Robert Garisto sent me a reply to my previous post; here it is. [A "vev" is shorthand for a non-zero value in the vacuum of space, what I call a "non-zero average value".]
Matt – Thanks for your extensive reply to my comment! Of course I agree that a scalar field without a vev can have a hard mass term. And I do agree that how the Higgs boson gets mass is at least somewhat different than how the W does.
Let’s agree to define a Higgs as a scalar field with a vev. Then I think you agree that the mass of the excitation about the vev, the Higgs boson, is not a hard mass term, one obtains it by finding the minimum of the potential as you did above. Now if there are other scalars with vevs, the mass of the Higgs boson we are concentrating on can depend on those too. But isn’t it correct to say that the mass of such a Higgs boson goes to zero in the limit that all of those vevs go to zero? If so, I would say that the Higgs boson mass is provided by the Higgs fields (all scalars with nonzero vevs).
Anyway, the main reason I made the comment is that for the purposes of explaining to the public electroweak symmetry breaking, I think it makes sense to say that the Higgs boson mass comes from the Higgs field, because it is, in the SM, proportional to the vev. It’s also kind of neat, I think.
We disagree, that’s all there is to it. What Garisto says about the Standard Model is a simple consequence of dimensional analysis, not a fundamental relation that applies widely. And no, it is not correct to say that the mass of a Higgs boson always goes to zero in the limit that all vevs go to zero; there can be first order phase transitions in which, as the parameters change, the Higgs field’s vev jumps from non-zero to zero abruptly, and the mass of the Higgs particle is never zero. So I think to tell the public that the Higgs particle gets its mass from the Higgs field is to confuse them into thinking that the Higgs particle gets its mass the same way the other known particles do — which is false.
But in any case, we agree it’s not that big a deal. The thing which is important for the public to understand is that the Higgs field does not give mass to all massive objects — such as atomic nuclei and black holes. And the thing which it is important for particle physics students to understand is that the Higgs mass is not generically proportional to the vev of the Higgs field.