Perturbative QCD below charm threshold: theory and tensions with $e^+e^-$ data

TL;DR

This work analyzes the theory prediction for the hadronic ratio $R_{uds}$ below open-charm threshold, confronting it with BES-III data and other measurements. By combining perturbative QCD with a renormalon-aware framework (including the renormalon-free GC scheme) and a Regge-inspired duality-violation model, the authors quantify theory uncertainties and show that DVs are meaningful up to $\sqrt{s}\approx 2.5$ GeV but negligible at higher energies. Data from various experiments are merged with a clustering method to test consistency, finding overall compatibility with theory within about $2\sigma$ for most intervals, though BES-III results above 3 GeV remain in tension at the $3\sigma$ level. The study also connects these findings to $a_\mu^{\rm HVP}$, highlighting the important role of accurate $R_{uds}$ modeling for the SM evaluation of the muon magnetic moment and for extractions of $\alpha_s$ and the charm mass.

Abstract

In this paper we carefully assess the theory prediction for $R(s)$ below charm threshold, $R_{uds}$, and address tensions with the existing data, notably with the 2021 BES-III results. We analyze the uncertainty of the perturbative QCD description in the light of renormalons making use of the large-$β_0$ limit and the renormalon-free gluon-condensate scheme. We provide a reliable estimate of the duality violation contributions; we show they are sizable up to $2.5$~GeV and improve the agreement between theory and data, but are negligible for higher energies. We then combine the available experimental data for $R_{uds}$ and find the data sets to be mutually compatible. Finally, we compare theory and data, both locally and in their contributions to the anomalous magnetic moment of the muon. Theory is compatible with the combined data but discrepancies with the BES-III data reach more than 3$σ$.

Figures (6)

• Figure 1: Results for $\delta_{\alpha_s}^{(0)}$ in FOPT (blue squares) and in RF-CIPT (red triangles) in the large-$\beta_0$ limit (left-hand panel) and in full QCD (right-hand panel) for $s=(2~{\rm GeV})^2$. The horizontal line in large-$\beta_0$ represents the Borel sum of the series (the imaginary ambiguity is very small in this case and cannot be seen in the plot). In QCD, results beyond $\mathcal{O}(\alpha_s^4)$ employ an estimate for the unknown series coefficients (see text for details). The band in the QCD plot shows the uncertainty due to the unknown coefficients in the FOPT result.
• Figure 2: Inclusive data for $R_{uds}$ from several experiments compared with pQCD (black line) and pQCD$+$DVs (blue-dashed line). The description including DVs is not a fit to the data shown in the figure and is obtained extrapolating from hadronic $\tau$-decay and exclusive $R_{uds}$ data analyses Boito:2025pwgBoito:2018yvlKeshavarzi:2018mgv. Both curves include the small strange-quark mass and EM corrections.
• Figure 3: Local discrepancies of data points from BES-III, KEDR, and BES with respect to pure pQCD (upper panel) and pQCD+DVs (lower panel). In both cases small $m_s$ and EM corrections are taken into account.
• Figure 4: Data combination (red triangles) with 19 clusters using the algorithm discussed in the text. The red error bars show errors before error inflation. The gray band gives the errors after error inflation is applied to account for local tensions between data points in the same cluster (the effect is small).
• Figure 5: Local $p$-value for each of the 19 clusters of the data combination shown in Fig. \ref{['fig:combined_R']}. The red-dashed line shows the value 5% to guide the eye.