Paper detail

Temporal breakdown and Borel resummation in the complex Langevin method

We reexamine the Parisi-Klauder conjecture for complex e^{iθ/2} ϕ^4 measures with a Wick rotation angle 0 <= θ/2 < π/2 interpolating between Euclidean and Lorentzian signature. Our main result is that the asymptotics for short stochastic times t encapsulates information also about the equilibrium aspects. The moments evaluated with the complex measure and with the real measure defined by the stochastic Langevin equation have the same t -> 0 asymptotic expansion which is shown to be Borel summable. The Borel transform correctly reproduces the time dependent moments of the complex measure for all t, including their t -> infinity equilibrium values. On the other hand the results of a direct numerical simulation of the Langevin moments are found to disagree from the `correct' result for t larger than a finite t_c. The breakdown time t_c increases powerlike for decreasing strength of the noise's imaginary part but cannot be excluded to be finite for purely real noise. To ascertain the discrepancy we also compute the real equilibrium distribution for complex noise explicitly and verify that its moments differ from those obtained with the complex measure.