The effective action of quarks

, (2.24)

where the quantities Σ^± are the regular self-energies, and Φ^± are the anomalous self-energies.

For Φ^±, also the term gap matrices is used. The gap matrices in connection with the quark self-energy (2.23) yield the so-called gap equations. By solving these gap equations, one obtains the gap parameters. In space-time, Σ⁺(X, Y ) [Σ⁻(X, Y )] has a quark [charge-conjugate quark]

entering at X and another quark [charge-conjugate quark] emerging at Y . The anomalous self-energies have to be interpreted as follows: a quark [charge-conjugate quark] enters Φ⁺(X, Y ) [Φ⁻(X, Y )] at X and, in contrast to the regular self-energies, here, a charge-conjugate quark [an ordinary quark] emerges at Y . This is typical for systems with a fermion-fermion condensate in the ground state [127]. The self-energies Φ^± symbolize this condensate. This is why the crucial quantities regarding color superconductivity are the gap matrices Φ^±. A nonzero value of Φ^± is equivalent to Cooper pairing, or, in other words, to a nonvanishing diquark expectation value.

The self-energies in Eq. (2.24) are related in the same way as the bilocal sources in Eq. (2.15), Σ⁻≡ C Σ⁺
T

C⁻¹ , Φ⁻≡ γ⁰ Φ⁺
_†

γ0. (2.25)

The quark propagator in Nambu-Gorkov space can be determined from the Dyson-Schwinger equation (2.22), see Sec. B.3 in the Appendix,

S =

G⁺ Ξ⁻ Ξ⁺ G⁻



, (2.26)

where

G^±=n

[G0^±]⁻¹+ Σ^±− Φ^∓ [G0^∓]⁻¹+ Σ^∓
₋₁ Φ^±o₋₁

, (2.27)

is the propagator for quasiquarks or charge-conjugate quasiquarks, respectively, and Ξ^± =− [G0^∓]⁻¹+ Σ^∓
₋₁

Φ^±G^±=− G^∓Φ^± [G0^±]⁻¹+ Σ^±
₋₁

, (2.28)

are the anomalous propagators. These anomalous propagators are typical for superconducting systems [127] and account for the possibility that in the presence of a Cooper-pair condensate, symbolized by Φ^±, a fermion can always be absorbed in the condensate, while its charge-conjugate counterpart is emitted from the condensate and continues to propagate.

The QCD pressure is, up to a prefactor T /V , equal to the QCD effective action (2.17) deter-mined at the stationary points (2.18) which is denoted by Γ^∗,

p = T

VΓ^∗, (2.29)

where T is the temperature and V the volume of the system.

2.1.2 The effective action of quarks

In this section, I investigate the phase diagram of massless neutral three-flavor quark matter. The quark spinor has the following color-flavor structure:

ψ = ψ_u^r, ψ^r_d, ψ_s^r, ψ^g_u, ψ_d^g, ψ^g_s, ψ^b_u, ψ^b_d, ψ_s^b
T

. (2.30)

Since I approximate the gluon-exchange interaction between quarks by a point-like four-fermion coupling, I only need to consider the contributions of quarks to the QCD effective action (2.17),

Γ [S] = 1

2Tr ln S⁻¹+1

2Tr S₀⁻¹S− 1

+ Γ2[S] . (2.31)

Here, the tree-level quark propagatorS0⁻¹ (2.13), which occurs in the QCD effective action (2.17), is replaced by the quark propagator,

S₀⁻¹≡

[G⁺₀]⁻¹ 0 0 [G⁻₀]⁻¹



, (2.32)

where

[G^±₀]⁻¹(X, Y )≡ (i∂_X± ˆµγ⁰) δ⁽⁴⁾(X− Y ) , (2.33) is the massless inverse Dirac propagator for quarks and charge-conjugate quarks, respectively, in which the constant background field A^a_µ disappears from the treatment, and the color chemical potentials µ3and µ8assume the role of the background field to ensure color neutrality. The quark chemical potential matrix in color-flavor space is defined as

ˆ

µ = diag µ^r_u, µ^r_d, µ^r_s, µ^g_u, µ^g_d, µ^g_s, µ^b_u, µ^b_d, µ^b_s

, (2.34)

where the chemical potential of each quark is given by Eq. (1.52) because quark matter inside neutron stars is in β equilibrium.

Since I shall present the phase diagram of massless neutral three-flavor quark matter in this section, the mass term of the inverse Dirac propagators (2.33) is omitted. At sufficiently large quark chemical potential, there is no need to take into account the small up and down quark masses because the dynamical effect of such masses around the quark Fermi surfaces is negligible.

