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I'm afraid this question could sound a little too vague. I don't even know if dark matter (DM) can be genuinely described by quantum field theory, or if quantum field theory should be somehow "modified" in order to include dark matter.

Assuming that ordinary QFT describes DM, what can be said (or what is known) about the number of degrees of freedom dark matter should have?

It is trivial to design a dark matter candidate that is compatible with quantum field theory: massive sterile neutrinos are a moderately popular possibility already.

But that doesn't prove anything, because it is just a dark matter candidate. Indeed the question is rather speculative until we know something about what the dark matter is rather than just things about what it isn't.

Dark matter candidates can quite naturally be described for example by supersymmetric quantum field theories. The MSSM (Minimal Supersymmetric Standard Model) is the simplest but not necessarily the most realistic version of these types of effective theories to describe nature, taking the so far obtained LHC results into account. Since R-parity is assumed to be conserved in the MSSM, the lightest supersymmetric particle (LSP) of its spectrum, for example a neutralino, is stable and could therfore serve as a dark matter candidate. The MSSM doubles the numbers of particles of the standard model and it contains a total number of five not eaten higgs particles (the graviton and the gravitino should not be included in the picture of the spectrum since gravity is not included in the MSSM).

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