[Paper Review] On the Determination of Elastic and Inelastic Nuclear Observables from Lattice QCD

Raúl A. Briceño|arXiv (Cornell University)|Nov 23, 2013

Quantum Chromodynamics and Particle Interactions135 references3 citations

TL;DR

This paper extends Lüscher's finite-volume formalism to compute elastic and inelastic nuclear observables from lattice QCD, deriving quantization conditions for two-nucleon systems with arbitrary partial waves and three-boson systems with non-zero total momentum. It provides a framework to extract infinite-volume scattering phase shifts by accounting for exponential and power-law volume corrections, enabling ab initio calculations of nuclear reactions and resonances from QCD.

ABSTRACT

One of the overarching goals of nuclear physics is to rigorously compute properties of hadronic systems directly from the fundamental theory of the strong interaction, Quantum Chromodynamics (QCD). Currently, lattice QCD (LQCD) provides the only reliable option for performing calculations of low-energy hadronic observables. LQCD calculations are necessarily performed in a finite Euclidean spacetime. As a result, it is necessary to construct formalism that maps the finite-volume observables determined via LQCD to the infinite-volume quantities of interest. This methodology is commonly referred to as the Luscher method, as it was Martin Luscher who first developed such formalism for scalar bosons with zero total momentum below inelastic thresholds. In this work, we review recent progress on the generalization of this formalism. We present a detailed derivation of the extension of Luscher's seminal work for multi-channel two-body scalar systems, two-nucleon non-relativistic systems, and three-body non-relativistic scalar systems. For all of these scenarios we allow for the total momenta of the systems of interest to be nonzero. We also present steps towards being able to study weak processes involving two-nucleon systems, in particular we show how to determine the transition amplitude for proton-proton fusion (pp->e^+ + nu_e) directly from LQCD.

Motivation & Objective

• To develop a systematic framework for extracting infinite-volume nuclear scattering observables from finite-volume lattice QCD calculations.
• To generalize Lüscher's two-body formalism to include arbitrary partial waves, spin, and parity in the nucleon-nucleon system.
• To derive a non-algebraic quantization condition for three identical bosons in a finite volume, accounting for coupled-channel effects and partial-wave mixing.
• To enable accurate determination of three-body scattering amplitudes and resonances by including non-perturbative volume corrections.
• To support future ab initio calculations of nuclear reactions, including fusion and decay processes, from QCD.

Proposed method

• Derives the quantization condition for two non-relativistic nucleons in a finite volume using the auxiliary field (dimer) formalism, valid for arbitrary partial waves and isospin symmetry.
• Applies the dimer formalism to construct a field-theoretic representation of two-nucleon systems, allowing derivation of the spectrum in finite volume via a determinant condition over angular momentum and eigenstate channels.
• Derives a non-algebraic quantization condition for three-boson systems in finite volume, expressed as a determinant condition involving dimer-boson relative motion and eigenstates, valid below the diboson breakup threshold.
• Incorporates exponential volume corrections from off-shell dimer excitations and power-law corrections above the threshold, ensuring non-perturbative treatment of finite-volume effects.
• Proposes a numerical fitting procedure to simultaneously extract energy eigenvalues from multiple irreducible representations of the cubic group and multiple boosts, enabling resolution of partial-wave mixing.
• Uses the resulting quantization condition to extract the three-body Bethe-Salpeter kernel and low-energy constants (LECs) that encode infinite-volume physics up to the four-particle threshold.

Experimental results

Research questions

• RQ1 How can the finite-volume spectrum of two-nucleon systems in lattice QCD be related to infinite-volume scattering phase shifts for arbitrary partial waves and spin states?
• RQ2 What is the correct quantization condition for three-boson systems in a finite volume, particularly when two-body bound states (dibosons) exist?
• RQ3 How do finite-volume corrections—exponential and power-law—modify the three-body spectrum, and how can they be systematically included in the quantization condition?
• RQ4 How does partial-wave mixing due to non-zero total momentum and cubic symmetry affect the extraction of scattering amplitudes in three-body systems?
• RQ5 What is the minimal set of lattice measurements (energies, boosts, volumes, irreps) required to reliably extract three-body scattering observables from lattice data?

Key findings

• The quantization condition for two-nucleon systems in a finite volume is derived for arbitrary partial waves, spin, and parity, using the auxiliary field formalism, enabling extraction of phase shifts from lattice spectra.
• For three-boson systems below the diboson breakup threshold, the quantization condition reduces to the Lüscher formula with exponential volume corrections scaling with the diboson binding momentum.
• Above the diboson breakup threshold, the formalism captures non-perturbative power-law volume corrections, which invalidate the standard Lüscher formula and require a coupled-channel treatment.
• The full quantization condition for three-boson systems is non-algebraic and requires solving an integral equation or determinant condition over dimer-boson relative angular momentum and eigenstates.
• The formalism accounts for partial-wave mixing induced by non-zero total momentum and cubic symmetry, showing that S-wave truncation introduces significant systematic errors.
• The framework enables the extraction of the three-body Bethe-Salpeter kernel and low-energy constants from lattice data, providing a path to ab initio nuclear physics from QCD.