[Paper Review] $π^0$ to two-photon decay in lattice QCD

Eigo Shintani, Sinya Aoki+3 more|arXiv (Cornell University)|Dec 2, 2009

Quantum Chromodynamics and Particle Interactions3 references3 citations

TL;DR

This lattice QCD study computes the $π^0 \to \gamma^*\gamma^*$ transition form factor using $N_f=2$ dynamical overlap fermions to preserve exact chiral symmetry. The form factor is extracted from three-point functions involving axial-vector and electromagnetic currents, and the Adler-Bell-Jackiw anomaly is successfully reproduced in the chiral and on-shell limits, validating the method for non-perturbative hadronic light-by-light scattering calculations.

ABSTRACT

We calculate the neutral pion ($π^0$) to off-shell two photon ($γ^*γ^*$) transition form factor in lattice QCD. The transition form factor can be extracted from the three-point function of the form (axial-vector)-(vector)-(vector) as a function of off-shell two-photon momentum. Since the axial-anomaly plays an important role in the $π^0 oγγ$ decay process, we employ the overlap fermion, which preserves the exact chiral symmetery on the lattice. After extrapolating to the chiral and the vanishing photon momentum limit with a fit function based on vector meson dominance (VMD) model, we find that the Adler-Bell-Jackiw anomaly is correctly reproduced.

Motivation & Objective

• To compute the $\pi^0 \to \gamma^*\gamma^*$ transition form factor in lattice QCD with exact chiral symmetry.
• To verify the reproduction of the Adler-Bell-Jackiw (ABJ) anomaly in the chiral and on-shell photon limits.
• To provide a model-independent input for the hadronic light-by-light scattering amplitude in the muon $g-2$ anomaly.
• To test the validity of the vector meson dominance (VMD) model and its extension including excited vector states in the lattice framework.

Proposed method

• Use of $N_f=2$ dynamical overlap fermion configurations on a $16^3 \times 32$ lattice at $\beta=2.3$ with $a^{-1} = 1.67$ GeV to preserve exact chiral symmetry.
• Computation of the three-point function $\langle AVV \rangle$ using a conserved axial-vector current and local electromagnetic currents to extract the form factor.
• Application of a VMD-inspired fit function with parameters $X_a$, $c_3$, $c_4$, and mass-dependent couplings to describe the momentum dependence.
• Incorporation of an excited vector meson state via a modified fit form $F^{\rm VMD+}$ to improve agreement at higher momenta.
• Chiral extrapolation using quark masses $m_q = 0.015, 0.025, 0.035, 0.05$ to reach the physical pion mass limit.
• Use of non-perturbative renormalization for the EM current with $Z = 1.3842(3)$ to reduce lattice artifacts.

Experimental results

Research questions

• RQ1 Can the Adler-Bell-Jackiw anomaly be accurately reproduced in lattice QCD using overlap fermions for the $\pi^0 \to \gamma^*\gamma^*$ transition form factor?
• RQ2 How well does the vector meson dominance (VMD) model describe the lattice data for the $\pi^0 \to \gamma^*\gamma^*$ form factor across different momentum transfers?
• RQ3 Does including an excited vector meson state in the fit function improve the description of the form factor at higher momentum transfers compared to the standard VMD model?
• RQ4 To what extent do lattice artifacts and disconnected quark diagrams affect the form factor at finite quark masses and momentum transfers?
• RQ5 Can the lattice result for the form factor be reliably extrapolated to the chiral and on-shell photon limits to match the anomaly prediction?

Key findings

• The Adler-Bell-Jackiw anomaly is successfully reproduced in the chiral and on-shell photon limit, with $X_a = 0.0243(29)$, consistent with the expected value $1/(4\pi^2) = 0.02533$.
• The VMD fit function provides a good description of the form factor at low momentum transfers, with the $c_4^0 = 1.20(21)$ parameter consistent with unity, indicating the validity of the VMD approximation.
• Incorporating an excited vector meson state via the $F^{\rm VMD+}$ fit improves agreement with lattice data at higher momenta, especially in the heavier quark mass region.
• The fit parameters $c_3^m = -8.4(8.6)$ and $c_4^m = -1.3(5.8)$ indicate that the excited state contribution is sub-leading, as expected.
• The physical point extrapolation shows negligible difference between VMD and VMD+ curves, suggesting that the VMD model is sufficient for the low-momentum region.
• The study demonstrates the feasibility of using lattice QCD with overlap fermions to compute the $\pi^0 \to \gamma^*\gamma^*$ form factor as a non-perturbative input for hadronic light-by-light scattering in the muon $g-2$.