BCS theory
Derivation of the gap equation
We start with a general Hamiltonian that describes many-fermion system interacting via a spin-independent interaction potential V(x),
H =X
σ Z
d3xψσ†(x)
−~
2∇2
2m −µ
ψσ(x)+ 1 2
X
σσ0
Z
d3xd3x0V(x−x0)ψ†σ(x)ψσ†0(x0)ψσ0(x0)ψσ(x).
Going to the momentum space in finite volume, i.e., substituting
ψ(x) = √1
Ω X
k
akeik·x, V(x) =
1 Ω
X
q
eiq·xV¯q,
we find
H =X
kσ ξka
†
kσakσ+ 1 2Ω
X
σσ0
X
k,k0,q
¯ Vqa
†
k+q,σa
†
k0−q,σ0ak0σ0akσ.
Restricting to pairing of fermions with zero total momentum and opposite spin, we get the BCS Hamiltonian,
H =X
kσ
ξka†kσakσ+ 1 Ω
X
k,k0
¯ Vk−k0a†
k0↑a
†
−k0↓a−k↓ak↑.
At finite temperature, we linearize the Hamiltonian by replacing the bilinear a−k↓ak↑ with its
thermal average,
ha−k↓ak↑iβ ≡ − ∆k
2Ek
tghβEk
2 . (1)
Introducing the coefficients
ck≡ −
1 2Ω
X
k0
¯ Vk−k0
∆k0
Ek0
tgh βEk0
2 , (2)
we find
H =X
kσ ξka
†
kσakσ + X
k
cka
†
k↑a †
−k↓ + H.c.
The spectrum of this Hamiltonian is found by the Bogolyubov transformation of the Nambu spinor, defined as
Ak =
ak↑
a†−k↓
.
Subtracting an irrelevant constant, the Hamiltonian reads
H =X
k
A†k
ξk ck
ck −ξk
The eigenvalues are ±Ek ≡ ±
p ξ2
k+c2k and the Hamiltonian is diagonalized by the unitary
transformation,
H =X
k
EkA
†
kUkτ3U
†
kAk, Uk=
cosθk −sinθk
sinθk cosθk
.
The eigenstate condition for the ‘+’ eigenstate readsξkcosθk+cksinθk =Ekcosθk. From here
we obtain tgθk= (Ek−ξk)/ck, or
tg 2θk=
ck
ξk
. (3)
The gap equation for the gap parameters ∆k is found by calculating the anomalous average in
Eq. (1). To that end, we write
U†Ak=
αk
β−†k
,
where αk, β−k are the annihilation operators of the particle/hole excitations above the BCS
ground state. From here we readily get
ak↑ =αkcosθk−β
†
−ksinθk, a−k↓ =α
†
ksinθk+β−kcosθk.
The inverse of these transformations define the annihilation operators of the Bogolyubov quasi-particles, which generate the BCS ground state from the Fock vacuum,
αk =ak↑cosθk+a
†
−k↓sinθk, β−k =−a
†
k↑sinθk+a−k↓cosθk.
The self-consistency condition (1) now becomes
−∆k
2Ek
tghβEk
2 =ha−k↓ak↑iβ = (−1 +n|α,k+{znβ,−k}
2f(Ek)
) sinθkcosθk =−
1
2sin 2θktgh βEk
2 .
Using Eq. (3) we find sin 2θk = ck/Ek and the gap equation becomes simply ck = ∆k, or
equivalently
∆k ≡ −
1 Ω
X
k0
¯ Vk−k0
∆k0
2Ek0
tghβEk0
2 , Ek =
q ξ2
k+ ∆
2
k. (4)
Solution of the gap equation
Take the infinite-volume limit and the high-density approximation,
1 Ω
X
k
→
Z d3k
(2π)3 =
Z
k2d|k|
2π2
dΩk
4π → N
Z +Λ
−Λ
dξk
dΩk
4π , where N ≡
mkF
2π2
is the density of states at the Fermi surface, and Λ is an UV cutoff on the radial integration.
