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

## Abstract

Hansen and Sharpe [Phys. Rev. D 92, 114509 (2015)PRVDAQ1550-799810.1103/PhysRevD.92.114509] derived a relation between the scattering amplitude of three identical bosons, M3, and a real function referred to as the divergence-free K matrix and denoted Kdf,3. The result arose in the context of a relation between finite-volume energies and Kdf,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 infinite-volume relation between Kdf,3 and M3. We show that, for any real choice of Kdf,3, M3 satisfies the three-particle unitarity constraint to all orders. Given that Kdf,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:

- Publication Date:

- Research Org.:
- Thomas Jefferson National Accelerator Facility (TJNAF), Newport News, VA (United States); Indiana Univ., Bloomington, IN (United States); Univ. of Washington, Seattle, WA (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), Nuclear Physics (NP); USDOE Office of Science (SC), High Energy Physics (HEP)

- Contributing Org.:
- Jefferson Lab, Indiana University

- OSTI Identifier:
- 1562572

- Alternate Identifier(s):
- OSTI ID: 1567050; OSTI ID: 1595125; OSTI ID: 1669366

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

Journal ID: ISSN 2470-0010; PRVDAQ; 054508

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

- Resource Type:
- Published Article

- Journal Name:
- Physical Review D

- Additional Journal Information:
- Journal Name: Physical Review D 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; Lattice QCD; Lattice field theory; Quantum field theory; Quantum field theory (low energy); Scattering amplitudes; 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; lattice QCD, scattering

### 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 [Phys. Rev. D 92, 114509 (2015)PRVDAQ1550-799810.1103/PhysRevD.92.114509] derived a relation between the scattering amplitude of three identical bosons, M3, and a real function referred to as the divergence-free K matrix and denoted Kdf,3. The result arose in the context of a relation between finite-volume energies and Kdf,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 infinite-volume relation between Kdf,3 and M3. We show that, for any real choice of Kdf,3, M3 satisfies the three-particle unitarity constraint to all orders. Given that Kdf,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

*Citation information provided by*

Web of Science

Web of Science

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.

##
Rummukainen-Gottlieb formula on a two-particle system with different masses

journal, January 2012

- Fu, Ziwen
- Physical Review D, Vol. 85, Issue 1

##
Recoupling Effects in the Isobar Model. I. General Formalism for Three-Pion Scattering

journal, July 1964

- Fleming, Gordon N.
- Physical Review, Vol. 135, Issue 2B

##
Finite-Volume Spectrum of ${\pi}^{+}{\pi}^{+}$ and ${\pi}^{+}{\pi}^{+}{\pi}^{+}$ Systems

journal, February 2019

- Mai, M.; Döring, M.
- Physical Review Letters, Vol. 122, Issue 6

##
Multiple-channel generalization of Lellouch-Lüscher formula

journal, July 2012

- Hansen, Maxwell T.; Sharpe, Stephen R.
- Physical Review D, Vol. 86, Issue 1

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

journal, April 2017

- Briceño, Raúl A.; Hansen, Maxwell T.; Sharpe, Stephen R.
- Physical Review D, Vol. 95, Issue 7

##
Analysis of Three-Hadron Final States

journal, October 1973

- Aaron, R.; Amado, R. D.
- Physical Review Letters, Vol. 31, Issue 18

##
The three-boson system with short-range interactions

journal, February 1999

- Bedaque, P. F.; Hammer, H. -W.; van Kolck, U.
- Nuclear Physics A, Vol. 646, Issue 4

##
Renormalization of the Three-Body System with Short-Range Interactions

journal, January 1999

- Bedaque, P. F.; Hammer, H. -W.; van Kolck, U.
- Physical Review Letters, Vol. 82, Issue 3

##
A method to measure the antikaon–nucleon scattering length in lattice QCD

journal, November 2009

- Lage, Michael; Meißner, Ulf-G.; Rusetsky, Akaki
- Physics Letters B, Vol. 681, Issue 5

