Comments on "Non-local Nucleon Matrix Elements in the Rest Frame"

Abstract

In a recent paper, "Non-local Nucleon Matrix Elements in the Rest Frame" (Phys. Rev. D 111, 5 (2025)), it was observed that the next-to-leading order calculations of the renormalization factor can describe, to a few percent accuracy, the logarithm of the lattice QCD rest frame matrix elements with separations up to distances of 0.6 fm on multiple lattice spacings. We argue that perturbative QCD breaks down at such a distance scale after resumming the associated large logarithms, while the ansatz used in the analysis there is not justified in perturbation theory. Besides, we explain the observation in Phys. Rev. D 111, 5 (2025) and demonstrate that the ansatz fails to describe the data for $z>0.3$ fm, showing an opposite trend. Finally, although Phys. Rev. D 111, 5 (2025) proposes multiplying the ansatz by a Gaussian correction model, which is shown to reduce the discrepancy with the data, this does not legitimize the use of perturbative QCD at such distance scales.

Figures (3)

• Figure 1: The exact and asymptotic solutions to $\Gamma(\pi z/a)$.
• Figure 2: The static pion matrix elements (data points), computed in Ref. LatticePartonLPC:2021gpi from the MILC ensembles, are compared to the fits (curves) with the ansatz in Eq. (\ref{['eq:ansatz']}), which corresponds to $n_f=4$ and $\Lambda_{\rm QCD}=0.12$ GeV. Upper panel: the full matrix elements. Lower panel: the matrix elements after subtracting $\exp[- \alpha_s C_F/(2\pi) \cdot \pi^2 |z|/a ]$.
• Figure 3: After subtracting the linear divergence, the static proton matrix elements (data points) in Ref. Karpie:2024bof are compared to the theory (curves) with $n_f=2$ and $\Lambda_{\rm QCD}$ fitted from each ensemble with $z\ge 2a$. Only the central values are plotted.