Monte Carlo studies of three-dimensional O(1) and O(4) \boldmath{$\phi^4$} theory related to BEC phase transition temperatures

The phase transition temperature for the Bose-Einstein condensation (BEC) of weakly-interacting Bose gases in three dimensions is known to be related to certain non-universal properties of the phase transition of three-dimensional O(2) symmetric $\phi^4$ theory. These properties have been measured previously in Monte Carlo lattice simulations. They have also been approximated analytically, with moderate success, by large $N$ approximations to O($N$) symmetric $\phi^4$ theory. To begin investigating the region of validity of the large $N$ approximation in this application, I have applied the same Monte Carlo technique developed for the O(2) model ([5]) to O(1) and O(4) theories. My results indicate that there might exist some theoretically unanticipated systematic errors in the extrapolation of the continuum value from lattice Monte Carlo results. The final results show that the difference between simulations and NLO large $N$ calculations does not improve significantly from N=2 to N=4. This suggests one would need to simulate yet larger $N$'s to see true large $N$ scaling of the difference. Quite unexpectedly (and presumably accidentally), my Monte Carlo result for N=1 seems to give the best agreement with the large $N$ approximation among the three cases.