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A simple approach to counterterms in N=8 supergravity

Henriette Elvang, Daniel Z. Freedman, Michael Kiermaier

TL;DR

The paper presents a practical, on-shell matrix-element method to test the supersymmetrization of local operators in ${ m N}=8$ supergravity, recasting potential counterterms as questions about locality and SUSY Ward identities of $n$-point matrix elements. By separating into MHV and NMHV sectors and enforcing permutation symmetry, the authors rule out many candidates (e.g., $R^n$, $D^2R^n$, $D^4R^n$, $D^6R^n$ for $n>4$ at MHV; $R^n$ and $D^2R^n$ at NMHV) and explicitly construct viable structures such as the NMHV $D^4R^6$ counterterm, as well as MHV candidates like $D^{8}R^4$ under certain loop-order restrictions. They also develop a gauge-theory–based construction for gravity NMHV operators and propose bounds that would imply finiteness for loop orders $L<7$, while outlining extensions to higher N$^K$MHV levels and potential connections to $E_{7,7}$ symmetry. Overall, the work provides a systematic, perspective-preserving framework for identifying which supersymmetric counterterms can exist in ${ m N}=8$ supergravity and how they relate to loop order and point-number.

Abstract

We present a simple systematic method to study candidate counterterms in N=8 supergravity. Complicated details of the counterterm operators are avoided because we work with the on-shell matrix elements they produce. All n-point matrix elements of an independent SUSY invariant operator of the form D^{2k} R^n +... must be local and satisfy SUSY Ward identities. These are strong constraints, and we test directly whether or not matrix elements with these properties can be constructed. If not, then the operator does not have a supersymmetrization, and it is excluded as a potential counterterm. For n>4, we find that R^n, D^2 R^n, D^4 R^n, and D^6 R^n are excluded as counterterms of MHV amplitudes, while only R^n and D^2 R^n are excluded at the NMHV level. As a consequence, for loop order L<7, there are no independent D^{2k}R^n counterterms with n>4. If an operator is not ruled out, our method constructs an explicit superamplitude for its matrix elements. This is done for the 7-loop D^4 R^6 operator at the NMHV level and in other cases. We also initiate the study of counterterms without leading pure-graviton matrix elements, which can occur beyond the MHV level. The landscape of excluded/allowed candidate counterterms is summarized in a colorful chart.

A simple approach to counterterms in N=8 supergravity

TL;DR

The paper presents a practical, on-shell matrix-element method to test the supersymmetrization of local operators in supergravity, recasting potential counterterms as questions about locality and SUSY Ward identities of -point matrix elements. By separating into MHV and NMHV sectors and enforcing permutation symmetry, the authors rule out many candidates (e.g., , , , for at MHV; and at NMHV) and explicitly construct viable structures such as the NMHV counterterm, as well as MHV candidates like under certain loop-order restrictions. They also develop a gauge-theory–based construction for gravity NMHV operators and propose bounds that would imply finiteness for loop orders , while outlining extensions to higher NMHV levels and potential connections to symmetry. Overall, the work provides a systematic, perspective-preserving framework for identifying which supersymmetric counterterms can exist in supergravity and how they relate to loop order and point-number.

Abstract

We present a simple systematic method to study candidate counterterms in N=8 supergravity. Complicated details of the counterterm operators are avoided because we work with the on-shell matrix elements they produce. All n-point matrix elements of an independent SUSY invariant operator of the form D^{2k} R^n +... must be local and satisfy SUSY Ward identities. These are strong constraints, and we test directly whether or not matrix elements with these properties can be constructed. If not, then the operator does not have a supersymmetrization, and it is excluded as a potential counterterm. For n>4, we find that R^n, D^2 R^n, D^4 R^n, and D^6 R^n are excluded as counterterms of MHV amplitudes, while only R^n and D^2 R^n are excluded at the NMHV level. As a consequence, for loop order L<7, there are no independent D^{2k}R^n counterterms with n>4. If an operator is not ruled out, our method constructs an explicit superamplitude for its matrix elements. This is done for the 7-loop D^4 R^6 operator at the NMHV level and in other cases. We also initiate the study of counterterms without leading pure-graviton matrix elements, which can occur beyond the MHV level. The landscape of excluded/allowed candidate counterterms is summarized in a colorful chart.

Paper Structure

This paper contains 23 sections, 40 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction and Summary of Results
  2. A matrix element approach to counterterms
  3. Local counterterms
  4. The method
  5. From permutation symmetry to locality
  6. Candidate MHV counterterms
  7. The $R^4$ counterterm
  8. No $D^{2}R^4$ counterterm
  9. No $R^n$ MHV counterterms for any $n \ge 5$
  10. No $D^{2}R^n$, $D^{4}R^n$, or $D^{6}R^n$ MHV counterterms for $n\ge 5$
  11. Candidate MHV counterterms $D^{2k} R^n$ for $k \ge 4$
  12. $D^{2k}R^4$ counterterms
  13. Candidate NMHV counterterms
  14. No $R^6$ and $D^2R^6$ NMHV counterterms
  15. No $R^n$ and $D^2R^n$ NMHV counterterms for $n \ge 6$
  16. ...and 8 more sections

Figures (1)

  • Figure 1: Results for candidate counterterms in ${\cal N}=8$ supergravity, organized by loop order $L$ and $n$-point level of their leading matrix elements. The color indicates whether a linearized supersymmetrization of the $D^{2k}R^n$ operator under consideration exists (green), is excluded (red), or is unknown (gray). Beyond the MHV level, there could be SUSY operators without a leading pure-graviton contribution. These could also "live" above the $R^n$ diagonal in this diagram. In section \ref{['secnograv']}, we rule out such operators at the NMHV level above the $D^4 R^n$ line.