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Iterative Structure Within The Five-Particle Two-Loop Amplitude

Freddy Cachazo, Marcus Spradlin, Anastasia Volovich

TL;DR

This work reveals an unexpected iterative structure in the two-loop five-gluon amplitude of ${\cal N}=4$ super Yang–Mills theory by isolating a parity-even subset $V_5^{(L)}(\epsilon)$ that satisfies an ABDK-type relation: $V_5^{(2)}(\epsilon)=\frac{1}{2}\big(V_5^{(1)}(\epsilon)\big)^2+f^{(2)}(\epsilon)V_5^{(1)}(2\epsilon)-\frac{\pi^4}{72}+{\cal O}(\epsilon)$. The authors prove this through ${\cal O}(\epsilon^{-1})$ using Mellin-Barnes techniques and identities among MB-building blocks, and verify the ${\cal O}(\epsilon^0)$ piece numerically via finite remainder analysis. They also show that a parity-odd piece $W_5^{(2)}(\epsilon)$ is necessary for unitarity, albeit with a milder infrared divergence ${\cal O}(\epsilon^{-1})$, consistent with Catani’s IR structure. Collectively, these results point to a partially iterative organization of the full two-loop amplitude and motivate further unitarity-based checks and computations of the missing pieces, potentially illuminating deeper iterative patterns in higher-point amplitudes.

Abstract

We find an unexpected iterative structure within the two-loop five-gluon amplitude in N = 4 supersymmetric Yang-Mills theory. Specifically, we show that a subset of diagrams contributing to the full amplitude, including a two-loop pentagon-box integral with nontrivial dependence on five kinematical variables, satisfies an iterative relation in terms of one-loop scalar box diagrams. The implications of this result for the possible iterative structure of the full two-loop amplitude are discussed.

Iterative Structure Within The Five-Particle Two-Loop Amplitude

TL;DR

This work reveals an unexpected iterative structure in the two-loop five-gluon amplitude of super Yang–Mills theory by isolating a parity-even subset that satisfies an ABDK-type relation: . The authors prove this through using Mellin-Barnes techniques and identities among MB-building blocks, and verify the piece numerically via finite remainder analysis. They also show that a parity-odd piece is necessary for unitarity, albeit with a milder infrared divergence , consistent with Catani’s IR structure. Collectively, these results point to a partially iterative organization of the full two-loop amplitude and motivate further unitarity-based checks and computations of the missing pieces, potentially illuminating deeper iterative patterns in higher-point amplitudes.

Abstract

We find an unexpected iterative structure within the two-loop five-gluon amplitude in N = 4 supersymmetric Yang-Mills theory. Specifically, we show that a subset of diagrams contributing to the full amplitude, including a two-loop pentagon-box integral with nontrivial dependence on five kinematical variables, satisfies an iterative relation in terms of one-loop scalar box diagrams. The implications of this result for the possible iterative structure of the full two-loop amplitude are discussed.

Paper Structure

This paper contains 13 sections, 58 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Five-Gluon Amplitudes at One and Two Loops
  3. Proof of the Main Result
  4. The ABDK Relation for $V_5^{(L)}(\epsilon)$ Through ${\cal O}(\epsilon^{-1})$
  5. The ABDK Relation at ${\cal O}(\epsilon^0)$ and The Finite Remainder
  6. Consistency Checks on the $W_5^{(2)}(\epsilon)$ Piece of $M^{(2)}_5(\epsilon)$
  7. Double-Double Cut
  8. Infrared Behaviour of $W_5^{(2)}(\epsilon)$
  9. Conclusion and Discussion
  10. Evaluation of Integrals
  11. Results
  12. Mellin-Barnes Representations
  13. Building Blocks

Figures (2)

  • Figure 1: The one-loop one-mass scalar box integral. See appendix A for details.
  • Figure 2: Cast of characters: here we define the five two-loop integrals which appear in this paper. All figures refer to the corresponding diagrams in a scalar field theory, i.e., a collection of propagators integrated over the loop momenta. The integrals $I^{(2) d}$ and $I^{(2) e}$ are multiplied by factors which depend on one of the loop momenta, as indicated explicitly above. These factors are meant to be included in the numerators of the corresponding scalar integrals, transforming them into tensor integrals. See appendix A for further details.