ ðNPÞ ¼−C^μμ₁₀ðNPÞ both in b→sl^þl⁻ and b→sνν¯ decays. Hence, the main aim of this work is to extract the common new physics that appear simultaneously in b→sl^þl⁻ andb→sν¯ν decays.

The layout of the present paper is as follows. In Sec.II, we add a theoretical framework that includes a brief discussion of effective Hamiltonian forb→sl^þl⁻andb→sνν¯parton level transition. In addition to this, we also present the differential decay distributions and other q² dependent observables ofB_c →D^ðÞ_s μ^þμ⁻andB_c →D^ðÞ_s νν¯processes.

In the context of new physics, we deal with the contributions arising due to the exchange ofLQandZ⁰particles in Sec.III.

In Sec.IV, we report and discuss our numerical analysis in the SM and in the presence of NP contributions. Finally, we end with our conclusion in Sec.V.

II. THEORETICAL FRAMEWORK A. Effective Hamiltonian

The effective Hamiltonian responsible for b→sll parton level transition in the presence of NP vector operator can be represented as[59]

Heff¼−αG_F ffiffiffi2 p πV_tbV_ts

2C^eff₇

q² ½sσ¯ ^μνq_νðm_sP_Lþm_bP_RÞbðlγ¯ _μlÞ þC^eff₉ ð¯sγ^μP_LbÞðlγ¯ _μlÞþC₁₀ð¯sγ^μP_LbÞðlγ¯ _μγ₅lÞ þC^ll₉ ðNPÞð¯sγ^μP_LbÞðlγ¯ μlÞ

þC^ll₁₀ðNPÞ¯sγ^μP_Lblγ¯ μγ5l

; ð1Þ

where G_F is the Fermi coupling constant, α is the fine structure constant, V_ij is the CKM matrix element, and C^eff₇ ; C^eff₉ , and C₁₀ are the relevant Wilson coefficients

(WC) evaluated at μ¼m^pole_b scale [59]. The Wilson coefficients C^ll₉ ðNPÞ and C^ll₁₀ðNPÞ are the effective cou- pling constants associated with the corresponding NP operators. The effective Hamiltonian describing b→sν¯ν decay processes is given by [60]

H^ν¯_eff^ν ¼ G_Fα 2 ffiffiffi

p2

πV_tbV_tsC^νν_LO^νν_L: ð2Þ Here, the effective four fermion operator O^νν_L ¼ ð¯sγ_μP_LbÞð¯νγ^μð1−γ₅ÞνÞ and the associated coupling strength C^νν_L ¼XðxtÞ=s²_W, which includes the Inami— Lim functionXðx_tÞgiven in Ref.[61]. In principle, several Lorentz structures in the form of chiral operators can be possible in the NP scenario, such as vector, axial vector, scalar, pseudoscalar, and tensor. However, among all the operators scalar, pseudoscalar, and tensor are severely constrained by B_s→μμ and b→sγ measurements [62].

Therefore, we consider the vector and axial vector con- tributions only. In our analysis, among possible NP operators we consider only the left chiral O^ll₉ ðNPÞ and O^ll₁₀ðNPÞ contributions and the associated Wilson coeffi- cients are assumed to be real. The effective Wilson coefficientsC^eff₇ andC^eff₉ are defined as [63]

C^eff₇ ¼C₇−C₅ 3 −C₆ C^eff₉ ¼C₉ðμÞ þhðmˆ_c;sÞCˆ ₀

−1

2hð1;sÞð4Cˆ ₃þ4C₄þ3C₅þC₆Þ

−1

2hð0;sÞðCˆ ₃þ3C₄Þ þ2

9ð3C₃þC₄þ3C₅þC₆Þ; ð3Þ where sˆ¼q²=m²_b, mˆ_c ¼m_c=m_b, and C₀¼3C₁þC₂þ 3C₃þC₄þ3C₅þC₆.

