Expressing the three-particle finite-volume spectrum in terms of the three-to-three scattering amplitude

Abstract

In this article we complete our formalism relating the finite-volume energy spectrum of a scalar quantum field theory to the three-to-three scattering amplitude, ${\mathcal{M}}_3$. In previous work we found a quantization condition relating the spectrum to a non-standard infinite-volume quantity, denoted ${\mathcal{K}}_{\mathrm{df},3}$. Here we present the relation between ${\mathcal{K}}_{\mathrm{df},3}$ and ${\mathcal{M}}_3$. We then discuss briefly how our now completed formalism can be practically implemented to extract ${\mathcal{M}}_3$ from the finite-volume energy spectrum.

24 pages, 6 figures, v2: added motivation in the introduction, added figure to clarify explanations, results unchanged---matches published version

References in corpus (4)

• A relativistic, model-independent, three-particle quantization condition
• Multichannel 0-to-2 and 1-to-2 transition amplitudes for arbitrary spin particles 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

Cited by in corpus (142)

• Hadronic molecules
• The $XYZ$ states: experimental and theoretical status and perspectives
• Non-Standard Heavy Mesons and Baryons, an Experimental Review
• Scattering processes and resonances from lattice QCD
• Relating the finite-volume spectrum and the two-and-three-particle $S$ matrix for relativistic systems of identical scalar particles
• 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
• Lattice QCD and Three-particle Decays of Resonances
• An $a_0$ resonance in strongly coupled $\pi \eta$, $K\overline{K}$ scattering from lattice QCD
• Three-body spectrum in a finite volume: the role of cubic symmetry
• Isoscalar $\pi \pi ,K\overline{K},\eta \eta$ scattering and the $\sigma ,f_0,f_2$ mesons from QCD
• $I=1$ and $I=2$ $\pi -\pi$ scattering phase shifts from $N_{\mathrm{f}}=2+1$ lattice QCD
• Finite-volume spectrum of ${\pi }^{+}{\pi }^{+}$ and ${\pi }^{+}{\pi }^{+}{\pi }^{+}$ systems
• Two- and three-pion finite-volume spectra at maximal isospin from lattice QCD
• Three-particle systems with resonant subprocesses in a finite volume
• $I=3$ three-pion scattering amplitude from lattice QCD
• 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
• The $I=1$ pion-pion scattering amplitude and timelike pion form factor from $N_{\mathrm{f}}=2+1$ lattice QCD
• Linear Response on a Quantum Computer
• Pion-nucleon scattering in the Roper channel from lattice QCD
• Implementing the three-particle quantization condition including higher partial waves
• Generalizing the relativistic quantization condition to include all three-pion isospin channels
• Three pion spectrum in the $I=3$ channel from lattice QCD
• Numerical exploration of three relativistic particles in a finite volume including two-particle resonances and bound states
• Numerical study of the relativistic three-body quantization condition in the isotropic approximation
• An solvable three-body model in finite volume
• Three-body unitarity versus finite-volume ${\pi }^{+}{\pi }^{+}{\pi }^{+}$ spectrum from lattice QCD
• $\rho$ and $K^{\ast }$ resonances on the lattice at nearly physical quark masses and $N_f=2$
• The $\pi \pi \to \pi {\gamma }^{\star }$ amplitude and the resonant $\rho \to \pi {\gamma }^{\star }$ transition from lattice QCD
• Three-body Unitarity with Isobars Revisited
• Relativistic, model-independent, multichannel $2\to 2$ transition amplitudes in a finite volume
• On the Equivalence of Three-Particle Scattering Formalisms
• Towards grounding nuclear physics in QCD
• Energy shift of the three-particle system in a finite volume
• Finite-volume energy spectrum of the $K^{-}{K}^{-}{K}^{-}$ system
• Hadro-quarkonium from Lattice QCD
• Hadrons and Nuclei
• Nuclear matrix elements from lattice QCD for electroweak and beyond-Standard-Model processes
• Unitarity of the infinite-volume three-particle scattering amplitude arising from a finite-volume formalism
• Scattering amplitudes from finite-volume spectral functions
• 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
• Determination of $s$- and $p$-wave $I=1/2$ $K\pi$ scattering amplitudes in $N_{\mathrm{f}}=2+1$ lattice QCD
• The $b_1$ resonance in coupled $\pi \omega$, $\pi \varphi$ scattering from lattice QCD
• Alternative derivation of the relativistic three-particle quantization condition
• Equivalence of relativistic three-particle quantization conditions
• Multi-particle systems on the lattice and chiral extrapolations: a brief review
• Scattering of two and three physical pions at maximal isospin from lattice QCD
• Perturbative results for two and three particle threshold energies in finite volume
• Variational approach to $N$-body interactions in finite volume
• Three-particle finite-volume formalism for ${\pi }^{+}{\pi }^{+}{K}^{+}$ and related systems
• Signatures of few-body resonances in finite volume
• One spatial dimensional finite volume three-body interaction for a short-range potential
• Two- and three-body interactions in ${\phi }^4$ theory from lattice simulations
• Hadron-Hadron Interactions from $N_f=2+1+1$ Lattice QCD: isospin-2 $\pi \pi$ scattering length
