Is the Higgs Field Essential for the Standard Model's Consistency?

[Lonewolf]

I had my first lecture on the Standard Model today. The lecturer presented the Lagrangian for the Standard Model, which took up two slides. However, it contained terms that referred explicitly to the Higgs field, whereas I was under the impression that the Standard Model didn't need the Higgs field, and that if the Higgs boson was shown not to exist, the Standard Model had alternate, albeit weird, ways of accounting for the masses of the W and Z bosons. Was I misled before?

 

[Haelfix]

You were not misled, though its customary to write the Higgs field in there as its the most natural thing to expect.

If the Higgs doesn't exist, look at the lagrangian you were given and scratch out every higgs vev you see and replace it with an unidentified constant (an as yet to be figured out parameter from a more fundamental theory) so to speak. Regardless, fit that to experiment (use say a chi squared method) and you still have the standard model in all its glory.

The only thing is you are left with a perplexing problem. Namely 'how the hell do I get this correct lagrangian in the first place', its not very satisfying to just cheat and have someone write it down for you.

From theoretical arguments, a lot of the mass terms you see have no earthly business being there from first principles, in fact if you really did start from scratch you would find a bunch of non gauge invariant terms and completelly massless fields.

Since that isn't what we observe, but there's good reason to trust our first principles, people invented the Higgs mechanism, which combines some of the problematic terms in a rather natural way (via spontaneous symmetry breaking) and that leads very simply to the lagrangian that you see. Any new theory that gives mass to particles would be required to reproduce (from scratch) this standard model lagrangian, in a very similar way (hence there's not much wiggle room).

 

[nrqed]

You were not misled, though its customary to write the Higgs field in there as its the most natural thing to expect.

If the Higgs doesn't exist, look at the lagrangian you were given and scratch out every higgs vev you see and replace it with an unidentified constant (an as yet to be figured out parameter from a more fundamental theory) so to speak. Regardless, fit that to experiment (use say a chi squared method) and you still have the standard model in all its glory.

I am a bit confused. Are you saying that this theory would be equivalent to the SM? But in that case doesn't the theory loses gauge invariance? And then, doesn't it become nonrenormalizable?

 

[Lonewolf]

Thanks for the illuminating answer. Was afraid no one was going to answer for a while. Looks like I'm going to be asking quite a few questions on this over the next few weeks...

 

[Haelfix]

Hmm let me say it again, just so we're clear.

The higgs has never been discovered, and the bare scalar field theory generates very few constraints (eg something like the weak mixing angle) so every SM coupling that involves say a Higgs vev are simply just unknown parameters that are just scaled to the electroweak scale. Eg they are still just things that are fit to experiment (read just constants), so scratching out the postulated scaling doesn't lose any information.

But like I said, one would have no idea how to *derive* that correct theory (the one we have written down in front of us) without the higgs mechanism. But as a phenomenological entity, that lagrangian doesn't possesses much critical dependancy on the finer aspects of the electroweak scale (at least at accessible energies). Which is why we make predictions everyday and the SM works, even with our complete ignorance about the higgs sector

But yes, any real theory (and not just phenomonology) that does not involve the Higgs mechanism, will have to explain why the myriad non gauge invariant mass terms suddenly have their gauge invariance restored, and why they 'gain' mass.

In fact, it would be nice to go beyond the vanilla higgs model, and actually find a working theory that predicts what each yukawa coupling would be, from first principles. Thats the holy grail of particle physics. To reduce the 26 or 27 undetermined, continously adjustable free parameters to say 2 or 3.

 

[nrqed]

...

But yes, any real theory (and not just phenomonology) that does not involve the Higgs mechanism, will have to explain why the myriad non gauge invariant mass terms suddenly have their gauge invariance restored, and why they 'gain' mass.

Just to make sure..

It is my understanding that if we were to simply replace the Higgs field by it's vev and drop it as a field, the theory left over would be nonrenormalizable. Am I wrong?

I agree that in terms of the values of the masses, the Higgs is not very predictive (since the Yukawa couplings are arbitrary anyway), so that this seems to suggest the need for a deeper theory. But my point was that even setting this issue aside, just replacing the Higgs field by its vev woul dlead to an inconsistent theory. Am I wrong?

Now, of course, if we treat the SM as an effective field theory, the issue of renormalizability is not crucial anymore, and the standard Higgs scenario is not as attractive as some extensions to the SM, like susy, etc. But I am talking of the SM treated as a renormalizable theory. Then, I thought, the Higgs was necessary.

Pat