Dispersive analysis of the $\pi^+ \pi^-$ production at the CMD3 experiment and the compatibility with muon pair production measurement by KLOE2 and the pion form factor by JLAB

This paper analyzes the compatibility of CMD3's $\pi^+ \pi^-$ production data with KLOE2's muon pair production and JLab's spacelike pion form factor measurements, finding that while CMD3 data shows no excess in analytical continuation, it remains largely incompatible with other recent measurements and the JLab results, impacting the derived QED running charge and hadronic vacuum polarization.

The Gist

Imagine the universe as a giant, complex machine, and physicists are the mechanics trying to understand how it runs. One of the most important parts of this machine is a tiny particle called the muon. Scientists have been trying to predict exactly how this muon spins and wobbles (its "magnetic moment") using math. But when they compare their math to real-world experiments, there's a tiny but stubborn mismatch.

To fix the math, they need to account for a "cloud" of virtual particles that pop in and out of existence around the muon. This cloud is called Hadronic Vacuum Polarization (HVP). Think of HVP like a foggy atmosphere around the muon. To measure the muon correctly, you have to know exactly how thick and dense that fog is.

The main way to measure this "fog" is to look at how electrons and positrons (matter and antimatter) smash together to create pairs of pions (another type of particle). This is like watching two cars crash to see what parts fly off, which tells you about the engine inside.

The Problem: The "CMD3" Discrepancy

For a long time, different labs (like BABAR, KLOE, BESIII) have been measuring these pion crashes. They mostly agreed, but recently, a new lab in Russia called CMD3 started measuring. Their results were like a loud shout in a quiet library: they were significantly different from everyone else, especially in a specific energy range called the "rho resonance" (think of this as a specific speed where the crash produces a lot of noise).

This created a crisis: Which data is right?

• If CMD3 is right, the "fog" (HVP) is different than we thought.
• If CMD3 is wrong (due to a hidden error in their equipment), then the old data is correct.

The Detective Work: Two Different Tests

The authors of this paper decided to play detective. They didn't just look at the pion crash data; they asked: "If we use the CMD3 data, does it break other parts of physics?" They ran two specific tests, like checking if a suspect's alibi holds up under scrutiny.

Test 1: The Time Travel Test (Spacelike vs. Timelike)

Imagine you have a map of a city drawn from a helicopter (the "timelike" data from the collider). You want to predict what the streets look like from a ground-level view (the "spacelike" data from a different experiment at Jefferson Lab).

• The Math: They used a mathematical rule called a Dispersion Relation. Think of this as a universal translator that converts the "helicopter map" into a "ground-level map."
• The Result: They translated the data with CMD3 and without CMD3. Surprisingly, both versions translated into a ground-level map that looked almost identical to the actual ground-level photos taken by Jefferson Lab.
• The Twist: Even though the CMD3 data looked weird in the crash zone, it didn't break the translation. However, when they looked closely, the version with CMD3 actually fit the Jefferson Lab photos slightly better. This suggests the CMD3 data might not be as "wrong" as it seemed, or at least, it doesn't create a contradiction with this other experiment.

Test 2: The Running Charge Test (The KLOE2 Check)

There is another famous experiment called KLOE2 that measures how the "strength" of electricity (the fine-structure constant) changes at different energies. This strength is directly affected by the "fog" (HVP).

• The Analogy: Imagine the "fog" (HVP) acts like a filter on a camera lens. If the fog is thick, the picture looks different. KLOE2 took a picture of the universe at a specific energy.
• The Result: The authors calculated what the picture should look like using the CMD3 data and compared it to the actual KLOE2 photo.
• The Verdict: The difference between the "CMD3 version" and the "Old Data version" was so tiny that the KLOE2 camera wasn't powerful enough to see it. It's like trying to tell the difference between two shades of blue using a black-and-white camera. The KLOE2 experiment needs to be about 10 times more precise to detect if the CMD3 data is actually causing a problem.

The Conclusion: A Calm After the Storm

So, what did they find?

1. The CMD3 data is weird: It definitely disagrees with other experiments in the raw crash data.
2. But it's not a disaster: When they used that weird data to predict other things (like the Jefferson Lab results or the KLOE2 results), it didn't cause a collapse. The math still worked.
3. The Mystery Remains: The fact that the CMD3 data is an "outlier" (a weird data point) but doesn't break other theories is strange. It suggests there might be a hidden systematic error in the CMD3 experiment that we haven't found yet, OR that our understanding of the "fog" is more complex than we thought.

In simple terms: The paper is like a mechanic saying, "This new part (CMD3 data) looks broken compared to the old ones. But if I install it, the car still drives fine and passes the safety inspection. So, either the part isn't actually broken, or the safety inspector (KLOE2) isn't sensitive enough to see the problem yet. We need a better inspector to solve the mystery."

The authors also fixed a small math error they made in a previous attempt, ensuring their current calculations are solid. They conclude that while the CMD3 data is an outlier, it doesn't currently create a major conflict between theory and experiment, but it definitely keeps the mystery of the muon's magnetic moment alive and waiting for more precise measurements.

Technical Summary

1. Problem Statement

The precise determination of the Hadronic Vacuum Polarization (HVP) is critical for comparing Standard Model theory with experimental results, particularly for the muon anomalous magnetic moment ($a_\mu$). A major source of systematic uncertainty arises from the $e^+e^- \to \pi^+\pi^-$ channel.

