Study of the ultraviolet behavior of an O (N) | φ → |6 theory in d=3 dimensions

We study the ultraviolet (UV) behavior of an O(N) |φ→|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 |φ→|6 coupling, g. The four-loop (4ℓ) beta function is known to have a (scheme-independent) UV zero at g=gUV,4ℓ, 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, gUV,6ℓ, if N>Nc, where Nc≃796, and we calculate gUV,6ℓ. To investigate the reliability of the result in the region of N≳Nc, we apply three methods: (i) calculation of the fractional difference between gUV,4ℓ and gUV,6ℓ, (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/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 gUV,4ℓ and gUV,6ℓ must be less than 15%, then our results show that this requirement is satisfied for N≳2×103. The possible role of nonperturbative effects is also noted.