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Random matrix theory of unquenched two-colour QCD with nonzero chemical potential

We solve a random two-matrix model with two real asymmetric matrices whose primary purpose is to describe certain aspects of quantum chromodynamics with two colours and dynamical fermions at nonzero quark chemical potential mu. In this symmetry class the determinant of the Dirac operator is real but not necessarily positive. Despite this sign problem the unquenched matrix model remains completely solvable and provides detailed predictions for the Dirac operator spectrum in two different physical scenarios/limits: (i) the epsilon-regime of chiral perturbation theory at small mu, where mu^2 multiplied by the volume remains fixed in the infinite-volume limit and (ii) the high-density regime where a BCS gap is formed and mu is unscaled. We give explicit examples for the complex, real, and imaginary eigenvalue densities including Nf=2 non-degenerate flavours. Whilst the limit of two degenerate masses has no sign problem and can be tested with standard lattice techniques, we analyse the severity of the sign problem for non-degenerate masses as a function of the mass split and of mu. On the mathematical side our new results include an analytical formula for the spectral density of real Wishart eigenvalues in the limit (i) of weak non-Hermiticity, thus completing the previous solution of the corresponding quenched model of two real asymmetric Wishart matrices.