We use the nonperturbative linear δ expansion method to evaluate analytically the coefficients c1 and c ′′ 2 which appear in the expansion for the transition temperature for a dilute, homogeneous, three dimensional Bose gas given by Tc = T0{1+c1an 1/3 +[c ′ 2 ln(an 1/3 )+c ′′ 2 ]a 2 n 2/3 +O(a 3 n)}, where T0 is the result for an ideal gas, a is the s-wave scattering length and n is the number density. In a previous work the same method has been used to evaluate c1 to order-δ 2 with the result c1 = 3.06. Here, we push the calculation to the next two orders obtaining c1 = 2.45 at order-δ 3 and c1 = 1.48 at order-δ 4 . Analysing the topology of the graphs involved we discuss how our results relate to other nonperturbative analytical methods such as the self-consistent resummation and the 1/N approximations. At the same orders we obtain c ′′ 2 = 101.4, c ′′ 2 = 98.2 and c ′′ 2 = 82.9. Our analytical results seem to support the recent Monte Carlo estimates c1 = 1.32 ± 0.02 and c ′′ 2 = 75.7 ± 0.4.

We use the optimized perturbation theory, or linear δ expansion, to evaluate the critical exponents in the critical 3d O(N ) invariant scalar field model. Regarding the implementation procedure, this is the first successful attempt to use the method in this type of evaluation. We present and discuss all the associated subtleties producing a prescription which can, in principle, be extended to higher orders in a consistent way. Numerically, our approach, taken at the lowest nontrivial order (second order) in the δ expansion produces a modest improvement in comparison to mean field values for the anomalous dimension η and correlation length ν critical exponents. However, it nevertheless points to the right direction of the values obtained with other methods, like the ǫ-expansion. We discuss the possibilities of improving over our lowest order results and on the convergence to the known values when extending the method to higher orders.