Abstract
We describe in detail the implementation of a systematic perturbative approach to observables in the QCD gradient-flow formalism. This includes a collection of all relevant Feynman rules of the five-dimensional field theory and the composite operators considered in this paper. Tools from standard perturbative calculations are used to obtain Green’s functions at finite flow time t at higher orders in perturbation theory. The three-loop results for the quark condensate at finite t and the conversion factor for the “ringed” quark fields to the overline{mathrm{MS}} scheme are presented as applications. We also re-evaluate an earlier result for the three-loop gluon condensate, improving on its accuracy.
Highlights
Parameters and fields, composite operators at positive flow time are finite [16]
One of the first possible cross-fertilizations among perturbative and lattice QCD is given by the definition of a new scheme for the strong coupling, defined by the gluon condensate (QCD action density) at finite flow time [3]
The various stages of automation in a perturbative multi-loop calculation of correlation functions are described in section 3: the generation of Feynman diagrams and insertion and evaluation of Feynman rules, followed by a reduction of the loop integrals to a set of master integrals, and the numerical evaluation of the latter
Summary
Are the regular covariant derivatives in the fundamental and adjoint representation, respectively, g is the gauge coupling, ξ the QCD gauge parameter, and nF the number of different quark flavors. Of mass dimensions 3 and 5/2 that otherwise carry the same quantum numbers as the flowed gluon and quark/antiquark fields Bμa and χ, χ, respectively. Note that the Lagrangian (2.11) does not include flowed ghost fields da(t, x) and da(t, x) They arise in the same way as the usual Faddeev-Popov ghosts ca(x) and ca(x) due to gauge-fixing, and obey the initial condition da(t, x)|t=0 = ca(x). As becomes clear later, closed loops of only flowed fields vanish, so that one can omit da and da already at the level of the Lagrangian
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