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 ${\mathbb{R}}^d$. Our aim is to sort out which ones of the previously published partly contradictory $ϵ$-expansion results to second order in $ϵ=4+\frac{m}{2}-d$ are correct. To this end, a field-theory calculation is performed directly in the position space of $d=4+\frac{m}{2}-ϵ$ 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 ${\eta }_{l2}$ and ${\eta }_{l4}$ and of the wave-vector exponent ${\beta }_q$ to order ${ϵ}^2$ are reduced to single integrals, which for general m=1,…,d-1 can be computed numerically, and for special values of m, 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 {\bf 17}, 3602 (1978)] and Mergulh{~a}o and Carneiro’s recent field-theory analysis [Phys. Rev. B {\bf 59},13954 (1999)].