Subleading Jet Functions in Inclusive B Decays

TL;DR

The paper develops a systematic two-step matching to extract subleading jet functions that enter the endpoint region factorization of inclusive B decays. It demonstrates the existence of these functions at ${\cal O}(\alpha_s/m_b)$ by separating soft and hard-collinear regions, with soft contributions accounted by known subleading shape functions and hard-collinear contributions arising from time-ordered products of subleading SCET currents. Eight subleading jet functions are defined and computed at one loop, with renormalization performed in the $\overline{DR}$ scheme; explicit results are provided for $B\to X_u\,l\,\bar{\nu}$ and the $Q_{7\gamma}-Q_{7\gamma}$ piece of $B\to X_s\,\gamma$. Together with recent two-loop results for the leading hard function and jet function, this work improves the theoretical description of inclusive B decays beyond leading power and offers insights applicable to other high-energy processes such as deep inelastic scattering near $x\to 1$.

Abstract

The contribution of subleading jet functions to inclusive decay distributions of $B$ mesons are derived from a systematic two-step matching of QCD current correlators onto soft collinear and heavy quark effective theory. Focusing on the tree level matching of QCD onto soft collinear effective theory, the subleading jet functions are defined to all orders in $α_s(μ_i)$ (with $μ_i^2\sim m_bΛ_{\rm QCD}$) and are calculated explicitly at first order in $α_s(μ_i)$. We present explicit expressions for the decay rates of $B\to X_u l \barν$ and the $Q_{7γ}-Q_{7γ}$ contribution to $B\to X_s γ$, where the subleading jet functions are multiplied by a tree level hard function and appear in a convolution with the leading order shape function. Together with the recent two loop calculation of the leading order hard function for $B\to X_u l \barν$, this paper will allow for a more precise description of inclusive B decays in the end point region.

Figures (2)

• Figure 1: One loop diagrams contributing to the hadronic tensor, top left: "Self energy" diagram, top right: "Box" diagram, bottom line: two "Vertex" diagrams. The letter next to each solid line denotes the flavor of the quark.
• Figure 2: One loop diagrams contributing to the parton level expression of $u(\omega)$.