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

This paper resolves tensions between perturbative QCD predictions and $e^+e^-$ data below the charm threshold by incorporating sizable duality violation contributions up to 2.5 GeV, which significantly improve global agreement despite persistent 3 $\sigma$ discrepancies with specific BES-III measurements.

Original authors: Diogo Boito, Marcelle Caram

Published 2026-08-26

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Diogo Boito, Marcelle Caram

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the subatomic world, particles called quarks are the fundamental building blocks of protons and neutrons, the matter that makes up our visible universe. However, quarks are never found alone; they are always bound together in groups by a powerful force known as the strong interaction. Physicists study this force using a theory called Quantum Chromodynamics, or QCD. To test if their theory is correct, scientists smash electrons and positrons together at high speeds. When these particles collide, they can transform into a shower of new particles made of quarks. By measuring how often this happens at different energy levels, researchers can calculate a specific ratio that acts as a fingerprint for the strong force. This fingerprint is crucial for understanding a mysterious property of the muon, a particle similar to an electron but much heavier. The muon's magnetic behavior is one of the most sensitive tests of the Standard Model, the current rulebook of particle physics. If the theory describing the strong force is slightly off, it could throw off the entire calculation of the muon's magnetic moment, potentially hiding signs of new, undiscovered physics.

For decades, physicists have been able to predict this fingerprint with high precision at very high energies, where the strong force behaves in a predictable, mathematical way. However, a puzzle emerged in 2021 when a team at the BES-III laboratory in China released new, highly precise measurements of this ratio in a specific energy range, just below the threshold where charm quarks begin to appear. These new measurements were consistently higher than what the standard theory predicted. This created a tension: either the theory was missing something important, or the new data contained a hidden error. The discrepancy was significant enough to worry researchers, as it sits in an energy zone that contributes to the calculation of the muon's magnetic moment. If the theory is wrong here, the entire Standard Model calculation for the muon could be flawed.

A team of researchers from the University of São Paulo in Brazil has now taken a deep dive into this problem to see if the theory itself could explain the gap. They carefully re-examined the mathematical description of the strong force in this energy region, looking for subtle effects that might have been overlooked. One of the main things they investigated was the behavior of the mathematical series used to calculate the force. In quantum physics, these calculations often involve adding up an infinite number of terms, but in practice, scientists must stop after a certain number. The researchers checked if stopping early was causing the error. They used advanced mathematical techniques to estimate what the missing terms would look like and found that the standard theory is actually very stable and reliable in this region. The uncertainty in the theory is tiny, far smaller than the difference seen in the new data.

The team then turned their attention to a phenomenon called duality violation. In simple terms, the theory assumes that at high enough energies, the chaotic behavior of individual quarks averages out to look smooth and predictable. However, in the real world, the energy levels are not perfectly smooth; they are influenced by the presence of short-lived particle resonances, which are like temporary, heavy versions of the particles being studied. These resonances cause small, wiggly oscillations in the data that the smooth theory does not capture. The researchers modeled these wiggles to see if they could account for the difference between the theory and the BES-III data. They found that these oscillations are indeed real and can be significant at lower energies, specifically below 2.5 GeV. When they added this effect to their theoretical prediction, the agreement with some of the older experimental data improved dramatically, and the theory began to match the wiggles seen in those measurements.

However, the story changes as the energy increases. The researchers found that these oscillating effects become negligible above 2.8 GeV. In the higher energy part of the range, where the BES-III data shows the largest disagreement with the theory, the "wiggles" are essentially gone. This means that the standard theory should be working perfectly well in that region, and there is no hidden mathematical effect left to explain the gap. When the team compared the theory, now including the small oscillation effects, against the combined data from all available experiments, a clear picture emerged. The older data sets, such as those from the KEDR collaboration, fit the theory very well, even without the oscillation corrections. The combined data from all experiments is also reasonably close to the theory, staying within a margin of about two and a half standard deviations, which is acceptable in this field.

The situation with the new BES-III data, however, remains unresolved. Even after accounting for the oscillations and all other known corrections, the BES-III measurements in the higher energy range, specifically above 3.4 GeV, still sit more than three standard deviations away from the theoretical prediction. This is a large gap that suggests a serious problem. The researchers checked if the different experimental data sets were compatible with each other and found that, while there are some local tensions, the data generally agree with one another. The issue is not that the experiments contradict each other, but that the new, precise BES-III results are systematically higher than what the theory predicts. The authors conclude that there is no known mechanism within the current understanding of the strong force that can explain why the BES-III data is so high in this specific energy range. The discrepancy is not a result of a missing piece of the theory, but rather a genuine conflict between the new measurements and the established laws of physics.

This finding leaves the scientific community with a difficult choice. The theory is robust, and the mathematical tools used to describe it are under control. The new data is precise and appears reliable. The fact that they do not agree suggests that either the BES-III measurements have an unaccounted-for systematic error, or there is something fundamental about the strong force in this energy region that is not yet understood. The researchers hope their work will encourage further experimental investigations to re-measure this energy range with different methods. Until then, the tension between the theory and the BES-III data remains one of the most intriguing puzzles in the study of the strong force, standing as a potential signpost for new physics or a call for a closer look at the experimental setup.

