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Simple loop integrals and amplitudes in N=4 SYM

J. M. Drummond, J. M. Henn

TL;DR

The paper develops a momentum-twistor framework combined with an AdS-inspired mass regulator to regulate infrared divergences in planar N=4 SYM, introducing a basis of loop integrals with twistor numerators that are computationally simpler than conventional ones. It applies this basis to the two-loop six-point MHV amplitude, obtains analytic remainder function results in special kinematic limits, and finds exact agreement with Wilson-loop calculations, while also demonstrating that the logarithm of MHV amplitudes can be expressed as simple twistor-space integrals. A central result is the reduction of the two-loop six-point integrand to a compact set of basis integrals, with favorable infrared behavior and tractable Mellin-Barnes representations. The work also shows that a single twistor integral with a magic numerator can capture the exponentiated four-point two-loop amplitude, highlighting the utility of twistor methods for verifying soft limits and guiding integrand construction.

Abstract

We use momentum twistors to evaluate planar loop integrals. Infrared divergences are regulated by the recently proposed AdS-inspired mass regulator. We show that two-loop amplitudes in N=4 super Yang-Mills can be expanded in terms of basis integrals having twistor numerators. We argue that these integrals are considerably simpler compared to the ones conventionally used. Our case in point is the two-loop six-point MHV amplitude. We present analytical results for the remainder function in a kinematical limit, and find agreement with a recent Wilson loop computation. We also provide two-loop evidence that the logarithm of MHV amplitudes can be written in terms of simple twistor space integrals.

Simple loop integrals and amplitudes in N=4 SYM

TL;DR

The paper develops a momentum-twistor framework combined with an AdS-inspired mass regulator to regulate infrared divergences in planar N=4 SYM, introducing a basis of loop integrals with twistor numerators that are computationally simpler than conventional ones. It applies this basis to the two-loop six-point MHV amplitude, obtains analytic remainder function results in special kinematic limits, and finds exact agreement with Wilson-loop calculations, while also demonstrating that the logarithm of MHV amplitudes can be expressed as simple twistor-space integrals. A central result is the reduction of the two-loop six-point integrand to a compact set of basis integrals, with favorable infrared behavior and tractable Mellin-Barnes representations. The work also shows that a single twistor integral with a magic numerator can capture the exponentiated four-point two-loop amplitude, highlighting the utility of twistor methods for verifying soft limits and guiding integrand construction.

Abstract

We use momentum twistors to evaluate planar loop integrals. Infrared divergences are regulated by the recently proposed AdS-inspired mass regulator. We show that two-loop amplitudes in N=4 super Yang-Mills can be expanded in terms of basis integrals having twistor numerators. We argue that these integrals are considerably simpler compared to the ones conventionally used. Our case in point is the two-loop six-point MHV amplitude. We present analytical results for the remainder function in a kinematical limit, and find agreement with a recent Wilson loop computation. We also provide two-loop evidence that the logarithm of MHV amplitudes can be written in terms of simple twistor space integrals.

Paper Structure

This paper contains 15 sections, 80 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Lightning review of momentum twistor space and massive loop integrals
  3. New representations of loop integrands
  4. Two-loop amplitudes in ${\cal N} = 4$ SYM
  5. Structure of the loop corrections for MHV amplitudes
  6. Evaluation of two-loop integrals and amplitudes
  7. Twistor space integral for exponentiated four-point two-loop amplitude
  8. Consistency of the loop integrand with soft limits
  9. Dual conformal symmetry of leading singularities
  10. Two-loop integrals
  11. Twistor identity to relate numerator factors
  12. Definition of the two-loop integrals
  13. Results for generic kinematics
  14. Analytic results in the limit $u\to 0$
  15. Box integrals for one-loop MHV amplitudes

Figures (4)

  • Figure 1: Pentagon integral (\ref{['twistorpentagon']}) with twistor numerator $({13AB})$. The figure on the right shows the same integral in the more familiar dual space notation, with the dashed line denoting a numerator connecting to the complex point $x_{a}$, see (\ref{['dualxpentagon']}).
  • Figure 2: Representative integrals appearing in two-loop MHV amplitudes. In the pictures the specific choice $n=6$ was made. The dashed lines stand for numerator factors that depend on the loop momentum.
  • Figure 3: Example of identity (\ref{['expansion']}) when used to expand $I_{6;2;9}$ in terms of our preferred basis. We do note display the (loop-momentum independent) prefactors and normalisations of the integrals.
  • Figure 4: Integrals appearing in the improved representation of the six-point two-loop MHV amplitude, c.f. (\ref{['MHV2loopimproved']}). The dashed lines indicate non-local twistor numerators, and a normalisation factor is not shown in the figure. Both are given explicitly in the Appendix.