# Numerical exploration of three relativistic particles in a finite volume including two-particle resonances and bound states

## Abstract

In this work, we use an extension of the quantization condition, given in ref. [1], to numerically explore the finite-volume spectrum of three relativistic particles, in the case that two-particle subsets are either resonant or bound. The original form of the relativistic three-particle quantization condition was derived under a technical assumption on the two- particle K matrix that required the absence of two-particle bound states or narrow two- particle resonances. Here we describe how this restriction can be lifted in a simple way using the freedom in the definition of the K-matrix-like quantity that enters the quantization condition. With this in hand, we extend previous numerical studies of the quantization condition to explore the finite-volume signature for a variety of two- and three-particle interactions. We determine the spectrum for parameters such that the system contains both dimers (two-particle bound states) and one or more trimers (in which all three particles are bound), and also for cases where the two-particle subchannel is resonant. We also show how the quantization condition provides a tool for determining infinite-volume dimer- particle scattering amplitudes for energies below the dimer breakup. We illustrate this for a series of examples, including one that parallels physical deuteron-nucleon scattering.more »

- Authors:

- CSIC-Univ. de Valéncia, Paterna (Spain)
- Univ. of Washington, Seattle, WA (United States). Dept. of Physics
- Thomas Jefferson National Accelerator Facility (TJNAF), Newport News, VA (United States); Old Dominion Univ., Norfolk, VA (United States). Dept. of Physics
- European Organization for Nuclear Research (CERN), Geneva (Switzerland)

- Publication Date:

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

- Sponsoring Org.:
- USDOE Office of Science (SC), Nuclear Physics (NP) (SC-26)

- OSTI Identifier:
- 1574401

- Report Number(s):
- JLAB-THY-19-3011; DOE/OR/-23177-4759; arXiv:1908.02411

Journal ID: ISSN 1029-8479

- Grant/Contract Number:
- AC05-06OR23177

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Journal of High Energy Physics (Online)

- Additional Journal Information:
- Journal Name: Journal of High Energy Physics (Online); Journal Volume: 2019; Journal Issue: 10; Journal ID: ISSN 1029-8479

- Publisher:
- Springer Berlin

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; Lattice QCD; Scattering Amplitudes

### Citation Formats

```
Romero-López, Fernando, Sharpe, Stephen R., Blanton, Tyler D., Briceño, Raúl A., and Hansen, Maxwell T. Numerical exploration of three relativistic particles in a finite volume including two-particle resonances and bound states. United States: N. p., 2019.
Web. doi:10.1007/JHEP10(2019)007.
```

```
Romero-López, Fernando, Sharpe, Stephen R., Blanton, Tyler D., Briceño, Raúl A., & Hansen, Maxwell T. Numerical exploration of three relativistic particles in a finite volume including two-particle resonances and bound states. United States. doi:10.1007/JHEP10(2019)007.
```

```
Romero-López, Fernando, Sharpe, Stephen R., Blanton, Tyler D., Briceño, Raúl A., and Hansen, Maxwell T. Thu .
"Numerical exploration of three relativistic particles in a finite volume including two-particle resonances and bound states". United States. doi:10.1007/JHEP10(2019)007. https://www.osti.gov/servlets/purl/1574401.
```

```
@article{osti_1574401,
```

title = {Numerical exploration of three relativistic particles in a finite volume including two-particle resonances and bound states},

author = {Romero-López, Fernando and Sharpe, Stephen R. and Blanton, Tyler D. and Briceño, Raúl A. and Hansen, Maxwell T.},

abstractNote = {In this work, we use an extension of the quantization condition, given in ref. [1], to numerically explore the finite-volume spectrum of three relativistic particles, in the case that two-particle subsets are either resonant or bound. The original form of the relativistic three-particle quantization condition was derived under a technical assumption on the two- particle K matrix that required the absence of two-particle bound states or narrow two- particle resonances. Here we describe how this restriction can be lifted in a simple way using the freedom in the definition of the K-matrix-like quantity that enters the quantization condition. With this in hand, we extend previous numerical studies of the quantization condition to explore the finite-volume signature for a variety of two- and three-particle interactions. We determine the spectrum for parameters such that the system contains both dimers (two-particle bound states) and one or more trimers (in which all three particles are bound), and also for cases where the two-particle subchannel is resonant. We also show how the quantization condition provides a tool for determining infinite-volume dimer- particle scattering amplitudes for energies below the dimer breakup. We illustrate this for a series of examples, including one that parallels physical deuteron-nucleon scattering. All calculations presented here are restricted to the case of three identical scalar particles.},

doi = {10.1007/JHEP10(2019)007},

journal = {Journal of High Energy Physics (Online)},

number = 10,

volume = 2019,

place = {United States},

year = {2019},

month = {10}

}

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