# Effects of a Single Universal Extra Dimension inBc→Ds,Dl+l- Decays

Authors: U. O. Yilmaz; E. Danapinar
Journal: Advances in High Energy Physics
 (2018)

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## Abstract

The rare semileptonicBc→Ds,dl+l- decays are studied in the universal extra dimension with a single extra dimension scenario. The sensitivity of differential and total branching ratios and polarization asymmetries of final state leptons to the compactification parameter are presented, for both muon and tau decay channels. Considering the ability of available experiments, it would be useful to study these effects.


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## Body

## 1. Introduction

The decays induced by flavor changing neutral current (FCNC)b→s,d transitions play an important role in testing the standard model (SM) and also they are very sensitive to the new physics. The decays of Bu,d,s mesons have been intensively studied since the observation of the rare radiative b→sγ decay [1], and it will make the B physics more complete if similar decays of Bc meson are included.TheBc meson is the ground state of a family of mesons containing two different heavy flavor quarks, b and c. Compared with the cc- and bb- bound states which have implicit flavors, the Bc meson has explicit flavor and decays weakly. Bu,d,s decays are described well in the framework of the heavy quark limit. The flavor and the spin symmetries must be reconsidered while discussing Bc. Bc was first observed by CDF Collaboration [2] and, due to those outstanding properties, Bc decays have received great attraction since then (for more about the Bc physics, see for example [3]).Considering the experimental facilities, the high centre of mass energies at the Large Hadron Collider (LHC) enables the experiments, such as LHCb, to study the production, properties, and decays of theBc meson. At the LHC in pp collisions, Bc is mainly produced through the gluon-gluon fusion process gg→Bc+b+c- and the production cross-sections of the Bc mesons have been calculated in the fragmentation approach and complete order αs4 approach. At s=8 TeV approximately 103Bc+ is obtained [4, 5]. In addition, at s=14 TeV and the design luminosity 1010Bc events per year are estimated [6, 7]. The available experimental facilities and their future improvements encourage studying the Bc meson phenomenology including the rare decays.In the rareB meson decays, new physics contributions appear through the modification of the Wilson coefficients existing in the SM or by adding new structures in the SM effective Hamiltonian. Among the various extensions of the SM, extra dimensions are especially attractive because of including gravity and other interactions, giving hints on the hierarchy problem and a connection with string theory.The models with universal extra dimensions (UED) allow the SM fields to propagate in all available dimensions [8–11]. The extra dimensions are compactified and the compactification scale allows Kaluza-Klein (KK) partners of the SM fields in the four-dimensional theory and also KK excitations without corresponding SM partners. Throughout the UED, a model including only a single universal extra dimension is the Appelquist-Cheng-Dobrescu (ACD) model [12]. The only additional free parameter with respect to the SM is the inverse of the compactification radius, 1/R. In particle spectrum of the ACD model, there are infinite towers of KK modes and the ordinary SM particles are presented in the zero mode.The experimental and theoretical discussions on the free parameter have been taken a significant part in the literature and the lower bound was commonly taken as1/R≥250GeV or 1/R≥350GeV [13–17]. An analysis of the electroweak precision tests with inclusive [18] and exclusive [19] rare radiative B decays is conducted, where the later one also includes the experimental results of B→K(∗)l+l- and the lower bound is estimated as 1/R>600GeV. Later analysis using the ATLAS and CMS data leads to 1/R>(700-715)GeV [20, 21] and another discussion on the UEDs including the 126GeV Higgs puts this limit as 1/R>500GeV [22]. Under above discussion on the compactification factor, we will take the 1/R>500GeV in this paper.The effective Hamiltonian of several FCNC processes [23, 24], semileptonic and radiative decays of B mesons [25–33], and FCNC baryonic decays [34–36] have been investigated in the ACD model. Polarization properties of final state particles in semileptonic decays, which is an powerful tool in searching new physics beyond the SM, have also been studied widely besides the other observables in these works, e.g., [17, 25, 29, 33].The main aim of this paper is to find the possible effects of the ACD model on some physical observables related to theBc→(Ds,D)l+l- decays. We study differential decay rate, branching ratio, and polarization of final state leptons, including resonance contributions in as many as possible cases. We analyze these observables in terms of the compactification factor and the form factors. The form factors for Bc→(Ds,D)l+l- processes have been calculated using different quark models [37–40] and three-point QCD sum rules [41]. In this work, we will use the form factors calculated in the constituent quark model [37].This paper is organized as follows. In Section2, we give the effective Hamiltonian for the quark level processes b→(s,d)l+l- with a brief discussion on the Wilson coefficients in the ACD model. We derive matrix element using the form factors and calculate the decay rate in Section 3. In Section 4, lepton polarizations are evaluated and the last two sections are dedicated to our numerical analysis, discussion on the obtained results, and conclusion.


