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

Abstract

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 R d . Utilizing dimensional regularization and minimal subtraction of poles in d=4+ m 2 −ϵ dimensions, we carry out a two-loop renormalization-group (RG) analysis of the field-theory models representing the corresponding universality classes. This gives the beta function β u ( u) to third order, and the required renormalization factors as well as the associated RG exponent functions to second order, in u. The coefficients of these series are reduced to m-dependent expressions involving single integrals, which for general (not necessarily integer) values of m∈(0,8) can be computed numerically, and for special values of m analytically. The ϵ expansions of the critical exponents η l2 , η l4 , ν l2 , ν l4 , the wave-vector exponent β q , and the correction-to-scaling exponent are obtained to order ϵ 2. These are used to estimate their values for d=3. The obtained series expansions are shown to encompass both isotropic limits m=0 and m= d.