hðz;sÞ ¼ˆ −8 9lnm_b

μ −8

9lnzþ 8 27þ4

9x−2

9ð2þxÞj1−xj¹⁼²

× 8<

: ln ffiffiffiffiffiffi

p1−x

ffiffiffiffiffiffiþ1 p1−x

−1−iπ; forx≡⁴_s^z_ˆ²<1 2arctan ffiffiffiffiffiffi¹

px−1; forx≡^4z_sˆ²>1 ð4Þ and

hð0;sÞ ¼ˆ −8 9lnm_b

μ −4

9lnsˆþ 8 27þ4

9iπ: ð5Þ Here, we have included the short distance perturbative contributions to C^eff₉ . The measurements of the B→ ðK^ðÞ;ϕÞll processes induced by b→sll quark level transition, in principle, include the available vector

resonances. These regions arise in the dimuon invariant mass resonance m_μμ around ϕð1020Þ; J=ψð3096Þ, and ψ_2Sð3686Þ along with broad charmonium states ψð3770Þ;ψð4040Þ;ψð4160Þ, andψð4415Þ. The amplitudes in these regions though are dominated but have a large theoretical uncertainty. However, this is performed with a model describing these vector resonances as a sum of the relativistic Breit-Wigner amplitudes. The explicit expres- sions of thecc¯ resonance part reads

YBWðq²Þ ¼3π α²

X

Vi¼J=ψ;ψ⁰

ΓðVi→l^þl⁻ÞMV_i

M²_Vi−q²−iM_ViΓVi

; ð6Þ where M_ViðJ=ψ;ψ⁰Þ is the mass of the vector resonance and ΓV_i is the total decay width of the vector mesons.

Moreover, in our study, we have taken care by avoiding all the c¯c resonances appearing at respective q² regions.

Hence, our all predictions concerned tob→sμ^þμ⁻ decay observables are done in the q² regions [0.1–0.98] and

½1.1–6 GeV². On the other hand, the case with b→sνν¯ quark level transitions are quite different. In principle, b→sνν¯ decays are free from various hadronic uncertain- ties beyond the form factors, such as the nonfactorizable corrections and photonic penguin contributions. In particu- lar, the vector charmonium states including ϕð1020Þ;

J=ψð3096Þ, and ψ_2Sð3686Þ etc. do not contribute to any dineutrino states in the concerned decays of b→sνν¯ channels. Hence, one can access the whole q² region in the prediction of various observables in B_c →D^ðÞs νν¯ decays. Hence, the b→sν¯ν decay channels are treated as theoretically cleaner than theb→slldecay processes.

Apart from this, additionally we do not even include any nonlocal effects in our analysis, which are important below the charmonium contributions that have been studied in detail in Refs. [64–67]. In principle, these hadronic non local effects are generally neglected in the study of the lepton flavor universality violation (LFUV) in various b→sll decays. In addition, the factorizable effects arising due to the spectator scattering may not affect severely to the LFU ratios, and hence, these effects are neglected in our present analysis[68,69].

B. Differential decay distribution and q² observables inBc→D^ðÞs μ⁺μ⁻

In analogy withB→Kl^þl⁻ decay mode, it is useful to note that the rare semileptonic B_c→D_sl^þl⁻ process is also mediated throughb→sl^þl⁻ transition in the parton level. In the standard model, we present the formula ofq² dependent differential branching ratio which is given as follows[70]:

dBR dq² ¼τB_c

ℏ

2a_lþ2 3c_l

; ð7Þ

where the parametersa_l andc_l are given by

a_l¼G²_Fα²EWjVtbV_tsj² 2⁹π⁵m³_B

c

β_l ffiffiffi pλ

q²jF_Pj² þλ

4ðjF_Aj²þ jF_Vj²Þ þ4m²_lm²_B

cjF_Aj² þ2m_lðm²_Bc−m²_Dsþq²ÞReðF_PF_AÞ

; ð8Þ

c_l¼−G²_Fα²_EWjV_tbV_tsj² 2⁹π⁵m³_B

c

βl

ffiffiffiλ p λβ²_l

4 ðjFAj²þ jFVj²Þ: ð9Þ Here, the kinematical factor λ and the mass correction factor β_l in the above equations are given by

λ¼q⁴þm⁴_B

cþm⁴_D

s −2ðm²_Bcm²_D

s þm²_B

cq²þm²_D

sq²Þ;

β_l ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1−4m²_l=q² q

: ð10Þ

However, the explicit expressions of the form factors such as F_P, F_V andF_A are given as follows:

F_P¼−m_lC₁₀

f_þ−m²_Bc−m²_Ds

q² ðf₀−f_þÞ

; ð11Þ

F_V¼C^eff₉ f_þþ 2m_b

m_Bcþm_DsC^eff₇ f_T; ð12Þ

F_A ¼C₁₀f_þ: ð13Þ We employ the Wilson coefficients at the renormalization scaleμ¼4.8GeV as reported in Ref. [70].

