Gravity. The entire dark matter theory is based around the observation that current mass/gravity models don't match our observations in the cosmos. As the Higgs field is what gives particles mass, it stands to reason that dark matter does interact with gravity -- it's pretty much the sole reason for its theoretic existence.
In other words, if dark matter didn't interact with/through the Higgs field, it wouldn't explain the observational anomalies and therefore wouldn't have a reason to exist at all, not even in theory.
Mass is a generic property of particles in a QFT. Unless there are symmetry considerations constraining them (gauge bosons like the photon, for instance, must be massless to avoid violating their corresponding gauge symmetries), masses are free parameters and zero is not particularly more natural than any other value.
What the Higgs mechanism does is add effective mass even to particles which are intrinsically massless, so long as they couple to the Higgs field. For the standard model, that means leptons, quarks, and the W and Z bosons. But there are other fundamental particles with intrinsic masses (the Higgs boson itself, for instance), and composite can acquire mass from their binding energy even if their constituents are massless.
> What the Higgs mechanism does is add effective mass even to particles which are intrinsically massless, so long as they couple to the Higgs field. For the standard model, that means leptons, quarks, and the W and Z bosons.
this was sloppy: the W and Z bosons have mass because they mix with the Higgs field (this is the Higgs mechanism, properly speaking), while the mass acquired by leptons and quarks is due to interacting with the remaining Higgs component in the usual way.
What exactly is mass from a Higgs boson point of view? I mean my layman definition is mass is the amount of matter, but that doesn’t seem like the thing that the Higgs boson is involved in
Characterizing what things "are" in a physical theory is not really possible: ultimately physics is only concerned with (and only has access to) what things do. When we say that "The X-ion has mass m", in the context of a quantum field theory, we mean that
- The magnitude of its 4-momentum is m (better known as E^2 = p^2 + m^2c^4)
- There's an X^2 m^2 term in the Lagrangian of our theory (there are some formal complications here but they're physically irrelevant), which in extremely loose terms means that in the absence of interactions X will behave like a collection of harmonic oscillators with mass m.
- The strength of its interaction with gravitational fields, neglecting relativistic effects, is proportional to m.
and so on for every other place "m" shows up. The meaning of mass, if you want to assign it one, is the correspondence between all these different quantities. And their meanings, in turn, are other correspondences, all the way down. To get actual physical meaning out, you need to find some place in the theory which you can identify with observational results (the typical case for particle physics is scattering amplitudes and collider experiments).
Ideally this is enough of a foothold to interpret everything: the readouts when we objects a and b together match the calculated scattering amplitudes for particles A and B, so we hypothesize that a is an A and b is a B. Now if we smash objects b and c together and get results that match theoretical predictions for B and C, we can be more confident b is a B and hypothesize c might be a C, and so on. But there's no law that says this must be possible, and no guarantee that every term in our theories points to some particular thing in the world.
> In other words, if dark matter didn't interact with/through the Higgs field, it wouldn't explain the observational anomalies
That conclusion seems premature to me. Not only because we lack a quantum theory of gravity but also because (classical) gravity doesn't care where your particle' mass comes from. Heck, it mostly looks at your particle's total energy (mass + kinetic energy + addition d.o.f.) to determine how spacetime should curve.
Besides, as the sibling pointed out, the Higgs mechanism is only responsible for generating the mass of the W± and Z bosons. Other masses are unaccounted for and have nothing to do with the Higgs.
An experimentalists would tell you its still an open possibility because we haven't seen enough Higgs particles to claim we know how they all decay. An anomaly in decay rates, lets say particles disappearing, or taking too long/short to decay might hint at some new physics.
