Study of the Ultraviolet Behavior of an O($N$) $|\vec \phi|^6$ Theory in $d=3$ Dimensions

We study the ultraviolet (UV) behavior of an O($N$) $|\vec \phi |^6$ theory in $d=3$ spacetime dimensions, focusing on the question of the range in $N$ over which the perturbative beta function exhibits robust evidence of a UV zero in the $|\vec \phi |^6$ coupling, $g$. The four-loop $(4\ell)$ beta function is known to have a (scheme-independent) UV zero at $g=g_{_{UV,4\ell}}$, which is reliably calculable for large $N$. For our analysis we use the six-loop beta function calculated in the minimal subtraction scheme. We find that this six-loop beta function has a UV zero, $g_{_{UV,6\ell}}$, if $N > N_c$, where $N_c \simeq 796$, and we calculate $g_{_{UV,6\ell}}$. To investigate the reliability of the result in the region of $N \gtrsim N_c$, we apply three methods: (i) calculation of the fractional difference between $g_{_{UV,4\ell}}$ and $g_{_{UV,6\ell}}$, (ii) a Pad\'e approximant, and (iii) an assessment of scheme dependence. Our results provide quantitative measures of the range of $N$ over which the six-loop beta function has a UV zero and of the $1/N$ corrections to the value of $g$ at the UV zero for large but finite $N$. If one imposes a benchmark requirement that the fractional difference between $g_{_{UV,4\ell}}$ and $g_{_{UV,6\ell}}$ must be less than 15 \%, then our results show that this requirement is satisfied for $N \gtrsim 2 \times 10^3$. The possible role of nonperturbative effects is also noted.