##
Moving multichannel systems in a finite volume with application to proton-proton fusion

journal, November 2013

- Briceño, Raúl A.; Davoudi, Zohreh
- Physical Review D, Vol. 88, Issue 9

##
Timelike pion form factor in lattice QCD

journal, March 2015

- Feng, Xu; Aoki, Sinya; Hashimoto, Shoji
- Physical Review D, Vol. 91, Issue 5

##
Weak Transition Matrix Elements¶from Finite-Volume Correlation Functions

journal, May 2001

- Lellouch, Laurent; Lüscher, Martin
- Communications in Mathematical Physics, Vol. 219, Issue 1

##
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.

##
Resonance scattering phase shifts on a non-rest-frame lattice

journal, September 1995

- Rummukainen, K.; Gottlieb, Steven
- Nuclear Physics B, Vol. 450, Issue 1-2

##
Relativistic, model-independent, multichannel $2\to 2$ transition amplitudes in a finite volume

journal, July 2016

- Briceño, Raúl A.; Hansen, Maxwell T.
- Physical Review D, Vol. 94, Issue 1

##
The I = 1 pion–pion scattering amplitude and timelike pion form factor from Nf = 2 + 1 lattice QCD

journal, February 2019

- Andersen, Christian; Bulava, John; Hörz, Ben
- Nuclear Physics B, Vol. 939

##
Modified Isobar Approximation for $\pi -\rho $ Scattering

journal, June 1965

- Holman, Wayne J.
- Physical Review, Vol. 138, Issue 5B

##
Relativistic, model-independent, three-particle quantization condition

journal, December 2014

- Hansen, Maxwell T.; Sharpe, Stephen R.
- Physical Review D, Vol. 90, Issue 11

##
Three particles in a finite volume

journal, May 2012

- Polejaeva, K.; Rusetsky, A.
- The European Physical Journal A, Vol. 48, Issue 5

##
A solvable three-body model in finite volume

journal, November 2017

- Guo, Peng; Gasparian, Vladimir
- Physics Letters B, Vol. 774

##
Coupled $\pi \pi $ , $K\overline{K}$ scattering in $P$ -wave and the $\rho $ resonance from lattice QCD

journal, November 2015

- Wilson, David J.; Briceño, Raúl A.; Dudek, Jozef J.
- Physical Review D, Vol. 92, Issue 9

##
Three-particle scattering amplitudes from a finite volume formalism

journal, May 2013

- Briceño, Raúl A.; Davoudi, Zohreh
- Physical Review D, Vol. 87, Issue 9

##
Lattice QCD and the Timelike Pion Form Factor

journal, August 2011

- Meyer, Harvey B.
- Physical Review Letters, Vol. 107, Issue 7

##
Two-particle multichannel systems in a finite volume with arbitrary spin

journal, April 2014

- Briceño, Raúl A.
- Physical Review D, Vol. 89, Issue 7

##
Expressing the three-particle finite-volume spectrum in terms of the three-to-three scattering amplitude

journal, December 2015

- Hansen, Maxwell T.; Sharpe, Stephen R.
- Physical Review D, Vol. 92, Issue 11

##
Minimal three-body equations with finite-range effects

journal, October 1975

- Amado, R. D.
- Physical Review C, Vol. 12, Issue 4

##
Towards three-body unitarity in ${D}^{+}\to {K}^{-}{\pi}^{+}{\pi}^{+}$

journal, November 2011

- Magalhães, P. C.; Robilotta, M. R.; Guimarães, K. S. F. F.
- Physical Review D, Vol. 84, Issue 9

##
Multichannel $1\to 2$ transition amplitudes in a finite volume

journal, February 2015

- Briceño, Raúl A.; Hansen, Maxwell T.; Walker-Loud, André
- Physical Review D, Vol. 91, Issue 3

##
Multichannel $0\to 2$ and $1\to 2$ transition amplitudes for arbitrary spin particles in a finite volume

journal, October 2015

- Briceño, Raúl A.; Hansen, Maxwell T.
- Physical Review D, Vol. 92, Issue 7

##
Volume dependence of the energy spectrum in massive quantum field theories: II. Scattering states

journal, June 1986

- Lüscher, M.
- Communications in Mathematical Physics, Vol. 105, Issue 2

##
Dynamically-coupled partial-waves in ρπ isospin-2 scattering from lattice QCD

journal, July 2018

- Woss, Antoni J.; Thomas, Christopher E.; Dudek, Jozef J.
- Journal of High Energy Physics, Vol. 2018, Issue 7

