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D = 5 maximally supersymmetric Yang-Mills theory diverges at six loops

Zvi Bern, John Joseph Carrasco, Lance J. Dixon, Michael R. Douglas, Matt von Hippel, Henrik Johansson

TL;DR

This work demonstrates that $D=5$ maximally supersymmetric Yang–Mills theory diverges at six loops in the planar limit, despite expectations tied to its connection with the $D=6$ $(2,0)$ theory. The authors construct the six-loop planar four-point integrand using generalized unitarity and dual conformal guidance, then reduce the UV divergence to vacuum integrals through a small-external-momentum expansion. They evaluate these vacuum integrals numerically with sector decomposition (FIESTA), including a thorough consistency-relations analysis to control uncertainties, and conclusively show a nonzero divergence coefficient. Remarkably, the divergences across loops up to six fit an approximate exponential form, enabling extrapolations to higher loops and offering a tantalizing hint of deeper all-loop UV structure, with potential implications for gravity-gauge relations and the UV completion mechanism in the $(2,0)$ theory context.

Abstract

The connection of maximally supersymmetric Yang-Mills theory to the (2,0) theory in six dimensions has raised the possibility that it might be perturbatively ultraviolet finite in five dimensions. We test this hypothesis by computing the coefficient of the first potential ultraviolet divergence of planar (large N_c) maximally supersymmetric Yang-Mills theory in D = 5, which occurs at six loops. We show that the coefficient is nonvanishing. Furthermore, the numerical value of the divergence falls very close to an approximate exponential formula based on the coefficients of the divergences through five loops. This formula predicts the approximate values of the ultraviolet divergence at loop orders L > 6 in the critical dimension D = 4 + 6/L. To obtain the six-loop divergence we first construct the planar six-loop four-point amplitude integrand using generalized unitarity. The ultraviolet divergence follows from a set of vacuum integrals, which are obtained by expanding the integrand in the external momenta. The vacuum integrals are integrated via sector decomposition, using a modified version of the FIESTA program.

D = 5 maximally supersymmetric Yang-Mills theory diverges at six loops

TL;DR

This work demonstrates that maximally supersymmetric Yang–Mills theory diverges at six loops in the planar limit, despite expectations tied to its connection with the theory. The authors construct the six-loop planar four-point integrand using generalized unitarity and dual conformal guidance, then reduce the UV divergence to vacuum integrals through a small-external-momentum expansion. They evaluate these vacuum integrals numerically with sector decomposition (FIESTA), including a thorough consistency-relations analysis to control uncertainties, and conclusively show a nonzero divergence coefficient. Remarkably, the divergences across loops up to six fit an approximate exponential form, enabling extrapolations to higher loops and offering a tantalizing hint of deeper all-loop UV structure, with potential implications for gravity-gauge relations and the UV completion mechanism in the theory context.

Abstract

The connection of maximally supersymmetric Yang-Mills theory to the (2,0) theory in six dimensions has raised the possibility that it might be perturbatively ultraviolet finite in five dimensions. We test this hypothesis by computing the coefficient of the first potential ultraviolet divergence of planar (large N_c) maximally supersymmetric Yang-Mills theory in D = 5, which occurs at six loops. We show that the coefficient is nonvanishing. Furthermore, the numerical value of the divergence falls very close to an approximate exponential formula based on the coefficients of the divergences through five loops. This formula predicts the approximate values of the ultraviolet divergence at loop orders L > 6 in the critical dimension D = 4 + 6/L. To obtain the six-loop divergence we first construct the planar six-loop four-point amplitude integrand using generalized unitarity. The ultraviolet divergence follows from a set of vacuum integrals, which are obtained by expanding the integrand in the external momenta. The vacuum integrals are integrated via sector decomposition, using a modified version of the FIESTA program.

Paper Structure

This paper contains 12 sections, 53 equations, 12 figures, 1 table.

Table of Contents

  1. Introduction
  2. Constructing the integrand
  3. From the amplitude to vacuum diagrams
  4. Obtaining the vacuum diagrams
  5. Simplifying the vacuum integrals via consistency relations
  6. Review of sector decomposition
  7. Sector decomposition overview
  8. Heuristic sector decomposition
  9. Numerical results and computational details
  10. Numerical integration
  11. Extrapolating in loop order
  12. Conclusions and outlook

Figures (12)

  • Figure 1: Graphs 1 through 35 for the planar six-loop four-point amplitude.
  • Figure 2: Graphs 36 through 68 for the planar six-loop four-point amplitude.
  • Figure 3: The six-loop cut evaluated in $D=6$ in ref. SixDCheck.
  • Figure 4: A few sample graphs with labels corresponding to the labels in eqs. (\ref{['SimpleSampleNumerator']}) and (\ref{['ComplexSampleNumerator']}) and in the ancillary file AttachedFile. The external momenta are outgoing.
  • Figure 5: The six distinct vacuum diagrams that appear in eqs. (\ref{['UVdivergence']}) and (\ref{['eq:integrand']}). Each dot indicates that the corresponding propagator should be squared (doubled) in the integrand. The five "tensor" integrals have numerator factors that are indicated by the prefactors. The numerator factors are built from momentum invariants involving a subset of the loop momenta, labeled by $l_1,l_2,l_3,l_4$.
  • ...and 7 more figures