ANALOGS OF THE C-THEOREM FOR 4-DIMENSIONAL RENORMALIZABLE FIELD-THEORIES

A perturbative analysis of products of composite operators on curved space is shown to give various consistency conditions which include a relation analogous to a generalisation of Zamolodchikov's c-theorem to four-dimensional renormalisable field theories. A detailed BRS analysis is given for gauge theories to ensure independence of gauge fixing and calculations of the various new counterterms required by this analysis are undertaken to two loops. Although positivity of the metric on the space of couplings is not demonstrated in general it is shown to be valid for weak coupling. The change in the C-function under renormalisation flow for gauge theories in the large-N limit with appropriate numbers of fermions, so that there is a perturbatively accessible infrared stable fixed point, is evaluated. The analysis is extended to take account of the lower dimension operators occuring in scalar field theories and the weak coupling metric is calculated for quartic scalar field and Yukawa interactions. The form of the β-function for the Yukawa coupling is shown to be constrained by the c-theorem demonstrated in this paper at two loops, in accord with previously calculated results. © 1990.