Results and techniques for higher order calculations within the gradient-flow formalism

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 MS scheme are presented as applications. We also re-evaluate an earlier result for the three-loop gluon condensate, improving on its accuracy.

The two-loop energy–momentum tensor within the gradient-flow formalism

“…In conclusion, we hope that our results will help to improve the studies of the EMT on the lattice. They are the first outcome of a systematic setup for higher-order perturbative calculations within the gradient-flow formalism [33], which should prove useful also in other applications of this theoretical framework.…”

“…A more detailed description of parts of our setup will be described in a forthcoming publication [33]. As a check, we evaluated the correlators G a µν G a µν , χχ and χ / Dχ through NLO.…”

“…(50). A detailed outline of this calculation is beyond the scope of this paper; it will be presented together with a more complete description of our setup in a forthcoming publication [33]. At this point, we simply quote the numerical value of this result up to three significant digits, which is more than sufficient in the light of the theoretical uncertainties to be discussed below:…”

“…We would also like to thank Tobias Neumann for fruitful communication, and for providing his tools which enabled us to obtainZ χ before the publication of Ref. [33]. This work was supported by Deutsche Forschungsgemeinschaft (DFG), project 386986591.…”

Hadronic vacuum polarization using gradient flow

“…We achieved this using the setup described in ref. [65], which employs qgraf [69,70] and q2e/exp [71,72] for the generation and subsequent categorization of the Feynman diagrams, FORM [73,74] JHEP08(2020)109 for the manipulation of the ensuing algebraic expressions, the color package [75] of FORM for the calculation of the gauge group factors, and Kira⊕FireFly [76][77][78][79][80] for the Feynman integral reduction using integration-by-parts identities and the Laporta algorithm [44][45][46] over finite fields [81][82][83][84]. For the evaluation of the master integrals, we adopt the method described in ref.…”