Critical $O(2)$ field theory near six dimensions beyond one loop

TL;DR

This study investigates an $O(2)$ tensorial field theory near six dimensions using multiple renormalization group methods, revealing a potential fixed point that becomes complex as the dimension approaches five, supporting triviality above four dimensions.

Contribution

It provides a detailed multi-method analysis of the $O(2)$ field theory near six dimensions, highlighting the fixed point behavior and its potential destabilization approaching five dimensions.

Findings

An interacting fixed point exists near six dimensions.

Second-order corrections may destabilize the fixed point at certain epsilon.

The fixed point collides with another and becomes complex near five dimensions.

Abstract

A tensorial representation of $\phi^4$ field theory introduced in Phys. Rev. D. 93, 085005 (2016) is studied close to six dimensions, with an eye towards a possible realization of an interacting conformal field theory in five dimensions. We employ the two-loop $\epsilon$-expansion, two-loop fixed-dimension renormalization group, and non-perturbative functional renormalization group. An interacting, real, infrared-stable fixed point is found near six dimensions, and the corresponding anomalous dimensions are computed to the second order in small parameter $\epsilon=6-d$. Two-loop epsilon-expansion indicates, however, that the second-order corrections may destabilize the fixed point at some critical $\epsilon_c <1$. A more detailed analysis within all three computational schemes suggests that the interacting, infrared-stable fixed point found previously collides with another fixed point and becomes complex when the dimension is lowered from six towards five. Such a result would conform to the expectation of triviality of $O(2)$ field theories above four dimensions.