  We propose two novel formulations of the hopping parameter expansion for
finite density QCD using Wilson fermions, while keeping the gauge action
intact. We use the complex Langevin equation to circumvent the sign problem in
the theory. We perform simulations at very high order of the expansion, such
that convergence is directly observable. We compare results to the full QCD
results, and see agreement at sufficiently high orders. These results provide
support for the use of complex Langevin dynamics to study QCD at nonzero
density, both in the full and the expanded theory, and for the convergence of
the latter.
