Couplings of fermions and bosons to the Higgs

[arestes]

Homework Statement

I have to show that the couplings to the Higgs ( W+ W- h , ZZh, hhh, and e+e-h) are proportional to the mass squared (for bosons) or mass (for fermions) of the particles. But according to this problem I don't have to explicitly construct the interaction terms in the lagrangian.

Homework Equations

According to Peskin & Schroeder, page 716 we can construct the interaction terms and explicitly read off the couplings but that's not the method I'm looking for. I am supposed to only show that they have to be proportional to mass or mass squared depending on the statistics of the particles involved

The Attempt at a Solution

I tried to use the mass dimensions of the wave functions involved using the lagrangian dimension = 4 but that didn't help to relate it to the couplings. It might be necessary to use the properties of the higgs (symmetry breaking mass) but I don't know how to do it

 

[Dick]

What's wrong with your idea of considering the mass dimension of the fields? The mass terms have to be quadratic in the fields, with the higgs terms providing the extra mass dimension to get it up to m^4.

 

[arestes]

Well, the thing is that I only get the dimensions of the wavefunctions only after I know what the interaction term in the lagrangian looks like! which is a pretty standard exercise in QFT not specifically related to the Higgs. But I don't know what the interactiont term looks like beforehand.

according to Peskin & Schroeder page 716 for fermions it is

$$L_f = -m_f \overline{f}f \left(1 + \frac{h}{v} \right)$$

where f and h are the wavefunctions of the fermion field and the higgs, respectively. v is the vacuum expectation value for a scalar field $$\phi$$ from which, after reparametrizing it with the h field we get a minimum at that point. (All this is in P&S page 715-716).


I don't think it will help me finding the couplings. By couplings I mean the vertex rules for Feynman diagrams, of course, which could be read off from the interaction terms.

Besides, knowing what the dimension of the wavefunction AND knowing that the whole term must be of mass dimension 4 doesn't account for other constants that may go with the masses in that term... so I think the approach must be different.

 

[Dick]

You can get the dimensions of the fields by looking at the kinetic terms, the free part of the Lagrangian that defines the propagators. Which basically come from the Klein-Gordon and Dirac equations. You don't need to peek at the interaction terms first. That will tell you what the dimensions of the couplings must be, but won't tell you their exact form, naturally. But you are only asked for the dimensions, right?

 

[arestes]

Thanks for answering...but I actually need to show that the couplings must be Proportional to the indicated masses. Not only the dimensions... but even so..

I found that the mass dimension of the fermion fields are 3/2 and for bosons it is 1 ... that means that for an interaction of the type $$f \overline{f} h$$ i have a mass dimension 4 already and the coupling that should be multiplying this should have mass dimension 0 ! how do I show it is supposed to be proportional to the mass of the fermions?? :S Also , for bosons like W in $$W+ W- h$$ the coupling that must multiply this has mass dimension 1 (and i must show that it should be proportional to the squared mass of the W). Working with mass dimensions only gives dimensions but doesn't tell me which masses go... ( for example in W+ W- h it is the mass squared of the W but not of the h). That's why i dropped this idea.

 

[Dick]

I've got to assume that the question they are asking is e.g. the boson mass term is W*W*m(...) where m(...) is some function of the other fields in the problem. They want to know the dimensions of m(...). There's no way you can figure out the exact form of (or even the fields involved in) m(...) without constructing a lagrangian.

 

[arestes]

Well. for the interaction vertices I need the term e.g. for fermions f like this : $$h \overline{f} f . m(...)$$

I actually don't think they want the exact form of m(...) what they want me to show is that $$m(...) = m_f . m'(...)$$ where m_f is the mass of the fermion (or antifermion) involved in the interaction $$h f\overline{f}$$ and m'(...) doesn't depend on this mass anymore. SImilarly for the bosons W and for h itself in the cases $$h W^-W^+$$ and $$h hh$$ where the function m(...) would be of the form m(...) = (m_W)^2 . m'(...) or = (m_h)^2 . m'(...) where again m'(..) doesn't depend of the masses factored out.

Maybe you could understand my context taking a look at the lecture notes where this problem came from and I'm trying to finish.
http://www.nat.vu.nl/~mulders/QFT-0E.pdf
It's exercise 12.5 in page 127 ... Many thanks again for your time.

 

[Dick]

I guess all I can think of here is that when the higgs acquires a VEV then whatever coupling is in front of the ff terms (for fermions) and BB terms for bosons is the effective mass. And it should be proportional to m_f for the fermions and m_B^2 for the bosons. Just by comparison with the Dirac lagrangian for f and the Klein-Gordon lagrangian for B. I've got to admit the phrasing of the question confuses me as well.

 

[turin]

I am very confused as well. (I have been lurking in this thread to see if anyone can sort it out, and I finally decided to pipe into agree with Dick that this is confusing.) I would assume that the context is Standard Model, which has a very specific Lagrangian. I think that the problem itself is not confusing, but the comment that you don't need to know the interaction terms is confusing. Maybe we're making this problem way more difficult than it should be. Maybe we should simply ignore the comment about knowning the interaction terms. Still, I'm curious to see if anyone can figure out how to do it without knowing the interaction terms.

 

[Avodyne]

The whole question is ambiguous. For example, the hWW coupling is of the form g²v, where g is the SU(2) gauge coupling and v is the Higgs VEV. Now, M_W=gv, so we can write g²v = gM_W = M_W²/v. Only the last form makes it proportional to M_W², but how are we supposed to know which form to use? There is an apparent implicit requirement that the gauge and Yukawa couplings be eliminated from the expressions for the couplings.

I suspect the "correct" answer makes use of the property that the Higgs VEV v and the physical Higgs field h only appear in the combination v+h, but the question is too poorly phrased to spend any more time on it.