[Paper Review] A study of the radiative transition $\pi \pi o \pi \gamma^{*}$ with lattice QCD

Luka Leskovec, Srijit Paul+8 more|arXiv (Cornell University)|Nov 1, 2016

Quantum Chromodynamics and Particle Interactions3 citations

TL;DR

This lattice QCD study computes the radiative transition ππ → πγ* using the Briceño-Hansen-Walker-Loud formalism to map finite-volume matrix elements to infinite-volume amplitudes, enabling the first ab initio determination of the ππ → πγ* amplitude with inclusion of ρ resonance effects. The key result is a finite-volume amplitude that exhibits a clear enhancement near the ρ meson mass (mρ = 798.2(5.3) MeV), consistent with expected resonance behavior.

ABSTRACT

Lattice QCD calculations of radiative transitions between hadrons have in the past been limited to processes of hadrons stable under the strong interaction. Recently developed methods for $1 o2$ transition matrix elements in a finite volume now enable the determination of radiative decay rates of strongly unstable particles. Our lattice QCD study focuses on the process $\pi \pi o \pi \gamma^{*}$, where the $ ho$ meson is present as an enhancement in the cross-section. We use $2+1$ flavors of clover fermions at a pion mass of approximately $320$ MeV and a lattice size of approximately $3.6$ fm. The required $2$-point and $3$-point correlation functions are constructed from a set of forward, sequential and stochastic light quark propagators. In addition to determining the $ ho$ meson resonance parameters via the L\"uscher method, the scattering phase shift is used in conjunction with the $1 o2$ transition matrix element formalism of Brice\~no, Hansen and Walker-Loud to compute the $\pi\pi o\pi\gamma^{*}$ amplitude at several values of the momentum transfer and $\pi\pi$ invariant mass.

Motivation & Objective

• To compute the radiative transition ππ → πγ* in lattice QCD, a process involving a strongly unstable resonance (ρ meson), which has been challenging due to finite-volume effects.
• To apply the recently developed Briceño-Hansen-Walker-Loud (BHWL) formalism for 1→2 transition matrix elements to map finite-volume correlation functions to infinite-volume amplitudes.
• To determine the ππ → πγ* transition amplitude at multiple values of momentum transfer and ππ invariant mass, including near the ρ resonance peak.
• To validate the method by comparing results with the Hadron Spectrum collaboration's earlier study, demonstrating consistency and computational efficiency.

Proposed method

• Use 2+1 flavor clover fermions on a 3.6 fm³ lattice with mπ ≈ 317 MeV to generate gauge ensembles for lattice calculations.
• Construct 2-point and 3-point correlation functions using forward, sequential, and stochastic light quark propagators to efficiently compute matrix elements.
• Apply the Lüscher method to extract the elastic ππ scattering phase shift and fit it to a Breit-Wigner form to extract ρ resonance parameters (mρ = 798.2(5.3) MeV, gρππ = 6.46(53)).
• Use the BHWL formalism to map finite-volume matrix elements to infinite-volume transition amplitudes via the Lellouch-Lüscher factor, incorporating phase shift derivatives and quantization conditions.
• Project 3-point functions onto definite states using optimized correlators based on irreducible representations and momentum quantum numbers.
• Perform Lorentz-invariant decomposition of the amplitude to extract the invariant form factor fππ,π(q², sππ), which describes the transition strength.

Experimental results

Research questions

• RQ1 How can the finite-volume effects in 1→2 transition matrix elements involving unstable hadrons be systematically treated in lattice QCD?
• RQ2 What is the behavior of the ππ → πγ* transition amplitude near the ρ resonance peak, and does it exhibit expected enhancement?
• RQ3 Can the BHWL formalism be successfully applied to compute the ππ → πγ* amplitude using a combination of forward, sequential, and stochastic propagators?
• RQ4 How do the extracted resonance parameters (mρ, gρππ) from the phase shift compare with experimental values and previous lattice studies?
• RQ5 What is the computational efficiency and data quality of the adapted correlation function construction method on large lattices?

Key findings

• The ππ → πγ* infinite-volume amplitude exhibits a clear enhancement near the ρ meson mass, with peak strength observed at √sππ ≈ mρ = 798.2(5.3) MeV.
• The extracted ρ resonance parameters are consistent with expectations: mρ = 798.2(5.3) MeV and gρππ = 6.46(53), obtained from fitting the ππ phase shift to a Breit-Wigner form.
• The computed amplitude shows a smooth dependence on momentum transfer and invariant mass, with no unphysical structures, indicating reliable extraction of the transition matrix element.
• The method using forward, sequential, and stochastic propagators enables efficient computation of 3-point functions on large lattices, yielding high-quality data with good statistical precision.
• The results are in good agreement with the Hadron Spectrum collaboration's earlier study, validating the BHWL formalism and the computational approach.
• The finite-volume to infinite-volume mapping via the Lellouch-Lüscher factor successfully accounts for volume effects, enabling reliable extraction of physical transition amplitudes.