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The complex Langevin method in conjunction with the gauge cooling is appliedto the two-dimensional lattice $SU(2)$ Yang-Mills theory that is analyticallysolvable. We obtain strong numerical evidence that at large Langevin time theexpectation value of the plaquette variable converges, but to a wrong valuewhen the complex phase of the gauge coupling is large.

Complex Langevin method applied to the 2D $SU(2)$ Yang-Mills theory