Lifshitz - point critical behaviour to O ( ǫ 2 )

We comment on a recent letter by L. C. de Albuquerque and M. M. in which results to second order in ǫ = 4 − d + m 2 were presented for the critical exponents ν L2 , η L2 and γ L2 of d-dimensional systems at m-axial Lifshitz points. We point out that their results are at variance with ours. The discrepancy is due to their incorrect computation of momentum-space integrals. Their speculation that the field-theoretic renormalization group approach, if performed in position space, might give results different from when it is performed in momentum space is refuted. In a recent letter [1] de Albuquerque and Leite (AL) presented results to second order in ǫ = 4 − d + m 2 for the critical exponents ν L2 , η L2 and γ L2 of d-dimensional systems with m-axial Lifshitz points. For the special case m = 1 of a uniaxial Lifshitz point, these results were previously given in a (so far apparently unpublished) preprint [2]. The ǫ 2 terms AL found are at variance with ours [3, 4]. As an explanation for these discrepancies AL suggest the following. Both we as well as AL employed a field-theoretic renormalization group approach based on dimensional regularization. To compute the residua of the ultraviolet poles at ǫ = 0, we found it convenient to perform (part of) the calculation in position space. By contrast, AL worked entirely in momentum space. They speculate [1] " that calculations performed in momentum space and coordinate space are inequivalent, as far as the Lifshitz critical behaviour is concerned ". This speculation is untenable and a serious misconception, a fact which should be obvious not only to readers with a background in field theory. The reason simply is: at each step of the calculation one can transform from momentum space to position space and vice versa.