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

TL;DR

This paper analyzes fixed points in multi-coupling scalar theories using $ abla$ expansion, revealing numerous irrational solutions near theoretical bounds but no new solutions saturating these bounds.

Contribution

It provides a combined analytic and numerical analysis of fixed points in multi-index tensor scalar theories at lowest order in $ abla$ expansion, identifying many irrational solutions.

Findings

Large number of irrational fixed points found numerically.

No new solutions saturating the known bound are identified.

For N ≥ 6, the stability matrix always has negative eigenvalues.

Abstract

The tensorial equations for non trivial fully interacting fixed points at lowest order in the $\varepsilon$ expansion in $4-\varepsilon$ and $3-\varepsilon$ 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 $N \geqslant 6$ 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 $N=5$.