I have a gauge theory with some rather strange covariant derivatives and I am wondering how they act on a composite field like $\psi= \phi\psi'$. In my setup, the covariant derivative acting on a scalar and on a fermion are defined as follows.
$$ D_\mu\phi = (\partial_\mu - \frac{1}{2}\kappa_\mu)\phi \quad , \quad D_\mu\psi = (\partial_\mu + \Gamma_\mu)\psi $$
I have made the field redefinition $\psi= \phi\psi'$ and I am wondering how the covariant derivative behaves in my fermion kinetic term $i\bar{\psi}\gamma^\mu D_\mu\psi$. Should we expand the derivative before we do the chain rule, or after? Explicitly, which of the following is correct?
(i) $D_\mu\psi = (\partial_\mu + \Gamma_\mu)\psi = (\partial_\mu + \Gamma_\mu)\phi\psi' = \phi\partial_\mu\psi' + \psi'\partial_\mu\phi + \Gamma_\mu\phi\psi'$
(ii) $D_\mu\psi = D_\mu(\phi\psi') = \phi(D_\mu\psi') + \psi'(D_\mu\phi) = \phi(\partial_\mu + \Gamma_\mu)\psi' + \psi'(\partial_\mu - \frac{1}{2}\kappa_\mu)\phi $
It seems that (ii) is correct if $D_\mu$ is truly a derivative in the sense of basic calculus, but I can also see why (i) might be right. I won't go into my hand-wavy reasoning here - is there a formal way to see which is correct?