Similarly, the transition amplitude for B_c →D_sl^þl⁻ decay channel can be obtained from the effective Hamiltonian given in the Eq. (1). The q² dependent differential branching ratio for B_c →D_sl^þl⁻ process is given as[71]

dBR=dq²¼dΓ=dq² ΓTotal

¼τB_c

ℏ 1

4½3I^c₁þ6I^s₁−I^c₂−2I^s₂; ð14Þ where the q² dependent angular coefficients are given in Appendix B and the corresponding SM WCs at the renormalization scale μ¼4.8GeV are taken from [63].

In addition to this, we also define other prominent observables such as the forward-backward asymmetry A_FB, the longitudinal polarization fraction F_L, and the angular observable P⁰₅, which are given by [12]

F_Lðq²Þ ¼ 3I^c₁−I^c₂ 3I^c₁þ6I^s₁−I^c₂−2I^s₂; A_FBðq²Þ ¼ 3I₆

3I^c₁þ6I^s₁−I^c₂−2I^s₂; hP⁰₅i ¼

R

bindq²I₅

2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

−R

bindq²I^c₂R

bindq²I^s₂

q : ð15Þ

To confirm the existence of the lepton universality violation, one can construct additional observables asso- ciated with the two different families of lepton pair which are quite sensitive to shed light into the windows of NP. The explicit expressions are given as below[13,72],

hQF_Li ¼ hF^μ_Li−hF^e_Li; hQA_FBi ¼ hA^μ_FBi−hA^e_FBi;

hQ⁰₅i ¼ hP^0μ₅i−hP^0e₅i: ð16Þ Also we define the ratio of the branching ratios ofμtoe transition inB_c→D^ðÞs l^þl⁻ decay modes as follows:

R_D^ðÞ

s ðq²Þ ¼BRðBc→D^ðÞs μ^þμ⁻Þ

BRðBc →D^ðÞs e^þe⁻Þ: ð17Þ

C. Differential decay distribution in B_c→D^ðÞs νν¯ The explicit study of B_c →D^ðÞs ν¯ν processes involved withb→sνν¯transitions are also quite important to search for NP beyond the SM as they are associated tob→sl^þl⁻ parton level bySUð2Þ_Lsymmetry group. From the effective Hamiltonian given in Eq.(2), the differential decay rate for B_c→D^ðÞs ν¯ν decay channel is given by [35,73]

dBRðBc→D_sν¯νÞSM

dq² ¼τB_c3jNj²X²_t

s⁴_wρD_sðq²Þ; ð18Þ dBRðBc →D_sν¯νÞ_SM

dq²

¼τB_c3jNj²X²_t

s⁴_w½ρA₁ðq²Þ þρA₁₂ðq²Þ þρVðq²Þ; ð19Þ where the factor 3 comes from the sum over neutrino flavors, and

N¼V_tbV_tsG_Fα 16π²

ffiffiffiffiffiffiffiffi m_Bc 3π r

ð20Þ

is the normalization factor. The relevant rescaled form factorsρi given in the above equations are given below,

ρD_sðq²Þ ¼λ³⁼²_Ds ðq²Þ m⁴_B

c

½f^D_þ^sðq²Þ²;

ρVðq²Þ ¼ 2q²λ³⁼²_Dsðq²Þ

ðmBcþm_DsÞ²m⁴_Bc½Vðq²Þ²; ρA₁ðq²Þ ¼2q²λ¹⁼²_Dsðq²Þðm_Bþm_DsÞ²

m⁴_B

c

½A₁ðq²Þ²;

ρA₁₂ðq²Þ ¼64m²_D

sλ¹⁼²_Dsðq²Þ m²_B

c

½A₁₂ðq²Þ²: ð21Þ The parameterλis already defined forB_c →D_stransition in Eq. (10) and the pseudoscalar D_s is replaced by the vector mesonD_s in B_c →D_s decays.

