gauge-invariant 6-quark order parameter

Question

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

Comments

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

Answer 1

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

Comments

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

---

how do one contracts? you said: "This is a Lorentz scalar if the spinors are contracted appropriately." Do we need some gamma matrices?

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

---

also, how about this one? physics.stackexchange.com/questions/376203

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

---

This is meant to be 2-component spinor notation. Then $\Psi_\alpha=\epsilon_{\beta\gamma}\psi_\alpha\psi_\beta\psi_\gamma$ is a spin 1/2 nucleon field (because $\Phi=\epsilon_{\beta\gamma}\psi_\beta\psi_\gamma$ is a spin 0 diquark). Taking a spin singlet combination of two nucleon fields gives a scalar order parameter.

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

---

@ Thomas, one more clarification, in this $ \epsilon_{\alpha\alpha'}\epsilon_{\beta\gamma}\epsilon_{\beta'\gamma'} (\psi_\alpha\psi_\beta\psi_\gamma)(\psi_{\alpha'}\psi_{\beta'}\psi_{\gamma'}) $, must I have $(\psi_\alpha\psi_\beta\psi_\gamma)$ as the 1st baryon and $(\psi_{\alpha'}\psi_{\beta'}\psi_{\gamma'})$ as the 2nd baryon?

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

---

Can I have instead $$ \epsilon_{\alpha\alpha'}\epsilon_{\beta \beta'}\epsilon_{\gamma\gamma'} (\psi_\alpha\psi_\beta\psi_\gamma)(\psi_{\alpha'}\psi_{\beta'}\psi_{\gamma'}) $$ while the color/flavor indices are given as above? (p.s. am going to accept you as the answer after this)

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

---

Or can I write these scalar in terms of $C \gamma^5$ in4-component spinor notations?

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

---

@ Thomas, I am also curious about the L and R handness indices and properties of parity here. Thanks a lot.

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

---

@ Thomas : In particular, I thought we are considering the Dirac spinor, how does $$\epsilon_{\alpha\alpha'} (\psi_\alpha )(\psi_{\alpha'})$$, $$Ψα=ϵ_{βγ}ψ_αψ_βψ_γ$$ represent which part of Dirac spinor? Or should the baryon composed by three of 2-component Weyl spinor $(\psi_\alpha )$ instead of 4-component Dirac spinor? Thanks!

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

---

@ Thomas, also how is the parity $P$ property of the condensate? (under $L \to R$?) Are there choices?

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

---

@annie heart Added a postscript

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

---

can it be negative parity, too? accepted, but please clarufy.

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