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The complex Langevin method is a promising approach to the complex-actionproblem based on a fictitious time evolution of complexified dynamicalvariables under the influence of a Gaussian noise. Although it is known to havea restricted range of applicability, the use of gauge cooling made itapplicable to various interesting cases including finite density QCD in certainparameter regions. In this paper, we revisit the argument for justification ofthe method. In particular, we point out a subtlety in the use of time-evolvedobservables, which play a crucial role in the previous argument. This requiresthat the probability of the drift term should fall off exponentially or fasterat large magnitude. We argue that this is actually a necessary and sufficientcondition for the method to be justified. Using two simple examples, we showthat our condition tells us clearly whether the results obtained by the methodare trustable or not. We also discuss a new possibility for the gauge cooling,which can reduce the magnitude of the drift term directly.

The argument for justification of the complex Langevin method and the condition for correct convergence