III. NEW PHYSICS ANALYSIS IN THE SCENARIOC^μμ₉ ðNPÞ=−C^μμ₁₀ðNPÞ

Assuming the NP exist only in the context ofμmode in b→sl^þl⁻transition, it will contribute to more number of Lorentz structures. The new physics scenarios for the parton level b→sμ^þμ⁻ transition that account for NP contributions are given as [28,71]

ðIÞ∶C^μμ₉ ðNPÞ<0;

ðIIÞ∶C^μμ₉ ðNPÞ ¼−C^μμ₁₀ðNPÞ<0;

ðIIIÞ∶C^μμ₉ ðNPÞ ¼−C^0μμ₉ ðNPÞ<0;

ðIVÞ∶C^μμ₉ ðNPÞ ¼−C^μμ₁₀ðNPÞ ¼C^0μμ₉ ðNPÞ ¼C^0μμ₁₀ðNPÞ<0; ð22Þ where the unprimed couplings differ from the primed Wilson coefficients by their corresponding chiral operator as discussed in the previous section. Keeping in mind, as from the Ref.[74], only three out of ten leptoquarks, such asS₃,U₁, andU₃can explain theb→sμ^þμ⁻data as they have good fits under certain scenario. AmongS₃,U₃, and U₁ leptoquarks, the U₁leptoquark has no contribution to the couplings corresponding to the NP operator responsible forb→sν¯ν processes, whereas other two LQs are differ- entiated with a definite contributions to it. Hence, the effect from U₁ LQ is not taken into account in the present analysis. It is important to say that the remainingS₃andU₃ LQs do not satisfy the scenarios I and III. On the other hand, in Ref.[75], it is also reported thatZ⁰can contribute to both scenario I and II, whereas a vast majority of this model use the scenario II. Hence, the feasible environment to study both LQs and Z⁰ simultaneously will be the scenario II∶C^μμ₉ ðNPÞ ¼−C^μμ₁₀ðNPÞ. Many works have been studied in these scenarios in LQs [76–84] and in the presence of Z⁰ [76,85–90]. Therefore, the purpose of this work is to concentrate on the scenario II:C^μμ₉ ðNPÞ ¼

−C^μμ₁₀ðNPÞ[75].

A. Leptoquark contribution

There are ten different leptoquark multiplets under the SM gauge group SUð3ÞC×SUð2ÞL×Uð1ÞY in the pres- ence of dimension ≤4 operators [91] in which five multiplets include scalar (spin 0) and other halves are vectorial (spin 1) in nature under the Lorentz transforma- tion. Among all, both the scalar tripletS³ (Y¼1=3) and vector isotriplet U³ (Y¼−2=3) can explain the b→ sμ^þμ⁻andb→sνν¯processes simultaneously. The relevant Lagrangian is given as follows[74]:

LS¼y⁰_lql¯^c_Liτ₂⃗τqLS³₁₌₃þH:c:;

LV ¼g⁰_lql¯Lγμ⃗τq_LU³₋₂₌₃þH:c:; ð23Þ where the fermion currents in the above Lagrangian include the SUð2Þ_L quark and lepton doublets “q_L” and “lL,” respectively, and τ represent the Pauli matrices. Most importantly, the parameters y⁰_lq and g⁰_lq are the quark— lepton couplings associated with the corresponding lepto- quarks. In particular, for this analysis,y^0μbðsÞ_lq is the coupling of the leptoquarkS³₁₌₃to the left-handedμorνμand a left- handed fermion fieldb(s). Similarly,g^0μbðsÞ_lq is the coupling correspond to the leptoquarkU³₋₂₌₃. On the other hand, for the parton level b→sνν¯ transitions, both S³₁₌₃ andU³₋₂₌₃ LQs contribute differently as reported in TableI. Hence, the Wilson coefficient C^νν_L associated with b→sνν¯ can be obtained by replacingC^νν_L →C^νν_L þC^νν_LðNPÞ. In our paper, we consider the couplingsy^0μb_lqðy^0μs_lqÞandg^0μb_lqðg^0μs_lqÞas real for theS³₁₌₃andU³₋₂₌₃LQs respectively with the assumption of same mass for both the leptoquarks. The tree level Feynman diagram for these processes (left panel) mediated via LQ is shown in Fig.1.

