Muon $g$$-$$2$: correlation-induced uncertainties in precision data combinations

This paper introduces a systematic framework to quantify uncertainties arising from imperfectly known systematic correlations in data combinations, applying it to $e^+e^- \rightarrow \mathrm{hadrons}$ cross section data to demonstrate that while these correlation-induced uncertainties are generally subdominant in the muon $g-2$ hadronic vacuum polarization determination, they are non-negligible and will be incorporated into the upcoming KNTW data combination.

The Gist

Imagine you are trying to bake the perfect cake, but you have to rely on recipes from three different chefs. Each chef has measured the amount of sugar, flour, and eggs slightly differently. To get the best result, you need to combine their measurements into one "super-recipe."

However, there's a catch: the chefs didn't work in isolation. They might have used the same scale, the same oven, or the same batch of ingredients. This means their errors are correlated. If Chef A's scale was off by 1%, Chef B's scale might be off by 1% too. If you ignore this connection, your final cake might be a disaster.

This paper is about a new, smarter way to handle these "shared errors" when combining scientific data, specifically for a famous physics mystery involving the muon (a tiny, heavy cousin of the electron).

The Problem: The "Trust" Factor

In physics, scientists often combine data from different experiments to get a precise answer. To do this, they use a mathematical tool called a covariance matrix. Think of this matrix as a "trust map." It tells the computer: "If this data point is wrong, that other data point is likely wrong in the same way."

The problem is that scientists don't always know exactly how "trustworthy" these connections are.

• The Old Way: Scientists had to guess. They might say, "Let's assume these two measurements are 100% linked," or "Let's assume they are totally independent."
• The Risk: If you guess wrong about how the data is linked, your final result could be biased. It's like assuming two friends are lying together when they are actually telling the truth, or vice versa.

The Solution: The "What-If" Simulator

The authors of this paper built a systematic framework (a new set of rules) to test how much their final answer changes if they change their assumptions about these connections.

Think of it like a flight simulator for data:

1. The Baseline: They start with the best guess of how the data is connected (the "standard flight path").
2. The Stress Test: They then deliberately "break" the connections in the simulator. They ask, "What if these two points are actually totally unrelated?" or "What if the connection is only half as strong as we thought?"
3. The Measure: They use a special ruler (called a "measure of deviation") to see how much the final result wobbles when they change these connections.
4. The Result: They calculate a new "safety margin" (uncertainty) that accounts for the fact that we aren't 100% sure about the connections.

The Muon Mystery (The "Why")

Why does this matter? Because of the Muon g-2 experiment.

• Scientists have measured how much a muon "wobbles" (its magnetic moment) in a magnetic field.
• They also have a theoretical prediction of what that wobble should be, based on the Standard Model of physics.
• The Tension: The measurement and the prediction don't quite match. This mismatch could mean we have discovered new physics (a new particle or force), or it could just mean our calculations are slightly off.

To calculate the theoretical prediction, scientists need to combine data from many different experiments measuring how electrons and positrons smash together to create hadrons (particles made of quarks). This data is messy and full of correlations.

What They Found

The authors applied their new "flight simulator" to the existing data combinations used to predict the muon's behavior.

1. The "Connection" Uncertainty is Real, but Small: They found that not knowing exactly how the data points are connected does add a little bit of extra uncertainty to the final answer. It's like adding a tiny extra pinch of salt to the cake because you aren't sure if the scale was perfect.
2. It's Not the Whole Story: This new uncertainty is not big enough to explain the huge gap between the different ways scientists have been combining data.
   • Analogy: Imagine two chefs arguing about the cake. One says, "We need more sugar!" and the other says, "Less sugar!" You might think the argument is just because they are using different scales (correlations). But this paper shows that even if you fix the scales perfectly, they would still argue. The disagreement comes from something deeper—like the chefs actually measuring different ingredients or using different methods.
3. The "BaBar vs. KLOE" Mystery: For a long time, two major experiments (BaBar and KLOE) gave very different results for the most important part of the calculation. People thought this difference was just because they handled their "trust maps" (correlations) differently. This paper proves that changing the trust maps alone cannot explain the difference. The disagreement is caused by more complex issues, including how the data was processed and the statistical quirks of the experiments themselves.

