Higher-order hadronic vacuum polarization contribution to the muon $g-2$ from lattice QCD

This paper presents the first sub-percent precision lattice QCD calculation of the next-to-leading order hadronic vacuum polarization contribution to the muon anomalous magnetic moment, yielding a result of $-101.69(25)_{\mathrm{stat}}(53)_{\mathrm{syst}}\times10^{-11}$ that is twice as precise as the 2025 White Paper estimate and shows a significant 4.8$\sigma$ tension with data-driven evaluations excluding the CMD-3 measurement.

The Gist

Imagine you are trying to weigh a feather with a scale that is incredibly sensitive, but the scale itself has a tiny, mysterious wobble. If the wobble is bigger than the feather, you can't trust the measurement.

This is essentially the situation physicists are in with the Muon, a subatomic particle that acts like a heavy, unstable cousin of the electron. Scientists have measured how much the muon "wobbles" (its magnetic moment) with incredible precision. However, when they try to predict that wobble using the Standard Model (our best rulebook for how the universe works), the numbers don't quite match up. The difference is tiny, but in the world of particle physics, it's a screaming alarm bell suggesting we might be missing a piece of the puzzle.

The biggest source of uncertainty in that prediction comes from something called Hadronic Vacuum Polarization (HVP). Think of the vacuum of space not as empty, but as a bubbling soup of virtual particles popping in and out of existence. When a muon moves through this soup, it gets jostled by these virtual particles, changing its wobble.

The Problem: Two Different Maps

To calculate this "jostling," physicists have two main ways to make a map:

1. The Data-Driven Map: They look at real-world experiments (colliding particles and measuring the debris) to guess how the soup behaves.
2. The Lattice QCD Map: They use supercomputers to simulate the soup from first principles, calculating it directly from the laws of quantum physics without needing experimental data.

For a long time, these two maps disagreed. The "Data-Driven" map suggested one value, while early "Lattice" maps suggested another. This disagreement made it hard to know if the muon's wobble was truly a sign of new physics or just a mistake in our calculations.

The New Breakthrough: A Sub-Percent Precision Map

This paper presents a major leap forward. A team of researchers (from Mainz, CERN, and Bonn) has created a brand new, ultra-precise Lattice QCD calculation for the next level of complexity in this problem.

Previously, they could only calculate the "main course" (the leading order). But to be sure, you also need to calculate the "side dishes" (the next-to-leading order, or NLO). These side dishes are much harder to calculate because they involve more complex interactions, like virtual photons and other particles interfering with each other.

Here is the analogy:
Imagine you are trying to calculate the total cost of a dinner party.

• Leading Order (LO): You calculate the cost of the main steak.
• Next-to-Leading Order (NLO): You calculate the cost of the wine, the appetizers, and the tax.

In the past, calculating the wine and tax was so messy and imprecise that it didn't matter much. But now, this team has calculated the "wine and tax" with sub-percent precision (better than 1% error). They did this by:

• Using a massive grid (a "lattice") to simulate space-time.
• Running simulations on 35 different "ensembles" (different versions of the grid with varying sizes and densities).
• Using a clever mathematical trick called the Time-Momentum Representation, which acts like a filter to separate the short-distance (high energy) noise from the long-distance (low energy) signal.

The "Magic Cancellation"

The most beautiful part of this work is a "happy accident" they discovered.
The calculation involves two main groups of diagrams (let's call them Team A and Team B).

• Team A pushes the muon's wobble in one direction (negative).
• Team B pushes it in the opposite direction (positive).

Usually, when you add two big numbers with opposite signs, you get a messy result with huge errors. But in this specific case, Team A and Team B cancel each other out almost perfectly at long distances. It's like two people pushing a heavy boulder from opposite sides with equal force; the boulder doesn't move, but the effort required to keep it still is very small and easy to measure.

Because of this cancellation, the "noise" that usually plagues these calculations disappears. This allowed the team to get a result that is twice as precise as the previous best estimate based on experimental data.

The Result: A Clearer Picture

Their final number for this "side dish" contribution is −101.69 (in very small units).

