# Three-body unitarity in the finite volume

## Abstract

We present the physical interpretation of lattice QCD simulations, performed in a small volume, requires an extrapolation to the infinite volume. A method is proposed to perform such an extrapolation for three interacting particles at energies above threshold. For this, a recently formulated relativistic $$3\to 3$$ amplitude based on the isobar formulation is adapted to the finite volume. The guiding principle is two- and three-body unitarity that imposes the imaginary parts of the amplitude in the infinite volume. In turn, these imaginary parts dictate the leading power-law finite-volume effects. It is demonstrated that finite-volume poles arising from the singular interaction, from the external two-body sub-amplitudes, and from the disconnected topology cancel exactly leaving only the genuine three-body eigenvalues. Lastly, the corresponding quantization condition is derived for the case of three identical scalar-isoscalar particles and its numerical implementation is demonstrated.

- Authors:

- George Washington Univ., Washington, DC (United States)
- George Washington Univ., Washington, DC (United States) ; 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:
- 1416327

- Report Number(s):
- JLAB-THY-17-2554; DOE/OR/23177-4221; arXiv:1709.08222

Journal ID: ISSN 1434-6001; PII: 12440; TRN: US1800924

- Grant/Contract Number:
- AC05-06OR23177; SC001658

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- European Physical Journal. A

- Additional Journal Information:
- Journal Volume: 53; Journal Issue: 12; Journal ID: ISSN 1434-6001

- Publisher:
- Springer

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS

### Citation Formats

```
Mai, M., and Döring, M.
```*Three-body unitarity in the finite volume*. United States: N. p., 2017.
Web. doi:10.1140/epja/i2017-12440-1.

```
Mai, M., & Döring, M.
```*Three-body unitarity in the finite volume*. United States. doi:10.1140/epja/i2017-12440-1.

```
Mai, M., and Döring, M. Mon .
"Three-body unitarity in the finite volume". United States.
doi:10.1140/epja/i2017-12440-1.
```

```
@article{osti_1416327,
```

title = {Three-body unitarity in the finite volume},

author = {Mai, M. and Döring, M.},

abstractNote = {We present the physical interpretation of lattice QCD simulations, performed in a small volume, requires an extrapolation to the infinite volume. A method is proposed to perform such an extrapolation for three interacting particles at energies above threshold. For this, a recently formulated relativistic $3\to 3$ amplitude based on the isobar formulation is adapted to the finite volume. The guiding principle is two- and three-body unitarity that imposes the imaginary parts of the amplitude in the infinite volume. In turn, these imaginary parts dictate the leading power-law finite-volume effects. It is demonstrated that finite-volume poles arising from the singular interaction, from the external two-body sub-amplitudes, and from the disconnected topology cancel exactly leaving only the genuine three-body eigenvalues. Lastly, the corresponding quantization condition is derived for the case of three identical scalar-isoscalar particles and its numerical implementation is demonstrated.},

doi = {10.1140/epja/i2017-12440-1},

journal = {European Physical Journal. A},

number = 12,

volume = 53,

place = {United States},

year = {Mon Dec 18 00:00:00 EST 2017},

month = {Mon Dec 18 00:00:00 EST 2017}

}