We evaluate analytically all previously unknown nonplanar master integrals for massless fiveparticle scattering at two loops, using the differential equations method. A canonical form of the differential equations is obtained by identifying integrals with constant leading singularities, in D space-time dimensions. These integrals evaluate to Q-linear combinations of multiple polylogarithms of uniform weight at each order in the expansion in the dimensional regularization parameter, and are in agreement with previous conjectures for nonplanar pentagon functions. Our results provide the complete set of two-loop Feynman integrals for any massless 2 → 3 scattering process, thereby opening up a new level of precision collider phenomenology.

We compute the full-color two-loop five-gluon amplitude for the all-plus helicity configuration. In order to achieve this, we calculate the required master integrals for all permutations of the external legs, in the physical scattering region. We verify the expected divergence structure of the amplitude, and extract the finite hard function. We further validate our result by checking the factorization properties in the collinear limit. Our result is fully analytic and valid in the physical scattering region. We express it in a compact form containing logarithms, dilogarithms and rational functions.

We compute the symbol of the full-color two-loop five-particle amplitude in N = 4 super Yang-Mills, including all non-planar subleading-color terms. The amplitude is written in terms of permutations of Parke-Taylor tree-level amplitudes and pure functions to all orders in the dimensional regularization parameter, in agreement with previous conjectures. The answer has the correct collinear limits and infrared factorization properties, allowing us to define a finite remainder function. We study the multi-Regge limit of the non-planar terms, analyze its subleading power corrections, and present analytically the leading logarithmic terms.

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We compute for the first time the two-loop five-particle amplitude in N = 8 supergravity. Starting from the known integrand, we perform an integration-by-parts reduction and express the answer in terms of uniform weight master integrals. The latter are known to evaluate to non-planar pentagon functions, described by a 31-letter symbol alphabet. We express the final result for the amplitude in terms of uniform weight four symbols, multiplied by a small set of rational factors. We observe that one of the symbol letters is absent from the amplitude. The latter satisfies the expected factorization properties when one external graviton becomes soft, and when two external gravitons become collinear. We verify that the soft divergences of the amplitude exponentiate. We extract the finite remainder function, which depends on fewer rational factors. By analyzing identities involving rational factors and symbols we find a remarkably compact representation in terms of a single seed function, summed over all permutations of external particles. Finally, we work out the multi-Regge limit, and present explicitly the leading logarithmic terms in the limit. The full symbol of the IR-subtracted hard function is provided as a supplementary file.