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

Abstract

We study the subleading-color (nonplanar) contributions to the four-gluon scattering amplitudes in = 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 = 8 supergravity. The exact one- and two-loop = 8 supergravity amplitudes can be expressed in terms of the one- and two-loop N-independent = 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 = 4 SYM theory have uniform transcendentality (as do the leading-color amplitudes). Moreover, the = 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.