Two-loop renormalization-group analysis of critical behavior at m-axial Lifshitz points

We investigate the critical behavior that d-dimensional systems with short- range forces and an n-component order parameter exhibit at Lifshitz points whose wave-vector instability occurs in an m-dimensional isotropic subspace of Rd. Utilizing dimensional regularization and minimal subtraction of pol es in d = 4 + m/2 - epsilon dimensions, we carry out a two-loop renormaliza tion-group (RG) analysis of the field-theory models representing the corres ponding universality classes. This gives the beta function beta (u)(u) to t hird order, and the required renormalization factors as well as the associa ted RG exponent functions to second order, in u. The coefficients of these series are reduced to m-dependent expressions involving single integrals, w hich for general (not necessarily integer) values of m is an element of (0, 8) can be computed numerically, and for special values of in analytically. The epsilon expansions of the critical exponents eta (l2), eta (l4), nu (l 2), nu (l4), the wave-vector exponent beta (q), and the correction-to-scali ng exponent are obtained to order epsilon (2). These are used to estimate t heir values for d = 3. The obtained series expansions are shown to encompas s both isotropic limits in = 0 and m = d. (C) 2001 Elsevier Science B.V. Al l rights reserved.