[Paper Review] Hadronic light-by-light scattering contribution to the muon g-2 from lattice QCD : Methodology

Masashi Hayakawa, Thomas Blum+2 more|ArXiv.org|Sep 7, 2005

Particle physics theoretical and experimental studies6 references4 citations

TL;DR

This paper proposes a novel lattice QCD method to compute the hadronic light-by-light scattering contribution to the muon g-2, a key source of theoretical uncertainty. By combining QCD and quenched QED simulations with a subtraction scheme that isolates the h-lbl amplitude through controlled averaging over U(1)em gauge configurations, the method enables a first-principles, non-perturbative evaluation of this elusive QCD contribution with reduced statistical noise.

ABSTRACT

The hadronic light-by-light scattering contribution to the muon g-2 is the most troublesome component of its theoretical prediction; (1) it cannot be determined from the other measurable quantities, (2) the dimensional argument and the estimation based on hadronic models imply that the magnitude of this contribution may be comparable to the discrepancy between the standard model prediction and the experimental value. The direct approach to evaluate the hadronic light-by-light scattering contribution requires the evaluation of the correlation function of {\it four} hadronic electromagnetic currents, and the summation of it over two independent four-momenta of off-shell photons, which is far from the reach of direct lattice simulation. Here we propose an alternative method using combined (QCD + QED) lattice simulations to evaluate the hadronic light-by-light scattering contribution.

Motivation & Objective

• To address the large theoretical uncertainty in the standard model prediction of the muon anomalous magnetic moment, particularly from the hadronic light-by-light scattering (h-lbl) contribution.
• To develop a practical lattice QCD approach for computing the h-lbl amplitude, which is infeasible with direct four-current correlation functions due to computational cost.
• To enable a non-perturbative, first-principles evaluation of the h-lbl contribution using combined (QCD + QED) simulations with controlled systematic errors.
• To isolate the h-lbl amplitude via a subtraction scheme that cancels leading-order QED contributions while preserving the non-perturbative h-lbl matrix element.
• To validate the feasibility of the method by ensuring that the required small electromagnetic coupling constant and perturbative interpretation are achievable in lattice simulations.

Proposed method

• The method uses combined (QCD + quenched QED) lattice simulations to compute the h-lbl contribution by evaluating a matrix element involving two electromagnetic currents in the quark loop and two muon vertices.
• It employs a subtraction scheme where the h-lbl amplitude is extracted as the difference between two expectation values: one with full U(1)em averaging and another with a modified averaging procedure (u_A vs. u_B) that isolates the h-lbl component.
• The key equation (4) defines the matrix element H_μ(t_F, x_c, t_I) as a nested average over gauge configurations, with the h-lbl contribution emerging from the difference in quark propagator contractions under different U(1)em configurations.
• The method ensures high cancellation of O(α_em²) terms by using the same QCD gauge configurations (U) and correlated U(1)em configurations (u_A, u_B), minimizing statistical noise.
• The calculation is efficiently performed in momentum space by Fourier transforming the gauge-invariant propagator D_ρλ(x,y), leveraging exact momentum-space expressions.
• The approach relies on the assumption that the electromagnetic splitting of pion masses is perturbative, supporting the validity of interpreting results at small α_em in a perturbative framework.

Experimental results

Research questions

• RQ1 Can the hadronic light-by-light scattering contribution to the muon g-2 be computed non-perturbatively using lattice QCD with a practical and feasible method?
• RQ2 How can the h-lbl amplitude be isolated from the full four-current correlation function without direct evaluation of the four-point function?
• RQ3 What is the optimal way to cancel the dominant O(α_em²) contributions while preserving the non-perturbative h-lbl matrix element in lattice simulations?
• RQ4 To what extent do multi-quark loop diagrams (e.g., two or four quark loops) contribute to the h-lbl amplitude, and can they be neglected in the quenched approximation?
• RQ5 Is the required small electromagnetic coupling constant α_em = 1/137 accessible in lattice simulations, and does the perturbative interpretation of results remain valid at such coupling?

Key findings

• The proposed method enables the first-principles, non-perturbative computation of the hadronic light-by-light scattering contribution to the muon g-2 using lattice QCD and quenched QED simulations.
• The h-lbl amplitude is extracted via a subtraction scheme that isolates the difference between two expectation values with distinct U(1)em averaging procedures, effectively canceling O(α_em²) terms.
• The method reduces statistical noise by ensuring high correlation between the O(α_em²) contributions in the two terms, as they share the same QCD gauge configurations.
• The use of quenched non-compact QED allows efficient generation of uncorrelated U(1)em gauge configurations, making the method computationally feasible.
• The approach is validated by the fact that electromagnetic splittings in pion masses are perturbative, supporting the reliability of the perturbative interpretation at small α_em.
• The method is generalizable to multi-quark loop diagrams, though their contributions are expected to be subdominant, especially in the flavor SU(3) limit.