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# PhD defence by Hjalte Frellesvig

**Title: **Generalized Unitarity Cuts and Integrand Reduction at Higher Loop Orders

**Abstract**: The huge amounts of data on particle scattering which have been collected by the LHC in the recent years, has enabled the investigation of interactions and other effects which so far have been too minuscule to be observed. A perfect example of this is the recent discovery of the Higgs particle.

The large amount of background to such processes has caused an ever increasing need for precise predictions by the fundamental theory. For most collision processes the leading contribution comes from QCD, the theory of the strong interactions, which is the main focus of this thesis. In practice the required precision corresponds to the next to next to leading order in the perturbative expansion of the cross section, which requires the calculation of Feynman diagrams with two or more loops.

For one-loop diagrams the corresponding problem has been solved to the extend that almost all one-loop amplitudes of physical interest are known, and their calculation automated. This is mainly due to the technique of generalized unitarity cuts combined with integrand reduction into what is known as the OPP method. This method is sufficiently easy and fast that the one-loop contributions can be incorporated into event-generation software on equal footing with the tree contributions.

More specifically, the OPP method calculates individually each topology in the expansion given by the integrand reduction. First it calculates the topologies with the most propagators, and when they are known one may calculate the lower topologies using the higher ones as subtraction terms. Combined with specialized methods to find the rational term, the OPP method provides a complete procedure for finding the one-loop corrections to any amplitude.

In this thesis we will develop a method to extend the OPP method to two loops and beyond. It is based on a categorization of the integrand using algebraic geometry, in which the set of propagators corresponding to each topology is identified with an algebraic ideal I. This allows for the identification of the set of terms which are allowed in the numerator corresponding to each topology, with those of the members of the quotient ring R/I which lives up to a set of renormalization constraints. The method works in both four and d space-time dimensions.

We start the thesis by an introduction to QCD, amplitudes, and unitarity cuts. Then we do a number of four-dimensional examples of our form of the OPP method in the context of the process gg -> gg We calculate the one-loop contribution in significant detail, the three two-loop, seven-propagator topologies called the double-box, the crossed box, and the pentagon-triangle, and finally we calculate the triple-box three-loop topology. We also take a brief look at another three-loop topology, the tennis court.

By the additional calculation of a number of six-propagator two-loop topologies, we illustrate two problems with our method in its four-dimensional version. The first is "the minor problem" in which a unitarity cut solutions for a topology coincide with a cut solution for one of its parent topologies causing infinities to appear in the calculation. The other is denoted "the major problem", and is characterized by the existence of terms which vanish on the cut without being a sum of terms proportional to products of the propagators.

It turns out that both of these problems can be solved by going to $d$ dimensions. We illustrate the d-dimensional method by repeating part of the one-loop gg -> gg calculation, showing how the higher dimensional contributions give rise to the rational term of the amplitude.

For doing d-dimensional two-loop calculations, we describe how to embed the higher-dimensional parts of the loop-momenta, denoted rho-parameters, into two extra dimensions, and we describe how to do calculations in these dimensions using the six-dimensional spinor-helicity formalism. As an example we calculate the planar two-loop contribution to gg -> gg for the case where all external gluons have the same helicity, and we get agreement with the known result.

The main calculation of these thesis is that of the planar part of gg -> ggg for the mentioned helicity configuration. There are eight topologies contributing to the amplitude and we find that the result agrees with numerical checks. We find a curious relation between the result and the MHV result for N=4 SYM, similar to a known relation at one-loop. We also calculate a few non-planar contributions to the amplitude.

We end the thesis by comparing the method to a number of alternative methods proposed by other groups, and by a number of appendices. Primary among the appendices are a detailed introduction to the six-dimensional spinor-helicity formalism, and a derivation of the method used to perform integrals over the higher dimensional rho-parameters.