##
Form factors of two-hadron states from a covariant finite-volume formalism

journal, August 2019

- Baroni, Alessandro; Briceño, Raúl A.; Hansen, Maxwell T.
- Physical Review D, Vol. 100, Issue 3

##
Tetraquark operators in lattice QCD and exotic flavour states in the charm sector

journal, November 2017

- Cheung, Gavin K. C.; Thomas, Christopher E.; Dudek, Jozef J.
- Journal of High Energy Physics, Vol. 2017, Issue 11

##
Minimal Three-Body Scattering Theory

journal, July 1974

- Amado, R. D.
- Physical Review Letters, Vol. 33, Issue 5

##
Three-particle systems with resonant subprocesses in a finite volume

journal, January 2019

- Briceño, Raúl A.; Hansen, Maxwell T.; Sharpe, Stephen R.
- Physical Review D, Vol. 99, Issue 1

##
Two particle states and the *S* -matrix elements in multi-channel scattering

journal, July 2005

- He, Song; Feng, Xu; Liu, Chuan
- Journal of High Energy Physics, Vol. 2005, Issue 07

##
Matrix elements of unstable states

journal, September 2012

- Bernard, V.; Hoja, D.; Meißner, U. -G.
- Journal of High Energy Physics, Vol. 2012, Issue 9

##
Three-body unitarity in the finite volume

journal, December 2017

- Mai, M.; Döring, M.
- The European Physical Journal A, Vol. 53, Issue 12

##
Scattering phases for meson and baryon resonances on general moving-frame lattices

journal, November 2012

- Göckeler, M.; Horsley, R.; Lage, M.
- Physical Review D, Vol. 86, Issue 9

##
Isoscalar $\pi \pi $ Scattering and the $\sigma $ Meson Resonance from QCD

journal, January 2017

- Briceño, Raul A.; Dudek, Jozef J.; Edwards, Robert G.
- Physical Review Letters, Vol. 118, Issue 2

##
A framework for the calculation of the <mml:math altimg="si1.gif" overflow="scroll" xmlns:xocs="http://www.elsevier.com/xml/xocs/dtd" xmlns:xs="http://www.w3.org/2001/XMLSchema" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns="http://www.elsevier.com/xml/ja/dtd" xmlns:ja="http://www.elsevier.com/xml/ja/dtd" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:tb="http://www.elsevier.com/xml/common/table/dtd" xmlns:sb="http://www.elsevier.com/xml/common/struct-bib/dtd" xmlns:ce="http://www.elsevier.com/xml/common/dtd" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:cals="http://www.elsevier.com/xml/common/cals/dtd" xmlns:sa="http://www.elsevier.com/xml/common/struct-aff/dtd"><mml:mi mathvariant="normal">Δ</mml:mi><mml:mi>N</mml:mi><mml:msup><mml:mrow><mml:mi>γ</mml:mi></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msup></mml:math> transition form factors on the lattice

journal, September 2014

- Agadjanov, Andria; Bernard, Véronique; Meißner, Ulf-G.
- Nuclear Physics B, Vol. 886

##
Resonances in Coupled $\pi K\text{\u2212}\eta K$ Scattering from Quantum Chromodynamics

journal, October 2014

- Dudek, Jozef J.; Edwards, Robert G.; Thomas, Christopher E.
- Physical Review Letters, Vol. 113, Issue 18

##
Finite volume corrections to the two-particle decay of states with nonzero momentum

journal, December 2005

- Christ, Norman H.; Kim, Changhoan; Yamazaki, Takeshi
- Physical Review D, Vol. 72, Issue 11

##
Scattering processes and resonances from lattice QCD

journal, April 2018

- Briceño, Raúl A.; Dudek, Jozef J.; Young, Ross D.
- Reviews of Modern Physics, Vol. 90, Issue 2

