Heavy Handed Quest for Fixed Points in Multiple Coupling Scalar Theories in the $\epsilon$ Expansion

Abstract

The tensorial equations for non trivial fully interacting fixed points at lowest order in the $\epsilon$ expansion in $4-\epsilon$ and $3-\epsilon$ dimensions are analysed for $N$-component fields and corresponding multi-index couplings $\lambda$ which are symmetric tensors with four or six indices. Both analytic and numerical methods are used. For $N=5,6,7$ in the four-index case large numbers of irrational fixed points are found numerically where $||\lambda |{|}^2$ is close to the bound found by Rychkov and Stergiou in arXiv:1810.10541. No solutions, other than those already known, are found which saturate the bound. These examples in general do not have unique quadratic invariants in the fields. For $N⩾6$ the stability matrix in the full space of couplings always has negative eigenvalues. In the six index case the numerical search generates a very large number of solutions for $N=5$.

50 pages, 6 figures. v2: 53 pages, 6 figures; Expanded discussion of N=4 case including splitting of fixed point at order ${\epsilon }^2$; v4: Minor corrections and additions

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• The bi-conical vector model at $1/N$
• Bootstrapping Mixed MN Correlators in 3D
• The search for the universality class of metric quantum gravity
• Coupled minimal models revisited II: Constraints from permutation symmetry
• Anomalous Dimensions in Hypercubic Theories
• Classifying irreducible fixed points of five scalar fields in perturbation theory
• Note on surface defects in multiscalar critical models