# gauge-invariant 6-quark order parameter

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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.

This post imported from StackExchange Physics at 2020-11-05 13:26 (UTC), posted by SE-user annie marie heart
It looks gauge invariant, because $\epsilon$ tensor is SU(3) invariant. About the Lorentz structure, there is a problem with your expression. You have $6$ $\psi$ fields and no $\bar \psi$. Therefore I think this correlation function vanishes. Perhaps you meant something like $\bar \psi^3 \psi^3$. In this case you still need to specify what you do with bispinor indices of $\psi$ and $\bar \psi$.

This post imported from StackExchange Physics at 2020-11-05 13:26 (UTC), posted by SE-user Blazej
can we show ϵ tensor is SU(3) singlet?

This post imported from StackExchange Physics at 2020-11-05 13:26 (UTC), posted by SE-user annie marie heart
Yes, the $\epsilon$ tensor is how one constructs a singlet out of fundamentals. Georgi's group theory book might be a useful place to check this out if it isn't familiar.

This post imported from StackExchange Physics at 2020-11-05 13:26 (UTC), posted by SE-user David Schaich

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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 $$\phi \sim \Psi C\gamma_5 \Psi, \qquad\Psi_\alpha = \psi_\alpha (\psi C\gamma_5\psi)$$

This post imported from StackExchange Physics at 2020-11-05 13:26 (UTC), posted by SE-user Thomas
answered Dec 25, 2017 by (280 points)
thanks +1, is this 6-quark term related to breaking the axial $Z_6$ of QGP to the $Z_2$ vector symmetry (same as axial) of CFL? or am I wrong???

This post imported from StackExchange Physics at 2020-11-05 13:26 (UTC), posted by SE-user annie marie heart
Instantons break $U(1)_A$ to $Z_6$ in both the QGP and CFL. CFL further breaks $Z_6$ to $Z_2$, but this is not seen from this order parameter. One way to see this is to observe that in CFL chiral symmetry is broken, and $\langle \bar\psi_L\psi_R\rangle$ is not zero.

This post imported from StackExchange Physics at 2020-11-05 13:26 (UTC), posted by SE-user Thomas
so what is the symmetry of this order parameter bring down? Is that $SU(3)_{L+R+C} \times Z_{6}$?

This post imported from StackExchange Physics at 2020-11-05 13:26 (UTC), posted by SE-user annie marie heart
Each baryon like operator either has spin 1/2 or spin 3/2. How should I understand your indices contraction in terms of spin objects? The total spin should be 0, but contracting from spin 1/2 and spin 1/2?

This post imported from StackExchange Physics at 2020-11-05 13:26 (UTC), posted by SE-user annie marie heart
@ Thomas, are you writing a 2-component spinor indices to contract with $\epsilon_{12}=-\epsilon_{21}=1$? And how is that correspond to spin 1/2 pairing mechanism?

This post imported from StackExchange Physics at 2020-11-05 13:26 (UTC), posted by SE-user annie marie heart
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