Higher dimension operator in free Dirac Lagrangian

Question

When discussing higher dimensional operators in a theory with fermions, why do I never see anyone ever talk about the dimension five operator $\partial_\mu\bar\psi\partial^\mu\psi$?

How does the Fermion field behave when such a strange operator is its Lagrangian?

$$\mathcal{L}=\bar\psi(i\partial_\mu\gamma^\mu-m)\psi+\frac{1}{\Lambda}\partial_\mu\bar\psi\partial^\mu\psi$$

Since it is still quadratic in $\psi$, I expect this to be fairly easy to analyze. What happens when such a term is added?

Answer 1

This particular extra term may be removed by a field redefinition $$\psi\to \psi' = \psi - K \cdot \gamma^\mu \partial_\mu \psi $$ for an appropriate value of $K\sim 1/\Lambda$, up to terms that are even higher dimension operators. This also modifies the mass. This field redefinition is an explicit off-shell way to realize Vibert's comment that one is just modifying the mass in which he assumed the equations of motion.

Mark Wayne is also right that (despite the equivalence, up to even higher-order terms), the propagator violates positivity of quantum mechanics (we really mean the positive norm of states with particle excitations which is linked to positive probabilities, a "must": excitations with negative norm are known as "[bad] ghosts"). If one writes down the full propagator for this (free) theory, one gets some additional poles which have a coefficient of the wrong sign. However, these pathologies occur at $p^2\sim \Lambda^2$ where we expect the theory to break down, anyway: the OP wrote the extra term as a correction in effective field theory that is meant to be used at energies $p^2\ll \Lambda^2$. This pathology with non-positivity may be fixed by new physics near $\Lambda$ and because of the equivalence, indeed, we can see that we may adjust the even higher-order terms (which may also arise from integrating out other fields and interactions) so that the full theory is exactly equivalent to the ordinary massive Dirac fermion and therefore has no ghosts.

That means that as long as we use this as an effective field theory, knowing that it may have further higher-order modification and new physics in general that kicks in near $p^2\sim \Lambda^2$, it is healthy as an effective field theory. Again, if we wanted to use it as a full theory even at energies of order $\Lambda$, it would be inconsistent.

In general, higher-order "free" terms like that – similarly $\square\phi\cdot \square\phi$ for bosons etc. – have been proposed as ways to make the propagators softer in the UV, like $1/p^4$ instead of $1/p^2$, which would improve the convergence of Feynman diagrams. The price is that if such a theory is to be taken seriously near $\Lambda$, the scale determining the size of the higher-dimensional operators, then this theory leads to new negative-norm excitations which is an inconsistency. So the addition of these new terms makes the theory break down "earlier", despite the intriguing softening of the UV problems.