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Calculating three-loop diagrams in Heavy Quark Effective Theory with integration-by-parts recurrence relations

A. G. Grozin

TL;DR

The paper presents an algorithm for calculating three-loop HQET propagator diagrams via integration-by-parts recurrence relations, implemented in Grinder (REDUCE) and re-implemented in AxiomJS. It systematically builds on lower-loop massless and HQET results to develop a comprehensive set of topologies and basis integrals, including several nontrivial boundary cases and new master-like integrals. The contributions include explicit recurrence relations, boundary conditions, and detailed topology-by-topology reduction strategies, enabling automated three-loop HQET computations and applications to HQET anomalous dimensions and fB determinations. The work provides a practical toolset for high-precision HQET calculations, with potential impact on lattice QCD and QCD sum-rule analyses.

Abstract

An algorithm for calculation of three-loop propagator diagrams in HQET, based on integration-by-parts recurrence relations, is constructed and implemented as a REDUCE package Grinder, and in Axiom.

Calculating three-loop diagrams in Heavy Quark Effective Theory with integration-by-parts recurrence relations

TL;DR

The paper presents an algorithm for calculating three-loop HQET propagator diagrams via integration-by-parts recurrence relations, implemented in Grinder (REDUCE) and re-implemented in AxiomJS. It systematically builds on lower-loop massless and HQET results to develop a comprehensive set of topologies and basis integrals, including several nontrivial boundary cases and new master-like integrals. The contributions include explicit recurrence relations, boundary conditions, and detailed topology-by-topology reduction strategies, enabling automated three-loop HQET computations and applications to HQET anomalous dimensions and fB determinations. The work provides a practical toolset for high-precision HQET calculations, with potential impact on lattice QCD and QCD sum-rule analyses.

Abstract

An algorithm for calculation of three-loop propagator diagrams in HQET, based on integration-by-parts recurrence relations, is constructed and implemented as a REDUCE package Grinder, and in Axiom.

Paper Structure

This paper contains 13 sections, 94 equations.

Table of Contents

  1. Introduction
  2. Diagrams with lower-loop propagator insertions
  3. Two-loop massless propagator diagrams
  4. Two-loop HQET propagator diagrams
  5. Three-loop HQET diagrams with lower-loop insertions
  6. Proper three-loop HQET propagator diagrams
  7. Diagram with two heavy-quark lines
  8. Mercedez with three heavy-quark lines
  9. Ladder with three heavy-quark lines
  10. Non-planar diagram with three heavy-quark lines
  11. Diagrams with four heavy-quark lines
  12. Diagrams with five heavy-quark lines
  13. Implementation and testing