Simplifying Multiloop Integrands and Ultraviolet Divergences of Gauge Theory and Gravity Amplitudes

TL;DR

The paper advances the computation of the four-loop four-point amplitudes in ${\cal N}=4$ sYM and ${\cal N}=8$ supergravity by exploiting the color-kinematics duality (BCJ) to express all graph numerators in terms of a small set of master graphs, and then constructing the gravity integrand via the double-copy relation. This approach yields a manifestly UV-counting-friendly representation, allowing a direct extraction of the UV divergences in the critical dimension ${D_c=11/2}$ and revealing that the ${\cal N}=8$ gravity divergence matches the ${1/N_c^2}$-suppressed single-trace divergence of ${\cal N}=4$ sYM. The calculation also clarifies the role of snail graphs and shows that double-copy cancels snail contributions in gravity, while double-trace divergences saturate their known bounds in $D=6-2\epsilon$. Collectively, the work provides strong loop-level evidence for color-kinematic duality, demonstrates a powerful method to import planar insights into nonplanar sectors, and deepens the gauge–gravity connection with tangible UV implications.

Abstract

We use the duality between color and kinematics to simplify the construction of the complete four-loop four-point amplitude of N=4 super-Yang-Mills theory, including the nonplanar contributions. The duality completely determines the amplitude's integrand in terms of just two planar graphs. The existence of a manifestly dual gauge-theory amplitude trivializes the construction of the corresponding N=8 supergravity integrand, whose graph numerators are double copies (squares) of the N=4 super-Yang-Mills numerators. The success of this procedure provides further nontrivial evidence that the duality and double-copy properties hold at loop level. The new form of the four-loop four-point supergravity amplitude makes manifest the same ultraviolet power counting as the corresponding N=4 super-Yang-Mills amplitude. We determine the amplitude's ultraviolet pole in the critical dimension of D=11/2, the same dimension as for N=4 super-Yang-Mills theory. Strikingly, exactly the same combination of vacuum integrals (after simplification) describes the ultraviolet divergence of N=8 supergravity as the subleading-in-1/N_c^2 single-trace divergence in N=4 super-Yang-Mills theory.

Figures (22)

• Figure 1: Pictorial Jacobi relation for a group of three graphs. The graphs can represent color factors or numerator factors. Except for the connections to the central (pink) lines, the graphs are identical in the three cases, as indicated by the common (momentum or color) labels $a,b,c,d$. The gray area represents some unspecified subgraph which is identical in all three graphs.
• Figure 2: The 12 nonvanishing graphs used in the construction of the ${{\cal N}=4}$ sYM and ${{\cal N}=8}$ supergravity three-loop four-point amplitude. The shaded (pink) lines mark the application of the duality relation used to determine the numerator of the graph. The external momenta are outgoing and the arrows mark the directions of the labeled loop momenta.
• Figure 3: Graphs (a) and (c) are generic propagator correction graphs that can appear at four loops and beyond if we have a cubic organization of graphs. Graphs (b) and (d) are rewritings of these graphs, which make explicit that in ${{\cal N}=4}$ sYM theory numerator factors always cancel the propagators that are external to the loops in the four-point amplitude.
• Figure 4: The planar master graphs, 18 and 28. The numerators and color factors of all other graphs are generated from the numerators and color factors of these two graphs through kinematic Jacobi relations.
• Figure 5: Cubic graphs 1 to 11 that contribute to the four-loop four-point amplitude of ${{\cal N}=4}$ sYM theory and ${{\cal N}=8}$ supergravity. The labels 1 to 4, indicate the legs carrying external momenta $k_1$ to $k_4$. The labels 5 to 8 indicate the propagators carrying the independent loop momenta $l_5$ to $l_8$. The arrows indicate the direction of the momentum. The graphs also specify the color factor of the graph, simply by dressing each cubic vertex with an $\tilde{f}^{abc}$, respecting the clockwise ordering of lines at each vertex.