Five-loop structure constants in N=4 SYM
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 (OPE), 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 hint that these problems could be tackled was given by the seminal paper by Prof. Joseph Minahan and Prof. Konstantin Zarembo, from Uppsala University. They discovered that the one-loop spectrum of long operators was given as the spectrum of an integrable spin chain. Since then, the spin-chain techniques have been extended to all loops and to finite size in the form of the Thermodynamic Bethe Ansatz. Finally, one is left with the problem of computing structure constants, which has seen great progress lately. Roughly one year ago a conjecture to compute structure constants at any value of the coupling has been proposed, and the prescription is to compute a two-point function of defect operators: the Hexagon form factors. In fact, very recently it was understood that this programme can be extended to compute correlation functions with any number of operators in N=4 SYM, effectively re-summing OPE decompositions. It has become obvious that the Hexagon form factors are the building blocks of correlation functions in conformal field theories. The aim of this project (collaboration with 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 that we understand how and if the regularization prescription holds at higher loops orders. In order to do so, one can compute four-point functions of protected operators perturbatively and check that the result obtained in the limit when two point approach each other is consistent with the conjectured Hexagon form factors. This perturbative approach implies however the computation of very complicated Feynman diagrams. Since it is only necessary to obtain their OPE limit, it is possible to use the technique of asymptotic expansions, which approximates each conformal integral by a combination of massless propagator integrals. We are left with the computation of propagator integrals at five loops which albeit feasible seems very demanding computationally.