Three-loop contribution of the Faddeev–Popov ghosts to the $$\beta $$-function of $$\mathcal{N}=1$$ supersymmetric gauge theories and the NSVZ relation

We find the three-loop contribution to the $$\beta $$β-function of $$\mathcal{N}=1$$N=1 supersymmetric gauge theories regularized by higher covariant derivatives produced by the supergraphs containing loops of the Faddeev–Popov ghosts. This is done using a recently proposed algorithm, which essentially simplifies such multiloop calculations. The result is presented in the form of an integral of double total derivatives in the momentum space. The considered contribution to the $$\beta $$β-function is compared with the two-loop anomalous dimension of the Faddeev–Popov ghosts. This allows verifying the validity of the NSVZ equation written as a relation between the $$\beta $$β-function and the anomalous dimensions of the quantum superfields. It is demonstrated that in the considered approximation the NSVZ equation is satisfied for the renormalization group functions defined in terms of the bare couplings. The necessity of the nonlinear renormalization for the quantum gauge superfield is also confirmed.

LECTURES ON MULTILOOP CALCULATIONS

I discuss methods of calculation of propagator diagrams (massless, those of Heavy Quark Effective Theory, and massive on-shell diagrams) up to three loops. Integration-by-parts recurrence relations are used to reduce them to linear combinations of basis integrals. Nontrivial basis integrals have to be calculated by some other method, e.g. using Gegenbauer polynomial technique. Many of them are expressed via hypergeometric functions; in the massless and HQET cases, their indices tend to integers at ε→0. I discuss the algorithm of their expansion in ε, in terms of multiple ζ values.