General properties of multiscalar RG Flows in d = 4 - ε

Fixed points of scalar field theories with quartic interactions in d = 4 - epsilon dimensions are considered in full generality. For such theories it is known that there exists a scalar function A of the couplings through which the leading-order beta-function can be expressed as a gradient. It is here proved that the fixed-point value of A is bounded from below by a simple expression linear in the dimension of the vector order parameter, N. Saturation of the bound requires a marginal deformation, and is shown to arise when fixed points with the same global symmetry coincide in coupling space. Several general results about scalar CFTs are discussed, and a review of known fixed points is given.