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Title: 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:
ORCiD logo; ORCiD logo
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}
}

Journal Article:
Free Publicly Available Full Text
Publisher's Version of Record
https://doi.org/10.1103/PhysRevD.102.054520

Figures / Tables:

FIG. 1 FIG. 1: Examples of time orderings in diagrams contributing to C3, 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 » these cuts are described in the text. Vertical columns divide contributions according to the number of relevant cuts. Horizontal rows contain the time orderings of (a) the leading-order Feynman diagram and (b) a Feynman diagram with a single four-point vertex.« less

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