Two- and Three-Pion Finite-Volume Spectra at Maximal Isospin from Lattice QCD

Blanton

et al. 2019

J. High Energ. Phys.

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.

Section: Introductionmentioning

confidence: 99%

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

Romero-López

Blanton

et al. 2019

J. High Energ. Phys.

“…In previous work, the three-particle coupling was extracted using the ground state energy in QCD [30,33,34], and also in ϕ 4 theory [35]. Going beyond this, the determination of complete spectra with quantum numbers of three pions has been achieved by multiple groups in the last two years [36][37][38]. In fact, very recently, a large number of three-π + levels (including those in moving frames) has been combined with the RFT formalism to constrain the three-particle scattering amplitude from first principles QCD [24].…”

Section: Jhep07(2020)047mentioning

confidence: 99%

“…Finally, for the J P = 3 − ω 3 (1670), the momentum-space amplitude must transform as J P = 3 + . One possible form is 36) where the second term is required to project against a J P = 1 + component. The corresponding cubic-group irrep, A − 2 , appears first in the same three-pion orbit as for the ω, for then the axial current A µ is nonzero.…”

Section: )mentioning

confidence: 99%

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

Hansen

Romero-López

2020

J. High Energ. Phys.

We present a generalization of the relativistic, finite-volume, three-particle quantization condition for non-identical pions in isosymmetric QCD. The resulting formalism allows one to use discrete finite-volume energies, determined using lattice QCD, to constrain scattering amplitudes for all possible values of two-and three-pion isospin. As for the case of identical pions considered previously, the result splits into two steps: the first defines a non-perturbative function with roots equal to the allowed energies, E n (L), in a given cubic volume with side-length L. This function depends on an intermediate three-body quantity, denoted K df,3 , which can thus be constrained from lattice QCD input. The second step is a set of integral equations relating K df,3 to the physical scattering amplitude, M 3. Both of the key relations, E n (L) ↔ K df,3 and K df,3 ↔ M 3 , are shown to be block-diagonal in the basis of definite three-pion isospin, I πππ , so that one in fact recovers four independent relations, corresponding to I πππ = 0, 1, 2, 3. We also provide the generalized threshold expansion of K df,3 for all channels, as well as parameterizations for all three-pion resonances present for I πππ = 0 and I πππ = 1. As an example of the utility of the generalized formalism, we present a toy implementation of the quantization condition for I πππ = 0, focusing on the quantum numbers of the ω and h 1 resonances.

“…Examples of the successful combination of these methods are in Refs. [1][2][3][4][5][6][7][8][9][10][11]. 1 A key output of the theoretical formalism is a quantization condition, an equation whose solutions give the finite-volume three-particle energy levels in terms of infinite-volume scattering quantities.…”

Section: Introductionmentioning

confidence: 99%

Alternative derivation of the relativistic three-particle quantization condition

Blanton

2020

Phys. Rev. D

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 Z 2 symmetry. The simplification is afforded by using a three-particle quasilocal K matrix that is not fully symmetrized, K ðu;uÞ 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, K df;3. The new derivation is fully explicit, allowing, for example, a closed-form expression for 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.