Higher-order evaluation of the critical temperature for interacting homogeneous dilute Bose gases

We use the nonperturbative linear $\delta$ expansion method to evaluate analytically the coefficients $c_1$ and $c_2^{″}$ that appear in the expansion for the transition temperature for a dilute, homogeneous, three-dimensional Bose gas given by $T_c{=T}_0({1+c}_1{\mathrm{an}}^{1/3}+[c_2^{\prime }\ln {(an}^{1/3}{)+c}_2^{″}\left]a^2{n}^{2/3}{+O(a}^{3}n\right)),$ where $T_0$ 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 $c_1$ to order ${\delta }^2$ with the result $c_1=\mathrm{3.06.}\mathrm{}$ Here, we push the calculation to the next two orders obtaining $c_1=2.45\mathrm{}$ at order ${\delta }^3$ and $c_1=1.48\mathrm{}$ at order ${\delta }^4.$ Analyzing 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 $c_2^{″}=\mathrm{82.9.}\mathrm{}$ Our analytical results seem to support the recent Monte Carlo estimates $c_1=1.32\mathrm{}\pm 0.02$ and $c_2^{″}=75.7\mathrm{}\pm \mathrm{0.4.}$