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Many presentations of the Higgs mechanism only explain it as giving mass to the $W$ and $Z$ gauge bosons, but don't mention the quarks or charged leptons. For example:

But it is equally responsible for the generation of the fermion mass terms via the Yukawa coupling of the fermion fields to the pre-symmetry-breaking Higgs field becoming a fermion mass term plus a new Yukawa coupling to the post-symmetry breaking Higgs field, correct? So, for example, I believe that during the electroweak epoch when the universe was hotter than 100 GeV and electroweak symmetry had not yet been broken, all fermions were completely massless.

I know that historically, Higgs et al were originally only trying to explain the masses of the gauge bosons, not fermions. Is the emphasis on the Higgs mechanism's granting mass to the gauge bosons just a historical relic?

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    $\begingroup$ 1. I'm not sure where the physics question is here, it seems to be more about personal idiosynracies in the presentation of the Higgs mechanism. 2. You have to be careful when talking about "pre-symmetry-breaking". In vacuum field theory at zero temperature (i.e. what is usually done when introducing Higgs), the theory is never unbroken, the breaking just becomes neglegible. You need thermal field theory to actually consider a "broken" and "unbroken" phase. $\endgroup$
    – ACuriousMind ♦
    Commented May 3, 2016 at 19:22
  • $\begingroup$ en.wikipedia.org/wiki/Higgs_mechanism#Consequences_for_fermions Like this? Both the wikipedia and scholarpedia articles mention the Yukawa terms/fermion masses. Beyond that, you can have massive spin-half particles without the Higgs mechanism, but you can't have massive spin-one particles without the Higgs mechanism. $\endgroup$
    – Luke Pritchett
    Commented May 3, 2016 at 19:22
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    $\begingroup$ @ACuriousMind 1. That is my question - is the presentation just a personal idiosyncrasy, or is there a fundamental difference that I'm not getting? 2. I know - by "pre-symmetry breaking" I simply meant expressing the Lagrangian in terms of the field where the symmetry is manifest, even if that field configuration isn't the physical ground state. I mentioned having to go to ~100 GeV temperatures to actually have the symmetry unbroken $\endgroup$
    – tparker
    Commented May 3, 2016 at 19:32
  • $\begingroup$ @LukePritchett I'm just wondering whether there's any important physically reason why they mention the gauge boson masses in the intro sentence but relegate the fermion masses to a subsection $\endgroup$
    – tparker
    Commented May 3, 2016 at 19:34
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    $\begingroup$ @tparker Here's my best guess: If you want a theory with massive spin-one particles you need the Higgs mechanism.* If you want a theory with massive spin-half particles you don't need the Higgs mechanism. In the SM you already know that your fermions have gauge quantum numbers, so you do need SSB for the mass, but you don't in general models of fermions. (* Not precisely true; there are other tricky ways to get spin-one massive states) $\endgroup$
    – Luke Pritchett
    Commented May 4, 2016 at 1:01

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The general problem that the Higgs mechanism solves is giving mass to spin-one particles. It turns out that finding relativistic, unitary theories of spin-one massive particles is non-trivial. There are a few known ways of doing it (this paper has a pretty good list of sources), but the oldest and easiest is probably the Higgs mechanism.

In contrast, there is no fundamental difficulty with theories with massive spin-1/2 particles. It's as easy as writing down the Dirac Lagrangian.

In the special case of the standard model we know that the fermions must couple to a chiral gauge field, and so the mass term must arise from spontaneous symmetry breaking, making the Higgs mechanism the answer to two problems instead of one. But in general, the Higgs mechanism is for giving mass to spin-one particles.

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Of course the SM Higgs gives mass to both fermions as well as the gauge bosons. However, the latter is much more fundamental and predictive than the former.

Point is, it is enough for a scalar to transform non-trivially under a gauge symmetry to contribute to its associated gauge bosons masses (after taking vev). This contribution is constrained by the gauge sector and you do not have a lot of freedom. However, in order to contibute to the fermions masses, you need to be in a very specific representation and the couplings are, a priori, arbitrary.

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