• This number is slightly lower than the previous "Data-Driven" estimate.
• It creates a tension (a disagreement) of nearly 5 standard deviations with the older data-driven results (which didn't include a recent, controversial experiment called CMD-3).

What does this mean?
It suggests that the "Data-Driven" maps might be flawed, or at least incomplete. The "Lattice" map, which is built from the ground up using pure math and supercomputers, is now so precise that it is starting to stand on its own.

Why Should You Care?

If the Standard Model is a perfect rulebook, the muon's wobble should match the prediction perfectly. If it doesn't, it means there is a new, undiscovered particle or force in the universe (like a "dark matter" particle) that is tugging on the muon.

By calculating the "side dishes" (NLO) with such high precision, this team has removed a major source of doubt. They have effectively said: "We have checked our math on the complex parts, and it is solid. The discrepancy we see is real."

This doesn't prove new physics exists yet, but it clears the fog. It tells us that if the muon is still wobbly, it's likely because the universe is stranger than we thought, not because our calculators are broken. It's a crucial step toward potentially rewriting the laws of physics.

Technical Summary

Here is a detailed technical summary of the paper "Higher-order hadronic vacuum polarization contribution to the muon g−2 from lattice QCD" (MITP-26-006).

1. Problem Statement

The muon anomalous magnetic moment ($a_\mu$) is a critical precision observable for testing the Standard Model (SM). While the experimental measurement by the Fermilab E989 experiment has reached a precision of 124 ppb, the theoretical prediction remains limited by uncertainties in the Hadronic Vacuum Polarization (HVP) contribution.

• Leading Order (LO): The LO HVP contribution ($a_\mu^{\text{hvp, LO}}$) dominates the uncertainty. Recent lattice QCD calculations and data-driven dispersive methods show significant tensions, particularly regarding the two-pion contribution.
• Next-to-Leading Order (NLO): The NLO HVP contribution ($a_\mu^{\text{hvp, NLO}}$) is a sub-dominant term of order $\alpha^3$. Currently, the SM prediction for this term relies entirely on data-driven dispersive methods (using experimental cross-section data).
• The Gap: To ensure consistency in the SM prediction, especially given the tensions in the LO sector, it is essential to calculate the NLO contribution using the same first-principles methodology (Lattice QCD) as the LO term. Prior to this work, lattice calculations of the NLO term were exploratory with limited precision (~13%).

2. Methodology

The authors present the first lattice QCD calculation of the NLO HVP contribution with sub-percent precision. The methodology involves several sophisticated components:

A. Theoretical Framework: Time-Momentum Representation (TMR)

The calculation utilizes the TMR formalism, expressing the HVP contribution as an integral over the convolution of a time-kernel function $\tilde{f}$ and the zero-momentum projected electromagnetic correlator $G(t)$.

• Kernel Functions: The authors derived analytic solutions and polynomial expansions for the NLO time-kernels. Specifically, they provided a closed-form solution for the NLOc diagram (two QCD insertions) and polynomial expansions for NLOa (photon lines/muon loops) and NLOb (electron/tau loops).
• Diagram Sets: The NLO contribution is decomposed into three topological sets:
  • NLOa: Corrections from photon lines and muon loops (dominant negative contribution).
  • NLOb: Corrections from electron and tau loops (large positive contribution).
  • NLOc: A single diagram with two QCD insertions (subleading).

B. Lattice Setup

• Ensembles: Calculations were performed on 35 gauge ensembles generated by the Coordinated Lattice Simulations (CLS) effort.
• Fermions: $N_f = 2+1$ flavors of $O(a)$-improved Wilson fermions.
• Parameters: The ensembles cover six lattice spacings ($a \in [0.039, 0.097]$ fm) and a range of pion masses including the physical point.
• Currents: Two discretizations of the electromagnetic current were used (local and conserved) with two different non-perturbative improvement sets to control systematic uncertainties.