Assuming that the interaction potential varies slowly enough within the layer of thickness 2Λ around the Fermi surface (i.e., V(x) has short range), we can treat ∆k merely as a function of
Considering now just spin-zero pairing, ∆n will actually be isotropic, ∆n = ∆. The angular
integration is then trivial and gives rise to the effective attractive coupling,
Veff =−
Z dΩ
n0
4π ¯ Vn−n0.
The gap equation thus becomes
1 =NVeff
Z Λ
0
dξ p
ξ2+ ∆2 tgh
βpξ2+ ∆2
2 . (5)
At zero temperature, the gap equation can be easily solved since
Z Λ
0
dξ p
ξ2 + ∆2 0
= arcsinh Λ ∆0
'log 2Λ ∆0
, (6)
so that the asymptotic form of the gap at weak coupling is
∆0 = 2Λ exp
− 1 NVeff
.
At the critical temperature, the gap equation (5) reduces to
1
NVeff
= Z Λ
0
dξ ξ tgh
βcξ
2 =
Z βc2Λ
0
dxtghx
x 'log
βcΛ
2 +γ+ log 4 π = log
2βcΛeγ
π , (7)
which is the formula (32) in the appendix. Comparing this to Eq. (6), we obtain the universal BCS relation between the gap at zero temperature and the critical temperature,
Tc= ∆0
eγ
π. (8)
Ginzburg–Landau free energy
The static part of the GL functional is derived (up to an overall factor) by expansion of the gap equation (5) in powers of ∆2,
tghβ
√
ξ2+∆2
2
p
ξ2+ ∆2 =
tghβξ2
ξ +
∆2
2|ξ|
d dx
tghβx2 x
x=|ξ|
,
the latter derivative being
d dx
tghβx2
x =−
1 x2 tgh
βx
2 +
β 2x
1
cosh2 βx2 =
βx−sinhβx
2x2cosh2 βx 2
.
The GL gap equation thus acquires the form,
1
NVeff
= Z β2Λ
0
dxtghx
x +
β∆
4
2Z β2Λ
0
dx2x−sinh 2x
which is rewritten using the above evaluated integrals as log T Tc = β∆ 4
2Z ∞
0
dx2x−sinh 2x x3cosh2x =
β∆ 4 2 2 Z ∞ 0 dx 1
x2cosh2x −
tghx x3
. (10)
The last integral is rewritten using integration per partes in such a way that the cancelation of the IR divergence at x→0 is manifest,
−
Z ∞
ε
dxtghx x3 =
tghx 2x2 ∞ ε − Z ∞ ε dx 1
2x2cosh2x =−
1 2ε −
Z ∞
ε
dx 1
2x2cosh2x +O(ε),
Z ∞
ε
dx 1
2x2cosh2x =
− 1
2xcosh2x ∞
ε
−
Z ∞
ε
dx sinhx xcosh3x =
1 2ε −
Z ∞
ε
dx sinhx
xcosh3x +O(ε),
so that we find
Z ∞
0
dx2x−sinh 2x x3cosh2x =−2
Z ∞
0
dx sinhx xcosh3x =−
14ζ(3) π2 .
Near the critical temperature, where the gap is expected to be small, we may approximate
log T Tc
' T
Tc
−1 = T −Tc Tc
≡t.
The GL equation then becomes
−t = 7ζ(3) 8π2T2
c
∆2. (11)
This GL equation derives from the GL free energy functional, determined up to an overall factor,
F ∝ Nt|∆|2+7ζ(3)mkF
32π4T2
c
|∆|4. (12)
Alternative derivation
Referring to the Matsubara formalism, we make use of the identity
T X
n 1 ν2
n+x2
= 1
2xtgh βx
2 , νn = (2n+ 1)πT, (13)
and rewrite the gap equation (5) as
1
NVeff
= 2T X
n Z Λ
0
dξ 1
ν2
n+ξ2+ ∆2 .
The expansion in powers of ∆2 is now straightforward,
1
NVeff
= 2T X
n Z Λ 0 dξ 1 ν2
n+ξ2
− ∆
2
(ν2
n+ξ2)2
.
