Coulomb force correction to the decay b -> c anti-c s in the threshold

Abstract

We study the physical origins of the $O(alpha_{s})$ and $O(alpha_{s}^{2})$ corrections to the $bar{c}-s$ current in the decay $b -> c bar{c} s$ in the threshold region $delta=(M_{b}-2m_{c})/2M_{b} << 1$. We obtain the corrections which are produced by the Coulomb force between the anti-charm and strange quarks. The Coulomb corrections $C_{F} pi^{2}$ at $O(alpha_{s})$ and $-C_{F}^{2} pi^{2} ln delta$ at $O(alpha_{s}^{2})$ account for 300% and 120% of the corresponding terms in the Abelian-type perturbative corrections respectively. The differences between the Coulomb and perturbative corrections imply that other corrections which have other physical origins exist. We also check that the Wilson coefficient for the anomalous dimension of the $bar{c}-s$ current reproduces the leading and the next-to-leading logarithmic terms in the perturbative corrections.