# Efficient solution of the multichannel Lüscher determinant condition through eigenvalue decomposition

## Abstract

We present a method for efficiently finding solutions of Lüscher’s quantization condition, the equation which relates two-particle scattering amplitudes to the discrete spectrum of states in a periodic spatial volume of finite extent such as that present in lattice QCD. The approach proposed is based on an eigenvalue decomposition in the space of coupled-channels and partial-waves, which proves to have several desirable and simplifying features that are of great benefit when considering problems beyond simple elastic scattering of spinless particles. We illustrate the method with a toy model of vector-vector scattering featuring a high density of solutions, and with an application to explicit lattice QCD energy level data describing $J^P=1^-$ and $1^+$ scattering in several coupled channels.

- Authors:

- Publication Date:

- Research Org.:
- Thomas Jefferson National Accelerator Facility (TJNAF), Newport News, VA (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR); USDOE Office of Science (SC), Nuclear Physics (NP); Science and Technology Facilities Council (STFC) (United Kingdom)

- Contributing Org.:
- for the Hadron Spectrum Collaboration

- OSTI Identifier:
- 1633699

- Alternate Identifier(s):
- OSTI ID: 1633980

- Report Number(s):
- JLAB-THY-20-3133; DOE/OR/23177-4898; arXiv:2001.08474

Journal ID: ISSN 2470-0010; PRVDAQ; 114505

- Grant/Contract Number:
- ST/P000681/1; ST/P002307/1; ST/R002452/1; ST/R00689X/1; SC0018416; AC05-06OR23177; AC05-00OR22725; AC02-05CH11231

- Resource Type:
- Published Article

- Journal Name:
- Physical Review D

- Additional Journal Information:
- Journal Name: Physical Review D Journal Volume: 101 Journal Issue: 11; Journal ID: ISSN 2470-0010

- Publisher:
- American Physical Society (APS)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; Lattice QCD; Quantum chromodynamics; Particles & Fields

### Citation Formats

```
Woss, Antoni J., Wilson, David J., and Dudek, Jozef J. Efficient solution of the multichannel Lüscher determinant condition through eigenvalue decomposition. United States: N. p., 2020.
Web. doi:10.1103/PhysRevD.101.114505.
```

```
Woss, Antoni J., Wilson, David J., & Dudek, Jozef J. Efficient solution of the multichannel Lüscher determinant condition through eigenvalue decomposition. United States. doi:https://doi.org/10.1103/PhysRevD.101.114505
```

```
Woss, Antoni J., Wilson, David J., and Dudek, Jozef J. Mon .
"Efficient solution of the multichannel Lüscher determinant condition through eigenvalue decomposition". United States. doi:https://doi.org/10.1103/PhysRevD.101.114505.
```

```
@article{osti_1633699,
```

title = {Efficient solution of the multichannel Lüscher determinant condition through eigenvalue decomposition},

author = {Woss, Antoni J. and Wilson, David J. and Dudek, Jozef J.},

abstractNote = {We present a method for efficiently finding solutions of Lüscher’s quantization condition, the equation which relates two-particle scattering amplitudes to the discrete spectrum of states in a periodic spatial volume of finite extent such as that present in lattice QCD. The approach proposed is based on an eigenvalue decomposition in the space of coupled-channels and partial-waves, which proves to have several desirable and simplifying features that are of great benefit when considering problems beyond simple elastic scattering of spinless particles. We illustrate the method with a toy model of vector-vector scattering featuring a high density of solutions, and with an application to explicit lattice QCD energy level data describing $J^P=1^-$ and $1^+$ scattering in several coupled channels.},

doi = {10.1103/PhysRevD.101.114505},

journal = {Physical Review D},

number = 11,

volume = 101,

place = {United States},

year = {2020},

month = {6}

}

DOI: https://doi.org/10.1103/PhysRevD.101.114505

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