Critical behavior at m-axial Lifshitz points:: Field-theory analysis and ε-expansion results

The critical behavior of d-dimensional systems with an n-component order parameter is reconsidered at (m,d,n)-lifshitz points, where a wave-vector instability occurs in an m-dimensional subspace of R-d. Our aim is to sort out which ones of the previously published partly contradictory epsilon -expansion results to second order in epsilon =4+ m/2-d are correct. To this end, a field-theory calculation is performed directly in the position space of d=4+m/2-epsilon dimensions, using dimensional regularization and minimal subtraction of ultraviolet poles. The residua of the dimensionally regularized integrals that are required to determine the series expansions of the correlation exponents eta (12), and eta (14) and of the wave-vector exponent beta (q) to order epsilon (2) are reduced to single integrals, which for general m= 1,...,d-1 can be computed numerically, and for special values of nt, analytically. Our results are at variance with the original predictions for general m. For m = 2 and m = 6, we confirm the results of Sak and Grest [Phys. Rev. B 17, 3602 (1978)] and Mergulhao and Carneiro's recent held-theory analysis [Phys. Rev. B 59, 13 954 (1999)].