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Massive amplitudes on the Coulomb branch of N=4 SYM

Nathaniel Craig, Henriette Elvang, Michael Kiermaier, Tracy Slatyer

TL;DR

This work develops a systematic framework for tree-level amplitudes with massive external states on the Coulomb branch of N=4 SYM, showing that leading small-mass Coulomb-branch amplitudes can be obtained from soft-scalar limits of massless amplitudes at the origin. It provides explicit all-n towers for massive W-boson plus gluon amplitudes, constructs Coulomb-branch superamplitudes organized into an MHV-band framework, and proves a CSW-like expansion for these massive amplitudes. The authors also establish dual conformal properties, explore 4d–6d connections, and discuss the broader implications for phenomenology and potential extensions to loops and QCD-like theories. Overall, the results indicate that much of the massless theory’s simplicity persists on the Coulomb branch and offer practical tools for computing and organizing massive amplitudes in highly symmetric settings.

Abstract

We initiate a systematic study of amplitudes with massive external particles on the Coulomb-branch of N=4 super Yang Mills theory: 1) We propose that (multi-)soft-scalar limits of massless amplitudes at the origin of moduli space can be used to determine Coulomb-branch amplitudes to leading order in the mass. This is demonstrated in numerous examples. 2) We find compact explicit expressions for several towers of tree-level amplitudes, including scattering of two massive W-bosons with any number of positive helicity gluons, valid for all values of the mass. 3) We present the general structure of superamplitudes on the Coulomb branch. For example, the n-point "MHV-band" superamplitude is proportional to a Grassmann polynomial of mixed degree 4 to 12, which is uniquely determined by supersymmetry. We find explicit tree-level superamplitudes for this MHV band and for other simple sectors of the theory. 4) Dual conformal generators are constructed, and we explore the dual conformal properties of the simplest massive amplitudes. Our compact expressions for amplitudes and superamplitudes should be of both theoretical and phenomenological interest; in particular the tree-level results carry over to truncations of the theory with less supersymmetry.

Massive amplitudes on the Coulomb branch of N=4 SYM

TL;DR

This work develops a systematic framework for tree-level amplitudes with massive external states on the Coulomb branch of N=4 SYM, showing that leading small-mass Coulomb-branch amplitudes can be obtained from soft-scalar limits of massless amplitudes at the origin. It provides explicit all-n towers for massive W-boson plus gluon amplitudes, constructs Coulomb-branch superamplitudes organized into an MHV-band framework, and proves a CSW-like expansion for these massive amplitudes. The authors also establish dual conformal properties, explore 4d–6d connections, and discuss the broader implications for phenomenology and potential extensions to loops and QCD-like theories. Overall, the results indicate that much of the massless theory’s simplicity persists on the Coulomb branch and offer practical tools for computing and organizing massive amplitudes in highly symmetric settings.

Abstract

We initiate a systematic study of amplitudes with massive external particles on the Coulomb-branch of N=4 super Yang Mills theory: 1) We propose that (multi-)soft-scalar limits of massless amplitudes at the origin of moduli space can be used to determine Coulomb-branch amplitudes to leading order in the mass. This is demonstrated in numerous examples. 2) We find compact explicit expressions for several towers of tree-level amplitudes, including scattering of two massive W-bosons with any number of positive helicity gluons, valid for all values of the mass. 3) We present the general structure of superamplitudes on the Coulomb branch. For example, the n-point "MHV-band" superamplitude is proportional to a Grassmann polynomial of mixed degree 4 to 12, which is uniquely determined by supersymmetry. We find explicit tree-level superamplitudes for this MHV band and for other simple sectors of the theory. 4) Dual conformal generators are constructed, and we explore the dual conformal properties of the simplest massive amplitudes. Our compact expressions for amplitudes and superamplitudes should be of both theoretical and phenomenological interest; in particular the tree-level results carry over to truncations of the theory with less supersymmetry.

Paper Structure

This paper contains 17 sections, 103 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Explicit amplitudes on the Coulomb branch
  3. Constructing massive amplitudes from massless ones
  4. $O(m)$ amplitudes from massless amplitudes
  5. $O(m^2)$ amplitudes from massless amplitudes
  6. General $O(m^s)$ amplitudes
  7. Coulomb-branch superamplitudes
  8. MHV superamplitudes for $n>3$
  9. Simplified superamplitudes for a smart choice of $q$-frame
  10. 3-point superamplitudes
  11. Single-soft limits and matching at the superamplitude level
  12. Multi-soft limits of N$^k$MHV superamplitudes
  13. The structure of superamplitudes with generic massive lines
  14. 4d-6d correspondence
  15. CSW expansion for all MHV$\!\times\!$N$^k$MHV superamplitudes
  16. ...and 2 more sections

Figures (1)

  • Figure 1: The structure of superamplitudes on the Coulomb branch. We can assign a separate N$^k$MHV level to the two factors of $SU(2)$ corresponding to $\eta_{i1},\eta_{i2}$ and $\eta_{i3},\eta_{i4}$. The full Coulomb-branch superamplitude can then be decomposed into N$^k$MHV$\times$ N$^{k'}$MHV superamplitudes.