Quark-mass effects in gradient-flow observables through next-to-next-to-leading order in QCD

TL;DR

This work delivers next-to-next-to-leading order (NNLO) QCD predictions for vacuum expectation values of flowed operators in the gradient-flow framework, incorporating the effects of a single massive quark among massless flavors to model charm and bottom quark dynamics. By formulating invariant combinations through a ringed renormalization scheme and providing detailed mass-dependent expansions for the scalar quark density, quark kinetic density, and action density, the authors establish a robust perturbative bridge to lattice calculations and potential mass extractions. The results show that mass effects can be substantial for $S_f$ and $E(t)$ and are increasingly controllable with higher perturbative orders, while radiative corrections generally reduce scale uncertainties; two-mass effects are found to be subdominant within the explored ranges. The accompanying ancillary data enable direct application to lattice-QCD analyses and the extraction of quark masses and $ SLA_ ext{QCD}$ from gradient-flow observables, advancing precision determinations of fundamental QCD parameters.

Abstract

We provide results for the vacuum expectation values of the flowed action density, the quark condensate, and the quark kinetic operator in the gradient-flow formalism. We work in $N_\text{F}$-flavor QCD, keeping the heaviest quark massive and all others massless. The vacuum expectation values of these operators are calculated numerically through next-to-next-to-leading order QCD, providing important input for the extraction of fundamental QCD parameters from lattice calculations. While the focus is on charm- and bottom-quark mass effects, we provide the results in a form that is independent of the specific quark mass.

Figures (16)

• Figure 1: Diagrams contributing to the vacuum expectation value of the quark condensate $S_f(t)$ (a), the quark kinetic operator $R_f(t)$ (a/b), and the gluon condensate $E(t)$ (c--e). The hatched bubble represents an arbitrary flowed-.9QCD sub-diagram. Some of the lines can also be flowlines. All Feynman diagrams in this paper were created with FeynGameBundgen:2025uttHarlander:2024qbnHarlander:2020cyh.
• Figure 2: Specific diagrams contributing to $S(t)$ (a), $R(t)$ (a/b) at the three-loop level and $E(t)$ at the two- (c) and three-loop level (d) and (e).
• Figure 3: $L_{\mu t}$-independent components of $\mathring{S}_f(t)$ in the form of \ref{['eq:Resu:jael', 'eq:Resu:epic']}.
• Figure 4: Derivatives of the $L_{\mu t}$-independent components of $\mathring{S}_f(t)$.
• Figure 5: $L_{\mu t}$-independent components of $\mathring{R}_f(t)$ in the form \ref{['eq:RtEXP']}.