Comparing AdS/QCD and Sum Rules predictions for $B\to K^*ν\barν$
Mohammad Ahmady Alexandre Leger Zoe McIntyre Alexander Morrison Ruben Sandapen



















    ḍ
            |||(){}→⟨⟩[ B  K^* l^+ l^-B  K^* μ^+ μ^-B  K^* τ^+ τ^-AdS/QCD vs SR predictions for B→ K*νν̅Using the form factors obtained from holographic AdS/QCD and QCD sum rules, we predict the differential branching ratio and longitudinal polarization fraction for the rare B→ K^* νν̅ decay.  This is an interesting decay channel as it does not suffer from hadronic uncertainties beyond the form factors.  We point out that the future measurement of the  with around 30% accuracy can discriminate between the two models.The International Conference on B-Physics at Frontier Machines - BEAUTY2018
            
            		6-11 May, 2018
            
            		La Biodola, Elba Island, Italy
            
            § INTRODUCTION
            
            
            The flavor changing neutral current (FCNC) b→ s is an excellent venue for searching signals of new physics (NP) beyond the Standard Model (SM).   The rare decay   is particularly interesting as, on the theory side, the presence of only one operator in the effective Hamiltonian for the b→ s νν̅ transition implies sensitivity to a minimal number of form factors (FF).  Experimentally, however, is challenging, as both leptons are detector eluding neutrinos. The only available data at this time are the upper bounds on the branching ratio (ℬℛ) set by the Belle Colaboration <cit.>:
            ℬℛ(B^+→ K^*+νν̅)   <    4.0× 10^-5   (90%   CL)   ,
            ℬℛ(B^0→ K^*0νν̅)   <    5.5× 10^-5   (90%   CL)   .
            
            However, the Belle-II experiment, with an integrated luminosity 50  ab^-1 that
            is expected to be collected by 2023, a measurement of the SM ℬℛs
            with 30% precision is expected <cit.>. Therefore, it is appropriate to have a closer look at the theoretical calculation of this decay rate focusing on the theoretical uncertainties in the SM predictions.
            Here, we present our results for the differential ℬℛ as well as the K^* longitudinal polarization fraction for  decay using the FFs parameterizing the B→ K^* hadronic matrix elements which are derived via light-cone sum rules (LCSR).  The required Distribution Amplitudes (DAs) for K^* are obtained from the holographic light-front wavefunctions (LFWFs) for vector mesons<cit.> and from QCD Sum Rules (SR) <cit.>.  The detailed calculations can be found in Ref <cit.>.  We show that the two models make distinct predictions for ℬℛ( ) specially at high K^* recoil kinematic region.
            
            
            
            
            § K^* DAS AND B→ K^* FFS
            
            In terms of the LFWFs, the twist-2 DAs, ϕ_K^*^∥ ,⊥ (x, μ ), for K^* are given as <cit.>:
            
                f_K^*ϕ_K^*^∥(x,μ) =√(N_c/π)∫ḅμ
                J_1(μ b) [1 + m_q̅ m_s -∇_b^2/M_K^*^2 x(1-x)] Ψ_L(x, ζ)/x(1-x) ,
            
            and 
            
                f_K^*^⊥(μ) ϕ_K^*^⊥(x,μ) =√(N_c /2 π)∫ḅμ
                J_1(μ b) [m_s - x(m_s-m_q̅)] Ψ_T(x,ζ)/x(1-x)  ,
            
            where f_K^* and f_K^*^⊥ are the longitudinal and transverse coupling constants, respectively. μ∼ 1GeV is the nonperturbative hadronic scale and
            Ψ_λ (x,ζ) = 𝒩_λ√(x (1-x))exp[ -κ^2 ζ^2   2] exp[ -(1-x)m_s^2+ xm_q̅^2  2 κ^2 x(1-x) ] ,
            
            are holographic meson wavefunctions obtained by solving the holographic light-front Schrödinger Equation for mesons <cit.>. λ =L,  T denotes the polarization and ζ=√(xx̅)b is the so-called holographic variable.  The longitudinal and transverse couplings are given by <cit.>
            f_K^*   =   √(%s/%s)N_cπ∫_0^1 d x  [ 1 +  m_q̅m_s-∇_b^2 x (1-x) M^2_K^*] . Ψ_L(ζ, x) |_ζ=0  ,
            
