The dilute interacting Bose gas : comparisons between Feynman diagrammatic expansion and the hierarchy equations for the imaginary time Green functions

We analyze the correspondence between two formalisms describing an interacting Bose gas, namely the standard Feynman diagrammatic expansion on the one hand, and the hierarchy equations for the imaginary-time Green functions on the other hand. We show that the Hartree-Fock approximation, as well as its first corrections at low density derived by Baym et al. [Eur. Phys. J. B, 24, pp. 107-124 (2001)], can be equivalently formulated in both formalisms. Within the hierarchy approach, the two-body correlations ${\mathcal{G}}^{(2,\mathrm{T})}$ are expressed at lowest order in the interactions in terms of the full one-body Green function. This sheds light on the physical content of the corrections to the Hartree-Fock theory. Moreover, this representation of ${\mathcal{G}}^{(2,\mathrm{T})}$ can be extended to higher orders, opening the way to systematic calculations of further corrections to the ideal critical temperature when the density increases.