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

## Abstract

Hansen and Sharpe 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 $$\mathcal K_{df,3}$$. The result arose in the context of a relation between finite-volume energies and $$\mathcal K_{df,3}$$, 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 relation between $$\mathcal K_{df,3}$$ and $$\mathcal M_3$$. We show that, for any real choice of $$\mathcal K_{df,3}$$, $$\mathcal M_3$$ satisfies the three-particle unitarity constraint to all orders. Given that $$\mathcal K_{df,3}$$ 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).

- Authors:

- Thomas Jefferson National Accelerator Facility (TJNAF), Newport News, VA (United States); Old Dominion Univ., Norfolk, VA (United States)
- European Organization for Nuclear Research (CERN), Geneva (Switzerland). Theoretical Physics Dept.
- Univ. of Washington, Seattle, WA (United States). Physics Dept.
- Indiana Univ., Bloomington, IN (United States). Physics Dept.; Indiana Univ., Bloomington, IN (United States). Center for Exploration of Energy and Matter; Thomas Jefferson National Accelerator Facility (TJNAF), Newport News, VA (United States)

- 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:
- 1562572

- Alternate Identifier(s):
- OSTI ID: 1567050

- Report Number(s):
- JLAB-THY-19-2945; DOE/OR/23177-4684; arXiv:1905.11188

Journal ID: ISSN 2470-0010; PRVDAQ

- Grant/Contract Number:
- SC0011637; FG02-87ER40365; AC05-06OR23177; SC0019229; PHY-1415459

- Resource Type:
- Published Article

- Journal Name:
- Physical Review D

- Additional Journal Information:
- Journal Volume: 100; Journal Issue: 5; Journal ID: ISSN 2470-0010

- Publisher:
- American Physical Society (APS)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 43 PARTICLE ACCELERATORS

### Citation Formats

```
Briceño, Raúl A., Hansen, Maxwell T., Sharpe, Stephen R., and Szczepaniak, Adam P. Unitarity of the infinite-volume three-particle scattering amplitude arising from a finite-volume formalism. United States: N. p., 2019.
Web. doi:10.1103/PhysRevD.100.054508.
```

```
Briceño, Raúl A., Hansen, Maxwell T., Sharpe, Stephen R., & Szczepaniak, Adam P. Unitarity of the infinite-volume three-particle scattering amplitude arising from a finite-volume formalism. United States. doi:10.1103/PhysRevD.100.054508.
```

```
Briceño, Raúl A., Hansen, Maxwell T., Sharpe, Stephen R., and Szczepaniak, Adam P. Wed .
"Unitarity of the infinite-volume three-particle scattering amplitude arising from a finite-volume formalism". United States. doi:10.1103/PhysRevD.100.054508.
```

```
@article{osti_1562572,
```

title = {Unitarity of the infinite-volume three-particle scattering amplitude arising from a finite-volume formalism},

author = {Briceño, Raúl A. and Hansen, Maxwell T. and Sharpe, Stephen R. and Szczepaniak, Adam P.},

abstractNote = {Hansen and Sharpe 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 $\mathcal K_{df,3}$. The result arose in the context of a relation between finite-volume energies and $\mathcal K_{df,3}$, 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 relation between $\mathcal K_{df,3}$ and $\mathcal M_3$. We show that, for any real choice of $\mathcal K_{df,3}$, $\mathcal M_3$ satisfies the three-particle unitarity constraint to all orders. Given that $\mathcal K_{df,3}$ 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).},

doi = {10.1103/PhysRevD.100.054508},

journal = {Physical Review D},

number = 5,

volume = 100,

place = {United States},

year = {2019},

month = {9}

}

DOI: 10.1103/PhysRevD.100.054508

Works referenced in this record:

##
Three-particle quantization condition in a finite volume: 1. The role of the three-particle force

journal, September 2017

- Hammer, Hans-Werner; Pang, Jin-Yi; Rusetsky, Akaki
- Journal of High Energy Physics, Vol. 2017, Issue 9, 23 p.

##
Three-particle quantization condition in a finite volume: 2. General formalism and the analysis of data

journal, October 2017

- Hammer, Hans-Werner; Pang, Jin-Yi; Rusetsky, Akaki
- Journal of High Energy Physics, Vol. 2017, Issue 10, 31 p.