Field-theoretic approach to the Lifshitz point

Abstract

We study the renormalization of the field theory that describes the Lifshitz point (LP). Our motivation was an old controversy on the order-${\ensuremath{\epsilon}}^{2}$ values of critical exponents for this multicritical point. First we analyze the Green functions at the LP where some simplifications occur. The primitively divergent diagrams are identified and renormalization prescriptions that eliminate ultraviolet divergences to all orders of perturbation are found. The Green functions in the neighborhood of the LP are expanded in terms of the Green functions calculated at the LP. This enables us to derive the renormalization-group equation satisfied by the renormalized Green functions and by analyzing its solutions we find expressions for the critical exponents that hold to all orders of perturbation. Finally, we obtain generalized scaling relations for the exponents associated with the LP.