5 Λ$_{α}$$_{α}$$_{′}$ ( x | x $^{′}$) = αx − α ′ x ′ a ) + $_{V}$2 2 − α ′ x ′ αx 1 2 b ) ... − α ′ x ′ 6 4 2 e ) + − αx 5 1 3 $_{V}$6 6 − − α ′ x ′ αx 1 2 c ) $_{V}$4 8 3 4 + + − α ′ x ′ αx 3 4 d ) $_{V}$6 48 5 6 1 2 FIG. 3: The skeleton diagrams for correlation function Λ $_{αα}$′ ( x | x $^{′}$). The thin dashed line is zero-order f − electron Green’s function. The rectangles depict the many-particles irreducible Green’s function. The double solid lines depict the renormalized conduction electron Green’s function G c $_{αα}$′ ( x | x $^{′}$). Λ ( x | x $^{′}$) = x x ′ + ( − 1) V 2 x ′ x 1 2 − − x ′ x 3 4 $_{V}$6 6 5 6 1 2 G f (0) G c G c G c G c x ′ 6 4 2 − x 5 1 3 $_{V}$6 6 G c G c G c ... + − x ′ x 1 2 $_{V}$4 2 3 4 G c G c FIG. 4: Correlation function Λ( x | x $^{′}$) in normal state. All the lines correspond to normal propagators and have a direction of propagators and of the arrays in the vertices. In normal state the diagrams of Fig. 5 are changed. The direction of propagator lines and of the arrays at vertices points appear together with new coefficients be- fore the diagrams. This changing is demonstrated on the Fig. 6, where some of the diagrams of Fig. 5 are demon- strated. Vacuum diagrams in superconducting and nor- mal states contain the factor 1 $_{n}$, where n is the order of perturbation theory in which given diagram appears. This factor makes difficult the investigation of this contri- butions. In order to remove this coefficient it is necessary to use the trick of integration by constant of interaction V . The result of such integration is depicted on the Fig. 7. By using the perturbation theory we have obtained for the connected part of evolution operator average the contributions depicted in the Fig. 5. The contributions of the first, third and fourth dia- grams of Fig. 5 are enumerated below: On the base of series expansions for renormalized prop- agators of the conduction c − electrons (see Fig. 1), lo- calized f − electrons (see Fig. 2) and definition of the correlation function Λ$_{αα}$ ′ ( x | x $^{′}$) we can prove that the contribution of the integrant of Fig. 7 in every order of perturbation theory can be presented itself as the prod- IV. VACUUM DIAGRAMS − V 2 ∑ α$_{1}$α$_{2}$ ∑ 12 α$_{1}$α$_{2}$G c (0) α$_{1}$$_{α$_{2}$}$(1 | 2) G f (0) α$_{2}$$_{α$_{1}$}$(2 | 1) , V 4 8 ∑ α$_{1}$...α$_{4}$ ∑ 1 ... 4 α$_{1}$α$_{2}$α$_{3}$α$_{4}$G c (0) α$_{1}$$_{α$_{2}$}$(1 | 2) G c (0) α$_{3}$$_{α$_{4}$}$(3 | 4) × × 〈 T f − α$_{1}$ 1 f α$_{2}$ 2 f − α$_{3}$ 3 f α$_{4}$ 4 〉 ir 0 , − V 6 48 ∑ α$_{1}$...α$_{6}$ ∑ 1 ... 6 α$_{1}$α$_{2}$α$_{3}$α$_{4}$α$_{5}$α$_{6}$G c (0) α$_{1}$$_{α$_{2}$}$(1 | 2) G c (0) α$_{3}$$_{α$_{4}$}$(3 | 4) × × G c (0) α$_{5}$$_{α$_{6}$}$(5 | 6) 〈 T f − α$_{1}$ 1 f α$_{2}$ 2 f − α$_{3}$ 3 f α$_{4}$ 4 f − α$_{5}$ 5 f α$_{6}$ 6 〉 ir 0 .