• $DDK$ system in finite volume
• Structure of the Roper Resonance from Lattice QCD Constraints
• New Ideas in Constraining Nuclear Forces
• Applying the relativistic quantization condition to a three-particle bound state in a periodic box
• The role of boundary conditions in quantum computations of scattering observables
• Novel approaches in Hadron Spectroscopy
• An numerical approach for finite volume three-body interaction
• Relativistic-invariant formulation of the NREFT three-particle quantization condition
• Solving relativistic three-body integral equations in the presence of bound states
• Pion scattering in the isospin I=2 channel from elongated lattices
• Nuclear Forces for Precision Nuclear Physics -- a collection of perspectives
• Long-range electroweak amplitudes of single hadrons from Euclidean finite-volume correlation functions
• Connecting physical resonant amplitudes and lattice QCD
• Finite-volume energy shift of the three-pion ground state
• Phenomenology of Relativistic $3\to 3$ Reaction Amplitudes within the Isobar Approximation
• Multiple-particle interaction in $1+1$ dimensional lattice model
• Relativistic $N$-particle energy shift in finite volume
• Three-particle bound states in a finite volume: unequal masses and higher partial waves
• Chiral Extrapolations of the $\mathbf{\rho }(770)$ Meson in ${\mathbf{N}}_{\mathbf{f}}=2+1$ Lattice QCD Simulations
• 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
• Multi-${\pi }^{+}$ systems in finite volume
• Dalitz plots and lineshape of $a_1(1260)$ from a relativistic three-body unitary approach
• Short-distance matrix elements for $D^0$-meson mixing for $N_f=2+1$ lattice QCD
• Decay amplitudes to three hadrons from finite-volume matrix elements
• Interactions of $\pi K$, $\pi \pi K$ and $KK\pi$ systems at maximal isospin from lattice QCD
• 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
• Testing the threshold expansion for three-particle energies at fourth order in ${\varphi }^4$ theory
• Quark-mass dependence in $\omega \to 3\pi$ decays
• Partial Wave Mixing in Hamiltonian Effective Field Theory
• On the effect of excited states in lattice calculations of the nucleon axial charge
• Propagation of particles on a torus
• Opportunities for lattice QCD in quark and lepton flavor physics
• A lattice model of heavy-light three-body system
• On the three-particle analog of the Lellouch-Lüscher formula
• Consistency checks for two-body finite-volume matrix elements: II. Perturbative systems
• Hamiltonian effective field theory in elongated or moving finite volume
• Threshold expansion of the three-particle quantization condition
• Six-pion amplitude
• 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
• Visualizing resonances in finite volume
• Threshold expansion formula of $N$-boson in finite volume from variational approach
• On-shell representations of two-body transition amplitudes: single external current
• Quantum Monte Carlo calculations of two neutrons in finite volume
• Nucleon resonance structure in the finite volume of lattice QCD
• The isospin-3 three-particle $K$-matrix at NLO in ChPT
• Particle-dimer approach for the Roper resonance in a finite volume
• Khuri-Treiman equations for $3\pi$ decays of particles with spin
• Few-body bound states and resonances in 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
• $I=2$ $\pi \pi$ scattering phase shift from the HAL QCD method with the LapH smearing
• Finite-volume energy shift of the three-nucleon ground state
• An alternative scheme for effective range corrections in pionless EFT
• Volume extrapolation via eigenvector continuation
• Axial-vector $D_1$ hadrons in $D^{\ast }\pi$ scattering from QCD
• Analytic expansions of multi-hadron finite-volume energies: I. Two-particle states
• Two-current transition amplitudes with two-body final states
• Three-particle formalism for multiple channels: the $\eta \pi \pi +K\overline{K}\pi$ system in isosymmetric QCD
• Charged-particle bound states in periodic boxes
• The three-pion $K$-matrix at NLO in ChPT
• Complex scaling in finite volume
• Spurious poles in a finite volume
• Six-meson amplitude in QCD-like theories
• Hadron Structure and Spectrum from the Lattice
• Prospects for ${\gamma }^{\star }{\gamma }^{\star }\to \pi \pi$ via lattice QCD
• Meson-Nucleon Scattering Amplitudes from Lattice QCD
• Finite density condensation and scattering data - a study in ${\varphi }^4$ lattice field theory
• Progress report on the relativistic three-particle quantization condition
• Progress in three-particle scattering from LQCD
• Pion-pion scattering and the timelike pion form factor from $N_{\mathrm{f}}=2+1$ lattice QCD simulations using the stochastic LapH method
• The $\pi \pi$ scattering amplitude at large $N_{\text{c}}$
• 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
• Meson electro-/photo-production from QCD
• From QCD to Physical Resonances
• 3-body quantization condition in a unitary formalism
• Three particles on the lattice
• Progress on the three-particle quantization condition
• Testing a new method for scattering in finite volume in the ${\varphi }^4$ theory
• Kaon decays and other hadronic processes in lattice QCD