  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
$\epsilon$-expansion results to second order in $\epsilon=4+\frac{m}{2}-d$ are
correct. To this end, a field-theory calculation is performed directly in the
position space of $d=4+\frac{m}{2}-\epsilon$ 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 $\eta_{l2}$ and $\eta_{l4}$ 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 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)].