B. Nonuniversal Z⁰ contribution

The extension of SM by an extra minimal Uð1Þ⁰ gauge symmetry produces a neutral gauge boson the so-calledZ⁰ boson. It is the most obvious candidate that represents b→sμ^þμ⁻ in the NP scenario. However the main attrac- tion of this model includes the flavor changing neutral current (FCNC) transition in the presence of new nonuni- versal gauge bosonZ⁰[92–94], which can contribute at tree TABLE I. Contributions of the LQs–S³₁₌₃; U³₋₂₌₃, and Z⁰ to the Wilson coefficients. The normalization RðMÞ≡ π=ð ffiffiffi

p2

αG_FV_tbV_tsðM_LQðM_Z⁰ÞÞ²and M_LQ¼M_Z⁰¼1TeV.

NP model C^μμ₉ ðNPÞ C^μμ₁₀ðNPÞ C^νν_LðNPÞ S³₁₌₃ Ry^0μ_lq^bðy^0μ_lq^sÞ −Ry^0μ_lq^bðy^0μ_lq^sÞ ¹₂Ry^0μ_lq^bðy^0μ_lq^sÞ U³₋₂₌₃ −Rg^0μ_lq^bðg^0μ_lq^sÞ Rg^0μ_lq^bðg^0μ_lq^sÞ −2Rg^0μ_lq^bðg^0μ_lq^sÞ Z⁰ −Mg^bs_Lg^μμ_L Mg^bs_Lg^μμ_L −Mg^bs_Lg^ν¯_L^ν

level given in Fig.1(right panel). After integrating out the heavyZ⁰the effective Lagrangian for 4 fermion operator is given as

L^eff_Z⁰ ¼− 1 2M²_Z0

J⁰_μJ^μ0; ð24Þ where the new current is given as

J^μ0¼−g^μμ_LLL¯γ^μP_LLþg^μμ_LðRÞμγ¯ ^μP_LðRÞμ

þg^ij_Lψ¯iγ^μP_LψjþH:c:; ð25Þ where i and j are family index, P_LðRÞ is the projection operator of left (right) chiral fermions andg^ij_Ldenote the left chiral coupling ofZ⁰gauge boson. Now relevant interaction Lagrangian is given as

L^eff_Z⁰ ¼−g^bs_L

m²_Z0ð¯sγ^μbÞð¯μγ^μðg^μμ_LP_Lþg^μμ_RP_RÞμÞ: ð26Þ Now, the modified Wilson coefficients in the presence of Z⁰ model can be written as [25]

C^μμ₉ ðNPÞ ¼−

ffiffiffi π

p2

G_FαV_tbV_ts

g^bs_Lðg^μμ_LÞ m²_Z0

; C^μμ₁₀ðNPÞ ¼

ffiffiffi π

p2

G_FαV_tbV_ts

g^bs_Lðg^μμ_LÞ m²_Z0

; ð27Þ whereg^bs_L is the coupling whenbquark couple tosquark and g^μμ_L is the μ^þ−μ⁻ coupling in the presence of new bosonZ⁰, and we have assumedg^μμ_R ¼0. Similarly for the b→sν¯νtransition, the NP contribution arising due toZ⁰is C^νν_LðNPÞ ¼C^μμ₉ ðNPÞ ¼−C^μμ₁₀ðNPÞ[73]. From the neutrino trident production, g^μμ_L ¼0.5 has been considered in our paper[25,74,75]. The new parameterg^bs_L is taken to be real in our analysis.

IV. NUMERICAL ANALYSIS AND DISCUSSIONS A. Input parameters

In this section, we report all the necessary input parameters used for our computational analysis. We con- sider the masses of mesons, quarks, Fermi coupling constant in the unit of GeV, and lifetime ofB_c meson in the unit of second, CKM matrix element, and fine structure constant from Ref.[95]. We have adopted the lattice QCD method[45]and the relativistic quark model[40]based on quasipotential approach for the form factors of B_c →D_s andB_c→D_s transitions, respectively.