The first term gives, after performing the Matsubara sum, back the first integral on the r.h.s. of Eq. (9). In the second term, we first integrate over ξ (taking Λ → ∞) and obtain
−2T∆2X n
Z ∞
0
dξ (ν2
n+ξ2)2
=−πT∆
2
2 X
n 1
|νn|3
=− ∆
2
π2T2
∞
X
n=0
1
(2n+ 1)3 =−
7ζ(3) 8π2T2∆
2. (14)
Derivation of the full Ginzburg–Landau functional
Free energy
Consider a theory defined by the (Euclidean) Lagrangian
L=X σ
ψσ†(x)
∂τ − ∇
2
2m −µ
ψσ(x)−Veffψ
† ↑(x)ψ
†
↓(x)ψ↓(x)ψ↑(x). (15)
The four-fermion interaction may be decoupled using the Hubbard–Stratonovich transforma-tion, i.e., by adding to the Lagrangian the term
∆L= 1
Veff
|φ−Veffψ↓ψ↑|2,
so that the Lagrangian becomes
L= |φ|
2
Veff
−Ψ†G−1Ψ,
where Ψ≡ tr (ψ↑, ψ †
↓) is the Nambu spinor, and G is the corresponding Nambu-space fermion
propagator, defined by G−1 =G−1
0 + Σ, where
G−01 =
−∂τ + ∇
2
2m +µ 0
0 −∂τ − ∇
2
2m −µ
, Σ =
0 φ
φ∗ 0
.
The mean-field expression for the free energy density is then obtained upon integrating out the fermions,
F = 1 Ω
1 Veff
Z
d3x|φ|2−T Tr logG−1
. (16)
Expansion near critical temperature
To derive the GL functional, we expand the logarithm in Eq. (16) in powers of the symmetry-breaking self-energy Σ and keep terms up to fourth order,
logG−1 = log(G−01+ Σ) = logG0−1+ log(1 +G0Σ) = logG−01+
∞
X
n=1
(−1)n+1
n (G0Σ) n.
The odd-nterms have vanishing trace over the Nambu space. After subtracting the free energy of the normal phase, Fn, we arrive at
F − Fn= 1 Ω
1 Veff
Z
d3x|φ|2+TTr
1 2(G0Σ)
2+1
4(G0Σ)
4
. (17)
may be performed straightforwardly in the momentum–frequency representation, using the free propagator
G−01(iνn,k) =
iνn−ξk 0
0 iνn+ξk
.
The quartic part of the GL potential thus becomes
T
4ΩTr(G0Σ)
4 = |φ|4
2 T X
n
Z d3k
(2π)3[G0,11(iνn,k)G0,22(iνn,k)]
2 = |φ|4
2 T X
n
Z d3k
(2π)3
1 (ν2
n+ξk2)2
.
In the high-density approximation, this integral is evaluated in Eq. (33) in the appendix, so that we can immediately write down the result,
T
4Ω Tr(G0Σ)
4
= 7ζ(3)N 8π2T2
|φ|4
2 =
7ζ(3)mkF
32π4T2
c
|φ|4. (18)
Now we concentrate on the quadratic part of the free energy (17), allowing the pairing field to vary in space. In this case, we get from the definition of the propagator in the coordinate space,
T
2ΩTr(G0Σ)
2 = T
ΩTr "
1
−∂τ+ ∇
2
2m +µ
φ(x) 1
−∂τ −∇
2
2m −µ φ∗(x)
#
.
Inserting two partitions of unity in terms of the momentum eigenstates normalized to one in finite volume, yields
T
2ΩTr(G0Σ)
2 = T
Ω X
n X
k,k0
|hk|φ|k0i|2
(iνn−ξk)(iνn+ξk0)
.
(Since the pairing field is assumed to be time-independent, the trace over the imaginary time may be simply performed by introducing a single Matsubara frequencyνn.) The matrix element of φ is merely its corresponding Fourier component,
hk|φ|k0i= 1 Ω
Z
d3xei(k0−k)·xφ(x)≡φk−k0. (19)
Next we switch to the new momentum variables,p= k+2k0,q =k−k0. Note that the Jacobian of
this transformation is unity, or equivalently, the number of discrete momentum states remains unchanged. The momentum summations now partially factorize,
T
2Ω Tr(G0Σ)
2 = T
Ω X
n X
q
|φq|2
X
p
1
(iνn−ξp+q2)(iνn+ξp−q
2)
.
