We compute systematically for the planar double box Feynman integral relevant to top pair production with a closed top loop the Laurent expansion in the dimensional regularisation parameter ε. This is done by transforming the system of differential equations for this integral and all its subtopologies to a form linear in ε, where the ε 0 -part is strictly lower triangular. This system is easily solved order by order in the dimensional regularisation parameter ε. This is an example of an elliptic multi-scale integral involving several elliptic sub-topologies. Our methods are applicable to similar problems.

In this article we give the details on the analytic calculation of the master integrals for the planar double box integral relevant to top-pair production with a closed top loop. We show that these integrals can be computed systematically to all order in the dimensional regularisation parameter ε. This is done by transforming the system of differential equations into a form linear in ε, where the ε 0 -part is a strictly lower triangular matrix. Explicit results in terms of iterated integrals are presented for the terms relevant to NNLO calculations. number of block sector master integrals master integrals kinematic propagators basis I basis J dependence 2 1 9 I 1001000 J 1 − 3 2 14 I 0111000 J 2 s 3 28 I 0011100 , I 0021100 J 3 , J 4 s 4 49 I 1000110 J 5 s 5 73 I 1001001 , I 200100123 = 4 G (1, 0, 0, 1; x) + 4 G (0, 1, 0, 1; x) + 4 G (0, 0, 1, 1; x) − 8 G (0, 0, 0, 1; x) +10 G (0, 0, −1, 0; x) − 4 G (0, −1, 0, 0; x) + 4 G (0, −1, 0, 1; x) + 4 G (0, −1, 1, 0; x) −6 G (0, 1, 0, 0; x) − 4 G (0, 0, 1, 0; x) + 4 G (0, 1, 1, 0; x) + 4 G (0, 1, −1, 0; x) +12 G (1, 0, 0, 0; x) − 24 G (1, 0, −1, 0; x) − 4 G (1, 0, 1, 0; x) + 2 ζ 2 G (0, 1; x) −4 ζ 2 G (1, 0; x) + 12 ζ 3 G (0; x) − 24 ζ 3 G (1; x) + 9 ζ 4 , J (0) 32 = 0, J (1) 32 = 0, J (2) 3237 = 0, J (1) 37 = 0, J (2) 37 = 0, J (3) 37 = 0, J (4) 37 = −6 G (0, 0, 0, 0; x) + 6 G (0, 0, 0, 1; x) + 4 G (0, 0, 1, 0; x) + 2 G (0, 1, 0, 0; x) −4 ζ 2 G (0, 0; x) , J (0) 40 = 0, J (1) 40 = 0, J (2) 40 = 0, J (3) 40 = 0, J (4) 40 = −G (0, 0, 0, 0; x) − 8 G (0, 0, −1, 0; x) + 6 G (0, 0, 0, 1; x) + 4 G (0, 1, 0, 0; x) −4 ζ 2 G (0, 0; x) − 2 ζ 3 G (0; x) − 3 ζ 4 . (190) 9.2.2 Integrals, which are expressed in the variable x ′ The integrals J 12 − J 13 are most naturally expressed in terms of harmonic polylogarithms in the variable x ′ . J (0) 12 = 0, J (1) 12 = 0, J (2) 12= G 0, 0; x ′ − 2 G 0, −1; x ′ + ζ 2 ,

In this paper we exploit factorisation properties of Picard-Fuchs operators to decouple differential equations for multi-scale Feynman integrals. The algorithm reduces the differential equations to blocks of the size of the order of the irreducible factors of the Picard-Fuchs operator. As a side product, our method can be used to easily convert the differential equations for Feynman integrals which evaluate to multiple polylogarithms to ε-form.

We present a complete set of analytic helicity amplitudes for top quark pair production via gluon fusion at two-loops in QCD. For the first time, we include corrections due to massive fermion loops which give rise to integrals over elliptic curves. We present the results of the missing master integrals needed to compute the amplitude and obtain an analytic form for the finite remainders in terms of iterated integrals using rationalised kinematics and finite field sampling. We also study the numerical evaluation of the iterated integrals.

Feynman diagrams constitute one of the essential ingredients for making precision predictions for collider experiments. Yet, while the simplest Feynman diagrams can be evaluated in terms of multiple polylogarithms-whose properties as special functions are well understood-more complex diagrams often involve integrals over complicated algebraic manifolds. Such diagrams already contribute at NNLO to the self-energy of the electron, t t production, γγ production, and Higgs decay, and appear at two loops in the planar limit of maximally supersymmetric Yang-Mills theory. This makes the study of these more complicated types of integrals of phenomenological as well as conceptual importance.In this white paper contribution to the Snowmass community planning exercise, we provide an overview of the state of research on Feynman diagrams that involve special functions beyond multiple polylogarithms, and highlight a number of research directions that constitute essential avenues for future investigation.