## The Higgs vev, the Higgs Mass, and WW-scattering

First of all, thank you to those who have helped me with questions; its been helpful and this forum is proving to be a useful learning device. This time, I'm just looking for some clarification.

From what I understand, the 'Hierarchy Problem' lies in the self-energy corrections to the Higgs boson (and its sensitivity to new physics at high scales). Basically, they make the Higgs very large and we have to fine-tune it back to the electroweak scale.

Now, the non-zero vev of the Higgs boson is around 246 GeV, and is determined basically by the masses of the quarks/leptons, and gauge bosons, which it gives mass to.

Question 1: Even if the Higgs mass was to be very large, would this affect the masses of the quarks, since it is the vev, not the mass, which is important?

Question 2: What is the relationship between the Higgs vev and the Higgs mass?

WW-scattering however, does 'require' the Higgs mass to be of the electroweak scale, since it violates unitarity at some TeV scale.

Question 3: What's wrong with breaking unitarity? Can't we just assume that the poor behaviour of WW-scattering is due to new physics at the scale in which it breaks unitarity? Must we fine-tune the Higgs mass to make things work? Perhaps, I already know the answer to this one..
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 Quote by shirosato From what I understand, the 'Hierarchy Problem' lies in the self-energy corrections to the Higgs boson (and its sensitivity to new physics at high scales). Basically, they make the Higgs very large and we have to fine-tune it back to the electroweak scale.
I've had fights with people about this, but I think it's more of semantics. Many high energy theorists would call that the "Fine Tuning Problem". The "Hierarchy Problem" is: Why is the electroweak scale (where W/Z physics is important, roughly 1 TeV) SO much less than the Planck scale (where quantum gravity is important). Of course, the two problems are related, but to be very picky, what you described is usually called the "Higgs Fine Tuning Problem".

 Now, the non-zero vev of the Higgs boson is around 246 GeV, and is determined basically by the masses of the quarks/leptons, and gauge bosons, which it gives mass to.
actually, it comes from $v=\frac{1}{\sqrt{2G_F}}$ which is measured in muon decay.

 Question 1: Even if the Higgs mass was to be very large, would this affect the masses of the quarks, since it is the vev, not the mass, which is important?
that is more or less correct: the mass of the Higgs is not so important for fixing the fermion or gauge boson masses, although there are indirect relationships between them.

 Question 2: What is the relationship between the Higgs vev and the Higgs mass?
strictly speaking, they are completely independent quantities. However, the Higgs quartic coupling is given by $m_h^2/v^2$, up to factors of 2, so if you don't want the Higgs quartic to be too large (so that perturbation theory breaks down) you can't let the Higgs be too heavy. That's the "triviality" bound, around 1 TeV or so.

 WW-scattering however, does 'require' the Higgs mass to be of the electroweak scale, since it violates unitarity at some TeV scale.
That's more or less what I was saying about the quartic coupling, in a different language.

 Question 3: What's wrong with breaking unitarity? Can't we just assume that the poor behaviour of WW-scattering is due to new physics at the scale in which it breaks unitarity? Must we fine-tune the Higgs mass to make things work? Perhaps, I already know the answer to this one..
Absolutely!! The problem is, then what?! If it really is strong dynamics at work, then what is the UV completion? Is it technicolor? That's more or less ruled out, except in very stretched circumstances. We will find out what happens when the LHC has enough data to probe WW scattering in the next few years...
 Thanks for the reply, it cleared things up a lot. Coincidentally, I believe you wrote a math primer I used when I first took QFT. See, that's what I never understood too well; the Hierarchy Problem stated as you stated it (as I often see it). Why is that unnatural? To me it seems equivalent to, "why is gravity so much weaker than the other forces?", as that is how the very large Planck scale is defined. Perhaps there is simply a rich landscape of physics from here on up to the Planck scale and that seems fine to me. I'm glad to read what you wrote last. Over the last year I think I finally began to understand EWSB. At that point, you start thinking that Higgs is a really odd/unnatural thing, whereas technicolor seems a lot better looking. Then you read about TC and realize it also has a bundle of problems. Then you think, "what else is there?" and "what the heck are we going to see at the LHC?"

Recognitions:

## The Higgs vev, the Higgs Mass, and WW-scattering

 Quote by shirosato Thanks for the reply, it cleared things up a lot. Coincidentally, I believe you wrote a math primer I used when I first took QFT.
 See, that's what I never understood too well; the Hierarchy Problem stated as you stated it (as I often see it). Why is that unnatural? To me it seems equivalent to, "why is gravity so much weaker than the other forces?", as that is how the very large Planck scale is defined. Perhaps there is simply a rich landscape of physics from here on up to the Planck scale and that seems fine to me.
we don't typically like hierarchies unless we can explain them. the fact that the scale of EW physics is so much smaller than the Planck mass has no explanation within the standard model. therefore it is not "natural" (i mean that in the TECHNICAL sense, not just in the "I don't like it." sense!).

compare this to Planck vs QCD scale: there is no problem there. Even if you try to set QCD at the Planck scale, dimensional transmutation and RG effects will force the relevant physics back down to 200 MeV (this is why "technicolor" was supposed to be the perfect solution!). That is why it does not bother us that there is such a large hierarchy between Planck and QCD scales.

However, there is no such effect that happens for the EW theory, at least not as it currently stands. That is why there is a problem.

 I'm glad to read what you wrote last. Over the last year I think I finally began to understand EWSB. At that point, you start thinking that Higgs is a really odd/unnatural thing, whereas technicolor seems a lot better looking. Then you read about TC and realize it also has a bundle of problems. Then you think, "what else is there?" and "what the heck are we going to see at the LHC?"
I buy you a virtual pint of beer, and toast you with it! Welcome to my world!
 ..and a toast to you too. Hopefully in a few years, we'll be amazed once more.

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