10 out first deriving the normal state self-energy microscop- ically (this is what we will do in the next section). The results of the calculations for the modified MFLI model are presented in Figs. 15 and 16. We clearly see that the behavior is now different and Δ W$_{K}$ < 0 for all Γ. This is the same behavior as we previously found in BCSI and EB models. So we argue that the ‘unconventional’ behavior exhibited by the original MFLI model is most likely the manifestation of a particular modeling incon- sistency. Still, Ref. 30 made a valid point that the fact that quasiparticles behave more close to free fermions in a SCS than in a NS, and this effect tends to reverse the signs of Δ W$_{K}$ and of the kinetic energy $^{43}$. It just hap- pens that in a modified MFLI model the optical integral is still larger in the NS. We now turn to a more microscopic model- the CB model. The model describes fermions interacting by ex- changing soft, overdamped collective bosons in a partic- ular, near-critical, spin or charge channel $^{31,44,45}$. This interaction is responsible for the normal state self-energy and also gives rise to a superconductivity. A peculiar feature of the CB model is that the propagator of a col- lective boson changes below T$_{c}$ because this boson is not an independent degree of freedom (as in EB model) but is made out of low-energy fermions which are affected by superconductivity $^{32}$. The most relevant point for our discussion is that this model contains the physics which we identified above as a source of a potential sign change of Δ W$_{K}$ . Namely, at strong coupling the fermionic self-energy in the NS is large because there exists strong scattering between low-energy fermions mediated by low-energy collective bosons. In the SCS, the density of low-energy fermions drops and a continuum collective excitations becomes gaped. Both effects reduce fermionic damping and lead to the increase of W$_{K}$ in a SCS. If this increase exceeds a conventional loss of W$_{K}$ due to a gap opening, the total Δ W$_{K}$ may become positive. The CB model has been applied numerous times to the cuprates, most often under the assumption that near- critical collective excitations are spin fluctuations with momenta near Q = ( π,π ). This version of a CB bo- son is commonly known as a spin-fermion model. This model yields d$_{x}$ $^{2}$− y 2 superconductivity and explains in a quantitative way a number of measured electronic fea- tures of the cuprates, in particular the near-absence of the quasiparticle peak in the NS of optimally doped and underdoped cuprates 39 and the peak-dip-hump structure in the ARPES profile in the SCS $^{31,32,46,47}$. In our analy- sis we assume that a CB is a spin fluctuation. The results for the conductivity within a spin-fermion model depend in quantitative (but not qualitative) way on the assumption for the momentum dispersion of a col- lective boson. This momentum dependence comes from model, where W$_{K}$ is larger in the NS for all Γ (see Fig. 4). In other words, the original MFLI model does not have the BCSI theory as its limiting case. We modified the MFLI model is a minimal way by changing the damping term in a SCS to Γ √ − ω $^{2}$+Δ2 to be consistent with BCSI model. We still use Eq. (18) for the MFL term simply because this term was introduced in the NS on phenomenological grounds and there is no way to guess how it gets modified in the SCS state with- FIG. 16: Evolution of the difference of the optical integrals between the SCS and the NS with the upper cut-off ω c for the “corrected” MFLI model. Now Δ W ( ω c ) is negative above some frequency. Parameters are same as in the Fig 15. FIG. 15: Top – σ ( ω ) in the NS and the SCS in the ‘corrected’ MFLI model with the feedback from SC on the quasiparticle damping: i Γ term transforms into Γ √ $_{−}$ω $^{2}$+Δ$_{2}$. In the SCS σ now begins at Ω = 2Δ. The parameters are same as in Fig. 10. Bottom – the behavior of Kubo sum with Γ. Observe that W ( ω c ) in the NS is larger than in the SCS. D. The collective boson model 0 0.5 1 0 0.4 0.8 Conductivities (Corrected MFLI) ( )  in eV NS SC 2  0 50 100 100 120 W K  (meV)  in meV Corrected MFLI SC NS 0.2 0.4 0.6 0.8 −10 0 10  $_{c}$ in eV W SC $_{(}$ω c ) − W$^{NS}$ $_{(}$ω c ) Corrected MFLI without lattice with lattice  W$_{K}$