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We propose two novel formulations of the hopping parameter expansion forfinite density QCD using Wilson fermions, while keeping the gauge actionintact. We use the complex Langevin equation to circumvent the sign problem inthe theory. We perform simulations at very high order of the expansion, suchthat convergence is directly observable. We compare results to the full QCDresults, and see agreement at sufficiently high orders. These results providesupport for the use of complex Langevin dynamics to study QCD at nonzerodensity, both in the full and the expanded theory, and for the convergence ofthe latter.

Hopping parameter expansion to all orders using the complex Langevin equation