Generalizing the relativistic quantization condition to include all three-pion isospin channels

• Published in: The journal of high energy physics 2020-07, Vol.2020 (7), p.1-49, Article 47
• Main Authors: Hansen, Maxwell T., Romero-López, Fernando, Sharpe, Stephen R.
• Format: Article
• Language: English
• Subjects: Channels; Classical and Quantum Gravitation; Elementary Particles; Formalism; High energy physics; Integral equations; Lattice QCD; Measurement; Physics; Physics and Astronomy; PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; Pions; Quantum chromodynamics; Quantum Field Theories; Quantum Field Theory; Quantum numbers; Quantum Physics; Regular Article - Theoretical Physics; Relativistic effects; Relativistic particles; Relativity Theory; Scattering amplitude; Scattering Amplitudes; String Theory
• ISSN: 1029-8479; 1029-8479
• DOI: 10.1007/JHEP07(2020)047

A bstract We present a generalization of the relativistic, finite-volume, three-particle quantization condition for non-identical pions in isosymmetric QCD. The resulting formalism allows one to use discrete finite-volume energies, determined using lattice QCD, to constrain scattering amplitudes for all possible values of two- and three-pion isospin. As for the case of identical pions considered previously, the result splits into two steps: the first defines a non-perturbative function with roots equal to the allowed energies, E n ( L ), in a given cubic volume with side-length L . This function depends on an intermediate three-body quantity, denoted K df , 3 , which can thus be constrained from lattice QCD in- put. The second step is a set of integral equations relating K df , 3 to the physical scattering amplitude, ℳ 3 . Both of the key relations, E n ( L ) ↔ K df , 3 and K df , 3 ↔ ℳ 3 , are shown to be block-diagonal in the basis of definite three-pion isospin, I πππ , so that one in fact recovers four independent relations, corresponding to I πππ = 0 , 1 , 2 , 3. We also provide the generalized threshold expansion of K df , 3 for all channels, as well as parameterizations for all three-pion resonances present for I πππ = 0 and I πππ = 1. As an example of the utility of the generalized formalism, we present a toy implementation of the quantization condition for I πππ = 0, focusing on the quantum numbers of the ω and h 1 resonances.