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

## TL;DR

This paper develops a systematic perturbative method within the QCD gradient-flow formalism, providing higher-order calculations for observables, including three-loop results for quark and gluon condensates, and tools for accurate scheme conversions.

## Contribution

It introduces a comprehensive perturbative framework for the gradient-flow formalism, including Feynman rules and higher-order calculations, enhancing precision in QCD observable evaluations.

## Key findings

- Three-loop results for the quark condensate at finite flow time.
- Conversion factor for ringed quark fields to the MS scheme.
- Re-evaluation and improved accuracy of the three-loop gluon condensate.

## 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 $\overline{\mbox{MS}}$ scheme are presented as applications. We also re-evaluate an earlier result for the three-loop gluon condensate, improving on its accuracy.