# Relating the finite-volume spectrum and the two-and-three-particle $S$ matrix for relativistic systems of identical scalar particles

## Abstract

Working in relativistic quantum field theory, we derive the quantization condition satisfied by coupled two- and three-particle systems of identical scalar particles confined to a cubic spatial volume with periodicity $L$. This gives the relation between the finite-volume spectrum and the infinite-volume $$\textbf 2 \to \textbf 2$$, $$\textbf 2 \to \textbf 3$$ and $$\textbf 3 \to \textbf 3$$ scattering amplitudes for such theories. The result holds for relativistic systems composed of scalar particles with nonzero mass $m$, whose center of mass energy lies below the four-particle threshold, and for which the two-particle K-matrix has no singularities below the three-particle threshold. Finally, the quantization condition is exact up to corrections of the order $$\mathcal{O}(e^{-mL})$$ and holds for any choice of total momenta satisfying the boundary conditions.

- Authors:

- Thomas Jefferson National Accelerator Facility (TJNAF), Newport News, VA (United States)
- Johannes Gutenberg-Univ. Mainz, Mainz (Germany)
- Univ. of Washington, Seattle, WA (United States)

- Publication Date:

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

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1353028

- Alternate Identifier(s):
- OSTI ID: 1352108

- Report Number(s):
- JLAB-THY-17-2400; arXiv:1701.07465; DOE/OR/23177-4058

Journal ID: ISSN 2470-0010; PRVDAQ; TRN: US1700533

- Grant/Contract Number:
- AC05-06OR23177; SC0011637

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Physical Review D

- Additional Journal Information:
- Journal Volume: 95; Journal Issue: 7; 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; finite volume; relativistic scattering theory; lattice QCD

### Citation Formats

```
Briceño, Raúl A., Hansen, Maxwell T., and Sharpe, Stephen R.
```*Relating the finite-volume spectrum and the two-and-three-particle S matrix for relativistic systems of identical scalar particles*. United States: N. p., 2017.
Web. doi:10.1103/PhysRevD.95.074510.

```
Briceño, Raúl A., Hansen, Maxwell T., & Sharpe, Stephen R.
```*Relating the finite-volume spectrum and the two-and-three-particle S matrix for relativistic systems of identical scalar particles*. United States. doi:10.1103/PhysRevD.95.074510.

```
Briceño, Raúl A., Hansen, Maxwell T., and Sharpe, Stephen R. Tue .
"Relating the finite-volume spectrum and the two-and-three-particle S matrix for relativistic systems of identical scalar particles". United States.
doi:10.1103/PhysRevD.95.074510. https://www.osti.gov/servlets/purl/1353028.
```

```
@article{osti_1353028,
```

title = {Relating the finite-volume spectrum and the two-and-three-particle S matrix for relativistic systems of identical scalar particles},

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

abstractNote = {Working in relativistic quantum field theory, we derive the quantization condition satisfied by coupled two- and three-particle systems of identical scalar particles confined to a cubic spatial volume with periodicity $L$. This gives the relation between the finite-volume spectrum and the infinite-volume $\textbf 2 \to \textbf 2$, $\textbf 2 \to \textbf 3$ and $\textbf 3 \to \textbf 3$ scattering amplitudes for such theories. The result holds for relativistic systems composed of scalar particles with nonzero mass $m$, whose center of mass energy lies below the four-particle threshold, and for which the two-particle K-matrix has no singularities below the three-particle threshold. Finally, the quantization condition is exact up to corrections of the order $\mathcal{O}(e^{-mL})$ and holds for any choice of total momenta satisfying the boundary conditions.},

doi = {10.1103/PhysRevD.95.074510},

journal = {Physical Review D},

number = 7,

volume = 95,

place = {United States},

year = {Tue Apr 18 00:00:00 EDT 2017},

month = {Tue Apr 18 00:00:00 EDT 2017}

}

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