Of course, the situation with the strange quark is different because its mass is not very small as compared to the quark chemical potential µ. The most important effect of a nonzero strange quark mass is, however, a shift of the strange quark chemical potential due to the reduction of the Fermi momentum,

(kF)ⁱ_s=p

(µⁱ_s)²− m²s≃ µⁱs− m²_s

2µⁱ_s ≃ µⁱs−m²_s

2µ , (2.35)

cf. Eq. (1.89). Here, I have approximated µⁱ_s by µ in the denominator. Quantitatively, this does not make a big difference.

Figure 2.1: Left panel: the sunset-type diagram. Right panel: the double-bubble diagram.

For the sum of all 2PI diagrams I only include the sunset-type diagram, which is shown in the left panel of Fig. 2.1,

Γ2[S] =−g² 4

Z

X,Y

Tr [Γ^µ_aS (X, Y ) Γ^ν_bS (Y, X)] D^ab_µν(X, Y ) . (2.36)

2.1 The phase diagram of massless quarks 47

The trace runs over Nambu-Gorkov, color-flavor, and Dirac indices. The Nambu-Gorkov vertex is defined as

The stationary point of the effective action of quarks (2.31), δΓ

δS = 0 , (2.38)

is the Dyson-Schwinger equation for the quark propagator,

S⁻¹(X, Y ) = S₀⁻¹(X, Y ) + Σ (X, Y ) , (2.39) where

Σ (X, Y ) = 2 δΓ2

δS (Y, X) =−g²Γ^µ_aS (X, Y ) Γ^ν_bD^ab_µν(Y, X) (2.40) is the quark self-energy. The effective action at the stationary point which is determined by the Dyson-Schwinger equation for the quark propagator (2.39) reads,

Γ^∗= 1

2Tr ln S⁻¹−1

4Tr (ΣS) . (2.41)

For translationally invariant systems, it is advantageous to work in energy-momentum space instead of in space-time,

where I assumed translational invariance of propagators and self-energies, Z≡ X − Y . In energy-momentum space, the Dyson-Schwinger equation for the quarks reads,

S⁻¹(K) = S⁻¹₀ (K) + Σ (K) . (2.43)

The inverse free quark propagator in energy-momentum space is given by S₀⁻¹(K) =

[G⁺₀]⁻¹ 0 0 [G⁻₀]⁻¹



, (2.44)

where the massless inverse Dirac propagator for quarks and charge-conjugate quarks, respectively, reads,

[G^±₀]⁻¹(K) = γ0(k0± ˆµ) − γ · k . (2.45) The quark self-energy is obtained by

Σ =

are the gap equations which are independent of K because I approximate the gluon-exchange interaction between quarks by a point-like four-fermion coupling. The inverse quark propagator is

S⁻¹(K) =

[G⁺₀]⁻¹ Φ⁻ Φ⁺ [G⁻₀]⁻¹



. (2.48)

The regular self-energies play an important role in the dynamics of chiral symmetry breaking, but they are of less importance in color-superconducting quark matter. (The effect of the regular self-energies was studied in Ref. [107].) Therefore, the regular self-self-energies in Eq. (2.46) are omitted.

The quark self-energy (2.46) contains the Feynman gauged gluon propagator, see Sec. B.4 in the Appendix,

D^ab_µν=−δ^abgµν

Λ² , (2.49)

which represents the gluon-exchange interaction between quarks by a point-like four-fermion cou-pling. In this approximation, the sunset-type diagram becomes a double-bubble diagram, which is shown in the right panel of Fig. 2.1.

The quark propagator may be obtained by inverting Eq. (2.48),

S (K) = is the propagator for quasiquarks or charge-conjugate quasiquarks, respectively, and

Ξ^±(K) =−G^∓0Φ^±G^±=−G^∓Φ^±G^±₀ , (2.52) are the anomalous propagators.

I want to study the most general ansatz of the gap matrix for the CFL phase in color-flavor space. Therefore, I use the following nine-parameter ansatz [110]:

Φ^±=

This ansatz for the gap matrix is indeed the most general one for the CFL phase because it can be obtained with some modifications from the antitriplet and sextet gap ansatz (1.20). The difference between the gap ansatz (1.20) and the gap matrix (2.53) is that the gap ansatz (2.53) has different entries for all nonzero color-flavor elements. The wavefunction requirements are still fulfilled because the gap matrix (2.53) is symmetric in color-flavor space.

The effective action of quarks at the stationary point (2.41) in energy-momentum space reads,

Γ^∗= 1

where I used Eqs. (2.42), the derivation of the Fourier transformed kinetic part of the QCD grand partition function which is shown in Sec. B.5 in the Appendix, and the relation ln det A = Tr ln A which is proven in Sec. B.6 in the Appendix.

2.1 The phase diagram of massless quarks 49