Non-perturbative renormalisation using dimensional regularisation: applications to the ϵ expansion

The authors give a prescription for the one-loop renormalisation of the imaginary parts of vertex functions in g phi 4 , which are generated when g<0. Within the dimensional regularisation scheme this involves another, non-perturbative, renormalisation over and above the usual one-loop perturbative renormalisation. The authors use an extended minimal subtraction scheme so that the resulting renormalisation group functions have a simple dependence on epsilon =4-d. In contrast to the usual case, however, both the beta function and the renormalised coupling g R are in general complex for real g, and the fixed point for epsilon , g R <0 is non-perturbative in epsilon . This fixed point determines the imaginary parts of the critical exponents which are generated when epsilon <0, and allows the determination of the high-order behaviour of the perturbation series in epsilon for these exponents. The generalisation of these ideas to the O(n) symmetric g( phi 2 ) 2 model is also given.