# Does the Higgs field give mass to all particles or only to gauge bosons?

## Main Question or Discussion Point

The title says it all.

I've seen an example worked out, and there mass was given to a gauge boson specifically. Also, I wouldn't know why the Higgs boson would want to give mass to the fermions, since they already have mass in the Yang-Mills theories; it's only the gauge bosons that initially lack mass whereas you would sometimes like them to be massive.

Based on that, I would expect the answer to be "the Higgs field (only) gives mass to the gauge bosons", however, I've always heard "the Higgs field gives particles mass", implying it's the origin of the mass for all particles.

So which of the two is it?

EDIT: or somewhere in between, which to me seems the most logical: strictly speaking it only gives mass to the massless gauge bosons, but it actually changes the mass of all particles.

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I can write down a theory where the fermions get masses de novo, and bosons get masses via the Higgs mechanism. This theory will have some calculational problems if I want to calculate quantum corrections to it, but it's not nearly as sick as the equivalent model where boson masses are put in by hand. That theory doesn't even get out of the gate - it predicts nonsensical results (like negative probabilities) even before you get to the quantum corrections.

In that theory, one discovers that in addition to the fermion masses that were put in by hand, the fermions also get a mass from the Higgs. And semi-miraculously, this mass has to be exactly proportional to the mass that was put in from the beginning.

Faced with this, most folks decide that the simplest thing to do is to avoid this impossible coincidence and start off with massless fermions, and assume that the same Higgs that gives masses to bosons gives masses to fermions. This has some calculational benefits as well, which I alluded to above. However, this is far from the only option.

Based on that, I would expect the answer to be "the Higgs field (only) gives mass to the gauge bosons", however, I've always heard "the Higgs field gives particles mass", implying it's the origin of the mass for all particles.

So which of the two is it?
The latter. In the standard model, all fermions "start out" massless and then get mass as a result of their interaction with the Higgs field. Left-handed and right-handed fermions are treated differently by the weak force: the weak force only couples to left-handed fermions. This means that a mass term for the fermions, which must couple the left and right-handed fermions, does not respect the electroweak SU(2)xU(1) gauge symmetry and so does not appear. Instead, there is a three-particle interaction term that couples the Higgs field, the left-handed fermions, and the right-handed fermions. When the Higgs field gets a vacuum expectation value, this interaction term can be rewritten to look like a fermion mass proportional to the Higgs VEV + an interaction term between the fermion and the Higgs boson.

Actually, there's a significant caveat to "the Higgs field gives all particles mass." Many strongly interacting particles, such as the proton and neutron, would still be massive even if all quarks had zero mass. In fact most of the mass of the proton and neutron comes from strong interaction effects and not the Higgs-produced quark masses. For instance the proton weighs almost 1 GeV, and only a small fraction of this comes from the three up and down quarks that compose it, which weigh only around 5 MeV each. If that 5 MeV was reduced to 0 the proton mass wouldn't change very much.

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Thank you.

I can write down a theory where the fermions get masses de novo, and bosons get masses via the Higgs mechanism. This theory will have some calculational problems if I want to calculate quantum corrections to it, but it's not nearly as sick as the equivalent model where boson masses are put in by hand. That theory doesn't even get out of the gate - it predicts nonsensical results (like negative probabilities) even before you get to the quantum corrections.

In that theory, one discovers that in addition to the fermion masses that were put in by hand, the fermions also get a mass from the Higgs. And semi-miraculously, this mass has to be exactly proportional to the mass that was put in from the beginning.

Faced with this, most folks decide that the simplest thing to do is to avoid this impossible coincidence and start off with massless fermions, and assume that the same Higgs that gives masses to bosons gives masses to fermions. This has some calculational benefits as well, which I alluded to above. However, this is far from the only option.
I'm afraid, this is simply not correct. You're ignoring issues of chiral symmetry. In a chiral theory, like the Standard Model, fundamental fermion masses break the gauge symmetry explicitly, causing the same sorts of problems that fundamental gauge boson masses would. But, because one of the chiral states is uncharged under the chiral force, it's possible to have an interaction between the charged fermion multiplet (in the fundamental representation), the uncharged fermion singlet, and the charged Higgs multiplet (in the anti-fundamental).

On the other hand, in a theory with chiral symmetry, fundamental fermion masses are just fine. But, in this case, there's no possible interaction term with the Higgs, as both chiral fermion states are in the fundamental (or anti-fundamental) representation, and the Higgs is as well, meaning that there's no way to construct a gauge singlet interaction term.