  We consider the initial value problem for the Dirac-Klein-Gordon equations in
two space dimensions. Global regularity for $C^\infty$ data was proved by
Gr\"unrock and Pecher. Here we consider analytic data, proving that if the
initial radius of analyticity is $\sigma_0 > 0$, then for later times $t > 0$
the radius of analyticity obeys a lower bound $\sigma(t) \ge \sigma_0
\exp(-At)$. This provides information about the possible dynamics of the
complex singularities of the holomorphic extension of the solution at time $t$.
The proof relies on an analytic version of Bourgain's Fourier restriction norm
method, multilinear space-time estimates of null form type and an approximate
conservation of charge.
