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One-Loop Gauge Theory Amplitudes in N=4 Super Yang-Mills from MHV Vertices

Andreas Brandhuber, Bill Spence, Gabriele Travaglini

TL;DR

The paper develops a twistor-inspired, MHV-vertex formalism to compute one-loop amplitudes in ${ m N}=4$ SYM, using the CSW off-shell prescription to assemble loop diagrams from MHV trees. It shows that the resulting amplitudes can be written as dispersion integrals and evaluates them exactly to reproduce the standard BD DK box-function expressions, providing a simplified four-dilogarithm representation for the two-mass easy box function. The approach leverages unitarity concepts via cuts, Nair superspace for ${ m N}=4$ states, and a careful off-shell momentum bookkeeping, linking twistor ideas with conventional field-theory results. The findings support the viability of MHV-vertex methods at loop level and point toward broader applications to NMHV and higher-loop amplitudes, with potential insights into the twistor-space formulation of ${ m N}=4$ SYM.

Abstract

We propose a new, twistor string theory inspired formalism to calculate loop amplitudes in N=4 super Yang-Mills theory. In this approach, maximal helicity violating (MHV) tree amplitudes of N=4 super Yang-Mills are used as vertices, using an off-shell prescription introduced by Cachazo, Svrcek and Witten, and combined into effective diagrams that incorporate large numbers of conventional Feynman diagrams. As an example, we apply this formalism to the particular class of MHV one-loop scattering amplitudes with an arbitrary number of external legs in N=4 super Yang-Mills. Remarkably, our approach naturally leads to a representation of the amplitudes as dispersion integrals, which we evaluate exactly. This yields a new, simplified form for the MHV amplitudes, which is equivalent to the expressions obtained previously by Bern, Dixon, Dunbar and Kosower using the cut-constructibility approach.

One-Loop Gauge Theory Amplitudes in N=4 Super Yang-Mills from MHV Vertices

TL;DR

The paper develops a twistor-inspired, MHV-vertex formalism to compute one-loop amplitudes in SYM, using the CSW off-shell prescription to assemble loop diagrams from MHV trees. It shows that the resulting amplitudes can be written as dispersion integrals and evaluates them exactly to reproduce the standard BD DK box-function expressions, providing a simplified four-dilogarithm representation for the two-mass easy box function. The approach leverages unitarity concepts via cuts, Nair superspace for states, and a careful off-shell momentum bookkeeping, linking twistor ideas with conventional field-theory results. The findings support the viability of MHV-vertex methods at loop level and point toward broader applications to NMHV and higher-loop amplitudes, with potential insights into the twistor-space formulation of SYM.

Abstract

We propose a new, twistor string theory inspired formalism to calculate loop amplitudes in N=4 super Yang-Mills theory. In this approach, maximal helicity violating (MHV) tree amplitudes of N=4 super Yang-Mills are used as vertices, using an off-shell prescription introduced by Cachazo, Svrcek and Witten, and combined into effective diagrams that incorporate large numbers of conventional Feynman diagrams. As an example, we apply this formalism to the particular class of MHV one-loop scattering amplitudes with an arbitrary number of external legs in N=4 super Yang-Mills. Remarkably, our approach naturally leads to a representation of the amplitudes as dispersion integrals, which we evaluate exactly. This yields a new, simplified form for the MHV amplitudes, which is equivalent to the expressions obtained previously by Bern, Dixon, Dunbar and Kosower using the cut-constructibility approach.

Paper Structure

This paper contains 7 sections, 108 equations, 3 figures.

Table of Contents

  1. Introduction
  2. One-loop MHV amplitudes from unitarity
  3. Off-shell derivation of the one-loop amplitudes
  4. Supersymmetric MHV amplitudes at one loop from MHV vertices
  5. The $n$-particle scattering amplitude at one loop
  6. Vanishing one-loop amplitudes
  7. Conclusions

Figures (3)

  • Figure 1: One-loop MHV Feynman diagram, computed in \ref{['mhv']} using MHV amplitudes as interaction vertices, with the CSW off-shell prescription. The scattering amplitudes with the desired helicities for the external particles are then obtained by expanding the supersymmetric scattering amplitude in powers of the ${\cal N}=4$ coordinates $\eta^{i}$.
  • Figure 2: The possible cuts of a box diagram, corresponding to the function $F$ defined in \ref{['boxcsw22']}. In the notation of that equation, $i$ and $j$ correspond to the momenta $p$ and $q$, respectively; and $P:= p_{i + 1} + \cdots + p_{j -1}$, $Q:= p_{j+1} + \cdots + p_{i-1}$ correspond to the two groups of momenta on the upper left and lower right corner, respectively. The vertical (horizontal) cuts correspond to the $s$-channel ($t$-channel) cuts respectively, and the upper left (lower right) corner cuts to the $P^2$-channel ($Q^2$-channel) cuts respectively.
  • Figure 3: One of the MHV diagrams for the $\langle \, -\, -\, -\, -\, -\, \rangle$ process. The tree-level diagrams on both side of the cuts are of the type $- \cdots -\, +$, and hence vanish.