Nonuniversal critical quantities from variational perturbation theory and their application to the Bose-Einstein condensation temperature shift

Abstract

For an $\mathrm{O}(N)$-symmetric scalar field theory with Euclidean action $\ensuremath{\int}{d}^{3}\phantom{\rule{0.3em}{0ex}}x[\frac{1}{2}\ensuremath{\mid}\ensuremath{\nabla}\ensuremath{\phi}{\ensuremath{\mid}}^{2}+\frac{1}{2}r{\ensuremath{\phi}}^{2}+\frac{1}{4!}u{\ensuremath{\phi}}^{4}]$, where $\ensuremath{\phi}=({\ensuremath{\phi}}_{1},\dots{},{\ensuremath{\phi}}_{N})$ is a vector of $N$ real-field components, variational perturbation theory through seven loops is employed for $N=0,1,2,3,4$ to compute the renormalized value of $r∕(N+2){u}^{2}$ at the phase transition. Its exact large-$N$ limit is determined as well. We also extend an earlier computation of the interaction-induced shift $\ensuremath{\Delta}⟨{\ensuremath{\phi}}^{2}⟩∕Nu$ from $N=1,2,4$ to $N=0,3$. For $N=2$, the results for the two quantities are used to compute the second-order shift of the condensation temperature of a dilute Bose gas, both in the homogenous case and for the wide limit of a harmonic trap. Our results are in agreement with earlier Monte Carlo simulations for $N=1,2,4$. The Appendix contains previously unpublished numerical seven-loop data.