Despite the early termination of the project, all the main scientific objectives have been achieved.
1) My collaborators and I computed moments of the splitting functions that determine the scale evolution of the parton densities. We extended a methodology based on the renormalisation of composite operators, which we had previously developed in the context of Yang-Mills theory, to the case of QCD. We first reproduced a set of moments of the known three-loop splitting functions. Finally, we computed the first ten moments of all the splitting functions at four loops, which corresponds to the Next-to-Next-to-Next-to-Leading Order (N3LO) in QCD perturbation theory.
2) We constructed an approximate four-loop scale evolution of the parton densities by using the ten moments of the splitting functions, which we had computed, and information on their kinematic limits. We estimated that the uncertainty of the approximation is of the order of 1% across the kinematic regions that are relevant for the description of the proton collisions at the LHC. We released the approximate splitting functions in a publicly available fortran code.
3) Using the same procedure adopted to construct the approximate splitting functions, we reconstructed the exact analytic result for one gauge invariant contribution to the quark-to-gluon splitting function at four loops.
4) We developed a technique to reconstruct the anomalous dimensions of classes of unphysical operators that are required in the procedure of composite operator renormalisation at the basis of the moment calculation. We obtain the one-loop anomalous dimensions, reproducing the results in the literature and providing new results for one class of operators.
In parallel I have been working on the high-energy limit of QCD amplitudes. Together with my collaborators
5) We formulated the high-energy factorisation through the Next-to-Next-to Leading Logarithms (NNLL), for the QCD amplitudes with five external partons. The factorisation formula involves two contributions, known as Regge pole and Regge cut. We developed a framework to disentangle the Regge cut from the Regge pole and we computed Regge cuts up to two loops.
6) We computed the two-loop QCD amplitudes with five external partons, with full information of the colour structure, in the high energy limit. We verified the high-energy factorisation formula, by removing the two-loop Regge cuts and isolating the Regge pole contribution. We verified the universality of the Regge pole, by comparing the results for three different processes, namely gluon-gluon to 3 gluon, quark-gluon to quark-gluon-gluon and quark-quark to quark-quark-gluon scattering. We determined the two-loop Lipatov vertex, which characterises the contribution of the Regge pole to the five-leg amplitudes and is a key ingredient for high-energy factorisation at NNLL.
7) We studied the high-energy factorisation, at the level of the NNLL, of the amplitudes for the production of one Higgs boson in association with one jet. We established that the Regge cuts must be absent through three-loop order. We verified that in the high energy limit the two-loop amplitudes, which are known in the literature, are indeed described purely by a Regge pole, involving a universal impact factor for the Higgs boson, which we determined at the two-loop level. We verified that the Regge-pole ansatz correctly reproduces the infrared singularities of the amplitudes at three loops.
