Can the bosonic coupling constant be extracted from the ARPES scattering rate in cuprate superconductors?

The recent ARPES results for the imaginary part of the self-energy obtained on a number of HTSC bismuthates \cite{Kordyuk} are analyzed. By accepting the ''Fermi-Bose'' {\it division-procedure} of the self-energy into the Fermi-liquid and bosonic parts - which is proposed in \cite{Kordyuk}, one obtains very small bosonic coupling constant $\lambda_{B, Im} <0.2$. If this procedure would be correct then the standard Eliashberg theory makes any bosonic mechanism of pairing irrelevant. As a consequence we are confronted with a trilemma: (1) to abandon the ``Fermi-Bose'' division-procedure \cite{Kordyuk}; (2) to abandon the Eliashberg theory; (3) to abandon the interpretation of ARPES data within the three-step model, where the ARPES intensity is proportional to the quasiparticle spectral function $A(\mathbf{k},\omega)$. However, since the bosonic coupling constant extracted from the ARPES nodal kink at 70 meV \cite{Lanzara}, which measures the real part of the self-energy is much larger than the one extracted from the ARPES line-width this means that the ``Fermi-Bose'' division procedure done in \cite{Kordyuk} is ambiguous.