            and  
            
                f_K^*^⊥(μ) =√(N_c/2π)∫_0^1 d x (xm_q̅ +(1-x)m_s)  ∫d b   μ J_1(μ b)  Ψ_T(ζ,x)/x(1-x)  ,
            
            respectively.  As shown in Table <ref>, the quark masses can be constrained by fitting to both the measured decay constant f_K^* and the lattice prediction for the ratio.  We observe that different sets of quark masses can be used to fit the measured decay constant with the larger quark masses being preferred in order to approach the lattice data for the ratio f_K^*^⊥/f_K^*. We shall use m_q̅=(195 ± 55) MeV and m_s=(300 ± 20) MeV in computing our predictions. 
            
            
            
            
            The twist-2 SR DAs are constructed as Gegenbauer expansions <cit.>
                ϕ_K^*^∥,⊥(x, μ) = 6 x x̅{ 1 + ∑_j=1^2
                a_j^∥,⊥ (μ) C_j^3/2(2x-1)} .
            
            where C_j^3/2 are the Gegenbauer polynomials and the coeffecients a_j^∥,⊥(μ) are given as a_1^∥ =0.06± 0.04, a_2^∥ =0.16± 0.09 for ϕ^∥(z, μ=1  GeV) and a_1^⊥ =0.04± 0.03, a_2^⊥ =0.10± 0.08 for ϕ^⊥(z, μ=1  GeV)<cit.>.
            
            
            
            Figure <ref> shows twist-2 DAs ϕ^∥ ,⊥(z, μ=1  GeV) for the K^* vector meson obtained from Eqs. <ref> and <ref> as compared to SR predictions as given by Eq. <ref>.  The error band for holographic DAs are due to the uncertainty in the quark masses but the error band in SR DAs are the result of the uncertainties in the Gegenbauer coefficients. 
            
            
            
            
            § B K^* TRANSITION FFS
            
            The FFs, computed via LCSR, are valid at low to intermediate q^2.  The extrapolation to high q^2 is performed via a two-parameter fit of the following form
            
            
                F(q^2)=F(0)/1-a(q^2/m_B^2)+b(q^4/m_B^4)
            
            to the LCSR predictions as well as form factor values obtained by the lattice QCD which are available at high q^2.
            
            Figures <ref> shows the AdS/QCD predictions including the lattice data points at high q^2 for the form factors V, A_1 and A_12.  The shaded bands in these figures represent the uncertainty due to the error band in the DAs.  Note that there is an additional uncertainly in the FFs inherent in the LCSR method (uncertainty in the Borel parameter, continuum threshold and other input parameters) which are the same in both models and are not included in our results.
            
            
            
            
            § DIFFERENTIAL DECAY RATE AND F_L
            
             Figure <ref> compare the AdS/QCD and SR predictions for the  differential decay rate and polarization fraction F_L.  The resulting uncertainties due to FFs are shown as the shaded bands.  We observe that, in general,  the AdS/QCD prediction is lower than SR prediction for all values of the momentum transfer q^2.  The difference between the two predictions is maximal ( ∼ 25%) for intermediate values of q^2.  Most interesting, the two predictions are quite distinct at low-to-intermediate q^2 where LCSR method is most reliable.    For the total branching ratio, we predict ℬℛ()_ AdS/QCD=(6.36^+0.59_-0.74)× 10^-6, compared to sum rules result ℬℛ()_ SR=(8.14^+0.16_- 0.17)× 10^-6.
             
            
              On the other hand, we observe that within error bands, the two model predictions for F_L are not distinguishable.  This confirms that the K^* longitudinal polarization fraction has little sensitivity to the non-perturbative FFs and is thus an excellent observable to probe New Physics signals.  Integrating over the whole kinematic region 0≤ q^2≤ (m_B-m_K^*)^2, we predict F_L()_ SM^ AdS/QCD=0.40^+0.02_-0.01 as compared to F_L()_ SM^ SR=0.41± 0.01.
            
            
            
            
            
            § CONCLUSION
            
            
            
            AdS/QCD predicts lower  decay rate than QCD sum rules.  We expect that a future measurement of this decay channel at BELLE II may be able to discriminate between the two models.
            
            
            JHEP ]
