Abstract
Numerical Stochastic Perturbation Theory (NSPT) allows for perturbative computations in quantum field theory. We present an implementation of NSPT that yields results for high orders in the perturbative expansion of lattice gauge theories coupled to fermions. The zero-momentum mode is removed by imposing twisted boundary conditions; in turn, twisted boundary conditions require us to introduce a smell degree of freedom in order to include fermions in the fundamental representation. As a first application, we compute the critical mass of two flavours of Wilson fermions up to order O(beta ^{-7}) in a {{mathrm{{mathrm {SU}}}}}(3) gauge theory. We also implement, for the first time, staggered fermions in NSPT. The residual chiral symmetry of staggered fermions protects the theory from an additive mass renormalisation. We compute the perturbative expansion of the plaquette with two flavours of massless staggered fermions up to order O(beta ^{-35}) in a {{mathrm{{mathrm {SU}}}}}(3) gauge theory, and investigate the renormalon behaviour of such series. We are able to subtract the power divergence in the Operator Product Expansion (OPE) for the plaquette and estimate the gluon condensate in massless QCD. Our results confirm that NSPT provides a viable way to probe systematically the asymptotic behaviour of perturbative series in QCD and, eventually, gauge theories with fermions in higher representations.
Highlights
With the Fourier transform described in Appendix C, the inverse free Wilson operator with twisted boundary conditions is diagonal in momentum space and can be expressed as
The inverse propagator is projected onto the identity in Dirac space. All these operations are performed order by order in perturbation theory keeping in mind that, after the measure of the propagator, all perturbative orders β−k/2 with an odd k are discarded, since the expansion in powers of β−1/2 is an artefact of Numerical Stochastic Perturbation Theory (NSPT)
Since our fermions are in the fundamental representation, we consistently provided them with a smell degree of freedom
Summary
Let us here summarise the main steps in defining NSPT for lattice gauge theories. Rather than trying to give a comprehensive review of the method, we aim here to introduce a consistent notation that will allow us to discuss the new developments in the rest of the paper. The idea of studying the convergence properties of a stochastic process order by order after an expansion in the coupling is quite general In this spirit different NSPT schemes can be set up, based on stochastic differential equations different from Langevin [22, 23]. A new, very effective second-order integration scheme for NSPT in lattice gauge theories has been introduced in Ref. Stochastic gauge fixing The zero modes of the gauge action do not generate a deterministic drift term, and their evolution in stochastic time is entirely driven by the stochastic noise, which gives rise to diverging fluctuations This phenomenon is well known since the early days of NSPT, see e.g. Ref.
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