Comparison of the hadronic vacuum polarization between hadronic $\tau$-decay data and lattice QCD

This paper compares isospin-symmetric lattice QCD calculations of hadronic vacuum polarization with dispersive results derived from corrected hadronic $\tau$-decay data, finding generally good agreement overall but revealing significant discrepancies in the $2\pi^-\pi^+\pi^0$ four-pion mode when evaluated against expectations from Pais relations and $e^+e^-$ cross sections.

The Gist

The Big Picture: Measuring a Tiny Wobble

Imagine the universe is a giant, complex machine. One of its most famous parts is the muon, a particle that acts like a tiny, spinning top. Scientists have measured how this top wobbles (its "anomalous magnetic moment") with incredible precision.

However, to predict exactly how much it should wobble based on our current rules of physics (the Standard Model), scientists need to account for a "fog" of virtual particles popping in and out of existence around the muon. This fog is called Hadronic Vacuum Polarization (HVP).

The problem is that calculating this fog is incredibly hard. There are two main ways scientists try to measure it:

1. The "Lattice" Method: Using supercomputers to simulate the laws of physics from scratch (like building a digital model of the fog).
2. The "Data" Method: Looking at real-world experiments where particles smash together to create this fog, then measuring the results.

For a long time, these two methods disagreed. The "Lattice" results and the "Data" results didn't match, creating a mystery in physics.

The New Experiment: Using a Different Camera

This paper tries to solve the mystery by using a different type of "camera" for the Data Method.

Usually, scientists look at data from electron-positron collisions (smashing an electron and a positron together). But this paper uses data from tau decays.

• The Analogy: Imagine you are trying to measure the shape of a specific type of cloud.
  • Method A (Electron collisions): You look at the cloud through a telescope that sometimes gets a little static interference (called "isospin breaking").
  • Method B (Tau decays): You look at the cloud through a different telescope that sees a slightly different angle.
  • The Goal: The authors take the "Tau" data, clean it up to remove the static (correcting for differences in physics between the two methods), and compare it to the "Lattice" computer simulation.

What They Did

The authors took a massive amount of data from tau particle decays (a heavy cousin of the electron). They focused on how these particles break apart into smaller pieces (like pions).

1. Cleaning the Data: The tau data isn't perfect; it has tiny differences compared to the ideal "pure" physics world used in the computer simulations. The authors built a mathematical "filter" to correct these differences, essentially translating the tau data into the language of the computer simulation.
2. The Comparison: They compared this cleaned-up tau data against the results from the Mainz and BMW supercomputer groups (the Lattice teams).

The Results: Good News and a Weird Glitch

1. The Good News (General Agreement)
For the most part, the two methods agreed very well.

• The Analogy: It's like two different weather stations measuring the temperature. Even though they use different thermometers, they both say it's 72°F.
• The Finding: When they looked at the total "fog" (the contribution to the muon's wobble) and the "middle-distance" parts of it, the tau-based data and the lattice computer simulations matched up nicely. This suggests that the computer simulations are likely correct and that the previous disagreements might have been due to issues with the electron-positron data, not the computer models.

2. The Weird Glitch (The Four-Pion Problem)
However, they found a specific spot where the data didn't match the rules of the universe.

• The Analogy: Imagine you are baking a cake. You have a recipe (the "Pais relations") that says if you mix 4 eggs and 2 cups of flour, you get a specific result.
  • When they looked at a specific type of cake (the 2π−π+π0 mode, or a specific way four particles break apart), the "Tau" data said the cake was one size, but the "Electron" data said it was a different size.
  • The authors checked this against the "recipe" (theoretical rules) and found a significant difference. The tau data for this specific four-particle combination didn't line up with what the electron data and the theoretical rules predicted.

The Conclusion

• Overall: The paper finds that when you use tau decay data (corrected properly), it agrees very well with lattice QCD (the supercomputer simulations). This supports the idea that the supercomputer results are likely the correct ones.
• The Caveat: There is a specific, complex part of the data (involving four particles breaking apart in a specific way) where the tau data and the electron data disagree significantly. This suggests there might be a problem with how we measure or understand that specific part of the particle breakdown, but it doesn't ruin the overall agreement for the main calculation.

In short: The authors used a new type of data (tau decays) to check the computer simulations. The check passed for the big picture, confirming the computer models, but it highlighted a specific, confusing detail in the data that still needs to be figured out.

Technical Summary

Technical Summary: Comparison of the Hadronic Vacuum Polarization between Hadronic $\tau$-Decay Data and Lattice QCD

Problem Statement
The determination of the Standard Model (SM) expectation for the muon anomalous magnetic moment, $a_\mu$, is currently limited by the uncertainty in the hadronic vacuum polarization (HVP) contribution, $a_\mu^{\text{HVP}}$. A significant discrepancy exists between lattice QCD calculations of the light-quark-connected (lqc) HVP contribution and dispersive estimates derived from the $R$-ratio measured in $e^+e^- \to \text{hadrons}(\gamma)$ processes. This tension is primarily attributed to the exclusive $e^+e^- \to \pi^+\pi^-$ channel around the $\rho$ resonance. Since the isospin-1 ($I=1$) component of the HVP is directly related to the lqc contribution in the isospin limit, and since $I=1$ spectral functions can also be accessed via non-strange hadronic $\tau$ decays (after correcting for isospin breaking), this paper investigates whether $\tau$-decay data can provide an independent dispersive determination that aligns with lattice QCD results.

