18 fact that one obtains 8 equivalent diagrams of the γ 4 0 order. With the usual definitions for the bare vibron Green’s functions D$_{0}$ : Taking the steady state limit and after Fourier trans- formation, the different Keldysh components of the double-exchange (DX) self-energy are given by The expression for the self-energy Σ DPH e -vib ( τ$_{1}$, τ$_{2}$ ) can actually be recast in a Fock-like diagram with a renor- malised vibron propagator D$_{0}$ $^{24,32}$: where 〈 N$_{ph}$ 〉 is the averaged number of excitations in the vibration mode of frequency ω$_{0}$ given by the Bose- Einstein distribution at temperature T$_{vib}$ . One can see that Σ H,<,> e -vib ( ω ) = 0 because D <,> 0 ( ω = 0) = 0 unless ω$_{0}$ = 0 which would be an odd case of study. And since D r,a 0 ( ω = 0) = − 2 /ω$_{0}$ , one has where the dressed vibron D$_{0}$ ( τ, τ $^{′}$) is given by Furthermore the lesser and greater Fock self-energies can be expressed in a more compact form: Σ DX,ζ$_{1}$ζ$_{2}$ e -vib ( ω ) = − γ 4 0 3 ∫ d u 2 π d v 2 π ∑ ζ$_{3}$,ζ$_{4}$ ζ$_{3}$ζ$_{4}$G ζ$_{1}$$^{ζ$_{3}$}$( v ) × D ζ i ζ$_{4}$ 0 ( ω − v ) G ζ$_{3}$$^{ζ$_{4}$}$( v − u ) D ζ$_{3}$ζ$_{2}$ 0 ( u ) G ζ$_{4}$$^{ζ$_{2}$}$( ω − u ) . (A19) D < 0 ( ω ) = − 2 πi [ 〈 N$_{ph}$ 〉 δ ( ω − ω$_{0}$ ) +( 〈 N$_{ph}$ 〉 + 1) δ ( ω + ω$_{0}$ )] , (A11) D > 0 ( ω ) = − 2 πi [ 〈 N$_{ph}$ 〉 δ ( ω + ω$_{0}$ ) +( 〈 N$_{ph}$ 〉 + 1) δ ( ω − ω$_{0}$ )] , (A12) D r $_{0}$( ω ) = 1 ω − ω$_{0}$ + iη − 1 ω + ω$_{0}$ + iη , η → 0 + (A13) D a $_{0}$( ω ) = [ D r $_{0}$( ω )] $^{∗}$, (A14) 3. Normalisation of the vibron propagators Σ F e $_{-vib}$( τ$_{1}$, τ$_{2}$ ) = i γ 2 0 D$_{0}$ ( τ$_{1}$, τ$_{2}$ ) G ( τ$_{1}$, τ$_{2}$ ) , (A20) D$_{0}$ ( τ, τ $^{′}$) = γ 2 0 ∫$_{C}$$_{K}$ d τ$_{2}$ d τ$_{3}$ D$_{0}$ ( τ, τ$_{3}$ ) Π( τ$_{3}$, τ$_{2}$ ) D$_{0}$ ( τ$_{2}$, τ $^{′}$) (A21) Σ H,r e -vib = Σ H,a e -vib = γ 2 0 ω$_{0}$ 2i ∫ dω ′ 2 π G $^{<}$( ω $^{′}$) . (A15) Σ F, { <,> } e -vib ( ω ) = γ 2 0 [ 〈 N$_{ph}$ 〉 G { <,> $^{}}$( ω ∓ ω$_{0}$ ) +( 〈 N$_{ph}$ 〉 + 1) G { <,> $^{}}$( ω ± ω$_{0}$ ) ] . (A16) Π( τ, τ $^{′}$) = − i G ( τ, τ $^{′}$) G ( τ $^{′}$, τ ) . (A22) D { r,a $^{}}$( ω ) = [ D { r,a } 0 ( ω )] − 1 − γ 2 0 Π { r,a $^{}}$( ω ) ] − 1 . (A23) and the polarization Π( τ, τ $^{′}$) is given by the electron-hole bubble diagram: So, in principle, we already have all the ingredients to perform calculations using the fully dressed vibron prop- agator D ( ω ) 24,32 whose advanced and retarded compo- nents are given by The two-vibron process self-energies correspond to the two diagrams shown in Figure 3, i.e. the so-called double exchange (DX) diagram and GW-like (dressed phonon/vibron DPH) diagram in which the vibron prop- agator is renormalized by a single electron-hole bubble polarization. The vibron spectrum is renormalized by the polariza- tion. Ther is a shift of the vibron energy related to γ 2 $_{0}$ℜ e Π { r,a } and a finite linewidth of the peaks related to γ 2 $_{0}$ℑ m Π { r,a $^{}}$, instead of Dirac δ peaks as in Eq.(B1). We have not performed the full vibron renormaliza- tion in the present work, because we want to study the effects of different diagrams of the same order, and order by order. D ( ω ) correspond to a partial resummation of all the bubble diagrams and contains all (even) orders of the electron-vibron coupling. The effect of this GW-like electron-vibron interaction will be presented in a forth- coming paper. The expressions for the self-energies for these diagrams are: and 2. Second-order self-energies Appendix B: Details of numerical calculations Σ DX e $_{-vib}$( τ$_{1}$, τ$_{2}$ ) = − γ 4 0 3 ∫$_{C}$$_{K}$ d τ$_{3}$ d τ$_{4}$G ( τ$_{1}$, τ$_{3}$ ) D$_{0}$ ( τ$_{1}$, τ$_{4}$ ) × G ( τ$_{3}$, τ$_{4}$ ) D$_{0}$ ( τ$_{3}$, τ$_{2}$ ) G ( τ$_{4}$, τ$_{2}$ ) , (A17) Σ DPH e -vib ( τ$_{1}$, τ$_{2}$ ) = + γ 4 0 3 G ( τ$_{1}$, τ$_{2}$ ) × ∫$_{C}$$_{K}$ d τ$_{3}$ d τ$_{4}$D$_{0}$ ( τ$_{1}$, τ$_{3}$ ) G ( τ$_{3}$, τ$_{4}$ ) G ( τ$_{4}$, τ$_{3}$ ) D$_{0}$ ( τ$_{3}$, τ$_{2}$ ) . (A18) In this section we provide more details about how cal- culations for the self-energies are performed. The factor 1 / 3 comes from the series expansion of the exponential in the time evolution operator S$_{C}$ K and the