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Lectures on perturbative HQET 1

A. G. Grozin

TL;DR

These lectures systematically develop perturbative HQET, deriving the static and 1/m-expanded Lagrangians, propagators, and vertices, and detailing renormalization, matching, and decoupling with QCD. A core focus is the controlled expansion in ω/m, with explicit one- and two-loop calculations in both HQET and QCD, using methods such as Feynman parameterization, integration-by-parts, and the background-field approach. Key results include the fixed kinetic-energy coefficient C_k = 1 by reparametrization invariance, the gauge-dependent yet renormalizable chromomagnetic operator with C_m(μ) and its RG evolution, and the structure of decoupling across heavy-flavor thresholds. The framework provides essential tools for precision heavy-quark phenomenology, including hyperfine splittings, lattice HQET applications, and systematic matching between full QCD and effective theories.

Abstract

Extended version of lectures given at the University of Karlsruhe and at Calc-2000 school in Dubna. Properties of HQET as a field theory, methods of calculation of HQET diagrams and some simple applications are explained in detail. These lectures can be used as an additional chapter with any modern QCD textbook.

Lectures on perturbative HQET 1

TL;DR

These lectures systematically develop perturbative HQET, deriving the static and 1/m-expanded Lagrangians, propagators, and vertices, and detailing renormalization, matching, and decoupling with QCD. A core focus is the controlled expansion in ω/m, with explicit one- and two-loop calculations in both HQET and QCD, using methods such as Feynman parameterization, integration-by-parts, and the background-field approach. Key results include the fixed kinetic-energy coefficient C_k = 1 by reparametrization invariance, the gauge-dependent yet renormalizable chromomagnetic operator with C_m(μ) and its RG evolution, and the structure of decoupling across heavy-flavor thresholds. The framework provides essential tools for precision heavy-quark phenomenology, including hyperfine splittings, lattice HQET applications, and systematic matching between full QCD and effective theories.

Abstract

Extended version of lectures given at the University of Karlsruhe and at Calc-2000 school in Dubna. Properties of HQET as a field theory, methods of calculation of HQET diagrams and some simple applications are explained in detail. These lectures can be used as an additional chapter with any modern QCD textbook.

Paper Structure

This paper contains 16 sections, 222 equations, 33 figures.

Table of Contents

  1. HQET Lagrangian
  2. One-loop massless propagator diagrams
  3. One-loop HQET propagator diagram
  4. Two-loop massless propagator diagrams
  5. Two-loop HQET propagator diagrams
  6. Renormalization of QCD
  7. Renormalization of HQET
  8. Heavy Electron Effective Theory
  9. $1/m$ corrections to HQET Lagrangian
  10. On-shell renormalization of QCD
  11. On-shell renormalization of HQET
  12. Scattering in external gluonic field in QCD
  13. Scattering in external gluonic field in HQET
  14. Chromomagnetic interaction
  15. Decoupling of heavy quark loops
  16. ...and 1 more sections

Figures (33)

  • Figure 1: One-loop massless propagator diagram
  • Figure 2: One-loop HQET propagator diagram
  • Figure 3: Two-loop massless propagator diagram
  • Figure 4: Two-loop HQET propagator diagram
  • Figure 5: Structure of diagrams for the gluon propagator
  • ...and 28 more figures