Subleading-color contributions to gluon-gluon scattering in N=4 SYM theory and relations to N=8 supergravity

TL;DR

This paper analyzes subleading-color (nonplanar) four-gluon amplitudes in ${\cal N}=4$ SYM up to three loops, focusing on IR-divergent terms and their organization via Catani and Sterman-Tejeda-Yeomans formalisms. It derives explicit IR structures for the subleading amplitudes, demonstrates uniform transcendentality through two loops, and reveals deep connections to ${\cal N}=8$ supergravity, including exact one- and two-loop relations for the $L$-loop, $N$-independent piece $A^{(L,L)}$. A key result is that simple all-loop relations between SYM and supergravity exist at low loops but fail at three loops, suggesting only a partial, loop-dependent correspondence and inviting refined duality pictures. The work also discusses an exponentiation pattern for IR divergences and shows that the Catani operators attain maximal transcendentality, aligning with QCD's maximal-transcendentality piece. Overall, the paper illuminates the structural harmony and limits of gauge–gravity connections in the subleading-color regime and advances the understanding of IR behavior and transcendentality in ${\cal N}=4$ SYM.

Abstract

We study the subleading-color (nonplanar) contributions to the four-gluon scattering amplitudes in N=4 supersymmetric SU(N) Yang-Mills theory. Using the formalisms of Catani and of Sterman and Tejeda-Yeomans, we develop explicit expressions for the infrared-divergent contributions of all the subleading-color L-loop amplitudes up to three loops, and make some conjectures for the IR behavior for arbitrary L. We also derive several intriguing relations between the subleading-color one- and two-loop four-gluon amplitudes and the four-graviton amplitudes of N=8 supergravity. The exact one- and two-loop N=8 supergravity amplitudes can be expressed in terms of the one- and two-loop N-independent N=4 SYM amplitudes respectively, but the natural generalization to higher loops fails, despite having a simple interpretation in terms of the 't Hooft picture. We also find that, at least through two loops, the subleading-color amplitudes of N=4 SYM theory have uniform transcendentality (as do the leading-color amplitudes). Moreover, the N=4 SYM Catani operators, which express the IR-divergent contributions of loop amplitudes in terms of lower-loop amplitudes, are also shown to have uniform transcendentality, and to be the maximum transcendentality piece of the QCD Catani operators.