Study of the question of an ultraviolet zero in the six-loop beta function of the $\mathrm{O}(N)$ $\lambda |\overrightarrow{\varphi }{|}^4$ theory

We study the possibility of an ultraviolet (UV) zero in the six-loop beta function of an $\mathrm{O}(N)$ $\lambda |\overrightarrow{\varphi }{|}^4$ field theory in $d=4$ spacetime dimensions. For general $N$, in the range of values of $\lambda$ where a perturbative calculation is reliable, we find evidence against such a UV zero in this six-loop beta function.

Figure 1

Plot of the ratio $R_n\equiv R_{a,n}$ defined in Eq. (2.6), as a function of $a$, for $N=10$ and (i) $n=2$ (red), (ii) $n=3$ (green), (iii) $n=4$ (cyan), (iv) $n=5$ (blue), and (v) $n=6$ (black) (colors in online version). Along a counterclockwise path around the point $(a,R_n)=(0,1)$ starting at the point $(a,R_n)=(0.1,0)$, the curves are for $n=6$, $n=4$, $n=2$, $n=3$, and $n=5$.