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
Σ ′′(ω) = − 1
2 λnωsf log
(
1 + ω2
ω2
sf
)
Σ ′(ω) = −λnωsf arctan ω
ωsf
(19)
where λn is the spin-fermion coupling constant, and ωsf
is a typical spin relaxation frequency of overdamped spin
collective excitations with a propagator
χ(q ∼ Q, Ω) = χQ
1 − i Ω
ωsf
(20)
where χQ 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
ωsf are related: λ scales as ξ2, where ξ is the bosonic
mass (the distance to a bosonic instability), and ωsf ∝
ξ−2 (see Ref. 49). For a ﬂat χ(q ∼ Q) the product λωsf
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 ∆.
The eﬀect of a d−wave pairing gap on a CB has been
discussed in a number of papers, most recently in 31. In
a SCS a gapless continuum described by Eq. (20) trans-
forms into a gaped continuum, with a gap about 2∆ and
a resonance atω = ω0 < 2∆, where for a d−wave gap we
deﬁne ∆ as a maximum of a d−wave gap.
The spin susceptibility near ( π, π) in a superconductor
can generally be written up as
χ(q ∼ Q, Ω) = χQ
1 − i Π(Ω)
ωsf
(21)
where Π is evaluated by adding up the bubbles made
out of two normal and two anomalous Green’s functions.
Below 2∆, Π(Ω) is real (∼ Ω 2/∆ for small Ω), and the
resonance emerges at Ω = ω0 at which Π( ω0) = ωsf . At
frequencies larger than 2∆, Π(Ω) has an imaginary part,
and this gives rise to a gaped continuum inχ(Ω).
The imaginary part of the spin susceptibility around
the resonance frequency ω0 is31
χ
′′
(q, Ω) = πZoω0
2 δ(Ω − ω0) (22)
where Zo ∼ 2 ωsf χ0/ ∂Π
∂ω |Ω= ω0
. The imaginary part
of the spin susceptibility describing a gaped continuum
exists for for Ω≥ 2∆ and is
χ
′′
(q, Ω) = Im
[
χ0
1 − 1
ωsf
( 4∆ 2
Ω D(4∆ 2
Ω 2 ) + iΩ K2(1 − 4∆ 2
Ω 2 )
)
]
≈ Im
[
χ0
1 − 1
ωsf
( π∆ 2
Ω + i π
2 Ω
)
]
for Ω >> 2∆ (23)
In Eq. (23) D(x) = K1(x)−K2(x)
x , and K1(x) and K2(x)
are Elliptic integrals of ﬁrst and second kind. The real
part ofχ is obtained by Kramers-Kr¨ onig transform of the
imaginary part.
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
Σ ′′(ω) = Σ ′′
A(ω) + Σ ′′
B(ω) (24)
where,
Σ ′′
A(ω) = πZo
2 λnωo Re
(
ω + ωo√
(ω + ωo)2 − ∆ 2
)
comes from the interaction with the resonance and
Σ ′′
B(ω) = −λn
∫ |E|
2∆
dx Re ω + x
√
(ω + x)2 − ∆ 2
x
ωsf
K2
(
1 − 4∆ 2
x2
)
[
1 − 4∆ 2
xωsf
D
( 4∆ 2
x2
) ] 2
+
[
x
ωsf
K2
(
1 − 4∆ 2
x2
) ] 2 (25)
comes from the interaction with the gaped continuum. The real par t of Σ is obtained by Kramers-Kr¨ onig trans-