Incorporating left-hand cuts into finite-volume scattering formalism

Abstract

Obtaining rigorous predictions from Quantum Chromodynamics (QCD), the theory of the strong interaction, is crucial for understanding hadronic and nuclear physics at a fundamental level and for the search for physics beyond the Standard Model. However, because the theory exhibits confinement at low energies, perturbative methods are not applicable in this regime. Lattice QCD currently provides the only reliable nonperturbative tool for ab-initio calculations from QCD at these energies. Such calculations necessarily use a finite discretised Euclidean spacetime, requiring additional procedures to extract the infinite-volume continuum physical observables of interest.

This thesis focuses on the formalism that permits the extraction of two-particle elastic scattering observables from the finite-volume spectrum, originally developed by M. L¨uscher. Recent lattice calculations have shown limitations in this standard method, specifically when applied to systems where the partial-wave-projected scattering amplitudes contain left-hand branch cuts below the elastic threshold. These cuts arise when the two scattering particles can exchange lighter mesons, two relevant examples being NN and DD* scattering, which receive contributions from single-pion exchanges. The presence of these cuts has so far been ignored in derivations of the formalism, leading to inconsistencies when it is applied to finite-volume energies that lie near to or on the cuts.

We address this issue by explicitly incorporating the effects of the left-hand cut into the formalism. Alternative quantisation conditions are presented that extend the standard result by L¨uscher and are applicable near to and on the cut, both for the case of identical particles (of arbitrary spin) and non-degenerate particles. These conditions allow the determination of intermediate infinite-volume quantities (K-matrices) from the finite-volume energy levels, which can then be used to obtain scattering amplitudes by solving integral equations also derived in this work.