Question of an ultraviolet zero of the beta function of the $\ensuremath{\lambda}({\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\phi}}}^{2}{)}_{4}^{2}$ theory

Question of an ultraviolet zero of the beta function of the $λ ({\vec{ϕ}}^{2})_{4}^{2}$ theory

Robert Shrock

Phys. Rev. D 90, 065023 – Published 19 September 2014

Abstract

We investigate the possibility of an ultraviolet (UV) zero in the $n$ -loop beta function of a $λ ({\vec{ϕ}}^{2})^{2}$ field theory with an $N$ -component scalar field, $\vec{ϕ}$ , in four spacetime dimensions, up to the level of $n = 5$ loops. Although the two-loop beta function has a UV zero, we find that the $n$ -loop beta function for $n = 3$ , 4, 5 either does not have a UV zero or does not have one at a value of $λ$ in approximate agreement with the two-loop calculation. Similar results are obtained after application of scheme transformations to the beta function and via calculation of Padé approximants. We thus conclude that in the range of $λ$ where the perturbative calculation of the $n$ -loop beta function is reliable, the theory does not exhibit robust evidence of a UV zero up to the level of $n = 5$ loops.