Massive 2- and 3-loop corrections to hard scattering processes in QCD

This thesis deals with calculations of higher-order corrections in perturbative quantum chromodynamics(QCD). The two-mass contributions to the 3-loop, polarized twist-two operator matrixelements (OMEs) A(3),PSQq and A(3)gg,Q are calculated. The N-space result for A(3)gg,Q is obtained analyticallyas a function of the quark mass ratio, which for A(3),PSQq is not yet possible. In thez-space representation, one obtains for both matrix elements semi-analytical representations interms of iterated integrals, whereby for reasons of efficiency an additional integral is necessaryfor some terms.These universal (process-independent) massive OMEs govern the asymptotic behaviour ofthe Wilson coefficients in deep-inelastic scattering at large virtualities Q2 ≫ m2c,b, with mc,bthe charm and bottom quark masses. These corrections are also required to define the variableflavour number scheme. This scheme describes the transition from massive quark corrections tothe massless ones for very high momentum scales, which is relevant to the description of colliderdata.In the single-mass, polarized case, we derive the logarithmic corrections for the Wilson coefficientsof the structure function g1 in the asymptotic region Q2 ≫ m2c,b. This is done using theknown OMEs and massless Wilson coefficients, using the renormalization group equations.For the non-singlet structure functions FNS2 and gNS1 we revisit the scheme-invariant evolutionoperator known for massless quarks and extend it to the massive case with single- and two-masscorrections. In this case, the evolution can effectively be described up to O(a3s) in the Wilsoncoefficients, where as = αs/(4π) denotes the strong coupling constant. The influence of thehitherto not fully known 4-loop non-singlet anomalous dimension can be described effectively.It turns out that the effect of the theory error in question can be completely controlled. Arepresentation by a Pad´e approximant proves to be sufficient.We consider the class of functions of multivariate hypergeometric series and study systemsof differential equations obeyed by them. We describe an algorithmic method to solve someclasses of such differential systems which delivers a hypergeometric series solution having nestedhypergeometric products as summand; we discuss the relationship between these products andPochhammer symbols. For a number of classical hypergeometric series we derive differentialsystems and their associated difference equations. We present some examples of series expansionsof such functions and of the mathematical objects which arise therein. We also present aMathematica package which implements algorithms related to the solution of partial linear differenceequations, focusing in particular on bounding the degree of the denominator of solutionswhich are rational functions. These methods are of particular importance when solving multi-legcalculations for Feynman diagrams, but also come into play when hypergeometric methods formulti-loop integrals are used.We describe a numerical implementation of an N-space library for the calculation of scalingviolations for structure functions, which can perform the evolution of parton distributionfunctions up to NNLO from a parametrization chosen by the user, and encodes massless andmassive Wilson coefficients for the structure functions F2 and g1 in the case of photon exchange,and for the structure functions FW+±W−3 in the case of charged-current exchange. The librarycontains analytic continuation of the relevant harmonic sums in Mellin-space up to weight 5 andmany weight-6 harmonic sums. The numerical representation in x space is performed by contourintegration around the singularities of the solution of the evolution equations in N space