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?
Answer
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.
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