## 2. Theoretical Framework

The effective Hamiltonian describing the quark levelb→(s,d)l+l- processes in the SM is given by the following [42]: (1)Heff=GFα2πVtq′Vtb∗C9effq-′γμLbl-γμl+C10q-′γμLbl-γμγ5l-2C7effmbq-′iσμνqνq2Rbl-γμl,where q=pBc-pDq′ is the momentum transfer and q′=s,d.New physics effects in the ACD model come out by the modification of the SM Wilson coefficients appearing in the above Hamiltonian. This process can be done by writing the Wilson coefficients in terms of1/R-dependent periodic functions, the details of which can be found in [23, 24]. That is F0(xt) in the SM is generalized by F(xt,1/R) accordingly: (2)Fxt,1R=F0xt+∑n=1∞Fnxt,xnwhere xt=mt2/mW2, xn=mn2/mW2 and mn=n/R. Here, mn is the mass of KK particles with n=0 for the ordinary SM particles. These modified Wilson coefficients can widely be found in the literature and the details are discussed in Appendix.In the ACD model, a normalization scheme independent effective coefficientC7eff can be written as follows: (3)C7effμb,1R=η16/23C7μW,1R+83η14/23-η16/23C8μW,1R+C2μW,1R∑i=18hiηai.The coefficientC9eff has perturbative part and we will also consider the resonance contributions coming from the conversion of the real cc- into lepton pair. So, C9eff is given by (4)C9effs′,1R=C9μ,1R1+αsμπws′+Yμ,s′+C9resμ,s′,where s′=q2/mb2. For C9, in the ACD model and in the naive dimensional regularization (NDR) scheme, we have (5)C9μ,1R=P0NDR+Yxt,1/Rsin2⁡θW-4Zxt,1R+PEExt,1Rwhere P0NDR=2.60±0.25 and the last term, PEE(xt,1/R), is numerically negligible.The perturbative part, coming from one-loop matrix elements of the four quark operators, is(6)Yμ=hy,s′3C1μ+C2μ+3C3μ+C4μ+3C5μ+C6μ-12h1,s′4C3μ+4C4μ+3C5μ+C6μ-12h0,s′C3μ+3C4μ+293C3μ+C4μ+3C5μ+C6μ,with y=mc/mb. The explicit forms of the functions appear above equations can be found in [43, 44]. The resonance contribution can be done by using a Breit-Wigner formula [45]: (7)C9res=-3αem2κ∑Vi=ψiπΓVi→l+l-mVismb2-mVi2+imViΓVi×3C1μ+C2μ+3C3μ+C4μ+3C5μ+C6μ.The normalization is fixed by the data in [46] and κ is taken 2.3.The Wilson coefficientC10 is independent of scale μ and given by (8)C10=-Yxt,1/Rsin2⁡θW.


## 3. Matrix Elements and Decay Rate

The matrix elements forBc→Dq′l+l- can be written in terms of the invariant form factors over Bc and Dq′. The parts of transition currents containing γ5 do not contribute, so the nonvanishing matrix elements are as follows [47]: (9)Dq′pDq′q′-iσμνqνbBcpBc=-fTq2mBc+mDq′PBc+PDq′μq2-qμmBc2-mDq′2,Dq′pDq′q′-γμbBcpBc=f+q2PBc+PDq′μ+f-q2qμ.The transition amplitude of theBc→Dq′l+l- decays can be written using the effective Hamiltonian and (9) as follows:(10)MBc⟶Dq′l+l-=Gα22πVtbVtq′∗l-γμlAPBc+PDq′μ+Bqμ+l-γμγ5lCPBc+PDq′μ+Dqμ,with(11)A=C9efff++2mbfTmBc+mDq′C7eff,B=C9efff--2mbmBc2-mDq′2fTq2mBc+mDq′C7eff,C=C10f+,D=C10f-.Finally, the following dilepton mass spectrum is obtained by eliminating angular dependence in the double differential decay rate:(12)dΓds=G2α2mBc212π5VtbVtq′∗2λvΔDq′where s=q2/mBc2, λ=1+r2+s2-2r-2s-2rs, r=mDs∗2/mBc2, v=1-4ml2/smBc2 and (13)ΔDq′=43mBc43-v2λA2+C2+4mBc4s2+r-s1-v2C2+16mBc2ml2sD2+32mBc2ml21-rReCD∗.


