[Paper Review] $(g-2)_\mu$ at four loops in QED

Peter Marquard, Alexander V. Smirnov+3 more|arXiv (Cornell University)|Jan 1, 2017

Particle physics theoretical and experimental studies15 references5 citations

TL;DR

This paper presents an independent, high-precision calculation of the four-loop quantum electrodynamics (QED) contribution to the anomalous magnetic moment of the muon, focusing on the universal (purely photonic and muon-loop) part. Using a fully automated reduction to master integrals and numerical evaluation with controlled precision, the authors cross-check the semi-analytic result from Laporta (2017), achieving agreement within uncertainties and confirming the dominant four-loop contribution to be approximately −1.87(12) + 132.86(48) × (α/π)⁴, which is of the same order as the current experimental-theoretical discrepancy.

ABSTRACT

We review the four-loop QED corrections to the anomalous magnetic moment of the muon. The fermionic contributions with closed electron and tau contributions are discussed. Furthermore, we report on a new independent calculation of the universal four-loop contribution and compare with existing results.

Motivation & Objective

• To provide an independent, automated calculation of the universal four-loop QED contribution to the anomalous magnetic moment of the muon (g−2)μ.
• To cross-check the semi-analytic result of Laporta (2017), which used high-precision numerical integration and the PSLQ algorithm to reconstruct transcendental constants.
• To improve the precision and reliability of the four-loop QED contribution, which is numerically dominant and comparable in magnitude to the current experimental-theoretical discrepancy.
• To validate the consistency of the four-loop result across multiple independent methods, including asymptotic expansions for electron and tau loops.

Proposed method

• The calculation uses the same integral families as in the MS-on-shell quark mass relation, reducing four-loop vertex integrals to master integrals via FIRE and Crusher.
• The reduction is performed for on-shell photon momentum transfer, with propagators expanded to handle the muon mass scale.
• Master integrals are evaluated numerically with high precision using FIESTA, and uncertainties are estimated from the ǫ-expansion coefficients of the master integrals.
• The universal contribution is split into six gauge-invariant subsets, each computed independently and combined with quadrature uncertainty propagation.
• For fermionic loops (electron and tau), asymptotic expansions in me/mμ and mμ/mτ are used, with numerical evaluation of complex integrals via FIESTA.
• The results are compared with prior calculations: [24] (semi-analytic), [19] (numerical), and [33] (previous cross-check), using consistent notation and error propagation.

Experimental results

Research questions

• RQ1 Can the universal four-loop QED contribution to (g−2)μ be independently computed using a fully automated reduction and numerical evaluation of master integrals?
• RQ2 How do the uncertainties in the current four-loop result compare to the experimental uncertainty and the hadronic contributions?
• RQ3 To what extent do the results from this work agree with the semi-analytic calculation of Laporta (2017), which used high-precision arithmetic and the PSLQ algorithm?
• RQ4 What is the numerical impact of the electron and tau loop contributions, particularly given their large logarithmic enhancements?
• RQ5 Is the four-loop QED contribution sufficiently precise to resolve the current 3σ discrepancy between experiment and the Standard Model?

Key findings

• The independent calculation confirms the semi-analytic result of Laporta (2017) for the universal four-loop contribution to (g−2)μ, with agreement within uncertainties.
• The final result for the universal part is a(8)μ = −1.87(12) + 132.86(48) × (α/π)⁴, with the electron/tau contribution being numerically dominant due to large logarithms log(mμ/me) ≈ 5.332.
• The uncertainty in the present work is about 12% for the universal part, which is two orders of magnitude larger than the uncertainty in Laporta’s result but still smaller than the current experimental uncertainty of 90 × 10−11.
• After multiplication by (α/π)⁴, the total four-loop contribution evaluates to approximately (−5.44(35) + 386.77(1.40)) × 10−11, which is of the same order as the observed (g−2)μ discrepancy of 250(90) × 10−11.
• The agreement across three independent calculations—this work, Laporta (2017), and [19]—confirms the reliability of the four-loop QED contribution, which is now cross-checked by at least two independent groups using different methods.
• The result for the universal part is also applicable to the electron’s anomalous magnetic moment, though it is less precise than the state-of-the-art result from [24] and thus not competitive for ae.