Unitarity of the infinite-volume three-particle scattering amplitude arising from a finite-volume formalism

Abstract

In a previous publication, two of us derived a relation between the scattering amplitude of three identical bosons, ${\mathcal{M}}_3$, and a real function referred to as the {divergence-free} K matrix and denoted ${\mathcal{K}}_{\text{df},3}$. The result arose in the context of a relation between finite-volume energies and ${\mathcal{K}}_{\text{df},3}$, derived to all orders in the perturbative expansion of a generic low-energy effective field theory. In this work we set aside the role of the finite volume and focus on the infinite-volume relation between ${\mathcal{K}}_{\text{df},3}$ and ${\mathcal{M}}_3$. We show that, for any real choice of ${\mathcal{K}}_{\text{df},3}$, ${\mathcal{M}}_3$ satisfies the three-particle unitarity constraint to all orders. Given that ${\mathcal{K}}_{\text{df},3}$ is also free of a class of kinematic divergences, the function may provide a useful tool for parametrizing three-body scattering data. Applications include the phenomenological analysis of experimental data (where the connection to the finite volume is irrelevant) as well as calculations in lattice quantum chromodynamics (where the volume plays a key role).

19 pages, 4 figures, JLAB-THY-19-2945, CERN-TH-2019-078

References in corpus (16)

• A relativistic, model-independent, three-particle quantization condition
• Light Nuclei and Hypernuclei from Quantum Chromodynamics in the Limit of SU(3) Flavor Symmetry
• Isoscalar $\pi \pi$ scattering and the $\sigma$ meson resonance from QCD
• Three particle quantization condition in a finite volume: 2. general formalism and the analysis of data
• Resonances in coupled πK,ηK scattering from quantum chromodynamics
• Three-body Unitarity in the Finite Volume
• Three-particle quantization condition in a finite volume: 1. The role of the three-particle force
• Multichannel one-to-two transition amplitudes in a finite volume
• Three-particle systems with resonant subprocesses in a finite volume
• Timelike pion form factor in lattice QCD
• The $I=1$ pion-pion scattering amplitude and timelike pion form factor from $N_{\mathrm{f}}=2+1$ lattice QCD
• Implementing the three-particle quantization condition including higher partial waves
• Three-body Unitarity with Isobars Revisited
• Tetraquark operators in lattice QCD and exotic flavour states in the charm sector
• Form factors of two-hadron states from a covariant finite-volume formalism
• Phenomenology of Relativistic $3\to 3$ Reaction Amplitudes within the Isobar Approximation

Cited by in corpus (36)

• Towards a theory of hadron resonances
• Generalizing the relativistic quantization condition to include all three-pion isospin channels
• Numerical exploration of three relativistic particles in a finite volume including two-particle resonances and bound states
• Three-body unitarity versus finite-volume ${\pi }^{+}{\pi }^{+}{\pi }^{+}$ spectrum from lattice QCD
• The energy-dependent ${\pi }^{+}{\pi }^{+}{\pi }^{+}$ scattering amplitude from QCD
• Three-body interactions from the finite-volume QCD spectrum
• Multi-particle systems on the lattice and chiral extrapolations: a brief review
• Equivalence of relativistic three-particle quantization conditions
• Three-particle finite-volume formalism for ${\pi }^{+}{\pi }^{+}{K}^{+}$ and related systems
• Novel approaches in Hadron Spectroscopy
• Relativistic-invariant formulation of the NREFT three-particle quantization condition
• Solving relativistic three-body integral equations in the presence of bound states
• Long-range electroweak amplitudes of single hadrons from Euclidean finite-volume correlation functions
• Finite-volume energy shift of the three-pion ground state
• Relativistic $N$-particle energy shift in finite volume
• Implementing the three-particle quantization condition for ${\pi }^{+}{\pi }^{+}{K}^{+}$ and related systems
• Three-body dynamics of the $a_1(1260)$ resonance from lattice QCD
• Decay amplitudes to three hadrons from finite-volume matrix elements
• Interactions of $\pi K$, $\pi \pi K$ and $KK\pi$ systems at maximal isospin from lattice QCD
• Efficient solution of the multi-channel Lüscher determinant condition through eigenvalue decomposition
• Light- and strange-quark mass dependence of the $\rho (770)$ meson revisited
• Variations on the Maiani-Testa approach and the inverse problem
• On-shell representations of two-body transition amplitudes: single external current
• The isospin-3 three-particle $K$-matrix at NLO in ChPT
• Few-body bound states and resonances in finite volume
• Bound states in the B-matrix formalism for the three-body scattering
• Finite-volume energy shift of the three-nucleon ground state
• Volume extrapolation via eigenvector continuation
• The $Z_b$ structures in a constituent quark model coupled-channels calculation
• Analytic expansions of multi-hadron finite-volume energies: I. Two-particle states
• Charged-particle bound states in periodic boxes
• Complex scaling in finite volume
• The three-pion $K$-matrix at NLO in ChPT
• Elastic Nucleon-Pion scattering amplitudes in the $\Delta$ channel at physical pion mass from Lattice QCD
• Spurious poles in a finite volume
• Multibody decay analyses -- a new phenomenological model for meson-meson subamplitudes