quantum field theory - gauge-invariant 6-quark order parameter

In this Review paper in p.1462, bottom left: Rev.Mod.Phys.80:1455-1515,2008 -- Color superconductivity in dense quark matter

It says that "There is an associated gauge-invariant 6-quark order parameter with the flavor and color structure of two Lambda baryons, $$ \langle\Lambda\Lambda\rangle $$ where this order parameter distinguishes the color flavor locking (CFL) phase from the quark gluon plsma QGP.

I suppose that it means the 6 quark condensate is $$ \bigl\langle(\epsilon^{abc}\epsilon_{ijk}\psi^a_i\psi^b_j\psi^c_k) (\epsilon^{a'b'c'}\epsilon_{i'j'k'}\psi'^a_i\psi'^b_j\psi'^c_k)\bigr\rangle, $$

1. 
   

   but how does this distinguish CFL from QGP?

   
   


2. 
   

   Is this operator precise? And is this gauge invariant under SU(3)???

   
   


3. 
   
   

   It is a Lorentz scalar or pseudo scalar?

   
   

It seems that the claim is not clear.

Answer

1. 
   

   It breaks $U(1)_B$, and therefore distinguished QGP from CFL.

   
   



2. 
   

   Yes, this is a gauge invariant operator.

   
   


3. 
   

   This is a Lorentz scalar if the spinors are contracted appropriately, for example $$ \phi \sim \epsilon_{\alpha\alpha'}\epsilon_{\beta\gamma}\epsilon_{\beta'\gamma'} (\psi_\alpha\psi_\beta\psi_\gamma)(\psi_{\alpha'}\psi_{\beta'}\psi_{\gamma'}) $$ In 4-component notation this can be written in terms of a (positive parity) baryon current $$ \phi \sim \Psi C\gamma_5 \Psi, \qquad\Psi_\alpha = \psi_\alpha (\psi C\gamma_5\psi) $$