### Conserving approximation from 2PI effective action?

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**How to derive the functional for a conserving (or "phi-derivable") approximation?**

The following is the expression from p. 75 of [1] for the phi-functional corresponding to the Hartree-Fock-Bogoliubov approximation:

Comparing this to the loop expansion of the so-called 2PI effective action from [2], one can clearly make out all of the terms in (4.94): the first and fourth terms in (4.94) correspond to the \(\Gamma_2^{\mathrm{HFB}}\) terms, whereas the other terms can be found easily in \(\Gamma^{\mathrm{1loop}}\).

In fact, in [2] they also say that already "the HFB dynamic equation [...] involve[s] a resummation of infinitely many graphs and therefore arbitrary high powers of the bare coupling".

Any comments?

References:

[1] Griffin, A., Nikuni, T. and Zaremba, E., 2009. Bose-condensed gases at finite temperatures. Cambridge University Press.

[2] Gasenzer, T., Berges, J., Schmidt, M.G. and Seco, M., 2005. Nonperturbative dynamical many-body theory of a Bose-Einstein condensate. Physical Review A, 72(6), p.063604. arXiv:cond-mat/0507480

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