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

Abstract

We investigate the critical behavior that d-dimensional systems with short-range forces and a n-component order parameter exhibit at Lifshitz points whose wave-vector instability occurs in a m-dimensional isotropic subspace of ${\mathbb R}^d$. Utilizing dimensional regularization and minimal subtraction of poles in $d=4+{m\over 2}-ε$ 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 $β_u(u)$ 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 $m\in (0,8)$ can be computed numerically, and for special values of m analytically. The $ε$ expansions of the critical exponents $η_{l2}$, $η_{l4}$, $ν_{l2}$, $ν_{l4}$, the wave-vector exponent $β_q$, and the correction-to-scaling exponent are obtained to order $ε^2$. 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.