∗Lattice Perturbation Theory by Langevin Dynamics

We present here an application of the standard Langevin dynamics to the problem of perturbative expansions on the Lattice QCD. This method can be applied in the computation of the most general observables. In this work we will concentrate in particular on the computation of the perturbative terms of the 1 × 1 Wilson loop, up to fourth order. It is shown that a stochastic gauge fixing is a possible solution to the problem of divergent fluctuations which affect higher order coefficients. Since its introduction in 1981 by Parisi and Wu [1], Langevin dynamics has been extensively used for Monte Carlo simulations. Basically it consists in a stochastic dynamical system on the field configuration space dictated by the general equation ∂φ(x, t) ∂t = − ∂S[φ] ∂φ(x, t) + η(x, t), (1) where φ is the field, S[φ] the action and η a Gaussian random noise satisfying to the normalization < η(x, t)η(x ′ , t ′ ) >= 2δ(x ′ − x)δ(t′ − t). (2) ∗We warmly thank P.Rossi, G.Parisi, G.C.Rossi, N.Cabibbo and C.Destri for interesting discussions and precious suggestions As a matter of fact, stochastic dynamics is devised in such a way that time averages on the noise converge to averages on the Gibbs measure