The form of the form factors for B_c→D_s transition in lattice QCD are given as follows:

fðq²Þ ¼Pðq²Þ⁻¹X^Nn

n¼0

c^ðnÞzˆ^ðn;NnÞ; ð28Þ whereN_n¼3and the q²dependent pole factor Pðq²Þ ¼ 1−q²=M²_res (M_res¼5.711(f₀)[96], 5.4158 (f_þ;T)[95]).

Using the Bourreley-Caprini-Lellouch (BCL) parametriza- tion[97], the expressions of ˆz^n;N_0;þ;Tⁿ are given as

ˆ

z^n;N₀ ⁿ ¼zⁿ; ˆz^n;N_þ;Tⁿ ¼zⁿ−nð−1Þ^Nnþ1−n

N_nþ1 z^Nnþ1: ð29Þ Here,zðq²Þis defined as

zðq²Þ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi t_þ−q²

p − ffiffiffiffiffiffiffiffiffiffiffiffiffiffi t_þ−t₀ p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi t_þ−q²

p þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi t_þ−t₀

p ; ð30Þ

wheret₀¼0andt_þ ¼ ðm_Bð0⁻ÞþM_Kð0⁻_ÞÞ²with the masses M_Bð0−Þ¼5.27964 andM_Kð0−Þ¼0.497611. The form fac- tor coefficientsc^ðnÞ are reported in TableII. For our error analysis, we employ 10% uncertainty in the form factor coefficientsc^ðnÞ. For all the omitted details, we refer to[45].

Similarly, the form factors for B_c→D_s transition are defined as

Fðq²Þ ¼ 8>

>>

>>

<

>>

>>

>:

Fð0Þ ð1−_M^q2₂Þ

1−σ1q2 M2 B

s

þσ2q4 M4 B

s

; forF¼ fV; A₀; T₁g Fð0Þ

1−σ1q2 M2 B

s

þσ2q4 M4 B

s

; forF¼ fA₁; A₂; T₂; T₃g:

ð31Þ TABLE II. The lattice QCD form factor coefficientsc^ðnÞforBc→Dstransition[45]and the relativistic quark model form factors at q²¼0 and the corresponding fitted parametersσ1 andσ2for B_c→D_s transition[40].

B_c→D_s c^ð0Þ c^ð1Þ c^ð2Þ c^ð3Þ B_c→D_s V A₀ A₁ A₂ T₁ T₂ T₃

f₀ 0.217 −0.220 1.300 −0.508 Fð0Þ 0.182 0.070 0.089 0.110 0.085 0.085 0.051

f_þ 0.217 −0.559 5.149 −0.217 σ1 2.133 1.561 2.479 2.833 1.540 2.577 2.783

f_T 0.299 −1.501 3.579 −0.348 σ2 1.183 0.192 1.686 2.167 0.248 1.859 2.170

FIG. 1. The tree level contribution of the LQs and theZ⁰for B_c→D^ðÞs ðμ^þμ⁻;ν¯νÞ.

Here,M¼M_BsforA₀ðq²Þ, whereasM¼M_Bsis considered for all other form factors. We useM_Bs ¼5.4254GeV from the Ref.[95]. The related form factor input parameters for B_c →D_sare reported in the TableII. Similarly, we employ 10% uncertainty in the zero recoil momentum functionFð0Þ for our theoretical error inB_c →D_s form factors.

B. Fit results

To obtain the NP parameter space in the presence ofZ⁰ and LQs, we perform a naiveχ²analysis with the available b→sll experimental data. In the fit, we consider specifically the LHCb measurements of five different observables, such as R_K, R_K, P⁰₅, BRðBs→ϕμμÞ, and BRðBs→μ^þμ⁻Þ. Our fit include the latest measurements of R_K, BRðB_s→ϕμμÞ, and BRðBs→μ^þμ⁻Þ as reported from LHCb in 2021. For our theoretical computation of the underlying observables, we refer to the lattice QCD form factors[70]forR_K and the form factors obtained from the combined analysis of LCSRþLQCD for B→K and B_s→ϕdecay processes[1]. We define theχ² as

