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 Schrödinger 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.
Highlights
Numerical stochastic perturbation theory (NSPT) based on the SMD algorithm [5,6,7] has recently been briefly looked at in Ref. [8] and was found to perform well
The viability of a variant of numerical stochastic perturbation theory, where the Langevin equation is replaced by the SMD algorithm, is examined
Numerical stochastic perturbation theory (NSPT) [1,2,3] is a powerful tool that allows many interesting calculations in QCD and other quantum field theories to be performed to high order in the interactions
Summary
The action S(q) is assumed to be differentiable and to have an expansion in powers of a coupling g of the form. 2.2 SMD algorithm to the current momenta and coordinates, with step sizes h proportional to. The SMD algorithm operates in the phase space of the theory and updates both the coordinates q and their canonical momenta p = An SMD update cycle consists of a momentum rotation followed by a molecular-dynamics evolution and, optionally, an acceptance–rejection step. Depend on the simulation time step > 0 and a parameter γ > 0 that controls the rotation angle. Are integrated from the current simulation time t to t + using a reversible symplectic integration scheme The algorithm (momentum rotation followed by the molecular-dynamics evolution) simulates the canonical distribution. Stochastic estimates of the expectation values (2.3) of the observables of interest are obtained by averaging their values over a range of simulation time
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