Not surprisingly, the large fraction (summed over pand n) is precisely the mean-field fermion-loop contribution to the propagator of the pairing field at zero frequency and momentum q. The ratio of the gradient and the static piece of the quadratic part of the GL functional could therefore be determined by inspection of the collective mode propagator. Starting from the free energy as here allows us to fix the overall numerical factor.
In order to account for slow variations of the order parameter, we next make a Taylor expansion in q up to second order. To this end, we write
1 iνn∓ξp±q2
= 1
iνn∓ξp− p
·q
2m ∓
q2
8m
= 1
iνn∓ξp
"
1 +
p·q
2m ±
q2
8m iνn∓ξp
+
p·q
2m 2
(iνn∓ξp)2
+O(q3) #
Multiplying the Taylor expansions of the two propagators gives, to second order in q,
T
2ΩTr(G0Σ)
2 =−T
Ω X
n X
q
|φq|2
X
p
1 ν2
n+ξp2
1 +
p·q
2m +
q2
8m iνn−ξp
| {z } +
p·q
2m 2
(iνn−ξp)2
+
+
p·q
2m −
q2
8m iνn+ξp
| {z } +
p·q
2m 2
(iνn+ξp)2
+
p·q
2m 2
(iνn−ξp)(iνn+ξp)
.
The underbraced expressions drop out from the problem: The terms linear in p vanish in the momentum integral, while the other two terms combine to an expression odd in ξp, i.e., they
vanish in the high-density approximation. In the remaining terms, we perform the angular average, which is effectively done by the replacement pipj → 13δijp2,
T
2ΩTr(G0Σ)
2 =−T
Ω X
n X
q
|φq|2
X
p
1 ν2
n+ξp2
1 + p
2q2
12m2
1 (iνn−ξp)2
+ 1
(iνn+ξp)2
− 1
ν2
n+ξp2
.
Now we analyze separately the static and gradient pieces, i.e., the zeroth and second order in
q. In the static part, we may directly perform the sum over q,
X
q
|φq|2 =
1 Ω
Z
d3x|φ(x)|2 =|φ|2.
The remaining sum-integral is already well known and is given by Eq. (32) in the appendix. Together with the condensate term in the free energy (17), this yields
|φ|2
1 Veff
− Nlog 2βΛe γ
π
=|φ|2N log T
Tc
' Nt|φ|2, (21)
where we used Eq. (7) to eliminate the coupling in favor of the critical temperature. In fact, Eq. (21) may be viewed as an alternative to our previous derivation of the critical temperature.
The gradient part of the quadratic term is, upon some manipulation and taking the high-density limit, cast in the form
T
2ΩTr(G0Σ)
2
grad =−
T 12m2
X
q
q2|φq|2
X
n Z
d3p
(2π)3 p 2
−
3 (ν2
n+ξp2)2
+ 4ξ
2
p
(ν2
n+ξ2p)3
=
=−Nk
2 F
12m2
X
q
q2|φq|2T
X
n
Z +∞
−∞
dξ
−3 (ν2
n+ξ2)2
+ 4ξ
2
(ν2
n+ξ2)3
.
The two integrals involved (one of which has actually already been evaluated in Eq. (14)) read
Z +∞
−∞
dξ 1
(ν2
n+ξ2)2
= π
2|νn|3 ,
Z +∞
−∞
dξ ξ
2
(ν2
n+ξ2)3
= π
8|νn|3 .
