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.

• Received 24 October 2022
• Accepted 13 December 2022

DOI:https://doi.org/10.1103/PhysRevD.107.034504

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP³.

Published by the American Physical Society