# SMD-based numerical stochastic perturbation theory

## TL;DR

This paper explores a variant of numerical stochastic perturbation theory using the SMD algorithm, demonstrating its convergence, efficiency with higher-order integrators, and successful two-loop calculations in SU(3) gauge theory.

## Contribution

It introduces and rigorously analyzes a new SMD-based approach to numerical stochastic perturbation theory, showing improved convergence and computational advantages.

## Key findings

- Proves convergence to a unique stationary state.
- Shows higher-order symplectic schemes are highly effective.
- Performs two-loop calculations in SU(3) gauge theory successfully.

## Abstract

The viability of a variant of numerical stochastic perturbation theory, where the Langevin equation is replaced by the SMD algorithm, is examined. In particular, the convergence of the process to a unique stationary state is rigorously established and the use of higher-order symplectic integration schemes is shown to be highly profitable in this context. For illustration, the gradient-flow coupling in finite volume with Schr\"odinger functional boundary conditions is computed to two-loop (i.e. NNL) order in the SU(3) gauge theory. The scaling behaviour of the algorithm turns out to be rather favourable in this case, which allows the computations to be driven close to the continuum limit.