[Paper Review] Position-space approach to hadronic light-by-light scattering in the muon $g-2$ on the lattice

Nils Asmussen, Jeremy Green+2 more|arXiv (Cornell University)|Sep 27, 2016

Particle physics theoretical and experimental studies4 citations

TL;DR

This paper presents a position-space approach to compute the hadronic light-by-light (HLbL) scattering contribution to the muon anomalous magnetic moment ($a_\mu^{\text{HLbL}}$) using lattice QCD. By expressing $a_\mu^{\text{HLbL}}$ as a multidimensional integral over a continuum QED kernel and a lattice QCD four-point correlator, the method avoids power-law finite-volume effects. The approach is validated by reproducing the pion-pole contribution with high accuracy, enabling future first-principles lattice calculations of $a_\mu^{\text{HLbL}}$.

ABSTRACT

The anomalous magnetic moment of the muon currently exhibits a discrepancy of about three standard deviations between the experimental value and recent Standard Model predictions. The theoretical uncertainty is dominated by the hadronic vacuum polarization and the hadronic light-by-light (HLbL) scattering contributions, where the latter has so far only been fully evaluated using different models. To pave the way for a lattice calculation of HLbL, we present an expression for the HLbL contribution to $g-2$ that involves a multidimensional integral over a position-space QED kernel function in the continuum and a lattice QCD four-point correlator. We describe our semi-analytic calculation of the kernel and test the approach by evaluating the $\\pi^0$-pole contribution in the continuum.

Motivation & Objective

• To develop a lattice QCD-compatible framework for computing the hadronic light-by-light scattering contribution to the muon anomalous magnetic moment ($a_\mu^{\text{HLbL}}$).
• To eliminate power-law finite-volume corrections by separating QED kernel calculations in position space from lattice QCD correlators.
• To enable first-principles lattice calculations of $a_\mu^{\text{HLbL}}$ by formulating the problem in terms of a multidimensional integral over position-space functions.
• To validate the method by reproducing the pion-pole contribution using a vector-dominance model (VMD) and comparing with momentum-space results.

Proposed method

• The method expresses $a_\mu^{\text{HLbL}}$ as a double integral over position-space coordinates $x$ and $y$, weighted by a QED kernel function $\mathcal{L}_{[\rho,\sigma];\mu\nu\lambda}(\hat{\epsilon},x,y)$ and the lattice QCD four-point correlator $\widehat{\Pi}_{\rho;\mu\nu\lambda\sigma}(x,y)$.
• The QED kernel is derived using continuum Euclidean position-space perturbation theory, with the muon momentum parametrized via a unit vector $\hat{\epsilon}$, ensuring rotational invariance.
• The kernel is computed semi-analytically using modified Bessel functions and position-space propagators $G_m(x)$, and stored on disk for fast lookup during simulations.
• The method avoids finite-volume effects by performing the QED kernel integral in infinite volume, while the QCD four-point function is computed on the lattice.
• The $a_\mu^{\text{HLbL}}$ contribution is projected via a trace over gamma matrices and the kernel, with the final integral reduced to a one-dimensional $|y|$-integral via angular averaging over $\hat{\epsilon}$.
• Numerical validation is performed by computing the pion-pole contribution using the VMD model for the pion transition form factor and comparing with known momentum-space results.

Experimental results

Research questions

• RQ1 Can a position-space formulation of the HLbL scattering amplitude be constructed that is compatible with lattice QCD simulations and avoids finite-volume artifacts?
• RQ2 How can the QED kernel for $a_\mu^{\text{HLbL}}$ be computed analytically in position space to enable efficient lattice evaluation?
• RQ3 To what extent does the method reproduce known results, such as the pion-pole contribution, when tested with model form factors?
• RQ4 Can the method be extended to include full QCD four-point functions with dynamical quarks on the lattice?
• RQ5 How does the computational cost scale with the number of $|y|$ integration points in a lattice implementation?

Key findings

• The position-space approach successfully reproduces the pion-pole contribution to $a_\mu^{\text{HLbL}}$ with high accuracy when using the VMD model for the pion transition form factor.
• The numerical evaluation of the $|y|$ integral converges only for $|y|^{\text{max}} \geq 2-3$ fm, even for large pion masses ($m_\pi = 600-900$ MeV), indicating slow convergence at large distances.
• The method avoids power-law finite-volume corrections by performing the QED kernel integral in infinite volume, preserving the accuracy of the lattice QCD four-point function.
• The kernel function $\mathcal{L}_{[\rho,\sigma];\mu\nu\lambda}(\hat{\epsilon},x,y)$ is computed semi-analytically and stored on disk, enabling fast evaluation during lattice simulations.
• The computational cost for a full evaluation is estimated at $1+N$ forward propagators and $6(1+N)$ sequential propagators, with $N \approx 20$ required for reliable $|y|$ integration.
• The approach is found to be consistent with existing momentum-space results, validating the kernel and the overall framework for future lattice QCD applications.