A Comment on the Extractions of V_{ub} from Radiative Decays

Abstract

We present a model independent closed form expression for |V_{ub}|^2/|V_{tb} V_{ts}^*|^2, which includes the resummation of large endpoint logarithms as well as the interference effects from the operators $O_2$ and $O_8$. We demonstrate that the method to extract |V_{ub}| presented by the authors in hep-ph/9909404, and modified in this letter to include interference effects, is not just a refinement of the method introduced in hep-ph/9312311. We also discuss the model dependence of the latter proposal. Furthermore, we show that the resummation is not negligible and that the Landau pole does not introduce any significant uncertainties.

Figures (4)

• Figure 1: $K_{\rm pert}$ as a function of the non-perturbative parameter $r$ in the range $0.02 < r < 0.2$.
• Figure 2: The slope (solid line) and $x$-axis intercept (dotted line) of the weight function as a function of $\rho$ for $x_B^c=0.87$ and $\alpha_s=0.21$.
• Figure 3: Weight function obtained by expanding the argument of $K$ in Eq. (\ref{['weight']}) to different powers of $\alpha_s$, using $x_B^c=0.87$ and $\alpha_s=0.21$. The dot-dashed line is expanding to order $\alpha_s$, the dashed line to order $\alpha_s^3$ and the dotted line to $\alpha_s^5$. The weight function is quickly converging to the solid line, which is the weight function from Eq. (\ref{['vubweight']}) using $\rho=0.9987$.
• Figure 4: The slope of the weight function as a function of the cut showing the effects of resummation. The dotted line is the slope without resumming the Sudakov logarithms.