Number of Feynman diagrams in arbitrary order of perturbation theory

Abstract

Recurrence relations are established to determine the number of Feynman diagrams in arbitrary order of perturbation theory for four expansions: (i) the Green's function $\mathcal{G}$ expanded in the noninteracting Green's function ${\mathcal{G}}^{(0)}$ and the bare interaction $V$, (ii) the proper self-energy $\ensuremath{\Sigma}$ expanded in ${\mathcal{G}}^{(0)}$ and $V$, (iii) $\ensuremath{\Sigma}$ expanded in $\mathcal{G}$ and $V$, and (iv) $\ensuremath{\Sigma}$ expanded in $\mathcal{G}$ and the particle-particle (hole-hole) ladder sum $\ensuremath{\Gamma}$. In each case, the number of diagrams has the asymptotic behavior $\mathrm{const}\ifmmode\times\else\texttimes\fi{}(2n+1)$!! for large $n$.