# The Higgs field (weak charge)

1. Jun 30, 2012

### robertjford80

This is from Lisa Randall's Knocking on Heaven's Door

Is weak charge something related to the weak force? What exactly is it?

2. Jun 30, 2012

### kloptok

3. Jun 30, 2012

### Einj

I'm not sure this is the correct interpretation (but I could be wrong :tongue:). I think the author is referring to the fact that Higgs boson have "weak charge" in the sense that it interacts with other particles weakly. The term "charge", in fact, is usually referred to electric charge that is the coupling constant that "tied togher" the electric current (for example electron's current) with the electromagnetic field. In the same way Higgs field is "tied togher" with other particles as far as weak interaction are concerned and the coupling constant of this binding coul be this "weak charge".

Now, Higg field's peculiarity is that its expectation value on the vaccum is different from zero. In some sense it means that vacuum is "filled" with it (the mean number of Higgs boson in vacuum is different from zero).

I hope my explanation has been clear :tongue: And mostly I hope it's correct!

Einj

Last edited: Jun 30, 2012
4. Jun 30, 2012

### kloptok

Actually I disagree. I'm by no means an expert on this, but I still think by "weak charge" Randall refers to weak hypercharge.
Yes, and in this case the fact that the Higgs particle interacts weakly comes from the fact that it has a non-zero weak hypercharge (and hence a non-zero weak isospin). Electric charge is zero for the Higgs though.

Actually, since electromagnetic and weak interactions have been unified one could perhaps see weak hypercharge as a sort of generalization of electric charge (I'm not sure about this though so I would be grateful for a comment on whether this is correct or not).
Charge can refer to different types of charges. For example colour charge of QCD and electric charge. Charge basically means a quantum number which tells if a particle interacts in a certain way or not, as the colour charge determines whether a particle interacts via the colour force or not. The charge is the generator of the symmetry group of the interaction, in the case of the weak force the weak hypercharge is the generator for the U(1) symmetry (and weak isospin are the generators of the SU(2) symmetry).

5. Jun 30, 2012

### Parlyne

In the symmetry-broken phase, it's not appropriate to talk specifically about weak hypercharge. The problem with doing so is that electric charge, which is conserved, is actually a linear combination of weak hypercharge and one of the components of weak isospin. My interpretation of the statement is that it refers to the other two components of weak isospin, as well as the linear combination of weak hypercharge and the third component of weak isospin which is orthogonal to electric charge. If looked at correctly, these can be seen as the charges related to couplings to the W and Z bosons, respectively. And, these are the charges of which the Higgs vacuum breaks the conservation.

6. Jun 30, 2012

### robertjford80

actually, here are some more quotes that I think clear things up

7. Jul 1, 2012

### Einj

This is exactly what I meant. On the other hand, kloptok, I'm still not sure it can be referred to the hypercharge as it is the generator of the U(1) group, while the Weinberg-Salam theory involves an SU(2)xU(1) gauge symmetry, so the weak hypercharge is not the only generator of the group (we have also the Pauli matrices). The fact that Higgs boson interact not only with Z boson but also with W boson is because it is not only a particle with +1 hypercharge, but also a weak isospin doublet.
So, if the "weak charge" is referred only to weak hypercharge it could not contain the interaction to the W boson.
So, I'm still convinced that the term "weak charge" refers generally to the capability of Higgs boson to interact weakly, thanks to the presence of the coupling constants (g and g') and not only to weak hypercharge. Actually I think this is what Parlyne meant.

Still, I could be wrong! :tongue:

8. Jul 1, 2012

### Naty1

robertjford: I see you posted an explanatory quote.....I have Randall's earlier 2005 book
WARPED PASSAGES [looks like Knocking on Heaven's Door is 2011]...in my earlier book, such explanations are spread far and wide.....

Her WARPED PASSAGES does an excellent job of describing particles and fields....not my favorite topics, but I got really engaged in her book.

Here are a few snippets on 'weak' charge from WARPED PASSAGES:

Elsewhere I have read of some 16 Higgs fields...so I'd suggest caution on assuming the model she uses in WARPED PASSAGES is precisely the same as robertjford references.....but at least you get an idea of some typical interactions.

Also of possible interest:

Last edited: Jul 1, 2012
9. Jul 1, 2012

### kloptok

Yes, I see what you mean and you may know more than me about this by all means. I got the impression from your previous post that you tried to convey that "charge" referred to electric charge, but apparently I misunderstood you. I am not an expert on this as I said, so you and Parlyne are probably correct.

With your comments in mind, am I correct in saying that Randall by "weak charge" refers more generally to weak hypercharge and/or weak isospin, signifying that a particle interacts weakly?

An SU(2)xU(1) symmetry should as far as I understand have four conserved charges (three for the SU(2) and one for the U(1)), is this correct? And in that case, what happens to the conservation of these when the symmetry is broken? And does the symmetry-breaking mean that there is no longer any symmetry or conserved charges left at all for the weak interaction?

10. Jul 1, 2012

### Einj

I think this could be the correct interpretation

As you said an SU(2)xU(1) symmetry leads to four currents. This set of currents contains two neutral current (the one belonging to SU(2) and the one belonging to U(1)). Usually one define the hypercharge (which itself generates an U(1) group) as a combination of this two currents in order to write the symmetry group as: $$SU(2)_L \times U(1)_Y$$.

However of this four currents only the electromagnetic one is strictly conserved in nature. In fact, when you develope an SU(2)xU(1) theory you make the hypotesis that leptons have zero mass, which is obviously not true and so the currents are not really conserved. :tongue:

Now for your last question, I admit not to be sure as I never asked myself something like that!! (Thanks to you for bring this problem ). However my idea is that the introduction of Higgs field doesn't destroy any conservation law. When we say that the symmetry is broken, in fact, we refer to the fact that Higgs fields has non unique ground state and so we are free to choose a particular one that brings the field to have non zero expectation value on vacuum.
However this not affects the symmetries of the lagrangian density (in this sense the symmetry is only spontaneously broken) which determines the conservation law. So, no, I think the introduction of Higgs field does not change the conservation laws. Still, I'm not sure :tongue2:

And for last, I'm sorry but I don't know very well the bibliography about the argument as I have studied it mainly on my teacher's lacture notes. However I have read something in Mandl-Shawn "Quantum Field Theory". It's just and introductory book but it could be very useful.

Oh god I have written so much! I'm sorry for that but I really love this topic!!