Non-universal Critical Quantities from Variational Perturbation Theory and Their Application to the BEC Temperature Shift

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For an O(N) symmetric scalar field theory with Euclidean action integral d³x [1/2 |nabla phi|² + 1/2 r phi² + 1/4! u phi⁴], where phi = (phi₁,...,phiₙ) 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)u² at the phase transition. Its exact large-N limit is determined as well. We also extend an earlier computation of the interaction-induced shift Delta<phi²>/Nu for 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 The appendix contains previously unpublished numerical seven-loop data provided to us by B.Nickel.