A relativistic, model-independent, three-particle quantization condition

Abstract

We present a generalization of Luescher's relation between the finite-volume spectrum and scattering amplitudes to the case of three particles. We consider a relativistic scalar field theory in which the couplings are arbitrary aside from a Z2 symmetry that removes vertices with an odd number of particles. The theory is assumed to have two-particle phase shifts that are bounded by π/2 in the regime of elastic scattering. We determine the spectrum of the finite-volume theory from the poles in the odd-particle-number finite-volume correlator, which we analyze to all orders in perturbation theory. We show that it depends on the infinite-volume two-to-two K-matrix as well as a nonstandard infinite-volume three-to-three K-matrix. A key feature of our result is the need to subtract physical singularities in the three-to-three amplitude and thus deal with a divergence-free quantity. This allows our initial, formal result to be truncated to a finite dimensional determinant equation. At present, the relation of the three-to-three K-matrix to the corresponding scattering amplitude is not known, although previous results in the non-relativistic limit suggest that such a relation exists.

Revtex, 59 pages, 20 figures (v2: style improved, typos corrected, no change in results) (v3: further typos corrected; consistent with published version)

References in corpus (4)

• Scalar mesons in a finite volume
• n-Boson Energies at Finite Volume and Three-Boson Interactions
• Three-boson problem at low energy and Implications for dilute Bose-Einstein condensates
• Signatures of S-wave bound-state formation in finite volume

Cited by in corpus (75)

• Three particle quantization condition in a finite volume: 2. general formalism and the analysis of data
• Three-body Unitarity in the Finite Volume
• Three-particle quantization condition in a finite volume: 1. The role of the three-particle force
• From deep inelastic scattering to heavy-flavor semi-leptonic decays: Total rates into multi-hadron final states from lattice QCD
• Towards a theory of hadron resonances
• Implementing the three-particle quantization condition including higher partial waves
• Three-body Unitarity with Isobars Revisited
• Finite-volume energy spectrum of the $K^{-}{K}^{-}{K}^{-}$ system
• Nuclear matrix elements from lattice QCD for electroweak and beyond-Standard-Model processes
• Flux Tube Spectra from Approximate Integrability at Low Energies
• Three-body interactions from the finite-volume QCD spectrum
• The energy-dependent ${\pi }^{+}{\pi }^{+}{\pi }^{+}$ scattering amplitude from QCD
• Relativistic three-particle quantization condition for nondegenerate scalars
• Alternative derivation of the relativistic three-particle quantization condition
• Multi-particle systems on the lattice and chiral extrapolations: a brief review
• Equivalence of relativistic three-particle quantization conditions
• Scattering of two and three physical pions at maximal isospin from lattice QCD
• Three-particle finite-volume formalism for ${\pi }^{+}{\pi }^{+}{K}^{+}$ and related systems
• One spatial dimensional finite volume three-body interaction for a short-range potential
• $DDK$ system in finite volume
• Novel approaches in Hadron Spectroscopy
• Applying the relativistic quantization condition to a three-particle bound state in a periodic box
• Multiplicity dependence of (multi-)strange baryons in the canonical ensemble with phase shift corrections
• The role of boundary conditions in quantum computations of scattering observables
• An numerical approach for finite volume three-body interaction
• Solving relativistic three-body integral equations in the presence of bound states
• Relativistic-invariant formulation of the NREFT three-particle quantization condition
• Long-range electroweak amplitudes of single hadrons from Euclidean finite-volume correlation functions
• Finite-volume energy shift of the three-pion ground state
• Relativistic $N$-particle energy shift in finite volume
• Three-body dynamics of the $a_1(1260)$ resonance from lattice QCD
• Implementing the three-particle quantization condition for ${\pi }^{+}{\pi }^{+}{K}^{+}$ and related systems
• Interactions of $\pi K$, $\pi \pi K$ and $KK\pi$ systems at maximal isospin from lattice QCD
• Decay amplitudes to three hadrons from finite-volume matrix elements
• Constraining $1+\mathcal{J}\to 2$ coupled-channel amplitudes in finite-volume
• Modeling few-body resonances in finite volume
• Consistency checks for two-body finite-volume matrix elements: I. Conserved currents and bound states
• Uncertainty Quantification in Lattice QCD Calculations for Nuclear Physics
• On the three-particle analog of the Lellouch-Lüscher formula
• A lattice model of heavy-light three-body system
• Light- and strange-quark mass dependence of the $\rho (770)$ meson revisited
• Coulomb corrections to two-particle interaction in artificial traps
• Hamiltonian effective field theory in elongated or moving finite volume
• Six-pion amplitude
• Charged particles interaction in both a finite volume and a uniform magnetic field
• Visualizing resonances in finite volume
• On the Extraction of Low-energy Constants of Single- and Double-$\beta$ Decays from Lattice QCD: A Sensitivity Analysis
• Variations on the Maiani-Testa approach and the inverse problem
• Two-neutrino double-beta decay in pionless effective field theory from a Euclidean finite-volume correlation function
• The isospin-3 three-particle $K$-matrix at NLO in ChPT
• Nuclear reactions in artificial traps
• Particle-dimer approach for the Roper resonance in a finite volume
• Bound states in the B-matrix formalism for the three-body scattering
• $DK/D\pi$ scattering and an exotic virtual bound state at the $SU(3)$ flavour symmetric point from lattice QCD
• Finite-volume energy shift of the three-nucleon ground state
• An alternative scheme for effective range corrections in pionless EFT
• Finite-volume matrix elements in multi-boson states
• Analytic expansions of multi-hadron finite-volume energies: I. Two-particle states
• Three-particle formalism for multiple channels: the $\eta \pi \pi +K\overline{K}\pi$ system in isosymmetric QCD
• The three-pion $K$-matrix at NLO in ChPT
• Charged particles interaction in both a finite volume and a uniform magnetic field II: topological and analytic properties of a magnetic system
• Spurious poles in a finite volume
• Spectrum of light nuclei in a finite volume
• Six-meson amplitude in QCD-like theories
• The $\pi \pi$ scattering amplitude at large $N_{\text{c}}$
• Quark Flavor Physics Review
• Toward extracting scattering phase shift from integrated correlation functions on quantum computers
• Perturbative application of next-to-leading order pionless EFT for $A\le 3$ nuclei in a finite volume
• Pion scattering in finite volume within the Inverse Amplitude Method
• Rediscovery of Numerical Lüscher's Formula from the Neural Network
• Three-particle quantization condition: an update
• Nuclear correlation functions using first-principle calculations of lattice quantum chromodynamics
• Testing a new method for scattering in finite volume in the ${\varphi }^4$ theory
• Three-pion scattering: From the chiral Lagrangian to the lattice
• Formal Developments for Lattice QCD with Applications to Hadronic Systems