Gradient-flowed operator product expansion without IR renormalons

TL;DR

This paper introduces a gradient-flowed operator product expansion (OPE) to resolve IR renormalon ambiguities in QCD perturbation theory by reexpressing local operator matrix elements in terms of gradient-flowed (flow-time dependent) counterparts. The Adler function serves as a testbed, where the gradient-flowed OPE combines a subtracted perturbative series with a nonperturbative, lattice-determined flowed gluon condensate $E(t)$, yielding improved convergence and nonperturbative accuracy down to low $Q^2$ while reducing theoretical uncertainties. A key result is the explicit cancellation of the leading IR renormalon (and, in extensions, of subleading renormalons) within the GF framework, removing the need for Stokes constants and enabling a Wilsonian-like hard cut-off without sacrificing gauge invariance. The approach also resolves the fixed-order vs contour-improved perturbation theory discrepancy in tau decays and offers a systematic path to incorporating dimension-six operators, with potential broad impact on SVZ sum rules and heavy-quark physics. Overall, gradient-flowed OPE provides a practical, lattice-friendly route to merge perturbative QCD with power corrections in a controlled, renormalon-free manner.

Abstract

A long-standing problem concerns the question how to consistently combine perturbative expansions in QCD with power corrections in the context of the operator product expansion (OPE), since the former exhibit ambiguities due to infrared renormalons, which are of the same order as the power corrections. We propose to use the gradient flow time $1/\sqrt{t}$ as a factorization scale and to express the OPE in terms of IR renormalon-free subtracted perturbative expansions and unambiguous matrix elements of gradient-flow regularized local operators. We show on the example of the Adler function and its leading power correction from the gluon condensate that this method dramatically improves the convergence of the perturbative expansion. We employ lattice data on the action density to estimate the gradient-flowed gluon condensate, and obtain the Adler function with non-perturbative accuracy and significantly reduced theoretical uncertainty, enlarging the predictivity at low $Q^2$. When applied to the hadronic decay width of the tau lepton, the method resolves the long-standing discrepancy between the fixed-order and contour-improved approach in favour of the fixed-order treatment.

Figures (8)

• Figure 1: The perturbative part $\Delta_D^{\rm pert}(Q^2,t)$ of the Adler function at $Q=m_{\tau}$, $\alpha_s(m_\tau)=0.314$ summed to perturbative order $N$. Shown are the unsubtracted and gradient-flow subtracted series (flow times $8 m_{\tau}^2 t=20, 30, 15$). The fainter colours indicate that these results rely on the estimated perturbative coefficients, i.e. beyond the 4th order for the "unsubtracted" series and beyond the 3rd order for the subtracted series.
• Figure 2: Upper panel: Non-perturbative action density. "Default model" and "Alt. model" refer to eq. \ref{['linearmodel']} and eq. \ref{['eq:alternativemodel']}, respectively. For the lattice data, see text. The reference scales associated with the action density are given in units of GeV${}^{-2}$ by $t_0 =0.525~\hbox{GeV}^{-2}$ and $w_0^2=0.765~\hbox{GeV}^{-2}$. Lower panel: Ratios of the lattice data to the linear model eq. \ref{['linearmodel']}.
• Figure 3: $t$-dependence of the Adler function at $Q=m_{\tau}$, $\alpha_s(m_\tau)=0.314$. Shown are the perturbative subtracted contribution (blue), the gradient-flowed gluon-condensate contribution (GFGC, red-dashed) and the sum (green) at the second, third and fourth order in perturbation theory. The lower panel shows a zoom-in that compares the non-perturbative to the standard (unsubtracted) perturbative approximation at second, third, and fourth order for the perturbative part.
• Figure 4: Dependence of the Adler function at $Q^2=m_{\tau}^2$ in standard perturbation theory (blue) and the gradient-flowed OPE (red) on the truncation order of the unit operator short-distance coefficients. The fainter points rely on the Borel function model to extrapolate the series expansions to higher order than exactly known. The black horizontal line/grey band display the Borel sum/ambiguity of the standard perturbative series.
• Figure 5: The Adler function $\Delta_D(Q^2)$ as a function of $Q$. The gradient-flow OPE result (red) with the NNNLO perturbative series and $E(t)$ compared to the standard perturbative results at N$^4$LO (blue) are shown. $\alpha_s(m_{\tau})=0.314$ is used. The error bars are computed as for the previous figure and described in the text.