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Register a new user$eta$-meson leading-twist distribution amplitude within QCD sum rule approach and its application to the semi-leptonic decay $ D_s^+ toeta{ell}^{+} u_{ell}$
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In this paper, we make a detailed discussion on the $eta$-meson leading-twist light-cone distribution amplitude (LCDA) $phi_{2;eta}(u,mu)$ by using the QCD sum rules approach under the background field theory. Taking both the non-perturbative condensates up to dimension-six and the next-to-leading order (NLO) QCD corrections to the perturbative part, its first three moments $langlexi^n_{2;eta}rangle|_{mu_0} $ with $n = (2, 4, 6)$ can be determined, where the initial scale $mu_0$ is set as the usual choice of $1$ GeV. Numerically, we obtain $langlexi_{2;eta}^2rangle|_{mu_0} =0.204_{-0.012}^{+0.008}$, $langlexi_{2;eta}^4 rangle|_{mu_0} =0.092_{ - 0.007}^{ + 0.006}$, and $langlexi_{2;eta}^6 rangle|_{mu_0} =0.054_{-0.005}^{+0.004}$. Next, we calculate the $D_stoeta$ transition form factor (TFF) $f_{+}(q^2)$ within the QCD light-cone sum rules approach up to NLO level. Its value at the large recoil region is $f_{+}(0) = 0.484_{-0.036}^{+0.039}$. After extrapolating the TFF to the allowable physical region, we then obtain the total decay widthes and the branching fractions of the semi-leptonic decay $D_s^+toetaell^+ u_ell$, i.e. $Gamma(D_s^+ toeta e^+ u_e)=(31.197_{-4.323}^{+5.456})times 10^{-15}~{rm GeV}$, ${cal B}(D_s^+ toeta e^+ u_e)=2.389_{-0.331}^{+0.418}$ for $D_s^+ toeta e^+ u_e$ channel, and $Gamma(D_s^+ toetamu^+ u_mu)=(30.849_{-4.273}^{+5.397})times 10^{-15}~{rm GeV}$, ${cal B}(D_s^+ toetamu^+ u_mu)=2.362_{-0.327}^{+0.413}$ for $D_s^+ toetamu^+ u_mu$ channel respectively. Those values show good agreement with the recent BES-III measurements.
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We make a detailed study on the $D_s$ meson leading-twist LCDA $phi_{2;D_s}$ by using the QCD sum rules within the framework of the background field theory. To improve the precision, its moments $langle xi^nrangle _{2;D_s}$ are calculated up to dimension-six condensates. At the scale $mu = 2{rm GeV}$, we obtain: $langle xi^1rangle _{2;D_s}= -0.261^{+0.020}_{-0.020}$, $langle xi^2rangle _{2;D_s} = 0.184^{+0.012}_{-0.012}$, $langle xi^3rangle _{2;D_s} = -0.111 ^{+0.007}_{-0.012}$ and $langle xi^4rangle _{2;D_s} = 0.075^{+0.005}_{-0.005}$. Using those moments, the $phi_{2;D_s}$ is then constructed by using the light-cone harmonic oscillator model. As an application, we calculate the transition form factor $f^{B_sto D_s}_+(q^2)$ within the light-cone sum rules (LCSR) approach by using a right-handed chiral current, in which the terms involving $phi_{2;D_s}$ dominates the LCSR. It is noted that the extrapolated $f^{B_sto D_s}_+(q^2)$ agrees with the Lattice QCD prediction. After extrapolating the transition form factor to the physically allowable $q^2$-region, we calculate the branching ratio and the CKM matrix element, which give $mathcal{B}(bar B_s^0 to D_s^+ ell u_ell) = (2.03^{+0.35}_{-0.49}) times 10^{-2}$ and $|V_{cb}|=(40.00_{-4.08}^{+4.93})times 10^{-3}$.
We reassess the $Btopiell u_{ell}$ differential branching ratio distribution experimental data released by the BaBar and Belle Collaborations supplemented with all lattice calculations of the $Btopi$ form factor shape available up to date obtained by the HPQCD, FNAL/MILC and RBC/UKQCD Collaborations. Our study is based on the method of Pad{e} approximants, and includes a detailed scrutiny of each individual data set that allow us to obtain $|V_{ub}|=3.53(8)_{rm{stat}}(6)_{rm{syst}}times10^{-3}$. The semileptonic $B^{+}toeta^{(prime)}ell^{+} u_{ell}$ decays are also addressed and the $eta$-$eta^{prime}$ mixing discussed.
We discuss the general properties of the amplitude of the $Bto l^+l^-l u$ decays and calculate the related kinematical distributions $d^2Gamma/dq^2dq^2$, $q$ the momentum of the $l^+l^-$ pair emitted from the electromagnetic vertex and $q$ the momentum of the $l u$ pair emitted from the weak vertex. We emphasize that electromagnetic gauge invariance imposes essential constraints on the $Bto gamma^*l u$ amplitude at small $q^2$ which in the end yield the behaviour of the differential branching fraction as $dGamma(Bto l^+l^-l u)/dq^2propto 1/q^2$ and a mild logarithmic dependence of $Gamma(Bto l^{+}l^{-}l u)$ on the lepton mass $m_l$. Consequently, (i) the main contribution to the decay rate $Gamma(Bto mu^+mu^-e u_e )$ comes from the region of light vector resonances $rho^0$ and $omega$, $q^2simeq M_rho^2, M_omega^2$ and (ii) the decay rate $Gamma(Bto e^{+}e^{-}mu u_mu)$ receives comparable contributions from the region of small $q^2$ and from the resonance region. As the result, the decay rate $Gamma(Bto e^+e^-mu u_mu)$ is only a factor $sim 2$ larger than $Gamma(Bto mu^+mu^-e u_e)$. We perform a detailed analysis of the uncertainties in the theoretical predictions for the decays $Bto l^+l^-l u$ in the Standard Model. We found that the theoretical expectations for such decays in the Standard Model are only marginally compatible with the recent upper limits of the LHCb collaboration.
Semi-leptonic $B_s to K ell u$ and $B_s to D_s ell u$ decays provide an alternative $b$-decay channel to determine the CKM matrix elements $|V_{ub}|$ and $|V_{cb}|$ or to obtain $R$-ratios to investigate lepton flavor universality violations. In addition, these decays may shed further light on the discrepancies seen in the analysis of inclusive vs. exclusive decays. Using the nonperturbative methods of lattice QCD, theoretical results are obtained with good precision and full control over systematic uncertainties. This talk will highlight ongoing efforts of the $B$-physics program by the RBC-UKQCD collaboration.
The $Bto gamma ell u_ell$ decay at large energies of the photon receives a numerically important soft-overlap contribution which is formally of the next-to-leading order in the expansion in the inverse photon energy. We point out that this contribution can be calculated within the framework of heavy-quark expansion and soft-collinear effective theory, making use of dispersion relations and quark-hadron duality. The soft-overlap contribution is obtained in a full analogy with the similar contribution to the $gamma^* gamma to pi$ transition form factor. This result strengthens the case for using the $Bto gamma ell u_ell$ decay to constrain the $B$-meson distribution amplitude and determine its most important parameter, the inverse moment $lambda_B$.
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