# A Mass of Higgs, W and Z bosons going to zero at high energies

1. Apr 24, 2018

### Renormalised

Hi Everyone! Hope everyone is really enjoying pushing the Physics frontiers!

Really need some help here. In the Electroweak sector of the Standard Model, it is apparently the case that the success of renormalisation to remove the Ultraviolet Divergence relies on the fact that at high energies the masses of the Higgs, W and Z bosons tend towards zero.

Could someone please provide me as many references to this as possible? Particularly references that would be accessible to a doctoral student, which would be greatly appreciated. I have not been able to find anything on the internet much about this at all, though I have always been aware of the fact.

Wold also greatly appreciate the answer to the question as to whether the Symmetry of the Higgs Lagrangian enables all this to happen? Is the symmetry of the Higgs Lagrangian driving this, and if so, how is this happening?

2. Apr 29, 2018

### PF_Help_Bot

Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.

3. May 7, 2018

### nrqed

Do you have a specific reference that you have in mind? It would make it easier to discuss your question. Usually, what is used is not that the masses tend to zero, it is rather that the ratio o the renormalized masses over the cutoff scale goes to zero, $M/\Lambda \rightarrow 0$. But again, it would be better to have a specific book/paper to focus on.

Last edited: May 29, 2018
4. May 29, 2018

### Renormalised

Thanks very much for your help. Happy for you to choose the reference. It would be handy if it were online. Below is the question which I am trying to answer:-

I have proposed a variant of the Standard model where the usual Higgs field (Lagrangian) expression breaks to a different expression than the usual one.

In the conventional theory (Standard Model), all particles lose their masses at high energies. The usual Higgs potential is symmetric, and at high energies the behaviour of the whole system becomes identical to that as if the vacuum Higgs condensate would be zero and the particles were massless. The latter fact helps to organize cancellation of ultra-violet divergences and makes the whole theory renormalizable. This is the key
point for which the Higgs mechanism was originally invented.

My question is whether, with this different scenario, will the theory still evolve to a massless W/Z boson regime at high energies?

5. May 31, 2018

### Renormalised

It would also be good to know the following: if you proposed a Higgs Lagrangian which is only different in that it has a different self-interaction to the usual one, but which is still symmetric under gauge transformations, and which breaks in the usual way as φ → v + H(x), will the EW theory still be renormalizable with this change?

That is, does the form of the potential for the Higgs matter a hoot in so far as renormalisation is concerned?

6. May 31, 2018

### Orodruin

Staff Emeritus
Yes. If you introduce any terms that have a mass dimension larger than 4 your theory will be superficially non-renormalisable.

7. Jun 2, 2018

### Renormalised

Thanks very much!

8. Jun 2, 2018

### Renormalised

So I take it that if I propose a Higgs field which is only different from the usual Higgs by having a different self-interaction with no terms in Φ of a higher power than 4, and it functions in exactly in the same way as the Higgs does in the usual Standard Model, then the EW theory with this change will still be renormalizable?

9. Jun 2, 2018

### Orodruin

Staff Emeritus
There is no other self-interaction than the $\phi^2$ and $\phi^4$ terms present in the SM that does not explicitly break SU(2) invariance. Also, please take note on the forum rules regarding personal theories.

10. Jun 2, 2018

### Renormalised

Very great thanks again. This is for a seminar and we have been asked a very tough question. Renormalisation is apparently a vast area with sections of it being also vast research areas in their own right!

Each of us has to propose some change to the Lagrangian of the Standard Model, which will not affect renormalizability. It does not have to be right of course! Each model will be discussed in the seminar and shown what problems it will produce (if any). We can stay in the Electroweak sector only, if we like. This is not for assessment of participants but rather, a sharp learning curve.

The original expression -μ2/2(ΦΦ) - λ/4(ΦΦ)2 is clearly unique. But after spontaneous symmetry breaking, this is expressed in terms of the field H, and there are variants of the self-interaction polynomial in H which will still be invariant under SU(2). Again, not asking whether this is right or not.

The suggestion I wish to pursue is that if you vary the SM Lagrangian now expressed in terms of H, by only changing the self-interaction of the Higgs ( H(x) ) and keep the order of the polynomial no higher than 4, and ensure that the interaction is still invariant under SU(2), will this still formally preserve renormalizability?

Thank you so much for your help.

11. Jun 3, 2018

### Orodruin

Staff Emeritus
The SM after symmetry breaking already includes other terms in the potential. This is, for example, where the fermion and gauge boson masses come from. In short, the SM already includes all of the terms it can while obeying its proposed gauge symmetries. If you introduce new terms after symmetry breaking you are going to have to explain what kind of term in the Lagrangian before symmetry breaking that it came from. In order to make alterations to the SM Lagrangian while satisfying gauge symmetry you would have to add new fields (with possibly one exception that I will not mention explicitly since it is your seminar question).