Numerical exploration of three relativistic particles in a finite volume including two-particle resonances and bound states

Abstract

In this work, we use an extension of the quantization condition, given in ref. [1], to numerically explore the finite-volume spectrum of three relativistic particles, in the case that two-particle subsets are either resonant or bound. The original form of the relativistic three-particle quantization condition was derived under a technical assumption on the twoparticle K matrix that required the absence of two-particle bound states or narrow twoparticle resonances. Here we describe how this restriction can be lifted in a simple way using the freedom in the definition of the K-matrix-like quantity that enters the quantization condition. With this in hand, we extend previous numerical studies of the quantization condition to explore the finite-volume signature for a variety of two-and three-particle interactions. We determine the spectrum for parameters such that the system contains both dimers (two-particle bound states) and one or more trimers (in which all three particles are bound), and also for cases where the two-particle subchannel is resonant. We also show how the quantization condition provides a tool for determining infinite-volume dimerparticle scattering amplitudes for energies below the dimer breakup. We illustrate this for a series of examples, including one that parallels physical deuteron-nucleon scattering. All calculations presented here are restricted to the case of three identical scalar particles.

Generalizing the relativistic quantization condition to include all three-pion isospin channels

“…[17,18], while those for I πππ = 0, 1, 2 are new. The implementation of the new quantization conditions is of similar complexity to the I πππ = 3 case, where there have been extensive previous studies [20,[22][23][24]. They do, however, exhibit some new features, such as the presence of odd partial waves and different relative signs between the finite-volume objects involved.…”

“…In the last few years, significant theoretical effort has been devoted to extensions and alternatives to the two-particle Lüscher formalism for more-than-two-particle systems. In particular, a three-particle quantization condition for identical (pseudo)scalars has been derived following three different approaches: 1 (i) generic relativistic effective field theory (RFT) [17][18][19][20][21][22][23][24], (ii) nonrelativistic effective field theory (NREFT) [25][26][27][28], and (iii) (relativistic) finite volume unitarity (FVU) [29][30][31]. (See ref.…”

“…The implementation of the isoscalar three-particle quantization condition requires only minor generalizations of the I πππ = 3 case implemented previously in refs. [20,[22][23][24]. Specifically, appendices A and B of ref.…”

Opportunities for Lattice QCD in quark and lepton flavor physics

“…[135] provides a parametrization of the infinite-volume three-particle scattering amplitude that is unitary [151], and has been shown to be equivalent to parametrizations used to analyze experimental scattering data [152]. Using simple parametrizations of the two-and three-particle amplitudes, as well as other well-motivated approximations, the quantization condition has been solved in simple examples [147,150,[153][154][155][156]. Because this formalism is needed to study most resonances in QCD-a topic of great interest in hadronic physics-it is likely that in the next few years a workable form of the three-particle quantization condition will be developed, including generalizations to nondegenerate particles and particles with spin.…”

Alternative derivation of the relativistic three-particle quantization condition

“…We refer to these papers in the following as HS1 and HS2, respectively. The formalism has been subsequently generalized to allow 2 ↔ 3 transitions [39], K matrix poles [40,41], and nonidentical but degenerate particles [42]. The numerical implementation of the formalism has been studied in Refs.…”