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 at energies never achieved before by mankind. With more and more data being accumulated over the next 15 years we will obtain measurements at previously unreached levels of precision. This data will put stringent new tests on the prevalent 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 abundances 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 physics may yet be hiding 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 still impossible to calculate general observables exactly in quantum field theory (the theoretical framework upon which the SM stands). 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. To calculate the higher order effects in QCD requires the calculation 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 establish and use a "generalised forest formula" for the subtraction of the troublesome IR divergences. While this proposition is far from trivial, a breakthrough which I have made in my recent research now gives strong evidence for its correctness. The future potential of this approach is great as it promises an in-principle general solution for calculating scattering rates of an arbitrary number of final-state particles and arbitrary orders in the perturbative expansion. One important objective of the proposed research is to implement this idea in a dedicated code-library and apply it in the calculations of higher-order QCD corrections of key importance for the LHC; such as the production of two and three jets at the respective 4th and 3rd order in the perturbative expansion.

The proposed research aims at developing new mathematical and computational methods for performing extremely challenging calculations in the framework of quantum field theory in order to predict the outcome of scattering events measured at CERN's Large Hadron Collider (LHC). The immediate beneficiaries of this research are therefore particle physicists (both experimental and theoretical) who are trying to understand whether the prevailing Standard Model (SM) correctly describes this data. Without accurate theoretical predictions it is indeed very hard, if not impossible, to tell the SM physics apart from beyond-the-SM (BSM) physics. Since there exists plenty of motivation for believing that the SM is incomplete, to understand and find out what this BSM physics is, is one of the highest priorities of the field as a whole. But also theoretical physicists more interested in in the mathematical structure of quantum field theory will profit from this research.

This research can also benefit mathematicians working in fields such as algebraic geometry and number theory. Feynman integrals, the building blocks of perturbative quantum field theory, exhibit many interesting and still poorly understood mathematical properties. This allows mathematicians to apply modern techniques and methods on interesting and highly non-trivial examples, which are otherwise hard for them to come by. Over the past decades this has lead to a fruitful synergy between theoretical physicists and mathematicians. In the proposed research new kinds of so-called forest formulas will be explored. Similar, although less involved, forest formulas are known to give rise to Hopf algebras. The study of these rich algebraic structures is mathematically interesting in its own right.

Also, computer scientists could potentially profit from this research. To accomplish the extremely complicated calculations proposed here, it is likely that new computational methods will have to be implemented in publicly available computer software. It is not unlikely that such methods could help solve problems in computer science too.

But also society as a whole ultimately profits from the kind of fundamental research proposed here. Fundamental physics, by extending the boundaries of our knowledge of nature, has been one of the driving forces for generating new ways of thinking for engineers of all disciplines, such as superconducting technology for the LHC, and the computing GRID. The importance of fundamental physics to society is thus very large. Since the particular research proposed here transcends the borders between physics, mathematics and computing, which ultimately make up the language and toolbox used by modern engineers and computer scientists to innovate new technologies and software, the proposed research may also influence the technology of the future. In this way, it will have an impact on the economy by providing the basis for creating new products and procedures.

There also exists at least one direct consequence from which society, as the main investor of the LHC, the biggest science project ever built by mankind, will profit from this project. That is to ensure that the experimental data of the LHC is properly interpreted with adequate theory predictions. It would be difficult to imagine the cost of the effect which misinterpreted data could have on the field. By providing some of the required predictions and developing methods with which to improve our abilities to make them in general, this project also aims to give insurance to this investment.