# Alternative derivation of the relativistic three-particle quantization condition

## Abstract

We present a simplified derivation of the relativistic three-particle quantization condition for identical, spinless particles described by a generic relativistic field theory satisfying a $$\mathbb{Z}$$2 symmetry. The simplification is afforded by using a three-particle quasilocal K matrix that is not fully symmetrized, $$\mathscr{K}$$˜$$^{(u,u)}_{df,3}$$, and makes extensive use of time-ordered perturbation theory (TOPT). We obtain a new form of the quantization condition. This new form can then be related algebraically to the standard quantization condition, which depends on a fully symmetric three-particle K matrix, $$\mathscr{K}$$_{df,3}. The new derivation is fully explicit, allowing, for example, a closed-form expression for $$\mathscr{K}$$_{df,3} to be given in terms of TOPT amplitudes. The new form of the quantization condition is similar in structure to that obtained in the "finite-volume unitarity" approach, and in a companion paper we make this connection concrete. Our simplified approach should also allow a more straightforward generalization of the quantization condition to nondegenerate particles, and perhaps also to more than three particles.

- Authors:

- Publication Date:

- Research Org.:
- Univ. of Washington, Seattle, WA (United States)

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

- OSTI Identifier:
- 1668893

- Alternate Identifier(s):
- OSTI ID: 1670147; OSTI ID: 1774030

- Grant/Contract Number:
- SC0011637

- Resource Type:
- Published Article

- Journal Name:
- Physical Review D

- Additional Journal Information:
- Journal Name: Physical Review D Journal Volume: 102 Journal Issue: 5; Journal ID: ISSN 2470-0010

- Publisher:
- American Physical Society (APS)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 73 NUCLEAR PHYSICS AND RADIATION PHYSICS; Lattice QCD; effective field theory; quantization condition; three-body scattering; few-body systems; lattice field theory

### Citation Formats

```
Blanton, Tyler D., and Sharpe, Stephen R. Alternative derivation of the relativistic three-particle quantization condition. United States: N. p., 2020.
Web. https://doi.org/10.1103/PhysRevD.102.054520.
```

```
Blanton, Tyler D., & Sharpe, Stephen R. Alternative derivation of the relativistic three-particle quantization condition. United States. https://doi.org/10.1103/PhysRevD.102.054520
```

```
Blanton, Tyler D., and Sharpe, Stephen R. Wed .
"Alternative derivation of the relativistic three-particle quantization condition". United States. https://doi.org/10.1103/PhysRevD.102.054520.
```

```
@article{osti_1668893,
```

title = {Alternative derivation of the relativistic three-particle quantization condition},

author = {Blanton, Tyler D. and Sharpe, Stephen R.},

abstractNote = {We present a simplified derivation of the relativistic three-particle quantization condition for identical, spinless particles described by a generic relativistic field theory satisfying a $\mathbb{Z}$2 symmetry. The simplification is afforded by using a three-particle quasilocal K matrix that is not fully symmetrized, $\mathscr{K}$˜$^{(u,u)}_{df,3}$, and makes extensive use of time-ordered perturbation theory (TOPT). We obtain a new form of the quantization condition. This new form can then be related algebraically to the standard quantization condition, which depends on a fully symmetric three-particle K matrix, $\mathscr{K}$df,3. The new derivation is fully explicit, allowing, for example, a closed-form expression for $\mathscr{K}$df,3 to be given in terms of TOPT amplitudes. The new form of the quantization condition is similar in structure to that obtained in the "finite-volume unitarity" approach, and in a companion paper we make this connection concrete. Our simplified approach should also allow a more straightforward generalization of the quantization condition to nondegenerate particles, and perhaps also to more than three particles.},

doi = {10.1103/PhysRevD.102.054520},

journal = {Physical Review D},

number = 5,

volume = 102,

place = {United States},

year = {2020},

month = {9}

}

https://doi.org/10.1103/PhysRevD.102.054520

#### Figures / Tables:

*C*

_{3, L}. Time flows from right to left, with the black circle (blue square) representing

*σ*

^{†}(

*σ*). Relevant cuts are shown by vertical (red) dashed lines, while irrelevant cuts are shown by solid (magenta) integral signs. The factors associated withmore »

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*Figures/Tables have been extracted from DOE-funded journal article accepted manuscripts.*