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High Power n of m_b in Beauty Widths and n=5 -> oo Limit

I. Bigi, M. Shifman, N. Uraltsev, A. Vainshtein

TL;DR

The paper introduces a large-$n$ expansion by treating the heavy-quark mass power $n$ in inclusive semileptonic widths as a free parameter and studying the $n\to\infty$ limit. This approach reveals a lower, short-distance scale $\mu\sim \Delta/n$ at which quark masses should be renormalized, enabling automatic resummation of leading $n$-enhanced perturbative corrections via low-scale Euclidean masses and clarifying the extended SV limit for $b\to c$ decays. It also defines a consistent low-scale heavy-quark mass $m_Q(\mu)$ and renormalized operator matrix elements, integrating perturbative and nonperturbative effects within the OPE framework. The results imply that inclusive widths are governed by short-distance dynamics up to ${\cal O}(1/m_Q^2)$, enabling more reliable extractions of CKM parameters $|V_{cb}|$ and $|V_{ub}|$, with controlled theoretical uncertainties and practical prescriptions for mass definitions and perturbative resummation.

Abstract

The leading term in the semileptonic width of heavy flavor hadrons depends on the fifth power of the heavy quark mass. We present an analysis where this power can be self-consistently treated as a free parameter n and the width can be studied in the limit n -> oo. The resulting expansion elucidates why the small velocity (SV) treatment is relevant for the inclusive semileptonic b->c transition. The extended SV limit (ESV limit) is introduced. The leading terms in the perturbative alpha_s expansion enhanced by powers of n are automatically resummed by using the low-scale Euclidean mass. The large-n treatment explains why the scales of order m_b/n are appropriate. On the other hand, the scale cannot be too small since the factorially divergent perturbative corrections associated with running of alpha_s show up. Both requirements are met if we use the short-distance mass normalized at a scale around m_b/n \sim 1 GeV. A convenient definition of such low-scale OPE-compatible masses is briefly discussed.

High Power n of m_b in Beauty Widths and n=5 -> oo Limit

TL;DR

The paper introduces a large- expansion by treating the heavy-quark mass power in inclusive semileptonic widths as a free parameter and studying the limit. This approach reveals a lower, short-distance scale at which quark masses should be renormalized, enabling automatic resummation of leading -enhanced perturbative corrections via low-scale Euclidean masses and clarifying the extended SV limit for decays. It also defines a consistent low-scale heavy-quark mass and renormalized operator matrix elements, integrating perturbative and nonperturbative effects within the OPE framework. The results imply that inclusive widths are governed by short-distance dynamics up to , enabling more reliable extractions of CKM parameters and , with controlled theoretical uncertainties and practical prescriptions for mass definitions and perturbative resummation.

Abstract

The leading term in the semileptonic width of heavy flavor hadrons depends on the fifth power of the heavy quark mass. We present an analysis where this power can be self-consistently treated as a free parameter n and the width can be studied in the limit n -> oo. The resulting expansion elucidates why the small velocity (SV) treatment is relevant for the inclusive semileptonic b->c transition. The extended SV limit (ESV limit) is introduced. The leading terms in the perturbative alpha_s expansion enhanced by powers of n are automatically resummed by using the low-scale Euclidean mass. The large-n treatment explains why the scales of order m_b/n are appropriate. On the other hand, the scale cannot be too small since the factorially divergent perturbative corrections associated with running of alpha_s show up. Both requirements are met if we use the short-distance mass normalized at a scale around m_b/n \sim 1 GeV. A convenient definition of such low-scale OPE-compatible masses is briefly discussed.

Paper Structure

This paper contains 11 sections, 79 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. The theoretical framework: large-$n$ expansion in inclusive semileptonic decays
  3. Cancelation of the infrared contribution
  4. The large $n$ limit
  5. Applications
  6. The extended SV limit
  7. Resummation of large perturbative corrections
  8. Power corrections and duality in inclusive widths
  9. Low-scale heavy quark mass; renormalized effective operators
  10. Determination of $|V_{cb}|\;$ ($|V_{ub}|$) from the inclusive widths
  11. Conclusions

Figures (1)

  • Figure 1: The decay distribution over invariant mass of the lepton pair and over energy release $E_{\rm r}=m_b-m_c-\left(q^2\right)^{1/2}$ in $b\rightarrow c\,\ell\nu$ at $m_c/m_b=0.3$.