General Properties of Multiscalar RG Flows in $d=4-\epsilon$

Abstract

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.

29 pages, 4 figures; see section 3 for a prize problem. v2: small correction in appendix, typos fixed. v3: minor additions. v4: some next-to-leading order results added, typos fixed