The calculation thus effectively reduces to twice the sum-integral in Eq. (32). We thus arrive at the final formula for the gradient part of the GL functional,
T
2Ω Tr(G0Σ)
2 grad=
Nk2 F
12m2
7ζ(3) 4π2T2
X
q
q2|φq|2 =
7ζ(3)k3 F
96π4mT2
c 1 Ω
Z
Putting all the pieces (18), (21), and (22) together, the complete Ginzburg–Landau free energy density acquires the form
F − Fn=
7ζ(3)kF3 96π4mT2
c
|∇φ|2 +Nt|φ|2+ 7ζ(3)mkF
32π4T2
c
|φ|4. (23)
Ginzburg–Landau functional for a relativistic superconductor
Now we will generalize the previous results to relativistic systems with several degrees of free-dom. Our starting assumption will be that the system can be described by a model of the Nambu–Jona-Lasinio type with a contact, momentum-independent interaction. The starting point will be the formula (16), which now becomes
F =− T
2ΩTr logG
−1
,
where G is the fermion propagator in the Nambu space, Ψ ≡ tr (ψ, ψC); the extra factor 12 comes from the doubling of the number of degrees of freedom. Note that we have also omitted the condensate contribution as this merely serves to make the coefficient of the “mass” term in the GL functional vanish at the critical temperature.
The inverse fermion propagator takes in general the form
G−1(p) =
/
p−m+µγ0 Φ˜
Φ /p−m−µγ0
,
where Φ is the gap matrix and ˜Φ = γ0Φ†γ0. (The interchange of Φ and ˜Φ as compared to the
nonrelativistic case is just a convention used in literature on color superconductivity.) According to our assumption, Φ can be written as Φ =φaTa, where Ta is a set of momentum independent matrices in the Dirac and flavor space. Slow variations of the order parametersφa(x) may then be taken into account by the same strategy as above.
In the high density approximation, we will now neglect the contributions of antiparticles. The free fermion propagator in the imaginary time thus takes the form
G0(iνn,k) =
Λ+kγ0
iνn−ξk 0
0 Λ
−
kγ0
iνn+ξk
!
.
The quasiparticle excitation energyξk is again measured with respect to the Fermi level and is
now defined asξk=εk−µ,εk=
√
k2+m2. The standard energy projectors Λ±
k are defined as
Λ±k = 1 2
1± 1
εk
γ0(γ·k+m)
. (24)
Finally, we just note that with the above modifications, Eq. (17) becomes simply
F − Fn= T 2ΩTr
1 2(G0Σ)
2+ 1
4(G0Σ)
4
, Σ =
0 Φ˜
Φ 0
and we are ready to set on the calculation.
We first evaluate the quartic term. The trace over the Nambu space is trivial and gives just a factor of two, the rest being
T 4
X
n
Z d3k
(2π)3 Tr(G0,11ΦG˜ 0,22Φ) 2 = T
4 X
n
Z d3k
(2π)3
Tr(Λ+kγ0γ0Φ†γ0Λ−kγ0Φ)2
(ν2
n+ξk2)2
.
Using Eq. (33) to evaluate the sum-integral, the property of the energy projectors,
γ0Λ±kγ0 = Λ±−k,
and inserting the definition of the gap matrix, we arrive at the most general result,
quartic term = 7ζ(3) 32π2T2
c
Nφ∗aφbφ∗cφd
Tr(Λ+kTa†Λ−−kTbΛ+kTc†Λ
− −kTd)
k. (25)
The angular brackets denote averaging over the direction of the indicated momentum. Note that the energy εk and momentum k in the projectors here are understood as being replaced
with their values on the Fermi surface, that is, µ and pFkˆ, where pF is the Fermi momentum.
The density of states in a relativistic system is given by
N = µpF 2π2.
