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Dimensionally regulated one-loop box scalar integrals with massless internal lines

G. Duplancic, B. Nizic

TL;DR

This work recalculates IR-divergent one-loop scalar box integrals with massless internal lines for all external-mass configurations (0–3 external masses) using Feynman parameterization and dimensional regularization while preserving the causal $i\epsilon$ prescription. Each integral $I_4^K$ is expressed in a compact form as $I_4^K = \kappa\,g^K\,(P^K+Q^K)$ and expanded in ${\varepsilon}_{IR}$ to extract double and single poles plus finite parts, with explicit closed-form results for the $P^K$ and $Q^K$ components. The authors compare to and align with the literature in the Euclidean region (Bern) and address analytic continuation to the physical region, clarifying issues with certain dilogarithm terms. The results extend the applicability of one-loop box integrals to arbitrary kinematics, aiding precise calculations of N-point amplitudes and QCD subdiagrams in perturbation theory.

Abstract

Using the Feynman parameter method, we have calculated in an elegant manner a set of one$-$loop box scalar integrals with massless internal lines, but containing 0, 1, 2, or 3 external massive lines. To treat IR divergences (both soft and collinear), the dimensional regularization method has been employed. The results for these integrals, which appear in the process of evaluating one$-$loop $(N\ge 5)-$point integrals and in subdiagrams in QCD loop calculations, have been obtained for arbitrary values of the relevant kinematic variables and presented in a simple and compact form.

Dimensionally regulated one-loop box scalar integrals with massless internal lines

TL;DR

This work recalculates IR-divergent one-loop scalar box integrals with massless internal lines for all external-mass configurations (0–3 external masses) using Feynman parameterization and dimensional regularization while preserving the causal prescription. Each integral is expressed in a compact form as and expanded in to extract double and single poles plus finite parts, with explicit closed-form results for the and components. The authors compare to and align with the literature in the Euclidean region (Bern) and address analytic continuation to the physical region, clarifying issues with certain dilogarithm terms. The results extend the applicability of one-loop box integrals to arbitrary kinematics, aiding precise calculations of N-point amplitudes and QCD subdiagrams in perturbation theory.

Abstract

Using the Feynman parameter method, we have calculated in an elegant manner a set of oneloop box scalar integrals with massless internal lines, but containing 0, 1, 2, or 3 external massive lines. To treat IR divergences (both soft and collinear), the dimensional regularization method has been employed. The results for these integrals, which appear in the process of evaluating oneloop point integrals and in subdiagrams in QCD loop calculations, have been obtained for arbitrary values of the relevant kinematic variables and presented in a simple and compact form.

Paper Structure

This paper contains 7 sections, 120 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Calculation and results
  4. Calculation of the integrals $P^{3m}$, $P^{2mh}$, and $P^{2me}$
  5. Calculation of the integral $Q^{3m}$
  6. Results
  7. Conclusion

Figures (2)

  • Figure 1: Basic one-loop box diagram.
  • Figure 2: One-loop box diagrams with massless internal lines but containing 0, 1, 2, 3, and 4 external massive lines; a) the four$-$mass box diagram; b) the three$-$mass box diagram; c) the two$-$mass box diagram with external masses at adjacent corners (legs 3 and 4); d) the two$-$mass box diagram with external masses at diagonally opposite corners (legs 2 and 4); e) the one$-$mass box diagram; f) the massless box diagram. Thick lines designate the massive external lines.