# Conservation of weak hypercharge and isospin

1. Sep 8, 2010

### michael879

So I've been playing around with different electroweak lagrangians (sourceless, higgsless, unbroken, broken, etc), and finding the equations of motion for them. Everything looks about right, but the issue I'm having is when you add in the masses of the weak bosons(no higgs though).This breaks the gauge symmetry of the U(1) and SU(2) fields, and should therefore also break the conservation law for weak hypercharge and isospin (of course there would still be a conservation law and gauge for the combination of hypercharge and isospin corresponding to the E&M field). And when I solve for the equations of motion, I do end up with the equations:
$$\partial^\mu{Y_\mu} = \partial^\mu(m_B^2B_\mu+M_a^2W_\mu^a)$$
$$\partial^\mu(I_\mu^a+J_\mu^a) = \partial^\mu(m_a^2W_\mu^a+M_a^2B_\mu)$$
where $$m_a$$ and $$M_a$$ are mass terms, and $$J_\mu^a$$ is the current of the a'th component of weak isospin due to the fields themselves.

These equations show that hypercharge and isospin are not necessarily conserved. However, from what I've read all 4 of these currents should be conserved. My question is whether this is just a postulate taken from experimental results or if I'm missing some theoretical reason that these currents must be conserved.

2. Sep 8, 2010

Staff Emeritus
They aren't conserved experimentally.

3. Sep 8, 2010

### michael879

Are you sure about that? I read from multiple sources that weak hypercharge and, at least the third component of, weak isospin are conserved..

4. Sep 8, 2010

Staff Emeritus
What is the weak hypercharge of a left-handed electron? What is the hypercharge of a right-handed electron?

5. Sep 8, 2010

### michael879

How does a left-handed electron become a right-handed electron...?

6. Sep 8, 2010

Staff Emeritus
Change of reference frame.

7. Sep 9, 2010

### michael879

Your talking about helicity not chirality... An electron with right handed helicity can still interact weakly.

8. Sep 9, 2010

Staff Emeritus
That's right. But what is the chirality of a physical electron? It doesn't have one - it's not in an eigenstate of chirality. So how can it be in an eigenstate of hypercharge?

9. Sep 9, 2010

### michael879

It doesn't have to be an eigenstate of chirality... Let's say the electron is in some superposition of left and right chilraliries. If we measure it interacting weakly that's just the same as it decohering into the left chirality state and the interacting weakly... You still havnt explained how weak charge conservation can be violated

10. Sep 9, 2010

Staff Emeritus
Take an electron - measure it's weak hypercharge somehow. Now measure it again. And again. Will you get the same answer every time?

11. Sep 9, 2010

### michael879

Umm yes... At least after the first time once it's collapsed into a hypercharge eigenstate...

12. Sep 9, 2010

### chrispb

Apparently without QCD, the EW Y is conserved: http://en.wikipedia.org/wiki/Weak_hypercharge

I don't know the answer to your original question, but I feel like the answer should jump out at you. In the expressions you've derived, it's clear that dJ is 0 in a particular gauge. I feel like it should be apparent in general, though.

13. Sep 10, 2010

### michael879

Thats not what they say in that article.. They say that the strong hypercharge is not a conserved quantity, but it is similar to electroweak hypercharge which IS a conserved quantity. However this is all according to wikipedia, which is not very reliable when you get this deep into physics. I couldn't find any reliable sources that claimed this conservation law, which is why I came here.

There is no gauge freedom in those potentials, so you can't just enforce the Lorentz gauge and say "charge is conserved" without some kind of justification. Jackson does this too when he brings up the Proca equations of a massive photon. He just says electric charge must be conserved therefore the lorentz gauge is the correct one. However, you can't just assume one or the other without justification (is charge even macroscopically "conserved" in superconductors??).

14. Sep 10, 2010

### chrispb

Well, both of those statements were actually what I intended to say. That's why I said I felt this should be apparent in general, which is not true from the expressions you've given.

Here, there is a conserved current and there IS a gauge symmetry. In the case of superconductivity, you're explicitly breaking local but not global gauge symmetry. Leaving global gauge symmetry intact ensures that charge is still conserved. That feels like a qualitatively different situation than the one you're describing.

15. Sep 10, 2010

### michael879

The expressions I've given can be trivially derived from the electroweak lagrangian, so I'm not sure what's wrong about them. In fact, I can't find anything in the lagrangian itself that would enforce either the Lorentz gauge or charge conservation.

Thats a good point, but doesn't global gauge symmetry only gaurantee global charge conservation? Thats a weaker statement than the local charge conservation derived from Maxwell's equations. And actually E&M in a superconductor is exactly the situation I'm talking about. The electromagnetic fields gain mass terms in a superconductor, and it is those mass terms (that are also in the weak fields) that destroy the local charge conservation law.

16. Sep 10, 2010

### michael879

Also, if there is a conserved current, please prove it! I see no way to get a conservation law out of the Proca lagrangian (in fact, I dont see a global gauge symmetry either...).

17. Sep 10, 2010

Staff Emeritus
No, because hypercharge doesn't commute with the total Hamiltonian. But since this isn't helping, let's try another direction.

Consider the vertex eL, H0, eR. The Higgs doesn't carry hypercharge, and the left and right handed electrons carry different hypercharge quantum numbers.

18. Sep 10, 2010

### michael879

Ok let me rephrase the question. Instead of dealing with the weak interaction, lets just talk about some hypothetical field that behaves like the E&M field in a superconductor. The lagrangian is given by:

$$L=-\frac{1}{4}F^{\mu\nu}F_{\mu\nu} - \frac{1}{2}mA^\mu A_\mu + J^\mu A_\mu$$

Solving for the equation of motion:

$$\partial^\nu F^{\mu\nu} + mA^\mu = J^\mu$$

And differentiating,

$$m\partial_\mu A^\mu = \partial_\mu J^\mu$$

Now for this field (which is analogous to the weak situation and the superconductor situation), can you please explain why (or why not) charge is locally conserved? Every source I've read says it is, but doesn't explain why.

Also, I still dont see any global symmetry. If you integrate over all space, you find that

$$\partial_0Q + \int\nabla\cdot\vec{J}d\tau = \int\partial_0\Phi d\tau + \int\nabla\cdot\vec{A}d\tau$$

And then assuming the vector potential and the current vanish at infinity (which is safe for a massive field),

$$\frac{\partial{Q}}{\partial{t}} = \int\frac{\partial{\Phi}}{\partial{t}}d\tau$$

So the total charge is only conserved for a static potential?

19. Sep 10, 2010

### michael879

The higgs has to carry hypercharge... It interacts with the B field.

20. Sep 10, 2010