Next we concentrate on the quadratic part of the free energy. Inserting as in the nonrelativis-tic calculation the momentum eigenstates, introducing the Fourier components of the order parameter as in Eq. (19), and performing the change of variables to the total and relative momentum, one obtains
T
4ΩTr(G0Σ)
2
= T
2Ω X
n X
k,k0
Tr(Λ+kγ0Φ˜k−k0Λ−
k0γ0Φk0−k)
(iνn−ξk)(iνn+ξk0)
=
= T
2Ω X
n X
q
φ∗a,qφb,−q
X
p
Tr Λ+p+q
2
Ta†Λ−−p+q
2
Tb
(iνn−ξp+q2)(iνn+ξp−q
2)
. (26)
The static mass term follows immediately upon settingq =0 and performing the sum-integral using Eq. (32),
mass term = 1 2Ntφ
∗
aφb
Tr(Λ+pTa†Λ−−pTb)
p. (27)
thus becomes
T 2Ω
X
n X
q
φ∗a,−qφb,q
X
p
Tr Λ+−p−q
2
γ0Ta†γ0Λ−p−q
2
γ0Tbγ0
(iνn−ξ−p−q2)(iνn+ξ−p+q2)
=
= T
2Ω X
n X
q
φ∗a,qφb,−q
X
p
Tr Λ+p−q
2
Ta†Λ−−p−q
2
Tb
(iνn−ξp+q2)(iνn+ξp−q2)
.
(In the second step, we changed the sign of the summation variable q and again used the symmetry of the denominator.) We thus conclude that for parity-even pairing channels, the trace in the numerator is also even in q. In the following, we shall neglect the second-order term coming from the expansion of the trace for the following reason. Such a term would be evaluated with the help of Eq. (32) and would therefore be logarithmically divergent. In spite of that, it can be seen that this “divergent” term is, at the critical temperature, negligible as compared to the “finite” term in Eq. (22). Indeed, the “divergent” term is proportional to logarithm of critical temperature, i.e., by Eq. (7) goes as 1/Veff in the weak-coupling limit. On
the other hand, the “finite term” goes as 1/Tc2 ∼exp NV2
eff
!
The dominant term coming from the denominator is evaluated along the same line as in the nonrelativistic case, only the Taylor expansion (20) becomes slightly more involved,
1 iνn∓ξp±q2
= 1
iνn∓ξp− 2pε·q
p ∓
q2
8εp ±
(p·q)2
8ε3
p
= 1
iνn∓ξp
1 +
p·q
2εp ±
q2
8εp ∓
(p·q)2 8ε3
p
iνn∓ξp
+
p·q
2εp
2
(iνn∓ξp)2
.
Multiplying the Taylor expansions of the two propagators, we realize that the extra term in the relativistic Taylor expansion does not contribute for the same reason as several other terms. We are thus left with
gradient term =− T
2Ω X
n X
q
φ∗a,qφb,−q
X
p
Tr(Λ+
pT
†
aΛ
− −pTb) ν2
n+ξp2
p·q
2εp
2 −
3 (ν2
n+ξp2)2
+ 4ξ
2
p
(ν2
n+ξp2)3
.
Using the sum-integral (33), this can be finally reexpressed as
gradient term = 7ζ(3)p
2 F
32π2µ2T2
c
N X
q
q2φ∗a,qφb,−q
( ˆp·qˆ)2Tr(Λ+pTa†Λ−−pTb)
p. (28)
Equations (25), (27), and (28) represent the general GL functional for a relativistic supercon-ductors. It is instructive to check them on a simple example. Let us therefore consider the relativistic generalization of the BCS model (15). It has a single order parameter φ and the Dirac structure T =γ5, corresponding to positive parity. Using the identity
γ5Λ±kγ5 = Λ∓−k,
we can see that all required Dirac traces are simply equal to two. The only nontrivial angular average will be h( ˆp·qˆ)2i= 13. The whole GL functional thus reduces to
F = 7ζ(3)p
3 F
96π4µT2
c
|∇φ|2+Nt|φ|2+7ζ(3)µpF 32π4T2
c
|φ|4,
GL function for spin-one color superconductors
As a nontrivial application of the above general formulas, we shall now calculate the traces and averages explicitly for spin-one pairing patterns. We will initially assume that the gap matrix has the structure
Φ =φaiQaγi,
without specifying the flavor matrices Qa. They will only be required to be normalized as Tr(QaQ
†
b) =δab.
