The $\pi\pi\to\pi\gamma^\star$ amplitude and the resonant $\rho\to\pi\gamma^\star$ transition from lattice QCD

TL;DR

This paper uses lattice QCD to determine the $ ho$ meson’s transition form factor to a pion and virtual photon, providing a first-principles calculation of an unstable hadron’s electromagnetic properties.

Contribution

It presents the first lattice QCD calculation of the $ ho o ext{pion} \, ext{gamma}^*$ transition form factor for an unstable hadron.

Findings

Observed dynamical enhancement due to the $ ho$ resonance.

Extracted the $ ho o ext{pion} \, ext{gamma}^*$ transition form factor.

Performed analytic continuation into the complex energy plane.

Abstract

We present a determination of the $P$-wave $\pi\pi\to\pi\gamma^\star$ transition amplitude from lattice quantum chromodynamics. Matrix elements of the vector current in a finite-volume are extracted from three-point correlation functions, and from these we determine the infinite-volume amplitude using a generalization of the Lellouch-L\"uscher formalism. We determine the amplitude for a range of discrete values of the $\pi\pi$ energy and virtuality of the photon, and observe the expected dynamical enhancement due to the $\rho$ resonance. Describing the energy dependence of the amplitude, we are able to analytically continue into the complex energy plane and from the residue at the $\rho$ pole extract the $\rho\to \pi \gamma^\star$ transition form factor. This calculation, at $m_\pi\approx 400$ MeV, is the first to determine the form factor of an unstable hadron within a first principles approach to QCD.