Three-body Unitarity in the Finite Volume

Abstract: The physical interpretation of lattice QCD simulations, performed in a small volume, requires an extrapolation to the infinite volume. A method is proposed to perform such an extrapolation for three interacting particles at energies above threshold. For this, a recently formulated relativistic $3\to 3$ amplitude based on the isobar formulation is adapted to the finite volume. The guiding principle is two- and three-body unitarity that imposes the imaginary parts of the amplitude in the infinite volume. In turn, these imaginary parts dictate the leading power-law finite-volume effects. It is demonstrated that finite-volume poles arising from the singular interaction, from the external two-body sub-amplitudes, and from the disconnected topology cancel exactly leaving only the genuine three-body eigenvalues. The corresponding quantization condition is derived for the case of three identical scalar-isoscalar particles and its numerical implementation is demonstrated.

• The paper develops a method to extrapolate finite-volume lattice QCD simulations of three interacting particles to infinite volume while ensuring two- and three-body unitarity.
• It adapts a relativistic 3→3 amplitude in the isobar framework to derive a quantization condition that cancels finite-volume singularities from two-body sub-amplitudes.
• The approach resolves discrepancies between continuous infinite-volume spectra and discrete lattice spectra, facilitating more accurate resonance extraction from lattice data.

Three-body Unitarity in the Finite Volume

This paper presents a method to extrapolate lattice QCD simulations of three interacting particles from a finite volume to the infinite volume, a process essential for accurate physical interpretations. The authors rely on a recently developed relativistic $3 \to 3$ amplitude within the isobar framework, which they adapt for finite volume studies, prioritizing two- and three-body unitarity to shape the amplitude's imaginary components in infinite volume. This ensures that the leading finite-volume corrections align with the finite-volume poles involved in the three-body system.

Methodological Development

The paper primarily addresses how to transform scattering problems in a finite cubic lattice with periodic boundary conditions into quantities applicable to infinite volume. This conversion is complex, given the distinction between continuous spectral functions and discrete lattice spectra. The core advancement lies in deriving a quantization condition for three identical scalar-isoscalar particles and demonstrating its numerical implementation.

Finite-volume effects become significant when the quark masses approximate physical values, where resonances appear and bound states dissolve. The authors implement a relevant adjustment to the $3 \to 3$ amplitude, ensuring it remains consistent with infinite-volume unitarity. They also delineate how finite-volume singularities arising in the interaction, external two-body sub-amplitudes, and disconnected topologies are negated to only retain genuine three-body eigenvalues.

Analytical Insights

The paper's approach relies on partial-wave expansion to reduce the multiplicitous nature of $3 \to 3$ scattering, simplifying the theoretical treatment via isobar and spectator quantum numbers. The formalism respects unitarity across energy thresholds beyond the three-body breakup. It shows that when all three interactions are on-mass-shell, the finite-volume effects manifest as power-law corrections, thereby supporting three-body quantization.

The development of the method aligns with the need for precise theoretical tools to interpret rapidly improving lattice QCD data, anticipating the role of three-hadron operators. By resolving the discrepancies between continuous energy spectra in three-body scattering and discrete lattice spectra, the paper contributes to a deeper understanding of nucleon and meson interactions and their compositional mechanics.

Numerical Implementation

In testing the derivations, the authors consider scalar-isoscalar particles, effectively setting up a system of equations representing the finite-volume three-body problem. A key facet of the implementation is how divergences in two-body interactions cancel out, isolating genuine three-body dynamics. This aspect is crucial for future studies facilitating resonance extraction from lattice QCD eigenvalues.

Future Directions

The implications for future developments in AI and computational lattice QCD are broad. This work provides a structured framework for handling complex interactions in finite lattice simulations, essential for achieving convergence with experimental findings. It demonstrates a scalable method with relevance for coupled-channel and, eventually, spin and isospin considerations. Future extensions could explore numerical stability enhancements and broaden its applicability to a diverse range of hadronic systems and quantum number configurations.

This paper significantly contributes to lattice QCD's methodological advancement, offering a refined tool for researchers to interpret increasingly accurate lattice simulation data while respecting the principles of unitarity and fidelity to infinite volume dynamics.