  We report a doping dependent study of the quasiparticles dynamics in
HgBa$_2$CuO$_{4+\delta}$ via Electronic Raman Scattering. A well-defined energy
scale is found in the normal state dynamics of the quasiparticles over a broad
doping range. It is interpreted as evidence for coupling between the
quasiparticles and a collective bosonic mode whose energy scale depend only
weakly with doping. We contrast this behavior with that of the superconducting
gap whose amplitude near the node continuously decreases towards the underdoped
regime. We discuss the implications of our findings on the nature of the
collective mode and argue that electron-phonon coupling is the most natural
explanation.