## 4. Lepton Polarization Asymmetries

Studying polarization asymmetries of final state leptons is an useful way of searching new physics. Therefore, we will discuss the possible effects of the ACD model in the lepton polarization. Using the convention in [48, 49], we define the orthogonal unit vectors Si- (i=L,T,N) for the longitudinal, transverse, and normal polarizations in the rest frame of l- as follows:(14)SL-≡0,e→L=0,p→lp→l,ST-≡0,e→T=0,e→N×e→L,SN-≡0,e→N=0,p→Dq′×p→lp→Dq′×p→l,where p→l and p→Dq′ are the three momenta of l- and Dq′ meson in the centre of mass (CM) frame of final state leptons, respectively. The longitudinal unit vector SL- is boosted by Lorentz transformation: (15)SL,CM-μ=p→lml,Elp→lmlp→l,while vectors of perpendicular directions remain unchanged under the Lorentz boost.The differential decay rate ofBc→Dq′l+l- for any spin direction n→- of the l- can be written in the following form: (16)dΓn→-ds=12dΓds01+PLe→L-+PN-e→N-+PT-e→T-·n→-,where (dΓ/ds)0 corresponds to the unpolarized decay rate, the explicit form of which is given in (13).The polarizationsPL-, PT-, and PN- in (7) are defined by the following: (17)Pi-s=dΓn-=ei-/ds-dΓn-=-ei-/dsdΓn-=ei-/ds+dΓn-=-ei-/ds.Here, PL- and PT- represent the longitudinal and transversal asymmetries, respectively, of the charged lepton l- in the decay plane, and PN- is the normal component to both of them.The straightforward calculations yield the explicit form of the longitudinal polarization forBc→Ds,dl+l- as follows:(18)PL-=163ΔmBc4vλReAC∗,and the transversal polarization is given by the following:(19)PT-=4mBc3mlπsλΔr-1sReAC∗+ReAD∗.The normal component of polarization is zero so we have not stated its explicit form here.


