The Resummed Rate for B -> X_s gamma

TL;DR

This paper analyzes threshold logarithms in the B→X_sγ decay rate with a photon-energy cut. It develops a next-to-leading-log resummation in x-space, connects it to a Mellin-space formalism, and assesses the impact of nonperturbative B-meson structure via a convolution with the structure function f(k_+). The results show that, for the current cut, resummation yields only modest corrections and the dominant uncertainties come from the heavy-quark parameters Λ̄ and λ1 and higher-order perturbative effects. Consequently, the resummation is not essential for accurate total-rate predictions at present, though it may become important if a precise extraction of the structure function is pursued. The study also highlights that two-loop contributions can be as significant as β0-driven pieces, underscoring the need for complete higher-order calculations to sharpen moment-based extractions of HQET parameters.

Abstract

In this paper we investigate the effect of the resummation of threshold logs on the rate for B -> X_s gamma. We calculate the differential rate dGamma/dE_gamma including the infinite set of terms of the form alpha_s^n log^{n+1}(1-x) and alpha_s^n log^n(1-x) in the Sudakov exponent. The resummation is potentially important since these logs turn into log(2E_{cut}/m_b), when the rate is integrated from the lower cut x=2E_{cut}/m_b to 1. The resummed rate is then convolved with models for the structure function to study whether or not the logs will be enhanced due to the fermi motion of the heavy quark. A detailed discussion of the accuracy of the calculation with and without the inclusion of the non-perturbative effects dictated by the B meson structure function is given. We also investigate the first moment with respect to (1-x), which can be used to measure \barΛand lambda_1. It is shown that there are some two loop corrections which are just as large as the alpha_s^2 beta_0 term, which are usually expected to dominate. We conclude that, for the present energy cut, the threshold logs do not form a dominant sub-series and therefore their resummation is unnecessary. Thus, the prospects for predicting the rate for B -> X_s gamma accurately, given the present energy cut, are promising.

Figures (5)

• Figure 1: The effects of resummation on the rate, integrated from $\delta$ to $1$. The solid curve is the leading log result. The sparse dotted curve is the numerically inverted next to leading log result. The tight dotted curve is the rate expanding $g_1$ to $O(\alpha_s)$. The long dashed curve is expanding $g_1$ to $O(\alpha_s^2)$, the dot-dashed curve is expanding $g_1$ to $O(\alpha_s^3)$, and the short dashed curve is expanding $g_1$ to $O(\alpha_s^4)$.
• Figure 2: The solid curve is the resummed double log rate, with the $O(\alpha_s)$ piece subtracted out. The dotted curve is the $O(\alpha_s^2)$ piece of the resummed double log rate. The dashed curve is the $O(\alpha_s^2)$ piece of the full resummed rate, including both $g_1$ and $g_2$.
• Figure 3: The rate integrated from $E_{cut}$. The solid curve is the resummed rate convoluted with the structure function in Eq. (\ref{['SF1']}), using the central values of $(\bar{\Lambda}, \lambda_1)$. The shaded region shows the uncertainty due to varying the values of $(\bar{\Lambda},\ \lambda_1)$ by one sigma. The dotted curve is the one loop rate without the structure function. The long dashed curve is the resummed rate without the structure function. The dot-dashed curve is the one loop rate convoluted with the structure function, using the central values of $(\bar{\Lambda},\ \lambda_1)$. The short dashed curve is the one loop rate and the $O(\alpha_s^2)$ piece from the resummation, convoluted with the structure function.
• Figure 4: Rate convoluted with the different structure functions. The solid curve is the rate convoluted with the Eq. (\ref{['SF2']}), and the dashed curve is the rate convoluted with Eq. (\ref{['SF2']}).
• Figure 5: Rate convoluted with the structure function Eq.(\ref{['SF1']}). The dotted curve corresponds to expanding $g_1$ to $O(\alpha_s)$, the dashed curve is for $g_1$ expanded to $O(\alpha_s^2)$, and the dot-dashed curve is expanded to $O(\alpha_s^3)$. The solid curve is the rate with the full $g_1$.