Forest Formulas for the LHC

One of the greatest scientific events of the century is the discovery of the Higgs boson by CERN's Large Hadron Collider (LHC). Yet the LHCs discovery potential has by no means been exhausted as collisions are now happening with an increasing rate at energies never achieved before by mankind. With more and more data being accumulated over the next 15 years we will obtain measurements at unprecedented levels of precision. This data will put stringent new tests on the Standard Model (SM) of particle physics. While the success of the SM is the greatest achievement of particle physics to date, it also poses many mysteries to physicists. For instance, the SM does not explain the observed matter-antimatter asymmetry, or the nature of dark matter and dark energy in the universe. To overcome these problems new models, featuring as exotic ideas as supersymmetry or extra dimensions, have been proposed. So far none of these models could be detected in experiments, but beyond-the-SM (BSM) physics may still be detected at the energy currently explored by the LHC.
To distinguish new physics from the SM, theoretical calculations must match the accuracy of the experimental measurements. This poses a tremendous challenge since it is impossible to calculate general observables exactly in quantum field theory. Instead, theoretical physicists resort to what is called the perturbative expansion; this is a systematic way to expand the complicated functions, which describe the scattering rates, in a series in the interaction strength, where each successive term is smaller than the preceding. By calculating enough terms in this expansion one can thus obtain increasingly reliable results. Especially in quantum chromodynamics (QCD), which governs the dynamics of the constituent quarks and gluons of the proton, the convergence of this expansion is relatively slow and in certain cases computations with three or four terms are required. The problem with this approach is that the Feynman diagrams, which appear in the individual terms of this expansion, rapidly increase in both number and complexity. To make matters worse, these Feynman diagrams also contain complicated infrared (IR) and ultraviolet (UV) divergences (singularities) which are of long- and short-distance origin.
While the problem of UV divergences has been solved already half a century ago by the procedure of renormalisation, the situation is very different for the IR divergences. Calculating higher-order effects in QCD requires the combination of two separate contributions: real corrections (due to emissions of observable particles) and virtual (loop or quantum) corrections. While it is well known that the divergences of the real emission corrections cancel with those of the virtual corrections, the cancellations only happen after all the different loop and phase-space integrals have been performed.
A rigorous approach to renormalisation is given by the Bogoliubov-Parasiuk-Hepp-Zimmermann (BPHZ) scheme also known as the "forest formula", where the term forest refers to sets of nested or disjoint divergent subgraphs. The key idea of this project is to develop and use "generalised forest formulas" for the subtraction of IR divergences. While this proposition is far from trivial, recent breakthroughs which I have made in my recent research have already proven the concept, obtaining a plethora of new results which could not have been achieved by other means. The future potential of this approach is great, as it opens the door for new ways of calculating quantities, which are desperately needed to improve the precision of current theory predictions, such as the 4-loop splitting functions, which govern the energy dependence of partons in the proton, and the 2-loop anomalous dimensions which govern the energy dependence of coupling parameters in the SM EFT, a general model-independent framework to BSM physics.