The gradient and mass terms contain the same Dirac trace which is evaluated using the explicit expression for the energy projectors (24),
Tr(Λ+pTai†Λ−−pTbj) =−δabTrD(Λ+pγiΛ−−pγj) =
=−1
4δab
TrD(γiγj) + TrD
1 εp
γ0(γ·p+m)γi
−1 εp
γ0(−γ·p+m)γj
=
=−1
4δab
−4δij + 1 ε2
p
TrD[(γ·p+m)γi(γ·p+m)γj]
=
=−δab
−δij+ 1 ε2
p
(2pipj−p2δij −m2δij)
→2δab
δij − p2F µ2pˆipˆj
.
With this trace, we immediately obtain upon angular averaging (Eq. (34))
mass term =Nt
1− p
2 F
3µ2
~
φ†a·φ~a. (29)
The gradient term follows similarly using both Eqs. (34) and (35),
gradient term = 7ζ(3)p
3 F
64π4µT2
c X
q
φ∗a,qφb,−qqkql
2ˆpkpˆlδab
δij − p2
F
µ2pˆipˆj
p
=
= 7ζ(3)p
3 F
32π4µT2
c X
q
φ∗a,qφa,−qqkql
1
3δijδkl− p2
F
15µ2(δijδkl+δikδjl+δilδjk)
,
gradient term = 7ζ(3)p
3 F
96π4µT2
c
1− p
2 F
5µ2
|∇iφa~ |2− 2p2F
5µ2|∇ ·~ φa~ | 2
. (30)
Note that formulas (29) and (30) are valid for arbitrary quark mass. To evaluate the Dirac trace in the quartic term, we resort to the ultrarelativistic limit. In this case, the energy projectors reduce the spatial γ-matrices to their transversal parts,
γ⊥i =Pijγj, Pij =δij −pˆipˆj,
thanks to the identities
Λ±kγ⊥=γ⊥Λ±k, Λ±kγk =γkΛ∓k. The Dirac trace thus simplifies to
1
Using the identity (35) we calculate the average of a product of two projectors,
hPijPklip =h(δij −pˆipˆj)(δkl−pˆkpˆl)ip=δijδkl− 1
3δijδkl− 1
3δijδkl+hpˆipˆjpˆkpˆlip =
= 1
15(6δijδkl+δikδjl+δilδjk), From here,
2hPijPkl− PikPjl+PilPjkip = 4
15(3δijδkl−2δikδjl+ 3δilδjk), and we finally get
quartic term = 7ζ(3)µpF 240π4T2
c
(3δijδkl−2δikδjl+ 3δilδjk)φ∗aiφbjφ∗ckφdlTrF(Q†aQbQ†cQd). (31)
Last, we calculate the remaining flavor trace for the single-flavor, color-antitriplet color super-conductor, for which the color matrices have the explicit form
(Qa)bc =− i
√
2εabc. After a simple manipulation, we get
TrF(Q†aQbQ
†
cQd) = 1
4εaijεbjkεcklεdli = 1
4(δakδib−δabδik)(δciδkd−δcdδik) =
= 1
4(δadδbc−δabδcd−δabδcd+ 3δabδcd) = 1
4(δabδcd+δadδbc). Inserting this in Eq. (31), we get the ultimate result
quartic term = 7ζ(3)µpF 480π4T2
c h
3(φ~†a·φ~a)2+ 3|φ~†a·φ~b|2−2|φ~a·φ~b|2 i
.
Appendix: Some useful formulas
Matsubara sums:
T X
n
log(νn2+x2) = x+ 2T log 1 +e−βx,
T X
n 1 ν2
n+x2
= 1
2xtgh βx
2 ,
T X
n
1 (ν2
n+x2)2
= 1
4x2
tghβx2
x −
β
2 cosh2 βx2 !
.
Sum-integrals in the high-density approximation:
T X
n
Z d3k
(2π)3
1 ν2
n+ξk2
=Nlog2βΛe γ
π , (32)
T X
n Z
d3k
(2π)3
1 (ν2
n+ξk2)2
= 7ζ(3)
8π2T2N, (33)
TX
n Z
d3k
(2π)3
1
(iνn−ξk)3(iνn+ξk)
= 7ζ(3)
Angular averages of products of momentum components:
hpˆipˆjip =
1
3δij, (34)
hpˆipˆjpˆkpˆlip =
1
