Tcfor homogeneous dilute Bose gases: A second-order result

The transition temperature for a dilute, homogeneous, three-dimensional Bose gas has the expansion $T_{\mathrm{c}}{=T}_0\{{1+c}_1{\mathrm{an}}^{1/3}+\left[c_2^{\prime }\ln {(an}^{1/3}{)+c}_2^{″}\right]a^2{n}^{2/3}{+O(a}^{3}n)\},$ where a is the scattering length, n the number density, and $T_0$ the ideal gas result. The first-order coefficient $c_1$ depends on nonperturbative physics. In this paper, we show that the coefficient $c_2^{\prime }$ may be computed perturbatively. We also show that the remaining second-order coefficient $c_2^{″}$ depends on non-perturbative physics but may be related, by a perturbative calculation, to quantities that have previously been measured using lattice simulations of three-dimensional $O\left(2\right)\mathrm{}$ scalar field theory. Making use of those simulation results, we find $T_{\mathrm{c}}\simeq T_0\{1+(1.32\pm 0.02){\mathrm{an}}^{1/3}+\left[19.7518\ln {(an}^{1/3}\right)+(75.7\mathrm{}\pm 0.4)\left]a^2{n}^{2/3}{+O(a}^{3}n\right)\}.$