Fixed points of scalar field theories with quartic interactions in $d=4-\varepsilon$ 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.
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- 1 Institut des Hautes Études Scientifiques [IHÉS]
- 2 École Normale Supérieure [ENS]
- 3 Los Alamos National Laboratory [LANL]
- 4 Organisation européenne pour la recherche nucléaire / European Organization for Nuclear Research [CERN]
- Engineering and Physical Sciences Research Council [EPSRC]
- Isaac Newton Institute for Mathematical Sciences [INI]
- Mitsubishi International Corporation (through Organization: 三菱重工業株式会社 / Mitsubishi Heavy Industries (Japan) [MHI])
- Simons Foundation