• The Conflict: Recent data from the CMD3 experiment (Novosibirsk) regarding the charged pion pair production cross-section ($\sigma_{\pi\pi}$) shows a significant deviation (exceeding 10 standard deviations at the $\rho$-meson peak) compared to other experiments like BABAR, BESIII, and KLOE.
• The Question: Does this incompatibility in the timelike region (production data) translate into observable discrepancies in related physical quantities? Specifically:
  1. Does it affect the pion electromagnetic form factor ($F_\pi$) when analytically continued to the spacelike region (compared to JLab data)?
  2. Does it alter the QED running coupling constant ($\alpha(s)$) derived from HVP, compared to KLOE2 measurements?

2. Methodology

The authors employ a dispersive analysis framework to extract spectral functions and calculate observables.

• Data Sets: Two distinct datasets were constructed:
  1. World Data (W): Combined data from KLOE, CMD/SND, BESIII, and BABAR, excluding CMD3.
  2. CMD3+ Data: The same world data including the CMD3 dataset.
• Spectral Function Extraction:
  • The authors extract the spectral function ($\rho_F$) of the pion form factor using an analyticized Gounaris-Sakurai (GS) model.
  • The fit includes standard resonances ($\rho, \omega, \phi$) and higher excitations. Notably, the model allows for "ghost" states (negative couplings) to account for complex non-perturbative structures and unitarity constraints.
  • Inflated Error (IE) Method: To handle the large systematic uncertainties and mutual incompatibilities between experiments, the authors introduce a "semi-stochastic inflated error" ($\bar{\sigma}_I$). This parameter is defined such that the $\chi^2$ of the fit equals 1, effectively treating unknown systematic errors as additional statistical noise.
• Analytical Continuation: Using the extracted spectral functions and the dispersion relation, the form factor is analytically continued from the timelike region ($q^2 > 0$) to the spacelike region ($q^2 < 0$).
• HVP and Running Coupling Calculation:
  • The HVP function $\Pi_h(s)$ is calculated via a dispersion integral over the total hadronic cross-section ($\sigma_h$).
  • The QED running coupling $\alpha(s)$ is derived from $\Pi_h(s)$ and compared against KLOE2 measurements of muon pair production ($\sigma_{\mu\mu}$).
• Correction: The authors correct a numerical error identified in their previous work (arXiv:1708.03616) regarding the "Holinde trick" used for principal value integration, which previously affected the spectral function near the threshold.

3. Key Contributions

• Resolution of CMD3 Incompatibility: The paper demonstrates that while CMD3 data is an outlier in the timelike cross-section, its inclusion in the dispersive analysis does not create a statistically significant tension with spacelike measurements (JLab) or KLOE2 running coupling data.
• Inflated Error Formalism: The authors propose and apply a robust statistical method (Inflated Error) to combine datasets with unknown systematic errors, allowing for a unified analysis without discarding high-quality data or over-weighting outliers.
• Correction of Numerical Artifacts: The paper rectifies a specific numerical error in the evaluation of the pion form factor near the threshold, ensuring higher precision in the spectral function extraction.
• Asymptotic Behavior Analysis: The study investigates the behavior of the pion form factor at high $Q^2$ (spacelike) and compares it with Perturbative QCD (pQCD) asymptotics.

4. Results

• Spacelike Pion Form Factor:
  • Both the CMD3-inclusive and CMD3-exclusive spectral functions yield form factors that are compatible with JLab experimental data.
  • Surprisingly, the CMD3-driven result is slightly closer to the JLab measurements than the world-average (without CMD3) result.
  • Asymptotics: The extracted form factors do not follow the naive pQCD asymptotic behavior ($F \sim 1/Q^2$) up to the TeV scale. The data suggests a deviation from simple pQCD predictions, and the JLab data hints at a constant behavior for $Q^2 F(Q^2)$ in the few GeV region.
• QED Running Coupling ($\alpha(s)$):
  • The calculated running coupling, derived from HVP using both datasets, shows no significant difference within the experimental errors of the KLOE2 measurement.
  • The incompatibility between the CMD3 and non-CMD3 datasets is too small to be detected by current KLOE2 precision. The authors estimate that a precision improvement of at least 10 times over KLOE2 would be required to distinguish between the two scenarios.
• Fit Quality:
  • The CMD3 data-driven fit achieves a $\chi^2 \approx 1$, indicating that the dispersion relation is well-satisfied even with the outlier data, suggesting no fundamental distortion of analyticity due to systematic errors in CMD3.

5. Significance

• Muon $g-2$ Context: The results suggest that the tension in $a_\mu$ calculations is not solely resolvable by simply switching between CMD3 and other datasets, as the impact on the running coupling and spacelike form factor is minimal within current experimental precision.
• Methodological Advance: The "Inflated Error" approach provides a practical tool for future global analyses of hadronic cross-sections where systematic uncertainties are dominant and poorly defined.
• Future Directions: The study highlights that resolving the CMD3 discrepancy requires either a re-evaluation of the CMD3 systematic errors or significantly higher precision in spacelike form factor measurements (JLab) and running coupling measurements (KLOE2 or future facilities).
• Theoretical Insight: The lack of agreement with naive pQCD asymptotics in the extracted form factors reinforces the complexity of non-perturbative QCD dynamics in the low-to-intermediate energy regime.

In conclusion, the paper argues that the CMD3 anomaly, while striking in the raw cross-section data, does not currently propagate into a detectable contradiction in other fundamental observables, implying that the "truth" of the HVP contribution to the muon $g-2$ remains elusive without further experimental refinement.