We present a calculation of the rational terms in two-loop all-plus gluon amplitudes using Ddimensional unitarity. We use a conjecture of separability of the two loops, and then a simple generalization of one-loop D-dimensional unitarity to perform calculations. We compute the fourand five-point rational terms analytically, and the six-and seven-point ones numerically. We find agreement with previous calculations of Dalgleish, Dunbar, Godwin, Jehu, Perkins, and Strong.For a special subleading-color amplitude, we compute the eight-and nine-point results numerically, and find agreement with an all-n conjecture of Dunbar, Perkins, and Strong.

We show that the differential equation for the three-loop equal-mass banana integral can be cast into an ε-factorised form with entries constructed from (meromorphic) modular forms and one special function, which can be given as an iterated integral of meromorphic modular forms. The ε-factorised form of the differential equation allows for a systematic solution to any order in the dimensional regularisation parameter ε. The alphabet of the iterated integrals contains six letters.

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We describe a systematic approach to cast the differential equation for the l-loop equal mass banana integral into an ε-factorised form. With the known boundary value at a specific point we obtain systematically the term of order j in the expansion in the dimensional regularisation parameter ε for any loop l. The approach is based on properties of Calabi-Yau operators, and in particular on self-duality.

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