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

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 poles in d=4+m2−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 βu(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 ηl2, ηl4, νl2, νl4, the wave-vector exponent βq, and the correction-to-scaling exponent are obtained to order ϵ2. 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.