I’m studying color superconductivity referring to “The Phases of Quantum Chromodynamics From Confinement to Extreme Environments” by John B. Kogut and Mikhail A. Stephanov (link). In chapter 9, the authors get an effective Lagrangian for the diquark condensate in momentum space around the Fermi surface in Euclidean space as follows (the notation may be wrong), \begin{align*} & \psi^\dagger_1(p)(ip_0+\epsilon_p)\psi_1(p)+ \psi^\dagger_2(-p)(-ip_0+\epsilon_p)\psi_2(-p) +\Delta_p\psi_1(p)\psi_2(-p)+\Delta_p\psi_2^\ast(-p)\psi_1^\ast(p) \\ & = \begin{bmatrix} \psi_1(p) \\ \psi_2^\ast(-p) \end{bmatrix}^\dagger \begin{bmatrix} ip_0+\epsilon_p & -\Delta_p \\ -\Delta_p & ip_0-\epsilon_p \end{bmatrix} \begin{bmatrix} \psi_1(p) \\ \psi_2^\ast(-p) \end{bmatrix} \end{align*} where $\epsilon_p= |p| -\mu$ and they assume the quark pairing between single component fermions $\psi_1$ and $\psi_2$ so that other quantum numbers are set appropriately (anti-symmetric color and flavor channel). And they introduce the quark and diquark propagators which they call normal and anomalous propagators,
\begin{align*} \langle\psi_1^\dagger(p)\psi_1(p)\rangle=\langle\psi_2^\dagger(p)\psi_2(p)\rangle=\frac{-ip_0+\epsilon_p}{p_0^2+\epsilon_p^2+\Delta_p^2} \end{align*} \begin{align*} \langle\psi_1(p)\psi_2(-p)\rangle=\frac{\Delta_p}{p_0^2+\epsilon_p^2+\Delta_p^2} . \end{align*} The problem is that there is no derivation for this and I'm confused because they are used in the rest of the discussion. It looks like BCS theory but do you have any ideas?