Methodology
The authors construct the isospin-1, vector-current subtracted vacuum polarization, $\Pi^{I=1}(Q^2)$, and the associated Euclidean time correlation function, $C(t)$, using a dispersive representation. The spectral function, $\rho_V^{I=1}(s)$, is constructed from three components:

1. Data Region ($s \leq m_\tau^2$): The inclusive vector isovector spectral function is derived from non-strange hadronic $\tau$ decays. The authors utilize a combination of data from ALEPH, OPAL, and Belle experiments, as compiled in Ref. [19]. This includes exclusive contributions from two-pion ($\pi^-\pi^0$), four-pion ($2\pi^-\pi^+\pi^0$ and $\pi^-3\pi^0$), and residual modes.
2. Perturbative Region ($s > m_\tau^2$): For invariant masses above the $\tau$ mass, where experimental data is unavailable, the spectral function is modeled using five-loop massless perturbative QCD (PT) with an estimated fifth-order coefficient.
3. Duality Violations (DV): A phenomenological model is employed to account for quark-hadron duality violations in the region above $m_\tau^2$.

To enable a direct comparison with lattice QCD results computed in "isoQCD" (isospin-symmetric QCD without electromagnetic corrections), the $\tau$-based spectral data undergoes a rigorous isospin-breaking (IB) correction. The authors assume IB effects are dominated by the two-pion mode. The correction factor $F_{\text{IB}}(s)$ accounts for:

• Differences in short-distance electroweak factors ($S_{\text{EW}}$ vs. $S_{\pi\pi}^{\text{EW}}$).
• Radiative corrections ($G_{\text{EM}}(s)$) derived from dispersive analyses.
• Phase-space adjustments due to pion mass differences ($m_{\pi^\pm} \neq m_{\pi^0}$).
• Isospin-breaking effects in the pion form factor $f_+(s)$, modeled using a chiral perturbation theory (ChPT)-inspired approach.

The study evaluates several quantities:

• The subtracted vacuum polarization $\Pi(Q^2)$ for $Q^2 \in [0.5, 12] \text{ GeV}^2$.
• The total lqc contribution to $a_\mu$, denoted $a_\mu^{\text{HVP, lqc}}$.
• The short-distance (SD), intermediate (W1), and long-distance (LD) window quantities defined by RBC/UKQCD.

Additionally, the authors perform alternative analyses where $\tau$ data is used only for specific exclusive modes (2$\pi$ or 2$\pi$+4$\pi$), while other contributions are taken from the KNT19 $e^+e^-$ compilation, to isolate the impact of specific channel discrepancies.

Key Contributions and Results

1. General Agreement: The paper finds generally good agreement between the $\tau$-based dispersive results and lattice QCD calculations for $a_\mu^{\text{HVP, lqc}}$ and the window quantities. The $\tau$-based results lie within $1\sigma$ of the lattice averages (specifically the WP25 averages from Ref. [3]) for the total HVP and the LD window, though the SD and W1 windows show slightly larger relative differences (up to $1.0\sigma$).
2. $\Pi(Q^2)$ Tension: For the subtracted vacuum polarization $\Pi(Q^2)$, the $\tau$-based results are systematically lower than the Mainz lattice results (Ref. [14]) by $2.1\sigma$ to $3.0\sigma$ across the range $Q^2 = 0.5$ to $12 \text{ GeV}^2$. However, these results show better agreement with the BMW lattice results (Ref. [15]).
3. Four-Pion Mode Discrepancies: A detailed comparison of the four-pion spectral distributions reveals significant tensions between $\tau$-based data and expectations derived from $e^+e^-$ data via Pais relations. Specifically, the $2\pi^-\pi^+\pi^0$ mode shows discrepancies ranging from $3.3\sigma$ (SD window) to $4.4\sigma$ (LD window) when comparing ALEPH $\tau$ data to KNT19 $e^+e^-$ combinations. The $\pi^-3\pi^0$ mode shows good agreement.
4. Impact of Mode Replacement: When the $\tau$-based 2$\pi$ spectral distribution is used to replace the pre-CMD-3 KNT19 $e^+e^-$ 2$\pi$ input, the resulting data-driven lqc contributions align well with lattice results. However, when the $\tau$-based 4$\pi$ data (specifically the $2\pi^-\pi^+\pi^0$ mode) is also used to replace the KNT19 input, the results remain compatible with lattice averages within errors, but the strong correlation between the 2$\pi$ and 4$\pi$ contributions leads to a $3.2\sigma$ tension between the "2$\pi$-only $\tau$" and "2$\pi$+4$\pi$ $\tau$" strategies for the LD window.
5. Error Budget: The authors note that for lower $Q^2$, the data-driven part of $\Pi(Q^2)$ dominates the error budget (approx. 91% at $Q^2=0.5 \text{ GeV}^2$), making the $\tau$-based errors competitive with lattice errors.

Significance
The paper demonstrates that using inclusive hadronic $\tau$-decay data, corrected for isospin breaking, provides a dispersive determination of the HVP that is largely consistent with recent lattice QCD calculations. This supports the hypothesis that the discrepancy between lattice and $e^+e^-$-based HVP estimates may stem from issues specific to the $e^+e^-$ data (particularly the $\pi^+\pi^-$ channel) rather than a fundamental failure of the lattice method or the dispersive approach itself.

However, the authors highlight that significant discrepancies persist in the four-pion channels between $\tau$ data and $e^+e^-$ expectations derived from Pais relations. While these discrepancies do not currently invalidate the overall agreement between the total $\tau$-based and lattice HVP values, they suggest that the four-pion sector requires further investigation. The work underscores the utility of $\tau$-decay data as an independent cross-check for the HVP contribution to $a_\mu$, particularly in the context of resolving the muon $g-2$ anomaly. The authors remain modest, noting that the systematic uncertainties in modeling isospin-breaking effects (particularly in the pion form factor) and the observed tensions in the four-pion modes prevent a definitive resolution of all discrepancies at this stage.