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The box integrals in momentum-twistor geometry

Andrew Hodges

TL;DR

This work reframes the massless one-loop box integrals within momentum-twistor geometry, using a compactified twistor-contour representation to recast Feynman parametrization in CP^5. By introducing a μ^2 regulator and generalizing to independent μ_i, the paper derives explicit, finite expressions for the box integrals across degenerate kinematics (3-mass to 0-mass and 2-mass-hard), expressed in terms of logs and dilogarithms of conformal cross-ratios. The approach reveals the twistor-geometric origin of the amplitudes, exposes conformal invariants, and connects with leading singularities and period contours, while aligning with traditional dimensional-regularization results. The results are positioned within a broader program linking momentum twistors, transversals, and Grassmannian methods in gauge theory amplitudes, with historical and complementary perspectives provided by related works.

Abstract

An account is given of how the 'box integrals', as used for one-loop calculations in massless field theory, appear in momentum-twistor geometry. Particular attention is paid to the role of compact contour integration in representing the Feynman propagator in twistor space. An explicit calculation of all the box integrals, using only elementary methods, is included.

The box integrals in momentum-twistor geometry

TL;DR

This work reframes the massless one-loop box integrals within momentum-twistor geometry, using a compactified twistor-contour representation to recast Feynman parametrization in CP^5. By introducing a μ^2 regulator and generalizing to independent μ_i, the paper derives explicit, finite expressions for the box integrals across degenerate kinematics (3-mass to 0-mass and 2-mass-hard), expressed in terms of logs and dilogarithms of conformal cross-ratios. The approach reveals the twistor-geometric origin of the amplitudes, exposes conformal invariants, and connects with leading singularities and period contours, while aligning with traditional dimensional-regularization results. The results are positioned within a broader program linking momentum twistors, transversals, and Grassmannian methods in gauge theory amplitudes, with historical and complementary perspectives provided by related works.

Abstract

An account is given of how the 'box integrals', as used for one-loop calculations in massless field theory, appear in momentum-twistor geometry. Particular attention is paid to the role of compact contour integration in representing the Feynman propagator in twistor space. An explicit calculation of all the box integrals, using only elementary methods, is included.

Paper Structure

This paper contains 18 sections, 104 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The box integrals
  3. Wick rotation and contours in twistor space
  4. Generalised Feynman parameters
  5. Evaluation of the 4-mass case
  6. Momentum-twistor parameters and transversals
  7. The box integrals in the degenerate cases
  8. 3-mass case
  9. 2-mass-easy case
  10. 0-mass case
  11. 1-mass case
  12. 2-mass-hard case
  13. Generalization to independent $\mu_i$
  14. Summary
  15. Historical note
  16. ...and 3 more sections

Figures (2)

  • Figure 1: 4-mass transversals
  • Figure 2: 3-mass transversals