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

Abstract: 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 ${\mathcal G}^{(2,\rm T)}$ 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 ${\mathcal G}^{(2,\rm T)}$ can be extended to higher orders, opening the way to systematic calculations of further corrections to the ideal critical temperature when the density increases.