Analysis of the Pion Electromagnetic Form Factor with Next-to-Next-to-Leading Order QCD Corrections

Abstract

NNLO QCD corrections for the pion electromagnetic form factor at large momentum transfer have been recently performed in [Phys. Rev. Lett. 132, 201901 (2024); Phys. Rev. Lett. 134, 221901 (2025)], revealing that the NLO and NNLO contributions are positive and sizeable. Unfortunately, these predictions have been obtained using the conventional scale-setting method and thus they are plagued by large renormalization scale ambiguities. In this paper, we analyze the pion electromagnetic form factor at NNLO by applying the Principle of Maximum Conformality (PMC), which is introduced with the aim of resolving renormalization scheme and scale ambiguities. By applying the PMC, a more precise perturbative QCD (pQCD) prediction for the pion EMFF \(Q^2F_π(Q^2)\) without conventional renormalization scale ambiguity can be achieved. This improved pQCD prediction is highly beneficial for the precise determination of the pion light-cone distribution amplitude. We then conduct a comprehensive comparison between theoretical predictions and experimental measurements of the pion EMFF \(Q^2F_π(Q^2)\).

Figures (8)

• Figure 1: The evolution of the Gegenbauer moment $a_2(\mu_f)$ from $a_2(2\,\rm{GeV})=0.116$ at the one-loop, two-loop and three-loop levels.
• Figure 2: The PMC scale $Q_{\star}$ versus $Q$ for the pion EMFF $F_\pi(Q^2)$, where the superscripts "LL" and "NLL" stand for the scale $Q_{\star}$ up to leading logarithmic order and next-to-leading logarithmic order accuracy, respectively.
• Figure 3: The renormalization scale uncertainties for the pion EMFF $Q^2F_\pi(Q^2)$ using conventional scale setting, where the dotted, dotdashed and dashed lines correspond to the central values for the LO, NLO and NNLO results, respectively. The factorization scale is fixed to $\mu_f=Q$, and the renormalization scale has been varied in the range of $\mu_r\in[Q/2,2Q]$. The red, green and blue bands correspond to the LO, NLO and NNLO scale errors, respectively.
• Figure 4: The factorization scale uncertainties for the pion EMFF $Q^2F_\pi(Q^2)$ using conventional scale setting, where the dotted, dotdashed and dashed lines correspond to the central values for the LO, NLO and NNLO results, respectively. The renormalization scale is fixed to $\mu_r=Q$, and the factorization scale has been varied in the range: $\mu_f\in[Q/2,2Q]$. The red, green and blue bands correspond to the LO, NLO and NNLO scale errors, respectively.
• Figure 5: The renormalization scale uncertainties for the pion EMFF $Q^2F_\pi(Q^2)$ using PMC scale setting, where the dotted, dotdashed and dashed lines correspond to the LO, NLO and NNLO results, respectively. The factorization scale is fixed to $\mu_f=Q$. The PMC prediction of pion EMFF $Q^2F_\pi(Q^2)$ is independent of the scale $\mu_r$.