[Paper Review] Lattice calculation of the $π^0$, $η$ and $η^{\prime}$ transition form factors and the hadronic light-by-light contribution to the muon $g-2$

Antoine Gérardin, Willem E. A. Verplanke+6 more|arXiv (Cornell University)|May 8, 2023

Particle physics theoretical and experimental studies9 citations

TL;DR

This paper reports an ab initio lattice QCD calculation of the π^0, η, and η′ transition form factors with physical light quarks and uses these form factors to compute the pseudoscalar-pole contribution to hadronic light-by-light scattering in the muon g-2, obtaining aμHLbL;P-pole = (85.1 ± 5.2) × 10^-11.

ABSTRACT

In this paper we present a first ab-initio calculation of the $π^0$, $η$ and $η^{\prime}$ transition form factors performed with physical light-quark masses. We provide a complete parametrization of the form factors that includes both single and double-virtual kinematics. Our results are compared with experimental measurements of the form factors in the space-like region and with the measured two-photon decay widths. In a second step, our parametrizations of the transition form factors are used to compute the dominant pseudoscalar-pole contributions to the hadronic light-by-light scattering in the muon $g-2$. Our final result reads $a_μ^{ m hlbl, ps-pole} = (85.1 \pm 5.2) imes 10^{-11}$. Although the pion-pole is dominant, we confirm that, together, the $η$ and $η^{\prime}$ provide roughly half of its contribution.

Motivation & Objective

• Compute the π^0, η, and η′ transition form factors (TFFs) with physical light-quark masses.
• Parametrize the TFFs for single- and double-virtual photon kinematics.
• Use TFFs to estimate the dominant pseudoscalar-pole HLbL contribution to the muon g-2.
• Compare lattice TFFs with experimental data in the space-like region and two-photon decay widths.
• Provide continuum-extrapolated results and assess systematic uncertainties.

Proposed method

• Lattice QCD calculation with Nf=2+1+1 staggered fermions on multiple ensembles with physical light quark masses.
• Extraction of TFFs from three-point and two-point correlation functions using a 2x2 mixing matrix for η and η′.
• Use of a 4-link taste-singlet pseudoscalar operator and a conserved one-link vector current to access connected and disconnected contributions.
• Form factor extraction via Euclidean-space amplitude Ãμν(τ) integrated over τ with a Simpson rule, and tail corrections via a lattice-momentum–dependent parametrization.
• Master equation for aμHLbL;P in terms of the pseudoscalar transition form factors, with analytic integration over angular variables (Gegenbauer technique).
• Finite-volume, finite-spacing, and tail-systematics studied and addressed.

Figure 1: Pseudoscalar-pole contribution to the hadronic light-by-light scattering diagram. The blob on the left of the equality represents the full HLbL four-point function. The blobs on the right-hand side represent the transition form factors of the light pseudoscalar mesons. The solid, dashed an

Experimental results

Research questions

• RQ1 What are the π^0, η, and η′ transition form factors across single- and double-virtual kinematics in lattice QCD with physical quark masses?
• RQ2 How do the lattice-determined TFFs compare with experimental measurements and decay width constraints?
• RQ3 What is the pseudoscalar-pole contribution to hadronic light-by-light scattering in the muon g-2 using these lattice TFFs?
• RQ4 What is the relative sharing of the HLbL contribution among π^0, η, and η′ states, and how do disconnected contributions affect η, η′?
• RQ5 What are the main systematic uncertainties and how can they be controlled in a continuum and infinite-volume limit?

Key findings

• The final pseudoscalar-pole HLbL contribution is aμHLbL;P = (85.1 ± 5.2) × 10^-11.
• Pion-pole dominates the HLbL contribution, but η and η′ together contribute about half of the pion’s portion.
• Lattice TFFs agree with experimental measurements in the space-like region and with two-photon decay width constraints.
• A complete parametrization of the TFFs is provided for both single- and double-virtual kinematics.
• Finite-volume and discretization effects are studied across multiple lattice spacings and volumes.

Figure 2: The four different Wick contraction topologies that are present in the three-point correlation function. Some topologies contains several diagrams that are not shown for brevity. The blue and green blobs represent a pseudoscalar or a vector current insertion respectively. Only the first tw