Relevance of space anisotropy in the critical behavior of m-axial Lifshitz points

The critical behavior of d-dimensional systems with n-component order parameter $\mathit{\phi }$ is studied at an m-axial Lifshitz point where a wave-vector instability occurs in an m-dimensional subspace $R^m$ $\mathrm{}(m>\mathrm{}1\mathrm{}).$ Field theoretic renormalization group techniques are exploited to examine the effects of terms in the Hamiltonian that break the rotational symmetry of the Euclidean group $E\mathrm{}\left(m\right).$ The framework for considering general operators of second order in $\mathit{\phi }$ and fourth order in the derivatives ${\partial }_{\alpha }$ with respect to the Cartesian coordinates $x_{\alpha }$ of $R^m$ is presented. For the specific case of systems with cubic anisotropy, the effects of having an additional term, ${\sum }_{\alpha =1\mathrm{}}^m({\partial }_{\alpha }^2\mathit{\phi }{)}^2,$ are investigated in an $\epsilon$ expansion about the upper critical dimension $d^{*}\mathrm{}\left(m\right)=4+m/2.\mathrm{}$ Its associated crossover exponent is computed to order ${\epsilon }^2$ and found to be positive, so that it is a relevant perturbation on a model isotropic in $R^m.$