Why is the QCD vacuum important in non-abelian gauge theories?

[QuantumCosmo]

Ah yeah I think I got it :)

But how does this B+L U(1) symmetry look?
I think U(1)_B is the vector symmetry of QCD:
\psi_L \rightarrow e^{i\alpha}\psi_L
and
\psi_R \rightarrow e^{i\alpha}\psi_R

and B is the conserved charge of that symmetry.

But I do not know the symmetries corresponding to L, B+L or B-L...

 

[blechman]

The Standard Model has quarks and leptons. The quarks have baryon number +1/3, and the leptons have lepton number +1, with their antiparticles the negative. This is a U(1) symmetry of the SM Lagrangian.

HOWEVER: both of these symmetries are anomalous: if you compute the triangle diagrams where you insert a B current on one vertex, and W bosons on the other two vertices, that triangle diagram is nonzero. Same with inserting an L current instead of a B current.

But it is also true that these anomalies (the B and the L) are equal. So that if you take the DIFFERENCE of the two, they cancel and that current is not anomalous. That is the B-L current. By a similar analysis, the sum of the two currents is 2x the anomaly of either one of them.

Notice however, that there is NO QCD anomaly: if you replace the W bosons with gluons, the triangle with the B current vanishes on its own. The one for L vanishes trivially, since leptons have no color charge.

So we have the equations:

\partial_\mu J_{B+L}^\mu = \frac{Ag^2}{16\pi^2}W\tilde{W}
\partial_\mu J_{B-L}^\mu =0

Notice that THERE ARE NO GLUONS IN THIS EQUATION! Just W bosons.

So by the arguments given above: performing a phase redefinition on all the fermions (quarks AND leptons) in accord with the B+L charge (using Weyl notation where the R-fermions are ANTIfermions; you can translate to Dirac easily by inserting \gamma^5s in the appropriate places):

q_L\rightarrow e^{i\alpha/3}q_L
q_R\rightarrow e^{-i\alpha/3}q_R
l_L\rightarrow e^{i\alpha}l_L
l_R\rightarrow e^{-i\alpha}l_R

will not change the lagrangian sans the theta terms (since it's a symmetry), and it won't change the QCD theta term, since there is no (B+L)-G-G anomaly. But it WILL shift the W-theta term by an amount proportional to \alpha. So if you choose \alpha just right, you can cancel the SU(2) theta angle without doing any more damage.

If you try to shift the QCD angle, you cannot use this phase redefinition I wrote down. You have to use another one (such as rotating both qL and qR by the SAME angle rather than opposite angle). Such a redefinition will reintroduce phases into the masses of the quarks, and so THAT is why \theta_{QCD} is physical. There will always be a phase there somewhere.

UNLESS one of the quark masses vanish. In that case, we're golden. Some people believe the up quark might have vanishing mass. It would mean that \theta_{QCD} is unphysical, and therefore solve the strong CP problem; but it also would go against various other results such as lattice calculations. Personally I am inclined to believe the up mass is nonzero, but plenty of perfectly respectable physicists disagree with me.

Anyway, I hope that helps. Does that make sense?

 

[QuantumCosmo]

Anyway, I hope that helps. Does that make sense?

Hm, yeah, I think I might have understood it :)

My whole problem is that my QFT basics aren't that good... and that I often have problems with (I guess) very easy parts, for example those U(1) transformations.

I now suspect (after what you just wrote) that
q_L\rightarrow e^{i\alpha/3}q_L
q_R\rightarrow e^{-i\alpha/3}q_R
is a (global) symmetry of the Lagrangian and it's corresponding charge is B.

And
l_L\rightarrow e^{i\alpha}l_L
l_R\rightarrow e^{-i\alpha}l_R
is another symmetry of the Lagrangian with charge L.

And if I want the symmetry with charge B+L, I simply apply both of them. Is that correct?

As for the results of applying that symmetry (the change in the weak \theta term), I think I have understood it :)

Thanks a million for that!

 

[QuantumCosmo]

Hi, I have found an article that explains why the \theta_{weak} term can be eliminated.
It seems, though, this is different from what we discussed here (although I don't really understood what they were doing...)
A. A. Anselm and A. A. Johansen, Nucl. Phys. B407(1993) 652

 

[ansgar]

see my presentation about this topic:

http://www.isv.uu.se/~wouda/axion-beamer-GW.pdf

 

[blechman]

And if I want the symmetry with charge B+L, I simply apply both of them. Is that correct?

yup!

As for the results of applying that symmetry (the change in the weak \theta term), I think I have understood it :)

Thanks a million for that!

As to that reference: that's just filling in all the details by doing the instanton calculation carefully, finding the relevant "zero modes" of the fermions, etc. I skipped all those details for the sake of sanity!

As to ansgar's presentation: thanks for sharing. It looks like a very interesting talk.