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We calculate the critical exponents $$\omega _\pm $$ ω± in the d-dimensional Gross–Neveu model in 1 / N expansion with $$1/N^2$$ 1/N2 accuracy. These exponents are related to the slopes of the $$\beta $$ β -functions at the critical point in the Gross–Neveu–Yukawa model. They have been computed recently to four loops accuracy. We checked that our results are in complete agreement with the results of the perturbative calculations.

In this thesis we present the evolution equation for non-singlet leading-twist operators in QCD to next-to-next-to-leading order in an expansion in the strong coupling constant. The method we use is based on conformal symmetry arguments. Using Ward identities we derive the conformal symmetry breaking in QCD in integer dimensions -- also known as conformal anomaly -- to two-loop accuracy. This result allows one to define a conformal invariant theory in d=4-2ε dimension by tuning the strong coupling to a certain (critical) value. The symmetry revealed in that way allows us to solve for the evolution kernel to the three-loop accuracy. This result is given both in the formulation of non-local light-ray operators as well as local operators. In the latter case we present an explicit analytic solution of the NNLO evolution equation on the example of the pion distribution amplitude., In dieser Arbeit präsentieren wir die Renormierungsgruppen-Gleichung für nicht-singlet twist-2 Operatoren in QCD bis zur dritten Ordnung in der störungstheoretischen Entwicklung. Die angewandte Methode beruht auf Symmetrieüberlegungen der konformen Symmetry-Gruppe. Mithilfe von Ward-Identitäten untersuchen wir die Anomalie der konformen Symmetriebrechung in QCD bis zur zweiten Ordnung in der Kopplungskonstante. Diese Anomalie erlaubt die Renormierungsgruppen-Gleichung zur dritten Ordnung in der Kopplungskonstanten zu bestimmen. Wir präsentieren die Ergebnisse sowohl für lokale als auch nicht-lokale Operatoren. Wir wenden unsere Ergebnisse an, um die Skalenabhängigkeit der Pion Distributionsamplitude zu bestimmen.

We calculate the critical exponents $\omega_\pm$ in the $d$-dimensional Gross-Neveu model in $1/N$ expansion with $1/N^2$ accuracy. These exponents are related to the slopes of the $\beta$-functions at the critical point in the Gross - Neveu - Yukawa model. They have been computed recently to four loops accuracy. We checked that our results are in complete agreement with the results of the perturbative calculations.