The Bottom Line

This paper doesn't solve the muon mystery, but it gives scientists a better, more honest ruler to measure their uncertainty.

• Before: "We aren't sure how the data is connected, so we'll just guess and hope for the best."
• Now: "We aren't sure how the data is connected, so we ran a simulation to see how much that guess could mess things up, and we added a specific 'safety margin' to our final number."

This makes the final calculation of the muon's behavior more robust and transparent, helping physicists get closer to the truth about whether we are on the verge of discovering new laws of the universe.

Technical Summary

1. Problem Statement

The determination of the Standard Model (SM) prediction for the muon anomalous magnetic moment ($a_\mu$) relies heavily on dispersive calculations of the hadronic vacuum polarization (HVP) contribution ($a_\mu^{\text{HVP}}$). These calculations require the precise combination of experimental cross-section data for $e^+e^- \to \text{hadrons}$ ($\sigma_{\text{had}}$).

The core challenge addressed in this paper is the quantification of uncertainties arising from imperfectly known systematic correlations between data points within these combinations.

• The Issue: Systematic uncertainties are estimates rather than directly measurable quantities. Consequently, the correlation structure (how errors in one bin affect another) is often based on assumptions (e.g., assuming 100% correlation for normalization errors).
• The Consequence: Different groups (e.g., KNTW vs. DHMZ) make different assumptions about these correlations, leading to different combined cross-sections and, subsequently, different values for $a_\mu^{\text{HVP}}$.
• The Gap: Previous approaches (such as the Muon g-2 Theory Initiative 2020 White Paper) introduced "ad hoc" systematic uncertainties to account for tensions between datasets (e.g., the BaBar vs. KLOE discrepancy in the $\pi^+\pi^-$ channel) based on differences in integrated results. This method lacked a systematic framework to quantify how much of the discrepancy was actually due to correlation assumptions versus intrinsic data tensions.

2. Methodology

The authors propose a general and systematic framework to quantify "correlation-induced uncertainties" directly at the level of the combined spectrum, rather than inferring them from final observables.

A. The Deviation Measure ($M$)

To quantify the impact of changing correlation assumptions, the authors define a measure of deviation, $M$, between a baseline combined spectrum ($O_i$) and a spectrum generated under modified correlation assumptions ($\tilde{O}_i$):
$$M = \sum_{i,j} \left[ (\tilde{O}_i - O_i) (\tilde{C}_{ij} + C_{ij})^{-1} (\tilde{O}_j - O_j) \right]$$

• This measure is normalized by the covariance matrices to be dimensionless.
• It depends on the absolute values of deviations and is symmetric.
• The goal is to maximize $M$ by varying the correlation structure to find the "worst-case" deviation.

B. Decorrelation Procedures

Since scanning the full parameter space of covariance matrices is impractical, the authors introduce decorrelation functions $F_{ij}(\alpha)$ that modify the assumed covariance matrix $C_{ij}$:
$$\tilde{C}_{ij} = F_{ij}(\alpha) C_{ij}$$
Two specific procedures are tested:

1. Global Decorrelation ($\alpha_G$): Reduces all off-diagonal correlation coefficients by a constant factor $\alpha_G$. $\alpha_G=1$ is the default; $\alpha_G=0$ implies full decorrelation.
2. Local Decorrelation ($\alpha_L$): Reduces correlations based on the distance between data points (energy bins). This simulates scenarios where correlations exist only over limited energy ranges (e.g., due to changing experimental conditions).

