Lattice QCD calculation of $\pi^0\rightarrow e^+ e^-$ decay

TL;DR

This paper uses lattice QCD methods to directly compute the complex decay amplitude of the rare $^0 ightarrow e^+ e^-$ process, improving theoretical understanding of this decay and related rare processes.

Contribution

It introduces a novel lattice QCD approach combining Minkowski and Euclidean methods to calculate the decay amplitude directly from fundamental theories.

Findings

Calculated the real and imaginary parts of the decay amplitude with improved precision.

Provided a more accurate ratio of the real to imaginary parts of the amplitude.

Estimated the partial width of the decay with systematic and statistical errors.

Abstract

We extend the application of lattice QCD to the two-photon-mediated, order $\alpha^2$ rare decay $\pi^0\rightarrow e^+ e^-$. By combining Minkowski- and Euclidean-space methods we are able to calculate the complex amplitude describing this decay directly from the underlying theories (QCD and QED) which predict this decay. The leading connected and disconnected diagrams are considered; a continuum limit is evaluated and the systematic errors are estimated. We find $\mathrm{Re} \mathcal{A} = 18.60(1.19)(1.04)\,$eV, $\mathrm{Im} \mathcal{A} = 32.59(1.50)(1.65)\,$eV, a more accurate value for the ratio $\frac{\mathrm{Re} \mathcal{A}}{\mathrm{Im} \mathcal{A}}=0.571(10)(4)$ and a result for the partial width $\Gamma(\pi^0\to\gamma\gamma) = 6.60(0.61)(0.67)\,$eV. Here the first errors are statistical and the second systematic. This calculation is the first step in determining the more challenging, two-photon-mediated decay amplitude that contributes to the rare decay $K\to\mu^+\mu^-$.