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

Problem Statement
The paper addresses a persistent tension between theoretical predictions and experimental measurements of the inclusive hadronic $R$ -ratio, $R_{uds}(s)$ , in the energy region below the open-charm threshold ( $\sqrt{s} \approx 1.8$ GeV to $3.7$ GeV). Specifically, recent high-precision measurements by the BES-III collaboration (2021) show values systematically larger than the perturbative QCD (pQCD) prediction, with discrepancies reaching over $3\sigma$ in certain energy windows. This tension is critical for two reasons:

Muon Anomalous Magnetic Moment ( $a_\mu$ ): $R_{uds}$ data are a primary input for the data-driven determination of the hadronic vacuum polarization (HVP) contribution to $a_\mu$ . While the contribution of this specific energy window to the total $a_\mu$ is smaller than the $\pi^+\pi^-$ region, a significant discrepancy challenges the consistency of the Standard Model (SM) assessment.
Fundamental QCD Parameters: Discrepancies affect the extraction of the strong coupling constant ( $\alpha_s$ ) and the charm-quark mass from relativistic sum rules, which rely on subtracting the $R_{uds}$ background from inclusive four-flavor measurements.

The authors aim to rigorously assess the theoretical uncertainties of the pQCD prediction, quantify non-perturbative effects (specifically duality violations), and determine if the theory can accommodate the experimental data, particularly the BES-III results.

Methodology
The authors employ a comprehensive theoretical framework combining fixed-order perturbation theory (FOPT), renormalon analysis, and models for duality violations (DVs).

Perturbative QCD Analysis:
- The calculation utilizes the massless Adler function and the operator product expansion (OPE). The perturbative series is known up to five loops ( $O(\alpha_s^4)$ ), with estimates for the sixth order ( $O(\alpha_s^5)$ ).
- To assess convergence and higher-order uncertainties, the authors analyze the series using the large- $\beta_0$ limit and the Renormalon-Free (RF) gluon-condensate scheme.
- They compare Fixed Order Perturbation Theory (FOPT) with Contour Improved Perturbation Theory (CIPT). The paper notes that standard CIPT is inconsistent with the OPE due to sensitivity to infrared (IR) renormalons. The RF scheme is employed to consistently remove the leading IR renormalon (associated with the gluon condensate), rendering the CIPT series (RF-CIPT) consistent with FOPT at higher orders.
Non-Perturbative Corrections:
- Duality Violations (DVs): The authors model DVs—residual oscillations around the pQCD prediction due to resonance tails—using an asymptotic parametrization derived from Regge trajectories and large- $N_c$ considerations. The DV contribution is modeled as an exponentially damped sine wave.
- Parameter Fixing: The isovector ( $I=1$ ) DV parameters are fixed using results from hadronic $\tau$ decays. The isoscalar ( $I=0$ ) parameters are constrained by fitting exclusive channel data in the $3.3 $–$ 4.0 $GeV$ ^2$ range.
- Mass and EM Corrections: Small corrections from the strange quark mass ( $m_s$ ) and electromagnetic effects are included up to $O(\alpha_s^2)$ and $O(\alpha_{EM})$ , respectively.
Data Combination:
- Experimental data from BES-III, KEDR, and previous BES generations are combined using the KNT algorithm (Refs. [46, 95]). This involves clustering data points, performing a $\chi^2$ fit, and applying error inflation to account for local tensions between datasets.

Key Contributions and Results

Validation of pQCD Convergence: The analysis confirms that the pQCD series for $R_{uds}$ is under control. While the FOPT series exhibits a peculiar sign change at $O(\alpha_s^3)$ , the large- $\beta_0$ limit and the RF-CIPT scheme demonstrate that the series converges to a stable value. The uncertainty from missing higher orders is found to be very small compared to experimental errors.
Quantification of Duality Violations: The study provides a reliable estimate of DV contributions.
- DVs are found to be sizable for $\sqrt{s} \leq 2.5$ GeV, where they can reach the percent level and exhibit oscillatory behavior.
- For $\sqrt{s} > 2.8$ GeV, DVs become negligible.
- Including DVs improves the agreement between theory and the KEDR data, particularly in reproducing the observed oscillations.
Comparison with Experimental Data:
- KEDR Data: The KEDR measurements are generally compatible with the pQCD prediction (with or without DVs) within $2\sigma$ . The inclusion of DVs reduces the discrepancy further.
- BES-III Data: The BES-III results remain systematically higher than the theoretical prediction. Even with the inclusion of DVs, the discrepancy persists, reaching $>3\sigma$ for the energy range $\sqrt{s} \geq 3.4$ GeV. The DV contribution only significantly affects the first two data points ( $\sqrt{s} \leq 2.4$ GeV) and has negligible impact on the higher-energy points where the tension is most severe.
- Combined Data: When all datasets are combined, the resulting $a_\mu$ contribution is compatible with theory within $2.4\sigma$ (pure pQCD) and $2.0\sigma$ (pQCD + DVs). However, the BES-III subset alone yields a $3.4\sigma$ (pure pQCD) and $3.2\sigma$ (pQCD + DVs) discrepancy.

Significance and Claims
The paper concludes that the theoretical description of $R_{uds}$ is robust, with pQCD uncertainties being negligible compared to experimental errors. The inclusion of duality violations resolves discrepancies for lower energies ( $\sqrt{s} \lesssim 2.5$ GeV) and aligns well with KEDR data.

However, the authors emphasize that no theoretical mechanism (including DVs or higher-order corrections) can account for the large values measured by BES-III at $\sqrt{s} \geq 3.4$ GeV. The discrepancy in this high-energy region remains unexplained and is described as "disconcerting." The authors state that this tension is not as critical for the total $a_\mu$ uncertainty as the $\pi^+\pi^-$ sector but represents a significant conceptual challenge to the validity of pQCD in this energy regime. They conclude that the current theory is unlikely to change significantly with unknown higher-order terms and call for further experimental investigation of $R_{uds}$ in the 3.4–3.7 GeV range to resolve the tension.

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