content_image/420015.jpg content_image/420016.jpg content_image/420028.jpg content_image/420029.jpg content_image/420008.jpg content_image/420009.jpg content_image/420010.jpg content_image/420011.jpg content_image/420007.jpg content_image/420013.jpg content_image/420014.jpg content_image/420018.jpg content_image/420017.jpg
With the usual definitions for the bare vibron Green’s functions D$_{0}$ : 

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$_{vib}$ . H,<,> <,> 

One can see that Σ 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 

<img src='content_image/420015.jpg'>

Furthermore the lesser and greater Fock self-energies can be expressed in a more compact form: 

<img src='content_image/420016.jpg'>

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 expressions for the self-energies for these diagrams are: 

<img src='content_image/420028.jpg'>

and 

<img src='content_image/420029.jpg'>

The factor 1 / 3 comes from the series expansion of the exponential in the time evolution operator S$_{C}$ K and the 

<img src='content_image/420008.jpg'>

<img src='content_image/420009.jpg'>

<img src='content_image/420010.jpg'>

## 2. Second-order self-energies 

<img src='content_image/420011.jpg'>

fact that one obtains 8 equivalent diagrams of the γ 4 0 order. 

Taking the steady state limit and after Fourier trans- formation, the different Keldysh components of the double-exchange (DX) self-energy are given by 

<img src='content_image/420007.jpg'>

## 3. Normalisation of the vibron propagators 

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$_{0}$ $^{24,32}$: 

<img src='content_image/420013.jpg'>

where the dressed vibron D$_{0}$ ( τ, τ $^{′}$) is given by 

<img src='content_image/420014.jpg'>

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 

<img src='content_image/420018.jpg'>

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. 

## Appendix B: Details of numerical calculations 

In this section we provide more details about how cal- culations for the self-energies are performed. 

<img src='content_image/420017.jpg'>

