11 high-energy fermions and is an input for the low-energy theory. Below we follow Refs. 31,33 and assume that the momentum dependence of a collective boson is flat near ( π, π ). The self energy within such model has been worked out consistently in Ref. 31,33. In the normal state a SCS a gapless continuum described by Eq. (20) trans- forms into a gaped continuum, with a gap about 2Δ and a resonance at ω = ω$_{0}$ < 2Δ, where for a d − wave gap we define Δ as a maximum of a d − wave gap. The spin susceptibility near ( π, π ) in a superconductor can generally be written up as where λ$_{n}$ is the spin-fermion coupling constant, and ω$_{sf}$ is a typical spin relaxation frequency of overdamped spin collective excitations with a propagator where χ$_{Q}$ is the uniform static susceptibility. If we use Ornstein-Zernike form of χ ( q ) and use either Eliashberg 45 or FLEX computational schemes $^{48}$, we get rather sim- ilar behavior of Σ as a function of frequency and rather similar behavior of optical integrals. The collective nature of spin fluctuations is reflected in the fact that the coupling λ and the bosonic frequency ω$_{sf}$ are related: λ scales as ξ $^{2}$, where ξ is the bosonic mass (the distance to a bosonic instability), and ω$_{sf}$ ∝ ξ − 2 (see Ref. 49). For a flat χ ( q ∼ Q ) the product λω$_{sf}$ does not depend on ξ and is the overall dimensional scale for boson-mediated interactions. In the SCS fermionic excitations acquire a gap. This gap affects fermionic self-energy in two ways: directly, via the change of the dispersion of an intermediate boson in the exchange process involving a CB, and indirectly, via the change of the propagator of a CB. We remind our- selves that the dynamics of a CB comes from a particle- hole bubble which is indeed affected by Δ. The effect of a d − wave pairing gap on a CB has been discussed in a number of papers, most recently in $^{31}$. In Substituting Eq 6 for χ ( q, Ω) into the formula for the self-energy one obtains Σ ′′ ( ω ) in a SCS state as a sum of two terms 31 In Eq. (23) D ( x ) = K 1 ( x ) − K 2 ( x x , and K$_{1}$ ( x ) and K$_{2}$ ( x ) are Elliptic integrals of first and second kind. The real part of χ is obtained by Kramers-Kr¨ onig transform of the imaginary part. where Z$_{o}$ ∼ 2 ω$_{sf}$ χ$_{0}$/ ∂ Π $^{∂ω}$| Ω= ω 0 . The imaginary part of the spin susceptibility describing a gaped continuum exists for for Ω ≥ 2Δ and is where Π is evaluated by adding up the bubbles made out of two normal and two anomalous Green’s functions. Below 2Δ, Π(Ω) is real ( ∼ Ω $^{2}$/ Δ for small Ω), and the resonance emerges at Ω = ω$_{0}$ at which Π( ω$_{0}$ ) = ω$_{sf}$ . At frequencies larger than 2Δ, Π(Ω) has an imaginary part, and this gives rise to a gaped continuum in χ (Ω). The imaginary part of the spin susceptibility around the resonance frequency ω$_{0}$ is 31 Σ ′′ ( ω ) = − 1 2 λ$_{n}$ω$_{sf}$ log ( 1 + ω 2 ω 2 sf ) Σ ′ ( ω ) = − λ$_{n}$ω$_{sf}$ arctan ω ω$_{sf}$ (19) χ ( q ∼ Q, Ω) = χ$_{Q}$ 1 − i Ω ω$_{sf}$ (20) χ ( q ∼ Q, Ω) = χ$_{Q}$ 1 − i Π(Ω) ω$_{sf}$ (21) χ $^{′′}$( q, Ω) = πZ$_{o}$ω$_{0}$ 2 δ (Ω − ω$_{0}$ ) (22) χ $^{′′}$( q, Ω) = Im [ χ$_{0}$ 1 − 1 ω$_{sf}$ ( 4Δ 2 Ω D ( 4Δ 2 Ω 2 ) + i Ω K$_{2}$ (1 − 4Δ 2 Ω 2 ) ] ≈ Im [ χ$_{0}$ 1 − 1 ω$_{sf}$ ( π Δ 2 Ω + i π $_{2}$Ω ) ] f or Ω >> 2Δ (23) Σ ′′ ( ω ) = Σ ′′ $_{A}$( ω ) + Σ ′′ $_{B}$( ω ) (24) Σ ′′ $_{A}$( ω ) = πZ$_{o}$ 2 λ$_{n}$ω$_{o}$ Re ( ω + ω$_{o}$ √ ( ω + ω$_{o}$ ) 2 − Δ 2 ) Σ ′′ $_{B}$( ω ) = − λ$_{n}$ ∫ | E | 2Δ dx Re ω + x √ ( ω + x ) 2 − Δ 2 x ω$_{sf}$ K$_{2}$ ( 1 − 4Δ 2 x 2 ) [ 1 − 4Δ 2 xω$_{sf}$ D ( 4Δ 2 x 2 ) ] 2 + [ x ω$_{sf}$ K$_{2}$ ( 1 − 4Δ 2 x 2 ) ] 2 (25) comes from the interaction with the resonance and comes from the interaction with the gaped continuum. The real part of Σ is obtained by Kramers-Kr¨ onig trans- where,