  We present a dynamical field theory for directed randomly branched polymers
and in particular their collapse transition. We develop a phenomenological
model in the form of a stochastic response functional that allows us to address
several interesting problems such as the scaling behavior of the swollen phase
and the collapse transition. For the swollen phase, we find that by choosing
model parameters appropriately, our stochastic functional reduces to the one
describing the relaxation dynamics near the Yang-Lee singularity edge. This
corroborates that the scaling behavior of swollen branched polymers is governed
by the Yang-Lee universality class as has been known for a long time. The main
focus of our paper lies on the collapse transition of directed branched
polymers. We show to arbitrary order in renormalized perturbation theory with
$\varepsilon$-expansion that this transition belongs to the same universality
class as directed percolation.