C. Analysis Strategy

• Window Observables: To control systematic effects, the integration domain is split into three time windows:
  • Short-Distance (SD): $t < 0.4$ fm. Dominated by high-energy physics; sensitive to lattice artifacts. The authors employed a subtraction technique to remove logarithmically enhanced cutoff effects ($O(a^2 \ln a)$ and $O(a^2 \ln^2 a)$) by redefining the kernel and computing the subtracted piece perturbatively.
  • Intermediate-Distance (ID): $0.4 < t < 1.0$ fm.
  • Long-Distance (LD): $t > 1.0$ fm. Dominated by low-energy physics; sensitive to finite-volume effects and signal-to-noise ratios.
• Cancellation Exploitation: A key feature of the analysis is the strong numerical cancellation between NLOa and NLOb contributions at large Euclidean times. By analyzing the combined NLOa&b contribution, the authors significantly reduced the sensitivity to statistical noise and finite-volume corrections in the LD window.
• Extrapolation: A global chiral-continuum extrapolation was performed using model averaging (Akaike Information Criterion) to handle higher-order lattice spacing and chiral mass dependencies.
• Corrections:
  • Finite-Volume (FV): Corrected using the Hansen-Patella formalism and Meyer-Lellouch-Lüscher method.
  • Isospin Breaking (IB): Electromagnetic and strong isospin-breaking effects were estimated using phenomenological models (Vector Meson Dominance) and spacelike representations, as direct lattice computation of NLO IB kernels was not feasible.
  • Subleading Terms: Contributions from bottom quarks, charm quark-disconnected diagrams, and tau loops were estimated and included.

3. Key Contributions

1. First High-Precision Lattice NLO Calculation: This is the first lattice determination of $a_\mu^{\text{hvp, NLO}}$ with sub-percent precision, moving beyond previous exploratory studies.
2. Analytic Kernel Solutions: The paper provides a closed-form analytic solution for the NLOc time-kernel and detailed polynomial expansions for NLOa and NLOb, facilitating future lattice calculations.
3. Systematic Control: The work introduces a robust subtraction method for logarithmically enhanced cutoff effects in the short-distance window and leverages the NLOa/NLOb cancellation to mitigate long-distance noise.
4. Consistency Check: The result provides a first-principles check of the NLO contribution, independent of experimental cross-section data, which is crucial given the tensions in the LO sector.

4. Results

The final continuum-extrapolated result for the NLO HVP contribution is:
$$a_\mu^{\text{hvp, nlo}} = -101.69(25)_{\text{stat}}(53)_{\text{syst}} \times 10^{-11}$$

• Total Precision: 0.6% (relative error).
• Breakdown:
  • NLOa: $-217.19(80)(85)... \times 10^{-11}$
  • NLOb: $+111.76(55)(62)... \times 10^{-11}$
  • NLOc: $+3.76(6)(6)... \times 10^{-11}$
  • Isospin Breaking correction: $+0.06(27) \times 10^{-11}$

Comparisons:

• vs. 2025 White Paper (WP25): The result is $1.5\sigma$below the WP25 estimate of$(-99.6 \pm 1.3) \times 10^{-11}$ but is two times more precise.
• vs. Data-Driven (Keshavarzi et al. 2019): The result exhibits a 4.8$\sigma$ tension with data-driven evaluations that exclude the recent CMD-3 measurement. This pattern of deviation mirrors the tension observed between lattice and data-driven methods for the LO HVP contribution.

5. Significance

• Independent Validation: The result offers an independent, first-principles verification of the NLO HVP contribution, free from the inconsistencies among experimental datasets that plague the dispersive method.
• Impact on SM Prediction: While the NLO term is smaller than the LO term, its high precision (0.6%) ensures it will not be a limiting factor as LO lattice calculations improve. The shift between lattice and data-driven NLO results suggests that the tension in the muon $g-2$ anomaly may be more complex than previously thought, potentially involving issues in the experimental input data used for dispersive integrals.
• Future Outlook: The authors provide dimensionless scheme dependencies for all observables, allowing the community to update these results as fundamental QCD scales (like $t_0$) are refined without re-running the lattice simulations. This work paves the way for a fully consistent lattice-based SM prediction for $a_\mu$.