## 5. Numerical Analysis

In this section, we will introduce numerical analysis of physical observables. Some of the input parameters used in this work aremBc=6.28GeV, mDs=1.968GeV, mD=1.870GeV, mb=4.8GeV, mμ=0.105GeV, mτ=1.77GeV, |VtbVts∗|=0.041, |VtbVtd∗|=0.008, GF=1.17×10-5GeV-2 and τBc=0.46×10-12s [46].To make numerical predictions, we also need the explicit forms of the form factorsf+, f-, and fT. In our analysis, we used the results of [37], calculated in the constituent quark model, and q2 parametrization is given by(20)Fq2=F01-aq2/mBc2+bq2/mBc22,where the values of parameters F(0), a and b for the Bc→(Ds,D) decays, are listed in Table 1.Table 1
B c → D s , d decays form factors calculated in the constitute quark model.
B c → D s l + l - F ( 0 ) a b f + 0.165 - 3.40 3.21 f - - 0.186 - 3.51 3.38 f T - 0.258 - 3.41 3.30  B c → D l + l - F ( 0 ) a b  f + 0.126 - 3.35 3.03 f - - 0.141 - 3.63 3.55 f T - 0.199 - 3.52 3.38In this work, we also took the long-distance contributions into account. While doing this, to minimize the hadronic uncertainties, we introduce some cuts aroundJ/ψ and ψ(2s) resonances as discussed in [17].In the analysis, first the differential branching ratios are calculated with and without resonance contributions and s dependence for the SM and1/R=500GeV are presented in Figures 1 and 2 for Bc→(Ds,D)l+l-. One can notice the change in the differential decay rate and difference between the SM results and new effects in the figures. The maximum deviation is around s=0.32(0.39) in Figure 1 and s=0.36(0.40) in Figure 2 for μ(τ). The deviation is ~10% and less for 1/R>500GeV. Considering the resonance effects, the differential decay rates also differ from their SM values.Figure 1
The dependence of differential branching ratio on s with and without resonance contributions forBc→Dsl+l- in the SM and the ACD Model for 1/R=500GeV. (The subscript R represents resonance contribution.)Figure 2
The dependence of differential branching ratio on s with and without resonance contributions forBc→Dl+l- in the SM and the ACD Model for 1/R=500GeV.To introduce the contributions of the ACD model on the branching ratio, we present1/R-dependent ratios with and without resonance cases in Figures 3 and 4. The common feature is that as 1/R increases, the branching ratios approach to their SM values and vary in the following ranges for 1/R≥500GeV: (21)BrBc⟶Dsμ+μ-=1.151-1.255×10-7BrBc⟶Dsτ+τ-=0.293-0.324×10-7BrBc⟶Dμ+μ-=0.290-0.317×10-8BrBc⟶Dτ+τ-=0.077-0.078×10-8.Figure 3
The dependence of branching ratio on 1/R with and without resonance contributions forBc→Dsl+l-. (The subscript R represents resonance contribution.)Figure 4
The dependence of branching ratio on 1/R with and without resonance contributions forBc→Dl+l-.Here, the first value in any branching ratios above is corresponding to the SM, while the second one is for1/R=500GeV, without resonance contributions. A similar behavior is valid for resonance case which can be followed by the figures.Adding the uncertainty on the form factors may influence the contribution range of the ACD model. However, the variation of the branching ratios, calculated with the central values of form factors, in the ACD model with the SM values, can be considered as a signal of new physics.The polarization properties of final state leptons give useful clues for new physics. Hence, the dependence of longitudinal polarization on s without resonance contributions is given by Figures5 and 6. The longitudinal polarization differ from the SM values slightly. The maximum deviation can be found to be less than 5%.Figure 5
The dependence of longitudinal polarization on s without resonance contributions forBc→Dsl+l-.Figure 6
The dependence of longitudinal polarization on s without resonance contributions forBc→Dl+l-.In order to clarify the dependence on1/R, we eliminate the dependence of the lepton polarizations on s, by considering the averaged forms over the allowed kinematical region as (22)Pi=∫2ml/mBc21-mDq′/mBc2PidB/dsds∫2ml/mBc21-mDq′/mBc2dB/dsds.The 1/R-dependent average longitudinal polarizations are given in Figures 7 and 8. As it can be seen from the figures, the maximum deviation is 2% for μ channels and 3% for Bc→(Ds,D)τ+τ-, respectively, at 1/R=500GeV.Figure 7
The dependence of longitudinal polarization on1/R with resonance contributions for Bc→Dsl+l-.Figure 8
The dependence of longitudinal polarization on1/R with resonance contributions for Bc→Dl+l-.The variation of transversal polarization with respect tos is given by Figures 9 and 10. In μ channels the difference is negligible. In τ channels up to s~0.46(0.48) for Bc→Ds(D)l+l- decays, respectively, the effects of the UED can be followed. Although the deviation with the SM values is very small, finally, the average transversal polarization can be observed by Figures 11 and 12.Figure 9
The dependence of transversal polarization on s without resonance contributions forBc→Dsl+l-.Figure 10
The dependence of transversal polarization on s without resonance contributions forBc→Dl+l-.Figure 11
The dependence of transversal polarization on1/R with resonance contributions for Bc→Dsl+l-.Figure 12
The dependence of transversal polarization on1/R with resonance contributions for Bc→Dl+l-.


## 6. Conclusion

In this work, we have studiedBc→Ds(D)l+l- in the framework of a single universal extra dimension and calculated the contributions to some physical observables.As an overall conclusion, considering the lower bound on the compactification factor which is1/R>500GeV, there still appears acceptable difference on the differential and integrated branching ratios compared with the SM results.The polarization effects inμ channels are either irrelevant nor negligible while in τ channels some small effects are obtained. Under the discussion in this work, studying these decays experimentally can be useful for understanding new physics.



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