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Foundation and generalization of the expansion by regions

Bernd Jantzen

TL;DR

This work lays a rigorous foundation for the expansion by regions by formulating a master identity that expresses the full asymptotic expansion as a sum over region contributions with overlap corrections. It clarifies when the traditional recipe suffices (commuting expansions with scaleless overlaps vanishing under dimensional/analytic regularization) and presents a generalized formalism to handle non-commuting regions via explicit overlap terms. The authors validate the approach with pedagogical one-loop examples (off-shell large-momentum, threshold, Sudakov, forward scattering), discuss connections to Mellin–Barnes and alpha-parameter techniques, and address convergence, completeness, and regulator dependence. Overall, the paper strengthens the theoretical underpinnings of expansion by regions and guides reliable application to multi-scale loop problems, including cases with finite boundaries or nonstandard regularization schemes.

Abstract

The "expansion by regions" is a method of asymptotic expansion developed by Beneke and Smirnov in 1997. It expands the integrand according to the scaling prescriptions of a set of regions and integrates all expanded terms over the whole integration domain. This method has been applied successfully to many complicated loop integrals, but a general proof for its correctness has still been missing. This paper shows how the expansion by regions manages to reproduce the exact result correctly in an expanded form and clarifies the conditions on the choice and completeness of the considered regions. A generalized expression for the full result is presented that involves additional overlap contributions. These extra pieces normally yield scaleless integrals which are consistently set to zero, but they may be needed depending on the choice of the regularization scheme. While the main proofs and formulae are presented in a general and concise form, a large portion of the paper is filled with simple, pedagogical one-loop examples which illustrate the peculiarities of the expansion by regions, explain its application and show how to evaluate contributions within this method.

Foundation and generalization of the expansion by regions

TL;DR

This work lays a rigorous foundation for the expansion by regions by formulating a master identity that expresses the full asymptotic expansion as a sum over region contributions with overlap corrections. It clarifies when the traditional recipe suffices (commuting expansions with scaleless overlaps vanishing under dimensional/analytic regularization) and presents a generalized formalism to handle non-commuting regions via explicit overlap terms. The authors validate the approach with pedagogical one-loop examples (off-shell large-momentum, threshold, Sudakov, forward scattering), discuss connections to Mellin–Barnes and alpha-parameter techniques, and address convergence, completeness, and regulator dependence. Overall, the paper strengthens the theoretical underpinnings of expansion by regions and guides reliable application to multi-scale loop problems, including cases with finite boundaries or nonstandard regularization schemes.

Abstract

The "expansion by regions" is a method of asymptotic expansion developed by Beneke and Smirnov in 1997. It expands the integrand according to the scaling prescriptions of a set of regions and integrates all expanded terms over the whole integration domain. This method has been applied successfully to many complicated loop integrals, but a general proof for its correctness has still been missing. This paper shows how the expansion by regions manages to reproduce the exact result correctly in an expanded form and clarifies the conditions on the choice and completeness of the considered regions. A generalized expression for the full result is presented that involves additional overlap contributions. These extra pieces normally yield scaleless integrals which are consistently set to zero, but they may be needed depending on the choice of the regularization scheme. While the main proofs and formulae are presented in a general and concise form, a large portion of the paper is filled with simple, pedagogical one-loop examples which illustrate the peculiarities of the expansion by regions, explain its application and show how to evaluate contributions within this method.

Paper Structure

This paper contains 50 sections, 276 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Example: off-shell large-momentum expansion
  3. Expansion by regions in the large-momentum limit
  4. Proof of the large-momentum expansion
  5. General formalism with commuting expansions
  6. Example: threshold expansion
  7. Formalism for non-commuting expansions
  8. Example: Sudakov form factor
  9. Example: forward scattering with small momentum exchange
  10. Regions for the forward-scattering integral
  11. Evaluation with analytic regulators
  12. Evaluation without analytic regulators
  13. Conclusions
  14. Technical details of the evaluations
  15. Large-momentum expansion
  16. ...and 35 more sections

Figures (6)

  • Figure 1: Two-point loop integral for off-shell large-momentum expansion.
  • Figure 2: Loop integral for the threshold expansion.
  • Figure 3: Vertex correction to the Sudakov form factor.
  • Figure 4: Convergence domains of the regions for the Sudakov form factor in the case $\vec{k}_\perp^2 \lesssim m^2$.
  • Figure 5: One-loop corrections to the forward-scattering amplitude with small momentum exchange $r$.
  • ...and 1 more figures