C. Application to KNT19 Data

The framework is applied to the KNT19 combination of $\sigma_{\text{had}}$ data. The authors:

1. Apply global and local decorrelation models to the systematic covariance matrices of the input data.
2. Re-combine the data to generate modified spectra ($\tilde{\sigma}$).
3. Calculate the maximum deviation ($M$) to derive a new systematic uncertainty component, denoted as $d\rho$.
4. Propagate these new covariance matrices to derived observables: $a_\mu^{\text{HVP}}$, $a_e^{\text{HVP}}$, $a_\tau^{\text{HVP}}$, $\Delta\alpha_{\text{had}}^{(5)}(M_Z^2)$, and the muonium hyperfine splitting.

3. Key Contributions

• Systematic Framework: Introduced a rigorous, non-ad hoc method to quantify uncertainties specifically caused by the assumptions made about systematic correlations.
• Spectrum-Level Analysis: Unlike previous methods that estimated uncertainties from differences in integrated values (like $a_\mu$), this method operates at the level of the combined cross-section spectrum. This preserves local variations and tensions, ensuring consistent propagation to any derived observable.
• Disentanglement of Effects: The framework allows researchers to separate uncertainties arising from correlation modeling from intrinsic discrepancies between datasets.
• Resolution of Historical Discrepancies: The paper investigates the long-standing difference between the KNTW and DHMZ combinations. It demonstrates that varying systematic correlations alone cannot explain the full magnitude of the difference between these groups.

4. Key Results

• Magnitude of Uncertainty: The new correlation-strength uncertainties ($d\rho$) are generally subdominant but non-negligible.
  • For $a_\mu^{\text{HVP}}$, the new uncertainty contributes approximately 16.6% to the total error budget.
  • For $\Delta\alpha_{\text{had}}^{(5)}(M_Z^2)$, the contribution is higher (34.9%) because this observable is more sensitive to high-energy inclusive channels where correlations are less constrained.
• The "BaBar-KLOE" Tension:
  • The authors tested whether varying systematic correlations could reproduce the "BaBar-KLOE" systematic uncertainty ($2.8 \times 10^{-10}$) introduced in the 2020 White Paper.
  • Result: The calculated uncertainty from correlation variation ($0.45 \times 10^{-10}$) is ~6 times smaller than the ad hoc uncertainty.
  • Conclusion: The discrepancy between KNTW and DHMZ is not primarily driven by different treatments of systematic correlations.
• Role of Statistical Correlations (Appendix A):
  • The paper reveals that the difference between KNTW and DHMZ is largely driven by the treatment of statistical correlations (specifically in the BaBar data), which arise from unfolding procedures and luminosity normalization.
  • The DHMZ method effectively reduces or eliminates these statistical correlations, whereas KNTW retains them. Since statistical correlations are derived from data properties (not assumptions), the authors argue that reducing them to mimic DHMZ results is physically unjustified for uncertainty estimation.

5. Significance and Impact

• Robustness for Future Analyses: This framework will be a core component of the upcoming KNTW data combination (the successor to KNT19). It provides a transparent, reproducible way to handle correlation uncertainties.
• Clarifying the Muon g-2 Anomaly: By showing that correlation assumptions alone do not account for the spread in $a_\mu^{\text{HVP}}$ values, the paper reinforces that the tension between dispersive results and Lattice QCD results (and the experimental measurement) is likely driven by intrinsic data tensions (e.g., the CMD-3 vs. BaBar/KLOE discrepancy) rather than just methodological choices.
• General Applicability: While demonstrated on $e^+e^- \to \text{hadrons}$, the methodology is broadly applicable to any precision measurement involving correlated data combinations, offering a new standard for quantifying systematic uncertainties in high-energy physics.

In summary, the paper provides a critical tool for precision physics, moving beyond "fudge factors" for data tensions to a rigorous quantification of how correlation modeling choices impact final results, ultimately clarifying the sources of uncertainty in the muon $g-2$ calculation.