Lattice Determination of the Anomalous Magnetic Moment of the Muon

Abstract: We compute the leading hadronic contribution to the anomalous magnetic moment of the muon a_mu^HLO using two dynamical flavours of non-perturbatively O(a) improved Wilson fermions. By applying partially twisted boundary conditions we are able to improve the momentum resolution of the vacuum polarisation, an important ingredient for the determination of the leading hadronic contribution. We check systematic uncertainties by studying several ensembles, which allows us to discuss finite size effects and lattice artefacts. The chiral behavior of a_mu^HLO turns out to be non-trivial, especially for small pion masses.

• The paper computes the leading hadronic vacuum polarization contribution to muon g‑2 using lattice QCD with CLS ensembles and partially twisted boundary conditions.
• The methodology employs current‑current correlators with multiple fit ansätze and NLO chiral perturbation theory estimates to control systematic uncertainties at around 12%.
• The results underscore the importance of chiral extrapolation and a partially quenched strange quark in refining Standard Model predictions and probing new physics.

Lattice Calculation of the Muon Anomalous Magnetic Moment: Leading Hadronic Contribution

Introduction

The computation of the muon's anomalous magnetic moment $a_\mu$, defined as $(g_\mu - 2)/2$, represents a stringent test of the Standard Model (SM). The persistent $3.2\sigma$ discrepancy between the experimental value and SM predictions primarily stems from uncertainties in the leading-order hadronic vacuum polarization (HVP) contribution, $a_\mu^\mathrm{HLO}$. Current SM estimates typically employ $e^+e^- \rightarrow \mathrm{hadrons}$ cross-section data together with the optical theorem, but these methods are susceptible to experimental and model-dependent uncertainties. A first-principles calculation using lattice QCD for $a_\mu^\mathrm{HLO}$ is therefore highly desirable, as it could enable reduction of theoretical uncertainties and scrutiny of potential physics beyond the SM.

Methodology

The calculation utilizes ensembles generated as part of the CLS project, with two dynamical flavors of non-perturbatively $\mathcal{O}(a)$ improved Wilson fermions and includes a partially quenched strange quark. Partially twisted boundary conditions are applied, following [14], to yield high-resolution sampling in momentum, particularly for low $q^2$ where the kernel of the defining integral for $a_\mu^\mathrm{HLO}$ peaks. Disconnected diagrams are neglected due to computational demands but are estimated using NLO chiral perturbation theory to contribute approximately $-10\%$ relative to the connected part [12].

The computation proceeds by evaluating the vacuum polarization $\Pi(q^2)$ and its subtraction $\hat\Pi(q^2) = 4\pi(\Pi(q^2) - \Pi(0))$, extracted from current-current correlators with twisted and periodic boundary conditions. To facilitate numerical integration and ensure control over the functional form of $\Pi(q^2)$, correlated least-squares fits are performed using four fit ansätze: Padé approximants, single and double vector dominance models (with variable/fixed vector masses), and matching to perturbative QCD at high momenta as per [16]. The fits are constrained both by non-perturbative determinations of renormalization factors (see [17]–[20]) and by enforcing smooth matching of fit and perturbative forms (including their derivatives) at high $q^2$.

Results

The use of twisted boundary conditions provides significantly improved momentum resolution, as shown by the density of low-$q^2$ data critical for the accurate determination of $a_\mu^\mathrm{HLO}$. All physically motivated fit ansätze yield statistically compatible results above $q^2 \gtrsim 1\,\mathrm{GeV}^2$, suggesting that systematic uncertainty from fit selection is subdominant in this regime.

Chiral extrapolation, crucial for obtaining phenomenologically relevant results, is performed with both linear and logarithmic components, motivated by chiral perturbation theory. A pronounced nonlinearity is observed for $m_\pi^2 \lesssim 0.2\,\mathrm{GeV}^2$, reinforcing the necessity of a controlled extrapolation and the inclusion of lighter pion ensembles in future studies.

The resulting values for the leading-order hadronic muon anomalous magnetic moment are:

• $$a_\mu^\mathrm{HLO} (N_f=2) = 537.1\,(53.8)_{\rm stat}\, (37.6)_{\rm chiral} \times 10^{-10}$$
• $$a_\mu^\mathrm{HLO} (N_f=2+1_Q) = 612.4\,(49.9)_{\rm stat}\, (48.5)_{\rm chiral} \times 10^{-10}$$

Statistical errors for individual ensembles range from 2% to 7%. The combined statistical, chiral, and systematic uncertainties yield a total error of approximately 12%, with dominant contributions from chiral extrapolation and statistical sampling in the low-$q^2$ regime. Cutoff and finite volume effects on $\hat\Pi(q^2)$ are observed to be sub-5% and sub-3%, respectively, in the region $q^2 < 1\,\mathrm{GeV}^2$.

Importantly, the value of $a_\mu^\mathrm{HLO}$ is expected to decrease with the inclusion of neglected disconnected diagrams.

Discussion and Implications

This work establishes that lattice QCD is able to access $a_\mu^\mathrm{HLO}$ at a precision of around 12%, with statistical and systematic controls sufficient for a meaningful comparison to other (model-dependent) determinations. The implementation of partially twisted boundary conditions and high-statistics CLS ensembles significantly mitigate discretization and finite-volume artifacts. The nontrivial chiral behavior, especially near the physical pion mass, demonstrates that naive extrapolation schemes are inadequate. The result that the inclusion of the strange quark (even partially quenched) drives $a_\mu^\mathrm{HLO}$ upward by approximately 14% is consistent with theoretical expectations.

From a theoretical perspective, the approach offers a path towards systematic improvement—finer lattice spacings, increased statistics (particularly at small $m_\pi$), and ultimately the inclusion of fully unquenched strange and charm quarks—each iteration bringing full ab initio SM calculations ever closer to experimental precision. The ability to directly compute hadronic contributions crucially impacts the search for new physics, as claims of discrepancies between experiment and theory become less susceptible to uncertainties inherent to hadronic models or input data.

Future Prospects

Further reduction of uncertainties will require:

• Simulations with lighter pions, at physical quark masses.
• Control over all sources of systematics, notably full treatment of disconnected diagrams.
• Dynamical inclusion of strange and charm quarks.

Progress in these domains may not only provide a SM prediction for $a_\mu$ at the level demanded by upcoming experimental efforts but will also be pivotal in clarifying whether the persistent experimental tension signals new physics outside the SM.

Conclusion

The lattice QCD determination of the leading hadronic vacuum polarization contribution to the muon anomalous magnetic moment is approaching phenomenological relevance. Improvements in statistical sampling, boundary condition techniques, and systematic control underscore the viability of the approach. Ultimate precision, and hence definitive theoretical predictions for $a_\mu$, will depend on ongoing efforts to address the remaining systematic effects—especially those from lighter quark masses, disconnected diagrams, and additional dynamical quark flavors. This work thus forms a substantial step toward an ab initio SM calculation necessary for utilizing the muon $g-2$ as a probe for physics beyond the Standard Model.