Prospects for $\gamma^\star \gamma^\star \to \pi \pi$ via lattice QCD

TL;DR

This paper develops a formalism to extract the $\gamma^ ext{star} \gamma^ ext{star} o \pi ext{pi}$ scattering amplitude from lattice QCD calculations, crucial for understanding scalar resonances and muon g-2 contributions.

Contribution

It provides an exact method to relate finite-volume Euclidean correlators to Minkowski amplitudes for $\gamma^ ext{star} o \pi ext{pi}$ processes, including all necessary auxiliary amplitudes.

Findings

Formalism relates Euclidean correlators to Minkowski amplitudes up to exponentially small corrections.

Requires calculation of multiple related amplitudes like $\pi ext{pi} o \pi ext{pi}$ and $\pi ext{gamma}^ ext{star} o \pi$.

Enables lattice QCD studies of photon-induced pion processes.

Abstract

The $\gamma^\star \gamma^\star \to \pi \pi$ scattering amplitude plays a key role in a wide range of phenomena, including understanding the inner structure of scalar resonances as well as constraining the hadronic contributions to the anomalous magnetic moment of the muon. In this work, we explain how the infinite-volume Minkowski amplitude can be constrained from finite-volume Euclidean correlation functions. The relationship between the finite-volume Euclidean correlation functions and the desired amplitude holds up to energies where $3\pi$ states can go on shell, and is exact up to exponentially small corrections that scale like $\mathcal{O}(e^{-m_\pi L})$, where $L$ is the spatial extent of the cubic volume and $m_\pi$ is the pion mass. In order to implement this formalism and remove all power-law finite volume errors, it is necessary to first obtain $\pi \pi \to \pi\pi$, $\pi \gamma^\star \to \pi$, $\gamma^\star \to\pi\pi$, and $\pi\pi\gamma^\star \to\pi\pi$ amplitudes; all of which can be determined via lattice quantum chromodynamic calculations.