In the last decade there has been great progress in the study of strongly interacting quantum field theories, mainly due to gauge-gravity dualities which relate conformal field theories to string theories in higher dimensional curved backgrounds. More in particular, the discovery of integrability for N=4 supersymmetric Yang-Mills (SYM) gives hope that this theory can be solved exactly in the planar limit. This would give great insights into non-perturbative features of real-world theories such as QCD, since they are closely related in some particular limits.
Being a conformal field theory, N=4 SYM has a convergent operator product expansion, which means that only two sets of numbers need to be understood in order to solve the theory: the spectrum of operators and their structure constants. The first has been solved in the past decade due to the discovery of an infinite amount of extra symmetry, while the latter is the subject of current research. Recently, a conjecture to compute structure constants at any value of the coupling has been proposed, and the prescription consists of a two-point function of twist operators: the Hexagon form factors.
The aim of this project (collaboration with Raul Pereira and Alessandro Georgoudis from Uppsala University and Vasco Gonçalves from ICTP-SAIFR Sao Paulo) is to understand precisely the validity of the Hexagon form factors at higher loops. It has been shown that divergences arise at four-loops and how to regularize them using a minimal subtraction scheme. It is however crucial to understand how and if the regularization prescription holds at higher loops orders. In order to do so, one needs to compute four-point functions of protected operators perturbatively and compare the result obtained in the OPE limit with the conjectured Hexagon form factors. This perturbative approach implies the computation of extremely complicated Feynman diagrams, which can be tackled with the technique of asymptotic expansions, where one approximates conformal integrals by simpler massless propagator integrals.
To compute propagator integrals, one needs to use Integration By Parts tools such as LiteRed (arXiv:1310.1145) and FIRE (arXiv:1408.2372) in order to reduce to a basis of simpler integrals. Once that reduction is complete one needs to evaluate the master integrals, which can be done using Dimensional Recurrence Relations (arXiv:0911.0252) and analysis of analytical properties of the integrals (arXiv:1511.03614), techniques that have proven extremely powerful already at four loops.