epsilon: A tool to find a canonical basis of master integrals

In 2013, Henn proposed a special basis for a certain class of master integrals, which are expressible in terms of iterated integrals. In this basis, the master integrals obey a differential equation, where the right hand side is proportional to $\epsilon$ in $d=4-2\epsilon$ space-time dimensions. An algorithmic approach to find such a basis was found by Lee. We present the tool epsilon, an efficient implementation of Lee's algorithm based on the Fermat computer algebra system as computational backend.Comment: 34 pages; changed reference to fuchsi

Towards light quark mass effects in Higgs production and decay at next-to-next-to-leading order

Even though the standard model of particle physics withstands the most precise tests so far, one hopes to find inconsistencies of experimental data from high precision calculations especially in the Higgs sector. Such deviations would guide the direction of theoretical efforts towards a quantum field theory that is phenomenologically more complete, including e.g. dark matter or gravity. We consider two highly relevant processes of the Higgs physics measured at the Large Hadron Collider at CERN, i.e. Higgs production via gluon fusion and the decay of a Higgs boson into two photons. Already at leading-order the contributions to both processes require a one-loop calculation. Nevertheless, the cross section of this Higgs production mode as well as the partial Higgs decay rate were already calculated to very high precision. However this calculation was done in the limit of a heavy top-quark and neglects effects from lighter quarks. This approximation increases the uncertainty of the gluon fusion cross section by about one percent. For both processes, a calculation with an exact quark-mass dependence is only known up to next-to-leading order. In this thesis, we discuss techniques to improve the next-to-next-to-leading order predictions. We show that a standard setup for multiloop calculations can provide a useful starting point for the computation of light quark mass effects. While expressing the Feynman amplitudes in terms of scalar integrals is straightforward, the reduction of those integrals to a small set of so-called master integrals is highly non-trivial. The major part of this thesis deals with calculation methods for master integrals. Recent progress in the method of differential equations motivated us to develop a tool for the construction of a so-called canonical basis of master integrals. Within this basis, the calculation of the master integrals becomes trivial. Using this technique, we were able to solve about half of the master integrals for the Higgs decay into two photons. For the remaining integrals, we use a different approach. Here, a calculation of the exact quark-mass dependence is not feasible at the moment. Therefore, we aim at an expansion in a small quark mass compared to the Higgs-boson mass. This expansion requires a combination of the method of differential equations and Mellin-Barnes techniques. In this process, we found a novel method for the construction of Mellin-Barnes representations, which complements existing approaches. We hope that the techniques discussed in this thesis are sufficient to obtain results for the light quark effects at the next-to-next-to-leading order of both processes. A complete calculation is still work in progress. Nevertheless, the techniques developed during the thesis can still be useful also for other problems

Mellin-Barnes meets Method of Brackets: a novel approach to Mellin-Barnes representations of Feynman integrals

In this paper, we present a new approach to the construction of Mellin-Barnes representations for Feynman integrals inspired by the Method of Brackets. The novel technique is helpful to lower the dimensionality of Mellin-Barnes representations in complicated cases, some examples are given.Comment: 22 pages, more details on the numerical evaluation, matches published versio