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

We study the ultraviolet (UV) behavior of an O($N$) $\mathrm{\mid }\vec{\phi }{\mathrm{\mid }}^6$ 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 $\mathrm{\mid }\vec{\phi }{\mathrm{\mid }}^6$ coupling, $g$. The four-loop $(4\mathrm{\ell })$ beta function is known to have a (scheme-independent) UV zero at $g=g_{{}_{UV,4\mathrm{\ell }}}$, 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, $g_{{}_{UV,6\mathrm{\ell }}}$, if $N>N_c$, where $N_c\simeq 796$, and we calculate $g_{{}_{UV,6\mathrm{\ell }}}$. 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 $g_{{}_{UV,4\mathrm{\ell }}}$ and $g_{{}_{UV,6\mathrm{\ell }}}$, (ii) a Padé 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 $1\mathrm{/}N$ 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 $g_{{}_{UV,4\mathrm{\ell }}}$ and $g_{{}_{UV,6\mathrm{\ell }}}$ 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.