Conservation of weak hypercharge and isospin

In summary: 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.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.
  • #1
michael879
698
7
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:
[tex]\partial^\mu{Y_\mu} = \partial^\mu(m_B^2B_\mu+M_a^2W_\mu^a)[/tex]
[tex]\partial^\mu(I_\mu^a+J_\mu^a) = \partial^\mu(m_a^2W_\mu^a+M_a^2B_\mu)[/tex]
where [tex]m_a[/tex] and [tex]M_a[/tex] are mass terms, and [tex]J_\mu^a[/tex] 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.
 
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  • #2
They aren't conserved experimentally.
 
  • #3
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
What is the weak hypercharge of a left-handed electron? What is the hypercharge of a right-handed electron?
 
  • #5
How does a left-handed electron become a right-handed electron...?
 
  • #6
Change of reference frame.
 
  • #7
Your talking about helicity not chirality... An electron with right handed helicity can still interact weakly.
 
  • #8
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
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
Take an electron - measure it's weak hypercharge somehow. Now measure it again. And again. Will you get the same answer every time?
 
  • #11
Umm yes... At least after the first time once it's collapsed into a hypercharge eigenstate...
 
  • #12
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
chrispb said:
Apparently without QCD, the EW Y is conserved: http://en.wikipedia.org/wiki/Weak_hypercharge
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.

chrispb said:
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.
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
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
chrispb said:
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.
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.

chrispb said:
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.
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
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 don't see a global gauge symmetry either...).
 
  • #17
michael879 said:
Umm yes... At least after the first time once it's collapsed into a hypercharge eigenstate...

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

[tex]L=-\frac{1}{4}F^{\mu\nu}F_{\mu\nu} - \frac{1}{2}mA^\mu A_\mu + J^\mu A_\mu[/tex]

Solving for the equation of motion:

[tex]\partial^\nu F^{\mu\nu} + mA^\mu = J^\mu[/tex]

And differentiating,

[tex]m\partial_\mu A^\mu = \partial_\mu J^\mu[/tex]

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 don't see any global symmetry. If you integrate over all space, you find that

[tex]\partial_0Q + \int\nabla\cdot\vec{J}d\tau = \int\partial_0\Phi d\tau + \int\nabla\cdot\vec{A}d\tau[/tex]

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

[tex]\frac{\partial{Q}}{\partial{t}} = \int\frac{\partial{\Phi}}{\partial{t}}d\tau[/tex]

So the total charge is only conserved for a static potential?
 
  • #19
Vanadium 50 said:
Consider the vertex eL, H0, eR. The Higgs doesn't carry hypercharge, and the left and right handed electrons carry different hypercharge quantum numbers.

The higgs has to carry hypercharge... It interacts with the B field.
 
  • #20
No, the Higgs doesn't carry weak hypercharge. It's its own antiparticle, so it can't.
 
  • #21
Umm 1) that rule would only be valid if hypercharge were conserved and 2) the Higgs doublet needs to have hypercharge or it wouldn't couple to the B field
 
  • #22
michael879 said:
Umm 1) that rule would only be valid if hypercharge were conserved

But that's the question.

The Higgs is C even. Therefore if there is a conserved hypercharge y carried by the Higgs, y = -y, so y = 0.

So let me try a third line of argument, and see if this helps. If hypercharge were conserved, there would be a long-range interaction associated with this conserved quantum number and the force carrier would be massless. The B is not only not massless, it's not even in a definite state of mass. Anyway, we would see a new long-range force, which we don't.
 
  • #23
Well that's exactly my point, the symmetry which gives rise to weak hypercharge is very broken but every source still says it's conserved... And sure, if it's conserved the Higgs has to have hypercharge 0, but are u sure that vertex is even possible? I don't have the resources to do te calculation on me right now, but if u replace the higgs with a photon that vertex is definitely forbidden
 
  • #24
And a great example of this, which I already brought up, is in superconductors. If weak hypercharge isn't conserved, then electric charge shouldn't be conserved in a superconductor!
 
  • #25
The Higgs couples left-handed guys to right-handed guys. That's it's job.
 
  • #26
michael879 said:
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:
[tex]\partial^\mu{Y_\mu} = \partial^\mu(m_B^2B_\mu+M_a^2W_\mu^a)[/tex]
[tex]\partial^\mu(I_\mu^a+J_\mu^a) = \partial^\mu(m_a^2W_\mu^a+M_a^2B_\mu)[/tex]
where [tex]m_a[/tex] and [tex]M_a[/tex] are mass terms, and [tex]J_\mu^a[/tex] 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.

I don't have a clear explanation for what's going on here, but I would be very cautious trying to relate these sorts of equations of motion to physical questions. Simply tossing gauge-boson mass terms into the lagrangian explicitly destroys the gauge invariance, and is most definitely not what occurs in nature, where the electroweak gauge invariance is maintained but hidden (or "spontaneously broken"). The same issue arises in superconductivity: electromagnetism is not "explicitly broken", but is "spontaneously broken" when the photons "eat" Cooper pairs.

This is exactly the problem Glashow ran into when trying to come up with an electroweak gauge theory, which was only resolved years later when Weinberg and Salam added "spontaneous symmetry-breaking" to the mix.
 
  • #27
Daschaich, thank you that makes perfect sense. I forgot I replaced the higgs terms with mass terms to simplify things. So if the gauge invariance remains intact after the spontaneous symmetry breaking, then the weak charges ARE conserved right?
 
  • #28
Also if that's the case, then if I were to find the equations of motion for the full electroweak lagrangian I should be able to derive charge conservation right?
 
  • #29
Ok so I'm working on deriving four conserved currents from the EW lagrangian, but if hypercharge isn't conserved, answer this vanadium. The EW lagrangian, with the higgs field, still has an SU(2)xSU(1) gauge symmetry. As long as you modify the higgs field in a way corresponding to the SU(2) and SU(1) independent gauge transformations, the lagrangian will remain unchanged. This alone should be proof that even after electroweak symmetry breaking there remain 4 conserved currents. Right?

They do clearly mix, so maybe weak hypercharge and weak isospin are no longer conserved, but there should be 4 combinations of those 4 charges that are conserved.
 
  • #30
Really? Noone has an answer for me??
 
  • #31
^bump?
 

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