χ²ðC^NP_i Þ ¼X

i

ðO^th_iðC^μμ_9;10ðNPÞÞ−O^exp_i Þ²

ðΔO^exp_i Þ²þ ðΔO^sm_i Þ² ; ð32Þ whereO^th_i represent the theoretical expressions, including the NP contributions andO^expi are the experimental central values. The denominator includes 1σ uncertainties asso- ciated with the theoretical and experimental results. From our analysis, we obtain the best fit values and the corresponding 1σ range of the NP coupling strengths associated with Z⁰, S³₁₌₃, and U³₋₂₌₃ LQs, respectively, as shown in TableIII. The best fit points for the NP couplings ofZ⁰ and LQs are obtained by minimizing theχ²variable.

Similarly, to obtain the allowed 1σ range of each NP coupling, we impose χ²≤9.488constraint corresponding to 95% C.L. The minimum and maximum value of the1σ range are given in TableIII.

C. Interpretation of Bc→D^ðÞs ðμ⁺μ⁻;ννÞ¯ decays in the standard model and beyond

1. Bc→D^ðÞs μ⁺μ⁻ decays

We perform NP studies ofB_c →D^ðÞs μ^þμ⁻decays in the presence of Z⁰, S³₁₌₃, and U³₋₂₌₃ LQs, which satisfy the

C^μμ₉ ðNPÞ ¼−C^μμ₁₀ðNPÞ new physics scenario. Although the NP coupling strengths associated with theZ⁰,S³₁₌₃, and U³₋₂₌₃ LQs are different from each other, the contribution from the C^μμ₉ ðNPÞ ¼−C^μμ₁₀ðNPÞ new Wilson coefficients inb→sl^þl⁻ decays are same. Hence, we expect similar NP signature fromZ⁰,S³₁₌₃, and U³₋₂₌₃ LQs in the under- lying B_c →D^ðÞs μ^þμ⁻ decays. We study various observ- ables such as the differential branching ratio, the forward backward asymmetry, the lepton polarization fraction, the LFU sensitive observables, including the ratio of the branching ratioR

D^ðÞs , and the difference of the observables associated withQparameters, such asQ_FL,Q_AFB, andQ⁰₅in the presence of SM as well as new physics. In TableIV, we report the central values and the corresponding standard deviation for all the observables in both SM and Z⁰=LQ new physics. Similarly in Figs. 2 and 3, we display the correspondingq²distribution plots as well asq²integrated bin wise plots for B_c →D_sμ^þμ⁻ and B_c→D_sμ^þμ⁻ processes, respectively. For the binned plots, we choose different bin sizes that 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]. Similarly, for the q² distribution plots, we display the central lines and the corresponding1σerror band for both SM andZ⁰=LQ new physics scenarios. The central lines are obtained by con- sidering only the central values of all the input parameters, and the corresponding 1σ error bands are obtained by varying the form factors and the CKM matrix element within1σ. In SM, we obtain the branching fraction to be Oð10⁻⁷ÞforB_c→D^ðÞs μ^þμ⁻decay channels. The detailed observations of our study are as follows:

(i) The q² dependency of the differential branching fraction forB_c→D^ðÞs μ^þμ⁻decays are shown in the top—left panel of Figs. 2 and 3, respectively. We notice that the differential branching ratio is reduced in the presence of Z⁰=LQ new physics and the NP central line lies away from the SM uncertainty band forB_c →D_sμ^þμ⁻ decay. Although theZ⁰=LQ new physics contribution in B_c →D_sμ^þμ⁻ decay devi- ates from the SM central curve, it cannot be distinguished beyond the SM uncertainty; however, slight more deviation can be found atq²>4GeV². Moreover, partial overlapping of the SM and NP uncertainties can be noticed over theq². Similarly, in the top—right panel of Fig. 2 and bottom middle panel Fig. 3, we display the corresponding binned plots, respectively, for both the decay modes.

We observe that for B_c→D_sμ^þμ⁻ decay in the all the bins the new physics contribution stand at >1σ away from the SM. For the decay B_c →D_sμ^þμ⁻, however, the NP central values differ from the SM, but no such significant obser- vations can be made.

TABLE III. The best-fit values and the corresponding1σranges of the NP couplings associated withZ⁰ and LQ models.

