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

Abstract

A tensorial representation of ${\varphi }^4$ 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.

8 pages, 7 figures