Do other particles besides scalars admit tachyonic solutions? For example fermions or gauge-boson tachyons? The picture in my head is that a tachyonic scalar simply rolls off some unstable potential until it finds a stable position in field space (so the higgs field is basically a tachyon until it condenses). But I don't see a similar picture for fermions or gauge bosons (they have higher dimensional operators, but not a potential in the same sense as a scalar particle). Furthermore for fermions I believe we can just perform a chiral phase transformation to get rid of any imaginary part of the mass term.
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