[Paper Review] Lattice Perturbation Theory by Langevin Dynamics

Francesco Di Renzo, Giuseppe Marchesini+2 more|arXiv (Cornell University)|Aug 7, 1993

Stochastic processes and statistical mechanics4 citations

TL;DR

This paper introduces a novel application of Langevin dynamics to lattice perturbation theory in QCD, enabling efficient computation of weak-coupling expansions for Wilson loops up to fourth order. By employing stochastic gauge fixing, the method successfully mitigates divergent fluctuations in higher-order coefficients, offering a robust framework for perturbative calculations on the lattice.

ABSTRACT

We present an application of the standard Langevin dynamics to the problem of weak coupling perturbative expansions for Lattice QCD. This method can be applied to the computation of the most general observables. In this preliminary work we will concentrate in particular on the computation of the perturbative terms of the $1 imes 1$ 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.

Motivation & Objective

• To develop a stochastic method for computing weak-coupling perturbative expansions in lattice QCD.
• To address the issue of divergent fluctuations that plague higher-order perturbative coefficients in lattice field theory.
• To apply the method specifically to the 1×1 Wilson loop up to fourth order in the coupling constant.
• To explore the feasibility and stability of stochastic gauge fixing in suppressing unphysical divergences during perturbative calculations.
• To lay the groundwork for extending the method to more general observables in lattice gauge theories.

Proposed method

• The authors employ standard Langevin dynamics to evolve gauge configurations stochastically, treating the gauge field as a dynamical variable subject to a stochastic force.
• The method uses a Fokker-Planck equation to describe the time evolution of the probability distribution of gauge fields, enabling sampling of configurations relevant to perturbation theory.
• Stochastic gauge fixing is introduced as a key component to control the growth of fluctuations in higher-order terms, particularly in the presence of gauge-dependent divergences.
• The approach is applied to compute perturbative coefficients of the 1×1 Wilson loop up to fourth order in the coupling constant.
• The Langevin equation is solved numerically with a discretized time evolution, ensuring stability and convergence of the perturbative series.
• The method is tested on a simple observable (the Wilson loop) to validate its effectiveness before generalization.

Experimental results

Research questions

• RQ1 Can Langevin dynamics be effectively used to compute weak-coupling perturbative expansions in lattice QCD?
• RQ2 How do divergent fluctuations in higher-order perturbative coefficients affect the reliability of standard stochastic methods in lattice field theory?
• RQ3 Can stochastic gauge fixing suppress these divergences and stabilize the computation of higher-order terms?
• RQ4 To what extent can this method be generalized to other observables beyond the Wilson loop?
• RQ5 What is the numerical stability and convergence behavior of the Langevin-based approach for perturbative lattice calculations?

Key findings

• The Langevin dynamics approach successfully computes perturbative coefficients of the 1×1 Wilson loop up to fourth order in the coupling constant.
• The method exhibits improved stability compared to standard stochastic approaches, particularly in handling higher-order terms.
• Stochastic gauge fixing is shown to be effective in suppressing divergent fluctuations that typically arise in higher-order perturbative expansions.
• The numerical implementation remains stable and convergent over multiple iterations, indicating robustness for perturbative calculations.
• The results suggest that this method is a viable alternative to traditional perturbation theory techniques in lattice QCD.
• The framework is generalizable to other observables, opening a path for broader application in lattice field theory.