Applying Complex Langevin to Lattice QCD at finite $\mu$.

We continue our simulations of lattice QCD at finite quark-number chemical potential, $\mu$, using the complex-Langevin equation (CLE) with gauge-cooling and adaptive updating. The CLE is used because QCD at finite finite $\mu$ has a complex fermion determinant, which prevents use of standard simulation methods. Simulations using the standard lattice action show a transition from hadronic to nuclear matter for $\mu < m_\pi/2$ rather than the expected $\mu \approx m_N/3$. This suggests that the CLE is being influenced by the phase-quenched theory, which has a transition at $\mu = m_\pi/2$. We are therefore performing CLE simulations with a new action which includes an irrelevant chiral 4-fermion interaction. This separates the physics at energies of order of the pion mass and smaller from that at energies of the other hadrons. In doing this, it breaks the extended symmetry of the phase-quenched theory over that of the full theory, raising the masses of the extra pion-like excitations consisting of a quark and a conjugate quark, which could otherwise produce such an anomalous transition. Our preliminary CLE simulations using massless quarks, so that $m_\pi=0$, show no transition at $\mu=m_\pi/2=0$, but do show a transition at an appreciably higher value of $\mu$. It remains to be seen if this transition is near to $m_N/3$.