[Paper Review] Hadronic light-by-light scattering in the muon g-2

Andreas Nyffeler|arXiv (Cornell University)|Oct 26, 2017

Particle physics theoretical and experimental studies4 references3 citations

TL;DR

This paper reviews the hadronic light-by-light scattering (HLbL) contribution to the muon's anomalous magnetic moment, synthesizing model-based estimates to derive $ a_{\mu}^{\mathrm{HLbL}} = (102 \pm 39) \times 10^{-11} $. It emphasizes emerging model-independent approaches using dispersion relations and lattice QCD as paths toward a more reliable uncertainty-controlled result, crucial for resolving the $ g-2 $ anomaly.

ABSTRACT

We briefly review the current status of the hadronic light-by-light scattering contribution to the anomalous magnetic moment of the muon. Based on various model calculations in the literature, we obtain the estimate a_{mu}^{HLbL} = (102 \pm 39) x 10^{-11}. Recent developments including more model-independent approaches using dispersion relations and lattice QCD, that could lead to a more reliable estimate, are also discussed.

Motivation & Objective

• To assess the current theoretical status of the hadronic light-by-light scattering (HLbL) contribution to the muon anomalous magnetic moment, $ a_\mu^{\mathrm{HLbL}} $, which is a major source of uncertainty in the Standard Model prediction.
• To evaluate the reliability of existing model-based estimates of $ a_\mu^{\mathrm{HLbL}} $, which suffer from uncontrollable uncertainties due to hadronic model dependence.
• To explore the potential of model-independent methods—specifically dispersion relations and lattice QCD—to reduce theoretical uncertainties and improve the precision of $ a_\mu^{\mathrm{HLbL}} $.
• To support the upcoming high-precision experiments at Fermilab and J-PARC by providing a refined theoretical estimate with controlled error bounds.

Proposed method

• Synthesizing results from multiple hadronic models (e.g., HKS, BPP, KN_02, MV_04) to derive a combined estimate of $ a_\mu^{\mathrm{HLbL}} $, with error combination via quadrature and linear methods considered.
• Analyzing the structure of the HLbL amplitude using chiral perturbation theory and large-$ N_c $ counting to classify contributions by their chiral and large-$ N_c $ power counting.
• Evaluating the role of various hadronic contributions: pion and axial-vector exchanges, scalar mesons, quark loops, and pion/K loops, with momentum-dependent form factors.
• Presenting a novel lattice QCD approach in position space using exact QED propagators and a master formula involving QED and QCD kernels, with finite-volume effects minimized via infinite-volume QED.
• Utilizing pre-computed QED weight functions depending on $ x^2, y^2, x\cdot y $ to maintain Lorentz invariance and enable efficient lattice computation.
• Validating the lattice method with numerical tests on the pion-pole contribution and lepton-loop results, achieving percent-level agreement with known analytical results.

Experimental results

Research questions

• RQ1 What is the current best-estimate value of the hadronic light-by-light scattering contribution to the muon $ g-2 $, and what is its uncertainty?
• RQ2 How do model-based estimates of $ a_\mu^{\mathrm{HLbL}} $ compare across different hadronic models, and what are the limitations of these approaches?
• RQ3 Can dispersion relation methods and lattice QCD provide a more reliable, model-independent estimate of $ a_\mu^{\mathrm{HLbL}} $ with controlled uncertainties?
• RQ4 What are the dominant contributions to $ a_\mu^{\mathrm{HLbL}} $, and how do they scale with momentum and hadronic structure?
• RQ5 To what extent do finite-volume and finite-lattice-spacing effects impact the accuracy of lattice QCD calculations of $ a_\mu^{\mathrm{HLbL}} $, and how are they mitigated?

Key findings

• The paper derives a combined estimate of $ a_{\mu}^{\mathrm{HLbL}} = (102 \pm 39) \times 10^{-11} $, based on a synthesis of model calculations, with the uncertainty reflecting model-dependent error propagation.
• The model-based estimates in the literature range from $ (105 \pm 26) \times 10^{-11} $ to $ (116 \pm 39) \times 10^{-11} $, indicating a significant spread due to model dependence and inconsistent error combination.
• Lattice QCD calculations in position space yield $ a_{\mu}^{\text{cHLbL}} = (116.0 \pm 9.6) \times 10^{-11} $ for connected diagrams and $ a_{\mu}^{\text{dHLbL}} = (-62.5 \pm 8.0) \times 10^{-11} $ for leading disconnected diagrams, showing agreement in magnitude with model estimates.
• The short-distance region ($ < 0.6\,\text{fm} $) dominates the HLbL integral, indicating that high-momentum physics is crucial for the total amplitude.
• Numerical tests confirm that large lattices ($ L \sim 5-10\,\text{fm} $) are required to reproduce known pion-pole results, validating the lattice method’s accuracy.
• The new lattice approach in position space avoids power-law finite-volume effects by using infinite-volume, continuum QED, and maintains Lorentz invariance via pre-computed weight functions.