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Radiative and semi-leptonic B-meson decay spectra: Sudakov resummation beyond logarithmic accuracy and the pole mass

Einan Gardi

TL;DR

The paper addresses inclusive B-meson decay spectra near the endpoint \(x\to 1\), where fixed-order perturbation theory fails due to Sudakov logs and infrared renormalons. It develops a Dressed Gluon Exponentiation (DGE) framework that resums running-coupling effects, factorizes the endpoint spectrum into hard, jet, and soft functions, and treats non-perturbative corrections at power accuracy. A central result is the explicit cancellation of the leading infrared renormalon with the pole-mass ambiguity, identifying the dominant non-perturbative shift as the meson–quark mass difference, \(M-m_{PV}\), which translates into a shift of the perturbative spectrum and a universal shape-function description for the soft dynamics. The approach applies to both radiative and semi-leptonic decays, establishing the universality of the soft and jet functions and providing a consistent, renormalon-aware path to incorporating HQET-based non-perturbative information with practical phenomenological impact.

Abstract

The inclusive spectra of radiative and semi-leptonic B-meson decays near the endpoint is computed taking into account renormalons in the Sudakov exponent (Dressed Gluon Exponentiation). In this framework we demonstrate the factorization of decay spectra into hard, jet and soft functions and discuss the universality of the latter two. Going beyond perturbation theory the soft function, which we identify as the longitudinal momentum distribution in an on-shell b quark, is replaced by the b-quark distribution in the B meson. The two differ by power corrections. We show how the resummation of running-coupling effects can be used to perform consistent separation to power accuracy between perturbative and non-perturbative contributions. In particular, we prove that the leading infrared renormalon ambiguity in the Sudakov exponent cancels against the one associated with the definition of the pole mass. This cancellation allows us to identify the non-perturbative parameter that controls the shift of the perturbative spectrum in the heavy-quark limit as the mass difference between the meson and the quark.

Radiative and semi-leptonic B-meson decay spectra: Sudakov resummation beyond logarithmic accuracy and the pole mass

TL;DR

The paper addresses inclusive B-meson decay spectra near the endpoint , where fixed-order perturbation theory fails due to Sudakov logs and infrared renormalons. It develops a Dressed Gluon Exponentiation (DGE) framework that resums running-coupling effects, factorizes the endpoint spectrum into hard, jet, and soft functions, and treats non-perturbative corrections at power accuracy. A central result is the explicit cancellation of the leading infrared renormalon with the pole-mass ambiguity, identifying the dominant non-perturbative shift as the meson–quark mass difference, , which translates into a shift of the perturbative spectrum and a universal shape-function description for the soft dynamics. The approach applies to both radiative and semi-leptonic decays, establishing the universality of the soft and jet functions and providing a consistent, renormalon-aware path to incorporating HQET-based non-perturbative information with practical phenomenological impact.

Abstract

The inclusive spectra of radiative and semi-leptonic B-meson decays near the endpoint is computed taking into account renormalons in the Sudakov exponent (Dressed Gluon Exponentiation). In this framework we demonstrate the factorization of decay spectra into hard, jet and soft functions and discuss the universality of the latter two. Going beyond perturbation theory the soft function, which we identify as the longitudinal momentum distribution in an on-shell b quark, is replaced by the b-quark distribution in the B meson. The two differ by power corrections. We show how the resummation of running-coupling effects can be used to perform consistent separation to power accuracy between perturbative and non-perturbative contributions. In particular, we prove that the leading infrared renormalon ambiguity in the Sudakov exponent cancels against the one associated with the definition of the pole mass. This cancellation allows us to identify the non-perturbative parameter that controls the shift of the perturbative spectrum in the heavy-quark limit as the mass difference between the meson and the quark.

Paper Structure

This paper contains 20 sections, 94 equations, 2 figures.

Table of Contents

  1. Introduction
  2. $\bar{B}\longrightarrow X_s\gamma$ decay
  3. Perturbation theory
  4. A renormalon calculation
  5. Borel representation of the Sudakov exponent
  6. Factorization into soft, jet and hard subprocesses
  7. Matching to fixed logarithmic accuracy and fixed--order calculations
  8. Principal value vs. fixed logarithmic accuracy
  9. Power corrections
  10. The shape function
  11. HQET and the OPE
  12. Cancellation of the $u=1/2$ renormalon
  13. Semi-leptonic decay
  14. Conclusions
  15. Extracting singular terms in $\bar{B}\longrightarrow X_s\gamma$
  16. ...and 5 more sections

Figures (2)

  • Figure 1: The three one--loop diagrams contributing to the squared decay matrix element of a heavy quark (thick line) into a light quark (horizontal, thin line).
  • Figure 2: The (normalized) moments $M_N$ of the photon spectrum as a function of $N$, as computed through Eq. (\ref{['BXg_logs_N_2loop']}), where $C_N (\alpha_s(m^2))$ is given by Eq. (\ref{['matching']}). The full line is the principal--value Borel sum. Dashed, dotdash and dotted lines correspond to truncation of the Sudakov exponent at increasing logarithmic accuracy -- the uppermost line is LL, then NLL, etc.