The Higgs mechanism allows massless fields to acquire mass through their coupling to a scalar field. But if the masses cannot be predicted because the couplings have to be fixed, what really is the utility of the Higgs mechanism? Instead of saying "Here are a priori couplings; the Higgs mechanism generates mass.", I could just as well say "Here are a priori masses. Period.".
I understand that the Higgs mechanism is crucial to electroweak unification, but I have the same question there. Why does electromagnetism and the weak force have to be unified? Even if the couplings of the photon, Z and W bosons became related on unification, this is still at the cost of introducing new parameters – so it's not really clear to me that something has been explained or tidied up.
Do either the Higgs mechanism or electroweak unification tell us something new? Do either make any predictions that don't come at the cost of extra parameters? (I'm not really challenging anything here; I'm sure the answer to both questions is 'yes' – I just want to fill the gaps in my understanding as I study the Standard Model)
Best Answer
Dbrane, aside from "beauty", the electroweak unification is actually needed for a finite theory of weak interactions. The need for all the fields found in the electroweak theory may be explained step by step, requiring the "tree unitarity".
This is explained e.g. in this book by Jiří Hořejší:
The sketch of the algorithm is as follows:
Beta-decay changes the neutron to a proton, electron, and an antineutrino; or a down-quark to an up-quark, electron, and an anti-neutrino. This requires a direct four-fermion interaction, originally sketched by Fermi in the 1930s, and improved - including the right vector indices and gamma matrices - by Gell-Mann and Feynman in the 1960s.
However, this 4-fermion interaction is immediately in trouble. It's non-renormalizable. You may see the problem by noticing that the tree-level probability instantly exceeds 100% when the energies of the four interacting fermions go above hundreds of GeV or so.
The only way to fix it is to regulate the theory at higher energies, and the only consistent way to regulate a contact interaction is to explain it as an exchange of another particle. The only right particle that can be exchanged to match basic experimental tests is a vector boson. Well, they could also exchange a massive scalar but that's not what Nature chose for the weak interactions.
So there has to be a massive gauge boson, the W boson.
One finds out inconsistency in other processes, and has to include the Z-bosons as well. One also has to add the partner quarks and leptons - to complete the doublets - otherwise there are problems with other processes (probabilities of interactions, calculated at the tree level, exceed 100 percent). It goes on and on.
At the end, one studies the scattering of two longitudinally polarized W-bosons at high energies, and again, it surpasses 100 percent. The only way to subtract the unwanted term is to add new diagrams where the W-bosons exchange a Higgs boson. That's how one completes the Standard Model, including the Higgs sector. Of course, the final result is physically equivalent to one that assumes the "beautiful" electroweak gauge symmetry to start with.
It's a matter of taste which approach is more fundamental and more logical. But it's certainly true that the form of the Standard Model isn't justified just by aesthetic criteria; it can be justified by the need for it to be consistent, too.
By the way, 3 generations of quarks are needed for CP-violation - if this were needed. There's not much other explanation why there are 3 generations. However, the form of the generations is tightly constrained, too - by anomalies. For example, a Standard Model with quarks and no leptons, or vice versa, would also be inconsistent (it would suffer from gauge anomalies).