Critical behavior at a Lifshitz point: Calculation of a universal amplitude ratio

The phase diagram in the vicinity of a Lifshitz point is analyzed using scaling arguments and renormalization-group calculations. Feynman-graph techniques in d=4+1/2m-ε dimensions are used to calculate the universal amplitude ratio governing the shape of the two branches of the critical line which intersect at an m-fold Lifshitz point. For $m<6$ the amplitude ratio is found to be AH(m)AF(m)=1-2/m+O(ε), where AH(AF) is the amplitude associated with the helical (ferromagnetic) branch, so that to zeroth order in ε, this ratio is independent of the number n of components of the order parameter. The calculation indicates that along the remainder ($6<~m<~8$) of the critical dimensionality line, ε=0, no helical long-range order is expected. We also calculate AHAF for a spherical model of a d-dimensional system exhibiting an m-fold Lifshitz point. Along the critical dimensionality line ε=0, the spherical model result reduces to that of the renormalization-group calculation.