T c for homogeneous dilute Bose gases : a second - order result

The transition temperature for a dilute, homogeneous, three-dimensional Bose gas has the expansion Tc = T0{1+c1an+[c2 ln(an1/3)+c′′ 2 ]a2n2/3+O(a3n)}, where a is the scattering length, n the number density, and T0 the ideal gas result. The first-order coefficient c1 depends on non-perturbative physics. In this paper, we show that the coefficient c′2 can be computed perturbatively. We also show that the remaining second-order coefficient c 2 depends on nonperturbative physics but can be related, by a perturbative calculation, to quantities that have previously been measured using lattice simulations of three-dimensional O(2) scalar field theory. Making use of those simulation results, we find Tc ≃ T0{1 + (1.32± 0.02) an1/3 + [19.7518 ln(an1/3) + (75.7± 0.4)]a2n2/3 +O(a3n)}.