##
Three-Body Unitarity and Khuri-Treiman Amplitudes

journal, December 1966

- Aitchison, I. J. R.; Pasquier, R.
- Physical Review, Vol. 152, Issue 4

##
Extraction of isoscalar $\pi \pi $ phase shifts from lattice QCD

journal, July 2018

- Guo, D.; Alexandru, A.; Molina, R.
- Physical Review D, Vol. 98, Issue 1

##
Elastic $I=3/2$ $p$ -wave nucleon-pion scattering amplitude and the $\mathrm{\Delta}(1232)$ resonance from ${N}_{f}=2+1$ lattice QCD

journal, January 2018

- Andersen, Christian Walther; Bulava, John; Hörz, Ben
- Physical Review D, Vol. 97, Issue 1

##
Numerical study of the relativistic three-body quantization condition in the isotropic approximation

journal, July 2018

- Briceño, Raúl A.; Hansen, Maxwell T.; Sharpe, Stephen R.
- Physical Review D, Vol. 98, Issue 1

##
Implementing the three-particle quantization condition including higher partial waves

journal, March 2019

- Blanton, Tyler D.; Romero-López, Fernando; Sharpe, Stephen R.
- Journal of High Energy Physics, Vol. 2019, Issue 3

##
Light nuclei and hypernuclei from quantum chromodynamics in the limit of SU(3) flavor symmetry

journal, February 2013

- Beane, S. R.; Chang, E.; Cohen, S. D.
- Physical Review D, Vol. 87, Issue 3

##
An ${a}_{0}$ resonance in strongly coupled $\pi \eta $ , $K\overline{K}$ scattering from lattice QCD

journal, May 2016

- Dudek, Jozef J.; Edwards, Robert G.; Wilson, David J.
- Physical Review D, Vol. 93, Issue 9

##
Two-particle states on a torus and their relation to the scattering matrix

journal, May 1991

- Lüscher, Martin
- Nuclear Physics B, Vol. 354, Issue 2-3

##
Three-body spectrum in a finite volume: The role of cubic symmetry

journal, June 2018

- Döring, M.; Hammer, H. -W.; Mai, M.
- Physical Review D, Vol. 97, Issue 11

##
Unitarity and the evaluation of discontinuities

journal, October 1962

- Olive, D. I.
- Il Nuovo Cimento, Vol. 26, Issue 1

##
Construction of Coupled Scattering and Production Amplitudes Satisfying Analyticity and Unitarity

journal, May 1960

- Bjorken, J. D.
- Physical Review Letters, Vol. 4, Issue 9

##
Determination of s- and p-wave I = 1/2 Kπ scattering amplitudes in Nf = 2 + 1 lattice QCD

journal, July 2018

- Brett, Ruairí; Bulava, John; Fallica, Jacob
- Nuclear Physics B, Vol. 932

##
Three-body unitarity with isobars revisited

journal, September 2017

- Mai, M.; Hu, B.; Döring, M.
- The European Physical Journal A, Vol. 53, Issue 9

##
Baryon masses at nonzero isospin/kaon density

journal, October 2013

- Detmold, William; Nicholson, Amy N.
- Physical Review D, Vol. 88, Issue 7

##
Unitary states for three pions

journal, July 1975

- Ascoli, G.; Wyld, H. W.
- Physical Review D, Vol. 12, Issue 1

##
Unitarity and Production Amplitudes

journal, July 1962

- Cook, L. F.; Lee, B. W.
- Physical Review, Vol. 127, Issue 1

##
Finite-volume effects for two-hadron states in moving frames

journal, October 2005

- Kim, C. H.; Sachrajda, C. T.; Sharpe, Stephen R.
- Nuclear Physics B, Vol. 727, Issue 1-2

##
Phenomenology of relativistic $$\mathbf {3}\rightarrow \mathbf {3}$$ 3 → 3 reaction amplitudes within the isobar approximation

journal, January 2019

- Jackura, A.; Fernández-Ramírez, C.; Mathieu, V.
- The European Physical Journal C, Vol. 79, Issue 1

##
Scattering and Production Amplitudes with Unstable Particles

journal, October 1962

- Ball, J. S.; Frazer, W. R.; Nauenberg, M.
- Physical Review, Vol. 128, Issue 1