Hjalte Frellesvig

Two-loop planar master integrals for Higgs → 3 partons with full heavy-quark mass dependence

A bstractWe present the analytic computation of all the planar master integrals which contribute to the two-loop scattering amplitudes for Higgs→ 3 partons, with full heavy-quark mass dependence. These are relevant for the NNLO corrections to fully inclusive Higgs production and to the NLO corrections to Higgs production in association with a jet, in the full theory. The computation is performed using the differential equations method. Whenever possible, a basis of master integrals that are pure functions of uniform weight is used. The result is expressed in terms of one-fold integrals of polylogarithms and elementary functions up to transcendental weight four. Two integral sectors are expressed in terms of elliptic integrals. We show that by introducing a one-dimensional parametrization of the integrals the relevant second order differential equation can be readily solved, and the solution can be expressed to all orders of the dimensional regularization parameter in terms of iterated integrals over elliptic kernels. We express the result for the elliptic sectors in terms of two and three-fold iterated integrals, which we find suitable for numerical evaluations. This is the first time that four-point multiscale Feynman integrals have been computed in a fully analytic way in terms of elliptic integrals.

Hepta-cuts of two-loop scattering amplitudes

A bstractWe present a method for the computation of hepta-cuts of two loop scattering amplitudes. Four dimensional unitarity cuts are used to factorise the integrand onto the product of six tree-level amplitudes evaluated at complex momentum values. Using Gram matrix constraints we derive a general parameterisation of the integrand which can be computed using polynomial fitting techniques. The resulting expression is further reduced to master integrals using conventional integration by parts methods. We consider both planar and non-planar topologies for 2 → 2 scattering processes and apply the method to compute hepta-cut contributions to gluon-gluon scattering in Yang-Mills theory with adjoint fermions and scalars.

A Two-Loop Five-Gluon Helicity Amplitude in QCD

A bstractWe compute the planar part of the two-loop five gluon amplitude with all helicities positive. To perform the calculation we develop a D-dimensional generalized unitarity procedure allowing us to reconstruct the amplitude by cutting into products of six-dimensional trees. We find a compact form for the integrand which only requires topologies with six or more propagators. We perform cross checks of the universal infra-red structure using numerical integration techniques.

An integrand reconstruction method for three-loop amplitudes

A bstractWe consider the maximal cut of a three-loop four point function with massless kinematics. By applying Gröbner bases and primary decomposition we develop a method which extracts all ten propagator master integral coefficients for an arbitrary triple-box configuration via generalized unitarity cuts. As an example we present analytic results for the three loop triple-box contribution to gluon-gluon scattering in Yang-Mills with adjoint fermions and scalars in terms of three master integrals.

Cuts of Feynman Integrals in Baikov representation

A bstractBased on the Baikov representation, we present a systematic approach to compute cuts of Feynman Integrals, appropriately defined in d dimensions. The information provided by these computations may be used to determine the class of functions needed to analytically express the full integrals.

Next-to-leading order QCD corrections to the decay width H → Zγ

A bstractWe present the analytic calculation of the two-loop QCD corrections to the decay width of a Higgs boson into a photon and a Z boson. The calculation is carried out using integration-by-parts identities for the reduction to master integrals of the scalar integrals, in terms of which we express the amplitude. The calculation of the master integrals is performed using differential equations applied to a set of functions suitably chosen to be of uniform weight. The final result is expressed in terms of logarithms and polylogarithmic functions Li2, Li3, Li4 and Li2,2.

On the reduction of generalized polylogarithms to Lin and Li2,2 and on the evaluation thereof

A bstractWe give expressions for all generalized polylogarithms up to weight four in terms of the functions log, Lin, and Li2,2, valid for arbitrary complex variables. Furthermore we provide algorithms for manipulation and numerical evaluation of Lin and Li2,2, and add codes in Mathematica and C++ implementing the results. With these results we calculate a number of previously unknown integrals, which we add in appendix C.

Multi-loop Integrand Reduction with Computational Algebraic Geometry

We discuss recent progress in multi-loop integrand reduction methods. Motivated by the possibility of an automated construction of multi-loop amplitudes via generalized unitarity cuts we describe a procedure to obtain a general parameterisation of any multi-loop integrand in a renormalizable gauge theory. The method relies on computational algebraic geometry techniques such as Grobner bases and primary decomposition of ideals. We present some results for two and three loop amplitudes obtained with the help of the MACAULAY2 computer algebra system and the Mathematica package BASISDET.

Multi-loop integrand reduction techniques

We review recent progress in D-dimensional integrand reduction algorithms for two loop amplitudes and give examples of their application to non-planar maximal cuts of the five-point all-plus helicity amplitude in QCD.

Two-Loop integrals for precision Higgs boson phenomenology

We report on the analytic computation of the planar master integrals relevant for the next-toleading order QCD corrections to the Higgs boson production in association with a jet in proton collisions, with exact top quark mass dependence. As a by product we compute the NLO QCD corrections to the decay width of the Higgs boson in a Z boson and a photon. The computation is performed with the differential equations method applied to a set of suitably chosen dimensionally regularized integrals.