Equivalence of relativistic three-particle quantization conditions

Tyler D Blanton,Stephen R Sharpe

doi:10.1103/physrevd.102.054515

Tyler D Blanton, Stephen R Sharpe

Open Access

https://doi.org/10.1103/physrevd.102.054515

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Abstract

We show that a recently derived alternative form of the relativistic three-particle quantization condition for identical particles can be rewritten in terms of the R matrix introduced to give a unitary representation of the infinite-volume three-particle scattering amplitude. Combined with earlier work, this shows the equivalence of the relativistic effective field theory approach of Hansen and Sharpe (Refs.[1,2]) and the "finite-volume unitarity" approach of Mai and D\"oring (Refs.[3,4]). It also provides a generalization of the latter approach to arbitrary angular momenta of two-particle subsystems.

Highlights

The study of resonant three-particle systems using lattice QCD (LQCD) is becoming feasible, due to advances in the underlying theoretical formalism [1,2,3,4,5,6,7,8,9,10,11,12,13] and its practical application [4,14,15,16], as well as in algorithmic and computational methods necessary to extract three-particle spectra
One of the key steps in the formalism is the derivation of three-particle quantization conditions, equations whose solutions give the finite-volume spectrum of three-particle states as functions of infinite-volume two- and threeparticle K matrices
Ref. [27] showed that, when restricted to s-wave dimers, and assuming a constant three-particle interaction, the RFT quantization condition could be manipulated into a form that agreed with that from the finite-volume unitarity” (FVU) approach

Summary

INTRODUCTION

The study of resonant three-particle systems using lattice QCD (LQCD) is becoming feasible, due to advances in the underlying theoretical formalism [1,2,3,4,5,6,7,8,9,10,11,12,13] and its practical application [4,14,15,16], as well as in algorithmic and computational methods necessary to extract three-particle spectra (see, for example, the recent results presented in Refs. [17,18,19]). The present frontier is the application to the 3πþ system [18,25,26]. The agreement required that the quantities describing three-particle interactions in the two approaches were restricted to their simplest, momentum-independent form [27] showed that, when restricted to s-wave dimers, and assuming a constant three-particle interaction, the RFT quantization condition could be manipulated into a form that agreed with that from the FVU approach (again aside from certain technical differences). We are able to derive the FVU form of the quantization condition starting from the RFT result, and to generalize the FVU approach to dimers in all partial waves. The key inputs here are, first, the new form of the RFT quantization condition that we obtained in BS1, and, second, a generalization we derive here of the relation between the K matrix of the RFT approach and the R matrix obtained in Ref. Appendix A summarizes notation and definitions, while Appendix B discusses subtleties concerning infinitevolume limits

RECAP OF PRIOR FORMS OF THE RELATIVISTIC QUANTIZATION CONDITION

The FVU quantization condition

K 2ðρPV

Combining results

SUMMARY AND OUTLOOK

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