Renormalized sum rules for structure functions of heavy mesons decays

TL;DR

The paper develops a renormalized framework for structure functions in inclusive B → X_c decays by representing them as Fourier-transformed Wilson-line matrix elements in two limiting c-quark mass regimes (heavy-heavy and heavy-light). It derives the distinct renormalization properties: multiplicative renormalization with cusp anomalous dimension for heavy-heavy transitions and GLAP-type evolution for heavy-light transitions, and introduces soft exponential cutoffs to generalize Bjorken, Voloshin, and third sum rules. Nonperturbative effects are analyzed through infrared renormalons, revealing cancellations with heavy-quark mass renormalization and leading μ_π^2-driven corrections, which motivate a Gaussian nonperturbative ansatz. The result is a coherent, renormalized description of heavy-meson structure functions that unifies perturbative and nonperturbative aspects with explicit evolution and sum-rule relations.

Abstract

We consider properties of the structure functions of inclusive heavy meson decays $B\to X_c$ and treat the $c$ quark mass as a free parameter. We show that in two extreme cases of heavy and light $c$ quark the structure functions of heavy--heavy and heavy--light transitions are given by a Fourier transform of the matrix elements of Wilson lines containing a time--like and a light--like segment, correspondingly. Using the renormalization properties of Wilson lines we find the dependence of the structure functions on the factorization scale, the structure function of heavy--heavy transition is renormalized multiplicatively while that of heavy-light transition obeys the GLAP--type evolution equation. We propose a generalization of the sum rules for the moments of the structure functions (Bjorken, Voloshin, and the ``third'' sum rules) with a soft exponential factorization cut--off, which correctly incorporates both perturbative and nonperturbative effects. We analyze nonperturbative corrections by first considering infrared renormalon contributions to the Wilson lines. Uncertainties induced by the leading renormalon pole at $u=\frac12$ are exactly cancelled by the similar uncertainty in the heavy quark pole mass. The leading nonperturbative corrections associated with the next renormalon at $u=1$ are parameterized by the matrix element $μ_π^2$ which is proportional to the heavy quark kinetic energy.