Manifest Ultraviolet Behavior for the Three-Loop Four-Point Amplitude of N=8 Supergravity

Abstract

Using the method of maximal cuts, we obtain a form of the three-loop four-point scattering amplitude of N=8 supergravity in which all ultraviolet cancellations are made manifest. The Feynman loop integrals that appear have a graphical representation with only cubic vertices, and numerator factors that are quadratic in the loop momenta, rather than quartic as in the previous form. This quadratic behavior reflects cancellations beyond those required for finiteness, and matches the quadratic behavior of the three-loop four-point scattering amplitude in N=4 super-Yang-Mills theory. By direct integration we confirm that no additional cancellations remain in the N=8 supergravity amplitude, thus demonstrating that the critical dimension in which the first ultraviolet divergence occurs at three loops is D_c=6. We also give the values of the three-loop divergences in D=7,9,11. In addition, we present the explicitly color-dressed three-loop four-point amplitude of N=4 super-Yang-Mills theory.

38 pages, 10 figures, 3 tables

References in corpus (27)

• New Relations for Gauge-Theory Amplitudes
• Gluon scattering amplitudes at strong coupling
• The Four-Loop Planar Amplitude and Cusp Anomalous Dimension in Maximally Supersymmetric Yang-Mills Theory
• The Two-Loop Six-Gluon MHV Amplitude in Maximally Supersymmetric Yang-Mills Theory
• Three-Loop Superfiniteness of N=8 Supergravity
• Maximally Supersymmetric Planar Yang-Mills Amplitudes at Five Loops
• The Two-loop Soft Anomalous Dimension Matrix and Resummation at Next-to-next-to Leading Pole
• The Two-loop Anomalous Dimension Matrix for Soft Gluon Exchange
• Is N = 8 Supergravity Ultraviolet Finite?
• Generating Tree Amplitudes in N=4 SYM and N=8 SG
• Ultraviolet properties of Maximal Supergravity
• Taming Tree Amplitudes In General Relativity
• Two-Loop Iteration of Five-Point N=4 Super-Yang-Mills Amplitudes
• The No-Triangle Hypothesis for N=8 Supergravity
• New Higher-Derivative $R^4$ Theorems
• Decoupling Supergravity from the Superstring
• KLT relations from the Einstein-Hilbert Lagrangian
• Iterative Structure Within The Five-Particle Two-Loop Amplitude
• Modular properties of two-loop maximal supergravity and connections with string theory
• Explicit Action of E7(7) on N=8 Supergravity Fields
• Three-Loop Leading Singularities and BDS Ansatz for Five Particles
• Two-loop graviton scattering relation and IR behavior in N=8 supergravity
• Note on graviton MHV amplitudes
• Explicit Cancellation of Triangles in One-loop Gravity Amplitudes
• E_7(7) on the Light Cone
• On Cancellations of Ultraviolet Divergences in Supergravity Amplitudes
• On-shell recursion relations for gravity

Cited by in corpus (33)

• Lorentz symmetry breaking as a quantum field theory regulator
• Enhanced Ultraviolet Cancellations in N = 5 Supergravity at Four Loop
• E7(7) constraints on counterterms in N=8 supergravity
• The Complete Four-Loop Four-Point Amplitude in N=4 Super-Yang-Mills Theory
• The Five-Loop Four-Point Integrand of N=8 Supergravity as a Generalized Double Copy
• Stringy KLT relations, global symmetries, and E_7(7) violation
• Matching the $D^6{\mathcal{R}}^4$ interaction at two-loops
• Four-gluon scattering at three loops, infrared structure and Regge limit
• Proof of the MHV vertex expansion for all tree amplitudes in N=4 SYM theory
• The two-loop five-point amplitude in $\mathcal{N}=8$ supergravity
• Nine-Propagator Master Integrals for Massless Three-Loop Form Factors
• Matter Dependence of the Three-Loop Soft Anomalous Dimension Matrix
• Tree-Level Amplitudes in N=8 Supergravity
• $D^6{R}^4$ amplitudes in various dimensions
• Four-graviton scattering to three loops in $\mathcal{N}=8$ supergravity
• The Six-Point NMHV amplitude in Maximally Supersymmetric Yang-Mills Theory
• N=8 Supergravity 4-point Amplitudes
• Higher Derivative Supergravities in Diverse Dimensions
• Higher-loop amplitudes in the non-minimal pure spinor formalism
• On UV Finiteness of the Four Loop N=8 Supergravity
• Gravitino Interactions from Yang-Mills Theory
• A super MHV vertex expansion for N=4 SYM theory
• The D^6 R^4 term from three loop maximal supergravity
• Cross-Order Integral Relations from Maximal Cuts
• The Structure of Multiloop Amplitudes in Gauge and Gravity Theories
• Relations Between Closed String Amplitudes at Higher-order Tree Level and Open String Amplitudes
• Illustrations of Integrand-Basis Building at Two Loops
• Obstruction to E_{7(7)} Deformation in N=8 Supergravity
• All-loop-orders relation between Regge limits of $\mathcal{N}=4$ SYM and $\mathcal{N}=8$ supergravity four-point amplitudes
• Non-analytic terms of string amplitudes from partial waves
• Perturbative Methods for Superconformal Quantum Field Theories in String - Gauge Theory Dualities
• Exceptional symmetries in light-cone superspace
• A Master Action for D=11 Supergravity in the Component Formulation