If dark matter forms halos around galaxies, it must have invariant mass. In the Standard Model, elementary particles acquire mass usually through the weak interactions with the Higgs field. Does this mean that dark matter must interact weakly? What are possible alternative mechanisms of dark matter particles to acquire invariant mass?


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First I'd like to correct a point of your question: Standard Model (SM) particles do not acquire mass through weak interaction with the Higgs field. You make the very common confusion between the Higgs field and the Higgs boson. But this is only tangential to your question and I won't elaborate on that.

So onto your question. In short, after which I will elaborate, we need the Higgs mechanism because the SM particles are subject to the electroweak interaction. Since as far as we know, dark matter particles may not interact with either SM particles or with dark matter particles (see this answer of mine for a highlight on the experimental evidences that dark matter behaves as a dust of non-interacting particles), there is actually no need of the Higgs mechanism and we could very well go for the simpler solution of just hardwiring the mass directly in the theory.

The simplest mechanism to give a mass to a particle is simply to hardwire that mass directly into the theory: technically we just add what is called a mass term to the Lagrangian. That works fine for Quantum Electrodynamics (QED) for example. As you surely know, QED is called a gauge theory, because it has an internal symmetry, called gauge symmetry, which is $U(1)$, since it consists in multiplying the electron field by a complex number of modulus 1, i.e. adding a phase to that field, while removing the derivative of that phase from the electromagnetic field. The essential point is that the electron mass term, which is conceptually like the electron field times its complex conjugate, is invariant under that gauge symmetry. So it's all fine and we can leave it at that. This explains why QED happily lived through the 50's and the 60's without any need of a Higgs mechanism. Note however that a mass term for the photon would not be gauge invariant, so QED only works with massless photon, and again, that's fine because it fits very well with experimental results.

Onto the electroweak theory. As you have surely heard, this is a gauge theory too, responding to the sweet name of $SU(2)_L\times U(1)_Y$ because we transform some components of the fields with $2\times 2$ unitary matrices and multiply others with a complex of modulus 1. The problem now is that a fermion mass term is not gauge invariant: this answer on this very site may be readable without too much formal background. This can't be right: we know electrons have a mass. Moreover, as for QED, any mass term for the new bosons $Z$ and $W$ is also forbidden. That can't be right either because experimental results require $Z$ and $W$ to have masses of the order of 100 GeV: we knew that before they were discovered, I mean. The solution of this conundrum is the Higgs mechanism: it kinds of save the day for the electroweak theory by giving masses to some of the fermions, to $Z$, to $W$ but not to the photon.

Now what happens with $SU(2)_L\times U(1)_Y$ could happen with other similarly twisted gauge theories, and if you assume one of those for your dark matter particles, then indeed a Higgs mechanism would be the simplest manner to get massive particles. But this is a very big "if" as you should be able to understand by now. First, there is no a priori reason to postulate such a mass-term-breaking gauge theory. If dark matter particles interact, this could be through theory like QED with a new dark electric charge. But they may actually not interact at all, and so far, experimental evidences lean in that direction actually.

This is the point at which I should then start talking about Supersymmetry (SUSY) as supersymmetry particles have been considered as some of the best candidates for dark matter for a long while, and then that would have me discuss about how supersymmetric particles get their mass, but that answer is already long enough. Feel free to ask another question about that as there are plenty of SUSY aficionados in these waters!


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