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Uplifting Amplitudes in Special Kinematics

Timothy Goddard, Paul Heslop, Valentin V. Khoze

TL;DR

The paper develops a universal uplift framework for planar N=4 SYM amplitudes in special 2d external kinematics, expressing the n-point remainder via a finite basis of collinearly vanishing blocks Sm that are dual-conformally invariant. By a detailed analysis of multi-collinear limits, it proves a general uplift formula that constructs all higher-point MHV amplitudes (and extends to N^kMHV) from lower-point building blocks, valid at any loop order. The construction explains how higher-point collinear-vanishing contributions are assembled from S8, S10, S12, ..., S4ell, and demonstrates consistency with known 2-loop and 3-loop results at 8 and 10 points, as well as detailed special-case checks (including the 10-point V10 piece). It also treats tree-level NMHV amplitudes in the same formalism and provides a clear pathway to extend the method to higher points and supersymmetric sectors. The work advances the understanding of the integrable structure of amplitudes in reduced kinematics and offers concrete uplift rules for analytic computations.

Abstract

We consider scattering amplitudes in planar N = 4 supersymmetric Yang-Mills theory in special kinematics where all external four-dimensional momenta are restricted to a (1+1)-dimensional subspace. The amplitudes are known to satisfy non-trivial factorisation properties arising from multi-collinear limits, which we further study here. We are able to find a general solution to these multi-collinear limits. This results in a simple formula which represents an n-point superamplitude in terms of a linear combination of functions S_m which are constrained to vanish in all appropriate multi-collinear limits. These collinear-vanishing building blocks, S_m, are dual-conformally-invariant functions which depend on the reduced m-point kinematics with 8 \leq m \leq 4l. For MHV amplitudes they can be constructed directly using, for example, the approach in Ref. [1]. This procedure provides a universal uplift of lower-point collinearly vanishing building blocks S_m to all higher-point amplitudes. It works at any loop-level l \geq 1 and for any MHV or N^kMHV amplitude. We compare this with explicit examples involving n-point MHV amplitudes at 2-loops and 10-point MHV amplitudes at 3-loops. Tree-level superamplitudes have different properties and are treated separately from loop-level amplitudes in our approach. To illustrate this we derive an expression for n-point tree-level NMHV amplitudes in special kinematics.

Uplifting Amplitudes in Special Kinematics

TL;DR

The paper develops a universal uplift framework for planar N=4 SYM amplitudes in special 2d external kinematics, expressing the n-point remainder via a finite basis of collinearly vanishing blocks Sm that are dual-conformally invariant. By a detailed analysis of multi-collinear limits, it proves a general uplift formula that constructs all higher-point MHV amplitudes (and extends to N^kMHV) from lower-point building blocks, valid at any loop order. The construction explains how higher-point collinear-vanishing contributions are assembled from S8, S10, S12, ..., S4ell, and demonstrates consistency with known 2-loop and 3-loop results at 8 and 10 points, as well as detailed special-case checks (including the 10-point V10 piece). It also treats tree-level NMHV amplitudes in the same formalism and provides a clear pathway to extend the method to higher points and supersymmetric sectors. The work advances the understanding of the integrable structure of amplitudes in reduced kinematics and offers concrete uplift rules for analytic computations.

Abstract

We consider scattering amplitudes in planar N = 4 supersymmetric Yang-Mills theory in special kinematics where all external four-dimensional momenta are restricted to a (1+1)-dimensional subspace. The amplitudes are known to satisfy non-trivial factorisation properties arising from multi-collinear limits, which we further study here. We are able to find a general solution to these multi-collinear limits. This results in a simple formula which represents an n-point superamplitude in terms of a linear combination of functions S_m which are constrained to vanish in all appropriate multi-collinear limits. These collinear-vanishing building blocks, S_m, are dual-conformally-invariant functions which depend on the reduced m-point kinematics with 8 \leq m \leq 4l. For MHV amplitudes they can be constructed directly using, for example, the approach in Ref. [1]. This procedure provides a universal uplift of lower-point collinearly vanishing building blocks S_m to all higher-point amplitudes. It works at any loop-level l \geq 1 and for any MHV or N^kMHV amplitude. We compare this with explicit examples involving n-point MHV amplitudes at 2-loops and 10-point MHV amplitudes at 3-loops. Tree-level superamplitudes have different properties and are treated separately from loop-level amplitudes in our approach. To illustrate this we derive an expression for n-point tree-level NMHV amplitudes in special kinematics.

Paper Structure

This paper contains 21 sections, 135 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Kinematics and collinear limits
  3. Multi-collinear limits
  4. Two-dimensional kinematics
  5. Collinear limits in the 2d kinematics
  6. $n$-point MHV amplitudes: Part 1
  7. n-points at 2 loops
  8. 8- and 10-point MHV amplitudes at 3 loops
  9. $8-$ and $10-$points recast
  10. $n$-point MHV amplitudes: Part 2
  11. The general formula for the $n$-point collinear uplift
  12. $\tilde{{\cal R}}_n$ has the correct collinear limits
  13. Discussion of the result
  14. Special cases
  15. $S_{10}$ contribution to $V_{10}$
  16. ...and 6 more sections

Figures (2)

  • Figure 1: Figure illustrating the zig-zag Wilson loop contour in 2d kinematics. Vertices $x_i$ are defined in terms of light-cone co-ordinates. In 2d the contour can also be specified by giving every other vertex $x_2, x_4, x_6, \dots$.
  • Figure 2: Figure illustrating the triple/soft collinear limit $z_n \to z_{n-2}$ while the variable $z_{n-1}$ remains free.