Seeking Fixed Points in Multiple Coupling Scalar Theories in the $\varepsilon$ Expansion

TL;DR

This paper explores fixed points in multi-coupling scalar theories across various dimensions using the $oldsymbol{ extit{ ext{epsilon}}}$-expansion, revealing how known fixed points emerge from a general framework and applying the $a$-theorem for classification.

Contribution

It introduces a unified framework with two couplings to derive fixed points in scalar theories across different dimensions, extending known results and analyzing symmetry breaking and the $a$-theorem.

Findings

Known fixed points are obtained from the general two-coupling framework.

Symmetry breaking patterns from $O(N)$ to subgroups are classified.

A lower bound for the $a$-function at fixed points is established.

Abstract

Fixed points for scalar theories in $4-\varepsilon$, $6-\varepsilon$ and $3-\varepsilon$ dimensions are discussed. It is shown how a large range of known fixed points for the four dimensional case can be obtained by using a general framework with two couplings. The original maximal symmetry, $O(N)$, is broken to various subgroups, both discrete and continuous. A similar discussion is applied to the six dimensional case. Perturbative applications of the $a$-theorem are used to help classify potential fixed points. At lowest order in the $\varepsilon$-expansion it is shown that at fixed points there is a lower bound for $a$ which is saturated at bifurcation points.