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SUSY Ward identities, Superamplitudes, and Counterterms

Henriette Elvang, Daniel Z. Freedman, Michael Kiermaier

TL;DR

This review explains how SUSY and R-symmetry Ward identities constrain on-shell scattering in N=4 SYM and N=8 supergravity, and shows how to solve these identities to express any N^KMHV superamplitude in terms of a minimal, representation-theoretically labeled basis. The basis elements are organized by semi-standard Young tableaux of rectangular diagrams, linking amplitude structure to SU(n-4) representations, and extending to higher N^KMHV via Young tableaux. The authors then apply this framework to ultraviolet counterterms in N=8 supergravity, constructing matrix elements of proposed operators and using locality and E7(7) symmetry to exclude many candidates, leaving no admissible counterterms below seven loops. Open-closed string relations (KLT) and soft scalar limits provide essential cross-checks with string theory and automorphism constraints, reinforcing the finiteness results and guiding the search for potential higher-loop structures. Collectively, the work fuses on-shell superspace, representation theory, and string-inspired consistency conditions to map the landscape of allowable counterterms and to illuminate the rich symmetry structure of maximally supersymmetric theories.

Abstract

Ward identities of SUSY and R-symmetry relate n-point amplitudes in supersymmetric theories. We review recent work in which these Ward identities are solved in N=4 SYM and N=8 supergravity. The solution, valid at both tree and loop level, expresses any (Next-to)^K MHV superamplitude in terms of a basis of ordinary amplitudes. Basis amplitudes are classified by semi-standard tableaux of rectangular N-by-K Young diagrams. The SUSY Ward identities also impose constraints on the matrix elements of candidate ultraviolet counterterms in N=8 supergravity, and they can be studied using superamplitude basis expansions. This leads to a novel and quite comprehensive matrix element approach to counterterms, which we also review. This article is an invited review for a special issue of Journal of Physics A devoted to "Scattering Amplitudes in Gauge Theories".

SUSY Ward identities, Superamplitudes, and Counterterms

TL;DR

This review explains how SUSY and R-symmetry Ward identities constrain on-shell scattering in N=4 SYM and N=8 supergravity, and shows how to solve these identities to express any N^KMHV superamplitude in terms of a minimal, representation-theoretically labeled basis. The basis elements are organized by semi-standard Young tableaux of rectangular diagrams, linking amplitude structure to SU(n-4) representations, and extending to higher N^KMHV via Young tableaux. The authors then apply this framework to ultraviolet counterterms in N=8 supergravity, constructing matrix elements of proposed operators and using locality and E7(7) symmetry to exclude many candidates, leaving no admissible counterterms below seven loops. Open-closed string relations (KLT) and soft scalar limits provide essential cross-checks with string theory and automorphism constraints, reinforcing the finiteness results and guiding the search for potential higher-loop structures. Collectively, the work fuses on-shell superspace, representation theory, and string-inspired consistency conditions to map the landscape of allowable counterterms and to illuminate the rich symmetry structure of maximally supersymmetric theories.

Abstract

Ward identities of SUSY and R-symmetry relate n-point amplitudes in supersymmetric theories. We review recent work in which these Ward identities are solved in N=4 SYM and N=8 supergravity. The solution, valid at both tree and loop level, expresses any (Next-to)^K MHV superamplitude in terms of a basis of ordinary amplitudes. Basis amplitudes are classified by semi-standard tableaux of rectangular N-by-K Young diagrams. The SUSY Ward identities also impose constraints on the matrix elements of candidate ultraviolet counterterms in N=8 supergravity, and they can be studied using superamplitude basis expansions. This leads to a novel and quite comprehensive matrix element approach to counterterms, which we also review. This article is an invited review for a special issue of Journal of Physics A devoted to "Scattering Amplitudes in Gauge Theories".

Paper Structure

This paper contains 30 sections, 93 equations, 1 table.

Table of Contents

  1. Introduction
  2. SUSY Ward identities
  3. Superamplitudes and their Symmetries
  4. Superamplitudes and supersymmetry constraints
  5. R-symmetry
  6. Basis expansion of superamplitudes in ${\cal N}=4$ SYM
  7. Strategy for solving the SUSY Ward identities
  8. Functional bases and single-trace amplitudes
  9. Beyond NMHV: superamplitudes and Young tableaux
  10. Basis expansion of superamplitudes in ${\cal N}=8$ supergravity
  11. Application: superamplitude approach to counterterms
  12. Candidate MHV counterterms
  13. Ruling out $R^n$ for $n \ge 5$:
  14. Ruling out $D^2 R^n$, $D^4R^n$, and $D^6R^n$ for $n\ge 5$:
  15. Ruling in $D^8R^n$
  16. ...and 15 more sections