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 two- particle K matrix that required the absence of two-particle bound states or narrow two- particle 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 dimer- particle 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.
Highlights
The first is based on a generic relativistic effective field theory [1, 33,34,35,36,37], the second uses non-relativistic effective field theory [38,39,40], and the third applies unitary constraints in finite volumes [41, 42]
In this work we have presented an extension of the formalism of ref. [1] that allows the study of three-body systems in the presence of two-body resonances or bound states
This removes a major shortcoming of the original formalism, which had previously only been resolved by a more complicated approach requiring the introduction of a fictitious two-body channel for each resonance [36]
Summary
This leads to a systematic truncation scheme in which ≤ max in all quantities entering the quantization condition, including the kinematic functions [1] Since both spectator momentum and angular-momentum index sums are truncated, the problem reduces to one involving finite matrices, suitable for numerical implementation. [35], Gs differs — here we use its relativistic form, since this leads to a Lorentz invariant Kdf,3 This invariance plays a role when expanding this function about three-particle threshold. Are the total squared energy and particle momentum in the c.m. frame of the nonspectator pair, and δs the s-wave phase shift Were it not for the second term in the denominator of eq (2.6), K2s would equal K2/(2ωk), where. In the case that K2 is restricted to the s-wave, this implies that Kdf, must be evaluated in the isotropic approximation, to which we turn
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