Revisiting the Annihilation Corrections in Non-leptonic ${\bar{B}}_s^0$ Decays within QCD Factorization

Abstract

Motivated by the recent measurements of non-leptonic ${\bar{B}}_s^0$ decays by CDF and LHCb collaborations, especially the large $\mathcal{B}({\bar{B}}_s^0\to \ddot{I}{€}^{+}\ddot{I}{€}^{-})$, we revisit the hard spectator and annihilation corrections in ${\bar{B}}_s^0$ decays within QCD factorization approach with two schemes for the possible parameters for the known end-point divergence appeared in the estimation of the hard spectator and annihilation diagrams. The first one is to conservatively estimate the possible contributions by parameterization (scheme I); another one uses an infrared finite gluon propagator (scheme II) to regulate the end-point singularity. In scheme I, with the constraints from the measured ${\bar{B}}_s^0\to PP(VV)$ decays, two (four) restricted solutions of the parameters spaces are found. In scheme II, we find that most of the theoretical predictions agree well with the experimental data with single parameter $m_g\sim 0.5\mathrm{GeV}$. However, within both schemes, $\mathcal{B}({\bar{B}}_s^0\to \ddot{I}•\ddot{I}•)$ are always much larger than $\mathcal{B}({\bar{B}}_s^0\to K^{\ast 0}{\bar{K}}^{\ast 0})$ in contrast to the experimental results $\mathcal{B}({\bar{B}}_s^0\to \ddot{I}•\ddot{I}•)\simeq \mathcal{B}({\bar{B}}_s^0\to K^{\ast 0}{\bar{K}}^{\ast 0})$. It is noted that the pattern $\mathcal{B}({\bar{B}}_s^0\to \ddot{I}•\ddot{I}•)>\mathcal{B}({\bar{B}}_s^0\to K^{\ast 0}{\bar{K}}^{\ast 0})$ also persists in other theoretical framework, thus the present experimental result $\mathcal{B}({\bar{B}}_s^0\to \ddot{I}•\ddot{I}•)\simeq \mathcal{B}({\bar{B}}_s^0\to K^{\ast 0}{\bar{K}}^{\ast 0})$ rises a challenge to theoretical approaches for B non-leptonic decays. Further refined measurements and theoretical studies are called for to resolve such a possible anomaly.

25pages, 6 figures; minor typos corrected; version to appear in Phys. Rev. D