Values Best fits 1σ range

NP models

Z⁰∶g^μμ_bs×10⁻³ 1.74 [0.11, 3.60]

S³₁₌₃∶y^0μb_lqðy^0μs_lqÞ×10⁻⁴ −8.70 [−15.50,−4.50]

U³₋₂₌₃∶ g^0μb_lqðg^0μs_lqÞ×10⁻⁴ 8.70 [4.50, 15.50]

(ii) The ratio of branching ratio R_DðÞ

s ðq²Þ is constant over the range q²∈½0.1;6.0 and is approximately equal to∼1. The uncertainties associated with this observable are almost zero both in SM as well as in

the presence of NP contribution. The NP contribu- tion fromZ⁰=LQ is easily distinguishable from the SM contribution beyond the uncertainties at more than5σas shown in Figs.2and3, respectively, for TABLE IV. The SM central value and the corresponding1σ standard deviation of various physical observables in SM and in the presence ofZ⁰=LQs forB_c→D^ðÞs μ^þμ⁻ decays.

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

B_c→D_sμ^þμ⁻

BR ×10⁻⁷ SM 0.0390.007 0.0720.014 0.0850.015 0.1260.024 0.2840.052 LQ=Z⁰ 0.0280.005 0.0530.011 0.0630.013 0.0950.014 0.2120.036 hR^μ_D^e

si SM 0.9930.025 1.0010.006 1.0010.004 1.0010.002 1.0010.003

LQ=Z⁰ 0.7200.020 0.7370.005 0.7450.004 0.7530.003 0.7460.003 B_c→D_sμ^þμ⁻

BR ×10⁻⁷ SM 0.0180.003 0.0170.007 0.0290.009 0.0700.024 0.1160.031 LQ=Z⁰ 0.0170.002 0.0130.005 0.0220.008 0.0530.013 0.0880.028 hFLi SM 0.3320.122 0.7070.111 0.5860.129 0.4540.101 0.5250.093 LQ=Z⁰ 0.2700.099 0.6820.113 0.5930.095 0.4610.082 0.5280.101 hAFBi SM 0.1630.026 0.0770.060 −0.1930.054 −0.3610.064 −0.2540.051 LQ=Z⁰ 0.1590.017 0.1370.085 −0.1510.037 −0.3410.049 −0.2200.045 hP⁰₅i SM 0.5280.082 −0.4770.124 −0.8690.100 −0.9360.085 −0.8420.094 LQ=Z⁰ 0.5730.085 −0.3370.134 −0.8250.084 −0.9240.087 −0.8030.070 hR^μe_D

si SM 0.9790.011 0.9880.005 0.9900.001 0.9930.000 0.9920.001

LQ=Z⁰ 0.9240.042 0.7830.020 0.7520.004 0.7530.004 0.7570.003 hQ_FLi LQ=Z⁰ −0.0570.027 −0.0180.009 0.0100.001 0.0080.001 0.0060.001 hQ_AFBi LQ=Z⁰ −0.0230.012 0.0580.015 0.0430.017 0.0200.007 0.0.0340.010 hQ⁰₅i LQ=Z⁰ 0.0730.009 0.1320.016 0.0380.015 0.0080.006 0.0320.009

FIG. 2. Theq² dependency and the bin wise distribution of the branching ratio and the ratio of branching ratio inB_c→D_sμ^þμ⁻ decays in SM and in the presence ofZ⁰=LQs.

both the decays. However, the claim of5σdeviation is observed only by considering the best fit points of Z⁰=LQ coupling strength and neglecting the corre- sponding experimental error of the measurement.

(iii) Theq²distribution of the forward backward asym- metryA_FBðq²Þhave a zero crossing at∼2.2 GeV²in SM, which is different from theZ⁰=LQ new physics contribution crossing nearly at∼2.5GeV²as shown FIG. 3. Theq²dependency and the bin wise distribution of various observables, such as the differential branching fraction, the ratio of branching ratioR_DðÞ

s , the forward backward asymmetry, the lepton polarization asymmetries, the angular observableP⁰₅, and theQ parameters forB_c→D_sμ^þμ⁻ decays in SM and in the presence ofZ⁰=LQs.