IS weak isospin conserved by all interactions?

1. metroplex021

133
Hi people: I keep reading one day that weak isospin is exactly conserved by all interactions; other days that sometimes weak isospin is *not* conserved. Can anyone clear this one up?!

Staff: Mentor

Ask there for processes that violate the conservation?

Maybe some BSM models violate it, no idea, the standard model does not as far as I know.
Edit: Okay, this is wrong, see Bill_K.

Last edited: Oct 4, 2013

4,160

4. metroplex021

133
Aargh! I don't understand... why would we have even introduced isospin symmetry if it were not conserved by any interaction?! Isn't the above phenomenon more indicative of parity violation than anything else?

5. The_Duck

929
Surely the issue is a little more subtle? If we write down the Standard Model Lagrangian before electroweak symmetry breaking, there is an exact SU(2)_isospin symmetry. Any process described by this Lagrangian should conserve weak isospin, period.

Yes, the gauge symmetry is then "spontaneously broken," but isn't it true that this is something of a misnomer and symmetries can't actually be "broken", only "hidden"? After spontaneous symmetry breaking, the vacuum is not invariant under weak isospin rotations, and so the particle spectrum does not consist of particles with definite weak isospin. But as far as I know that doesn't mean that weak isospin isn't conserved, contrary to what Bill_K's quote implies.

For example, consider a 1D double-well potential with an ##x \to -x## parity symmetry, and let the barrier between the two wells be essentially infinite. This system will exhibit "spontaneous breaking" of the parity symmetry: you can construct energy eigenstates that localized to one well only, and so are not parity eigenstates. Nevertheless parity is still conserved in this system.

Last edited: Oct 4, 2013
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6. Bill_K

4,160
Spontaneous symmetry breaking is not what's involved. As the book points out, any interaction that fails to preserve handedness, fails to conserve weak isospin.

Also, the electromagnetic interaction disconserves weak isospin, the photon being a mixture of Tw = 1 and Tw = 0.

7. dauto

Isospin is exactly conserved before symmetry breaking but it is not conserved after symmetry breaking because the Higgs field is not an isospin singlet and it forms a condensate. Particles are moving through this condensate at all times and interacting with it. Why then introduce the weak isospin at all as a symmetry, someone asked?
That's the only way to renormalize the weak interaction correctly.

8. Bill_K

4,160
The Higgs field is chosen to be an isospin doublet specifically to guarantee that the interaction term eR v EL = (singet) (doublet) (doublet) comes out to be invariant under isospin, and consequently conserves isospin. The vacuum state spontaneously breaks the symmetry, but that's unrelated to the effect of the interaction term with fermions.

9. The_Duck

929
So how do you reconcile this claim with the fact that if you write down the Standard Model Lagrangian before electroweak symmetry breaking, weak isospin is an exact symmetry and so must be exactly conserved?

Surely spontaneous symmetry breaking is at the heart of the matter here. Before electroweak symmetry breaking, the mass term for the electron (say) is actually a three-particle interaction between the left-handed electron, the Higgs doublet, and the right-handed electron. So the electron can change from left-handed to right-handed as long as it emits a Higgs boson, which carries away the conserved weak isospin. Here is an interaction that changes the handedness of an electron but manifestly conserves isospin.

If you write down the Lagrangian after spontaneous symmetry breaking things are more confusing, to me. Then there is an electron mass term that seems to violate weak isospin. There is also an electron-Higgs interaction term that violates weak isospin. But if you add these terms together the sum conserves weak isospin.

I think the electromagnetic interaction definitely doesn't violate weak isospin. Electromagnetism is the unbroken part of weak isospin and weak hypercharge; how can it break weak isospin or weak hypercharge?

It's true that the photon isn't an *eigenstate* of weak isospin. But despite the claims of the book you cited, I don't see how this proves anything about whether weak isospin is conserved. A spontaneously broken symmetry isn't manifest in the particle spectrum, but the corresponding current is still conserved.