Dimensional continuation, regularization, minimal subtraction (MS). Renormalization group (RG) functions

Abstract

Abstract In this chapter, the notions of dimensional continuation and dimensional regularization are introduced, by defining a continuation of Feynman diagrams to analytic functions of the space dimension. Dimensional continuation, which is essential for generating Wilson–Fisher's famous ϵexpansion in the theory of critical phenomena, and dimensional regularization seem to have no meaning outside the perturbative expansion of quantum field theory (QFT). Dimensional regularization is a powerful regularization technique, which is often used, when applicable because it leads to much simpler perturbative calculations. Dimensional regularization performs a partial renormalization, cancelling what would show up as power-law divergences in momentum or lattice regularization. In particular it cancels the commutator of quantum operators in local QFTs. These cancellations may be convenient but may also, occasionally, remove divergences that have an important physical meaning. It is not applicable when some essential property of the field theory is specific to the initial dimension. For example, in even space dimensions, the relation between γS (identical to γ5 in four dimensions) and the other γ matrices involving the completely antisymmetric tensor ϵμ1···μd, may be needed in theories violating parity symmetry. Its use requires some care in massless theories because its rules may lead to unwanted cancellations between ultraviolet and infrared logarithmic divergences. Explicit calculations at two-loop order in a scalar QFT with a general four-field interaction are performed.