Renormalization-scheme ambiguity and perturbation theory near a fixed point

Abstract

We consider the perturbative calculation of critical exponents in massless, renormalizable theories having a nontrivial fixed point. In conventional perturbation theory, all results depend on the arbitrary renormalization scheme used. We show how to resolve this problem, following the "principle of minimal sensitivity" approach. At least three orders of perturbation theory are required for quantitative results. We give scheme-independent criteria for determining the presence or absence of a fixed point in $n\mathrm{th}$ order, and discuss the conditions under which perturbative results might be reliable. As illustrations we discuss QED with many flavors, and ${({\ensuremath{\varphi}}^{4})}_{4}$ theory. In neither case do we find a fixed point, in contrast to naive perturbative expectations.