  We consider the Dirac Operator acting on the Clifford Algebra ${C\ell}_{m}$.
We show that under critical assumptions on the potential and the spinor field
the equation is subject to an integrability by compensation phenomenon and has
a sub-critical behaviour below some positive energy threshold (i.e.
$\epsilon-$regularity theorem). This extends in 4 space dimensions as well as
in 3 dimensions a similar result obtained previously by the two first authors
in 2 D in \cite{DLR1}.
