Study of the ultraviolet behavior of an O(N) |()over-right-arrow|⁶ theory in d=3 dimensions

We study the ultraviolet (UV) behavior of an O(N) vertical bar(phi) over right arrow vertical bar(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 vertical bar(phi) over right arrow vertical bar(6) coupling, g. The four-loop (4l) beta function is known to have a (scheme-independent) UV zero at g = g(UV;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, g(UV;6l), if N > N-c, where N-c similar or equal to 796, and we calculate g(UV,6l). To investigate the reliability of the result in the region of N greater than or similar to N-c, we apply three methods: (i) calculation of the fractional difference between g(UV,4l) and g(UV,6l), (ii) a Pade 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,4l) and g(UV),(6l) must be less than 15%, then our results show that this requirement is satisfied for N greater than or similar to 2 x 10(3). The possible role of nonperturbative effects is also noted.