# Applying Complex Langevin Simulations to Lattice QCD at Finite Density

## TL;DR

This paper investigates the use of complex Langevin equations for simulating lattice QCD at finite density, assessing conditions for accurate results and exploring how parameters affect the simulation's fidelity.

## Contribution

It demonstrates that complex Langevin simulations can produce correct results at certain parameters and explores how proximity to the SU(3) manifold influences accuracy.

## Key findings

- Good agreement with expected results at small and large chemical potentials.
- Accuracy improves with weaker coupling and smaller quark mass.
- Proximity to the SU(3) manifold correlates with correct physics outcomes.

## Abstract

We study the use of the complex-Langevin equation (CLE) to simulate lattice QCD at a finite chemical potential ($\mu$) for quark-number, which has a complex fermion determinant that prevents the use of standard simulation methods based on importance sampling. Recent enhancements to the CLE specific to lattice QCD inhibit runaway solutions which had foiled earlier attempts to use it for such simulations. However, it is not guaranteed to produce correct results. Our goal is to determine under what conditions the CLE yields correct values for the observables of interest. Zero temperature simulations indicate that for moderate couplings, good agreement with expected results is obtained for small $\mu$ and for $\mu$ large enough to reach saturation, and that this agreement improves as we go to weaker coupling. For intermediate $\mu$ values these simulations do not produce the correct physics. We compare our results with those of the phase-quenched approximation. Since there are indications that correct results might be obtained if the CLE trajectories remain close to the $SU(3)$ manifold, we study how the distance from this manifold depends on the quark mass and on the coupling. We find that this distance decreases with decreasing quark mass and as the coupling decreases, i.e. as the simulations approach the continuum limit.